Nastran Primer

NASTRAN Primer Static and Normal Modes Analysis

By Harry G. Schaeffer

NASTRAN Primer Static and Normal Modes Analysis

by Harry G. Schaeffer Copyright © 2015 by Harry G. Schaeffer All rights reserved. No part of this book shall be reproduced, transmitted by any means without the express written consent of the Copyright owner. No patent liability is assumed with respect to the information contained herein. Although every precaution has been exercised exercised by the author author of this book, the the author assumes assumes no responsibility responsibility for errors errors or emissions. emissions. Neither is any liability liability assumed assumed for any damages damages resulting from the use of the information contained contained herein. herein.

Trademarks Al items in this book that are known to be trademarks or service marks are Capitalized. In particular, MSC.Nastran is a trademark of MSC Software and NX NASTRAN is a trademark of Siemens Corp. The author cannot attest to accuracy of this information. Use of a term in this book should not be regarded as affecting the viability of and trademark or serviceman.

Warning and Disclaimer Every effort has been made to make this book as complete and accurate as possible but no warranty or fitness is implied. The information provided is on an “as is” basis. The author shall have neither liability or responsibility to any person or entity with respect to any loss or damages arising from use of the information in this book.

! Harry G. Schaeffer, 2015. All Rights Reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means without prior written permission of Harry G. Schaeffer.

Chapter 1 Getting Started

1.1

Historical Introduction ................................................ 1 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5

1.2

The Finite Element Method .................................................. 1 Solving NASA’s Simulation Needs..................................... 2 NASA NASTRAN.................................................................. 3 MSC NASTRAN..................................................................... 3 NX NASTRAN ....................................................................... 4

NASTRAN® Architecture ........................................... 4 1.2.1 Independent Modules ........................................................... 4 1.2.2 Database .................................................................................. 4 1.2.3 Executive System ................................................................... 4

1.3

NASTRAN Technology ............................................... 5 1.3.1 1.3.2 1.3.3 1.3.4

1.4

Engineering Mechanics ......................................................... 5 Numerical Analysis ............................................................... 5 Computer Science .................................................................. 5 Computer Graphics ............................................................... 6

Design of the Finite Element Model........................... 6 1.4.1 Basic Modeling objects .......................................................... 6 1.4.2 The Design Procedure ........................................................... 7

1.5

Processing the Finite Element Model ........................ 8 1.5.1 1.5.2 1.5.3 1.5.4 1.5.5 1.5.6 1.5.7 1.5.8 1.5.9 1.5.10 1.5.11 1.5.12 1.5.13

1.6

Marshall the Model Data ...................................................... 8 Generation of Element Stiffness Matrices .......................... 8 Assemble the Unconstrained Stiffness Matrix................... 8 Detect and remove essential singularities.......................... 8 Apply Constraints.................................................................. 8 Generate Loads....................................................................... 8 Solve for Constrained Displacements................................. 8 Recover Unconstrained Displacements.............................. 9 Recover Element Behavior.................................................... 9 Stress Averaging at the Grid Points .................................... 9 Strain Energy Density............................................................ 9 Results Files ............................................................................ 9 Post Processing....................................................................... 9

User Program Interface................................................ 9 1.6.1 Input File ................................................................................. 9 1.6.2 Types of Solutions.................................................................. 9

1.7

The Input File .............................................................. 10 1.7.1 NASTRAN Statement.......................................................... 10 1.7.2 Problem Definition Section................................................. 11 1.7.3 Getting Started with MSC.Nastran ................................... 11

Nastran Primer i

1.8

Specifying the Finite Element Model - Bulk Data.. 13

1.8.1 Free Field Bulk Data ............................................................ 14 1.8.2 Replication and Field Generation...................................... 14 1.8.3 Summary of Structurally Oriented Bulk Data ................. 15 1.8.4 Data Selection ....................................................................... 20 1.8.5 Load Selection....................................................................... 20 1.8.6 Example: Load Specification .............................................. 20 1.8.7 Thermal Field Selection....................................................... 21 1.8.8 Example: Thermal Field Specification .............................. 21 1.8.9 Example: Specification of Constraints .............................. 21 1.8.10 Specification of Eigenvalue Method.................................. 21 1.8.11 Output Control..................................................................... 22 1.8.12 Titling Directives.................................................................. 22 1.8.13 Output Line Control ............................................................ 22 1.8.14 Bulk Data Echo ..................................................................... 22 1.8.15 Set Specification.................................................................... 22 1.8.16 Element Output Requests ................................................... 22 1.8.17 Grid and Scalar Point Requests.......................................... 23 1.8.18 Subcases................................................................................. 23 1.8.19 Case Control Examples ....................................................... 26

1.9

Executive Control ....................................................... 28 1.9.1 1.9.2 1.9.3 1.9.4 1.9.5

Solution Sequence................................................................ 28 Specification of Execution Time......................................... 29 Diagnostic Print Requests................................................... 29 Termination of Executive Control ..................................... 29 Executive Control Examples .............................................. 29

1.10 TRUSS EXAMPLE ...................................................... 30 1.10.1 1.10.2 1.10.3 1.10.4 1.10.5

Components of the File. ...................................................... 30 The Input File........................................................................ 31 Truss Schematic Diagram ................................................... 33 Running the Truss Example ............................................... 33 Examining the OUT File...................................................... 33

1.11 Problems ...................................................................... 35 1.12 References .................................................................... 35 Chapter 2 Basic Relationships from Engineering Mechanics

2.1 2.2 2.3 2.4 2.5 ii Nastran Primer

Theoretical Foundations ............................................ 37 Continuum Mechanics............................................... 37 Kinematic Relations.................................................... 38 Strains ........................................................................... 38 Stress Vector on a Surface.......................................... 40

2.6 2.7 2.8 2.9 2.10

Components of Stress................................................. 42 Constitutive Behavior ................................................ 44 Equilibrium Conditions - Newtonian Mechanics .. 45 Principle of Virtual Work .......................................... 45 Solutions from the Theory of Elasticity ................... 46

2.10.1 Uniform Stress...................................................................... 46 2.10.2 Stretching of a Prismatic Bar by Its Own Weight............ 47

2.11 Strategies for Solving Real Problems ....................... 47 2.12 Two Dimensional Theory of Elasticity .................... 48 2.12.1 Plane Stress ........................................................................... 48 2.12.2 Plane Strain ........................................................................... 48 2.12.3 Equilibrium Equations ........................................................ 48 2.12.4 Strain Displacement Relations ........................................... 49 2.12.5 Constitutive Relations ......................................................... 49 2.12.6 Stress Functions.................................................................... 49 2.12.7 Solutions based on Stress Functions ................................. 50

2.13 Beam Theory................................................................ 52 2.13.1 2.13.2 2.13.3 2.13.4

Stress Resultants................................................................... 52 Stresses due to Extension and Bending ............................ 53 Equilibrium Equations ........................................................ 54 Solution to Beam Bending Equation ................................. 55

2.14 Plate theory.................................................................. 55 2.14.1 Kinematic Relationships ..................................................... 56 2.14.2 Kirchhoff Hypothesis .......................................................... 57 2.14.3 Moment and Force Resultants ........................................... 58 2.14.4 Constitutive Relations ......................................................... 59 2.14.5 Equilibrium Equation.......................................................... 60

2.15 Problems ...................................................................... 60 2.16 References .................................................................... 61 Chapter 3 Variation Principles and Approximation Theory

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Principle of Virtual Work .......................................... 63 Complementary Virtual Work.................................. 64 Minimum Potential Energy....................................... 64 Minimum Total Complementary Potential Energy65 Generalized Principle................................................. 66 Hellinger-Reissner Principle ..................................... 66 Approximation Theory .............................................. 67 Rayleigh-Ritz Soution of a Beam.............................. 68 Nastran Primer iii

3.9

References .................................................................... 69

Chapter 4 Finite Element Formulations

4.1 4.2 4.3 4.4 4.5 4.6

Finite Element Formulation ...................................... 71 Rod Element ................................................................ 71 Steps in Determining the Stiffness Equation .......... 73 Stiffness Matrix for a Beam Bending Element ........ 74 Basis Functions for Finite Elements ......................... 77 Isoparametric Transformation .................................. 79

4.6.1 Basis Functions and the Jacobian of Transformation...... 80 4.6.2 Example - Calculating the Area of a Square .................... 81

4.7

Numerical Integration................................................ 82 4.7.1 Integration for the Membrane Problem............................ 83

4.8

Plate and Shell Elements............................................ 83 4.8.1 4.8.2 4.8.3 4.8.4 4.8.5

4.9

Curvature Approximations ................................................ 84 Transverse Shear Strain Approximation .......................... 84 Curved Shell Elements........................................................ 85 Normal Rotations................................................................. 86 Non-Planar Nodes for a Flat Shell Element ..................... 86

References .................................................................... 88

Chapter 5 Structural Elements in NASTRAN

5.1 5.2

Introduction................................................................. 89 Defining Element Objects .......................................... 90 5.2.1 Defining Element Connectivity.......................................... 91 5.2.2 Defining Element Properties .............................................. 91 5.2.3 Standard Notation ............................................................... 92

5.3

Scalar Elastic Elements............................................... 92 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 5.3.6

5.4

Spring Connecting Grid Point Degrees of Freedom....... 93 Spring Connecting Scalar Points........................................ 93 Specification of Connected Degrees of Freedom............. 93 Properties .............................................................................. 93 Stress Recovery..................................................................... 94 Modeling with the ELAS Element..................................... 94

Rod and Truss Elements ............................................ 95 5.4.1 Description of Input Data for Truss .................................. 96 5.4.2 Stiffness Matrix..................................................................... 96 5.4.3 Stress Recovery..................................................................... 97

5.5

The BAR Beam Bending Element ............................. 99 5.5.1 Description of BAR Input Data........................................ 100

iv Nastran Primer

5.5.2 BAR Connectivity .............................................................. 101 5.5.3 BAR Element Coordinate System .................................... 101 5.5.4 Defining BAR Offset .......................................................... 103 5.5.5 Pinned Connections........................................................... 103 5.5.6 Defining BAR Properties................................................... 104 5.5.7 Area Properties................................................................... 107 5.5.8 Nonstructural Mass ........................................................... 108 5.5.9 Stress Recovery................................................................... 109 5.5.10 Element Forces - Use of CBARAO................................... 109 5.5.11 Data Recovery Points - CBARAO Form 1 ...................... 110 5.5.12 Data Recovery Points - CBARAO Form 2 ...................... 111 5.5.13 BAR-Element Examples .................................................... 111 5.5.14 Modeling Considerations ................................................. 116

5.6

The BEAM Bending Element .................................. 117

5.6.1 5.6.2 5.6.3 5.6.4 5.6.5 5.6.6

5.7

Curved Beam Element ............................................. 127 5.7.1 5.7.2 5.7.3 5.7.4 5.7.5 5.7.6

5.8

Defining The BEND Element ........................................... 128 Bend Element Connectivity and Geometry ................... 130 BEND Element Coordinate System................................. 130 Specifying the Arc of Geometric Centrolds ................... 132 BEND Element Properties ................................................ 132 Element Forces and Stresses............................................. 136

Shear Panels............................................................... 136 5.8.1 5.8.2 5.8.3 5.8.4

5.9

Degrees of Freedom........................................................... 118 BEAM Description ............................................................. 118 CBEAM Data Statement.................................................... 119 Local Coordinate System for the BEAM......................... 119 PBEAM Data Statement .................................................... 120 BEAM Examples ................................................................ 122

SHEAR Definition.............................................................. 137 Description of Shear Panel Input..................................... 137 Element Properties............................................................. 137 Recovery of Forces and Stresses ...................................... 138

Shell Elements ........................................................... 138 5.9.1 5.9.2 5.9.3 5.9.4 5.9.5 5.9.6 5.9.7 5.9.8

Flat Shell Elements - OUAD4 and TRIA3....................... 139 Defining Connectivity and Properites ............................ 140 Element Connectivity........................................................ 140 Element Coordinate System ............................................. 141 Material Coordinate........................................................... 141 Reference Surface Offset ................................................... 142 Element Properties - PSHELL .......................................... 142 Specifying Element Behavior ........................................... 142 Nastran Primer v

5.9.9 Element Constitutive Relations........................................ 143 5.9.10 Solid Homogeneous Symmetric Cross-Section ............. 143 5.9.11 Sandwich Type Cross-Section.......................................... 144 5.9.12 Effect of Warping ............................................................... 146 5.9.13 Elastic Stiffness Matrices................................................... 146 5.9.14 Mass Matrix ........................................................................ 146 5.9.15 Stress, Strain and Element Force Recovery .................... 147

5.10 Curved Shell Elements............................................. 148 5.10.1 TRIA6 and QUAD8 Connectivity.................................... 149 5.10.2 Element Connectivity ........................................................ 149 5.10.3 Element Coordinate Systems ........................................... 149 5.10.4 Material Orientation .......................................................... 149 5.10.5 Element Thickness ............................................................. 150 5.10.6 Specifying Element Behavior ........................................... 150 5.10.7 Element Stiffness Matrices................................................ 150 5.10.8 Element Mass Matrices ..................................................... 150 5.10.9 Stress, Strain and Element Force Recovery .................... 150

5.11 Solid Elements........................................................... 150 5.11.1 Defining Solid Elements.................................................... 152 5.11.2 Element Connectivity ........................................................ 153 5.11.3 The TETRA Element .......................................................... 153 5.11.4 The PENTA Element ......................................................... 153 5.11.5 HEXA Element ................................................................... 153 5.11.6 Properties of Solid Element .............................................. 153 5.11.7 Hexa Element Coordinate System................................... 155 5.11.8 PENTA Element Coordinate System .............................. 156 5.11.9 TETRA Element Coordinate Syatem............................... 157 5.11.10Elastic Stiffness Matrix ...................................................... 157 5.11.11Mass ..................................................................................... 158 5.11.12Stress and Strain Recovery ............................................... 158

5.12 Congruent Elements................................................. 158 5.13 References .................................................................. 159 Chapter 6 Global Analysis Procedures

6.1 6.2 6.3 6.4 6.5 6.6

The Global Stiffness Matrix ..................................... 161 Local and Global Coordinate Systems................... 164 Transformation of Element Stiffness Matrices ..... 164 Specifying Structural Degrees of Freedom ........... 165 Defining Scalar Degrees of Freedom - SPOINT ... 167 Coordinate Systems - CORD................................... 167

6.6.1 Subsidiary Coordinate System......................................... 168 vi Nastran Primer

6.7

Grid Points -GRID .................................................... 171 6.7.1 6.7.2 6.7.3 6.7.4 6.7.5

6.8 6.9

Grid Identification Number.............................................. 172 Geometric Coordinates ..................................................... 172 Displacement Coordinates................................................ 172 Permanent Constraints...................................................... 173 Default Values - GRDSET................................................. 173

External and Internal Degrees of Freedom ........... 173 Displacement Sets..................................................... 174 6.9.1 Merged Data Sets ............................................................... 175 6.9.2 Mutually Independent Data Sets..................................... 175

6.10 Multipoint Constraints - MPC ................................ 175 6.10.1 Uses of MPC........................................................................ 175 6.10.2 Reduction to the n-Set ....................................................... 177 6.10.3 MPC Data Entity ................................................................ 180 6.10.4 Combining MPC Sets - MPCADD................................... 180 6.10.5 Specifying the m-set Degrees of Freedom ...................... 181 6.10.6 Recovering and Printing MPC Forces............................. 181 6.10.7 Defining the Constraint Equation.................................... 181

6.11 Single Point Constraints - SPC................................ 184 6.11.1 Reduction to the f -Set ........................................................ 184 6.11.2 Single Point Constraint Forces ......................................... 184 6.11.3 SPC Data Entity.................................................................. 185 6.11.4 Selecting SPC Sets in Case Control.................................. 186 6.11.5 Specifying s-set Degrees of Freedom and Value ........... 186 6.11.6 Purging Degrees of Freedom ........................................... 186 6.11.7 AUTOSPC - Automatic Purging...................................... 187 6.11.8 Example: Specification of Single Point Constraints ...... 188

6.12 Static Condensation -OMIT and ASET.................. 189 6.12.1 Reduction to the a-Set........................................................ 189 6.12.2 Recovery of Omitted Degrees of Freedom..................... 190 6.12.3 Physical Interpretation of [Goa]....................................... 190 6.12.4 Why Use Static Condensation? ........................................ 190 6.12.5 The OMIT and ASET Data Entities ................................. 191 6.12.6 Specifying Degrees of Freedom ....................................... 192 6.12.7 Example: Static Condensation.......................................... 193

6.13 Support for Free Bodies - SUPORT........................ 193 6.13.1 Reduction to the l-Set ........................................................ 194 6.13.2 Rigid Body Transformation Matrix................................. 194 6.13.3 Rigid Body Stiffness Matrix.............................................. 195 6.13.4 SUPORT Bulk Data............................................................ 196 6.13.5 Specifying Degrees of Freedom ....................................... 196 Nastran Primer vii

6.14 Flexibility to Stiffness Transformation .................. 196 6.14.1 GENEL Bulk Data .............................................................. 200 6.14.2 Example:Beam Stiffness Using GENEL.......................... 201 6.14.3 Direct Specification of Element Stiffness ........................ 202

6.15 References .................................................................. 202 6.16 Problems .................................................................... 203 Chapter 7 Accuracy and Performance of Finite Elements

7.1 7.2 7.3

Element Performance............................................... 205 Interpolation Failure................................................. 205 Effect of Element Shape ........................................... 206 7.3.1 7.3.2 7.3.3 7.3.4 7.3.5 7.3.6 7.3.7 7.3.8

7.4 7.5 7.6

Two Dimensional Patch Test............................................ 208 Three Dimensional Patch Test.......................................... 210 Patch Test Results .............................................................. 211 MacNeal Beam Tests.......................................................... 212 Rectangular Plate Tests ..................................................... 215 Cylindrical Shell Tests....................................................... 217 Pinched Sphere................................................................... 218 Incompressible Material.................................................... 219

Summary of Test Results ......................................... 220 Problems .................................................................... 222 References .................................................................. 223

Chapter 8 Rigid and Constraint Elements

8.1 8.2 8.3

Rigid and Constraint Elements............................... 225 RROD - Extensional Constraint.............................. 226 RBAR - Rigid Link Connecting Two Grids........... 227 8.3.1 RBAR Example: Rigid Link Connecting Two Points.... 228

8.4 8.5

RTRPLT - Rigid Triangular Constraint Element.. 228 Rigid Constraint Element - RBE1 and RBE2......... 229

8.5.1 Example: Rigid Inclusion .................................................. 231

8.6 8.7

Elastic Constraint Element - RSPLINE .................. 231 Weighted Average Constraint Element - RBE3.... 234

8.7.1 Example -"Beaming" Loads and Masses......................... 235

8.8

References .................................................................. 236

Chapter 9 Material Properties

9.1

Isotropic Material...................................................... 240 9.1.1 One-Dimensional Elements .............................................. 241 9.1.2 Two-Dimensional Elements - Plane Stress..................... 241 9.1.3 Two-Dimensional Elements - Plane Strain..................... 242

viii Nastran Primer

9.1.4 Three- Dimensional Elements .......................................... 242

9.2

Anisotropic Material for Surface Elements........... 243 9.2.1 Orthotropic Material Properties ...................................... 243 9.2.2 Example: Orthotropic Material ........................................ 244

9.3 9.4

Orthotropic Material for Surface Elements........... 245 Anisotropic Material for Solid Elements ............... 246 9.4.1 Orthotropic Constants for Solid Elements ..................... 247

9.5 9.6 9.7

Transformation of Elastic Constants...................... 248 Temperature- Dependent Materials....................... 249 Material Property Table........................................... 250 9.7.1 Example: Temperature- Dependent Elasticity............... 252

9.8

Modeling Composite Materials .............................. 253 9.8.1 9.8.2 9.8.3 9.8.4

9.9

Lamina Properties.............................................................. 254 Laminate Properties........................................................... 255 Defining a Composite Laminate Properties................... 257 Lamina Failure Criteria ..................................................... 259

Reference.................................................................... 260

Chapter 10 External Loads

10.1 10.2 10.3 10.4 10.5 10.6

Concentrated External Forces at Grid Points ....... 264 Force Vector Defined by Components................... 265 Force Direction Defined by Two Points ................ 266 Vector Normal to a Surface ..................................... 267 Load Applied to Scalar Point - SLOAD................. 268 Distributed Load on Beam - PLOAD1................... 269

10.6.1 Load Type and Direction .................................................. 269 10.6.2 Load and SCALE Fields .................................................... 269

10.7 Uniform Normal Pressure - PLOAD2 ................... 271 10.7.1 Sign Convention for Positive Pressure ........................... 271

10.8 Nonuniform Surface Tractions - PLOAD4............ 272 10.8.1 Direction of Surface Traction............................................ 272 10.8.2 Specifying Load Intensity ................................................. 272 10.8.3 Specifying the Loaded Face of Solid Elements .............. 273

10.9 Gravity Loads - GRAV............................................. 274 10.10 Rotation About an Axis- RFORCE ......................... 275 10.10.1Remarks............................................................................... 277

10.11 Combined Loading - LOAD.................................... 277 10.12 Enforced Displacements - SPCD ........................... 278

Nastran Primer ix

10.13 Enforced Deformation - DEFORM........................ 279 10.14 Thermal Loading ...................................................... 279 10.15 Grid Point Temperatures - TEMP and TEMPD ... 281 10.16 . Thermal Field for Axial Elements - TEMPRB..... 281 10.17 Thermal Field for Surface Elements....................... 282 Chapter 11 Static Analysis

11.1 System Matrices ........................................................ 285 11.1.1 Direct Specification of System Matrices.......................... 285

11.2 11.3 11.4 11.5 11.6

Constraint and Static Condensation ...................... 289 Static Loads................................................................ 289 Inertia Relief .............................................................. 290 Data Recovery ........................................................... 291 Input Specifications .................................................. 291

11.6.1 Executive Control Section................................................. 291 11.6.2 Case Control Section.......................................................... 292 11.6.3 Parameters........................................................................... 293

11.7 Solution Sequence Output....................................... 293 11.7.1 Automatic Output.............................................................. 293 11.7.2 Output of System Response Variables............................ 294

11.8 Fatal Errors ................................................................ 294 11.9 Grid Point Singularity Processing.......................... 295 11.9.1 Legacy NASTRAN............................................................. 296 11.9.2 MSC and NX NASTRAN.................................................. 296 11.9.3 Displacement Set Membership ........................................ 298

11.10 Grid Point Weight Generator.................................. 300 11.10.1Rigid Body Transformation Matrix................................. 300 11.10.2Rigid Body Mass Matrix ................................................... 300 11.10.3Principal Mass Axes........................................................... 300 11.10.4Centroid............................................................................... 301 11.10.5Moments of Inertia............................................................. 301

11.11 Example Problems.................................................... 301 11.12 Cantilever Beam with Uniform Load .................... 302 11.12.1Element Properties............................................................. 303 11.12.2Distributed Line Load ....................................................... 303 11.12.3The Input File...................................................................... 304 11.12.4Running NASTRAN.......................................................... 306 11.12.5The Output File .................................................................. 306 11.12.6Discussion of Results......................................................... 310 x Nastran Primer

11.13 Simply-Supported Rib-Stiffened Plate................... 312 11.14 Problems .................................................................... 312 11.15 References .................................................................. 313 Chapter 12 Normal Modes Analysis

12.1 12.2 12.3 12.4

Dynamic Motion ....................................................... 315 The Eigenvalue Problem.......................................... 316 Standard Form of Eigenvalue Problem ................. 319 Specification of Element Inertia Properties........... 320

12.4.1 Structural Mass................................................................... 320 12.4.2 Nonstructural Mass ........................................................... 320 12.4.3 Consistent and Lumped Formulations........................... 320 12.4.4 Mass Elements.................................................................... 321 12.4.5 Scalar Mass - CMASS ........................................................ 321 12.4.6 Concentrated Grid Point Mass - CONM1 ...................... 322 12.4.7 Offset Concentrated Grid Point Mass - CONM2........... 322

12.5 Real Eigenvalue Extraction Techniques ................ 323 12.6 The Inverse Power Method ..................................... 325 12.6.1 Specification of EIGR for Inverse Power with Shifts .... 325 12.6.2 Summary for the Inverse Iteration Method ................... 326 12.6.3 Finding Lowest Eigenvalue with Inverse Method........ 328

12.7 The Lanczos Method ................................................ 329 12.7.1 Description of the Lanczos Procedure ............................ 329 12.7.2 The Specification of the Lanczos Procedure - EIGRL ... 331

12.8 Tridiagonal Methods................................................ 332 12.8.1 EIGR for the Triadiagonalization Methods.................... 333

12.9 Dynamic Degrees of Freedom ................................ 334 12.10 Reducing Dynamic Degrees of Freedom .............. 335 12.10.1Guyan Reduction ............................................................... 335

12.11 Removing Matrix Singularities............................... 336 12.11.1Detect Singularities and Exit - ASING = -1..................... 337 12.11.2Remove Massless Degrees of Freedom ASING = 0....... 337

12.12 Input File for Normal Modes Analysis.................. 338 12.12.1Executive Control Section................................................. 338 12.12.2Case Control Section.......................................................... 338

12.13 Parameters ................................................................. 339 12.14 Example Problems.................................................... 339 12.14.1Cantilever Beam ................................................................. 339 12.14.2Simply Supported Plate .................................................... 340 Nastran Primer xi

12.15 Problems .................................................................... 341 12.16 References .................................................................. 342 Chapter A Use of Parameters

xii Nastran Primer

1 1.1

Getting Started

Historical Introduction Designers and builders have sought guidance from the sciences throughout recorded history. Before the development of science as we know it, builders used intuition, rules of thumb and design rules that became trade secrets passed from generation to generation. However to design modern structures for the industrial revolution engineers first had to develop the concepts of physics and mathematics that lead to understand the factors associated with the design of simple structures. With the ability to predict the behavior of built-up structures comprised of rods, beams, and plates, engineers can design bridges and steam engines that altered trade routes and shortening the time required to transport products to ever-widening markets. Early research and development activities focused on improving the analytical sciences providing predictive tools that furthered technological advances. In hindsight, it is remarkable that the engineering technology of the 1920s and 1930s was adequate for designing systems such as jet engines and subsonic aircraft. However, predictive tools that supported World War II aerospace technology were not adequate for the design challenge of high-performance aircraft with swept wings. It was this challenge that lead to the revolution in predictive analysis which was developed at Boeing by Turner and his colleagues. This technology that Ray Clough termed “finite elements,” together with the parallel development of digital and analog computers, provided designers with new insight on how loads are carried in a swept wing. This understanding, in turn, lead to the detailed design of weight efficient structures for transonic and supersonic flight. Finite element technology was the result of a need and of a focused research program to develop the enabling technology for high performance aircraft. Other technology sectors also drove the development of predictive technology used to design jet engines, automobiles, nuclear power plants, and safe toys. These industries benefited from the predictive power of the finite element method.

1.1.1

The Finite Element Method

The finite element method synthesizes complicated structural systems as a connected collection of objects, called finite elements, that embody local physical laws. General purpose analysis systems containing a library of finite elements can then be developed. Engineers can then used these modeling elements to create a mathematical model of complicated structural assemblies. The computer then solves the resulting set of linear algebraic equations to determine the predicted behavior. In a very short time after the appearance of Turner’s seminal paper[1] every aerospace company in the United States, if not in the world, developed its own proprietary finite element program. These programs had strange-sounding names like SAMUS, ASKA, MAGIC, and STRUDEL. Each of these programs was promoted by its developers but none was really available as a commercial product.

1 Nastran Primer

1

Getting Started Historical Introduction

1.1.2

Solving NASA’s Simulation Needs

In the early 1960s, the National Aeronautics and Space Administration (NASA) determined that a large scale finite element-based program was an enabling technology for designing the space system for lunar exploration. NASA’s goal was to develop a structural analysis system to solve large problems that would employ the finite element method, use the best currently available algorithms for solving matrix equations and eigenvalue analysis and run on all major computers of that era. To satisfy this need, a NASA team headed by a visionary named Tom Butler developed a set of specifications for an innovative general purpose analysis system. In hindsight NASA was asking for the moon, literally speaking. Finite element technology was in its infancy. Fortran, the only high level language of that era, was not mature. The solution algorithms for equation solving and eigenvalue analysis were very inefficient. Computers were slow, had small memories, did not have disk storage, and were expensive. In other words, the development of a system to satisfy NASA’s needs was quite a challenge. The analysis system was called the NASA Structural Analysis System, which was of course called NASTRAN®, a name that NASA registered as a trademark The development team which included the Computer Science Corporation and The MacNeal-Schwendler Corporation (MSC)1 was working in uncharted territory and made significant contributions to computer science, numerical analysis, and finite element approximation theory. When a linker was required for one of the target computers, they designed and built one. When they needed a technique for allocation of memory at run time, they developed the “open core” concept. When they needed new finite elements, they designed them. And they designed a complete operating system with file allocation tables and I/O utilities, a modular concept for organizing data structures and algorithms, which is now called object oriented programming, and an internal language called DMAP for binding objects into an executable code. The original release of NASTRAN was enthusiastically received by the user community, which at the time consisted of only the large aerospace companies and government laboratories that could afford the multi-million dollar computers supported by MSC.Nastran. In years immediately following the release of the program, NASA hosted a series of NASTRAN Colloquia that provided a forum for a lively interchange of ideas between the user community and MSC, which developed the program and then provided maintenance and development services. The NASTRAN system was released to NASA contractors in 1968, and users were invited by NASA to share their experience in a series of NASA-sponsored Colloquia, starting in 1971. The resulting colloqium proceedings are of historical interest since they chronicle: 1.The diffusion of finite element technology from the aerospace industry to other industries,including automotive, chemicals, and electronics. 2.The growth of user understanding of the technology and its innovative application to design. 3.The explosion of problem size and the collateral need for database procedures for automated substructural analysis. 4.The development of computer graphics and the collateral development of pre- and post processors. 5.The establishment of the finite element method on a firm variational foundation on which the modern element library was built.

1.In 1999, The MacNeal-Schwendler Corp. changed its name to The MSC Software Cor poration. The company is still commonly referred to as MSC.

2 Nastran Primer

Getting Started Historical Introduction

The early adapters of finite element analysis were engineering design analysts well versed in that day’s analysis technology, which included design handbooks and specialty analysis programs based on strength of materials approximations. These users were skeptical of finite elements and accepted programs like NASTRAN, only after correlating MSC.Nastran solutions with those given by Roark’s Handbook and solutions from Timoshenko’s Plates and Shells. These early adapters found that the use of finite elements lead to reasonable results, but accurate results depended on: 1.Understanding the physics of the problem. 2.Understanding the behavior of the elements. 3.Selecting the correct element, the number of elements. and their distribution. 4.Critically evaluating the results and making modification in the conceptual model to improve the accuracy. To a large extent, these same factors should influence today’s user in creating an analysis model. In the time since the introduction of commercial finite element programs in the late 1960s and early 1970s, element behavior has improved and automated procedures for improving accuracy, using either a refined mesh or higher ordered polynomial functions, have been developed and implemented. But today’s engineer should be just as skeptical of the results as the early adapters since the finite element method is simply a tool that can be misused and abused. Today we approach the use of finite elements from a practical rather than a theoretical point of view. Our attitude is to ask the program to explain itself using computer aided empiricalism. That is, let’s find out by doing or at least looking at the results from experiments that give guidance in producing acceptable finite element analyses.

1.1.3

NASA NASTRAN

NASA continued to maintain the public domain version until 1995. After that date the program was available by requesting a copy from the NASA Technology Transfer Office at Langley Research Center. This version was recently released as open source software at the NASA/NASTRAN GitHub site. This version can be downloaded and built on your computer using a fortran computer. I have successfully built using Visual Studio Fortran 6.5 under Windows XP; and GNU gfortran using MinGW and MSYS on a Windows 8.1 computer. More on that later. The public domain version clearly represents the state of the art in 1995. It needs new element technology, the addition of sparse technology for solving matrix equations and for eigenvalue extraction to name a few.

1.1.4

MSC NASTRAN

In the late 1960s, MSC started to market and support it’s own version of NASTRAN, called MSC.Nastran. At first, it was similar to the public domain version. Over the years, MSC has made a large investment in research and development. Dr. MacNeal, the founder of MSC was responsible for a great deal of this technology which included: designing and adding robust elements; embracing p-element technology, incorporating best of class algorithms for sparse matrix equation solvers and Lanczos eigenvalue methods; incorporating multi-physics capability; developing superelement technology; and, improving the basic architecture of the program by using modern database technology. MSC also consolidated the NASTRAN marketplace by purchasing two competitors, Computerized Structural Analysis and Research Corporation (CSAR) and Universal Analytics, Inc. (UAI).

Nastran Primer 3

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Getting Started NASTRAN® Architecture

1.1.5

NX NASTRAN

The FTC soon took notice of the reduced competition in the NASTRAN submarket which FTC identified as programs which included the DMAP language. Their action caused MSC to sell a copy of MSC NASTRAN to UGS Corporation that was subsequently acquired by Siemens. Siemens markets this version as NX NASTRAN.

1.2

NASTRAN ® Architecture 1.2.1

Independent Modules

Conceptually NASTRAN is a collection of independent program united called Modules. These Modules were further categorized as: • Structural Modules - Performed tasks associated with: element generation and assembly; and, applying constraints and loads. • Matrix Modules - Performed matrix addition, multiplication solution, eigen-extraction, etc. • Utility Modules - Performed task involving interaction with the database

1.2.2

Database

The database evolved from a collection of tape reels in the early versions of NASTRAN to a robust relational database in NASA NASTRAN and in MSC.Nastran which are implemented using high speed disks. The need for a persistent database in the early versions of NASTRAN was supported by a feature called Checkpoint and Restart. Modules communicate through the database. The program input is parsed and validated and then written to the database as appropriate tables such as the Basic Grid Point Definition Table (BGPDT), the Grid point Element Connection Table (GPECT) and the Equivalence between Internal and External points (EQEXIN) to name a few.

1.2.3

Executive System

The Executive System employs anevent loop to run the modules appropriate for the solution. It is driven by a table called the Operation Sequence Array (OSCAR) that defines order of modules to be executed and the resources required to perform the operation. The OSCAR, in turn is created from user input that defines the process steps to be performed using a NASTRAN problem orient language called Direct Matrix Abstraction Programming (DMAP). It is interesting to note that the FTC identified DMAP as a distinguishing feature of the NASTRAN genre. The Exec extracts the resource requirement, allocates input, output and scratch files, loads parameter values and executes the module. The next OSCAR entry is read and the process repeats until the OSCAR is terminated.

4 Nastran Primer

Getting Started NASTRAN Technology

1.3

NASTRAN Technology 1.3.1

Engineering Mechanics

Although NASTRAN can perform multi-physics simulations, we consider only its capability for determining the behavior of solid elastic bodies. Solid elastic bodies are a class of continuous media whose behavior is governed by the a set of partial differential equations collectively called the theory of elasticity. The foundations for this theory were established in the late 19th century. However, it is one thing to have a theory and another matter to solve the field equations for meaningful design problems. Since designers needed insight about the behavior of structural elements, simplifications were introduced into the theory of elasticity to make solutions tractable. One avenue of research at that time was to consider the stress or strain field to be either two- or onedimensional. One dimensional stress theory lead to the rod elements Two dimensional theory lead to plane stress and plane strain solutions. The two-dimensional theory lead to an understanding of important engineering design problems. Perhaps most notably are the stress concentration about a hole in a rectangular membrane and with that an understanding of fatigue failure in early jet transports; and the exact solution for a cantilever beam. One-dimensional theory lead to the rod element widely used in bridge design. However, many structural components were thin in either one or two dimensions. Using appropriate assumptions for the variation of in plane displacement through the thickness for resulted in beam, plate, and shells theories. Each of these theories has its place in structural design. For example, large bridges and skyscrapers were designed using rod, beam and plate elements, and the DC-3 and Boeing B29 were designed using rod and shear panel approximations. However, the need for lowweight, high-performance structures, especially in the aerospace industry, required better analysis support to design. So a talented group of engineers at Boeing invented a new procedure for representing and solving complicated structures using an assemblage of simple small structural components now called the finite element method.

1.3.2

Numerical Analysis

Engineering mechanics and continuum mechanics together with approximation theory led to a plethora of finite elements types. Numerical analysis provides the algorithms for solving systems of equations, the eigenvalue or single value problems The algorithms for solving has improved significantly from early releases of NASTRAN. I recall seeing a comment in one of the real eigenvalues routines called Inverse Power with Shifts that said, “OK Lets eat up the budget”: Computers in those days were slow and expensive. Fortunately great strides have been made reducing the time required to perform solutions and the cost of computers is a small fraction of it was when only large mainfromas could process simulation taks such as NASTRAN. Sparse matrix techniques changed the economincs of solving large sets of equations. Now all compertive FEA programs include sparse solvers and eigenvalue routines.

1.3.3

Computer Science

Computer Science is responsible for the basic arcitecture, and the data structures associated with a particular computational genre. The data structures used by NASTRAN are meant to optimize the data processing requirements, These same data structures might not work at all for annother application.

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Getting Started Design of the Finite Element Model

Computer science also develops efficient algoritms for soring and finding things. These algorithms are always improving so that a cutting edge program must continue to implement the curent state of the art.

1.3.4

Computer Graphics

When NASTRAN was started the use of computers to create graphic images did not compute (pun intended). There were no products for Computer Graphics. There was no CAD. Computervision and Autocad were not on anyone’s radar screen. But there was plotting. NASTRAN included a capability for creating structural plots and for XY plots. The use of a graphic data base had to wait until someone invented Computer Aided Design Drafting (CADD). This had the unintended consequenc of creating a geometric data base; and soon someone with a tectronix scope figured out how to use the CADD data base to create a finite element model. This led to the development of programs, independent of the FEA program, called pre and post processors. PATRAN was developed by PDA Engineering, since purchased by MSC; and, about the same time Desktop was developed by ANSYS, Inc. Another program started about the same time was FEMAP, now a Siemens product.

1.4

Design of the Finite Element Model The finite element model is a conceptual model of the part or assembly that embodies the physics associated with its behavior and the displacement and force boundary conditions. This model leads to an approximate solution for the partial differential equations governing the behavior of the part. The physics of the behavior is, in turn, encapsulated by the finite element objects. The design of the finite element model thus begins by selecting the finite element object that incorporates the desired behavior. Since the finite element object approximates the local behavior, the level of approximation for solving the governing partial differential equations must be selected by distributing a finite number of elements over the geometry of the part. This requires that the user understand how element objects perform as a function of element shape and boundary conditions. The finite element objects generate a set of linear equations representing the differential equations. The particular solution of these equations is dependent on the displacement and force boundary conditions.

1.4.1

Basic Modeling objects

The finite element analysis program generates and solves a set of equations based on modeling objects that we define. The modeling objects include: 1..A material point object, called a GRID point, that specifies a point in the undeformed structure that displaces with the structure. The grid point has both a geometric attribute and a physics attribute. The geometric attribute defines the location of the point in the undeformed media, and the physics attribute is the displacement vector that defines where the point is in the deformed media. 2.An element object defining the local physics in a volume defined by connecting two or more GRID points. 3.A material object that specifies the material properties for an element object. 4.A load object that specifies a load either with reference to a GRID or an element object. 5.A constraint object that specifies displacement boundary conditions at a GRID point, or a linear relationship among displacements at GRID points. 6 Nastran Primer

Getting Started Design of the Finite Element Model

The design objective of the finite element model is to represent the physics of a component or a structural system and to obtain its behavior to the accuracy required by the design team. To accomplish this task, finite element analysts must understand the design tools and how structures behave.

1.4.2

The Design Procedure

The design procedure uses the basic principles of mathematical physics. The accuracy of the solution is a function of: 1.The number and location of the discrete material points called grid points. 2.The design of the finite element that represents the average behavior in a region. The conceptual analysis model is designed by: 1.Understanding the strength of materials assumptions appropriate for the analysis. 2.Creating the analysis geometry that reflects the strength of materials assumptions. 3.Choosing the finite elements used to represent the behavior. 4.Choosing an adequate mesh refinement to represent the critical behavior of interest. 5.Creating a field of grid points on the analysis geometry that will lead to the desired mesh density. 6.Connecting the grid points with the appropriate elements. 7.Describing the material properties. 8.Describing the loads over the analysis geometry that will generate the appropriate loads at the grid points. 9.Describing the constraints, with respect to the analysis geometry, that will generate the appropriate set of constraints at the grid points. 10.Describing special relationships that must exist between loads and/or displacements using a set of linear constraint equations. Except for the simplest of cases the creation of the finite element model is a major task and should not be undertaken without the aid of graphic-oriented pre-processors. Creating a finite element model is far different than creating a geometrically faithful rendering of the geometry. In some cases the structural component can be much simpler than the geometry model of a component. For example, a finite element model of a beam having a very complicated geometric cross section can be modeled with a simple beam element. The complicated geometry is captured by the cross sectional area and the moments of inertia. The resulting solution provides the analyst with element forces that can be used to evaluate the bending stresses at critical points in the cross section. The beam could also be modeled by solid 10 node tetrahedral elements. The beam is the better solution, if it is appropriate, for the following reasons: 1.For a static simulation, one beam element will give the exact result as long as beam theory is adequate. 2.The solid elements represent local behavior approximately. A large number of Tet10 elements may be required to accurately predict the behavior. 3.The density of elements must be sufficient to represent local strain and stress gradients. The bottom line is that geometric fidelity is not necessarily the objective of the finite element model. The objective is behavioral fidelity, and the creation of a physics model requires an understanding of physics. Nastran Primer 7

1

1.5

Getting Started Processing the Finite Element Model

Processing the Finite Element Model The finite element model is represented by a set of modeling objects stored on a file. The finite element analysis program processes these objects and generates a solution. The principal steps performed for a linear static analysis are described below.

1.5.1

Marshall the Model Data

The NASTRAN preface reads and verifies the modeling objects in the input file, and creates data structures used in later stages of the solution process.

1.5.2

Generation of Element Stiffness Matrices

Element matrices are generated by the EMG Module using the connectivity defined by the element modeling object, the geometry defined by the grid objects, the material defined by the material objects, and properties defined by property objects. The element stiffness is always generated in a local element coordinate system and then transformed to the displacement coordinate system. NASTRAN allows the reference coordinate system for the displacement vector for each grid object to be defined by the user.

1.5.3

Assemble the Unconstrained Stiffness Matrix

The element matrix assembler (EMA) nodule generates an assembled stiffness matrix by appropriately adding the stiffness contributions of each element object connected at a grid point. The assembler also looks for zeros on the diagonal of the stiffness matrix which, if not removed by later processing, would lead to a singular matrix.

1.5.4

Detect and remove essential singularities

The assembled stiffness matrix is singular for two reasons. First, the displacement functions for all elements include rigid body motion so the assembled stiffness contains rigid body motion. Second, the element objects used in the model might not connect all displacement degrees of freedom at the grid objects. The rigid body motion must be removed by constraint objects. However, the removal of unconnected degrees of freedom is essentially a data processing tasks included in most commercial finite element programs. This procedure is called Autospc in NASTRAN.

1.5.5

Apply Constraints

After removing the unconnected degrees of freedom from the analysis, the constraints, which are defined by constraint objects, are processed. The stiffness matrix and load vectors are reduced by the resulting transformation matrices.

1.5.6

Generate Loads

The loads associated with the grid point displacements are generated using the load modeling objects. The resulting load vector(s) are transformed to incorporate the effect of the constraints.

1.5.7

Solve for Constrained Displacements

The solution is performed using Gauss elimination. The procedure has two phases. The first phase is called decomposition since the stiffness matrix is transformed to lower and upper triangular factors. This is followed the second phase called forward backward substitution. The end result is the displacement vector for the constrained set of equations.

8 Nastran Primer

Getting Started User Program Interface

1.5.8

Recover Unconstrained Displacements

The unconstrained displacements are recovered using the transformations generated by the constraints.

1.5.9

Recover Element Behavior

MSC.Nastran recovers element strains sing the strain-displacement relations. After determining the strains, the stresses are recovered and, if appropriate for the element, the element forces are recovered.

1.5.10

Stress Averaging at the Grid Points

The stresses are evaluated at internal points in the element called the Barlow points, at which the stress calculations are most accurate. The components of the stress tensors must be averaged in some way and presented as either an average element stress or as grid point stresses. The stresses at the grid points are not continuous, so grid point stresses must be further processed to provide useful stress values to the analyst.

1.5.11

Strain Energy Density

The variation strain energy density provides the analyst with important insight about the distribution of the element objects. It is generally calculated and plotted as part of the model verification procedure.

1.5.12

Results Files

After generating all user requested behavioral variables NASTRAN generates several files, including the output file whose file extension. is “out”. (This file has a “f06” extension in MSC and NX NASTRAN).

1.5.13

Post Processing

The output file containing all the requested output may be very large. A graphic-oriented post-processor provides the analyst with a visual display of the results.

1.6

User Program Interface 1.6.1

Input File

Using the text editor utility you can create a model file containing modeling directives. An input text file is always required to execute the program. Later in this chapter a simple example is presented to demonstrate the format of the input file.

1.6.2

Types of Solutions.

MSC.Nastran includes a number of solution sequences for performing various types of analyses, as shown in Table 1-1. Solution Number

Description of Solution

1

Static analysis

2

Static Analysis with Inertia Relief

3

Normal modes analysis

5

Buckling

Nastran Primer 9

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Getting Started The Input File

Solution Number


7

Direct complex eigenvalue analysis

8

Direct frequency response

9

Direct transient analysis

10

Modal complex eigenvalue analysis

11

Modal frequency response

12

Modal transient analysis

14*

Cyclic symmetry static analysis

15*

Cyclic symmetry normal modes analysis

18

Cyclic symmetry frequency response

29

Nonlinear transient analysis

44

Static aeroelastic response

45

Aerodynamic flutter

46

Aeroelastic response

53

Static structural/steady state heat transfer

59

Transient structural/steady state heat transfer

Table 1-1 Solution Sequences in MSC.Nastran

Many of the algorithms in MSC.Nastran are beyond the scope of this text. Only solutions for linear static, normal modes, and buckling, identified with an asterisk, are described. These solution sequences are written in DMAP. NASTRAN performs a simulation using any of the included solution sequences. However, an understanding of DMAP programming can increase the scope of the problems that can be solved. In addition, unnecessary steps in a solution sequence can be bypassed and the solution can be selectively stopped and started. DMAP is described in Appendix E. The DMAP language allows you to specify symbolic names for matrices in just like any higher level programming language. DMAP takes care of all of the internal interface to the data base.

1.7

The Input File The input file begins with an optional NASTRAN statement followed by) 1.A required Executive Control section 2.A required Case Control section 3.A required Bulk Data section

1.7.1

NASTRAN Statement

The NASTRAN statement is used to specify values for certain system parameters which are called system cells. The default values for the systems cells are usually appropriate so the NASTRAN statement is only used in exceptional cases. The format of the NASTRAN statement is NASTRAN cellname1=value1, ..., callnamen = valuen or NASTRAN system(i)=value1, ..., system(n) = value n A complete list of parameters which can be set by the NASTRAN statement is presented in the NASTRAN User Manual Vol. I. 10 Nastran Primer


1.7.2

Problem Definition Section

The the problem definition is completely controlled by user-specified commands and data statements in the input file. The problem is specified by 1.Defining the physical problems or the system of equations to be solved. 2.Providing user control over input and output. 3.Providing user control over executive functions. The remaining sections of the input file performs one of the functions described above. The problem definition section consists of 1.The EXECUTIVE CONTROL section performs function 3 above and is the first section of the input file. 2.The SUBSTRUCTURE CONTROL section. 3.The CASE CONTROL section performs function 2 above and is the second section of the input file. 4.The BULK DATA section performs function 1 above and is physically the third section of the input file. The subsection sections of the problem definition will be discussed in what appears to be reverse order since the user typically defines the finite element model first in the Bulk Data section, then considers Case Control, and then defines the Executive Control necessary to accomplish a specific task. The various sections of the problem definition section are separated by delimiter statements described in Table 1-2. These delimiters are required in every MSC.Nastran input data file even if the corresponding section is an empty set. These delimiters are Delimiter Name

Purpose

CEND

Terminates the Executive Control section.

BEGIN BULK

Terminates the Case Control section

ENDDATA

Terminates the Bulk Data section

Table 1-2 Delimiter Statements

1.7.3

Getting Started with MSC.Nastran

Let us suppose that you, the user, have a background in matrix structural analysis and the finite element method and perhaps have used another general purpose finite element program. The question then isn't “what does the program do?” but “how do I pose my problem so that MSC.Nastran will understand it?”. A reasonable approach is to first define the physical problem then worry about the rest of the MSC.Nastran input deck. It thus seems to make sense to think first about that portion of the file called Bulk Data in which the finite element models us described by specifying • The locations of the mesh points • The element connectivity • The element properties • The material properties • The constraints

Nastran Primer 11

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• The loads • Eigenvalue extraction routines, if appropriate • etc.

After defining the problem in the Bulk Data section it is reasonable to think about selecting data items that are defined in Bulk Data that are also to be included in the present analysis and, in addition, about specifying the behavioral variables to be generated. The concept of sets of input data items is important since this allows the user to define any number of logical sets of loads or constraints in the Bulk Data section and then, by using appropriate Case Control directives, to specify which of the logical sets are to be included when the program is executed. Loads and constraints are given a numerical tag, called a set identification number, which is specified by an appropriate Case Control directive. For example, suppose that a set of forces has been given the set identification number 101. Then this set of loads would be specified in static analysis by the following Case Control directive LOAD = 101 In addition to selecting input data sets from Bulk Data the Case Control can be used to • Print or not print the Bulk Data • Generate and write element behavioral variables such as element forces, stresses, and strain energy to the output file • Write displacements, velocities, accelerations, forces of constraint, etc. to the output file. • Define Subcases • Add the results of subcases

Finally, after completely defining the problem the type of analysis must be specified. This is done by means of Executive Control directives in the Executive Control section which • Specifies the solution sequence by referring to one of the solution sequence numbers from Table 1-1. • Specifies the maximum time for MSC.Nastran execution • Requests that certain executive tables such as displacement set membership be written to the output file.

The Primer includes a description of all of the MSC.Nastran capability associated with statics, normal modes and buckling analyses. The organization of the Primer is as follows: • Description of the physical format of Bulk Data in Appendix A • Summary of all structurally-oriented Bulk Data by function with section references to detailed description in the text in Sec. 1.10 • Description of Case Control in Sec. 1.11 • Description of Executive Control in Sec. 1.12

Examples of static, normal modes and buckling solutions using MSC.Nastran are included in Chapters 12, 15 and 16, respectively. In addition the MSC.Nastran Demonstration Manual includes examples of the use of all the solution sequences and most, if not all, of the MSC.Nastran modeling capability.

12 Nastran Primer

Getting Started Specifying the Finite Element Model - Bulk Data

After preparing the complete input file the user may still have a dilemma … how to interface with the computer in order to execute MSC.Nastran. The execution of MSC.Nastran on any one of the computer systems on which it is installed is controlled by machine-dependent operating system commands. The computer command structure required for a particular installation is described in the MSC.Nastran Application Manual for that particular computer but the best course of action for a new user is to ask a colleague who is familiar with the system. The new user should run small example problems, the examples in the Primer for instance, to make sure that the program actually runs and gives the right answers. Then it is time to think about the problems associated with defining the model of a real-life structure. After mastering the MSC.Nastran capability described in the Primer the user is to move an to non linear analysis. At that time it is appropriate to become familiar with the full set of MSC.Nastran documentation. A selected menu of the documentation that will be of interest is as follows:

1.8

MSC.Nastran User's Manual

Section

Complete description of MSC.Nastran input file

2.0

Description of all Solution sequences

3.0

Description of the MSC.Nastran plotting

4.0

Description of DMAP

5.0

Specifying the Finite Element Model - Bulk Data The finite element model is specified in the Bulk Data section of the input data file. The finite element model is then processed to generate the matrix equations that represent the behavior of the model. A large proportion of the Bulk Data section is applicable to all problems in structural mechanics, independent of whether a linear static or non linear dynamic analysis is to be employed. All the Bulk Data which are independent of analysis type as well as applicable to linear static, normal modes and buckling analysis will be considered in this book. The Bulk Data describes the structural model, specifies sets of constraints and/or loads, and the values of parameters used in the solution sequences. The concept of data sets is of importance because it allows several load or constraint conditions to be included in Bulk Data. The particular set of loads and constraints to be utilized at execution time is specified by a Case Control Directive in the Case Control section. Each logical Bulk Data statement may consist of one or more physical statements. Each data statement must be either prepared according to formats described in Appendix A or by using a free field format described below. Bulk Data statements have the following important attributes or restrictions: 1.Each statement has 80 characters and is separated into ten fields 2.Each logical statement is given a tag which defines its attributes. This tag is a mnemonic that must appear in the first field. The use of a mnemonic field allows data statements to be inserted in Bulk Data in any order. The Bulk Data is then sorted alpha numerically before interpreting the data fields. 3.Each type of data statement is described in alphanumerical order in the MSC.Nastran User's Manual and the Quick Reference Guide. The Bulk Data appropriate for linear static, normal modes and buckling are summarized by function in Summary of Structurally Oriented Bulk Data (p. 15) of the present text, and a detailed description and the associated capability is presented in later chapters. The user should note that there are restrictions on the format (i.e., real or integer) and range that the data elements must satisfy.

Nastran Primer 13

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4.The datum may be placed anywhere within the appropriate fields. The input interpreter assumes that blanks or a commas are a termination characters; thus, imbedded blanks within the field are not allowed. 5.A data statement may consist of one or more physical lines so that continuation is allowed. A continuation is effected by a unique continuation mnemonic in the last field of the “parent” statement and the same mnemonic is preceded by a continuation character in the first field of the “child”. The construction of continuations is discussed in more detail in Appendix A, but it should be emphasized at this point that the continuation mnemonic must be unique among all Bulk Data mnemonics in the entire input file. When Bulk Data is sorted, the existence of a special continuation character in the mnemonic field implies that a “chiId”/"parent" relation exists, and MSC.Nastran then searches for a "parent" with the same continuation mnemonic as that found on the continuation card.

1.8.1

Free Field Bulk Data

MSC.Nastran provides the user with an alternative to the Bulk Data card format which is described in Appendix A. The alternate form allows the user to separate field entries for small field cards by a comma (,) thereby greatly simplifying the task of preparing Bulk Data input. The form of the free field Bulk Data card is f 1, f 2, f 3, f 4, f 5, f 6, f 7, f 8, f 9, f 10 where the datum, f i, are separated by commas. Successive commas define blank fields so that the free field Bulk Data statement TABLED1,,,,,,,,,+T1 would be interpreted as TABLED1 in field one, blanks in fields two through nine and ‘+T1’ in field ten.

1.8.2

Replication and Field Generation

A simple, but useful, set of duplication and generation commands can be imbedded in Bulk Data. These commands are 1.Duplication of field from previous statement is defined by an equal sign, (=), in the associated field. 2.Duplication of all remaining fields is defined by two equal signs, in the first of the fields to be repeated. 3.Generation of an incremented value of a field from a statement is defined by the construct, *(w), in the associated field where w is the real or integer value of the increment. 4.Repeated duplication and incrementation is defined by including a duplication control statement after the last statement definition having the form =(n), where n is the number of additional statements to generate. The rules which govern replication features are as follows: 1.Only small-field data statements can be generated 2.Continuations cannot be generated 3.Data items must be separated with one or more blanks or a comma

14 Nastran Primer


1.8.3

Summary of Structurally Oriented Bulk Data

The Bulk Data statements of interest for static and normal modes analyses of structures are grouped by function and are described below Data Statement 1.8.3.1 SPOINT

1.8.3.2

Description

Reference

Scalar Degrees of freedom Defines scalar point

Defining Scalar Degrees of Freedom SPOINT

Coordinate Systems

CORD1C

Cylindrical Coordinate System defined by point coordinates

Subsidiary Coordinate System

CORD1R

Rectangular Coordinate System defined by point coordinates


CORD1S

Spherical Coordinate System defined by point coordinates


CORD2C

Cylindrical Coordinate System defined by grid points


CORD2R

Rectangular Coordinate System defined by grid points


CORD2S

Spherical Coordinate System defined by grid points


1.8.3.3

Grid Points

GRDSET

Default values for Grid data fields

Grid Points -GRID

GRID

Defines Grid point

Grid Points -GRID

1.8.3.4

Constraint and Partitioning

MPC

Specify linear relationship between degrees of freedom

Multipoint Constraints - MPC

MPCADD

Forms the union of MPC sets

Multipoint Constraints - MPC

SPC

Specify constraints to displacement degrees of freedom

Single Point Constraints - SPC

SPC1

Specify constraints to displacement degrees of freedom


SPCADD

Forms the union of SPC sets


Nastran Primer 15

1


OMIT

Specify degrees of freedom to be removed from the analysis set by static condensation

Static Condensation -OMIT and ASET

OMIT1

Specify degrees of freedom to be removed from the analysis set by static condensation


ASET

Specify degrees of freedom to be retained in analysis set


ASET1

Specify degrees of freedom to be retained in analysis set


SUPORT

Specify statically determinate set of degrees of freedom Support for Free to remove rigid body motion Bodies - SUPORT

1.8.3.5 GENEL

1.8.3.6

Flexibility or stiffness for a general element

Support for Free Bodies - SUPORT

Scalar Elements

CELAS1,2

Springs connecting Grid Point degrees of freedom

Scalar Elastic Elements

PELAS

Property for elastic spring


CELAS3,4

Springs connecting scalar degrees of freedom


1.8.3.7

16 Nastran Primer

General Element

One Dimensional Elements

CROD

Axial rod element

Rod and Truss Elements

PROD

Property for the ROD element


CTUBE

An axial rod having a cross section of a thin walled tube Rod and Truss Elements

PTUBE

Property for the TUBE element

CBAR

A beam bending element having a uniform cross section The BAR Beam Bending Element

PBAR

Property for the BAR element

The BAR Beam Bending Element

BEAM

A beam bending element having non uniform cross section

The BEAM Bending Element

PBEAM

Property for the BEAM element

The BEAM Bending Element

CBEND

A curved beam element

Curved Beam Element

PBEND

Property for the BEND element

Curved Beam Element


Getting Started Specifying the Finite Element Model - Bulk Data 1.8.3.8

Two-Dimensional Shear Element

CSHEAR

Quadrilateral Shear Panel

Shear Panels

PSHEAR

Property for the SHEAR element

Shear Panels

1.8.3.9

Two-Dimensional Shell Elements

CQUAD4

Flat isoparametric quadrilateral shell connection

Shell Elements

CQUAD8

Curved isoparametric quadrilateral connection

Shell Elements

CTRIA3

Flat isoparametric triangular shell connection

Shell Elements

CTRIA6

Curved isoparametric triangular shell connection

Shell Elements

PSHELL

Common property card for QUAD4, QUAD8, TRIA3, andTRIA6

Shell Elements

PCOMP

Layered composite element property

1.8.3.10

PCOMP Layered Composite

Three Dimensional Elements

CHEXA

Six-sided isoparametric element with variable number of node points

Defining Solid Elements

CPENTA

Five-sided isoparametric element with variable number of node points


CTETRA

Four sided isoparametric element with variable number of node points


PSOLID

Common properties for HEXA and PENTA


1.8.3.11

Constraint Elements

RROD

Extensional constraint in direction of line segment between two points

RBAR

Rigid link between two points

RBAR - Rigid Link Connecting Two Grids

RTRPLT

Rigid connection between three points

RTRPLT - Rigid Triangular Constraint Element

RBE1

Rigid connection between an arbitrary

Rigid Constraint Element - RBE1 and RBE2

RBE2

number of points

Rigid Constraint Element - RBE1 and RBE2

RSPLINE

Elastic spline-fit constraint element

RBE3

Constraint element with dependent degrees of freedom WeightedAverage equal to a weighted average of other degrees of freedom Constraint Element - RBE3

1.8.3.12 MAT1

RROD Extensional Constraint

Elastic Constraint Element RSPLINE

Material Specification Isotropic material coefficients

Isotropic Material

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MAT2

Anisotropic Material for Surface Elements

MAT8

Orthotropic material coefficients

Orthotropic Material for Surface Elements

MAT9

Anisotropic material for three-dimensional stress state

Anisotropic Material for Solid Elements

MATT1

Specifies temperature dependence of material coefficients on MAT1

TemperatureDependent Materials

MATT2

Specifies temperature dependence of coefficients specified on MAT2


MATT9

Specifies temperature dependence of material coefficients on MAT9


TABLEM1,2,3,4

Material Property Tables

1.8.3.13

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Anisotropic coefficients for two-dimensional elements

Material Property Table

Specification of Static Loads

FORCE

Concentrated force at geometric grid point defined by vector components

Concentrated External Forces at Grid Points

FORCE1

Concentrated force at geometric grid point defined two Concentrated grid points External Forces at Grid Points

FORCE2

Concentrated force at geometric grid point defined cross Concentrated product External Forces at Grid Points

MOMENT

Concentrated moment at geometric grid point defined by vector components


MOMENT1

Concentrated moment at geometric grid point defined two grid points


MOMENT2

Concentrated moment at geometric grid point defined cross product


SLOAD

Load applied to scalar point

Load Applied to Scalar Point SLOAD

PLOAD1

Defines load distribution along length of BAR or BEAM elements

Distributed Load on Beam PLOAD1

PLOAD2

Defines a normal pressure load on two-dimensional elements

Uniform Normal Pressure PLOAD2

PLOAD4

Defines a surface load on two and three dimensional isoparametric elements

Nonuniform Surface Tractions - PLOAD4

Getting Started Specifying the Finite Element Model - Bulk Data GRAV

Specifies gravity vector for entire structure

RFORCE

Specifies rotational velocity vector for calculating centrifugal forces

Rotation About an Axis- RFORCE

LOAD

Load combination

Combined Loading - LOAD

SPCD

Specify an enforced displacement for static analysis which is treated as a load rather than as a constraint

Enforced Displacements SPCD

DEFORM

Initial misfit of lineal elements

Enforced Deformation DEFORM

TEMP

Grid point temperature

Grid Point Temperatures TEMP and TEMPD

TEMPD

Default grid point temperature

Grid Point Temperatures TEMP and TEMPD

TEMPRB

Specify temperature distribution for truss and beam elements

. Thermal Field for Axial Elements TEMPRB

TEMPP1

Specify temperature distribution in plate and membrane Thermal Field for elements Surface Elements

1.8.3.14

Gravity Loads GRAV

Mass Elements

CMASS1

Scalar mass connection

Scalar Mass CMASS

CMASS2


Scalar Mass CMASS

CMASS3


Scalar Mass CMASS

CMASS4


Scalar Mass CMASS

PMASS

Scalar mass property

Scalar Mass CMASS

CONM1

Specifies a 6 x 6 symmetric mass matrix partition on the Concentrated Grid diagonal of the system mass matrix Point Mass CONM1

CONM2

Concentrated mass at (or optionally offset from) GRID Offset point Concentrated Grid Point Mass CONM2

1.8.3.15

Eigenvalue Extraction

EIGR

Eigenvalue extraction method, Inverse Power

The Inverse Power Method

EIGR

Eigenvalue extraction method, Tridiagonal Methods

EIGR for the Triadiagonalizatio n Methods

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EIGL

Lanczos method of eigenvalue extraction

1.8.3.16

The Specification of the Lanczos Procedure EIGRL

Miscellaneous Data Statements

DMIG

Direct input of matrix coefficients

Direct Specification of System Matrices

PARAM

Specifies parametric input data

Bulk Data Image A-1

The Case Control Directives perform the following functions: 1.Specify set of Bulk Data input that are to be included in the analysis at execution time. 2.Select solution techniques as appropriate. 3.Control the calculation and display of derived quantities. 4.Control subcases. The Case Control Directives are free-field. The directive is separated from the specification by one or more blanks or by an equal sign.

1.8.4

Data Selection

The concept of data sets in Bulk Data allows the user to define any number of different load and constraint sets. The particular set to be used at execution time is then specified by an appropriate Case Control Directive. The form of the data selection Case Control directive is = SID where indicates the particular type of data to be included in the analysis and SID is the set identification number associated with the data in Bulk Data. In this book, the following convention is to be used for variable names or quantities to be filled in by the user: The name or quantity will be lowercase, preceded by ‘<‘, and succeeded by ‘>’. The angle brackets(< and>) are not part of the name or quantity to be added by the user. All Case Control directives may be shortened to the first four characters provided they are unique.

1.8.5

Load Selection

The following Case Control Directives are used to select sets of load-type Bulk Data: DEFORM LOAD

1.8.6

Selects initial element deformations specified by a set of DEFORM Bulk Data. Selects a set of static loads defined by LOAD, PLOAD, PLOAD1, PLOAD2, PLOAD4, FORCE, MOMENT, FORCE1, MOMENT1, FORCE2, MOMENT2, GRAV, RFORCE, or SPCD data statements.

Example: Load Specification

The Case Control Directives DEFORM = 2 LOAD = 13

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specify that the set of initial deformations defined by DEFORM Bulk Data with set identification number 12 and the set of static loads defined by appropriate Bulk Data having set identification number 13 are to be combined to define the set of grid point loads.

1.8.7

Thermal Field Selection

The following Case Control Directives are used to specify thermal fields defined by temperature-type Bulk Data: TEMP(LOAD)

Selects a temperature set defined by TEMP, TEMPD, TEMPRB, TEMPP1 Bulk Data to be used for calculating equivalent thermal loads.

TEMP(MATERIAL)

Selects a temperature set defined in Bulk Data to be used only for determining temperature-dependent material! properties.

TEMPERATURE

Selects a temperature set from Bulk Data for calculation of thermal loads and temperature-dependent properties.

1.8.8

Example: Thermal Field Specification

The Case Control Directive TEMP = 25

specifies that the temperatures defined by suitable Bulk Data with set identification 25 are to be used to calculate both thermal loads and temperature-dependent properties. Selection of Constraints MPC

Selects a multipoint constraint set defined by MPC or MPCADD Bulk Data.

SPC

Selects a single point constraint set defined by SPC, SPC1, or SPCADD Bulk Data.

1.8.9

Example: Specification of Constraints

The directives MPC = 10 SPC = 10

specify that multipoint constraints defined by MPC Bulk Data with set identification number 10 and single point constraints defined by SPC Bulk Data with set identification number 10 are to be used to define constraints.

1.8.10

Specification of Eigenvalue Method

The user specifies an eigenvalue extraction algorithm for normal modes analysis with the following Case Control Directive: METHOD

Selects a real eigenvalue extraction method defined by an EIGR, EIGRL or EIGB Bulk Data.

Example: Specification of Eigenvalue Routine

The Case Control Directive METHOD = 10

specifies that the EIGR Bulk Data with set identification number 10 defines the real eigenvalue extraction algorithm to be used.

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1.8.11

Output Control

The Case Control Directives that are used for output control are listed below in functional groups.

1.8.12

Titling Directives

TITLE

Defines text to be printed on first line of each page of output.

SUBTITLE

Defines text to be printed on second line of each page of output.

LABEL

Defines text to be printed on third line of each page of output.

1.8.13

Output Line Control

LINE

Defines the number of data lines per printed page, default is 50 lines for 11-inch paper.

MAXLINES

Defines the maximum number of output lines, default is 100,000.

1.8.14

Bulk Data Echo

ECHO options for Bulk Data SORT

Prints only sorted Bulk Data (default)

UNSORT

Prints only the unsorted Bulk Data

BOTH

Prints both unsorted and sorted Bulk Data

NONE

Prints neither

PUNCH

Writes the sorted Bulk Data on the system punch file

NOSORT

Prints only Bulk Data cards which were changed on a "check pointed" run (see Sec. 1.9.3)

SORT(t1, t2)

Prints sorted echo of only card types t1, t2,...along with their line numbers.

The user can selectively control the echo of selected portions of the unsorted Bulk Data deck by requesting ECHO = UNSORT

and including the desired Bulk Data between pairs of ECHOON and ECHOOFF statements in Bulk Data.

1.8.15

Set Specification

SET

1.8.16

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Defines set of point numbers, element numbers, times, or frequencies for use in point and element output directives.

Element Output Requests

ELFORCE

Requests forces in a set of structural elements.

STRESS

Requests the stresses in a set of structural elements.

ESE

Requests strain energy for set of elements (see associated PARAMeter "TINY" in Appendix B).


1.8.17

Grid and Scalar Point Requests

SPCFORCES

Requests single point constraint forces at a set of points.

OLOAD

Selects applied loads at a set of grid or scalar points.

ACCELERATION

Requests accelerations for a selected set of grid or scalar points.

DISPLACEMENT

Requests displacements for a selected set of grid or scalar points.

VELOCITY

Requests velocities for a selected set of grid or scalar points.

VECTOR

Equivalent to DISPLACEMENT.

GPFORCE

Requests grid point force balance for a selected set of physical points.

MPCFORCE

Requests multi-point constraint forces at a set of points

The form of the output directives is = N

where is one of the output directives and N is the number of a SET that defines the elements or grid points for which the calculated quantity is to be displayed. If N is ALL, then the calculated quantity will be displayed for ALL elements or ALL points. A subset of ALL must be defined before it can be referenced by an output control directive. For example, the specification ELFORCE = ALL

specifies that the element forces in all the elements in the finite element model are to be printed. The specification SET 5 = 5, 6, 10 ELFORCE = 5

specifies that element forces are to be printed for set five which consists of elements five, six, and ten. The specification SET 10 = 1 THRU 25, EXCEPT 10 DISP = 10

specifies that the displacements be printed at grid or scalar point 1 through 25, except for point 10.

1.8.18

Subcases

In general, a separate subcase may be defined for each loading condition. In static analysis, separate subcases are also allowed for each set of constraints. Subcases may be used in connection with output directives, such as requesting different output for each mode in a real eigenvalue solution. Case Control is structured so that a minimum amount of repetition is required when using subcases. Only one level of subcase definition is provided and all items placed above the subcase level (ahead of the first subcase) will be used for all following subcases, unless overridden within the individual subcase. In static problems, provision has been made for the combination of the results of several subcases. This capability is convenient for combinations of individual loading conditions and for superposition of solutions for symmetrical and antisymmetrical boundary conditions. Typical examples of subcase definition are given, following a brief description of the subcase Case Control Directives. SUBCASE

Defines the beginning of a subcase that is terminated by the next SUBCASE Case Directive.

SUBCOM

Defines a combination of two or more immediately preceding subcases in statics problems. Output requests above the subcase level are used.

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SUBSEQ

Must appear in a subcase defined by SUBCOM to give coefficients for making the linear combination of the preceding subcases.

SYM

Defines a subcase in statics problems for which only output requests within the subcase will be honored. Primarily for use with symmetry problems where the individual parts of the solution may not be of interest.

SYMCOM

Defines a combination of two or more immediately preceding SYM subcases in static problems. Output requests above the subcase level are used.

SYMSEQ

May appear in a subcase defined by SYMCOM to give the coefficient for making the linear combination of the preceding SYM subcases. A default value of 1.0 is used if no SYMSEQ card appears.

REPCASE

Defines a subcase in statics problems that is used to make additional output requests for the previously run subcases. This capability is required because multiple output requests for the same item are not permitted in the same subcase. Output requests above the subcase level are still used.

MODES

Repeats the subcase in which it appears MODES times for eigenvalue problems. Used to repeat the same output request for several consecutive modes.

The following examples of the use of subcases in the Case Control deck have been included: 1.Static Analysis and Multiple Loads DISPLACEMENT = ALL MPC = 3 SUBCASE 1 SPC = 2 TEMP(LOAD) = 101 LOAD = 11 SUBCASE2 SPC = 2 DEFORM = 52 LOAD = 12 SUBCASE3 SPC = 4 LOAD = 12 SUBCASE4 MPC 4 SPC 4

Four subcases are defined by this example. The output request for displacements will be honored for all subcases and MPC set three will be used for all subcases except subcase four where MPC = 4 overrides the specification above the subcase level. Since subcase one and two have the same constraint sets the solutions will be performed simultaneously. The thermal load set 101 and external load set 11 will be combined for subcase one, and the external load set 12 will be combined with deformation set 52 for subcase two. Since there is no load set specified in subcase four, the SPC set four must include enforced grid point displacements. 2.Linear Combination of Subcases SPC = 2 OUTPUT SET 1 1 THRU 10, 20,30 DISP ALL

24 Nastran Primer

Getting Started Specifying the Finite Element Model - Bulk Data STRESS =1 SUBCASE1 LOAD = 101 OLOAD = ALL SUBCASE2 LOAD = 201 OLOAD ALL SUBCOM 51 SUBSEQ 1.0, 1.0 SUBCOM 52 SUBSEQ 2.5, 1.5

Two static load cases are defined. SPC 2 is used for each subcase because the specification is made above the subcase level. The subcase combination SUBCOM = 51 is a linear combination of one times the result of subcase one and one times the result of subcase two. The subcase combination SUBCOM = 52 consists of 2.5 times subcase 1 plus 1.5 times the result of subcase 2. The displacements at all grid points and the stresses in the elements defined by set one will be printed. 3.Statics Problem with One Plane of Symmetry SET 1 = 1, 11, 21, 31, 41 SET 2 = 1 THRU 50 DISP = 1 ELFORCE = 2 SYM 1 SPC = 11 LOAD = 21 OLOAD = ALL SYM 2 SPC = 12 LOAD = 22 SYMCOM 3 SYMSEQ 1., 1. SYMCOM 4 SYMSEQ 1., - 1.

Two symmetry subcases are defined by subcases one and two. The symmetric subcase combination SYMCOM 3 defines the sum of the previous subcases while SYMCOM 4 define the difference. The nonzero components of static load will be printed for subcase one; and the displacements associated with set one and the element forces for set two will be printed for symmetric combination three and four. Note that only output requests defined within the symmetric subcase are honored while output requests above the subcase level are honored by SYMCOM. 4.Use of REPCASE in Statics: SET 1 = 1 THRU 10 SET 2 = 11 THRU 20 SET 3 = 21THRU30 SUBCASE1 LOAD = 100 SPC 101 DISP ALL SPCF = ALL ELFO = 1 REPCASE 2 ELFO = 2 REPCASE 3 ELFO = 3

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One subcase is requested for solution and with two additional subcases defined for output. The displacements and SPC forces at all grid points and the element forces in SET 1 will be displayed for subcase one. The element forces in SET 2 will be printed for REPCASE 2 and those in SET 3 will be printed for REPCASE 3. 5.Use of MODES in Eigenvalue Problem METHOD = 12 SUBCASE 1 MODES 5 STRESS = ALL SUBCASE 6 DISPLACEMENT = ALL

The METHOD Case Control Directive points to an EIGR data statement that defines the real eigenvalue method to be used and associated parameters. The MODES Case Control Directive causes the results for each eigenvalue to be considered as a separate subcase starting with the subcase number containing the MODES request. The stress will be printed in all elements for the first five modes, and the displacements will then be printed for all additional modes. If the MODES request is not included in a subcase, the output requests will apply to all eigenvalues starting with the subcase number. The following output request could then be used to print the displacements associated with the first three modes. METHOD = 12 SUBCASE1 MODES = 3 DISP = ALL SUBCASE 4 DISP = NONE

1.8.19

Case Control Examples

The following examples illustrate the use of Case Control Directives for static and normal modes analysis. 1.8.19.1

Static Analysis

The following tasks 1.Print both sorted and unsorted bulk data. 2.Specify maximum number of output lines to be 75000. 3.Specify 35 lines of output per page. 4.Two sets of loads have been defined; a temperature set five, and a force set six. Print the results for each and then combine the results by multiplying thermal loads by a factor of one and the external loads by two. 5.Multipoint constraints for both load sets are defined by MPC set 14. 6.Single point constraints for both load sets are defined by SPC set 12. 7.Print external loads, displacements, and forces of reaction at all grid points. 8.Print element type output for elements 100 through 200, except for element 151. could be performed by the following Case Control directives: TITLE = STATIC ANALYSIS OF TRUSS SUBTITLE = CASE CONTROL EXAMPLE ECHO =BOTH MAXLINES = 75000 LINES =35 SET 100 = 100 THRU 200, EXCEPT 151 MPC 14

26 Nastran Primer

Getting Started Specifying the Finite Element Model - Bulk Data SPC 12 SPCFORCE = ALL OLOAD = ALL DISPLACEMENT = ALL STRESS = 100 ELFORCE = 100 SUBCASE 1 LABEL = THERMAL STRESSES TEMP(LOAD) = 5 SUBCASE 2 LABEL STRESSES DUE TO EXTERNAL LOADS LOAD 6 OLOAD ALL SUBCOM 12 LABEL COMBINED OUTPUT SUBSEQ 1.0, 2.0 TEMP(LOAD) = 5 BEGIN BULK

It should be noted that the TEMP(LOAD) = 5 must be included under the SUBCOM Case Control directive in order to properly account for thermal strains when element stresses and element forces are calculated for the combined loading condition. 1.8.19.2

Normal Modes Analysis

The following tasks 1.Print out sorted and unsorted Bulk Data. 2.Specify maximum number of output lines to be 125,000. 3.Specify 35 lines of output per page. 4.Print nodal displacements for all degrees of freedom for all nodes. (10 eigenmodes are expected) 5.Print stress output for elements 2, 5, and 10 for eigenmodes four and five. could be performed by the following Case Control directives:

10

TITLE = NORMAL MODE ANALYSIS OF TRUSS SUBTITLE = CASE CONTROL EXAMPLE ECHO = BOTH METHOD = 12 MAXILINES = 125000 LINES = 35 SET 5 = 2, 5, 10 DISP = ALL SUBCASE 1 LABEL = OUTPUT FOR MODES 1 THRU 3 MODES = 3 SUBCASE 4 LABEL = OUTPUT FOR MODES 4 AND 5 STRESS = 5 MODES = 2 SUBCASE 6 LABEL= OUTPUT FOR MODES 6 SUBCASE 11THRU MODES = 5 SUBCASE 11 DISP = NONE VELO = NONE ACCEL = NONE BEGIN BULK

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1.9

Getting Started Executive Control

Executive Control The Executive Control section is the first of the four subsections that comprise the input file. The Executive Control Directives perform the following functions: 1.Provides user control over the computer operating system by an optional NASTRAN data statement. 2.Specifies the solution sequence number from Table 1-1 using the SOL directive. 3.Provides a means of modifying the DMAP for a altering solution sequences by means of ALTER directive. 4.Provides a means of printing internal data sets. The Executive Control Directives that accomplish these functions are freeform, which means that the directive may start at any place on the data card. The directive is separated from the specification by one or more blanks.

1.9.1

Solution Sequence

The solution sequence is specified by the SOL Executive Control statement SOL K

where the allowable solution numbers are defined in Table 1-1 which is repeated here for convenience. Solution Number

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101*

Static analysis

103*

Normal modes analysis

105*

Buckling

106

Non linear static analysis

107

Direct complex eigenvalue analysis

108

Direct frequency response

109

Direct transient analysis

110

Modal complex eigenvalue analysis

111

Modal frequency response

112

Modal transient analysis

114*

Cyclic symmetry static analysis

115*

Cyclic symmetry normal modes analysis

118

Cyclic symmetry frequency response

129

Non linear transient analysis

144

Static aeroelastic response

145

Aerodynamic flutter

146

Aeroelastic response

153

Static structural/steady state heat transfer

159

Transient structural/steady state heat transfer

Getting Started Executive Control

1.9.2

Specification of Execution Time

The time specified in Executive Control is MSC.Nastran time and is used in conjunction with timing algorithms for each of the computer models. The system estimates the time required for a matrix operation and then checks to see if there is sufficient time remaining, based on the current time used and the MSC.Nastran time estimated. If sufficient time remains, MSC.Nastran will terminate normally. The Executive Control directive is TIME K

where K is the maximum allowable cpu execution time in minutes. The default is 60 minutes.

1.9.3

Diagnostic Print Requests

The DIAG Executive request provides the user with a means of requesting printout of executive tables or for requesting override of defaults. The form of the request is DIAG

The DIAGs of general interest to the structural analyst are described below. Complete tabulation of all DIAGs can be found in Section 3 of the MSC.Nastran Quick Reference Guide. DIAGs which are of general interest are. DIAG Meaning Number 4

Print cross reference table which is produced during DMAP compilation.

5, 6

Print BEGIN and END time for each DMAP module

8

Print matrix summary data, called matrix trailers, as matrices are generated. Matrix trailers contain information that may be helpful in determining the cause of fatal errors.

14

Print the DMAP source compilation.

17

Writes the compiled DMAP sequence to the system PUNCH file.

31

Writes the Link Specification Table and Module Property Table to the output file

Multiple options may be selected by using multiple integers separated by commas. Other options and other rules associated with the DIAG directive that primarily concern the programmer can be found in the NASTRAN User’s Manual Vol. I.

1.9.4

Termination of Executive Control CEND

1.9.5 1.9.5.1

(Required) Indicates end of Executive Control deck.

Executive Control Examples Static Analysis TIME 3 SOL 101 CEND

SOL 101 specifies that the static analysis is to be used. TIME specifies that the total MSC.Nastran execution time is expected to be three minutes or less. 1.9.5.2

Static Analysis, Optional Output TIME 3 SOL 101 DIAG 14

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Getting Started TRUSS EXAMPLE

CEND

The Executive Control is the same as the previous case except for the inclusion of the DIAG directive. As a result of this statement the DMAP listing for the SOL 101 will be printed. Recommended practice if DMAP ALTERS or user-supplied DMAP program is used. 1.9.5.3


The previous examples are applicable to the normal modes analysis as well as static analysis. The only change is that the SOL directive is changed to specify normal modes: SOL 103

1.10

TRUSS EXAMPLE Consider the following input file: Line 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 38 29 30 31

1.10.1

Entry SOL 1 CEND TITLE = STATIC ANALYSIS OF A PLANE TRUSS LOAD = 101 SPC = 201 DISPLACEMENTS = ALL ELFORCE = ALL STRESS = ALL BEGIN BULK $...1...*...2...*...3...*...4...*...5...*...6...*...7.... GRID 1 0 0. 0. 0. 0 GRID 2 0 0. 180. 0. 0 GRID 3 0 0. 360. 0. 0 GRID 4 0 240. 0. 0. 0 GRID 5 0 240. 180. 0. 0 CROD 1 8 1 2 CROD 2 4 2 3 CROD 3 6 1 4 CROD 4 4 2 4 CROD 5 6 2 5 CROD 6 4 3 5 CROD 7 4 4 5 PROD 4 4. 30 PROD 6 6. 30 PROD 8 8. 30 MAT1 30 30.E6 0.3 FORCE1 101 3 60000. 4 1 SPC1 201 2 1 SPC1 201 12 4 PARAM AUTOSPC YES ENDDATA

Components of the File.

The input file contains the three sections of the Problem Definition Section described above. The Bulk Data Section completely describes the simulation model and it is by far the largest part of the input file. The bulk data statements used in the above examples are described in the following chapters of the Primer: Chapter 6 6 5 9

30 Nastran Primer

Modeling Component Coordinate systems Grid and scalar points The element library Constraints and partitioning

Getting Started TRUSS EXAMPLE 9 10

1.10.2

Material properties Loads

The Input File

The following section describes each line of the input file: Statement Number

1

Description

SOL 1 This statement specifies the solution algorithm that is to be run. You may prefer to use the name of the algorithm, such as STATICS , rather than a solution number.

2

CEND A required termination for the "Executive Control" section of the input file. The next section, which starts after CEND, is the "Case Control Section."

3

TITLE An optional Case Control directive specifying a title that is to be printed at the top of every page of output.

4

LOAD A required Case Control directive that selects a set of loads (set 101 in this case, from the Bulk Data Section).

5

SPC A Case Control directive that selects a set of single point constraints (set 201 in this case).

6

DISPLACEMENTS A Case Control directive that requests the r ecovery and output of displacements at specified grid points (ALL) in the model.

7

ELFORCES A Case Control directive requesting that element forces be calculated and output for specified elements (ALL) in the model.

8

STRESS A Case Control directive requesting that the calculation and output of element stress f or specified elements (ALL) in the model.

9

BEGIN BULK A required Case Control directive terminating the Case Control Section.

10

Comment Comment lines always begin with a dollar sign, “$”; everything to the right of the $-sign is a comment. This comment helps align data in the Bulk Data fields.

11-15

The GRID Bulk Data described in Chapter 5 specifies the location of material points in the analysis. The location of the grid points is shown by Table 1-4

16-22

CROD The CROD, which is described in Chapter 6, specifies the use of the ROD element, identifies the connected grid points G 1 and G2, and points to a PROD property as described by Table 1-5:

23-25

PROD The PROD described in Chapter 6 defines the properties for each ROD element and points to a material set as described Table 1-6

26

MAT1 The MAT1 described in Chapter 8 defines the physical properties for an isotropic material having a MID = 30, E = 3.e7 and ν = 0.3

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Statement Number

27

Description

FORCE1 The FORCE1 described in Chapter 10 specifies a concentrated force at grid point 3 with a magnitude of 60,000 and in the direction of the line drawn from grid point 4 to grid point 1 which is the negative X -direction.

28-29

SPC1 The SPC1 described in Chapter 7 specifies the displacement constraints to u y at Grid 1 and to u x and u y at Grid 4:

30

The PARAM data entity defines a scalar parameter named AUTOSPC whose value is YES in the solution algorithm. This parameter controls the action of a processor which detects singularities in the assembled stiffness matrix as described in Chapter 12.

31

ENDDATA Required terminator of the Bulk Data Section.

Table 1-1 Grid Point Locations Grid ID

Reference Coordinate

X

Y

Z

System

1

Basic

0.

0.

0.

2

Basic

0.

180.

0.

3

Basic

0.

360.

0.

4

Basic

240.

0.

0.

5

Basic

240.

180.

0.

Table 1-2 ROD Elements ROD ID

PID

G1

G2

1

8

1

2

2

4

2

3

3

6

1

4

4

4

2

4

5

6

2

5

6

4

3

5

7

4

4

5

Table 1-3 PROD Property definitions

32 Nastran Primer

PID

Material ID

Area

4

30

4.

6

30

6.

8

30

8.


1.10.3

Truss Schematic Diagram

With the interpretation provided above for each Bulk Data statement, the physical model can now sketched, including grid point locations, element connections, loads, constraints, etc., as shown in Figure 1-1. y

60,000 lbs. 3 2

6 5 5

2 4

7

1

4 1

x

3

Figure 1-1 Truss Example.

1.10.4

Running the Truss Example

NASTRAN can be executed from a command window. The actual form of the command is dependent on the version of NASTRAN. That being said, MSC Nastran is executed from the command line by the following command: Nastran filename [keyword1=value1 keyword2=value2 ... ]

where filename is the name of an input file having a “.dat” extension and where keyword is one or more optional keywords which are described in Chapter 1 of the MSC.Nastran Quick Reference Guide. Assuming you have saved the truss example input in file ‘truss.dat’, the example would be executed by Nastran TRUSS.DAT

The output file is given the name of the data file with an ".OUT" extension. The truss output which will appear in the same directory containing TRUSS.DAT. After the program is finished executing you can now use the editor to examine the output file.

1.10.5

Examining the OUT File

As you scroll through the output file using your editor, you will find the following sections: 1.10.5.1

The Title Page:

Tells you the current version number and the creation date. 1.10.5.2

Executive Control Section Echo

This section is an echo of the Executive Control submitted for this example.

Nastran Primer 33

1


1.10.5.3

The Case Control Section Echo

This section is an echo of the Case Control submitted for this example. 1.10.5.4

The Unsorted Bulk Data Echo

The Bulk Data echoed exactly as it appears in the input file. Note that there is a format bar on the top of the page that helps justify the data fields of the Bulk Data. 1.10.5.5

The Sorted Bulk Data Echo

After reading the data file the Bulk Data statements are first sorted and then interpreted. The interpretation phase makes sure that MSC.Nastran knows about the Bulk Data statement name, such as GRID. The remaining datum must satisfy type and range criteria. A type is associated with each data item, i.e., real, integer, or literal string and several datum have specific ranges of values. Each Bulk Data item has a line number in the sorted bulk data that is used to reference detected errors. No errors exist in this example, so MSC.Nastran will go to the next section. 1.10.5.6

Grid Point Singularity Table

This table is automatically printed if there are any singularizes. The presence of the AUTOSPC parameter with a value YES tells MSC.Nastran to remove any detected similarities using either single or multipoint constraints as appropriate and to proceed with the analysis. 1.10.5.7

Solver message - Epsilon

The solver will print diagnostic information that you should look at for every analysis. The information includes the value of normalized virtual work done by the calculated set of displacements. If the displacements are the true equilibrium set, the virtual work should be zero. Generally, the value will be small, usually 10.E-8 or less. A warning message will appear if the value is greater than 10.E-5. The displacements at the grid points.

Printed because of the DISP = ALL Case Control directive. You should see the following nonzero values.

D I S P L A C E M E N T V E C T O R POINTTYPE 1 2 3 4 5

G G G G G

1.10.5.8

T1

T2

T3

R1

R2

R3

0.0 0.0 0.0 0.0 -6.75E-2-2.850E-1 0.0 -1.35E-1-7.512E-1 0.0 0.0 0.0 0.0 6.75E-2-3.650E-1

0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0

The forces of constraint.

Printed because of the SPCFORCE Case Control directive. You should see the following non zero values.

F OR CE S O F S IN GL E- PO IN T C ON ST RA IN T POINTTYPE 1 4

34 Nastran Primer

G G

T1

T2

T3

R1

R2

R3

0.0 9.00E+4 0.0 -9.00E+4

0.0 6.00E+4

0.0 0.0

0.0 0.0

0.0 0.0

Getting Started Problems 1.10.5.9

END OF JOB.

An End of Job statement is printed at the end of program execution.

1.11

Problems [1] Run the truss example and compare the results in the output file, truss.f06, with those presented in Chapter 1. [2] For the truss shown by Figure 1-1, determine the exact node point displacements and element forces and compare with the calculated results.

1.12

References [1] M.Turner, R. Clough, H. Martin and L. Topp, “Stiffness and deflection analysis of complex structures”, J. Aero. Sci., Vol 23, No. 9, Sept. 1956, pp. 805-823

Nastran Primer 35

1

36 Nastran Primer

Getting Started References

2 2.1

Basic Relationships from Engineering Mechanics

Theoretical Foundations The primary focus of this book is understanding the theory and use of finite elements for simulating the linear behavior of structures in support of engineering design. The theoretical foundation for this area of engineering mechanics is called the linear theory of elasticity. The complete nonlinear theory would include the effects of nonlinear geometry including large displacements and non linear strain-displacement relations and non linear material behavior. In the present study we will consider only linear material behavior but we will include non linear strain-displacement relations which are required to present the theory of buckling. In this chapter we will state the relevant principles that will allow us to study the linear theory of finite elements. In later chapters we will present general concepts for matrix structural analysis and the solution of linear sets of equations, the inclusion of mass effects and the study of structural dynamics and the inclusion of first order non linear effects in the element formulation and the study of linear buckling.

2.2

Continuum Mechanics The structure is considered to be a continuum wherein the material is distributed without gaps and empty spaces. Continuum mechanics is then employes to study the behavior of the solid continuum under the effect of external forces. The continuum is assumed to have certain material properties including the mass density per unit volume, ρ , and certain other properties which will be identified and described in subsequent sections

37 Nastran Primer

2

2.3

Basic Relationships from Engineering Mechanics Kinematic Relations Relations

Kinematic Re Relations In the continuum mechanics the deformed state is defined by determining how each material point moves under the action of external forces. The position of a material point, A, is identified by a position vector, R which is also also represented represented in the text text by the notation notation { R}= { R x , R y , R z } as shown by Figure 2-1 where the components are understood to be defined with reference to the cartesian reference system.

x 3

A

u r

A*

x 3

x 1

x 2

R y 3

x 1 y 2

x 2 y 1

Figure 2-1 Displacement Vector

As a result of applied loads and enforced boundary displacements, displacements, the material point moves to a nearby point, A* having a position vector, { R}, in the deformed configuration. The displacement displacement of the point from { r } to { R} is represented by the displacement vector, {u} having components u, v, and w in the x, y and z directions, directions, respectively. The displacement at a point P then then defines as: {u} = { R} -{r }

2.4

(Eq. 2-1)

Strains As the continuum moves as a result of displacements, an elemental volume in the neighborhood of the point A translates and its neighborhood rotates as a rigid body, and strains. The internal strain energy of the body is associated with strains which cause volumetric changes. The linear components strain at a point are given by: {ε} = [ L L]{u} Where [ L L] is the linear differential operator:

38 Nastran Primer

(Eq. 2-2)

Basic Relationships from Engineering Mechanics Strains

∂ 0 0 ∂ x ∂ 0 0 ∂ y ∂ 0 0 ∂ z [ L ] = ∂ ∂ 0 ∂ y ∂ x ∂ ∂ 0 ∂ z ∂ y ∂ ∂ 0 ∂ z ∂ x

(Eq. (Eq. 2-3) 2-3)

The strains produced by [ L] are called the engineering strains. We should note that the six independent independent components of strain are represented as the following pseudo vector:

' $ $ $ $ ε = % $ $ $ $ "

ε xx ( $ ε yy $ $ ε zz $ & γ xy $ $ γ yz $ $ γ zx #

(Eq. (Eq. 2-4) 2-4)

The components ε xx , ε yy , ε zz are called the extensional strains and the components γ xy , γ yz , γ zx are called the shear strains. The extensional strains are changes in length per unit of original length in the coordinate directions; and, the shear strains are changes of the ninety degree angle formed by the two coordinate axes associated with its idices. It should be noted that the order of the strain components is rather arbitrary and different authors prefer alternate orders. The same same order will be used used for the components components of stress. stress. The engineering strains are related to, but are not equal to, the components of a second order strain tensor which we will designate in indicial notation as l ij where the indices have a range of i, j = x, y, z . It is convenient to use the tensor strain components in tensor transformation equations. equations. The two sets of strains are related as follows: l xx = ε xx

l yy = ε yy

l yy = ε zz

an d

γ xy

l xy = ------2

γ yz

l yz = -----2

γ zx

(Eq. (Eq. 2-5) 2-5)

l zx = -----2

The components of the tensor are symmetric under interchange interchange of the subscripts as are the components of the engineering strains. strains. Combining equations (Eq. 2-4) and (Eq. 2-5) gives the following strain-displacement relations:

Nastran Primer 39

2

Basic Relationships from Engineering Mechanics Stress Vector on a Surface

' $ $ $ $ $ $ $ $ $ {ε} = % $ $ $ $ $ $ $ $ $ "

( $ $ $ $ ∂v $ ∂ y $ $ ∂w $ ∂ z $ & ∂u + ∂v $ $ ∂ y ∂ x $ ∂v + ∂w $$ ∂ z ∂ y $ $ ∂w + ∂u $ ∂ x ∂ z $ # ∂u ∂ x

(Eq. (Eq. 2-6) 2-6)

The set of three displacements displacements and six strain components represent represent the kinematic behavior of the element. In order order to complete the definition of the behavior of a deformable deformable solid we need to describe the internal forces and the associated equilibrium conditions. conditions.

2.5

Stress Vector on a Surface The intensity of a force, {∆ F }, }, acting at a point P on on a surface with having an outward unit normal, {n}, is represented by the stress vector,

' ( $ τ x $ $ $ { τ } = % τ y & $ $ $ τ z $ " #

(Eq. (Eq. 2-7) 2-7)

where:

{τ} =

40 Nastran Primer

{ ∆ F } lim -------------∆ A ) 0 ∆ A

(Eq. (Eq. 2-8) 2-8)

Basic Relationships from Engineering Mechanics Stress Vector on a Surface

x 3

Deformed Body n

∆ F

P

τ

∆ A

x 2

x 1 Figure 2-2 Stress Vector

The stress vector is a function of position, time and the orientation of the unit normal. The stress vector can be resolved into a normal components in the direction of the normal and two orthogonal shear components acting in the surface.

Nastran Primer 41

2

2.6

Basic Relationships from Engineering Mechanics Components of Stress

Components of of St Stress We now consider a differential volume at an arbitrary point P in in the continuum where stress vectors act an each face as shown by Figure 2-3 The stress vector acting on faces having positive normals in the x, y and z directions directions are {τ}x, {τ}y, and {τ}z, respectively where the subscript on the vector defines the face on which the vector acts by denoting the coordinate associated with the normal to the face.

{ τ } z

z

i3 i2

i1

y

{ τ } y

x

{ τ } x Figure 2-3 Stress vectors acting on the faces

σ zz

τ zy τ zx

τ yz

τ xz

τ xy

τ yx

σ xx Figure 2-4 Components of Stress

42 Nastran Primer

σ yy

Basic Relationships from Engineering Mechanics Components of Stress

Each stress vector acting on a faces can be resolved into components in the coordinate directions. The set of all components of the stress vectors acting on all faces is called the components of stress and is represented by a three by three matrix [ σ]. Each row of the matrix represents components of the stress vector acting on a face.

σ xx σ xy σ xz

= σ xx τ xy τ xz =

σ yx σ yy σ yz

=

σ zx σ zy σ zz

= τ zx τ zy σ zz =

τ yx σ yy τ yz

τ x τ y τ z =

x

τ x τ y τ z τ x τ y τ z

y

z

Where the shear stress components are represented in the following by either the τij or σij since they are equivalent. The first subscript of the general matrix element, σij, defines the face on which the stress vector acts and the second defines the direction. The set [ σ] is called the components of stress and transforms as a second order tensor as described in Appendix D. The components of stress acting normal to a surface and are called the normal stresses and are denoted by σ xx , σ yy, and σ xx. The other components of stress which act tangent to the surface are called the shear stresses and denoted by τxy, τyz and τzx. The components of stress are actually components of a second order tensor that has nine components. Under most conditions the tensor is symmetric and there are then only six independent components of stress. The six independent componen components ts of stress are represented by the pseudo pseudo vector, {σ}.

' $ $ $ $ {σ } = % $ $ $ $ "

σ xx ( $ σ yy $ $ σ zz $ & τ xy $ $ τ yz $ $ τ zx #

(Eq. (Eq. 2-9) 2-9)

It should be noted that the order of the components of stress in the pseudo vector is rather arbitrary, and different authors prefer different orders. The components of stress are positive in the positive coordinate directions when acting on a face having a positive normal; and, are positive in in the negative coordinate coordinate directions when acting acting on a face having a negative normal. In Figure 2-4, we only show the stress components on faces with positive normals. It is not hard to show that the stress vector acting on a face with a normal, { n}, is:

' ( ' ( $ τ x $ $ σ xx n x + σ yx n y + σ zx n z $ $ $ $ $ % τ y & = % σ xy n x + σ yy n y + σ zy n z & $ $ $ $ $ τ z $ $ σ xz n x + σ yz n y + σ zz n z $ " # " #

(Eq. (Eq. 2-10) 2-10)

Nastran Primer 43

2

2.7

Basic Relationships from Engineering Mechanics Constitutive Behavior

Constitutive Behavior The components of stress and components of strain are related by a material law. Empirical evidence has shown that, for a wide r ange of materials having engineering significance, a linear equation relating the six components of stress to the six components of strain compares well with test data. The most general linear relation is of the form:

{ σ } = [ E ] ( { ε } – { ε0 } ) + { σ0 }

(Eq. (Eq. 2-11) 2-11)

where {ε0} are thermal strains and {σ0}are initial stresses. E ], The matrix of elastic coefficients, [ E ], can be shown to be symmetric symmetri c and thus has at most 21 independent elastic coefficients. Using various material symmetry conditions the number of independent coefficients can be reduced. An orthotropic material that has reflective symmetry about three mutually orthogonal planes has nine independent material coefficients; coefficients; and, an isotropic material, for which material properties are independent of direction or orientation of axes at a point, has two independent material coefficients. The two independent coefficients are generally taken to be the so-called engineering constants the modulus of elasticity, or Young’s modulus, E , and and Poisson’s Poisson’s ratio, ν . These These two two constants constants are found from tension test data. Young’s modulus is the slope of the load deflection curve; and, Poisson’s ration is the strain normal to the direction of loading measured as a fraction of the direct strain. Poisson’s effect is such that the strains in the transverse direction are the negative of those in the direction of the loads.

Based on test data the extensional strains are related to the stresses as follows for an isotropic material: 1 E

ε xx = --- [ σ xx – ν ( σ yy + σ zz ) ] 1 E

ε yy = --- [ σ yy – ν ( σ zz + σ xx ) ]

(Eq. (Eq. 2-12) 2-12)

1 ε zz = --- [ σ zz – ν ( σ xx + σ yy ) ] E

The shear strains are then given by: 1 γ xy = ---- τ xy G

1 γ yz = ---- τ yz G

(Eq. (Eq. 2-13) 2-13)

1 G

γ zx = ---- τ zx where G = E ⁄ ( 1 + 2 ν ) . It is rather straight-forward, straight-forward, for an isotropic material, to obtain the inverse relation:

44 Nastran Primer

Basic Relationships from Engineering Mechanics Equilibrium Conditions - Newtonian Mechanics

' $ $ $ $ % $ $ $ $ "

2.8

σ xx ( $ σ yy $ $ σ zz $ E & = -------------------2( 1 + ν) τ xy $ $ τ yz $ $ τ zy #

2 ( 1 – ν ) -------------------1 – 2 ν 2 ν --------------1 – 2 ν 2 ν --------------1 – 2 ν 0 0 0

2 ν --------------1 – 2 ν 2 ( 1 – ν ) -------------------1 – 2 ν 2 ν --------------1 – 2 ν 0 0 0

2 ν --------------1 – 2 ν 2 ν --------------1 – 2 ν 2 ( 1 – ν ) -------------------1 – 2 ν 0 0 0

0 0 0 ' ε ( xx 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1

$ $ $ $ % $ $ $ $ "

$ ε yy $ $ ε zz $ & γ xy $ $ γ yz $ $ γ zy #

(Eq. 2-14)

Equilibrium Conditions - Newtonian Mechanics The stress components must satisfy equilibrium. Employing Newtonian mechanics we can write the equilibrium conditions as: T

[ L ] { σ } – { X } = 0

(Eq. 2-15)

Where [ L] is the same operator that was used in the strain-displacement equations and where { X } are the external body forces. The equilibrium conditions in expanded form are:

∂σ xx ∂τ yx ∂τ zx + + – X x = 0 ∂ x ∂ y ∂ z ∂τ xy ∂σ yy ∂τ zy + + – X y = 0 ∂ x ∂ y ∂ z

(Eq. 2-16)

∂τ xz ∂τ yz ∂σ zz + + – X z = 0 ∂ x ∂ y ∂ z

2.9

Principle of Virtual Work The principle of virtual work is an alternative statement of equilibrium. The virtual work can be expressed as the sum of the virtual work of the internal and external forces. Since the virtual work of the internal forces is the negative of the virtual change of the internal strain energy, U , the virtual work performed by all of the forces acting on a body moving through a virtual displacement is expressed as:

δ W = – δ U + δ W ex

(Eq. 2-17)

where:

δ U =

* {σ}

T

{ δε } d Ω

Ω

δ W ex =

(Eq. 2-18)

* {τ}

T

*

T

{ δ u } d Γ + { X } { δ u } d Ω

Γ

Ω

where Ω is the volume of the continuum and Γ is its surface. The principle of virtual work states that:

Nastran Primer 45

2

Basic Relationships from Engineering Mechanics Solutions from the Theory of Elasticity

If the virtual work is equal to zero for any arbitrary kinematically admissible virtual displacement then the system is in equilibrium.

This is a powerful statement that can be used to obtain the equilibrium equations presented in the previous section directly by substituting the strain displacement relations, which are kinematic constraints on the displacements, into the virtual work expression. The equilibrium equations, which result after employing integration by parts and noting the arbitrary nature of the virtual displacements, can then be interpreted as a set of Lagrange operators which are the forces of constraint required to enforce the kinematic conditions. The principle of virtual work and other energy functionals are a very important foundation for developing approximate solutions since equilibrium is ‘weakly’ enforced in some integrated sense rather than being enforced point-wise which is the case if we insist on satisfying the equilibrium equations everywhere in the continuum. The principle of virtual work and energy functionals will be used in the next Chapter to obtain the solutions for certain continuum problems using a Rayleigh Ritz approach; and, the equilibrium equations for finite elements.

2.10

Solutions from the Theory of Elasticity The theory of elasticity problem is to find a set of three displacements, six strains and six stress components that satisfy the strain displacement equations, the equilibrium equations and the equation of state on the boundary and in the interior of the deformable body and satisfy specified boundary conditions. Not too surprisingly, there a very few solutions to the three dimensional field equations that are of practical engineering importance. However, those solutions which are available are of interest to finite element practitioners since they provide exact solutions which can be used to test elements for which the behavior is three dimensional. We provide two examples in the following sections.

2.10.1

Uniform Stress

We consider a uniform tension force uniformly distributed over the end of a prismatic bar so that the stress vector at the ends at x = 0 and x = L of the solid is a vector in the x - direction. The magnitude of the stress vector is P/A where P is the applies load and A is the cross sectional area. The only non zero components of stress on the end face is σ xx, whose magnitude is P/A. The equilibrium equations and stress boundary conditions are satisfies by taking:

σ xx= P/A, σ yy = σ xx = τ xy = τ yz = τ zx = 0

(Eq. 2-19)

For an isotropic material the strains are given by: 1 E

ε xx = --- σ xx – ν ε yy = ------ σ xx E

(Eq. 2-20)

– ν ε zz = ------ σ xx E

With all shear strains equal to zero. The strains are constant and therefore satisfy the compatibility equations. The displacements can be determined by integrating the strain displacement relations.Taking the origin of the coordinate at the left face and the x - axis to be at the center of the solid as shown hives the following displacements:

46 Nastran Primer

Basic Relationships from Engineering Mechanics Strategies for Solving Real Problems

σ xx Px u = -------- x = ------ E AE νσ xx ν Py v = – ---------- y = – --------- E AE

(Eq. 2-21)

νσ xx ν Pz w = – ---------- z = – -------- E AE 2.10.2

Stretching of a Prismatic Bar by Its Own Weight

Letting ρ g be the weight per unit volume of the bar the body forces are: p z = – ρ g

; p x = 0

; p y = 0

(Eq. 2-22)

It is shown in [2] that the components of stress which satisfy the equilibrium equations and the boundary conditions are:

σ zz = ρ gz , and σ xx = σ yy = τ xy = τ yz = τ zx = 0

2.11

(Eq. 2-23)

Strategies for Solving Real Problems The three dimensional field equations were formulated early in the 19th century but were of little use in supporting the design of structures such as bridges, pressure vessels and steam ships. The needs to support design with simulation lead engineers to seek approximate solutions based on simplifying assumptions. Two major classes of approximations were used. One class, called plane stress and plane strain approximations, assumed strains or stresses were functions of two rather than three dimensions and lead to two dimensional theory of elasticity. The other major class is based on assumed displacements for plate and beam bending. Both classes of approximations lead to solutions which were of great use to design engineers. For example: 1.Determining the stress concentration due to hole in a stretched plate, 2.The fact that a concentrated load led to infinite local stresses, 3.Determining the exact solution for beam bending using two dimensional theory of elasicity 4.Determining strength of material solutions for beam bending. 5.Stresses due to torsion using two dimensional theory of elasticity 6.Plate bending for classical boundary conditions and general loading using strength of materials approximations 7.Stresses in cylindrical and spherical shells However, the needs of modern technology required more accurate analysis support to design. For example, stress analysis techniques bases on simple design formulas that had been adequate for the aircraft from the Wright Flyer through the Boeing B-29 were no longer adequate for the new swept wing B-47. New techniques were required and, at this time a great deal of research was devoted to the behavior of skewed plates which lead to a better understanding of the behavior of the bending of non rectangular plates. However, more accurate results analysis were required to design an efficient swept wing.

Nastran Primer 47

2

Basic Relationships from Engineering Mechanics Two Dimensional Theory of Elasticity

Fortunately, a group of talented engineers at Boeing, developed a new approximation technique that lead to a much better representation of the structure. Their original work [1] was widely recognized as a revolutionary way of modeling and analyzing structures and lead, in a very short time, to an industry-wide development of general purpose analysis engines based on this new technology which Clough called the finite element method. The early development of two and three dimensional elements was creative, to say the least. These early developments were based on direct formulations using the field equations for one, two and three dimensional elasticity; and plate, shell and beam theory. Only the beam bending element performed well by today’s standards but the early two and three dimensional elements gave tremendous insight to load paths in the complex structures being developed in the late 1950s and early 1960s. The field equations for two dimensional theory of elasticity, plates and beams will be summarized in succeeding sections. The development of the finite element method will be discussed in Chapter 4.

2.12

Two Dimensional Theory of Elasticity 2.12.1

Plane Stress

The basic assumption associated with two dimensional theory is that the behavioral variables do not vary in one dimension. In addition, if a thin plate is loaded only by forces at the boundaries parallel to the plane of the plate all components of stress in the z -direction are assumed to vanish with those in the x and y-directions then the only non-zero components of stress are:

' $ σ xx $ { σ } = % σ yy $ $ σ xy "

2.12.2

( $ $ & $ $ #

(Eq. 2-24)

Plane Strain

Consider bodies for which one dimension is large compared to the other two, and that the loading is perpendicular to and does not vary in the long dimension. In this case the strains in the z -direction are assumed to vanish so that the only non zero components of strain are:

' $ ε xx $ { ε } = % ε yy $ $ ε xy "

2.12.3

( $ $ & $ $ #

Equilibrium Equations

For either plane stress or plane strain the equilibrium equations are:

48 Nastran Primer

(Eq. 2-25)


∂σ xx ∂τ xy ----------- + ---------- + X = 0 ∂ x ∂ y

(Eq. 2-26)

∂σ yy ∂τ xy ----------- + ---------- + Y = 0 ∂ y ∂ x 2.12.4

Strain Displacement Relations

For plane stress and plane strain, the strain displacement relations reduce to: u ε xx = ∂ ----∂ x

∂v ε yy = ----∂ y

(Eq. 2-27)

∂u ∂v γ xy = ----- + ----∂ y ∂ x 2.12.5

Constitutive Relations

For plane stress the constitutive relations for an isotropic material are:

' $ ε xx $ % ε yy $ $ γ xy "

( $ 1 –ν 0 $ 1 -& = E – ν 1 0 $ 0 0 2( 1 + ν ) $ #

' $ σ xx $ % σ yy $ $ τ xy "

( $ $ & $ $ #

(Eq. 2-28)

For plane strain we have:

σ zz = ν ( σ xx + σ yy )

(Eq. 2-29)

Hook’s law for this case is then:

' $ ε xx $ % ε yy $ $ γ xy "

2.12.6

( 2 $ ( 1 – ν ) – ν ( 1 + ν ) 0 $ 1 --2 & = E – ν ( 1 + ν ) ( 1 – ν ) 0 $ $ 0 0 2( 1 + ν ) #

' $ σ xx $ % σ yy $ $ τ xy "

( $ $ & $ $ #

(Eq. 2-30)

Stress Functions

If the body forces are derivable from a potential, V as:

∂V X = – -----∂ x ∂V Y = – -----∂ y

(Eq. 2-31)

Then, the stresses and body forces are derivable from a stress function, φ(x,y), as follows:

Nastran Primer 49

2


2

σ xx – V = ∂ ϕ 2 ∂ y 2

σ yy – V = ∂ ϕ 2 ∂ x

(Eq. 2-32)

2

τ xy

∂φ = – ∂ x ∂y

Substituting (Eq. 2-32) into (Eq. 2-26) then leads to: 4

4

4

2

2

∂ φ + 2∂ φ ∂ φ = – ( 1 – ν ) / ∂ V + ∂ V 0 + - 2 . 2 2 4 4 2 + ∂ x ∂ x ∂ y ∂ x ∂ y ∂ y ,

(Eq. 2-33)

Where the right hand side is zero for the case of no body forces. The solution to two dimensional is thus reduced to solving for a stress function which satisfies (Eq. 2-30) and an appropriate set of boundary conditions.

2.12.7

Solutions based on Stress Functions

Using the so-called semi-inverse method several interesting problems can be solved by noting that second order polynomial functions automatically satisfy (Eq. 2-30). Using this fact the several problems of interest can be solved. These solutions, taken from Timoshenko[2] are summarized below. Bending of a Cantilever Loaded at One End

The cantilever is shown by Figure 2-5 l

l c

A x

c

P y Figure 2-5 Cantilever Beam

The stresses are given by: Px y I

σ xx = – -------- P 2 I

y

2

τ xy = – ----- 1 – ----c

(Eq. 2-34)

2

σ yy = 0 Where I is the area moment of inertia of the cross section. It is interesting to note that the axial stress at a given position x varies linearly over the cross section; and, that the transverse shear stress varies quadratically over the cross section. The displacements are given by:

50 Nastran Primer

Basic Relationships from Engineering Mechanics Two Dimensional Theory of Elasticity 3

2

3

2

2

Py Px y Py Pl Pc u = – ------------ – ν --------- + ---------- + / --------- – ----------0 y + 6 EI 6 GI 2 EI 2 GI , 2 EI 2

3

2

(Eq. 2-35)

3

Py x Px Pl x Pl v = ν ------------ + --------- – ----------- + --------2 EI 6 EI 2 EI 3 EI

Evaluating v(x,y) at y = 0 gives the deflection curve: 3

2

3

Px Pl x Pl v = --------- – ----------- + --------6 EI 2 EI 3 EI

(Eq. 2-36)

Further, evaluating (Eq. 2-36) at x = 0, the loaded end gives the maximum tip displacement: 3

Pl v ma x = --------3 EI

(Eq. 2-37)

Effect of a Circular Hole On Stress Distribution

A rectangular plate with a circular hole is subject to a uniform tension of magnitude S in the x-direction as shown by Figure 2-6. m1

2a

m p

b q

n

x

θ

n1

S

y Figure 2-6 Hole in a Rectangular Plate

In terms of radial coordinates, the components of stress are found to be: 2 4 2 S / 3a 4a 0 a 0 S / σ rr = --- - 1 – -----. + --- - 1 + -------- – --------. cos 2 θ 2 4 2 2+ 2+ r , r r , 2 4 S / 3a 0 a 0 S / σ θθ = --- - 1 + -----. – --- - 1 + --------. cos 2 θ 2 4 2+ r , 2 + r ,

(Eq. 2-38)

S / 3a 2a 0 τr θ = --- - 1 – -------- + --------. sin 2 θ 4 2 2+ r r , 4

2

It can be seen that σtt is greatest at θ = π/2 or θ = 3π/2. At these points:

( σ θθ ) ma x = 3 S

(Eq. 2-39)

Nastran Primer 51

2

2.13

Basic Relationships from Engineering Mechanics Beam Theory

Beam Theory 2.13.1

Stress Resultants

In beam theory it is convenient to use the force and moment resultants shown by Figure 2-7 rather than stress components. z y

y B

σ xy

σ xx

σ xz z

A x

O dx

x

z V z M y

y

V y P

M x

x

x

O

M z

Figure 2-7 Stress components and Stress Resultants on a Beam Cross Section

The forces and moments stress resultants acting on the cross section are obtained by integrating the stress components over the cross section as follows: It should be noted that the surface has a negative normal in the x-direction and that the sign of the bending moment resultants is chosen so that a positive moment will produce a positive curvature of the elastic curve associated with the beam reference axis:

P =

* σ xx d A

V y = – τ xy d A

*

V z = – τ xz d A

A

A

A

M t =

*

* ( τ xz y – τ xy z )d A

(Eq. 2-40)

(Eq. 2-41)

A

M y = – σ xx z d A

*

M z = – σ xx y d A

A

A

where A is the area of the cross section.

52 Nastran Primer

*

(Eq. 2-42)


If the beam is statically determinate then the stress resultants can be found using only equilibrium considerations. Otherwise it is necessary to consider the deformations.

2.13.2

Stresses due to Extension and Bending

Classical beam theory is based on the observation that for a long prismatic bar loaded with transverse loads the displacement in the axial direction varies in accordance with the BernoulliEuler theory of bending. The displacement at point B in Figure 2-7 is assumed to be: u B = u ( x ) – θ y z – θ z y

(Eq. 2-43)

where O is a point on a reference axis, and where u( x) is the axial displacement along the xaxis and θ y and θ z are the angles by which the plane of the cross section rotates about the y and z axes, respectively. Using the first of (Eq. 2-3), the extensional strain in the axial direction is then given by:

ε xx =

du d θ z d θ y – y – z = C 1 + C 2 y + C 2 z d x d x d x

(Eq. 2-44)

The Bernoulli-Euler displacement assumption is equivalent to postulating a linear strain distribution over the cross section and that the shear stress components are small compared to the axial stress, σ xx. The axial stress is then given by the one dimensional stress strain relation:

σ xx = E ( ε xx – ε 0 )

(Eq. 2-45)

The substitution of the stress strain relation, (Eq. 2-45) and the strain displacement relation, (Eq. 2-44) into the expressions for stress resultants then gives: P --- = AC 1 + yA C 2 + zA C 3 E M y – ------= zA C 1 + I yz C 2 + I yy C 3 E

(Eq. 2-46)

M y – ------= yA C 1 + I zz C 2 + I yz C 3 E

where A =

* d A A

1 y = --- y d A A

*

A

1 2 I yy = --- z d A A

*

A

1 z = --- z d A A

*

(Eq. 2-47)

A

1 2 I zz = --- y d A A

*

A

1 I yz = --- yz d A A

*

A

The solution for the coefficients in (Eq. 2-36) is simplified if the y and z axes are chosen so that y = z = 0 . The x-axis is therefore taken as the axis of centroids of the cross section. The coefficients are then found to be:

Nastran Primer 53

2


P C 1 = ------ EA

1 ( M z I yy – M y I yz ) C 2 = – --- -------------------------------------- E ( I I – I 2 ) yy zz

(Eq. 2-48)

1 ( M y I zz – M z I yz ) C 3 = – --- ------------------------------------- E ( I I – I 2 )

yz

yy zz

yz

where I yy, I zz and I yz are the area moments of inertia of the cross section. Using these values the stress becomes P / M z I yy – M y I yz 0 / M y I zz – M z I yz 0 σ xx = --- – - ---------------------------------. y – - --------------------------------- . z A + I I – I 2 , + I I – I 2 , yy zz yz yy zz yz

(Eq. 2-49)

If the axes coincide with principle axes then the stress is: M y P M z σ xx = --- – /+ -------0, y – /+ -------0, z A I zz I yy

(Eq. 2-50)

Comparing (Eq. 2-49) and (Eq. 2-50) it would appear that the use of principal axes results in a great simplification of the stress. However, for those cases where the principle axes are not apparent, the calculation of principal axes, principal moments of inertial, and then the transformation of loads and stress recovery points to the principal axes takes much more time than the use of (Eq. 2-49)directly. And of course there is always the possibility of making computational errors in process.

2.13.3

Equilibrium Equations

We assume the beam is subjected to an applied force, { p} and moment, {m}, per unit length as shown by Figure 2-8

dP P + -------ds ds

P z

V z

A ′

P y V y

dM z M z + ----------ds ds m z

dV y V y + ---------ds ds

P x

O′

P

m y M y

dV z V z + ---------ds ds

A ′ m t

O′

dM t M t + --------- ds ds dM y M y + ---------- ds ds

M t M z

(a)

(b) Figure 2-8

Using standard procedures described by Rivello[3] the equilibrium equations are dV y

dP = – p x d x dM t d x

54 Nastran Primer

= – m t

d x dM y d x

= p y

= V z + m y

dV z d x

= p z

dM z d x

(Eq. 2-51) = V y – m z

Basic Relationships from Engineering Mechanics Plate theory

where ϕ is the rotation about the x axis. For a beam loaded by a line load in the z -direction and using principal axes for the moment of inertia we obtain: 2

d M y d x

2

= p z (Eq. 2-52)

2

2/

d w0 EI yy . = p z 2 2 d x , d x + d

2.13.4

Solution to Beam Bending Equation

Integrating (Eq. 2-52) for the case where the load is zero results in: 3

2

w ( x ) = a 3 x + a 2 x + a 1 x + a 0

(Eq. 2-53)

The coefficients can then be found using specific boundary conditions on displacement or load. The solution for the cantilever beam which is fixed at the end x = 0 and for which the boundary conditions are: w(0) = w' (0) = 0; V z (L) = P ; My(L) = 0 is: 3 2 P w ( x ) = --------------( x – 3 L x ) 6 EI yy

The maximum displacement at x = L is 3

PL w ma x = -------------3 EI yy

(Eq. 2-54)

which agrees with the exact solution from two dimensional theory given by (Eq. 2-37).

2.14

Plate theory Plate theory is obtained from the three-dimensional theory of elasticity for a continuum which is planar and which has one dimension, t, which small compared to a characteristic length of the plate as shown by Figure 2-9 z ,w

y,v

α

t x,u,β

Nastran Primer 55

2


Figure 2-9

The plate is embedded in a reference coordinate system as shown below such that the x y plane coincides with the reference surface of the plate and the z -axis is in the direction of the thickness.

2.14.1

Kinematic Relationships

The three dimensional plate is replace by a two dimensional reference surface to which we will ascribe properties associated with the thickness. We denote the displacement at any point in the plate by (u, v, w) which are defined in terms of displacements and rotations of the reference surface. We will represent the displacement at any point in terms of the displacements, (u0 , v0 , w0)and mean rotations, (α, β) of the reference surface where, following MacNeal’s notation[4]:

α = – θ y

β = θ x

(Eq. 2-55)

where the reference surface displacements and rotations are functions of x and y only. The in plane displacements are then assumed to vary linearly over the thickness and so the displacements at any point in the cross section is represented in terms of reference surface displacements and rotations as follows: u = u 0 ( x, y ) – z α ( x, y ) v = v 0 ( x, y ) – z β ( x, y )

(Eq. 2-56)

w = w0

In addition, the normal stress in the direction normal to the plane of the plate is assumed equal to the mean surface pressure:

σ zz = p z ( x, y )

(Eq. 2-57)

where p z is the symmetrical part of the transverse load applied to the top and bottom surfaces. The strains at any point in the cross section can then be expressed in terms of reference surface displacements and rotations as:

ε xx =

∂u0 ∂α 0 ∂u = – z = ε xx – z χ x ∂ x ∂ x ∂ x

∂v = ∂v 0 – z ∂β = 0 – z ε yy = ε yy χ y ∂ x ∂ x ∂ x ε zz =

(Eq. 2-58)

∂w = 0 ∂ z

and

∂w – γ xz = α ∂ x γ yz =

∂w – β ∂ y

0 0 ∂α ∂β γ xy = γ xy – z / + 0 = γ xy – z χ xy + ∂ y ∂ x,

56 Nastran Primer

(Eq. 2-59)


where the membrane strains are denoted by a superscript and the curvatures of the reverence surface are:

∂α χ x = ∂ x 2.14.2

∂β χ y = ∂ y

∂α + ∂β χ xy = ∂ y ∂ x

(Eq. 2-60)

Kirchhoff Hypothesis

The Kirchhoff hypothesis is that the transverse shear components γ xz and γ yz are zero. The curvatures are then related to the transverse displacements of the reference surface as follows: 2

χ x =

2

∂w ∂ x

χ y =

2

2

∂w ∂ y

χ xy

2

∂w = ∂ x ∂y

(Eq. 2-61)

Using this assumptions Timoshenko[5] shows that the behavior of the plate is governed by the following fourth order differential equation: 4

4

4

∂ w + 2∂ w ∂ w = p + --- z -2 2 4 4 D ∂ x ∂ y ∂ x ∂ y

(Eq. 2-62)

where D for a solid cross section of thickness, t , having an isotropic material is: 3

Et D = ------------------------2 12 ( 1 – ν )

(Eq. 2-63)

Solutions to (Eq. 2-62) can be obtained for rectangular plates with various boundary conditions and loads as described by Timoshenko. For the case of a simply supported boundary loaded by a sine pressure distribution:

π x π y p z = p 0 sin ------ sin -----2a 2b where a and b are the length and width of the plate, respectively. A displacement function of the form:

π x π y w ( x, y ) = w0 sin ------ sin -----2a 2b satisfies the simple support boundary conditions. The coefficient w 0 can then be determined by substituting the expression for w(x, y) into (Eq. 2-62): p 0 w 0 = ---------------------------------4 1 1 π D /+ ----- + -----0, 2 2 a b

Nastran Primer 57

2


Several solutions obtained by Timoshenko are of particular interest for validating the behavior of plate-type finite elements.The following table presents the central displacement for a square plate having a Poisson’s ratio, ν = 0.3, is presented for clamped and simply supported edges for both a uniform pressure load and a concentrated load at the center. Table 2-1 Central Deflection of a Square Plate Uniform Load Simple Supports Clamped

Concentrated Load

4

3

0.00416( ( p z a ) ⁄ D )

0.01 ( ( Pa ) ⁄ D )

4

3

0.00140( ( p z a ) ⁄ D )

0.00560( ( Pa ) ⁄ D )

Early plate-type finite elements based on the Kirchhoff hypothesis were found to give poor performance. A large literature has been devoted to improving the performance of both Kirchhoff plate finite elements as well as those based on Mindlin plate theory.

2.14.3

Moment and Force Resultants

In plate and shell theory the stress components are integrated through the thickness to obtain a set of transverse shear force and moment resultants shown by Figure 2-10.

Q y

z y

m y

x m x

m xy

m xy

m xy

Q x

Q x

m xy m x

m y

Q y Figure 2-10 Shear and Moment Resultants

The membrane force resultants, { N } ,(not shown), are related to the symmetric part of the stress which is called the membrane stress and is designated by { σ}m:

' $ N x $ { N } = % N y $ $ N xy "

( $ $ & = $ $ #

The transverse shear resultants, { Q}, are:

58 Nastran Primer

' $ σ xx $ * – --t - %$ σ yy 2 $ τ xy " t --2

( $ $ & dz $ $ #m

(Eq. 2-64)


' ( $ Q x $ {Q} = % & = $ Q y $ " #

' $ τ * – --t - %$ τ xz yz 2 " t --2

( $ &dz $ #

(Eq. 2-65)

The moment resultants, { M }, are the integrated moment of the bending stress which is the antisymmetrical part of the stress components:

' $ M x $ { M } = % M y $ $ M xy "

2.14.4

( ' $ t $ σ xx --$ 2 $ & = – * t % σ yy – --- $ $ 2 $ $ τ xy # "

( $ $ & zd z $ $ #b

(Eq. 2-66)

Constitutive Relations

The plate cross section is assumed to be represented by n layers of Anisotropic material. This implies that the integration presented in the previous section is to be replaced by a piecewise integration over each individual layer having a thickness, t i. Substituting the strain displacement relation into(Eq. 2-64) and integrating over the thickness then leads to: { N}=[A]{ ε0 }+[B]{ χ}

(Eq. 2-67)

where Jones[6] shows the matrices [ A] and [ B] are obtained as follows for a layered composite material: n

[ A ] =

t i+1

1 *

[ E m ] i dz

(Eq. 2-68)

1 * z [ E b ]i dz

(Eq. 2-69)

i = 1 t i

and n

[ B ] =

t i+1

i = 1 t i

Similarly, the moment resultant is: { M }=[ D]{c}+[ B]{ε0}

(Eq. 2-70)

where [ D] is: n

[ D ] =

t i+1

2

1 * z [ E b ]i dz i=1

(Eq. 2-71)

t i

In the same manner, the transverse shear, {Q}, is related to the shear strains as follows:

{ Q } = k [ G ] { γ }

(Eq. 2-72)

where the parameter, k, is a transverse shear factor for the cross section and where [ G] is: Nastran Primer 59

2

Basic Relationships from Engineering Mechanics Problems

n

[G] =

t i+1

1 * i=1

2.14.5

[ E s ]i dz

(Eq. 2-73)

t i

Equilibrium Equation

The moment and resultants are shown by Figure 2-10. The equilibrium equations for a flat pate are then found in the usual way to be:

∂ M x ∂ M xy + + Q x + m x = 0 ∂ x ∂ y ∂ M y ∂ M xy + + Q y + m y = 0 ∂ y ∂ x

(Eq. 2-74)

∂Q x ∂Q y + + p z = 0 ∂ x ∂ y where m x and m y are applied moments.

2.15

Problems [1] Considering (Eq. 2-14), what are the limiting values of Poisson’s ratio for an isotropic material [2] For an isotropic material, verify the matrix of elastic coefficients , [ E ] , given by (Eq. 2-30) [3] Given the displacement functions – 3

u = 10 ( x + y ⁄ 2 ) – 3

v = 10 ( y + x ⁄ 2 ) w = 0

Determine the components of strain for for the case of plane stress. [4] Using the components of strain detrmined from problem 3, determine the components of stress for an isotropic material [5] Given the following relations for transverse displacment and mean rotations for plate bending, determine the extensional and shear strains. – 3

10 2 2 u = ----------- ( x + xy + y ) 2

∂w = 10 – 3 x + y 2 θ x = ( ⁄ ) ∂ y θ y = –∂ = – 10 ( y + x ⁄ 2 ) ∂ x w

– 3

[6] Given the following relations for displacements, determine the components of strain.

60 Nastran Primer

Basic Relationships from Engineering Mechanics References – 3

u = 10 ( x + y ⁄ 2 + z ⁄ 2 ) – 3

v = 10 ( x ⁄ 2 + y + z ⁄ 2 ) – 3

w = 10 ( x ⁄ 2 + y ⁄ 2 + z )

[7] Given the components of strain from Problem 6, determine the components of stress for an isotropic material [8] Given the transverse displacement for a uniform beam P z 3 2 3 w ( x ) = – --------------( x – 3 L x + 2 L ) 6 EI yy

Determine the moment, transverse shear and distributed load at any point along the length of the beam.

2.16

References [1] M.Turner, R. Clough, H. Martin and L. Topp, “Stiffness and deflection analysis of complex structures”, J. Aero. Sci., Vol 23, No. 9, Sept. 1956, pp. 805-823 [2] S. Timoshenko and J.N. Goodier, Theory of Elasticity, McGraw-Hill, 1951. [3] R.M. Rivello, Theory of Flight Structures , McGraw-Hill, 1969, p.156. [4] R.H. MacNeal, Finite Elements: Their Design and Performance, Marcel Decker, New York, 1994, p.371. [5] S. Timoshenko and J.N. Goodier, Theory of Elasticity, McGraw-Hill, 1951, p.82. [6] R.M. Jones, Mechanics of Composite Materials, McGraw-Hill, New York, 1975, pp. 152156.

Nastran Primer 61

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62 Nastran Primer

Basic Relationships from Engineering Mechanics References

3

Variation Principles and Approximation Theory

Early finite elements developed using direct formulations served as a useful bridge to modern computational analysis. However, truly useful finite element formulations for two and three dimensional continuum elements are based on variational principles and approximation theory. Variational principles are based on work and energy considerations, and the concept of admissible behavioral functions. Admissible displacements, for example, are those that satisfy the kinematic constraints, and admissible stresses are those that satisfy the equilibrium conditions in the interior and the stress boundary conditions on that part of the boundary having prescribed stresses. These two sets of behavioral variables are related by constitutive relations. Variational methods allow us to determine the equations associated with the theory of elasticity and, by employing Lagrange multipliers, yield the Hellinger-Reissner principle and the principle of minimum complementary energy. We will see that the principle of minimum potential is associated with the displacement formulation of finite element while the HellingerReissner and Principle of Minimum Complementary potential are useful for mixed and stress formulations. When the problem of elasticity cannot be solved exactly, and that is generally the case, the use of a variational principle will yield an approximate solutions. However, as we will see in a simple example, care must be taken when applying the Raleigh-Ritz method since the stress distribution is not nearly as accurate as the displacement approximation

3.1

Principle of Virtual Work The principle of virtual work states that if the virtual work is zero for any admissible displacement then the system is in equilibrium. The virtual work is the sum of the virtual work of the internal forces plus the virtual work of the external forces. Since the virtual work of the internal forces is the negative of the variation of the internal energy we can write: – δ U + δ W ex t = 0

(Eq. 3-1)

where δU is the variation of the internal strain energy and δW ext is the virtual work of all external forces. The variation of the internal strain energy for a virtual strain variation is:

δ U =

* ( σ xx δε xx + σ yy δε yy + σ zz δε zz + τ xy δγ xy + τ yz δγ yz + τ zx δγ zx )d Ω

(Eq. 3-2)

Ω

Using the components of stress and strain as pseudo vectors we can write:

δ U =

* {σ} Ω

The virtual work is then:

63 Nastran Primer

T

{ δε } d Ω

(Eq. 3-3)

3

Variation Principles and Approximation Theory Complementary Virtual Work

*

T

*

T

*

T

– { σ } { δε } d Ω + { X } { δ u } d Ω + { t } { δ u } d Γ = 0 Ω

Ω

(Eq. 3-4)

Γ

The equilibrium equations and the boundary conditions can then be obtained by using the strain displacement relations and using integration by parts.

3.2

Complementary Virtual Work The principle of complementary virtual work is obtained by considering a body in equilibrium. We then consider a variation of stress, { δσ}such that the equilibrium equations are satisfied. During this process the strains are given by the strain displacement relations so that the virtual work done by the virtual change of stress is:

* ( { ε } – [ L ] { u } )

T

{ δσ } d Ω +

* ( { u } – { u } )

T

{ t } dS = 0

(Eq. 3-5)

Γ 2

Ω

Where Γ 2 is that portion of the boundary on which the displacement is prescribed. Using integration by parts and noting that the stress components satisfy equilibrium in the interior and that the variation of the stress vector must vanish on the part of the boundary on which the stress is prescribed leads to the following equation which is called the principle of complementary virtual work:

* {ε } Ω

3.3

T

{ δσ } d Ω –

* {u}

T

{ δ t } d Γ = 0

(Eq. 3-6)

Γ

2

Minimum Potential Energy In stating the principle of virtual work we expressed the virtual change of the work of the internal forces in terms of the variation of the internal strain energy. In developing the principle of minimum potential energy we assume the existence of a positive definite strain energy density function, A, of the following form: 1 T A = --- { ε } [ E ] { ε } 2

(Eq. 3-7)

Taking the variation of A with respect to the strains and using the constitutive relations for a linearly elastic body then gives: T

δ A = { σ } { δε }

(Eq. 3-8)

Assuming the body and surface forces are derivable from potential functions Φ(x,y,z) and Ψ(x,y,z), respectively we can transform the principle of virtual work to: – δ U + δΦ + δΨ = δΠ = 0 where

64 Nastran Primer

(Eq. 3-9)

Variation Principles and Approximation Theory Minimum Total Complementary Potential Energy

* δ Ad Ω

δ U =

Ω

* { X }

δΦ =

T

{ δ u } d Ω

(Eq. 3-10)

Ω

* { t }

δΨ =

T

{ δ u } d Γ

Γ

1

and,

Π =

* ( A – Φ ) d Ω – * { t } Ω

T

{ u } d Γ

(Eq. 3-11)

Γ

1

The principle of minimum total potential energy states: Among all states of strain which satisfy the kinematic constraints within and on the boundary of the body, that which makes the total potential an absolute minimum also satisfies equilibrium.

It is noted that the principle of total strain energy is restricted to linearly elastic bodies whereas the principle of virtual work is valid for any constitutive relation.

3.4

Minimum Total Complementary Potential Energy The principle of minimum total complementary energy can be derived from the principle of virtual complementary work by expressing the equation of state in terms of a stress rather than a strain as: T 1 B = --- { σ } [ S ] { σ } 2

(Eq. 3-12)

where [S ] = [ E ]-1. The principle of complementary work can then be transformed to:

* δ B d Ω – * { t } Ω

T

{ δ u } d Γ = δΠ c

(Eq. 3-13)

Γ

2

where

Π =

* ( A – Φ ) d Ω – * { t } Ω

T

{ u } d Γ

(Eq. 3-14)

Γ

1

The principle of minimum total complementary energy states: Among all states of stress satisfying equilibrium, that state which makes the total complementary potential energy a minimum satisfies the kinematic constraints in the interior and on the boundary.

Nastran Primer 65

3

3.5

Variation Principles and Approximation Theory Generalized Principle

Generalized Principle Modern element formulations use multi-field approximations; i.e., displacements are assumed on the boundary and either an assumed stress or strain field is assumed in the interior. Other forms of a minimum principle are thus desirable. A very general form is presented by Washizu[1], as follows, which is termed a generalized principle.

Π I =

* [ A – { X }

T

T

*

{ u } ] d Ω – ( { ε } – [ L ] { u } ) { σ } d Ω –

Ω

(Eq. 3-15)

Ω

* { t }

T

T

*

{ u } d Γ – ( { u } – { u } ) { p } d Γ

Γ

Γ

In this generalized principle { σ} and { p} are Lagrange multipliers, which are to be determined, which are required to hold the constraints of the strain-displacement relations in the interior and the imposed displacement on Γ 2, respectively. The actual solution is found by determining the stationary conditions for the functional, ΠΙ. Taking the variation with respect to the displacement, { u}, the strains, {ε} and the Lagrange multipliers, { σ} and { p} leads to:

δΠ I =

* [ ( [ E ] { ε } – { σ } )

T

T

{ δε } – ( { ε } – [ L ] { u } ) { δσ }

(Eq. 3-16)

Ω T

T

* [ ( { X } – { X } )

– ( [ L ] { σ } + { X } ) { δ u } ] d Ω +

T

{ δ u } ] d Γ

Γ

1

–

* [ ( { u } – { u } )

T

{ δ p } ] d Γ +

Γ

* [ ( { X } – { p }

T

) ] { δ u } d Γ +

Γ

2

2

with the following stationary conditions:

{ σ } = [ E ] { ε } in Ω

(Eq. 3-17)

{ ε } = [ L ] { u } in Ω

(Eq. 3-18)

T

3.6

[ L ] { σ } + { X } = 0 in Ω

(Eq. 3-19)

{ t } = { t } on Γ 1

(Eq. 3-20)

{ u } = { u } on Γ 2

(Eq. 3-21)

{ p } = { t } on Γ 2

(Eq. 3-22)

Hellinger-Reissner Principle The Hellinger-Reissner principle is a special case of the generalized principle, (Eq. 3-14), for which the strains are no longer independent. The coefficients of the variations of strain, {δε}, in (Eq. 3-16) must vanish so that the strains must be determined from the stress strain relations. The Hellinger-Reissner principle is then expressed as:

66 Nastran Primer

Variation Principles and Approximation Theory Approximation Theory

Π R =

* ( – [ B + { X }

T

T

{ u } ] + { σ } [ L ] { u } ) d Ω –

(Eq. 3-23)

Ω

* { t }

T

T

*

{ u } d Γ – ( { u } – { u } ) { p } d Γ

Γ

Γ

where B is: T

B = { σ } { ε } – A

(Eq. 3-24)

Elimination of the strain components using the stress strain relations then gives the following relation for the variation of B: T

δ B = { ε } { δσ }

(Eq. 3-25)

The independent quantities open to variation are the stress components, { σ} and the stress vector, { p}, and the displacement, { u}. Taking variation with respect to these quantities leads to the following stationary conditions

[ L ] { u } = [ E ] { σ } in Ω

(Eq. 3-26)

together with (Eq. 3-19) through (Eq. 3-22). Another form of the functional (Eq. 3-23) obtained by using integration by parts is: *

– Π R =

*

T

T

T

*

*

T

[ B + ( [ L ] { σ } ) { u } ] d Ω – ( { t } – { t } ) { u } d Γ – { t } { u } d Γ (Eq. 3-27)

Ω

Γ

Γ

1

2

where (Eq. 3-22) has been used to eliminate { p}. The independent quantities open to variation are then the components of stress, { σ}, and the displacements, {u}.

3.7

Approximation Theory Approximation theory lets us choose a reasonable approximation to the solution defined in terms of a one or more generalized coordinates and then employ one of the variation principles to determine the generalized coordinate. The resulting approximation will then satisfy the subsidiary relations in some mean square sense. In the principle of virtual work the displacements must satisfy the strain displacement relations in Ω and the displacement boundary conditions on Γ 2, that portion of the boundary on which displacements are prescribed. We take the displacements to be: N

u ( x, y, z ) = u 0 ( x, y, z ) +

1 an un ( x, y, z ) n N

v ( x, y, z ) = v 0 ( x, y, z ) +

1 bn vn ( x, y, z )

(Eq. 3-28)

n N

w ( x, y, z ) = w 0 ( x, y, z ) +

1 cn wn ( x, y, z ) n

Nastran Primer 67

3

Variation Principles and Approximation Theory Rayleigh-Ritz Soution of a Beam

where u0, v0 and w0 are chosen such that they satisfy the enforced displacements on Γ 2; and where un, vn, and wn are chosen to vanish on Γ 2 for all indices. The displacements are now functions of the undermined parameters so the variation of w, for example, is: N

δ w ( x, y, z ) = w 0 ( x, y, z ) +

1 δcn wn ( x, y, z ) n

After substituting the assumed displacements and into the strain displacement relations the virtual work is seen to be a function of the variations of the independent coefficients. Since the variations of an, bn and cn are arbitrary it follows that the coefficients of each variation must vanish independently resulting in 3 N equations for the 3 N independent coefficients.

3.8

Rayleigh-Ritz Soution of a Beam As an example, consider the case of a simply supported beam of length, L, a cross section area moment of inertia, I , and modulus of elasticity, E , with a concentrated load at the center. We consider an approximate solution of the form:

π x w ( x ) = c 1 sin ----- L

(Eq. 3-29)

which satisfies the displacement boundary conditions at x = 0, L, where w( x) is the transverse displacement in the z -direction. It can be shown that the principle of virtual work for the beam bending in the x-z plane is: L

L – w xx x + P δ w / ---0 = 0 , EI δ w xx , d + 2,

*

(Eq. 3-30)

0

where the second term accounts for the virtual work of the applied load. Taking the variation of the assumed displacement function,(Eq. 3-29) and substituting into (Eq. 3-30)then gives:

/ L 0 4 EI π x0 2 π / -------------c sin ------ d x + P . = 0 δ c 1 - – * 4 1+ , . 2 L + 0 ( 2 L ) ,

(Eq. 3-31)

Since the variation of the parameter, δ c 1 , is arbitrary, the coefficient must be equal to zero. Evaluating of the integrals and solving for c 1 then gives: 3

2 PL c 1 = ------------4 π EI 3

(Eq. 3-32)

2 PL π x w = ------------- sin -----4 π EI L The difference between the approximate and exact displacement at the center of the beam is less than 3%. However, when the moment is evaluated using the moment curvature relationship the error is found to be as great as 25%.

68 Nastran Primer

Variation Principles and Approximation Theory References

This example shows the general result found for assumed approximate displacements: The displacements are reasonably accurate but the force quantities obtained by differentiating the displacement function may vary widely from the exact solution. And, of course it is generally the stress type behavior that is used as a failure criterion in design. In the next Chapter we use approximation theory to obtain the equilibrium equations in a finite region of the continuum. The result will be the equations for the finite element method.

3.9

References [1] K. Washizu, Variational Methods in Elasticity and Plasticity , Second Edition, Pergamon Press, Oxford, 1974, p.32.

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3

70 Nastran Primer

Variation Principles and Approximation Theory References

4

Finite Element Formulations

Successful finite element formulations for two and two dimensional elements are based on variational principles. The use of the variational principle allows us to satisfy equilibrium in some average sense and is termed a weak formulation. The objectives of this chapter are to: • Develop a procedure for determining the equilibrium equations for finite elements by considering the axial rod and beam bending elements • Introduce basis functions for interpolating behavioral variables in a region • Introduce the concept of mapping from metric space to parameter space • Develop the finite element equations for two dimensional elasticity for a displacement and mixed formulations • Develop the finite element equations for the dynamic response of a quadrilateral shell element

4.1

Finite Element Formulation In the formulation of the approximate equilibrium equations for an element the displacement field within the element is approximated in a Rayleigh-Ritz sense. The number of generalized coordinates in the approximation must be the same number as the number of node point displacements for the element so that the generalizes coordinates can be related to the node point degrees of freedom.

4.2

Rod Element In order to describe the procedure let us consider the uniform rod shown by Figure 4-1 .

y

u1, F 1

u2, F 2

1

z

71 Nastran Primer

2

x

4

Finite Element Formulations Rod Element

Figure 4-1 Axial Rod Element

The axial displacements and forces at the node points are:

{u} = { F } =

T

u1 u2

(Eq. 4-1)

T

F 1 F 2

The only non zero stress component is σ xx so the principle of virtual work becomes:

*

T

T

δ W = – { σ xx } { δε xx } d Ω + { F } { δ u } = 0

(Eq. 4-2)

Ω

The axial displacement is assumed to vary linearly along the axis of the rod so that the following polynomial is assumed to represent the displacement u = a + bx

(Eq. 4-3)

The coefficients a and b are now related to the node point displacements by noting that u ( 0 ) = u1

(Eq. 4-4)

u ( L ) = u2

Solving for a and b gives a = u1

(Eq. 4-5)

u 2 – u 1 b = ---------------- L

The assumed displacement function, written in terms of the node point displacements as generalized coordinates, is x x u ( x ) = / 1 – ---0 u 1 + / ---0 u 2 = N 1 u 1 + N 2 u 2 + L, + L, x N 1 = / 1 – ---0 + L,

(Eq. 4-6)

x N 2 = / ---0 + L,

or u ( x ) = N { u }

(Eq. 4-7)

The strain is then

'

ε xx

d = u(x ) = d x

(

dN 1 dN 2 $ u 1 $ % & $ u2 $ d x d x

"

(Eq. 4-8)

#

The node point displacements are to be determined such that the principle of virtual work is satisfied. Taking the variation of the displacement gives

72 Nastran Primer

Finite Element Formulations Steps in Determining the Stiffness Equation

δ u ( x ) = N 1 δ u 1 + N 2 δ u2 =

' ( $ δ u1 $ N 1 N 2 % & $ δ u2 $ " #

(Eq. 4-9)

Using Hooke’s Law, the principle of virtual work becomes: T

*

– { u } Ω

T

dN 1 dN 2 d x

d x

T E dN 1 dN 2 { δ u } d Ω + { F } { δ u } = 0 d x d x

(Eq. 4-10)

Since the variations, { δ u } , are arbitrary it follows that

[ k ] { u } = { F }

(Eq. 4-11)

where [k ] is T

[ k ] =

* Ω

dN 1 dN 2 d x

d x

E dN 1 dN 2 d Ω d x d x

(Eq. 4-12)

AE [ k ] = ------- 1 – 1 L –1 1

The equilibrium equation for a rod element is then

'

(

'

(

$ F $ AE 1 – 1 $ u 1 $ ------= % 1 & % & L –1 1 $ u $ $ F 2 $ 2 "

#

"

(Eq. 4-13)

#

It is important to note that the stiffness equation for the element is singular since the displacement field includes rigid body motion. As a consequence, the equilibrium equations cannot be solved until sufficient constraints are applied to remove the rigid body motion. Setting the displacement u 1 = 0 then gives F 2 L u 2 = -------- AE

(Eq. 4-14)

which is the well known expression for the displacement of the end of an axial rod subject to and end force.

4.3

Steps in Determining the Stiffness Equation The procedure for determining the stiffness equation for any finite element is essentially the same as that employed for determining the stiffness matrix for the axial rod element:

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4

Finite Element Formulations Stiffness Matrix for a Beam Bending Element

1.Represent the displacement field in terms of a set of basis functions where the number of generalized parameters must be the same as the number of node point degrees of freedom. The displacement function must include all rigid body modes for the element. In the case of the rod, there is only one rigid body mode: translation of the rod in the axial direction. However, in the case of a three dimensional element, the displacement field must include three rigid body translations and three rigid body rotations. 2.Relate the generalized parameters in the displacement approximation to the node point displacements 3.Determine the strains using the strain-displacement relations 4.Use Hooke’s law to express the stress in terms of node point displacements 5.Substitute the expressions for stress and virtual strains into the principle of virtual work 6.Evaluate the stiffness matrix.

4.4

Stiffness Matrix for a Beam Bending Element We now apply this procedure to obtain the equilibrium equations for a beam subject to a distributed line load as shown by Figure 4-2 .

y p( x) v2, P 2

v1, P 1

h 2

x

1

θ 1, M 1

z

θ 2, M 2

b

L

Figure 4-2 Beam Bending in the x-y Plane

The behavior of the beam is represented by the displacement, v , and the rotation, θ , at each end of the beam as shown in the figure. The set displacements and associated force and moment resultants at the node points is represented by

74 Nastran Primer


' $ $ $ {u} = % $ $ $ "

(

' $ P 1 $ $ M {Q} = % 1 $ P 2 $ $ M 2 "

v1 $

$ θ1 $ & v2 $ $ θ2 $ #

( $ $ $ & $ $ $ #

(Eq. 4-15)

In the displacement approach we now represent the displacement using a cubic polynomial which has four generalized parameters as 2

v ( x ) = a + b ξ + c ξ + d ξ

3

(Eq. 4-16)

where ξ = x ⁄ L . We note that the displacement function includes rigid body translation and rotation. In order to determine the generalized parameters in terms of the node point displacements we need the derivative: c d 2 d b v ( x ) = --- + 2 --- ξ + 3 --- ξ L L d x L

(Eq. 4-17)

The generalized parameters can now be determined in terms of the node point displacements from the following relations v ( 0 ) = v1

v' ( 0 ) = θ 1

v ( 1 ) = v2

v' ( 1 ) = θ 2

(Eq. 4-18)

After determining the generalized parameters the displacement can be expressed as: v ( x ) = N 1 v 1 + N 2 θ 1 + N 3 v 2 + N 4 θ 2 v ( x ) = [ N ] { u }

(Eq. 4-19)

where 2

N 1 = 1 – 3 ξ + 2 ξ N 2 = x ( 1 – ξ ) 2

3

2

N 3 = 3 ξ – 2 ξ

(Eq. 4-20)

3

2

N 4 = x ( ξ – ξ )

The functions, [ N ] are called shape functions. It is easy to verify that N 1 and N 3 are equal to unity when evaluated at the position of the node point displacement and equal to zero otherwise; and, that N 2 and N 4 have the same property when evaluating the derivative of the displacement function. The virtual work expression for the beam, (Eq. 3-30) in Chapter 1, modified to account for bending in the x-y plane and the virtual work of the applied line load and concentrated node point forces is: L

L

*

*

0

0

T

– v xx x + p ( x )δ v d x + { Q } { δ u } = 0 , EI δ v xx , d

(Eq. 4-21)

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4


Using (Eq. 4-19) to represent the displacement and taking the indicated derivatives then leads to the following virtual work expression: L

*

– { u } 0

T

2

T

d N d x

2

L

2

EI zz

d N d x

2

T

*

{ δ u } d x + p ( x ) [ N ] { δ u } d x + { Q } { δ u } = 0

(Eq. 4-22)

0

Since the virtual displacement, { δ u } , is arbitrary it follows that the equilibrium equation for the beam is

[ k ] { u } = { P } = { P we } + { Q }

(Eq. 4-23)

where { P we } are called work equivalent forces since they perform the same amount of virtual work as the distributed loads and the stiffness matrix for the beam is L

[ k ] =

* 0

Noting that [ N ] = (Eq. 4-20) then leads to

[ k ] = EI zz L

* 0

d x

2

2

EI zz

2

d N d x

2

(Eq. 4-24)

d x

and that the shape functions are given by

N 1 N 2 N 3 N 4

( N '' 1 ) L

T

2

d N

( N '' 1 ) ( N '' 2 ) ( N '' 1 ) ( N '' 3 ) ( N '' 1 ) ( N '' 4 )

( N '' 2 ) ( N '' 1 )

( N '' 2 )

2

( N '' 2 ) ( N '' 3 ) ( N '' 2 ) ( N '' 4 )

( N '' 3 ) ( N '' 1 ) ( N '' 3 ) ( N '' 2 )

( N '' 3 )

2

( N '' 4 ) ( N '' 1 ) ( N '' 4 ) ( N '' 2 ) ( N '' 4 ) ( N '' 3 )

d x

(Eq. 4-25)

( N '' 3 ) ( N '' 4 ) ( N '' 4 )

2

where 2

N '' i ≡

d N i d x

2

i = 1, 2, 3 ,4

Evaluating the derivatives and integrating then leads to the stiffness matrix for the beam 12 6 L –12 6 L 2

EI zz 6 L 4 L – 6 L 2 L [ k ] = ---------3 L

2

–12 –6 L 12 – 6 L 2

6 L 2 L – 6 L 4 L

(Eq. 4-26)

2

The determination of the work equivalent loads is left as an exercise. However, it should be noted that the use of work equivalent loads leads to the exact solution of the beam equations for the case of a distributed line load. The use of ad hoc load lumping does not lead to the exact result.

76 Nastran Primer

Finite Element Formulations Basis Functions for Finite Elements

4.5

Basis Functions for Finite Elements The one dimensional elements considered in the previous sections are not finite elements in the true sense of the word since the results are exact within the restrictions of one dimensional theory of elasticity and beam theory. In this section we consider the formulation of elements which represent two and three dimensional states of stress and which lead to an approximate representation of the behavior except for states of uniform stress. There are two element formulations called the h- and p-element formulations, respectively. In the h-formulation the behavior within a finite volume is represented using basis functions that are polynomial functions such that we can express a displacement component, u, for example as: N

u =

1 N i ui

(Eq. 4-27)

i=1

where the functions, N i, are the basis functions and ui are the displacements at the node points. As we found for the cases of the rod and beam elements, the N i have the property that they are equal to 1 when evaluated at the node point associated with ui and are equal to zero at all other node points. Suitable basis functions for a two dimensional rectangular membrane shown by Figure 4-3, referred to the coordinates of metric space are: 1 x y N 1 = --- / 1 – ---0 / 1 – ---0 , + 4+ a b, 1 x y N 2 = --- / 1 + ---0 / 1 – ---0 4+ a, + b, 1 x y N 3 = --- / 1 + ---0 / 1 + ---0 + , + 4 a b,

(Eq. 4-28)

1 x y N 4 = --- / 1 – ---0 / 1 + ---0 4+ a, + b,

2a

y 2b

x

Figure 4-3 Rectangular Membrane Element

A similar expression for the displacement, v, can be written using the same basis functions so the vector of displacement components can be written as:

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4

Finite Element Formulations Basis Functions for Finite Elements

' $ $ $ $ $ N 1 0 N 2 0 N 3 0 N 4 0 $ ' u ( {u} = % & = % 0 N 1 0 N 2 0 N 3 0 N 4 $ " v # $ $ $ $ $ "

u1 (

$

v1 $

$

u2 $

$

v2 $

& = [ N e ] { ue }

u3 $

(Eq. 4-29)

$

v3 $

$

u4 $

$

v4 #

Basis functions for other geometric regions can also be determined using polynomial functions as will be shown later. At this point, let us proceed to determine the equation of motion using the principle of virtual work. The virtual work expression is:

* {σ}

T

T

*

T

{ δε } d Ω – ( { X } – ρ { u··} ) { δ u } t d A – { P Ce } { δ u e } = 0

Ω

(Eq. 4-30)

A

where ρ is the mass density and the body forces, { X } , are augmented with D’Alembert forces representing the inertia forces associated with dynamic motion of the element. The stresses and strains are represented in terms of the shape functions by substituting (Eq. 4-29) into (Eq. 2-2) in Chapter 2 and ((Eq. 2-11) in Chapter 2, respectively

{ ε } = [ L ] { u } = [ L ] [ N e ] { δ ue } = [ B ] { δ ue } { σ } = [ E ] { ε } = [ E ] [ L ] [ N e ] { ue } = [ E ] [ B ] { u e }

(Eq. 4-31)

Substituting (Eq. 4-31) into (Eq. 4-30) then gives:

{ u··e }

T

* [ N ]

T

ρ [ N ] { δ ue } t d A + { u e }

T

A

* [ B ]

T

[ E ] [ B ] { δ ue } t d A

A

*

T

T

– { X } [ N e ] { δ ue } t d A – { P Ce } { δ u e } = 0

(Eq. 4-32)

A

or T

T

T

T

( { u··e } [ m ee ] + { u e } [ k ee ] – { P Xe } – { P Ce } ) { δ u e } = 0 Where:

[ m ee ] =

* [ N ]

T

ρ [ N ] t d A

A

[ k ee ] =

* [ B ]

T

[ E ] [ B ] t d A

(Eq. 4-33)

A T

{ P Xe } =

* { X }

T

[ N e ] t d A

A

where [ mee ] is thhe consistent mass matrix for the element and { P Xe } are the work equivalent forces associated with the distributed loads.

78 Nastran Primer

Finite Element Formulations Isoparametric Transformation

The discrete equilibrium expression for the element is determined from (Eq. 4-32) by noting that, since the variation of the generalized coordinates is arbitrary, the coefficient matrix must be zero, so that:

[ m ee ] { u··e } + [ k ee ] { ue } – { P Xe } – { P Ce } = 0

(Eq. 4-34)

[ m ee ] { u··e } + [ k ee ] { ue } = { P Xe } + { P Ce } = { P e }

For a rectangular shaped element the integration is straight forward but tedious. The resulting element performs well as long as the elements are indeed rectangular. But accuracy drops off significantly for even small deviations from rectangular shape. For general modeling a more geometrically robust formulation is required where the element shape is a general quadrilateral.

4.6

Isoparametric Transformation The basis functions described in the previous section are appropriate for a rectangular or square region. We therefore define a mapping from metric space to a parametric space that is bi-square as shown by Figure 4-4: (1, 1)

(-1, 1) 4

3

η

4 y

3

ξ

x 1

2

1

(-1, -1)

2 (1, -1)

Figure 4-4 Mapping from Metric to Parameter Space

Since the variational formulation is a scalar equation it applies to any coordinate system so that the stiffness matrix is still given by the second of (Eq. 4-32) but where the integration is to be taken with respect to the parametric coordinates. The stiffness matrix is then given by:

[ k ee ] =

* [ B ]

T

[ E ] [ B ] t d A

(Eq. 4-35)

A ′

where A' is the area in parametric space. The differential area in metric space is related to the differential area in parametric space by: dA = Jd ξ d η

(Eq. 4-36)

where J is the determinant of the Jacobian of transformation which will be described below. The stiffness matrix in parameter space is then given by: 1 1

[ k ee ] =

* * [ B ]

T

[ E ] [ B ] tJ d ξ d η

(Eq. 4-37)

– 1 – 1

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4


4.6.1

Basis Functions and the Jacobian of Transformation

The basis functions in parameter space are: 1 N 1 = --- ( 1 – ξ ) ( 1 – η ) 4 1 N 2 = --- ( 1 + ξ ) ( 1 – η ) 4 1 N 3 = --- ( 1 + ξ ) ( 1 + η ) 4

(Eq. 4-38)

1 N 4 = --- ( 1 – ξ ) ( 1 + η ) 4 As we showed in the previous section, the strains are then determined using(Eq. 4-31). However, since we are integrating with respect to the parametric coordinates and the differential operators in the matrix, [ L], are with respect to metric space. We therefore need to determine the relation between the differential operators in metric and parametric space. Assuming three dimensional metric and parametric spaces we have, using the chain rule:

' $ $ $ % $ $ $ "

∂ ( ∂ $ ∂ x $ ∂ξ ξ x , η x , ζ x , $ ∂ ∂ & = ξ y , η y , ζ y , ∂ y $ ∂η ξ η ζ $ z z z , , , ∂ $ ∂ ∂ z # ∂ζ

(Eq. 4-39)

where a comma indicates partial differentiation with respect to the parameter following the comma as indicated by (Eq. 4-41). In order to evaluate the partial derivatives of metric coordinates with respect to the parametric coordinates we need a transformation between metric and parameter space. Using the same basis functions used to represent displacements leads to an isoparametric transformation. For three dimensions in both metric and parametric space we have: N

x =

1 N i xi i=1 N

y =

1 N i yi

(Eq. 4-40)

i=1 N

z =

1 N i z i i=1

where xi, yi, z i are the coordinates of the nodes. The partial of x with respect to the parametric coordinates then gives: N

x ,ξ =

∂ x = ∂ξ

1 ∂ ξ xi i=1

80 Nastran Primer

∂ N i

(Eq. 4-41)


which are the reciprocal of the partial derivatives required for the transformation of the partial derivatives with respect to metric space. We therefore express the partial with respect to parameter space and invert:

' $ $ $ % $ $ $ "

∂ ∂ξ ∂ ∂η ∂ ∂ζ

( $ x ,ξ y ,ξ z ξ $ $ & = x ,η y ,η z ,η $ x ,ζ y ,ζ z ,ζ $ $ #

' $ $ $ % $ $ $ "

∂ ∂ x ∂ ∂ y ∂ ∂ x

( ' ∂ ( $ $ $ $ $ ∂ x $ $ $ ∂ $ & = [ J ] % & $ $ ∂ y $ $ $ ∂ $ $ $ $ # " ∂ x #

(Eq. 4-42)

where [ J ] is the Jacobian of the transformation. The inverse relationship is:

' ∂ ( – 1 ' ∂ ( % ∂ { x } & = [ J ] % ∂ { ξ } & " # " #

(Eq. 4-43)

Comparing (Eq. 4-39) and (Eq. 4-42) shows that:

– 1

[ J ]

ξ x , η x , ζ x , = ξ y , η y , ζ y ,

(Eq. 4-44)

ξ z , η z , ζ z , The inverse of the Jacobian for two dimensional space is:

– 1

[ J ]

y ,η 1 y – = --- ,ξ J – x ,ξ x ,η

(Eq. 4-45)

J = de t [ J ] = x ,ξ y ,η – y ,ξ x ,η

4.6.2

Example - Calculating the Area of a Square

It may seem trivial but it is instructive to calculate the area of a unit square where: T 1 1 1 { x e } = x 1 x 2 x 3 x 4 = – 1 --- --- --- – ---

2 2 2

2

T 1 1 1 { y e } = y 1 y 2 y 3 y 4 = – 1 --- – --- --- ---

2

2 2 2

Then x = N 1 x 1 + N 2 x 2 + N 3 x 3 + N 4 x 4 y = N 1 y 1 + N 2 y 2 + N 3 y 3 + N 4 y 4

The partial derivatives are:

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4

Finite Element Formulations Numerical Integration

1 1 ∂ x = – 1 1 --- ( 1 – η ) x 1 + --- ( 1 – η ) x 2 + --- ( 1 + η ) x 3 – --- ( 1 + η ) x 4 4 4 4 4 ∂ξ 1 1 1 ∂ x = – 1 --- ( 1 – ξ ) x 1 – --- ( 1 + ξ ) x 2 + --- ( 1 + ξ ) x 3 + --- ( 1 – ξ ) x 4 4 4 4 4 ∂η 1 1 ∂ y = – 1 1 --- ( 1 – η ) y 1 + --- ( 1 – η ) y 2 + --- ( 1 + η ) y 3 – --- ( 1 + η ) y 4 4 4 ∂ξ 4 4 1 1 1 ∂ y = – 1 --- ( 1 – ξ ) y 1 – --- ( 1 + ξ ) y 2 + --- ( 1 + ξ ) y 3 + --- ( 1 – ξ ) y 4 4 4 4 4 ∂η Substituting the positions of the vertices in metric space gives, J = 1/4. Integrating over the parametric coordinates the gives A = 1. You will note the Jacobian is a constant for a rectangular element. However, J is generally a function of the parametric coordinates making the strains an irrational function of the parametric criminates.

4.7

Numerical Integration Now that we have transformed [ B] to parametric coordinates the stiffness and other element matrices can be determined for arbitrary quadrilateral geometry, provided metric space transforms to parameter space. As we noted in the previous section, the Jacobian will be a function of the parametric coordinates for the general case so that it will be impractical to integrate in closed form. Instead we use Gauss integration since it requires fewer function evaluations that other numerical integration procedures such as Euler or trapezoidal integration. Gauss integration evaluates an integral by summing the values of the function determined at specific points, called Gauss points, that are multiplied by weighting factors. Using Gauss integration the evaluation of the stiffness matrix is: G

[ k ee ] =

1 f ( ξ g , η g ) w g

(Eq. 4-46)

g = 1

where G is the number of Gauss points, ( ξ g , η g ) , is the location of the Gauss point and w g is the weighting factor. The accuracy of the numerical integration is a function of the number of Gauss points. Since both the interpolation point and weighting factor are known, a linear polynomial can be 2 integrated exactly using a single Gauss point so the error is O ( ξ ) . Similarly two point Gauss 4 can integrate a cubic exactly so the order of error is O ( ξ ) . The following table shows the location of the Gauss points, the weighting factors and order of error for Gauss integration through Gauss integration of order 3. Table 4-1 One Dimensional Gauss Integration

82 Nastran Primer

Integration Order

ξ g

w g

1

0

2

2

( ± 3)

1

3

( ± 0.6, 0 )

/ 5---, 8---0 + 9 9,

Order of Error 2

O( ξ ) 4

O( ξ ) 6

O( ξ )

Finite Element Formulations Plate and Shell Elements

4.7.1

Integration for the Membrane Problem

The integration of the stiffness and other system matrices requires that we know the number of Gauss points and their location. The number of Gauss points is determined by the polynomial order of the integrand. For the stiffness matrix the polynomial order is: 2

P k = ( P B ) P E P J

(Eq. 4-47)

The typical term in [ B] is P B = X ,ξ ξ x ,

(Eq. 4-48)

ξ x , = y ,η ⁄ P J The polynomial order of [ B] is therefore P B = X ,ξ y ,η ⁄ P J

(Eq. 4-49)

And, the polynomial order of the element stiffness matrix [k ee] is: 2

P k = ( X ,ξ y ,η ⁄ P J ) P E P J

(Eq. 4-50)

For the case when the J is a constant P k = 4 , where we take the elastic matrix to be constant in the element. This will require a two by two Gauss integration where the Gauss points and weighting factors are shown by Figure 4-5. 1 ------3 w2

Figure 4-5 Two by Two Gauss Integration

It is interesting to note that the integration of the mass matrix requires 3x3 Gauss integration for the general case; and, that the work equivalent forces required 2x2 Gauss integration.

4.8

Plate and Shell Elements The field equations for a flat plate were developed in Plate theory for both Mindlin and Kirchhoff theories. It was noted in that section that plate elements based on Kirchhoff hypothesis did not perform well in practice. Later studies revealed that the problem was related to the need to represent the second derivative terms such as w, xx at the nodes. To do so the displacement functions must be complete through quadratic terms. This is a much more difficult requirement that requiring only continuous displacements for elements based on two and three dimensional theory of elasticity. Robust plate bending elements are based on Mindlin theory which retains α and β as well w to represent the displacements u and v.

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4.8.1

Curvature Approximations

The shape functions, (Eq. 4-28) are used to represent the rotations { θ } , so that the curvatures, (Eq. 2-55) in Chapter 2 are:

' $ χ x $ % χ y $ $ χ xy "

∂ 0 ( ∂ x $ $ ∂ ' α ( & = 0 % & ∂ y " β # $ $ ∂ ∂ # ∂ y ∂ x

' ' α ( 0 1 $ θ x % & = % – 1 0 $ θ y " β # "

(Eq. 4-51)

( $ & = [ T ] { θ } $ #

Representing the rotations in terms of the shape functions than gives:

{ χ } = [ L ] ( [ T ] [ N ] ) { θ } = [ B ] { θ } [ B ] = [ L ] ( [ T ] [ N ] ) 4.8.2

(Eq. 4-52)

Transverse Shear Strain Approximation

The transverse shear strains are:

γ xz =

b ∂u ∂w + = w x , – α ∂ z ∂ x

∂v + ∂w = γ yz = ∂ z ∂ y

(Eq. 4-53)

b w y , – β

so that, if the same order interpolents are used for both w and the rotations, the rotations will contain higher ordered terms than those present in w, x and w, y. This leads to very large shear stiffness and to an undesirable phenomenon called transverse shear locking . In order to eliminate transverse shear locking MacNeal[4]used a multi-field approach where the transverse shears were represented using the following approximation:

' $ ' ( $ $ γ xz $ 1 y 0 0 $ % & = % 0 0 1 x $ $ γ yz $ " # $ $ "

(

a1 $

$

a2 $

&

a3 $

(Eq. 4-54)

$

a4 $

#

The coefficients, {a}, are evaluated by evaluating the transverse shears in parametric coordinates at the center of the edges of the element. The shear components in parametric coordinates are related to those in metric coordinates by:

γ sz = γ xz cos δe + γ yz sin δ e

84 Nastran Primer

(Eq. 4-55)


The shear strains are then evaluated at the Gauss points. The performance of this and other elements in NASTRAN and those presented in the literature is discussed and compared in Chapter 7.

4.8.3

Curved Shell Elements

There is an extensive literature on shell theory and curved shell elements [5]-[6]. Shell equations are based on modifying the field equations for three dimensional theory of elasticity for structures which are both thin and one or more principle radii of curvature. Shells may be of any geometric shape, such as the Opera House in Sydney Australia. However a large number of shells using in industry are shells of revolution for which the revolved line segment can be a general shape. If the revolved line segment is a straight line the shell has the shape of either a cone or a cylinder. If the revolved shape is a semicircle then the shell is a sphere. Curved shell elements are complicated by the fact that the surface of the element is curved rather than flat. The curvature terms in shell theory couple the in-plane and bending behavior, much the same way that the material [ B] matrix couples membrane and bending in flat elements which include both in-plane and bending behavior. MSC.Nastran includes the Tria6 and Quad8 shell elements whose formulations are based on shell theory. In order to represent the curvature of the shell these elements include nodes on the edges of the shell element and are termed higher order elements. The shell equations for these elements is beyond the scope of the primer and the interested reader is referred to References [5]-[6]. 4.8.3.1

Modeling Shells with Flat Elements

Shells can be represented to very good accuracy by an assemblage of flat elements as long as the elements include both in-plane and bending behavior. This is a two dimensional analogy to representing a curved arch with straight beam sections. Let’s see what is happening by considering two such elements connecting a common node as shown byFigure 4-1.

m1 N 1

θ Q1 l 1 N 2

N 1 = N 2 cos θ Q 1 = – N 2 cos θ m 1 = – N 2 l 1 cos θ

Figure 4-1 Connecting Flat Shell Elements

The element stiffness matrices are generated in element coordinates. After generating the stiffness it is transformed from element to the Basic coordinate system at each connected node and the stiffness coefficients are associated with the g -set degrees of freedom at the node points. As we show later in Transformation of Element Stiffness Matrices, the element stiffness is given by: e

T

[ k gg ] = [ T eg ] [ k ee ] [ T eg ]

(Eq. 4-1)

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where [ T eg ] is the set of transformation matrices for each node point. The effect of the transformation is to couple the in-plane and bending modes due to the relative angles between the surfaces of the elements. The Quad4 element in MSC.Nastran includes both in-plane and bending modes and is widely used to model shells in model modeling situations. As we will see in Chapter 7, the flat Quad4 element generally gives results which are as accurate as those given by the higher order element which include curvature.

4.8.4

Normal Rotations

In plate and shell theory the rotation about the normal to the shell reference surface vanishes. For flat shell elements this means that there are only five degrees of freedom at a grid point that are elastically coupled by the stiffness matrix. As we note in Chapter 6, this leads to problems in the solution phase since a matrix having a row or columns of zeros cannot be solved. We mention this here since the transformation, (Eq. 4-1) will result in a stiffness about the normal which will, of course, become vanishingly small as the angle between element surfaces becomes very small. NASTRAN incorporates a processor controlled by the AUTOSPC Parameter that detects small stiffness and corrects the numerical problem automatically. However, there is a test problem called the Raasch challenge, Knight [7], that shows the inadequacy of the Autospc processor for certain geometries. This problem shows that a Autospc is not appropriate to remove the singularities in the problem and that another technique is required. The technique is to generate the set of linear constraint equations at each node point that will set the stiffness for the normal rotation to zero. MSC has included the modifications at the element level which remove the difficulty identified by Raasch.

4.8.5

Non-Planar Nodes for a Flat Shell Element

When using the flat shell element to model curved surfaces the node points might not lie on a flat plane. In this case the element experiences unbalanced forces and moments. The stiffness calculated for the flat surface must be transformed to the actual nodal geometry. MacNeal [8] shows the Transformation is of the form: T

[ k ]nodes = [ R ] [ k ] pl an e [ R ]

(Eq. 4-2)

T

where [ R ] is transformation matrix which relates forces at the nodes to forces in the plane: T

{ F } nodes = [ R ] { F } pl an e

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(Eq. 4-3)


The desires transformation for the forces is obtained by considering the geometry associated with non-planar nodes as shown by Figure 4-2 3 h F 21

z

F 41

h

1

l 14

h

2

h

4 F 12

F 14

Figure 4-2 Non-Planar Nodes for Flat Shell Element

The corrective force at node 1 and 4 for example is a vertical force h l 14

∆ F 1 z = – ∆ F 4 z = ------ ( F 14 – F 41 )

(Eq. 4-4)

In addition to the force correction MacNeal [8] also shows a moment correction is required as shown by Figure 4-3

α M z M 1

B

α

A M 2

Figure 4-3 Moment due to Twisted Element

The moment M 1 must be equilibrated by a force directed along the node line: sin α F 1 = – F 2 = ----------- M 1 l 12

(Eq. 4-5)

Using these modifications the matrix can be constructed and used to transform the stiffness matrix. It is interesting to note that the result of not putting the element into proper equilibrium can result in results that are orders of magnitude too large.

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4.9

Finite Element Formulations References

References [1] T.H.H. Pian, “Derivation of element stiffness matrices by assumed stress distribution”, AIAA J., 2, 1333-1336, 1964 [2] T. Pian and S.W. Lee, “Improvement of plate shell finite elements by mixed formulation”, AIAA J. 16(1), 29-34, 1978 [3] T.H.H. Pian and K. Sumihara, “Rational approach for assumed stress finite elements”, Int. J. Numer. Methods Engrg., 20(9), 1547-1569, 1984 [4] R.H. MacNeal, “Derivation of element stiffness matrices by assumed strain distributions”, Nuc. Engr. Design, 70, 3-12, 1982. [5] R.H. MacNeal, “Specifications for the QUAD8 quadrilateral curved shell element’, MacNeal-Schwendler Corp. Memo RHM-46B, 1980 [6] R.H. MacNeal, Finite Elements: Their Design and Performance, Marcel Dekker, New York, p 419-481, 1994. [7] N.F. Knight, “Raasch challenge for shell elements”, AIAA J . 35(2), 375-381, 1997 [8] R.H. MacNeal, Finite Elements: Their Design and Performance, Marcel Dekker, New York, p 438-440, 1994.

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5 5.1

Structural Elements in NASTRAN

Introduction The element libraries of Legacy and MSC Nastran share some common elements, noted by “Both” in the Version column in the following table. However, except for the addition of QUAD4 and TRIA3, Legacy Nastran’s element library for shell and solid elements is obsolete or, at best in the case of the isoparametric IHEX i elements, not supported by modern pre processor programs. I have. therefore not included details for elements whose Version is Legacy. For those elements, Legacy Nastran users are referred to the Nastran User’s Manual, Vol I (NUM) that is include in the GitHub download site for Legacy Nastran. You will note that the page reference for legacy elements is given in the form “NUM . The joint NASTRAN library includes the following types of elements: Element Type

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Descriptions

Version

Reference

BAR

A constant section beam bending element

Both

Bulk Data Image 1-3

ELAS

An elastic spring connecting two displacement degrees of freedom

Both

Bulk Data Image 1-1

QUAD4

A flat quadrilateral shell element connecting four Both vertex nodes

Bulk Data Image 1-7

ROD

A one-dimensional theory of elasticity element

Both

Bulk Data Image 1-2

SHEAR

A for node shear panel

Both

Bulk Data Image 1-6

TRIA3

A flat triangular element connecting three vertex Both nodes

Bulk Data Image 1-7

ELBOW

Curved beam or elebow element

Legacy

NUM 2.4-44

HEXA1

Six sided element having 8 vertex nodes formed by 5 Legacy TETRA elements

Legacy

NUM 2.4-105

HEXA2

Six sided element having 8 vertex nodes formed by 10 overlapping Legacy TETRA elements

Legacy

NUM 2.4-105

IHEX1

Eight node isoparametric solid. Uses same numbering scheme as the HEXA element

Legacy

NUM 2.4-56

IHEX2

Twento node isoparametric solid.

Legacy

NUM 2.4-57

IHEX3

32 node isoparametric solid

Legacy

NUM 2.4-58

QDMEM

Four node membrane quad formed by four overlapping TRMEM elements

Legacy

NUM 2.4-82

QDMEM1

Four node isoparametric quad

Legacy

NUM 2.4-83

QDMEM2

Four node membrane quad formed by four overlapping TRMEM elements

Legacy

NUM 2.4-84

5

Structural Elements in NASTRAN Defining Element Objects

Element Type

Descriptions

Version

Reference

QDPLT

Four node bending element

Legacy

NUM 2.4-85

QUAD1

Four node membrane and bending

Legacy

NUM 2.4-86

QUAD2

Four node membrane and bending

Legacy

NUM 2.4-87

TETRA

A tetrahedral element connecting four vertex nodes and six midsides nodes

Legacy

NUM 2.4-96

TRIA1

Three node triangular membrane and bending

Legacy

NUM 2.4-101

TRIA2

Three node triangular membrane and bending

Legacy

NUM 2.4-102

TRIAM6

Six node lineat strain membrane

Legacy

NUM 2.4-105

TRMEM

Three node triangular membrane

Legacy

NUM 2.4-106

TRPLT

Three node triangular plate

Legacy

NUM 2.4-107

TRPLT1

Six node lineat strain membrane and bending

Legacy

NUM 2.4-108

TRSHL

Six node shell

Legacy

NUM 2.4-109

WEDGE

Six node element havinf three quadralateral and two triangular faces. Same numbering as the PENTA element

Legacy

NUM 2.4-113

BEAM

A beam bending element with variable cross section

MSC

Bulk Data Image 1-4

BEND

A curved beam element

MSC

Bulk Data Image 1-5

HEXA

A hexahedral element connecting eight vertex nodes and twelve midsides nodes

MSC

Bulk Data Image 1-9

PENTA

A pentahedral element connecting six vertex nodes and nine midsides nodes

MSC

Bulk Data Image 1-9

QUAD8

A curved shell element connecting four vertex and four midside nodes

MSC

Bulk Data Image 1-8

TETRA

A tetrahedral element connecting four vertex nodes and six midsides nodes

MSC

Bulk Data Image 1-9

TRIA6

A curved shell element connecting three vertex and three midside nodes

MSC

Bulk Data Image 1-8

The elements are described in this chapter by describing: • The appropriate Bulk Data statements • The degrees of freedom for each element • Special modeling features • Stress recovery features

5.2

Defining Element Objects The evaluation of the element stiffness and mass matrices requires: a description of the local geometry; element properties for all but solid elements; and, the material properties. Since the data base required to define a finite element model may include thousands of elements it is important that a minimum set of data be specified for the element definition. Typical structural systems are generally composed of relatively few materials. Since several parameters may be required to define the properties of a material it makes sense to attach a set number, i.e., a numerical flag, to the data associated with each individual material. Then, instead of repeating the material parameters for each element, the appropriate material

90 Nastran Primer

Structural Elements in NASTRAN Defining Element Objects

can be specified by pointing to the correct material set number. Furthermore, area properties such as plate thickness and beam cross-sectional area moments of inertia tend to be the same over several elements. So once again the concept of attaching a numerical flag to a related set of data is useful in defining a minimum data set. NASTRAN uses the set concept for the definition of the element data base by defining • Connectivity information on a connection data statement that includes a pointer to a set of element properties. • Element properties on a property data statement that, in turn, includes a pointer to a set of material properties. • Material properties on a material data statement.

The data statements associated with the element connectivity and properties are described in this chapter. The Bulk Data associated with the definition of material properties are described in Chapter 10. The form of the connection and property statements is similar for all the elements. The general characteristics of these data statements and standard notation that will be used throughout this chapter are described in subsequent sections.

5.2.1

Defining Element Connectivity

A connection statement specifies the region of the continuum represented by the object and, based on the gird point locations, the geometry of the object. Members of the finite element library are characterized by a name such as ROD, BAR, or HEXA for truss, beam, and solid hexagonal elements, respectively. The mnemonic for a connection statement is formed by prefixing a 'C' to the element name. For example, CROD, CBAR, and CHEXA are the data statement names for specifying connectivity data for the truss, beam, and solid elements. The physics in a local region is defined by the connection statement name, and a list of grid points on the connection statement appropriate for the element. The connection statements have the following general characteristics: 1.The type of element is specified by the mnemonic in the first field of the data statement. 2.The second field is the Element Identification Number (EID). The EID is the external tag for the element and must be a unique integer number for each element in the input data file. 3.The third field is a Property Identification Number (PID). The PID is an integer number that must be unique for a specific property data statement. 4.The number and the format of additional fields are dependent on the specific data statement. The format rules for each data statement are defined in an internal table called the IFP table that is used to interpret each Bulk Data statement. Data statements that violate the format rules are flagged in the output and an error flag is set to terminate the program prior to execution.

5.2.2

Defining Element Properties

The property statement always includes a pointer to a material statement. The mnemonic name of a property data statement is formed by a ‘P’ preceding the element mnemonic or, for solids and shell elements by generic properties statements beginning with a P. For example, PROD, PBAR, and PBEAM are associated with the ROD, BEAM and BAR elements. PSOLID and PSHELL are generic property statements for solid and shell elements,

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Structural Elements in NASTRAN Scalar Elastic Elements

respectively. The property statement includes all the geometric properties appropriate for a given type of element. A single property statement may be referenced by all connection statements which define elements with the same property. The use of the pointer system thus eliminates repetition of data. Property statements have the following general characteristics: • The type of property is specified by the mnemonic in the first field. • The second field is the Property Identification, which must be an integer number. The PID must be unique among the property numbers for a specific type of element. • The third field is a Material Identification (MID), which is an integer number of the MID of a material data statement. • The remainder of the property data statement is dependent on the specific property data statement.

5.2.3

Standard Notation

NASTRAN input data statements must satisfy specific restrictions on the type of number (real or integer) and permissible range of values. In the remainder of this chapter the following standard notation will be used:

5.3

EID

Element identification number, (integer, > 0).

PID

Property identification number, (integer, > 0).

MID

Material identification number, (integer, > 0).

G

Grid or scalar point number, (integer, > 0).

C

Degree of freedom (integer, > 0 or blank) code at grid point, equal to zero or blank for a scalar point.

CID

Coordinate identification number, (integer, > 0).

NSM

Non structural mass defined per unit length for lineal elements and per unit area for two-dimensional elements, (real).

Scalar Elastic Elements A scalar element having a single elastic constant that relates the displacements of two degrees of freedom. z

k

G2 G1

y x Figure 1-1 Scalar Elastic Element

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Scalar elements may be defined between two grid point degrees of freedom by the ELAS1 and ELAS2 elements and between two scalar points by the ELAS3 and ELAS4 elements as shown byBulk Data Image 1-1.

5.3.1

Spring Connecting Grid Point Degrees of Freedom

1

2

3

CELAS1 EID

PID

G1

C1

PELAS

K

GE

S1

K

G1

C1

PID

CELAS2 EID

5.3.2

4

5

6

7

G2

C2

G2

C2

8

Ge

9

10

9

10

S

Spring Connecting Scalar Points

1

2

3

4

5

CELAS3

EID

PID

S1

S2

CELAS4

EID

K

S1

S2

6

7

8

Bulk Data Image 1-1 Elastic Spring, CELAS

5.3.3

Specification of Connected Degrees of Freedom

The connected degrees of freedom at a grid point are defined on the ELAS1 and ELAS2 data statements by specifying the external degree of freedom code using the pair G, C where G

is a grid or scalar point

and C

is a degree of freedom code that may be zero or blank if G is a scalar point. If G is a grid point, then 1 < C< 6, where the degree of freedom codes are interpreted in terms of the displacement coordinate system of G as specified on the GRID data statement.

If the second entry for the connected degree of freedom is left completely blank, then the displacement of the spring is connected to ground and the displacement at the second end is zero. If the element is to connect only scalar points, then the CELAS3 and CELAS4 data statements can be used where the degree of freedom is specified by the scalar point number. It is interesting to note that the scalar elements imply the existence of the scalar points so that they do not need to be explicitly defined using the SPOINT data statement.

5.3.4

Properties

The properties associated with the scalar spring may be defined directly on the connection data statement, if CELAS2 or CELAS4 are used, or on a separate PELAS property data statement. The properties in either case are defined as K

The spring rate (real)

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Ge

Damping coefficient (real)

S

Stress recovery coefficient (real)

5.3.5

Stress Recovery

The stress recovery factor defined on the PELAS or CELAS data statement is used to calculate the stress in the scalar element by P S

σ = ---

(5-1)

where P is the force in the element and S is the stress recovery coefficient. The stress is printed by including an appropriate STRESS data statement in Case Control.

5.3.6

Modeling with the ELAS Element

The ELAS element can be used to • Define elastic constraint for generalized coordinates. • Represent boundary flexibilities. • Define elastic connections between degrees of freedom.

This general capability could be used to model an axial elastic element between two grid points as shown by Figure 1-2. However, the ELAS element is not recommended as a substitute for the ROD element, which is described in Rod and Truss Elements (p. 95), because even a slight misalignment between the spring displacement and the coordinate axis can lead to serious spurious internal constraints. In spite of the potential problems involved with using the ELAS element to represent ROD behavior we will consider modeling the elastic spring described below to illustrate the modeling procedures involved in Its specification. The spring has a spring rate of 5000 lb f inch and connects the points G1 and G2 that have the following coordinates G1 = (5, 3, 4) inch G2 = (2, 5, 3) inch The most straightforward way of specifying the spring is to define and connect displacement degrees of freedom which are in the direction of the line segment which connects the two grid points. This can be accomplished by: • Defining a new coordinate system which has an axis parallel to the line between the two points. • Referring the displacement degrees of freedom at the two grid points to the new coordinate system.

This modeling procedure has the advantage that the element and coordinate axes are aligned so that no spurious constraints will be produced. The spring can be represented by the following data statements.

1 CORD1R

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2 100

3 1

4 2

5 3

6

7

8

9

10

Structural Elements in NASTRAN Rod and Truss Elements

1

2

3

4

5

6

7

GRID

1

0

5.

3.

4.

100

GRID

2

0

2.

5.

3.

100

GRID

3

0

0.

0.

0.

CELAS1

1

5

1

3

2

PELAS

5

5.E3

8

9

10

3

where: 1.Coordinate system CID = 100 is a right-handed system whose origin is at point 1, whose X3-axis is along the line joining points 1 and 2. 2.Specification of CID = 100 on the GRID data statements means that grid point displacements and forces at these points will be interpreted relative to the CID = 100 coordinate system. 3.The elastic spring defined by the CELAS1data statement defines a scalar spring in the X3-direction of the CID = 100 coordinate system.

5.4

Rod and Truss Elements A truss element is a lineal structural element whose properties and behavioral variables are continuously distributed along a line that joins two grid points and that provides resistance to axial displacement and torsion, as shown in Figure 1-2.

θ x1 z u x1

G1

y

A,J

θ x2

G2 u x2

x Figure 1-2 Truss Element

The element has a uniform cross-sectional area, A, and torsional constant, J , and connects the two grid points G1 and G2, as shown. The element x-axis is defined by the directed line drawn from G1 to G2.

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The element may be defined either by CONROD, which includes connective as well as property data on one data statement, or by the connectivity, property pairs CROD/PROD or CTUBE/PTUBE, as shown in Bulk Data Image 1-2 The difference between the TUBE and ROD elements is that the cross-section of the tube is assumed to be a circular cylinder of diameter DIAM and wall thickness, THICK, while the rod has a cross-sectional area, AREA, and torsional constant, 1

2

3

4

5

CONROD EID

G1

G2

MID

CROD

EID

PID

G1

G2

PROD

PID

MID

A

J

6

7

A

J

C

NSM

8 C

9

10

NSM

Bulk Data Image 1-2 Specification of Truss Element 1

2

3

4

5

CTUBE

EID

PID

G1

G2

PTUBE

PID

MID

D

T

5.4.1

6

7

8

9

10

NSM

Description of Input Data for Truss

The input data items included on the connect and property data statements are: Field

Description

EID

Element identification number, which must be unique among all elements defined in Bulk Data. (integer, > 0)

G1, G2

Geometric grid point joined by the element. (integer, >0, G 1 G2)

MID

Material identification number that points to a set of material properties. (integer, > 0)

AREA

Cross-sectional area. (real)

J

Torsional constant. The torsional constant is equal to the polar moment of inertia only for circular cross-section. In general, J must be determined experimentally, from a handbook such as the American Institute for Steel Construction, or by means of torsion theory. (real)

D

Outside diameter of the tube. (real)

T

Thickness of the tube. (real, THICK < 1/2DIAM)

C

Stress recovery coefficient, which is the distance from the centroidal axis to the point at which the shear stress is desired. (real)

NSM

Non structural mass per unit length of the element.

5.4.2

Stiffness Matrix

The stiffness matrix for the truss element is based on the assumption that the axial displacement and the rotation about the local x-axis vary linearly along the length of the beam. The displacement and rotation are then taken to be:

96 Nastran Primer


' $ $ N 1 N 2 0 0 $ ' u ( % & = % 0 0 N 1 N 2 $ " θ # $ $ "

(

u x 1 $

$

u x 2 $

& θ x 1 $ $ θ x 2 $ #

(Eq. 5-2)

where: x N 1 = / 1 – ---0 + L,

x N 2 = -- L

(Eq. 5-3)

and L is the length of the element between grid points G 1 and G2. The resulting stiffness equation for the element is given by: AE AE ------- – ------- 0 L L AE AE – ------- ------ L L

0 0

0

0

0 0

JG JG ------- – ------ L L

JG JG 0 – ------- ------ L L

' $ $ $ % $ $ $ "

(

' $ F x 1 $ $ $ $ F x 2 & = % $ $ M x 1 $ $ $ $ M x 2 # "

u x 1 $ u x 2

θ x 1 θ x 2

( $ $ $ & $ $ $ #

(Eq. 5-4)

where A is the cross sectional area, J is the torsional constant, E is the Modulus of Elasticity and G is the shear modulus.

5.4.3

Stress Recovery

The axial and shear stress are calculated as: P σ xx = -- A

σ xy

TC = ------ J

(Eq. 5-5)

The stress recovery coefficient, C , is specified directly on the PROD data statement and is taken to be D/2 for the TUBE. If A or J is not specified the associated stress recovery is bypassed. Stress recovery is requested by the STRESS Case Control Directive.

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5.4.3.1

Example: Plane Truss

Consider the truss shown in Figure 1-3. The Bulk Data statements required to specify the grid point locations, the element connections, purge of the unconnected degree of freedom, and the specification of geometric constraints using permanent single point constraints are shown below. y

Note: 15 ft.

1.Element numbers are shown in circles. 1.The areas for all elements is 3in 2. 1.Only loads in the x, y-plane are considered.

4

1.Simply supported at nodes 1 and 2.

2

15 ft.

3

1

3 4 5

1

10 ft.

2

x

Figure 1-3 Plane Truss 1

2

GRDSET

3

4

5

6

7

0

8

9

10

3456

GRID

1

0

0.

0.

0.

0

123456

GRID

2

0

180.

0.

0.

0

23456

GRID

3

0

180.

120.

0.

0

GRID

4

0

0.

180.

0.

0

CROD

1

1

1

4

CROD

2

1

4

3

CROD

3

1

1

3

CROD

4

1

3

2

CROD

5

1

1

2

PROD

1

100

3.

MAT1

100

3.E7

where 1.A GRDSET data statement specifies a default for the permanent single point constraint field of the Grid data statement to purge degrees of freedom 3, 4, 5, and 6. This allows each grid point to move only in the x- and y-directions. 2.The coordinates of the grid points and the area on the PROD data statement are expressed in consistent units (inches). 3.The fields associated with the torsional constant, J , the torsional stress recovery coefficient, C , and the non structural mass, NSM, have been left blank. The values of these fields are set equal to zero. 4.The PROD points to a MAT1 material data statement whose MID is 100. Since only the modulus of elasticity is used to calculate the stiffness for the rod element, the entries for other material coefficients have been omitted. 98 Nastran Primer

Structural Elements in NASTRAN The BAR Beam Bending Element

5.The structure is restrained against rigid body motion by setting the x and y components of displacement at Grid point one and the y component at Grid point two equal to zero by means of the PSPC field. Since any entry in the PSPC field of the GRID data statement overrides the default value on GRDSET, it is necessary to include all the degrees of freedom as shown.

5.5

The BAR Beam Bending Element A beam is a lineal element whose properties and behavioral variables are continuously distributed along a line segment that joins two grid points. The BAR can represent bending behavior about two perpendicular planes, axial behavior and torsion about the axis as shown in Figure 1-4. u 6

z

u5

u3

u4

P 6

u2

u1

A Z a

y

P 3

P 5 P 2

B P 1

GA

Z b

P 4

GB

x Figure 1-4 The BAR Element

Legacy NASTRAN element library includes the BAR while MSC and NX include two straight beam-type elements, called the BAR and the BEAM, as well as a curved beam and pipe element called the BEND. The BEAM allows an offset shear center and axially varying area properties while the BAR represents a subclass of the BEAM element capability without these features. The BAR element is described in this section while the additional capabilities of the BEAM are described in The BEAM Bending Element (p. 117). The BEND element is described in Curved Beam Element (p. 127). Referring to Figure 1-4, the BAR element may be offset from the connected grid points to allow the modeling of eccentrically stiffened plates and shells. The element connects the offset ends A and B and the element x-axis is taken as the directed line segment from offset end A to offset end B of the beam. The centroidal axis of the element is assumed to coincide with the local x-axis. This assumption precludes the use of the BAR element to model a beam whose centroidal axis and shear center do not coincide. The BEAM element described in Sec. 6.6 must be used in this case.

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The BAR element has the following modeling features and limitations: 1.The element represents bending behavior in two perpendicular planes in addition to axial and torsional behavior. 2.The centroidal axis may be offset from the grid points. 3.The centroidal axis and axis of shear centers coincide. 4.Transverse shear flexibility may be included. 5.Pinned connections may be defined. 6.The area properties are constant. 7.The principal axes of inertia need not coincide with the element coordinate axes. 8.Stress can be recovered at up to four points on the cross-section at each end.

5.5.1

Description of BAR Input Data

The Bulk Data for describing the BAR element include the CBAR, CBARAO, PBAR, and BAROR data statements shown on Bulk Data Image 1-3 1

2

BAROR

CBAR

PBAR

3

4

5

PID

6

7

8

V1,G0

V2

V3

9

EID

PID

GA

GB

V1,G0

V2

V3

PA

PB

Z1A

Z2A

Z3A

Z1B

Z2B

PID

MID

A

Izz

Iyy

J

NSM

Cy

Cz

Dy

Dz

Ey

Ez

Fy

Fz

K y

K z

Iyz

SCALE

X1

X2

X3

X4

X5

X6

SCALE

NPTS

X1

∆X

CBARAO EID

10

Z3B

or CBARAO EID

where: CBAR

Defines the connectivity, the beam offset, the element coordinate system, and pinned connections.

CBARAO

Defines a series of points along the BAR element at which stress and/or internal element forces may be recovered. (MSC only)

PBAR

Defines the area properties relative to the element coordinate system, the non structural mass, the points on the cross section at which stresses are to be recovered, and the shear flexibility factors.

BAROR

Provides default values for the PID and the orientation of the element. Bulk Data Image 1-3 BAR Beam Bending Element

The fields of the CBAR and PBAR data statements are described in later sections.

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5.5.2

BAR Connectivity

The CBAR Bulk Data statement defines connectivity for the BAR element, including the definition of the offsets at grid points GA and GB; and, the element coordinate system. The fields of the CBAR data statement are described by Table 5-1. Table 5-1 Fields of CBAR Field

Description

Grid point identification numbers of connection points. (integer, > 0,

GA, GB

G A ≠ G B )

V1, V2, V3

Components of a free vector at end A defined with respect to the displacement (i.e. CD) coordinate system at grid point G A. The vector is used to determine the orientation of the element coordinate system. (real)

G0

Identification number of third grid point to optionally define the free vector. (integer, >0)

PA, PB

Pin flags for bar ends A and B. Used to specify forceless degrees of freedom corresponding to the pin flag number at the appropriate end of the bar. (Up to 5 of the unique digits 1-6 can be put anywhere in the field with no imbedded blanks, integer, > 0).

(Z1A,Z2A,Z3A)

Components of offset vectors defined with respect to the displacement coordinate systems at grid points G A and GB that define the position of

(Z1B,Z2B,Z3B)

5.5.3

the centroidal axis relative to the grid points. (real)

BAR Element Coordinate System

The features of offset centroidal axis and the three-dimensional behavior of the element complicate the specification of the element coordinate system. Points A and B are points at the ends of the centroidal axis of the BAR. They coincide with the grid points only if there are no offsets. The element coordinate system is shown by Figure 1-5

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z V

y Z

A

Y

Z a

B

x

Z b

GA

GB

X Figure 1-5 Local Coordinates for BAR Element

where 1.The element x-axis is coincident with the line segment between points A and B, positive from A to B. (If these is no offset then the x axis is the line segment drawn from Grid GA to Grid GB.) 2.The element xy-plane is defined by the element x-axis and a user-defined vector, V , having its origin at end A. The vector, V , may not be coincident with x-axis. The user may define the vector in either of two ways 1.By specifying the components ( V 1 ,V 2 , V 3) of a vector, V , that are defined in terms of the displacement coordinate system at Grid point, G A, and that has its origin at end A. (Note: The displacement coordinate system is specified by the CD coordinate system field of the Grid data statement which specifies Grid point G A.) 2.By specifying a third Grid point, G 0. The vector is taken as the directed line segment from GA to G0 even if offsets are present. If a third grid point, G0, is defined only for the purpose of specifying the vector, then the degrees of freedom at G0 are not connected to the structure and will be removed automatically. The element coordinate system is then defined using the element x-axis and V z element = x element × V e lement

and y element = z element × x element

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5.5.4

Defining BAR Offset

The BAR element may be offset from the connected Grid points by specifying offset vectors Z a and Z b as shown in Figure 1-5. The position of points A and B is then given by r a = r G + Z a a

r b = r G + Z b b

where 1.The offset vectors Z a and Z b are defined by components ( Z 1a , Z 2a, Z 3a) and (Z1b , Z2 B, Z3b), respectively, on the CBAR data statement. These are components referred to the displacement coordinate system for the respective points. Thus, if the displacement coordinate systems associated with G A and G B are rectangular and cylindrical, respectively, the components of Z a and Z b will be interpreted in the sense of unit vectors in the ( X, Y, Z ) and ( R, θ , Z ) directions, respectively. 2.The offset is treated as a rigid link between the Grid points and the associated ends of the BAR. 3.The element coordinate system is defined with r espect to the offset centroidal axis of the element.

5.5.5

Pinned Connections

The pin flags P A and P B are used to specify one or more degrees of freedom, in the element coordinate system, that does not transmit a force from the connected grid point to the element. The pin flag field is a packed set of degree of freedom code numbers The pin flag degree of freedom codes are: Pin Flag

Meaning

1

No force in element x-direction

2

No force in element y-direction

3

No force in element z -direction

4

No moment about element x-axis

5

No moment about element y-axis

6

No moment about element z -axis

The specification of a pinned connection results in an additional degree of freedom at a grid point since the element and grid point displacements are not compatible as indicated in Figure 1-6, which shows two BAR elements that are attached at grid point two. z

a

θ y2 θ y1

b

θ y2 1

u z1

(a)

2

(b)

u z2

x

3

θ y3 uz3

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Figure 1-6 Independent Rotations at a Hinge

A momentless hinge associated with the rotation about the z -axis at Grid point two is specified using either P B = 5 for element (a) or PA = 5 for element (b). Since the pin flag defines a forceless degree of freedom, the order of the stiffness matrix for the element in which the pin flag is specified can be reduced by noting that the stiffness equation for the pinned degree of freedom is equal to zero and that the associated row defines a constraint equation. For example, the stiffness equation for element (a) in Figure 1-6 is given by:

'

(

' $ F 1 z $ $ θ 1 y $ $ M 1 y & = % u 2 z $ $ F 2 z $ $ θ 2 y $ $ M 2 y # "

k 11 k 12 k 13 k 14 $ u 1 z $

$

k 21 k 22 k 23 k 24 $

%

k 31 k 32 k 33 k 34 $

$

k 41 k 42 k 43 k 44 $

"

( $ $ $ & $ $ $ #

(Eq. 5-1)

Setting the moment M2y equal to zero and solving for θ2y gives: 1 k 44

θ 2 y = – ------- k 41 u1 z + k 42 θ 1 y + k 43 u2 z

(Eq. 5-2)

This equation is of the form of linear constraint relation for the pinned degree of freedom in terms of the remaining degrees of freedom that is used to reduce the order of the stiffness matrix for the element in which the pin connection is specified. The stiffness coefficient for the pinned degree of freedom must therefore be nonzero. Because the pinned degree of freedom is reduces the stiffness matrix for the element we see that the hinge may be specified in either element but not in both. If the same degree of freedom were specified in both elements it would be completely uncoupled (i.e. it would have no stiffness). The displacement degree of freedom associated with the pin are not recovered. This fact may influence the user's choice of the element in which the pin is defined.

5.5.6

Defining BAR Properties

The properties of the BAR element are specified on the PBAR data statement, whose fields are described below Field

DESCRIPTION

AREA

Area of bar cross-section. (real)

Iyy, Izz, Iyz

Area moments of inertia about the element y- and z -axes and the associated cross product of inertia, respectively. (real)

J

Torsional constant. (real)

NSM

Non structural mass per unit length. (real)

K y, K z

Transverse shear factors in the element y- and z -directions, respectively. The shear factor is dependent on the beam cross-section.

(Cy, Cz)

Element y and z -coordinates for four points on beam cross-section at which stresses are to be recovered at each end of the BAR. (real). (These points are identified in the output as points 1 through 4, respectively.)

Table 5-2 Description of PBAR Fields

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The area properties specified on the PBAR data statement are calculated with respect to the element coordinate system. In order to specify the area properties we thus consider the cross-section of a beam element as shown in Figure 1-7. z ′ z dA y ′

y

P z ′

θ

y ′

O

Figure 1-7 BAR cross section

where O

is a point which lies on the centroidal axis

P

is a general point on the cross section with an area dA

The second area moments of inertial relative to an arbitrary set of axes, ( y',z' ), are given by: 2

I y' y' =

* ( z ' ) d A

I z ' z ' =

* ( y' ) d A

2

( y,z )

(Eq. 5-4)

*

(Eq. 5-5)

* d A

(Eq. 5-6)

I y' z ' = y ' z ' d A

A =

(Eq. 5-3)

is the set of local coordinates which is shown by Figure 1-5 and which is to be defined on CBAR data statement. The moments of inertia of the cross section relative to ( y, z ) are defined in terms of those found with respect to ( y', z' ) as follows

The area moments of inertia can be transformed to a set of axes, ( y, z ) which are related to the ( y', z' ) coordinates by a rotation, θ, as follows: 2

2

I yy = I y' y' cos θ + I z ' z ' sin θ – I y' z ' sin 2 θ

(Eq. 5-7)

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2

2

I zz = I z ' z ' cos θ + I y' y' sin θ – I y' z ' sin 2 θ

(Eq. 5-8)

2 1 ) sin 2 θ I yz = I y' z ' cos θ + --- ( I y' y – ' I z ' z ' 2

(Eq. 5-9)

Since the moments of inertia for the cross-section are defined relative to a specific coordinate system the analyst must specify the moments of inertia on the PBAR that are calculated with respect to the element coordinates specified on the CBAR. The other definitions of cross-sectional properties, J , the torsional constant and the transverse shear coefficients, K y and K z , are not as straight forward since both depend on the distribution of area in a more complex way than the bending moments of inertia and area. The torsional constant, J , relates the angle of twist, θ x, to the torsional moment, M x, the length, L, and the shear constant, G, as follows TL θ x = ------ JG

(Eq. 5-10)

The torsional constant is equal to the polar moment of inertia only for the case of a circular cross-section. For other cross-sections the analyst should consult handbooks such as [1]. The shear flexibility coefficients, K y and K z , define the shear-effectiveness of the beam. These shear coefficients define the shear displacements v ys and v zs, respectively. The total displacement of the reference axis is then given by v y= v yb + v ys

(Eq. 5-11)

v z = v zb +v zs

(Eq. 5-12)

and

where v yb and v zb are bending displacements. The total shear displacement of a beam of length L in the z -direction is related to the shear strain γ xz as follows: v zs = γ L xz

(Eq. 5-13)

where the shear strain and shear stress are related by:

σ xz γ xz = -------G

(Eq. 5-14)

Then, since σ xz = V z /A where V z is the transverse shear force, we have: V Q

z σ xz = ----------

I zz b

(Eq. 5-15)

where V z is the transverse shear force, Q, is the moment of the area beyond that value of z for which the shear stress is to be determined, and b is the width of the cross section at that point. Since Q/I zz b has the units of area we can represent(Eq. 5-15) as: V

z σ xz = ----------

K z A

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(Eq. 5-16)


where K z is a factor which depends on the cross-section and accounts for the of shear distribution over the cross-section. The substitution of (Eq. 5-16) and ((Eq. 5-14) into (Eq. 5-13) then leads to the following relationship for the shear displacement V z L v zs = ------------- K z AG

(Eq. 5-17)

where the factor 1 /KAG is called the shear flexibility. The shear factor, K , which is the reciprocal of that defined in [5] is: K = 5/6 for rectangular cross-sections, K = 9/10 for circular cross-sections, K ≅ A f ⁄ ( 1.2 A ) and K ≅ A w ⁄ A for bending of wide flange I-beam about minor and major axes of inertia, respectively, where A f is the flange area and Aw is the web area. The analyst should consult handbooks such as [5] for additional material.

5.5.7

Area Properties

The BAR element relates the six grid point displacement degrees of freedom at each of its ends to the associated stress and moment resultants as indicated by P 1 through P6 on Figure 1-4. The components of this force vector are defined as P1

Axial force in direction of the element x-axis

P2

Axial force in direction of the element y-axis

P3

Axial force in direction of the element z -axis

P4

Moment about the element x-axis

P5

Moment about the element y-axis

P6

Moment about the element z -axis

The following area coefficients relate the element displacement degree of freedom { ue}, and force resultants, { P e} A

Cross-sectional area defines resistance to an axial load P 1.

K y

Shear area coefficient, resists the transverse load P 2.

K z

Shear area coefficient, resists the transverse load P 3.

J

Torsional constant, resists the torsional moment P 4.

I yy

Area moment of inertia about the element y-axis, resists the bending moment P 5 and the transverse load P 3.

I zz

Area moment of inertia about the element z -axis, resists the bending moment P 6 and the transverse load P 2.

I yz

Cross product of inertia, nonzero unless the element y- and z -axes are principal axes of the cross section; for non-principal axes, resists all loads except P 1 and P3.

The area coefficients generate non zero terms in the stiffness matrix associated with the element degrees of freedom as shown by Table 5-3 Connected Displacements Area Coefficient

Description

u1

A

Area

K y

Shear Coefficient

u 2

u 3

u4

u 5

u6

X X

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Connected Displacements Area Coefficient

Description

u1

u 2

K z

Shear Coefficient

J

Torsional Constant

I yy

Area moment of inertia about y-axis

I zz

Area moment of inertia about z -axis

X

I yz

Cross product of inertia

X

u 3

u4

u 5

u6

X X X

X X

X

X

X

Table 5-3 Connected Displacements

The absence of one of the area coefficients on the PBAR data statement will result in zero coefficients in the rows and columns of the element stiffness matrix for the associated degrees of freedom. For example, if I yy is the only nonzero area coefficient specified on the PBAR statement, then the stiffness matrix will provide resistance to only the u3 and u5 degrees of freedom (the displacement in the z direction and the rotation about the y-axis) in the element coordinate system. The shear coefficients K y and K z can be specified only if the element coordinate system coincides with the principal axis for the cross-sectional inertia. If I yz ≠ 0,the BAR element is assumed to be infinitely rigid to shear deformations.

5.5.8

Nonstructural Mass

The mass of the beam per unit length may be entered by the parameter NSM in field eight of the PBAR data statement. The calculation of mass properties is discussed in Chapter 12, Normal Modes Analysis.

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5.5.9

Stress Recovery

The location of up to four points labeled C, D, E and F at which stress is recovered on the cross-section at each end of the bar element can be specified relative to the centroidal axis as shown by Figure 1-8. The location of these points is specified by element y- and z -coordinates at from one to four points on the PBAR data statement. y

C

Fz

F Fy z

D

E

Figure 1-8 Stress Recovery Points on the BAR

The following stress output recovered at ends A and B of the BAR element by STRESS Case Control Directives: 1.The bending stress at points C, D, E and F on the cross-section 2.The axial stress 3.The maximum stress 4.The margins of safety based on stress limits prescribed on a MAT1 material data statement.

5.5.10

Element Forces - Use of CBARAO

The CBARAO data statement is a feature that is available in MSC Nastran that allows the user to recover element forces and stresses at intermediate points along the length of the BAR element. The CBARAO data statement is typically used in conjunction with a load distribution defined by a PLOAD1 data statement. The inclusion of internal data recovery points may lead to some confusion concerning the sign convention used to interpret the element forces since either of the following sign conventions can be used. • Beam Convention • Coordinate Convention

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5

The beam sign convention which is shown by Figure 1-9 is that positive forces and moments cause positive curvature of the beam element. The coordinate sign convention, on the other hand, is that positive forces and moments are in the direction of positive displacements and rotations, respectively. y z

Vyb Mza

Mya

Mzb

Myb

P

P

P

P

Vza

Vya x

x b) Plane 2 (xz)

a) Plans 1 (xy)

Figure 1-9 Beam Forces and Moments

The sign convention used for element forces for the BAR element is: • Coordinate convention if PLOAD1 and/or CBARAO data statements exist • Beam sign convention otherwise

For other NASTRAN beam-type elements, i.e., the BEAM and BEND elements the element forces are always interpreted using the coordinate convention. There are two forms of the CBARAO Bulk Data data statement shown on Bulk Data Image 1-3 that provide the user with two different ways of prescribing internal data recovery points. Form 1-

Allows the user to prescribe up to six unique positions on the BAR

Form 2-

Allows the user to specify positions by means of a length increment and number of increments.

5.5.11

Data Recovery Points - CBARAO Form 1

The data fields on the first form of the CBARAO data statement allow the user to define internal points by at up to six interior points along the length of the bar. The data fields shown on Bulk Data Image 1-3 are described as follows. EID

Element identification number of a specific BAR element (integer > 0)

SCALE

A literal parameter that defines how the axial coordinates of interior points are to be scaled as follows LE - The values xi are actual distances. FR - The values xi are normalized distances where element length is the normalizing factor.

xi

The positions of up to six interior points along the element axis. The end points are not to be included since the forces and stresses are normally recovered at these points. (Real > 0.0)

If the literal value 'LE' is supplied for the SCALE field then a PLOAD1 Bulk Data data statement for the element must be present.

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5.5.12

Data Recovery Points - CBARAO Form 2

The second form allows the user to specify a number of equally spaced internal points with a minimum input. The specific form of the data statement is implied by the type of number in the fourth field as follows Real

-Form 1

Integer

-Form 2

The fields of the second form which are different from the first form are as follows NPTS

Number of interior data recovery points. (Integer 0 < NPTS < 19)

xi

Position of first point. (Real > 0.0)

∆ x

Incremental distance along element axis. (Real > 0.0)

The series interior points x1 = xi - 1 + ∆ x

(i = 1, 2 ....NPTS)

is then generated.

5.5.13 5.5.13.1

BAR-Element Examples Specifying the Element Coordinate System

Consider the structure shown in Figure 1-10, which consists of two beams at right angles. The structure is fully constrained at grid point one, and may be loaded with arbitrary forces and moments at grid points two and three. Each of the beams has the same cross-sectional inertia properties, but the moments of inertia about the two orthogonal axes are not equal. Assuming that the element coordinates are principal axes, we wish to orient the axes of the two beams such that each beam provides maximum resistance to bending in the Basic X, Y -plane. y

200 in. 2 50 in.

2

2

3 1

1 2

1

I 22 = 420 in4

100 in. 1

I 12 = 0 I 11 = 1170in4 A = 30.3 in2

G0

K 1 = K 2 = 0 J = 10 in4

50 in. x

E = 30 X 106 psi

Figure 1-10 Example of Specifying Local Axes

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The orientation vector, V , and the element x-axis, define the element x-y plane. The vector can be specified either by a third geometric grid point or by its components. The same vector can be used for both elements if it is not colinear with the element x-axis for either element. The vector can then be specified, in terms of components, by the following BAROR data statement. 1

2

3

4

5

BAROR

6 1.

7 1.

8

9

10

0.

where the components define a vector in the Basic X-Y plane at an angle of 45 degrees to the Basic X -axis. The element x,y-plane is then coincident with the Basic X,Y -plane, since V lies in the Basic X,Y -plane and, by definition, also lies in the element x,y-plane. Alternately, grid points three and one could be used to specify G 0 for elements one and two, respectively, or an additional grid point G 0, as shown, could be defined and be referenced by both elements. Using the BAROR specification of the orientation vector, the element coordinate system is known and the element properties can be specified using the PBAR statement. The data set required to define the grid points, the element properties, and the constraints is: 1

2

3

4

5

6

7

GRID

1

0

50.

50.

0.

0

GRID

2

0

50.

100.

0.

0

GRID

3

0

200.

100.

0.

0

1.

1.

420.

10.

BAROR

20

CBAR

1

1

2

CBAR

2

2

3

PBAR

20

30

30.3

1170.

MAT1

30

3.E7

8

9

10

123456

0.

.3

where 1.The constraints at grid point 1 have been specified by means of permanent single point constraints on the Grid statement 2.(The unconnected degrees of freedom will be removed during the analysis by the AUTOSPC procedure described in Chapter 11. 3.A consistent set of units is used in specifying grid point coordinates and element properties. 4.The BAROR data statement has been used to define the components of V for each element. Note that the vector lies in the Basic X,Y -plane. The element x,y-plane for each element is thus coplanar with the Basic X,Y -plane. 5.The same PBAR statement is referenced by both CBAR data statements 6.The PBAR statement references a MAT1 Bulk Data statement described in Chapter 9 having a material set 30.

112 Nastran Primer

Structural Elements in NASTRAN The BAR Beam Bending Element 5.5.13.2

Use of Pin Flags

Consider the structural system shown in Figure 1-11, which consists of two cantilevered beams attached attached with a momentless momentless hinge at point A, loaded in the Basic Y -direction. -direction. y

P a I zz = 7c m

4

x A

2.5m

2.5m

Figure 1-11 Example of Forceless Hinge

The system is modeled using two BAR elements as shown: y

1

2

1

x

2

3

The condition that no moment M z is transmitted from element one to element two is specified by specifying specifying the pin f lag to be 6 in element two at grid point two. This modeling specification means that the rotation of element one will be different than the rotation of element two at grid point two. This degree of freedom is not recovered by the output modules. The displacement recovered at grid point two will be the rotation of element one and the common displacement displacement u2y. The, data statements describing the system are 1

2

3

4

5

6

7

8

GRID

1

0

0.

0.

0.

0

123456

GRID

2

0

250.

0.

0.

0

1345

GRID

3

0

500.

0.

0.

0

123456

CBAR

1

15

1

2

0.

1.

0.

CBAR

2

15

2

3

0.

1.

0.

PBAR

15

10

MAT1

10

1.E8

+C1

6

9

10

+C1

7.

where 1.A consistent set of metric units (cgs) has been used so that the load would be specified in a Newtons 2.The constraints are specified on the grid statement. 3.The element x,y-axes of the BARs have been defined to be coincident with the Basic X,Y -axes. -axes.

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4.The pin flag is specified in element two. The pin flag could be specified in either element but not in both. The rotational degree of freedom at grid point two in element two is not recovered since the displacement degree of freedom associated with the pin flag is removed from the analysis set of displacements displacements 5.The pin flag is specified in a continuation whose parent is the CBAR statement for element two. The continuation, C1, has been placed after the MAT1 statement to emphasize that the Bulk Data statements may be placed in any order. 5.5.13.3

Use of Offset Vector

Consider the case of the eccentrically stiffened stiffened plate shown in Figure 1-12 with geometric parameters for for the plate and beam given as follows L = 10 cm

I yy = 1 cm4

a = b = 20 cm

I zz = 0.25 cm4

h = 3 cm

= 0.5 cm4 J =

= 0.5 cm t =

A = 0.75 cm2

z

L

h x b a

t Figure 1-12 Eccentrically Stiffened Plate

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y


The load is assumed to be symmetric so that the system can be modeled by using one-quarter of the structure and appropriate symmetry boundary conditions. The finite element model for the eccentrically stiffened plate is shown in Figure 1-13, where the quarter panel has been discretized discretized by using nine nine Quad4 shell shell elements and and three BAR elements as shown. The BAR centroidal axis is to be offset from the centroidal surface of the plate through a distance of (h+t/2).

Z4 Z3 Z2

4 3

8

2

Z1

7

1

x

Beam centroidal

12

6

11

5

16

10

y

15

9 14 13 Figure 1-13 Finite Element Model of Stiffened Plate

The Quad4 shell elements, which are described in a later section, must include both bending and membrane action. action. Only the input for for modeling the offset beam beam is included included in the the data set shown below: 1

2

3

4

5

6

7

8

9

10

CBAR

1

1

1

2

0.

1.

0.

+C1

CBAR

2

1

2

3

0.

1.

0.

+C2

CBAR

3

1

3

4

0.

1.

0.

+C3

+C1

0.

0.

1.75

0.

0.

1.75

+C2

0.

0.

1.75

0.

0.

1.75

+C3

0.

0.

1.75

0.

0.

1.75

.3725

.125

.5

.25

PBAR

1

1

In this Bulk Data we have defined three BAR elements that include the offset vectors. 1.The offset vector Z is the same at at both ends of each BAR. The vector vector is the Basic Basic -direction and has a magnitude of (t + h)/2. Z -direction 2.The area properties are one-half those for the physical beam because the other half of the system is accounted for by symmetry conditions. 3.The oreintation vector, V , is taken as a unit vector in the y-direction for each BAR. This vector has its origin at the offset points lying on the centroidal axis. The vector thus lies in a plane z = = 1.75. 4.Consistent metric units have been used.

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5.5. 5. 5.14 14

Mode Mo delin ling g Con Consi side dera rati tion onss

The stiffness matrix for the BAR is exact, within the context of beam theory, for the case where the beam cross-section is constant and the element is loaded only by concentrated forces and moments at the ends. More often than not the structural analysis problem involves distributed loads and nonuniform cross-sections, cross-sections, and the question of modeling errors naturally arises. 5.5.14.1

Distributed Lo Loads

Some indication of the number of elements required to model adequately a distributed load acting on a beam can be obtained by considering the cantilevered beam subject to a uniform load as shown in Figure 1-14. z po

EI

y

L Figure 1-14 Uniformly Loaded Cantilever

The beam is represented as a collection of beam elements where the lumped load at the ends of each element is given by 1 L p i = -- p o --2 N where N is is the number of elements used to discretize the beam. The approximate tip displacement, normalized with respect to the exact solution, is presented as a function of the number of elements elements in Figure 1-15. 1-15. For this particular particular case of of load distribution and boundary conditions it can be seen that four elements are required r equired to give a reasonable solution for the tip displacement when load lumping is used. One would expect that even more elements would be required to provide adequate resolution to the stress distribution, especially especially in the area of the base where there are high curvature gradients. The correct procedure for distributed loads is using work equivalent loads which leads to the exact solution by using one element. This load, which is is available in MSC Nastran called PLOAD1 is described in Distributed Load on Beam - PLOAD1 (p. 269).

δ δ exact

ti p --------------

1

1

2

3

4

5

6

Discretization n Error in Modeling Uniform Load Figure 1-15 Discretizatio

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Structural Elements in NASTRAN The BEAM Bending Element 5.5.14.2

Nonuniform Cross-Section

In order to investigate the effect of nonuniform cross-section, a cantilever beam is considered with a variable moment of inertia loaded by a concentrated concentrated load in Fig. Figure 1-16. y P x I ( x ) = I o / 1 – -- 0 + L, x

EI ( x x) L Figure 1-16 Variable Cross-Section Beam

The nondimensional tip displacement is presented as a function of the number of elements in Figure 1-17. The cross-sectional moment moment of inertia for each element is taken to be that at the midpoint of the element. It can be seen that for this particular case of element geometry and external loads, approximately four elements are required to obtain reasonable convergence convergence to the exact solution. In NASTRAN a variable cross-section bending element is more properly defined by using the BEAM element which is described in 5.6. Both these examples indicate that the knowledgeable user will perform simple analyses of this kind to evaluate the effect of parametric changes on the number and distribution of structural elements.

δ

ti p --------------

δ exact

1

1

2

3

4

5

Tip Displacement Displacement for Variable Variable Cross-Section Cross-Section Beam Figure 1-17 Normalized Tip

5.6

The BEAM Bending Element The BEAM element is a generalization of the BAR element that is availble in MSC Nastran that includes includes all of the the modeling capability capability of the BAR BAR and the following following additional additional features. 1.The centroidal axis, the axis of shear centers, and the axis of centers of nonstructural mass may all be different. 2.Section area properties and nonstructural mass may vary arbitrarily along the beam.

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5

Structural Elements in NASTRAN The BEAM Bending Element

3.Contribution of cross-section warping to torsional stiffness may be included. 4.Distribution of mass polar moment of inertia can be specified. 5.Shear relief due to taper can be defined.

5.6.1

Degrees of Freedom

The degrees of freedom associated with the BEAM element are the three displacements and three rotations, as shown in Figure 1-4 for the BAR element and one additional degree of freedom at each end, the twist φ. The twist is related to the rotation about the axis of the beam as follows: d θ x

ϕ =

5.6.2

(Eq. (Eq. 5-1) 5-1)

d x

BEAM De Description

The BEAM element connects two offset points A and B as shown by Figure 1-18. z Z M yA A′

A

N N yA zA

A ′′ M zA

y

Centroidal Axis

v

Z A

M yB N yB

GA

Axis of Shear Centers Z B

B ′′

B ′ M zB N zB Y

B X

Axis of Nonstructural Mass Centers of Gravity

x

GB Figure 1-18 BEAM Tapered Beam Bending Element

The geometry of the BEAM is similar to that for the BAR element, but the element x-axis, which is the line segment drawn between A and B, is now assumed to be coincident with the axis of shear centers rather than the centroidal axis. The BEAM is specified by the CBEAM and PBEAM data statements shown show n by Bulk Data Image 1-4. The connectivity for the BEAM is defined on a CBEAM data statement which is similar to the CBAR data statement except for the additional scalar degrees of freedom at each end that represent the twist.

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The BEAM properties are specified by the PBEAM data statement. The PBEAM statement is similar to the PBAR statement but, since variable area properties and internal stress recovery points can be specified, the form of the PBEAM must support these options. Because the BEAM represents an extension of the BAR modeling capability only those BEAM features that are not supported by the BAR element will be described. Note that the continuation mnemonics, mnemonics, 001, 002 etc. are referred to subsequently when describing the input data fields.

1

2

3

4

5

6

7

8

9

PID

GA

GB

V1

V2

V3

PA

PB

Z1A

Z2A

Z3A

Z1B

Z2B

Z3B

SA

SB

MID

AA

IzzA

IzzA

IyzA

JA

NSMA

CyA

CzA

DyA

DzA

EyA

EzA

FyA

FzA

SO

X/XB

A

IZZ

IYY

IYZ

J

NSM

Cy

Cz

Dy

Dz

Ey

Ez

Fy

Fz

K y

K z

Sy

Sz

NSIA

NSI b

CWA

CW b

MyA

MzA

MyB

MzB

NyA

NzA

NyB

NzB

CBEAM EID

PBEAM PID

10

Bulk Data Image 1-4 Specification of BEAM Element

5.6.3

CBEAM Da Data St Statement

The CBEAM Bulk Data statement includes all of the data described on Bulk Data Image 1-3 for the BAR element and an optional second continuation (+CB2) containing the fields S A and SB where SA, SB

Scalar or grid point identification numbers for the warping variables at ends A and B, respectively. (integer > 0 or blank)

The warping degrees of freedom S A and SB must be defined using either SPOINT or GRID data statements. If GRID data statements are used then the first degree of freedom is considered to be the warpage.

5.6. 5. 6.4 4

Loca Lo call Coo Coord rdin inat atee Syst System em fo forr the the BEAM BEAM

The definition of the local coordinate system for the BEAM element is complicated by the ability to define noncoincident noncoincident centroidal axis, axis of shear centers, and axis of centers of nonstructural mass in addition to defining the orientation of the local beam axes with respect to the global coordinates. The element coordinate system for the BEAM is shown by Figure 1-18 where 1.The element x-axis is taken to be along the line from end A to end B. 2.The axis of shear centers coincides with the element x-axis. 3.Ends A and B may be offset from the geometric grid points G A and GB by vectors Z A and Z B as described for the BAR element. 4.The element x,y-plane is defined defined by the vector, vector, V , as shown by Fig. 6-5 where where the vector can be specified in either of two ways.

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• Components of a free vector ( V 1, V 2, V 3) taken from end A • The vector from end A to a third grid point GO

5.The centroidal axis is along the line from points A' to B' which are defined relative to A and B by specifying the components of the offset vectors, N A and N B , respectively. The components of these vectors ( N yA , N zA) and ( N yB , N zB) are specified in element coordinates on the PBEAM statement (+PB5) 6.The axis of nonstructural mass centers of gravity is along the line from points A" to B" which are defined relative to A and B by the components of the offset vectors M A and M B respectively. The components of these vectors ( M yA , M zA) and ( M yB , M zB) are specified in the element coordinates on the PBEAM (+PB5).

5.6.5

PBEAM Data Statement

The properties of the BEAM element are specified on the PBEAM data statement shown on Bulk Data Image 1-4. The PBEAM statement is designed to allow the specification of a significant amount of descriptive data for the most general case but for the case where the modeling capability required coincides with the BAR capability the PBEAM data fields are closely related to the PBAR fields. The user may thus consider using the BEAM exclusively since it contains the BAR capability as a subset. The PBEAM together with all of the continuation statements looks formidable but the detail is necessary to allow the analyst to use all of the BEAM capability. The PBEAM data fields are described as follows, where the reference is made to the sequence number in field 10 of the data statements in Bulk Data Image 1-4. The PBEAM is described by numbers 004 through 008. These statements are: Statement Name

Description

PBEAM

The first statement of PBEAM is almost identical to the PBAR and is the only required input. The only essential difference is the inclusion of the Iyz field in the parent data statement.

004

An optional continuation that specifies the position of up to four stress recovery points on the cross section at end A. This continuation statement can be omitted even if additional PBEAM continuations are required.

005 - 006

Optional continuations that allow the user to specify area properties at specific location as along the length of the BEAM. Up to nine internal points can be defined using pairs of statements of this form.

007

An optional continuation statement defining shear factors, ( K y , K z ), Shear relief for taper, ( S y , S z ), nonstructural mass moment of inertia, NSI, and warping coefficients, CW.

008

An optional continuation on which offset axes of non structural mass and offset centroidal axes are defined

PBEAM Includes a pointer to a MAT1-type Bulk Data set and specifies the cross-sectional area properties and the non structural mass where the fields are defined as follows

120 Nastran Primer

A A

Cross-sectional area at end A (real > 0., no default)

(I yyA , I zzA )

Cross-sectional moment of inertias about axes parallel to y and z axes which pass through centroidal axis (real > 0., no default)

I yzA

Cross product of inertia at end A. (real or blank, default is 0.)

J A

Torsional constant at end A. (real or blank, default is 0.)


NSMA

Nonstructural mass per unit length at end A. (real or blank, default is 0.)

Statement 004

An optional continuation statement that specifies the location of four stress recovery points (C, D, E, and F) on the cross section at end A. It may be omitted if stress recovery is not required and if data fields on subsequent continuations do not require its use.

Sequence 005

Successive packets of statements having the form of statements 005 and 006 that specify area properties and nonstructural mass and/or stress recovery points at internal positions, x/xB along the length of the BEAM. The second field, SOPT, of 005 is a character string

and Sequence 006

which does two things: it identifies the continuation as the specification of a internal position; and, it controls the calculation of stress and internal forces at intermediate points depending on the value of the parameter as follows

YES

Calculate stresses at the points defined on the immediately following 0006-type continuation

YESA

Calculate stresses at the same points defined at end A by a 004-type statement which is then required if this option is specified

NO

No intermediate stress output is desired.

The 005-type statement is allowed only if a 005-type statement has a value, YES, in field 2. The normalized position along the beam is defined by the x/x B-field on a 005-type statement followed by fields defining the area coefficients and non-structural mass at that position. Up to nine points can be defined but one of them must have a value of 1.0 for x/x B to define the properties of end B.

Statement 006

An optional continuation statement defining

( K y , K z )

Shear stiffness factors, K , which appears in the relation, KAG, in the y and z directions. (real or blank, note that the default is 1, whereas the default is zero for the PBAR)

(S y , S z )

Shear relief coefficient due to taper in the y and z directions. (real or blank, default is 0.)

NSIA, NSIB) Nonstructural mass moment of inertia per unit length about the nonstructural mass center of gravity at ends A and B. (real or blank, default for NSIA is 0., that for NSIB is the value of NSIA) (CWA, CWB) Warping coefficients at ends A and B. (real or blank, default for CWA is 0., that for CW B is the value of CW A) Statement 007

An optional continuation statement defining the components of the offset vectors (MyA, MzA) Offset of axis of nonstructural mass center of gravity from the axis (MyB, MzB) of shear centers at end A and B. (real or blank, default for M y is 0., that for Mz is the value of M y) NyA, NzA) (NyB, NzB)

Offset of centroidal axis from the axis of shear centers at ends A and B. (real or blank, default is N y is 0., that for Nz is the value of Ny)

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5.6.6

BEAM Examples

5.6.6.1

Specifying Noncoincident Shear and Centroidal Axes.

Consider the channel beam shown in Figure 1-19. L A

x

A N y

z n

Section Properties Relative to Centroidal Axis e 2

3b e = ---------------- = 0.469in. 6b + h

t

3

y

N

S

h

t 4 J = ---- ( 2b + h ) = 0.009in 3 3 2

tb h 3b + 2h 6 C w = -------------/ -------------------0 = 0.301in + , 12 6b + h A = 0.986in

b

I yy = 1.58in

2

2

I zz = 0.172in t = 0. 17 in. b = 1.315 in.

4

N y = e + n = 0.766in

h = 3.17 in. E = 30 x lb/in.2 I = 20.0 in. Figure 1-19 Channel Beam

The BEAM modeling capability provides a great deal of freedom in defining the shear and centroidal axes. By using the various offset vectors we can define either the shear and centroidal axes to be coincident with the line segment between two geometric grid points. We will proceed to define the BEAM both ways, but first we need to obtain the geometric properties of the cross-section. The position of the shear center and the centroidal axes must be calculated based on the cross-sectional geometric parameters. The position of the shear center relative to the center of the web is given by e. The position of the shear center can be calculated using the techniques outlined in [4] or can be looked up in handbooks such as [5]. The centroidal axis offset from the center of the web, n, can also be determined from the condition that the first area moment about the centroidal axis must be zero. The offset of the centroidal axis from the shear center as well as other cross-sectional properties for the channel can then be found and are shown on Figure 1-19.

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Structural Elements in NASTRAN The BEAM Bending Element 5.6.6.2

Shear Axis on Line Joining Grid Points

The axis of shear centers lies on the line joining offset points A and B as shown by Figure 1-18. If the offset vectors of each end of the beam are null then the axis of shear centers lies on the line joining the geometric grid points. The channel could be modeled with a single BEAM element which connects the structural degrees of freedom associated with geometric grid points 1 and 2 at each end and two scalar degrees of freedom, points 3 and 4, which represent the twist at the left and right ends, respectively, by the following Bulk Data 1

2

3

4

5

6

7

8

9

10

GRID

1

0

0.

0.

0.

0

0001

GRID

2

0

20.

0.

0.

0

0002

SPOINT

3

4

CBEAM

1

2

0003 1

2

0.

1.

0.

0004

+003

0005

+004

3

4

PBEAM

2

101

0006 0.986

0.172

1.58

0.

0.009

0007

+007 +008

0008 0.

0.

+009

0.766

0.

0.301

0.301

0009

0.766

0.

0010

where sequence 0001 and 0002 in field 10 define grid points along the X -axis of the Basic coordinate system and sequence 0003 defines two scalar points which are associated with warping. The degrees of freedom associated with these points are connected by the CBEAM, sequence 0004 through 0006 where: 1.Grid points 1 and 2 are associated with ends A and B, respectively 2.The element coordinate system is defined by vector components (0., 1., 0.) on 0004 in terms of the Basic coordinate system so that the coordinates of the BEAM element coincide with the coordinate system shown in Fig. 6-19 3.Sequence 0005 is required even though the fields are null because the twist degrees CBEAM. 4.The continuation, sequence 0006, specifies scalar degrees of freedom 3 and 4 are associated with the warpage at ends A and B, respectively. 5.6.6.3

Non Coincident Shear and Centroidal Axes

The properties are defined on the PBEAM, sequence 0007 and its continuations 0008, 0009 and 0010 where: 1.The specification of moments of inertia is appropriate for the element coordinate system defined by the connection statement, where I zz has been transferred to the shear axis. 2.The PBEAM continuation 0008 is required in this case even though stress recovery is not desired because data fields in subsequent continuations must be defined. 3.The continuation 0009 specifies the warping coefficients. 4.The continuation 0010 specifies the component of the offset vector from the shear center to the centroidal axis. The vector components (0.766, 0.) at each end are interpreted in element coordinates so that the centroidal axis is offset 0.766 inch along the element y-axis at each end.

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5.6.6.4

Centroidal Axis on Line Joining Grid Points

The centroidal axis can be made coincident with the line joining grid points GA and GB if points A' and B' as shown by Figure 1-18 coincide with G A and GB. This can be accomplished by defining Z A = – N A and Z B = – N B . The channel could then be modeled using the following Bulk Data

GRID

1

0

0.

0.

0.

0

0001

GRID

2

0

20.

0.

0.

0

0002

SPOINT

3

4

CBEAM

1

2

0003

+0004 +0005

3

4

PBEAM

2

101

1

2

0.

1.

0.

-0.766

0.

0.

-0.766

0.

0004 0.

0005 0006

0.986

0.172

1.58

0.

0.009

0.

+007

0007 0008

+008 +009

0.766

0.

0.301

0.301

0009

0.766

0.

0010

Sequence 0005 specifies the components of the offset vectors Z A and Z B . The components (-0.766, 0., 0.) define points A and B to be offset from the geometric grid points G A and GB by 0.766 inch in the negative Basic Y-direction. Since the element and basic coordinate systems are coincident, the offset defined by 0010 makes points A' and B' coincident with G A and GB as desired. 5.6.6.5

Tapered Beam with Shear Relief

Consider the tapered beam with heavy flange shown by Figure 1-20.

Figure 1-20 Tapered Beam

The beam is to be modeled using five Beam elements connecting the grid points as shown. The element properties are presented by Table 5-1 where: 1.The local element x-axis is parallel to the Basic X axis 2.The cross sectional area is equal to the web area

124 Nastran Primer


3.The bending moment of inertia about the y-axis is calculated using only the flange area 4.The web depth varies linearly along the length of the beam 5.The bending moment of inertia about the z -axis is equal to 1. Table 5-1 Area Properties for Tapered Beam

Element No.

Area Properties A A (in2)

I yyA (in4)

I zzA (in4)

1

2.

240.

1.

2

3.

540.

1.

3

4.

960.

1.

4

5.

1500.

1.

5

6.

2160.

1.

The shear factors which account for the effect of beam taper can be determined by considering the idealization of the Beam element shown byFigure 1-21 P z 1

P 1

z

P x1 M A

B

A

M B x

h A

h B

V

V

α P x2 L

P z 2

P 2

Figure 1-21 Tapered Beam Element

The shear, Q B, at cross section B is found to be Q B = V + sin α (P 1 - P 2 )

(Eq. 5-1)

where P 1 and P 2 are the forces in upper and lower flanges, respectively. The flange forces are related to the moment, M B, by the equilibrium of moments so that 2 M B P 2 – P 1 = ------------------h B cos α

(Eq. 5-2)

The substitution of (Eq. 5-2) into(Eq. 5-1) then gives

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5

2 M B tan α Q B = V – ----------------------h B

(Eq. 5-3)

The transverse shear in the element y and z axes can then be written as S y Q y = V y – ----- M z l

(Eq. 5-4)

S z Q z = V z – ----- M y l

where S y and S z are the average shear relief coefficients for taper. The shear coefficient, S z , is given by: 2 ( h A – h B ) S z = -------------------------( h A + h B )

(Eq. 5-5)

The values for the stress relief factors for the tapered beam shown by Figure 1-20, calculated using (Eq. 5-5), are presented by Table 5-2 which can be generated in the Patran model using the Element Properties form shown below . Table 5-2 Shear Coefficients for Taped Beam Example Web Depth

hA

hB

Shear Relief Coefficient S z

1

20.

10.

0.667

2

30.

20.

0.400

3

40.

30.

0.286

4

50.

40.

0.222

5

60.

50.

0.182

Element Number

The tapered beam presented by Figure 1-20 could be modeled using the following data statements 1

2

3

4

5

6

7

8

9

10

GRID

1

0

100.

0.

0.

0

0001

GRID

2

0

80.

0.

0.

0

0002

GRID

3

0

60.

0.

0.

0

0003

GRID

4

0

40.

0.

0.

0

0004

GRID

5

0

20.

0.

0.

0

0005

GRID

6

0

0.

0.

0.

0

0006

CBEAM

1

1

2

1

0.

1.

0.

0007

CBEAM

2

2

3

2

0.

1.

0.

0008

CBEAM

3

3

4

3

0.

1.

0.

0009

CBEAM

4

4

5

4

0.

1.

0.

0010

CBEAM

5

5

6

5

0.

1.

0.

0011

PBEAM

1

101

2.

240.

1.

126 Nastran Primer

0012

Structural Elements in NASTRAN Curved Beam Element

1

2

3

4

5

6

7

8

9

10

+012

NO

1.

1.

60.

1.

0013

+013

0.

0.

0.

0.667

0.

0014

PBEAM

2

101

3.

540.

1.

0015

+015

NO

1.

2.

240.

1.

0016

+016

0.

0.

0.

0.400

0.

0017

PBEAM

3

101

4.

960.

1.

0018

+018

NO

1.

3.

540.

1.

0019

+019

0.

0.

0.

0.286

0.

0020

PBEAM

4

101

5.

1500.

1.

0021

+021

NO

1.

4.

960.

1.

0022

+022

0.

0.

0.

0.222

0.

0023

PBEAM

5

101

6.

2160.

1.

0024

+024

NO

1.

5.

1500.

1.

0025

+025

0.

0.

0.

0.182

0.

0026

where 1.The grid point sequence and orientation vector defined on CBEAM Bulk Data statements define a element coordinate system in same sense as the Basic coordinate system shown byFigure 1-20. 2.Since the properties for each element are different, each element connection must reference a different PBEAM. 3.The second optional continuation as shown by Bulk Data Image 1-4 is not required because SOPT=NO

5.7

Curved Beam Element The cureved beam element in MSC Nastran is called the BEND and is decribed below. The comparable element is Legacy Nastran is called the ELBOW and is described in [10].

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A curved beam element, which is convenient for modeling piping systems, is shown by Figure 1-22. Arc of the Neutral Axis Arc of Geometric Centers u z

B

∆ N

uθ u r

θ z θ r

A

θθ GB

Center of Curvature

Z C

GA RC

R B

O′ Figure 1-22 Curved Pipe Element - CBEND

The NASTRAN BEND element has the following characteristics 1.The area properties and radius of curvature are constant. 2.The element coordinate axes ( r, z ) must coincide with the principal axes of inertia for the cross section. 3.Flexibility and stress intensification factors can be specified in several ways. 4.Bending about and transverse shear in the direction of two perpendicular axes as well as extension and torsion. 5.Only the consistent mass matrix can be calculated for structural or nonstructural mass. The arc of geometric centers can be offset from the arc joining the geometric grid points. The offset is the same at both ends.

5.7.1

Defining The BEND Element

The Bulk Data for the connective and properties of the BEND element are shown by Bulk Data Image 1-5. 1

2

CBEND EID

128 Nastran Primer

3 PID

4 GA

5 GB

6 V1, G0

7 V2

8 V3

9 GEOM

10


1

2

3

4

5

6

7

8

MID

A

Izz

Ir r

J

R B

θB

Cr

Cz

Dr

Dz

Er

Ez

Fr

Fz

K r

K z

NSM

R C

ZC

∆ N

PBEND PID

9

10

9

10

Alternate form of PBEND for Elbows and Curved Pipe 1 PBEND

2

3

PID

MID

4

5

6

FSI

R M

T

NSM

R C

ZC

7 P

8 R B

θB

where Field

Description

EID

Element identification number, integer

PID

Property identification number of a PBEND data statements, integer

GA, GB

Connected grid identification numbers, integer

G0

Identification number of third grid point used to define the element local coordinate system as described below, integer.

V1, V2, V3

Components of the oreintation vector used to define the element local coordinate system as described below, real.

GEOM

Flag used to select the specify the option used for the element local coordinate system as described below, integer.

MID

Material identification number, integer.

A

Cross sectional area of element, real.

Izz

Area moment of inertia about the element local z axis.

Irr

Area moment of inertia about the element local r axis.

J

Torsional constant, real.

R B

Bend radius of the line of centroids, real.

θB

Optional arc angle, in degrees, of the element, real

Ci, Di, Ei, Fi, i = r,z

The r,z locations, in the element local coordinate system, of up to four points at which stresses will be calculated, real.

K r , K z

Shear factors in the r and z directions, real.

NSM

Nonstructural mass per unit length, real

R c

Radial offset of the geometric centroid from the connected grid points, real.

Zc

Offset of the geometric Centroid in the direction perpendicular to the plane of the connected points and the orientation vector, real.

∆ N

Radial offset of the neutral axis f rom the geometric centroid, positive toward the center of curvature, real.

FSI

Flag selecting the flexibility and stress intensification factors, integer=1,2,3.

R M

Mean cross sectional radius of the curved pipe, real.

T

Wall thickness of the pipe, real

P

Internal pipe pressure, real.

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Bulk Data Image 1-5 Specification of BEND element

5.7.2

Bend Element Connectivity and Geometry

A curved beam or pipe element is specified by 1.Defining the connected grid points, G A and GB. 2.Defining the plane containing the arc from the two connected grid points. 3.Defining the curvature and center of curvature of the BEND element. 4.Defining the position of the arc of geometric centers and the arc of neutral axes relative to the arc connecting the grid points. In general the geometry of the BEND element cannot be completely defined by the data on the CBEND statement. CBEND is sufficient only if the following conditions are satisfied. 1.The element is an elbow or curved pipe and is described by the alternate form of the PBEND data statement. In this case the offset of the arc of the neutral axis is calculated. 2.The arc of geometric centers coincide with the arc joining the grid points. 3.The radius and centers of curvatures are defined using the GEOM field by • Specifying a line which contains the center of curvature, (GEOM = 1). • Specifying a line which is tangent to the arc of the element, (GEOM = 2).

For all other cases the additional data required to specify the geometry of the BEND element are defined on the PBEND as follows:

5.7.3

R B

The radius of the arc of the geometric axis. (real)

θB

The arc angle of the element. (real > 0.)

R c

The radial offset of the arc of geometric centers from the arc joining the geometric grid points. (real)

Zc

Offset of the axis of geometric centroids in a direction perpendicular to plane containing points GA, GB and the vector, V . (real)

∆ N

Radial offset of the arc of the neutral axis from the geometric centroid, positive toward the center of curvature. (real)

BEND Element Coordinate System

The element coordinate system for the BEND element that is shown by Figure 1-23 is defined by one of four methods using the GEOM field on the CBEND statement. The GEOM options define: 1.How the vector, V , is to be used in determining the orientation element coordinate system.

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2.Whether radius of curvature or arc angle for the element are specified on an associated PBEND statement. center of curvature

B GB z

arc of geometric centroids

θ Z C

GA

R B

A

RC

r

Figure 1-23 Local Coordinate System for the BEND Element

The geometric configurations associated with the GEOM-options are shown by Figure 124 GO V

GA

GA

GB

GB

V

GO b) Specification of Tangent to Arc at GA (GEOM = 2)

a) Specification of Center of Curvature (GEOM = 1)

GA A

R B

θ B

GB B

GA

GB

RC V V

c) Specification of Radius of Curvature GEOM = 3)

d) Specification of Arc Angle (GEOM = 4)

Figure 1-24 Specification of Curvature for BEND Element

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where GEOM

Meaning

(integer > 0) 1

The center of curvature lies on a line which is coincident with the vector, V , or the line connecting GA and G0.

2

The tangent to the arc of the element at GA is coincident with the vector, V , or the line connecting GA and G0.

3

The BEND radius, RB , is specified on PBEND. The center of curvature lies in a plane which is parallel to that containing the line connecting G A and GB and the vector, V , and lies on the opposite side of the line AB from GO or the vector, V .

4

The arc angle is specified on PBEND. The center of curvature is then located as described above for GEOM = 3.

The positive sense of the element curvilinear coordinates system ( r, θ , z ) shown on Figure 1-24 depends on the GEOM option chosen by the analyst. The element z-axis is taken in the sense of a vector, z , which is defined by the cross product AB × V for GEOM = 1 and V × AB for all other GEOM options, where AB is the vector from GA to GB. The components of the vector, V , can be defined in two ways on the CBEND statement 1.Three components of a free vector ( V 1 , V 2 , V 3) originating at GA and defined relative to the displacement coordinate system at G A. (real) 2.The directed line segment from GA to a third grid point, G0. (integer)

5.7.4

Specifying the Arc of Geometric Centrolds

The offset of the arc of the axis of geometric centroids is specified with respect to the element coordinate system by the dimensions Rc and Z c, using either form of the PBEND statement as shown by Bulk Data Image 1-5. The element (r, θ, z) coordinate system is defined in the offset plane as shown by Figure 1-22 where 1.The r-θ plane is parallel to the plane which contains the vector, V , and the line joining the geometric grid points GA and GB. 2.The element angular coordinate, θ, has its origin at end A of the curved beam or pipe element.

5.7.5

BEND Element Properties

The BEND element properties are defined using one of the two forms for the PBEND statement shown by Figure 1-22. The first of these two alternate forms can be used to define the properties of a curved beam element having an arbitrary cross section. The second can be used to define elbow or pipe elements having circular cross sections. For either form, fields two and three of the PBEND statement are PID

Property identification number (integer > 0)

MID

Identification number of an isotropic material (MAT1) (integer > 0)

5.7.5.1

Geometric Parameters for Curved Beam

A BEND element having an arbitrary cross section can be described by the first form of the PBEND statement shown on Bulk Data Image 1-5 where the fields are defined as: A

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Cross Sectional area (Real > 0.0)


I zz , I rr

Area moments about the element z and r axes, respectively (Real > 0.)

J

Torsional constant (Real > 0.0)

The offset of the arc of the neutral axis from the arc of the axis of geometric centroids can be specified by the parameter, ∆ N (real or blank) which is positive towards the center of curvature. If the field associated with ∆ N is left blank then an approximate offset will be calculated using the relation I

zz ∆ N = ----------

AR B

(Eq. 5-1)

where the parameters in (Eq. 5-1) are defined on PBEND statement. The default value is recommended if (R2 B A)/I zz >15 in which case the calculated value is within five percent of the exact expression for an element having either a circular or square cross section. Alternatively, the user may choose to calculate the exact value for the offset using the following analytical expression R

B ∆ N = --------------------

2 R B A

(Eq. 5-2)

1 + --------- Z where 2

r Z = ---------------- d A r 1 + ----- A R B

*

(Eq. 5-3)

and where the integration is taken over the cross section, and r is the radius to a point on the cross section of the element.

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5.7.5.2

Circular Cross Section

An elbow or a pipe element having a circular cross section as shown by Figure 1-25 z

Center of curvature

R r

θ A

Zc

t

GB GA

θ B

Rc

R B

Figure 1-25 Curved Pipe Element

can be described by using the alternate or second form of the PBEND data statement as shown by Bulk Data Image 1-5. The following properties are then calculated: 1.Cross sectional area, area moments and torsional constant 2.Neutral axis offset 3.Shear factors The cross section of the elbow or pipe element is completely defined by the following fields on the alternate form of the PBEND data statement R

Mean radius of elbow or pipe. (Real > 0.)

T

Wall thickness. (Real > 0.; R + T /2 < R B)

If T =0. then a solid cross-section of radius R is assumed. 5.7.5.3

Intensification Factors

The flexibility and stress intensification factors are defined by an integer value for the FSI field on the alternate form of the PBEND as follows FSI

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Meaning


1 The flexibility factors which multiply the bending terms of the flexibility matrix are set to unity. The stress intensification factor for bending about the z-axis is set to unity and that associated with bending about the r-axis is set equal to S i where I zz 1 R B – ∆ N S i = ---------- --- + ----------------------------------- AR B r i ∆ N + ( R B + r i )

(5-1)

and r i is the radius of the ith stress recovery point relative to the element coordinate system. 2 ASME Code Section III, NB-3687.2., NB-3685.2.,1977. 3 Empirical factors from Welding Research Council Bulletin 179 by Dodge and Moore. Finally, an internal pipe pressure can be specified by the P (Real) field on the alternate form of the PBEND statement. 5.7.5.4

Shear Factors

The shear factors are automatically calculated if the alternate form of the PBEND data statement is used. If the first form is used to describe a curved beam then the shear factors are specified by the fields K r and K z (Real) which are terms in the relation KAG. A blank or zero value specifies that shear flexibility is to be set to zero where the factors are calculated automatically for the alternate form. 5.7.5.5

Stress Recovery Points

The position of four stress recovery points (C, D, E and F) can be specified by defining the position of the four points in element coordinates. For example (C r , Cz) define r and z coordinates of point C relative to the θ-axis. If the alternate form of the PBEND is used then the stress recovery points are automatically located as shown by Figure 1-26. z

F

C

E

r

D Figure 1-26 Stress Recovery Points for Elbow and Pipe Element

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Structural Elements in NASTRAN Shear Panels

5.7.6

Element Forces and Stresses

The element forces for the BEND element are interpreted as internal forces at each end of the beam. The positive sign convention for internal forces in the bend element are shown by Figure 1-27 at an arbitrary angular position, θ. Center of curvature

V Z V r

M Z

A

M r F

θ

M

B

θ

GA

θ

GB RC

R B

Figure 1-27 Positive Element Forces for BEND Element

The following element forces are requested by an 'ELFORCE' Case Control Directive 1.Bending moments M z and M r , (identified as M 1 and M 2, respectively, in the output file, and Shear forces V r and V z . (identified as V 1 and V 2, respectively, in the printout) 2.The average axial force, F θ. 3.The torque about the axis of geometric centroids, M θ. The following stresses are requested by a STRESS Case Control Directive. 1.Longitudinal stress at the four stress recovery points at each end. 2.Maximum and minimum longitudinal stress. 3.Margins of safety in tension and compression if stress limits are defined on the MAT1 Bulk Data data statement. The stresses are modified if the alternate form of the PBEND is used to account for internal pressure and stress intensification due to curvature as indicated by the FSI field. Tensile stress is considered positive.

5.8

Shear Panels The shear panel is an essential element for modeling aerospace as well as other structures characterized by very thin elastic sheets and stiffeners. This class of structures has been traditionally modeled by a combination of ROD and SHEAR elements where the RODs account for the extensional behavior of the stiffeners and the SHEARs account for the load carried by the thin elastic sheets.

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Structural Elements in NASTRAN Shear Panels

5.8.1

SHEAR Definition

The shear panel is shown in Figure 1-28. k 4 F 43

F 41

k 3 4 q3

k 2

k 1

q4

q2 3

F 12 q1

1

F 32

F 34

F 21

2 F 23

F 14

Figure 1-28 Shear Panel Showing Corner Forces and Shear Flows

where the points 1, 2, 3, and 4 refer to the order in which the connected grid points are specified using the CSHEAR data statement shown by Bulk Data Image 1-6 1

2

3

4

5

CSHEAR

EID

PID

G1

G2

PSHEAR

PID

MID

T

NSM

PID

MID

T

NSM

6

7

G3

G4

F1

F2

8

9

10

or PSHEAR

Bulk Data Image 1-6 Shear Panel

5.8.2

Description of Shear Panel Input

The Bulk Data for the shear panel description consists of a CSHEAR statement having a unique element identification number that references an appropriate PSHEAR. The shear panel connects four grid points which must be ordered consecutively around the perimeter of the element, as shown byFigure 1-28.

5.8.3

Element Properties

The element properties are defined on the PSHEAR Bulk Data data statement. The material property for the element is defined by reference to a material specification, which must be a MAT1-type data statement. The other element properties are T

Shear panel thickness. (real, ≠ 0.)

NSM

Nonstructural mass per unit area. (real)

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Structural Elements in NASTRAN Shell Elements

The alternate form of the PSHEAR allows the user to define equivalent axial RODs on the periphery of the element by means of the fields F 1 and F2 which are area effectiveness parameters. Field F1 defines RODs along edges 1-2 and 3-4 while F 2 is associated with edges 1-4 and 2-3. Real entries in these fields will cause the generation of an effective extensional area as follows Fi < 1.01

Set effective ROD areas equal to 1/2Fi (Twi) where w, is the effective width in the i direction and T is the panel thickness.

Fi > 1.01

Set effective ROD areas equal to 1/2 FiT2.

5.8.4

Recovery of Forces and Stresses

The element forces may be output by the Case Control Directive ELFORCE =

where is a set of elements defined in Case Control. The shear panel forces consist of the force applied to the element at the node points, kick forces at the corners in a direction normal to the plane formed by the two adjacent sides, and shear flows (force per unit length) along the four edges as shown by Figure 1-28 The element stresses consist of the average of the shear stress calculated at the corners and the maximum stress. The shear stresses as well as margin of safety may be output by the Case Control Directive STRESS =

5.9

Shell Elements The NASTRAN element library includes both flat and curved shell elements which support membrane and bending behavior. The form of the element specification allows the analyst to control the generation of stiffness coefficients for • Membrane-only behavior • Bending-only behavior • Bending and membrane behavior • Materially-coupled bending-membrane behavior

The modern shell elements in the NASTRAN include TRIA3

A three node isoparametric flat element

QUAD4

A four node isoparametric flat element

TRIA6

A six node triangular isoparametric curved thin shell element (MSC and NX only)

QUAD8

An eight node isoparametric quadrilateral curved thin shell element (MSC and NX only)

where the four grid points associated with a QUAD4 shell element need not lie in a plane.

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5.9.1

Flat Shell Elements - OUAD4 and TRIA3

The triangular and quadrilateral flat shell elements are shown by Figure 1-29a and Figure 1-29b, respectively, and are defined using the data statements shown on Bulk Data Image 6-7. z z

y

G4

u z

θ y

G3

G1

u y

y u z

G1 O

θ y

β θ x

θm

u x

xm

θm

α

G3

α

γ

xm

G2 G2

u y

θ x u x

x

x

a) Triangular Element - TRIA3

b) Quadrilateral Element - QUAD4

Figure 1-29 Isoparametric Flat Thin Shell Elements

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5.9.2

Defining Connectivity and Properites

5.9.2.1

Triangular Element Connectivity -- TRIA3

1

2

CTRIA3 EID

5.9.2.2

3 PID

4

5

6

G1

G2

G3

T1

T2

T3

7 θm or MCID

8

9

10

9

10

ZOFFS

Quadrilateral Element - Connectivity- QUAD4

1

2

CQUAD4 EID

5.9.2.3

3 PID

4

5

6

7

G1

G2

G3

G4

T1

T2

T3

T4

8

θm or MCID

ZOOFS

Common Propery - PSHEL

The mechanical behavior of the shell elements is defined by a common PSHELL property Bulk Data statement that is shown below. In addition, material properties for composiste layups can be defined by PCOMP Layered Composite (p. 258), as described in Laminate Properties (p. 255) in Chapter 9. 1

2

PSHELL PID Z1

3

4

MID1

T

Z2

MID4

5 MID2

6 R I

7 MID3

8 R T

9

10

NSM

Bulk Data Image 1-7 Specification of Flat Shell Elements - QUAD4 and TRIA3

The general capabilities of the TRIA3 and QUAD4 are 1.Thickness may vary isoparametrically over the surface of the element. 2.Either coupled and lumped mass formulations can be requested; however, the coupled mass matrix does not include inertias associated with rotational degrees of freedom. 3.The anisotropic elastic coefficients associated with in-plane, bending, transverse shear, and coupling between in-plane and bending behavior may be defined independently. These elements can thus be used to model unbalanced composite materials or plates offset from node points.

5.9.3

Element Connectivity

The triangular and quadrilateral elements connect three and four grid points, respectively, as shown by Figure 1-29a and Figure 1-29b. The grid point numbers associated with the fields G1, G2, G3, (and G4) must be unique and must be ordered consecutively around the perimeter. All interior angles for the QUAD4 must be less than 180 degrees. The flat shell elements connect the five degrees of freedom in element coordinates including the two in-plane displacements, u x and u y, the normal displacement, u z , and the two bending rotations, θ x and θ y. There is no rotational degree of freedom about the normal for these elements so that there will be a row and column of zeros in the six-by-six stiffness matrix

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associated with each connected grid point. The assembled system stiffness matrix, [ K gg ], will thus be singular unless the normal rotation is connected by another type of element such as a BAR or if the flat plates joining at the grid point do not lie in a plane so that there is a geometry-related stiffness.

5.9.4

Element Coordinate System

The definition of the local element coordinate system is of interest to the analyst in determining the direction and positive sense of the normal to the surface. The positive normal direction then defines the sense of positive normal pressure on the PLOAD2 and PLOAD4 Bulk Data and the positive element face for stress recovery. The coordinate systems for the triangular and quadrilateral flat shell elements are shown by Figure 1-29a and Figure 1-29b, respectively. The element coordinate system for the TRIA3 is defined such that 1.The x-axis coincides with the line drawn from grid point G1 to G2. 2.The three grid points define the x-y plane 3.The z -direction is normal to the plane containing the three grid points and is positive in the sense of the right hand rule when applied to the order of grid point specification. 4.The y-direction is found by: y = z × x The local coordinate system for the QUAD4 element, shown byFigure 1-29b, has the desirable properties of 1.Being uniquely defined, independent of the starting grid point number within even multiples of 90 degrees. 2.Being continuous with respect to changes in element shape. 3.Being parallel to the edges for a rectangular element. 4.Having the x-axis nearly parallel to the longest bisector of opposite sides in the general case. The local z -axis is perpendicular to the element x-y plane and is positive in the sense of the right hand rule applied to the grid point numbering scheme.

5.9.5

Material Coordinate

The shell elements support anisotropic materials. Since nonisotropic material properties are defined relative to a set of material coordinates the analyst must specify the orientation of one of the material axes relative to a reference axis in the element. The material coordinate axis can be specified by either a material angle, θm, or by the a material coordinate system which is identified by the MCID. The field θm (Real), on the connection statements is the orientation angle, in degrees, between the reference line between G1 and G2 and the material axis. If MCID is specifies then the x-axis of the material coordinate system is determined by projecting the x-axis of the MCID coordinate system on the surface of the element.

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5.9.6

Reference Surface Offset

Elements can be offset from the surface of the connected grid points by the ZOFFS field of the connection statement. Other parameters such as material matrices and stress fiber distances are given relative to the offset reference plane. A positive value of ZOFFS implies that the element reference surface is offset a distance along the positive normal of the element. If the reference surface is offset then bending and inplane forces are coupled and both MID1 and MID2 must be specified.

5.9.7

Element Properties - PSHELL

The element properties are defined using the PSHELL statement for all thin shell elements. The fields on the property data statement for isoparametric shell elements are defined as follows

5.9.8

MID1

Material identification number for membrane behavior (integer >0 or blank)

T

Membrane Thickness (Real)

MID2

Material identification number for bending behavior (Integer > 0 or blank)

R I

Normalized bending inertia per unit length. R I = I/I o where I o = T 3 /12 and I is the actual bending moment of inertia per unit length of the cross-section. (Real or blank, default = 1.)

MID3

Material identification number for transverse shear behavior. (integer > 0 or blank)

RT

Normalized shear thickness, RT = T S ⁄ T , where T S is the shear thickness and T is the membrane thickness. (Real or blank, default = 0.833333)

NSM

Nonstructural mass per unit area (Real)

Z 1, Z 2

Stress recovery distances for bending (Real, default Z 1 = Z 2 = T/ 2 thickness)

MID4

Material identification number for membrane-bending coupling (integer > 0 or blank)

Specifying Element Behavior

The shell elements include in-plane, bending and transverse shear behavior. The presence or absence of the corresponding entry of a material identification number in the appropriate field then specifies whether of not the associated behavior is to be included in the element formulation. If an entry is present, its value is the material set to be used in determining the material properties associated with the behavior. The behavior associated with each of the MID fields is: Material Field

Type of Behavior

MID1

Membrane (MAT1 or MAT2)

MID2

Bending (MAT1 or MAT2)

MID3

Transverse shear (MAT1 or MAT2)

MID4

Bending-Membrane Coupling (MAT2 only)

Material fields are not exclusive so entries for MID1 and MID2, and MID4 would specify that coupled membrane-bending behavior without transverse shear is desired, while entries in only MID1 would specify membrane-only behavior.

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5.9.9

Element Constitutive Relations

In order to clarify what is meant by the various types of behavior and their associated material identification numbers it is worthwhile to consider the constitutive relations that are supported by the NASTRAN shell elements. The general form of the constitutive relation is

2 / { N } 0 T [ G1 ] T [ G 4 ] 0 . - { M } . = T 2 [ G ] T I [ G ] 0 4 2 . { } Q + , 0 0 T S [ G 3 ]

/ o o - { ε } – { ε I } - { χ o } – { χ o } I o { γ } +

0 . . . . . ,

(Eq. 5-1)

o

where { N } and{ M } are the in-plane force and moment resultants, respectively, { ε } are o o o reference surface strains, { χ } are reference surface curvatures, { ε I } , { χ I } are initial strains o and curvatures, and { Q } and { γ } are defined as

' ( $ Q x $ {Q} = % & $ Q y $ " #

(Eq. 5-2)

' o ( $ γ $ { γ } = % xz & $ γ o $ " yz #

(Eq. 5-3)

o

o

o

The quantities Qx and Qy are the transverse shear forces and γ xz , are γ yz the corresponding transverse shear strains of the reference surface. The cross section properties T, I, and T s are the membrane thickness, second area moment for the cross section and shear thickness respectively, referred to the reference surface. The matrices [ G1], [G2], [G3], and [G4] are elastic coefficients defined by appropriate material statements having material IDs MID1, MID2, MID3, and MID4, respectively. The form of the relation (6-33) allows for the coupling of bending and membrane action through the material coupling terms T 2 [G4]. The coupling terms do not exist for symmetric cross-sections when the middle surface is taken as the reference surface. However, they have been included in the element formulation to allow modeling composite materials having unsymmetrical layups and offset plates. In the usual case of a solid homogeneous plate the field associated with material coupling, MID4, would be blank and the cross sectional properties would be evaluated with respect to the middle surface.

5.9.10

Solid Homogeneous Symmetric Cross-Section

The desired behavior is specified by entries in the material identification fields, as appropriate. We can define, for example, uncoupled membrane or bending behavior and include transverse shear effects by using the MID1, MID2, and MID3 fields. Since there is no material coupling of bending and membrane behavior for a solid homogeneous symmetric cross section if the middle surface is used as the reference surface, it follows that MID4 must be blank. The membrane thickness can vary isoparametrically over the surface of the element. The fields T1, T2, T3, (and T4) on the CTRIA3 and CQUAD4 statements define the membrane thickness at the associated grid points. The T-field on the PSHELL defines the default value for membrane thickness.

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The bending moment of inertia per unit length of the edge and the transverse shear thickness are defined in a manner such that the default values are correct for solid cross sections. The bending moment of inertia is specified as the ratio, R I, which is equal to 1.0 for a solid homogeneous cross section. If bending behavior is requested (i.e. an entry in the MID2-field which points to a material data statement) and if the plate has a solid homogeneous cross section then the R I-field can be left blank since the default is 1.0. Transverse shear behavior is specified by an entry in the MID3 field whose value is the material ID of a material data statement defining the material shear coefficients. If the ratio R T is blank then the default value, R T = 0.833333, is taken. It a MAT2 data statement is used to define transverse shear material coefficient only the G 33 is used. For example, a membrane having a uniform thickness of 0.5 in. could be defined by the following property statement 1

2

PSHELL 1

3 10

4

5

6

7

8

9

10

0.5

where the material membrane behavior is defined by material set number 10. Similarly, an element having membrane, bending and transverse shear behavior could be defined by the following 1

2

PSHELL 1

3 10

4 0.5

5 10

6

7

8

9

10

10

where material set 10, which we suppose is a MAT1 statement, has been specified in the fields associated with membrane, bending, and shear behavior and where the default values for the fields R l and R T have been taken.

5.9.11

Sandwich Type Cross-Section

A sandwich plate is constructed of a core and two face sheets as shown by Figure 1-30. The core is typically a light-weight material such as metal or paper honeycomb, or wood, whose purpose is to separate the facesheets in order to obtain a desired bending moment of inertia. The core is assumed to exhibit only shear behavior while the face sheets provide membrane and bending behavior. Generally the shear modulus of the core is an order-of-magnitude smaller than the elastic constants for the face sheets so that appreciable shear deformation can be expected. reference surface t 1 z

t 2

h

zo

t 3

Figure 1-30 Sandwich Plate

The constitutive relations for the sandwich plate are then formulated by noting that the moment and force resultants are defined by (2-61) and (2-59), respectively. The components of stress in these equations can then be written in terms of strains and curvatures of a reference surface located at a distance z o from the bottom of the cross section as follows:

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{ σ } = [ E ] ( { ε } – z { χ } )

(Eq. 5-4)

where z is the distance from the reference surface. The substitution of (Eq. 5-4) into (Eq. 2-58) and (Eq. 2-60)then gives the following expressions for moment and force resultants. o

*

o

{ M } = – z [ E ] ( { ε } – z { χ } ) d z

{ N } =

*

o

(Eq. 5-5)

o

[ E ] ( { ε } – z { χ } ) d z

(Eq. 5-6)

where the integration is taken through the thickness of the plate. In general, the elastic modulae are piecewise constant functions of the shell depth. In the present case the elastic coefficients are constant over the core depth, t 2, and may be different but constant over the cover plate thicknesses t 1 and t 3 The force resultants can then be found o o by piecewise integration provided that we are careful to define { ε } and { χ } to be the strains and curvatures of the reference surface, which is defined at a distance z 0 from the bottom of the cross section. The expression for the force resultants for the sandwich plate shown by Figure 1-30 then becomes

h – z o

{ N } =

* [ E 1 ] ( { ε

o

o

} + z { χ } ) d z

(Eq. 5-7)

h – z – t o 1 h – z – t o 1

+

* [ E 2 ]( { ε

o

o

} + z { χ } ) d z

h – z – t – t o 1 2 h – z – t – t o 1 2

+

* [ E 3 ] ( { ε

o

o

} + z { χ } ) d z

h – z – t – t – t o 1 2 3

For a symmetric cross section we have: t 1 = t 3 = t ; [ E 1] = [ E 3] = [ E f ]; t 2 = c; and [ E 2] = [ E c] so that the integration of (Eq. 5-7) leads to the following expression. o

1 2

o

{ N } = { ε } [ 2 t [ E f ] + c [ E c ] ] – --- { χ } ( 2 th [ E f ] + ch [ E c ] ) ( 1 – 2 a )

(Eq. 5-8)

where z o = ah and where a is a parameter that can be chosen by the analyst. Inspection of the coefficient of the curvature term shows that the in-plane forces can be made independent of the curvature if the parameter, a, is set equal to one-half. The strains and curvatures are then associated with the reference surface located at h/2 which is the middle surface for a symmetric layup. However, the NASTRAN shell elements allow for the coupling terms so that the analyst has the freedom to choose any convenient reference surface. The force and moment resultants associated with a reference surface at the middle of a symmetric cross section can now be determined by using (Eq. 5-5) and (Eq. 5-6) and by noting that the strains and curvatures are defined at the middle surface. The coupling term in the moment relation then disappears so that the moment and force resultants are the moment given by:

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o

o

{ N } = 2 t [ E f ] { ε } = T [ G 1 ] { ε }

(Eq. 5-9)

3

o o 2 2 t { M } = [ E f ] /+ c t + ct + ----0, { χ } = I [ G2 ] { χ } 3

(Eq. 5-10)

where we have assumed that the core carries no membrane forces. As a specific example consider the case where t = 0.25 in. and c = 0.50 in. The face sheets are isotropic aluminum and the core has a shear modulus of 300,000 lb./in. 2. The element properties could then be defined using the following data statements: 1

2

3

PSHELL 1

101

MAT1

101

10.e6

MAT1

102

4 0.5

5 101

6 9.5

7 102

8

9

10

1.0

0.3 3.+5

0.3

where the bending moment of inertia and shear thickness have been scaled appropriately by the membrane thickness. The subject of unsymmetrical sandwich construction and composite materials is beyond the scope of the present text. The interested reader should consult Refs. 5-8 and 5-9 for more details.

5.9.12

Effect of Warping

The QUAD4 element connects four grid points that need not lie in a plane. The element matrix for a flat element is modified in this case by appropriate pre- and post-multiplications in order to satisfy rigid body properties. The procedure which is used is described in NonPlanar Nodes for a Flat Shell Element (p. 86) and is only satisfactory for small deviations from flatness and accounts for the unbalanced nodal forces and moments.

5.9.13

Elastic Stiffness Matrices

Both the TRIA3 and the QUAD4 use the same shape functions used to interpolate displacements to transform the element geometry from physical to parametric space. The basic theory is straight-forward and follows the isoparametric development outlined in Chapter 4. The formulation of the shear stiffness for the QUAD4 element is handled quite differently than the extensional stiffness. The extensional strains are calculated at the usual Gauss points but the shear strains are evaluated at special points as described by MacNeal [4-8] for the QUAD4 element. A reduced integration scheme is then employed for the shear strains that generally results in improved element performance over a broad of element geometry and loads.

5.9.14

Mass Matrix

Lumped and consistent mass matrices are calculated at user option. The lumped mass is the default; the consistent mass is requested using the COUPMASS parameter which is described in Appendix B. Mass coefficients are calculated for only translational degrees of freedom for both the lumped and consistent formulations.

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The structural mass is calculated using the density associated with the membrane material data statement. Therefore, even if bending-only behavior is desired for dynamics, the membrane properties must be specified in order to generate the structural mass matrix. The nonstructural mass is calculated using the value for NSM on the PSHELL. If bending-only behavior is desired the mass matrix for transverse displacements can be calculated by using NSM without specifying membrane action.

5.9.15

Stress, Strain and Element Force Recovery

Components of stress, strain and element forces can be recovered using STRESS, STRAIN and ELFORCE Case Control directives. Strain and stress components are computed at distances z 1 and z 2 from the reference plane along a line perpendicular to the reference plane at the centroid of the element. The stress is given by z

i

o

{ σ } = – ---i { M } + { σ } + { ∆σ }

(Eq. 5-11)

I

where the i takes the values 1 and 2 for the bottom and top surface of the element, respectively and where the membrane stress is given by: o

o

o

o

o

{ σ } = [ G 1 ] ( { ε } – { ε1 } ) + T [ G4 ] ( { χ } – { χ1 } )

(Eq. 5-12)

The thermal stress, { ∆σ } , results from the difference between the outer fiber temperature and a linear fit between the temperatures at the plate surfaces; and the strains and curvatures of the reference surface are determined using the grid point displacements. Stresses and strains are tensor quantities which are calculated in the element local coordinate system. We recall from Chapter 3 that the stress components for the two dimensional flat shell element are:

{ σ } = σ xx σ xx τ xy

T

In addition to these quantities, the following quantities are calculated for the element: 1.Principal stresses and principal stress directions 2.Von Mises stress The interpretation of stress (and strain) quantities is complicated since 1.The calculated stress and strain components are not continuous between elements 2.The stress and strain components are most accurate at the Gauss points used for calculating the strain matrix, [ B], but are actually determine at the element centroid. Furthermore, the orientation of the element coordinate system changes from element to element. For simple structures, such as a flat rectangular plate, it is possible to align the element coordinate systems with the Basic coordinate system by using a consistent ordering procedure for the element connection list. But, for an arbitrary component which may be curved, the element coordinate system for each element will have a different orientation relative to a Basic coordinate system. In order to interpret the components of stress at the grid points they must be averaged in some way. However, since the stress components for each element connected to grid point is defined in the local element coordinate system, the components of stress must first be transformed to a common coordinate system prior to stress averaging.

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Structural Elements in NASTRAN Curved Shell Elements

5

In addition to the stresses, the force resultants { M }, { N }, and {Q} are calculated at the Grid points. The user may request stress and force output by the STRESS and ELFORCE Case Control Directives, respectively.

5.10

Curved Shell Elements The QUAD8 and TRIA6 curved shell elements have all of the general modeling capability associated with the flat shell elements. including: 1.lsoparametric thickness variation over the element 2.Element thermal field 3.Anisotropictemperature-dependent material properties 4.Elastic coupling of membrane and bending behavior The major differences between the curved and the flat shell elements are 1.The shape functions used for the QUAD8 and TRIA6 for displacements and geometry are of a higher order than those used for the QUAD4 and TRIA3. 2.The QUAD8 and TRIA6 may connect up to eight and six node points, respectively. The TRIA6 and QUAD8 elements are shown by Figure 1-31a and Figure 1-31b, respectively, where it can be seen that the elements connect all degrees of freedom except the normal rotation in element coordinates at the grid point. z

z

G3 G3 y

G7 x

m

G6

θ

G4

m

G5

y

η = const.

z l

ξ = const.

G4

θ xl

a) Six-Node Triangle

η = const.

l

G5

x

θ yl l

G1 l

y

y x

b) Eight-Node Quadrilateral

Figure 1-31 Isoparametric Curved Thin Shell Elements

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y

G2

θ xl

θ yl x

x

θ

m

G8 y

G2

G1

x

m

z l

G6

l

Structural Elements in NASTRAN Curved Shell Elements

5.10.1

TRIA6 and QUAD8 Connectivity

The connectivity for the elements is defined by the CQUAD8 and the CTRIA6 Bulk Data which are shown in Bulk Data Image 1-8. The PSHELL property data statement which is shown on Bulk Data Image 1-7 and which was described in the previous section is used to define the element properties for the QUAD8 and TRIA6 as well as the TRIA3 and QUAD4. In addition layered compoite properties can be defined using PCOMP-type Bulk Data as decribed by Bulk Data Image 9-7 1 CTRIA6

2

4

5

6

PID

G1

G2

G3

ZOFFS

T1

T2

T3

EID

PID

G1

G2

G7

G8

T1

T2

EID

θm or MCID

CQUAD8

3

7

8

9

G4

G5

G6

G3

G4

G5

G6

T3

T4

θm or MCID

10

ZOFFS

Bulk Data Image 1-8 TRIA6 and QUAD8 Thin Shell Elements

5.10.2


The TRIA6 and QUAD8 element connect up to six and eight nodes as shown by Figure 1-31a and Figure 1-31b, respectively. The mid side nodes may be selectively deleted so that the TRIA6 can connect from 3 to 6 nodes and the QUAD8 can connect from 4 to 8 nodes. The connected node points are defined by G1 through G6 on the CTRIA6 and G 1 through G8 on the CQUAD8. The grid point entries must be ordered as shown. All of the corner node points must be defined. The edge nodes should be at or near the center of the edges; however, any of the edge nodes can be deleted by leaving the appropriate field blank. For example, a blank in the G6-field of the CTRIA6 data statement would imply that there is no point on the edge connecting nodes G 1 and G3.

5.10.3

Element Coordinate Systems

The coordinate system for the curved shell elements is taken to be an orthogonal curvilinear system in geometric space as shown by Figure 1-31a and Figure 1-31b for the TRIA6 and QUAD8 elements, respectively. The normal coordinate is taken to be perpendicular to the surface with a positive sense determined by applying the right hand rule to the corner node number sequence.

5.10.4

Material Orientation

The orientation of the material coordinate system relative to the element coordinates is defined by specifying either θm or the MCID. The angle θ (real) in degrees is taken to be constant over the element and is used at each Gauss integration point to define the orientation of the xm material axis as a line in the plane tangent to the surface of the element and at an angle θ to the intrinsic element coordinate η = constant as shown. If MCID is specified the x-axis of the material coordinate system is determined by projecting the x-axis of the MCID coordinate system onto the surface of the element.

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Structural Elements in NASTRAN Solid Elements

5.10.5

Element Thickness

The membrane thickness can be defined as a constant over the element using the T-field on the PSHELL data statement; or to vary over the surface by defining the thickness at the corner node points on the appropriate connection statement using the fields T 1, T2, T3, (and T4) (Real). If the thickness fields on the connection statement are left blank the thickness on the PSHELL is used.

5.10.6

Specifying Element Behavior

The TRIA6 and QUAD8 are capable of selectively representing membrane, bending, or coupled membrane-bending behavior depending on the entries in the MID-type fields of the PSHELL, Bulk Data Image 1-7, and by PCOMP, PCOMP1 and PCOMP2 Bulk Data Image 9-7. The resulting constitutive equations are then defined by (Eq. 5-1)

5.10.7

Element Stiffness Matrices

The TRIA6 and QUAD8 are isoparametric elements using quadratic interpolation for displacement as well as mapping the geometry to parametric space. The stiffness and mass matrices are then formulated in parametric space using the techniques described in Chapter 4. Both the TRIA6 and QUAD8 use reduced integration of the shear energy. As noted in Chapter 4, reduced integration generally results in a more flexible element having superior convergence characteristics compared the standard approach of evaluating all strain components at the appropriate Gauss integration points. The resulting element is non conformable in the sense that the displacement fields do not match along common element edges and the behavior does not converge monotonically to theoretical solution, but it does converge.

5.10.8

Element Mass Matrices

The lumped mass formulation is the default for both the TRIA6 and QUAD8 but the analyst can request the consistent mass by means of the COUPMASS parameter described in Chapter 12. Mass coefficients for rotational degrees of freedom are not calculated in either case so that the elemental mass matrix associated with bending behavior is singular.

5.10.9

Stress, Strain and Element Force Recovery

The element strains and curvatures, and therefore the stress components, are calculated in element coordinates as described for the flat shell elements in the previous section. The components of stress are computed using (Eq. 5-11) at distances z 1 and z 2 from the reference surface which are defined on the PSHELL statement. The stresses are recovered at the vertices and the centroid for each element..

5.11

Solid Elements The NASTRAN solid elements include: a four-sided element called the TETRA; a five-sided element called the PENTA; and, a six sided element called the HEXA. These elements are shown by Figure 1-32, Figure 1-33 and Figure 1-34, respectively. Each of these elements can include optional edge nodes. However, the 4-node TETRA and the 6-node PENTA result in very stiff elements and should not be used for modeling. The 10-node TETRA and the 15-node PENTA perform quite well, especially compared to their lower order counterparts. The 8-node HEXA performs well and can be used for general modeling situations calling for elements capable of representing a three dimensional stress field. These elements were developed to

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1.Support Anisotropic material 2.Improve stress output 3.Improve performance in representation of curved shell behavior 4.Improve treatment of differential stiffness G4

G8

G10

G9 G3

G7 G1 G6 G5 G2 Figure 1-32 Four Sided TETRA Element

.

6 15 4 14 13

12

5

10 11 3 9 1

8 7

2

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Figure 1-33 Five-Sided Isoparametric Solid Element - PENTA

8 20

19

5

7

18 16

17 6

13

15

14

4

12

11 3

1

10

9 2

Figure 1-34 Six-Sided Isoparametric Element - HEXA

5.11.1


The Bulk data statements for defining solid elements representing three dimensional stress states are as follows: 1

2

CTETRA EID G7

1

3

G7

5

PID

G1

G2

G8

G9

G10

2

CPENTA EID

4

3

4

6 G3

5

7 G4

6

8 G5

7

9

10

9

10

G6

8

PID

G1

G2

G3

G4

G5

G6

G8

G9

G10

G11

G12

G13

G14

G15

1

2

4

5

6

7

8

9

PID

G1

G2

G3

G4

G5

G6

G7

G8

G9

G10

G11

G12

G13

G14

G15

G16

G17

G18

G19

G20

G13

G14

CHEXA EID

1

3

2

PSOLID PID

3 MID

4 CORDM IN

5

6

7

STRESS ISOP

8

9

FCTN

Bulk Data Image 1-9 Isoparametric Solids; PENTA and HEXA

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10

10


5.11.2


The TETRA, PENTA and HEXA elements connect the three displacement degrees of freedom at each node point on the element. The node points to be connected are identified by: G1 through G10 on the CTETRA; G1 through G15 on the CPENTA; and G1 through G20 on the CHEXA data statements shown by Bulk Data Image 1-9. The property field, PID, refers to a common PSOLID data statement.

5.11.3

The TETRA Element

The TETRA element four vertex nodes and six edge nodes. The node number sequence shown on Figure 1-32 must be preserved, i.e. nodes G1 through G4 must define the vertices. The edge nodes, G5 through G10, are optional and positional. That is: G5 is the node on the edge between G1 and G2, G6 is the node on the edge between G2 and G3, etc. If the element is to be modeled without the edge node, G 5, then the field associated with that node is left blank. The continuation is not required if all of the edge nodes are omitted. The 4-node TETRA element performs poorly in almost all modeling situations and is not recommended. However, the 10-node TETRA performs well and is widely used.

5.11.4

The PENTA Element

The PENTA element has six vertex nodes and nine edge nodes. The node number sequence shown on Figure 1-33 must be preserved, i.e. nodes G 1 through G3 must define the vertices of a triangular face; and, nodes G 4 through G6 must then define the opposite triangular face with G4, G5, and G6 on the edges connecting G1, G2, and G3, respectively. The edge nodes G 7 through G15 are optional and positional. That is: G7 is the node on the edge between G1 and G2, G8 is the node on the edge between G2 and G3, etc. If the element is to be modeled without an edge node, G 7 for example, then the field associated with that node, field 2 on the first continuation of the CPENTA statement, contains no entry and is thus blank. Continuations are not required if all edge nodes are omitted.

5.11.5

HEXA Element

The HEXA element has eight vertex nodes and twelve edge nodes. The node number sequence shown on Figure 1-34 must be preserved. Nodes G 1 through G4 must define one face and nodes G5 through G8 must then define the opposite face with G5 lying on the edge connecting G1 etc. The edge nodes G 9 through G20 are optional and positional; and, any or all may be defined. As in the case of the TETRA and PENTA elements, the edge nodes are positionally defined with node G9 on the line segment connecting G 1 and G2 and G17 on the line segment between G5 and G6, for example. The absence of an entry for G 9, field 4 of the first continuation of the CHEXA, implies there is no node on the edge connecting G 1 and G2. If no edge nodes are defined then the second continuation is not required.

5.11.6

Properties of Solid Element

The properties of the TETRA, PENTA and HEXA solid elements are defined by a common property data statement, the PSOLID, as shown on Bulk Data Image 6-11. The fields of PSOLID are defined as: Field

Description

PID

The property ID, integer > 0.

MID

The material set ID, integer > 0.

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Field

Description

CORDM

The set ID of a coordinate system to be used as the reference system f or a non-isotropic material as described below, integer, blank or > 0.

IN

Specifies the integration procedure to be used, integer or character as described below.

STRESS

Specifies location for stress output, integer or character as described below.

ISOP

Integration scheme, integer, character or blank as described below.

FCTN

Fluid element flag, blank for structural solution.

5.11.6.1

Material Coordinate System

The material identification refers to either an isotropic or an anisotropic material defined by a MAT1 or MAT9 material statement, respectively. If the material is anisotropic then the orientation of the material axis relative to element coordinates must be defined by an entry in the CORDM field as follows CORDM

Meaning

Blank or - 1

Align material coordinates with element coordinates.

0

Material axes are aligned with Basic coordinates.

>0

Material axes are aligned with coordinates defined by coordinate identification number appearing in the field.

5.11.6.2

Gauss Integration Procedure

The IN and ISOP fields provide control of • Gauss integration order • Use of Bubble functions • Use of reduced integration

Two sets of integration points are provided for each element. The user can specify either of the sets by including an integer or character value for the IN field as shown by Table 5-1. Table 5-1 Specifying Gauss Integration Order Element Type IN HEXA

PENTA

Tetra

2 or TWO

8 points

6 points

1 point

3 or THREE

27 points

21 points

5 point

If the elements have no midside nodes, IN can also be set equal to BUBBLE and ISOP = REDUCED, which are the defaults, which minimize locking due to shear and Poisson’s ratio effects. The defaults are recommended.

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Structural Elements in NASTRAN Solid Elements 5.11.6.3

Stress Locations

Stress components are calculated at the vertex nodes by default. Alternatively, the stress components can be calculated at the Gauss integration points for elements having no edge nodes by setting STRESS to either ‘GAUSS’ or 1. The locations of the Gauss points must, of course, be known in order to process the stress components. These locations are described in Section 15.3 of the NASTRAN Reference Manual .

5.11.7

Hexa Element Coordinate System

The orientation of the element coordinate system is of importance to the analyst if anisotropic material axes are to be aligned with the element axes. The element coordinates for the HEXA element are chosen using the following procedure. 1.Let a1 , a2 , and a3 be the directed line segments joining the centroids of faces 1 and 2, faces 3 and 4, and faces 5 and 6, respectively where the faces contain nodes as follows Face

Node1

Node2

Node3

Node4

1

1

2

3

4

2

5

6

7

8

3

1

2

6

5

4

4

3

7

8

5

1

5

8

4

6

2

6

7

4

2.Choose the unit vector, a i such that | ai | (i = 1, 2, 3) is a maximum. 3.Among the remaining directed line segments choose the unit vector v in the direction of the next longest directed line segment. 4.Find the unit vector z perpendicular to the plane containing ( i ,v ) by forming the cross product k = i × v . 5.Define the unit vector to be perpendicular to the plane containing ( i ,k ) by forming the cross product j = k × i . 6.Permute the directions ( i ,j ,k ) so that i' is approximately parallel to the line segment from G1 to G2 and j' is approximately parallel to the line segment joining G1 and G4.

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Consider the example shown by Figure 1-35 8 7 a1 a2

v i 4

5 6

3

a 3 , k

1 2 Figure 1-35 Orientation of Local Coordinates for HEXA Element

Apparently, a2 is the longest segment so that the unit vector, i , is in that direction. The line segment a 1 is the next longest segment so that v is a unit vector in this direction. The unit vector k (assumed to be coincident with a ) is then perpendicular to the plane containing i and v while (here assumed to be coincident with v ) is perpendicular to the plane containing ( i and k ). The direction k is seen to be approximately parallel to the edge 1-2 while i is approximately parallel to the edge 1-4. The local coordinate system is then taken to be x = k , v = i , and z = j

5.11.8

PENTA Element Coordinate System

The local coordinate system for the PENTA element is formed as shown by Figure 1-36 which shows only the vertices. z

6

4 v

y

5

3 1 2

x

Figure 1-36 PENTA Local Coordinate System

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The local x-axis is taken to be the line segment that joins the centroids of the straight line segments which join vertices 1-4 and 2-5, respectively. A vector, v , is then defined by the directed line segment that joins the centroids of the straight line segments which join vertices 1-4 and 3-6. The vector, v , is then taken to be in the local x-y plane so that z = x × v and y = z × x .

5.11.9

TETRA Element Coordinate Syatem

The element coordinate system for the TETRA element is generated using the three vectors, R , S and T shown by Figure 1-37. G4

S

R

G3

G1 T

G2 Figure 1-37 TETRA Element Coordinate System

The element coordinate system is obtained from the three vectors, R , S and T which join the midpoints of the opposite edges as shown by Figure 1-37. The element coordinate system is chosen as close as possible to these vectors and points in the same general direction.

5.11.10 Elastic Stiffness Matrix HEXA and PENTA and TETRA elements may have any or all of the edge nodes deleted. The calculation of the stiffness matrix is then formulated by taking the deleted nodes into account. 5.11.10.1

The TETRA Element

The element is transformed to parametric space to a right isosceles tetrahedron. The stiffness matrix is then calculated using Gaussian integration with either of two integration schemes which correspond to 4- and 10-node elements. The 4-noded element is the three dimensional analogy of the two dimensional linar triangle element for which one Gauss point is appropriate. Elements having edge nodes use the higher order integration scheme. 5.11.10.2

The HEXA Element

The element is transformed to parametric space as described in Chapter 4. The stiffness matrix is then calculated using Gaussian integration with either of two integration schemes which correspond to the 8- and 20-noded elements.

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Structural Elements in NASTRAN Congruent Elements

The 8-noded element is the three dimensional analogy of the two dimensional bilinear quadratic element. The same type of shear correction is thus used in the three dimensional element that is used to improve the membrane shear behavior of the QUAD4 element. Thus, for the 8-noded element the extensional strains are evaluated at a 2 × 2 × 2 array of Gauss points but the shear strain is evaluated at a reduced number of special points. The second integration scheme corresponds to elements having one or more undeleted edge nodes. The extensional strains are evaluated at a 3 × 3 × 3 array of Gauss points with reduced integration at special points for shear strains. 5.11.10.3

The PENTA Element

The PENTA element is transformed to a right isosceles prism in parameter space by using appropriate shape functions. Two integration schemes are provided that use six or nine integration points. The six point scheme is the default when all edge nodes are deleted. In all other cases the nine node scheme is the default. Reduced integration is used for shear strains in either integration scheme.

5.11.11 Mass The solid elements support both the coupled and lumped mass formulations. The lumped formulation is default only for the four point TETRA, the six node PENTA and the eight node HEXA elements. If the elements have edge nodes the coupled formulation is used. The coupled formulation can be specified independent of the default by including a COUPMASS parameter statement as described in Chapter 12.

5.11.12 Stress and Strain Recovery The components of stress and strain are calculated in the material coordinate system; i.e., the system specified by the CORDM field on the associated PSOLID statement. The recovery of stress and strain quantities is controlled by the STRESS and STRAIN Case Control directives. The components of stress (strain) include: 1.The six components of stress 2.The three principal stresses and principal stress directions 3.The von Mises stress The components of stress are recovered at the centroid of each element and at the vertex nodes. The post processing procedures used to transform and average the stress components by Patran are described in Chapter 12.

5.12

Congruent Elements Congruent elements can be defined to designate secondary elements that are identical to a primary element. The congruent elements are defined using the CNGRNT data statement, which is shown in Bulk Data Image 1-10

1

2

3

4

5

CNGRNT

PRID

SECID1

SECID2

SECID3

+001

SECID8

SECID9

etc.

SECID N

PRID

SECID1

THRU

6 SECID4

7 SECID5

8 SECID6

9 SECID7

or CNGRNT

Bulk Data Image 1-10 Specification of Identical Elements

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10 0001

Structural Elements in NASTRAN References

where the fields are defined as

5.13

PRID

Identification number of the primary element for which elemental properties will be calculated

SECIDi

Element identification numbers of secondary elements whose elemental properties will be identical to the primary element

References [1] R.J. Roark, Formulas for Stress and Strain, McGraw-Hill, Third Edition 1954, pp. 119-121. [2] J.S. Przemieniecki, Theory of Matrix Structural Analysis , McGraw-Hill,1965, pp. 79. [3] NASTRAN Programmers Manual , NASA SP-223(01) pp. 4.87-21 through -23 [4] R.M. Rivello, Theory and Analysis of Flight Structures, McGraw-Hill, 1969, pp. 241-246. [5] R.J. Roark, Formulas for Stress and Strain, McGraw-Hill, Third Edition 1954, pp. 129-130. [6] NASTRAN Application Manual, M acNeal-Schwendler Corp., 1977, p. 2.6-19. [7] J.E. Ashton, J.C. Halpin, and P.H. Petit, Primer on Composite Materials: Analysis, Technomic, 1969, pp. 37-45. [8] R.M. Jones, Mechanics of Composite Material , McGraw-Hill, 1975, pp. 147-173. [9] R.H. MacNeal, "A Simple Quadrilateral Shell Element", Computers & Structures, Vol. 8, pp. 175-183, 1978. [10] NASTRAN USER’S MANUAL, NASA SP-222(08) Vol. I, pp. 2.4-44.

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6

Global Analysis Procedures

A structural system is composed of a collection of structural components that are attached to each other in some defined way. The purpose of this chapter is to consider those operations that lead to the global stiffness matrix, together with all operations that are performed on the global stiffness matrix. These operations include the specification of constraints, partitioning, and flexibility-stiffness transformations.

6.1

The Global Stiffness Matrix The equilibrium equation for each element obtained using the finite element method relates element displacements and element forces as follows

[ k ee ] { ue } = { P e } where {ue} represents the vector of discrete node point displacements and rotations, { P e} represents the vector of external forces acting on t he element and [ k ee] is the element stiffness matrix. The notation used in this text is to represent matrices using a subscript to indicate a specific set and, implicitly, to specify the size and form of the matrix. The single subscript on {ue} and { P e} defines column vectors having e rows as appropriate for a specific element. The double subscript on [ k ee] indicates it is a square matrix having e rows and e columns where e is the number of degrees of freedom in the element. The set of element displacements is a subset of the set of all the Grid point degrees of freedom in the model which is designated by {u g }. An element, ‘n’, connects a subset of the {u g }degrees of freedom. The element degrees of freedom { ue} for element n can be expressed in terms of {u g } using a Boolean transformation as: n

n

{ ue } = [ Beg ] { u g } n

(Eq. 6-1)

where [ B eg ] is an e-by- g Boolean matrix whose elements are either one or zero and where it is also assumed that the stiffness matrix has been transformed from element coordinates to the global set of coordinates as will be described later in this section. The result of representing all element displacements in terms of the global set of displacements is to enforce displacement compatibility at all Grid points in the model.

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Global Analysis Procedures The Global Stiffness Matrix

As an example of the construction of the Boolean transformation, consider the structure shown by Figure 6-1 6

5 5 6

4

1

7 1

4 3

2

2

3

Figure 6-1 Idealized Structure

where the structural system is composed of six triangular elements that connect the seven node points. Assuming, for purposes of explanation, that there is a single degree of freedom of each node, the g -set is given by:

{ u g } =

T

u 1 u 2 u3 u4 u5 u 6 u7

(Eq. 6-2)

The set of displacement degrees of freedom connected by element one includes displacements at Grids one, two, and seven so that the set of displacements for element one is:

' ( $ u1 $ 1 0 0 0 0 0 0 $ $ 1 { u e } = % u 2 & = 0 1 0 0 0 0 0 { u g } $ $ 0 0 0 0 0 0 1 $ u3 $ " #

(Eq. 6-3)

The global stiffness may now be defined using either an energy or a direct approach. Both methods will be described in following sections. After the element stiffness matrices have been transformed to the global set of displacements, the procedure for assembling the global stiffness matrix can be deduced by noting that the total strain energy for the structure is the sum of the strain energies for the individual elements. The virtual change of the internal strain energy is thus: N

δ U =

1 δ U n

(Eq. 6-4)

n=1

where the scalar quantity, δU n is the virtual change in the internal strain energy for element ‘n’ undergoing a virtual displacement, { δue}, and δU is the virtual strain energy for entire system due to a virtual change in the g -set displacements, { δu g }. The virtual strain energy can be expressed in terms of the appropriate element stiffness so that (Eq. 6-4) can be written as:

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Global Analysis Procedures The Global Stiffness Matrix N T

δ U = { δ u g } [ K gg ] { u g } =

1 { δ ue }

T

[ k ee ] { ue }

(Eq. 6-5)

e=1

Since the element displacements, after transformation from element to global coordinates, are subsets of { u g } set, the virtual strain energy becomes: N T

1

δ U = { δ u g } [ K gg ] { u g } =

T

n

T

n

n

{ δ u g } [ B eg ] [ k ee ] [ B eg ] { u g }

(Eq. 6-6)

n=1

where the superscript on element-related matrices is the element number; and, the sum is taken over all N elements in the model. The global stiffness matrix, by inspection: N

[ K gg ] =

1

n

T

n

n

[ B eg ] [ k ee ] [ Beg ]

(Eq. 6-7)

n=1

The effect of the transformation, (Eq. 6-7), is to embed each of the [ k ee] matrices in a g -size matrix. The stiffness coefficients are not changed in value as a result of this transformation, they are placed in appropriate rows and columns of g -sized matrix. The resulting g-size elemental stiffness matrices are then added to obtain the global stiffness matrix [ K gg ]. The procedure is similar to the process of adding springs in parallel. For example, the transformation of element two of the structural system shown in Figure 6-1 leads to: 0 0

0 0 0 0 0

0 k 222 k 223 0 0 0 k 227 n T n [ B eg ] [ k 2ee ] [ Beg ]

=

0 k 232 k 233 0 0 0 k 237 0 0 0 0 0 0

(Eq. 6-8)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 k 272 k 273 0 0 0 k 277 where a superscript on [ k ee] and its coefficients is the element number. The explicit Boolean transformation, (Eq. 6-7), is not actually used in a finite element program. Instead, logic is used in the assembly phase to generate the global stiffness coefficients without formally performing the matrix algebra. The procedure is: n

1.Generate the stiffness coefficients for all elements [ k ee ] ( n = 1, 2, ... , N ) and a data structure that relates element vertices to external grid points. This data structure is essentially the connectivity list for an element. This phase is performed by the Element Matrix Generation (EMG) module. 2.Relate element degrees of freedom to global degrees of freedom using internal tables called the Equivalence Table (EQEXIN) and the Scalar Index List (SIL). The EQEXIN relates internal grid IDs to external grid IDs; and the SIL relates grid point degrees of freedom to the degrees of freedom the g -set. 3.Add all elemental stiffness coefficients with the same row and column designations. This phase is accomplished by the Element Matrix Assembly (EMA) module.

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6.2

Global Analysis Procedures Local and Global Coordinate Systems

Local and Global Coordinate Systems The assembly of the system stiffness matrix in the previous section requires that all the Grid point degrees of freedom for an element be transformed to the displacement coordinate system which is specified at each Grid point. This is required since the effect of the Boolean transformation from element to global coordinates is to enforce displacement compatibility for all elements connected to a Grid point. NASTRAN allows the user to define a different displacement coordinate system at each grid point. The procedure used in the EMG module is to: • Generate a local coordinate system for the element based on the order in which the grid points are defined. The procedure for defining the local coordinate system is elementdependent and is described as a part of the element descriptions in the previous chapter. • Generate the element stiffness relative to the element local coordinate system. • Determine the set of direction cosines between the element and displacement coordinate system for each Grid point as defined by the CD field on the Grid Bulk Data entity. • Transform the displacement and rotation stiffness partitions of the stiffness matrix associated with the Grid points to the displacement coordinate system.

6.3

Transformation of Element Stiffness Matrices The global set of coordinates is the set of all coordinate systems defined by Cord - type data statements. The coordinate transformation for the global set of coordinates is available in the Coordinate System Transformation Matrices (CSTM) table in NASTRAN which is available to the EMG module. Let { ue' } be the set of element displacements defined relative to the element coordinate system, ( x', y', z ' ) ,and let [ k e' e' ] be the element stiffness generated in the element coordinate system. Then, calculate the transformation matrix from global to the element coordinate system, [ λ e' e ] so that':

{ u e' } = [ λ e' e ] { u e }

(Eq. 6-9)

The virtual work for the element is a scalar quantity which is independent of coordinate system where the virtual work of the element in element coordinates is: T

δ W ' = { δ ue' } [ k e' e' ] { u e' } – { δ ue' } { P e' }

(Eq. 6-10)

Substituting(Eq. 6-9) into (Eq. 6-10) leads to: T

T

T

T

δ W = δ W ' = { δ u e' } { λ e' e } [ k e' e' ] { λ e' e } { u e' } – { δ u e' } { λ e' e } { P e' }

(Eq. 6-11)

The element equilibrium equation in global coordinates is:

[ k ee ] { ue } = { P e }

(Eq. 6-12)

where the transformed element stiffness and load are defined to be: T

[ k ee ] = [ λe' e ] [ k e' e' ] [ λ e' e ]

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(Eq. 6-13)

Global Analysis Procedures Specifying Structural Degrees of Freedom

and T

{ P e } = [ λe' e ] { P e' }

(Eq. 6-14)

The transformation of displacements, loads, and stiffness from local element to global coordinates is completely automated in finite element programs such as NASTRAN. Generally the user is not concerned with the local element coordinate system; however, there are cases that the user must understand the how the local element coordinate system is defined relative to the global set of coordinates. These are 1.The specification of the local element axes for the three-dimensional beam 2.The specification of a normal pressure on plate bending elements 3.The interpretation of stress output.

6.4

Specifying Structural Degrees of Freedom We saw in earlier chapters that the finite element method represents a structural system as an assemblage of elements that are connected at discrete points in the structure. The behavior of the structure is characterized by the value of the generalized displacements at these discrete points. The accuracy of the solution is therefore dependent on the number of node points used to describe the system as well as the distribution of the node points throughout the structure. Suitable criteria for choosing the number and the distribution of node points is discussed in later chapters of this text. Node points serve four distinct purposes: 1.The specification of structural topology 2.The specification of degrees of freedom 3.Accurately locate points at which displacements are constrained or at which loads are applied 4.Define locations where behavioral variables are to be calculated It is interesting to note the structural topology, which is the most important characteristic of a node point from a design point of view, is used only during the calculation of the stiffness matrix. From the analysis point of view the displacement degrees of freedom at the node is the most important characteristic since they are associated with the behavior of the structure. For structurally oriented finite element models six degrees of freedom are defined at each node, or Grid, point defined on the structure. These degrees of freedom at the geometric point are shown by Figure 6-2 which show the associated forces as well.

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Global Analysis Procedures Specifying Structural Degrees of Freedom

z

z

u zi

P zi M zi

θ zi

θ xi

i

θ yi

u xi

u yi

M xi

i

P xi

y

P yi

M yi

y

x

x

b) Grid Point Forces

a) Grid Point Degrees of Freedom

Figure 6-2 Displacements and Forces at a Grid Point

The displacements are represented by the set

{ u˜ i } =

u 1 u 1 u 3 θ 1 θ 2 θ 3

T

(6-15)

where { u˜ i } is vector of degrees of freedom at node i. Grid point degree of freedom are identified in NASTRAN by a pair of integer numbers, (G,C) where G

Identification number of a geometric grid point

C

A component number having a value from one to six that identifies a degree of freedom using the following correspondence table Table 6-1 Degree of Freedom Codes Physical Degree of Freedom

Degree of Freedom Code

u1

1

u2

2

u3

3

θ1

4

θ2

5

θ3

6

where the subscripts on the physical degree of freedom variables indicates the coordinate sense of the displacement, ui, or the axis associated with the rotation, θi. Geometric grid points are defined using the Grid data statement described in Grid Points -GRID (p. 171). The coordinate systems used to define the location of the geometric grid points are defined using Cord-type data statements as described in Coordinate Systems CORD (p. 167).

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Global Analysis Procedures Defining Scalar Degrees of Freedom - SPOINT

Degrees of freedom may be associated with non structural behavior such as control system variables, temperatures, and scalar spring mass systems. Scalar degrees of freedom may be defined directly in NASTRAN using Spoint as described in Defining Scalar Degrees of Freedom - SPOINT (p. 167).

6.5

Defining Scalar Degrees of Freedom - SPOINT Scalar degrees of freedom are useful for representing non structural degrees of freedom such as the average of several displacements. However, it is also used to specify the cross sectional warping for the Beam element. Scalar degrees of freedom maybe defined in two different ways: 1.Implicitly by reference on a scalar element data statement (i.e. CELAS, CMASS, etc.) 2.Explicitly by an SPOINT Bulk Data statement In either case, a single degree of freedom in defines in the g-set for each scalar point. Scalar elements are defined explicitly by using the SPOINT Bulk Data entity shown by Bulk Data Image 6-1. 1

2

SPOINT SPID 1

3

4

SPID2

SPID3

THRU

SPID2

5 SPID4

6 SPID5

7

8

9

10

etc.

*alternatively SPOINT SPID 1

Bulk Data Image 6-1 Scalar Point - SPOINT

where Field SPID

6.6

Description Scalar point identification number that must be unique among all GRID of SPOINT in Bulk Data. (Integer > 0)

Coordinate Systems - CORD Coordinate systems define reference systems for geometry and for vector quantities such as displacements and forces. We know from experience that different types of coordinate systems are convenient for different geometric topologies. Therefore, we would expect a finite element analysis program to provide the user with several types of coordinate systems. The default coordinate system, which exists and needs no definition, is called the Basic Coordinate system. An unlimited number of number of rectangular, cylindrical and spherical coordinate systems can be defined. These subsidiary coordinate systems are defined relative to the Basic Coordinate system or another defined coordinate system by Cord-type data statements as described below. Coordinate systems are identified by a coordinate identification (CID) number, which must be a positive integer. The coordinate identification CID= 0 refers to the Basic Coordinate system which is a rectangular Cartesian coordinate system. The set of all coordinate systems defined for the analysis is called the Global Set of Coordinates. It must be noted that several authors have used the term “global coordinate system” to define what is termed the Basic Coordinate System in NASTRAN.

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Global Analysis Procedures Coordinate Systems - CORD

A subsidiary coordinate system ( X 1', X 2', X 3' ) is shown by Figure 6-3.

X ' 3

X 3

B

X ' 2

C V

A

X 1

X 2 X ' 1

Figure 6-3 Subsidiary Coordinate System

The new coordinate system is defined in terms of a reference coordinate system ( X 1 , X 2 , X 3) by specifying the position of three points A, B, and C as follows: 1.Point A is the origin of the new coordinate system 2.The directed line segment between point A and B defines the X ' 1 -axis. 3.Point C lies in the X ' 1 , X ' 3 -plane. The X ' 2 -axis is thus given by i ′ 2 = i ′ 3 × V

(6-16)

i ′1 = i ′ 2 × i ′ 3

(6-17)

and

where V is the vector from point A to C.

6.6.1


The location of the three points can be defined two ways: The coordinates of points A, B, and C can be specified directly by the CORD2-type statements, which are shown in Bulk Data Image 6-2; or, three geometric Grid points, which correspond to points A, B, and C as shown by Bulk Data Image 6-3.

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Global Analysis Procedures Coordinate Systems - CORD Cylindrical. 1

2

3

4

CORD2C

CID

RID

A1

+1

C1

C2

C3

5 A2

6 A3

7 B1

8

9

B2

B3

10 +1

Rectangular. 1

2

3

CORD2R CID

RID

A1

C1

C2

C3

2

3

4

CORD2 CID S

RID

A1

C2

C3

+1

4

5

6

7

8

9

A2

A3

B1

B2

B3

5

6

7

8

9

10 +1

Spherical. 1

C1

+1

A2

A3

B1

B2

B3

10 +1

Bulk Data Image 6-2 Coordinate Systems Defined by Coordinates of Three Points

The logical CORD2-type statements described by Bulk Data Image 7-2 each require two lines of input. The coordinate definitions for each type of coordinate system are identical except for the mnemonic name in the first field of the input statement where CORD2C, CORD2R, and CORD2S specify cylindrical, rectangular, and spherical coordinates, respectively. The other fields of the CORD2-type data statements are Field

Description

CID

Coordinate system identification number (integer > 0)

RID

The CID of the reference coordinate system. A blank or zero entry in this field indicates that the Basic Coordinate system is the reference. However, any coordinate system defined in the Bulk Data section may be used. The coordinates of the points A, B, and C are then interpreted for the various allowable types of coordinate systems as shown by Table 7-1 (integer > 0)

(A1, A2, A3)

Coordinates of point A that defines the origin of coordinate system (real)

(B1, B2, B3)

Coordinates of point B that lies on the X 3' -axis (real)

(C1, C2, C3)

Coordinates of point C that lies in the X 3', X 1' -plane (real)

Cylindrical. 1 CORD1C

2 CID

3 GA

4 GB

5 GC

6 CID

7 GA

8 GB

9

10

9

10

GC

Rectangular. 1 CORD1R

2 CID

3 GA

4 GB

5 GC

6 CID

7 GA

8 GB

GC

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Global Analysis Procedures Coordinate Systems - CORD

6

Spherical. 1

2

CORD1S

3

4

GA

CID

5

GB

6

GC

7

8

GA

CID

9

GB

10

GC

Bulk Data Image 6-3 Coordinate System Defined by Three Grid Points

The coordinate definition data statements shown on Bulk Data Image 6-3 are identical for each type of coordinate system except for the data statement mnemonic in the first field where CORD1C, CORD1R, and CORD2S specify cylindrical, rectangular, and spherical coordinates, respectively. The other field designations are Field

Description

CID

Coordinate system identification number (integer, > 0)

GA,GB,GC

Identification numbers of three GRID points (integer, > 0) where:

GA defines Grid point A at the origin GB defines Grid point B lying on the X 3' -axis, and GC defines Grid point C lying in the X 3'- X 1' -plane The cylindrical, rectangular, and spherical coordinate systems defined by the CORD-type Bulk Data are shown by Figure 6-4a, Figure 6-4b, and Figure 6-4c, respectively. The components of the points A, B, and C specified on the CORD2-type statements are interpreted in terms of the coordinate system defined by coordinate set identification number RID. For example, if a spherical coordinate system is associated with this set identification number, then the components would be the R, θ , and φ coordinates of the point as shown by Figure 6-4c. z

u1

u3

P

u2

z p

y

x p

y p

x

a) Rectangular

z

z

u3

u1 P

R

b) Cylindrical

u3 y

y

φ

θ x

P

R u1

z

θ

u2

x

u2

c) Spherical

Figure 6-4 Coordinate Definitions and Displacement Components

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Global Analysis Procedures Grid Points -GRID

The components of displacement referred to rectangular, cylindrical, and spherical coordinates are also shown by Figure 6-4a,Figure 6-4b, andFigure 6-4c. NASTRAN identifies the Grid point degrees of freedom in the output by labeling the components of displacement T1, T2, and T3 and the components of the rotation R1, R2, and R3. These components are interpreted as components of displacement in the displacement coordinate system specified on the Grid data statement, which is defined in the next section, associated with the point, using Table 6-4. The displacement coordinate system at each geometric grid point can be different from the system used to define the location of the point. Coordinate Component

Coordinate

1

2

3

Rectangular

X

Y

Z

Cylindrical

R

θ(degrees)

Z

Spherical

R

φ(degrees)

θ(degrees)

Type

Table 6-4 Definition of Coordinate Components

6.7

Grid Points -GRID A Grid point data statement shown by Bulk Data Image 6-5 defines. 1.The coordinates of a point in space relative to a specified coordinate system 2.six degrees of freedom in the g -set of displacements 3.The reference coordinate system for displacements and forces 4.Degrees of freedom which are to be either constrained or removed from the analysis. 1

2

GRDSET

3

4

5

6

CP

GRID

GID

CP

X1

X2

X3

7

8

CD

PSPC

CD

PSPC

9

10

SEID

Bulk Data Image 6-5 Geometric Grid Point Definition - GRID

where Field

Description

GID

Grid point identification number (integer, >0).

CP

Identification number of the coordinate system used as reference f or the coordinates (integer, > 0).

(X1, X2, X3)

Coordinates of the grid point with respect to the CP-coordinate system (real). See Table 7-2 in the previous section for definition of coordinate directions in non rectangular coordinate systems.

CD

Identification number of the displacement coordinate system (integer, > 0).

PSPC

Purged permanent single point constraints (integer, > or blank)

The Grid point is a material point at which a displacement vector is defined. The displacement vector represents the six displacement degrees of freedom at the point: three components of the displacement vector ( u1, u 2, u3 ) and three components of the rotation vector ( θ 1, θ 2, θ 3 ) . Six degrees of freedom are added to the g -set for each GRID point. Nastran Primer 171

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Global Analysis Procedures Grid Points -GRID

Since the number of rows and columns in the system matrices is determined by the number of degrees of freedom in the g -set it may turn out that one or more of the degrees of freedom are not connected by elastic elements.This would result in a singular stiffness matrix. For example, suppose the model is to represent in-plane stretching of a plate represented using membrane elements. The membrane element only provides connectivity for in-plane displacements so the out of plane displacement and all rotations have no connectivity. This, in turn, means that the associated rows and columns are null and that the stiffness matrix is singular. NASTRAN incorporates a process called AUTOSPC for detecting and removing singular degrees of freedom. However, they are more properly removed by using the PSPC field since the size of the assembled stiffness matrix is greatly reduced for some models.

6.7.1

Grid Identification Number

Each Grid point must have a Grid identification (GID) number that is unique among all Grid and Spoint identification numbers. The GID is called the external Grid point number. The external GID are converted to internal numbers to simplify internal calculations. Two tables are maintained that allow conversion from one form to another. Results are always identified using the external GID. The GIDs need not form a consecutive numerical sequence. If a structural model were to be represented by three geometric grid points, for example, the sequences (100, 200, 300), (1, 2, 3), or (7, 12, 17) would produce the same internal sequence of equations and would result in system matrices with the same matrix topology.

6.7.2

Geometric Coordinates

The location of the Grid point is specified by the coordinates ( X 1 , X 2 , X 3) defined relative to the reference coordinate system specified by the CP field. The value of the entry in the CP field is the coordinate system identification (CID) of the reference system. If the CP field is blank the Basic Coordinate system is used. If a non blank entry exists in the CP field the coordinate system associated with the entry must be defined or else the system will generate a fatal error message during execution. If CP is zero, then ( X 1 , X 2 , X 3) are taken to be the coordinates of the point referenced in the Basic Coordinate system. If CP is the number of a user-defined coordinate system, then the coordinates are interpreted according to Table 6-4.

6.7.3

Displacement Coordinates

The CD field specifies the reference coordinate system for the displacement and rotation vectors associated with the Grid point. If CD is zero, then the Basic Coordinate system is used. If the displacement and rotation vectors are to be defined in some other coordinate system then the CD field is the CID of a user-defined coordinate system. The components of the displacement and rotation vectors are interpreted for the various coordinate systems using Figure 6-4. The rotations ( θ1, θ2, θ3) are components of the rotation vector in the directions using the right-hand rule.

6.7.4

Permanent Constraints

The function of the PSPC field is to specify displacement degrees of freedom for which permanent constraints are to be prescribed. If no stiffness is associated with degrees of freedom identified by the PSPC field the resulting action is to remove the them from the analysis during the assembly of the g -sized stiffness matrix. Since there is no stiffness the associated constraint force will be zero.

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Global Analysis Procedures External and Internal Degrees of Freedom

PSPC can also be used to specify displacement constraints for connected degrees of freedom. However, it is highly recommended that displacement boundary conditions be applied using SPC data entities as described in a later section. Constraints applied using PSPC have a constraint value of zero. The entry in the PSPC field is a packed set of degree of freedom codes which specify degrees of freedom included in the permanent constraint set. These degrees of freedom are interpreted as displacement components in the coordinate system specified by the CD field on the GRID entity. The codes for the six displacement at the geometric grid point are given by Figure 6-1 presented in a previous section. The use and specification of constraints is described in succeeding sections of this chapter. It is worth repeating that the constraints specified on Grid data entities are applied during the assembly of the stiffness matrix and cannot be changed. In addition, the intended use of PSPC is to remove one or more of the six degrees of freedom that are automatically generated but that do not connected by elastic elements from the analysis.

6.7.5

Default Values - GRDSET

The GRDSET, which is also shown by Bulk Data Image 6-5, is used to define default values for the CP, CD, and PSPC fields. It is a user convenience for eliminating repetitious data. However, a word of caution is in order regarding the use of GRDSET: The values CP, CD, and PSPC on the GRDSET are default values and are used only if the corresponding field of the Grid is blank. The default value is not used if an integer, including the integer zero, exists in the associated field of the GRID entity. A common mistake is assuming that permanent single point constraints placed in the GRID statement are added to those specified in the GRDSET. On the contrary, a PSPC specified on GRID overrides the default value on GRDSET. Only one GRDSET may appear in Bulk Data; and, at least one of the fields, CID, CD, or PSPC must be different from zero. The inclusion of a GRDSET statement with only zero entries will result in a fatal error.

6.8

External and Internal Degrees of Freedom The program logic is simplified if the degrees of freedom are identified by a set of consecutive numbers starting at one. All user references to degrees of freedom are in terms of the external Grid numbers and degree of freedom codes and internal computations use internal codes. The conversion from external to internal degrees of freedom is automatically performed by NASTRAN as follows: • The lowest point number is found. If it is a SPOINT then the first internal degree of freedom corresponds to the SPOINT; if it is a GRID then the first six internal degrees of freedom correspond to the GRID point. • The next highest point number is found and one internal degree of freedom is generated if it is an SPOINT and six internal degrees of freedom if it is a GRID, etc.

The set of all GRID point and SPOINT scalar degrees of freedom is called the g -set and is denoted by { u g } , The total number of degrees of freedom in the g -set is N g = 6N + S

where N is the number of GRID points and S is the number of SPOINTs. The parameter called LUSET (length of the USET) is given the value of N g in the solution sequences.

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6.9

Global Analysis Procedures Displacement Sets

Displacement Sets It is convenient to consider the displacement degrees of freedom as the elements of a displacement set. The set of all grid point degrees of freedom including scalar degrees of freedom is called the g -set and is of order N g as described above. The stiffness matrix associated with the g -set is denoted by [ K gg ] and is normally generated from a finite element model. The system matrix associated with the g -set of degrees of freedom may be singular, i.e., the rank of [ K gg ] is less than N g , either because the internal degree of freedom does not physically exist, or because the elements of the g -set are linearly dependent. As noted in the description of the GRID entity, six internal degrees of freedom are defined at each Grid point implying six rows and six columns in the global stiffness matrix that are initially zero. If the finite element model does not define elements in the stiffness matrix associated with a degree of freedom, then the degree of freedom is not connected and must be purged from the g -set. After purging all unconnected degrees of freedom, the resulting subset of { u g } may still include rigid body motion that must be restrained to make [ K gg ] non singular. NASTRAN allows the user to define a number of independent subsets of { u g } associated with the following types of constraints: 1.Specification of values for specific degrees of freedom, which is termed a single point constraint (SPC) 2.Specification of linear relation between two or more degrees of freedom, which is termed a multi point constraint (MPC) 3.Specification of reactionless constraints, which are used to model unsupported structures (SUPORT) In addition, static condensation can be specified using ASET and OMIT. The data sets that are removed by a constraint or reduction operation are mutually exclusive. A degree of freedom can only exist in one of the mutually exclusive sets so that a degree of freedom specified in any one of the sets cannot be specified in another removed data set. Each of these constraint and reduction operations acts on a merged set of displacement degrees of freedom to produce partitioned data sets, one of which is retained in the analysis and one of which is removed. These operations are performed in the sequence shown by the following table that identifies the combined set, the data set removed, and the data set retained by the operation. Constraint Operation

Operates On Merged Set

Partitioned Data Sets Set Retained

Set Removed

MPC

{u g }

{un}

{um}

SPC

{un}

{u f }

{u s}

{u f }

{ua}

{uo}

{ua}

{ul }

{ur }

OMIT or ASET SUPORT

Table 6-6 Displacement Data Sets for Static Analysis

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Global Analysis Procedures Multipoint Constraints - MPC

6.9.1

Merged Data Sets

The merged data sets are {u g } Set of all g rid point degrees of freedom including extra points. {un} Set of all degrees of freedom not eliminated by multi point constraints. {u f }

Set of f ree degrees of freedom that remain after specification of single point constraints.

{ua} Analysis set is solution set for real eigenvalue analysis.

6.9.2

Mutually Independent Data Sets

The merged sets are partitioned and one of the sets is removed from the analysis. The set removed is called a mutually independent set since degrees of freedom cannot be members of another removed set. The mutually independent displacement sets identified as removed data sets in Table 6-6 are {um} Set removed by multi point constraints {u s} Set removed by single point constraints {uo} Set omitted by static condensation {ur } Set of forceless degrees of freedom, which remove r igid body motion and {ul }

6.10

Set l eft over, which is the solution set in static analysis

Multipoint Constraints - MPC A multi point constraint is a linear equation relating displacement degrees of freedom. Relations of this type can be used to define ad hoc connectivity between degrees of freedom. Using MPCs we can model rigid links between Grid points or to specify constraints that are a linear combination of Grid point displacements.

6.10.1

Uses of MPC

There are a number of modeling situations that require the use of ad hoc relations between degrees-of-freedom including: 6.10.1.1

Representing very stiff structural members.

Let's assume that you are modeling a structural assembly that includes one or more components which are extremely stiff compared with the rest of the structure. Examples would be a vehicle model that includes the engine, or a spacecraft model that includes a very stiff interface flange between booster stages. In each of these cases the stiff component defines the load path through that part of the structure, and in each case the stiff component deforms very little compared to the deformations in the more flexible components of the structure. They have the property of being "Rigid" compared to the flexible components. Before the need for “Rigid” elements was discovered, stiff elements in a finite element model were represented by stiff elastic elements. After all, what's wrong with faithfully modeling reality? Unfortunately, early practitioners found the solutions were invalid because of something called matrix ill conditioning.

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Without going into the mathematics, stiff elastic components lead to large differences in magnitude between the off-diagonal coefficients of the stiffness matrix. These large differences, in turn, lead to numerical problems in the solution phase. Research into the reasons for wrong answers to, what appeared to be, well posed finite elements models lead to the recognition that stiff members must be represented by using proper constraint equations rather than very stiff elastic elements. All of this discussion leads to two conclusions: 1..Indicators of the accuracy of the solution process are required. NASTRAN provides several indicators including the approximate conditioning numbers and a norm called epsilon whose value tells us how many significant figures in the result are valid. 2.Finite element programs must support linear constraint equations. Linear constraints must be incorporated in finite element programs. NASTRAN implements MPC Bulk Data and a number of rigid elements. 6.10.1.2

Load transfer between dissimilar elements.

Let's suppose that you wanted to model one portion of the structure using three dimensional continuum elements and another portion using a strength of materials shell approximation. The model looks great, and it contains no apparent singularities. However the solution fails in decomposition because the reduced stiffness matrix is singular. Why? Because the model contains a “mechanism”. There is no way to transfer a moment from the shell elements to the solids because the solids have no rotational degrees-of-freedom. In order to eliminate the singularity an ad hoc relation must be defined that will, in effect, create a couple in the solid to resist the moment in the shell. This is an interesting exercise which will be discussed later. 6.10.1.3

Constraints in an arbitrary direction.

You will recall that the displacement coordinate system, which is defined by the CD field on the GRID entity, is used for all references to degrees-of-freedom in constraint equations. Now, how would you describe the constraint equation for a grid point which is constrained to move parallel to a boundary, where the boundary in the x- direction is not aligned with a coordinate direction? A set of linear relations that describes kinematic constraints between degrees of freedom can be specified directly by Multi Point Constraints (MPC). The prescribed value for a single degree of freedom is called a Single Point Constraint (SPC) and is discussed in the following section. The combined capability of MPCs, SPCs and scalar degrees of freedom is sufficient to specify any linear relationship between degrees of freedom.

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Global Analysis Procedures Multipoint Constraints - MPC 6.10.1.4

4. Rigid Links

For example, consider the rigid link that connects grid points A and B as shown by Figure 6-1 x AB

u 2 B y u 6 B

B

u 1 B y AB

A

u 1 A u 6 A

u 2 A x Figure 6-1 Planar Rigid Link

Since the link between points A and B is rigid, the following kinematic relations exist between displacement degrees of freedom at A and B u 1 B = u 1 A – y AB u 6 A u 2 B = u 2 A + x AB u 6 A

(Eq. 6-1)

u 6 B = u 6 A

where x AB and y AB represent the distances from point A to B in the x and y directions, respectively.

6.10.2

Reduction to the n-Set

The general form of the MPC relation is:

[ G mg ] { u g } = 0

(Eq. 6-2)

(Eq. 6-2) represents a set of m constraint equations where [ Gmg ] is a matrix of coefficients specified by the user having m rows and g columns. The solution set { u g } must, therefore, satisfy(Eq. 6-2) in addition to the following set of equilibrium equations for the g -set:

[ K gg ] { u g } = { P g } + { c g }

(Eq. 6-3)

where { P g }are external forces applied to the g-set degrees of freedom and { c g } are forces of constraint. There are two procedures for incorporating the constraint equations into the solution: a reduction procedure or augmenting the stiffness to incorporate the constraints using Lagrange multipliers.

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NASTRAN uses the elimination procedure, and there’s a story there. The g -set is first partitioned into two complementary sets; the m-set which contains exactly m degrees of freedom where m is the number of constraint equations, and the remaining degrees of freedom which is called the n-set. The question is how to determine the m-set degrees of freedom, since once the m-set is known the process for determining the transformation from the u g set to the n-set is straight forward as we will show below. For a simple set of constraint equations the choice of the m-set degrees of freedom can be fairly straight forward. But for a general set of equations there must be a better way: and there is. It’s all about the rank of the matrix G mg . If the rank is less than m the equations are not linearly independent. If the equations are linearly dependent than we need to find the m best pivot columns. It turns out that is not to hard to do; and MSC has included the automated determination of the m-set degrees of freedom. Legacy NASTRAN doesn’t include it yet but hopefully it is on the to-do list as a future enhancement. So, once the m-set is known, then the g-set is represented in partitioned form as:

{ u g } =

{ um } { un }

(Eq. 6-4)

The constraint equations are then used to defines a transformation between the m- and nset that is used to reduce(Eq. 6-3)to the n-set. In order to reduce the set of equations to the n-set, we proceed by writing the set of constraint equations (Eq. 6-2) in terms of the m- and n partitions as follows:

[ R m m ] { um } + [ R mn ] { un } = 0

(Eq. 6-5)

The partitioned constraint equation can now be used to define a transformation between the mutually independent set, { um } , and the merged set, { un } , as follows:

{ u m } = [ G mn ] { un }

(Eq. 6-6)

where – 1

[ G mn ] = – [ R mm ] [ R mn ]

(Eq. 6-7)

The value of the mutually independent set, { um } , is seen to be dependent on the value of the retained set of displacements, { u n } . Since (Eq. 6-6) shows that { u m } is dependent on the { un } displacements the term “mutually independent” can lead to some confusion. However, the term means that { um } is eliminated from and is therefore independent of other eliminated sets of displacements including the { u s } , { u o } and { ua } sets as we will see later. We can now use the constraint relation (Eq. 6-6)to obtain a transformation between { u g } and { un } by writing (Eq. 6-4) as follows:

/ [G ] 0 mn . { u g } = -. { u n } = [ G gn ] { u n } I [ ] nn + ,

(Eq. 6-8)

where [ I nn ] is an identity matrix of n-size. The reduced stiffness matrix is then found by noting that the constraint forces { c g } perform no work during a virtual displacement, δ u g }, so that the virtual work expression associated with(Eq. 6-3)is

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Global Analysis Procedures Multipoint Constraints - MPC T

{ δ u g } ( [ K gg ] { u g } + { P g } ) = 0

(Eq. 6-9)

Admissible virtual displacements must satisfy the constraints on the displacement function (Eq. 6-8), so (Eq. 6-9) becomes T

T

{ δ u n } [ G gn ] ( [ K gg ] [ G gn ] { un } + { P g } ) = 0

(Eq. 6-10)

The virtual displacements, δ u g }, are arbitrary the satisfaction of (Eq. 6-10) requires that T

T

[ G gn ] [ K gg ] [ G gn ] { un } – [ G gn ] { P g } = 0

(Eq. 6-11)

[ K nn ] { un } = { P n }

(Eq. 6-12)

or

where the reduced stiffness and load are given by T

[ K nn ] = [ G gn ] [ K gg ] [ G gn ]

(Eq. 6-13)

T

{ P n } = [ G gn ] { P g }

(Eq. 6-14)

The multi point constraint specification must, therefore, perform two tasks: 1.Define the matrix of coefficients [ G mg ] 2.Specify the degrees of freedom in the m-set The form of the MPC Bulk Data thus requires you to: 1.Define each equation in the set of m constraint equations with a single logical MPC data statement. This means that if there are six constraint equations there will be six logical MPC Bulk Data, all with the same MPC Identification, MPCID. 2.define an m-set degree of freedom. This degree-of-freedom is then partitioned into the m-set and thus must be unique among all MPC definitions. Since m MPC Bulk Data are required to represent the m equations of constraint, this means that the mset contains exactly m unique degrees-of-freedom. 3.Assure that [ R mm ] is not singular so that [ G mn ] can be formed.

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6.10.3

MPC Data Entity

The MPC data entity is designed to accomplish both tasks. Each constraint equation is defined by one logical MPC entity. In order to specify ‘m’ constraint equations ‘m’ logical MPC entities are required. The MPC entity shown by Bulk Data Image 6-1defines a degree of freedom to be included in the m-set as well as nonzero coefficients of the constraint. 1

2

MPC

MPCSID

MPCADD MPCSID S8

3

4

5

6

7

8

CN1

AN1

S5

S6

GM

CM

Am

GN1

GN2

CN2

AN2

-etc.-

S1

S2

S3

S4

S9

-etc.-

9

10

S7

Bulk Data Image 6-1 Multi point Constraint Specification

wherethe data fields are described as follows: Field Name MPCID

GM, CM

Description The MPC set identification number that is referenced by an MPC Case Control directive to select a set of constraint equations from Bulk Data A Grid point number and degree of freedom code for a single degree of freedom that is partitioned from the g -set into the m-set (GM, integer > 0, 0 < CM < 6)

AM

The constraint coefficient of degree of freedom (GM i, CMi) in the current constraint equation. (real)

GN j, CN j

A Grid point number and degree of freedom code defining a single g -set degree of freedom in the current constraint equation which has a non zero coefficient, integer > 0

GN j, CN j

A Grid point number and degree of freedom code defining a single g -set degree of freedom in the current constraint equation which has a non zero coefficient, integer > 0

AN j

The constraint coefficient of degree of freedom (GNj, CNj) in the current constraint equation. (real)

Sk

MPC set identification number of MPC sets being unioned by MPCADD Bulk Data

The set identification number in field two of the MPC data statement allows several logical MPC sets to be included in Bulk Data. The particular MPC set which is to be used at execution time is specified by the MPC Case Control Directive as shown below MPC =

where MPCSID is the integer number assigned to an MPC set in Bulk Data.

6.10.4

Combining MPC Sets - MPCADD

Multiple constraint equations can be combined into a single logical set by the MPCADD entity. The MPC sets specified by sets S 1, S2,...,S N are combined and given a unique MPCSID on the MPCADD entity that is then specified by an MPC Case Control Directive. All the MPC and MPCADD set identification numbers must be unique.

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6.10.5

Specifying the m-set Degrees of Freedom

One logical MPC data statement is required for each degree of freedom to be included in the m-set. The degrees of freedom are identified by the external degree of freedom number, which consists of the pair (GM, CM) where GM

is the identification number of a GRID or SPOINT and

CM

6.10.6

is a degree of freedom code at G. If G is a scalar point, then C is either blank or equal to integer zero.

Recovering and Printing MPC Forces

The forces of associated with the MPC constraints are often of great interest. If MPCs are used to model a spot weld, for example, the associated MPC forces are those that the weld must transmit. The forces associated with MPCs are requested using the MPCFORCES case control directive as described in Chapter 1. All MPC forces could be requested using the following Case Control directive: MPCFORCES = ALL

6.10.7

Defining the Constraint Equation

The first degree of freedom, identified by (GM, CM) in fields three and four, is a g-set degree of freedom which will be partitioned into the m-set. The coefficient Am is the associated coefficient of [ G mg ] . All subsequent degrees of freedom defined by point number and degree of freedom code pairs, (GN j, CN j) specify a column in the current constraint equation having a non zero coefficient A j in the [ G mg ] matrix. The Grid point degrees of freedom specified on the MPC data statement are defined with respect to the displacement coordinate system specified by the CD field on the associated GRID entity. 6.10.7.1

General Rigid Link

It is common to have relatively stiff structure connecting flexible components. One may be tempted to use a stiff elastic element to connect such points, but experience has shown an ill conditioned stiffness matrix may result due to orders-of-magnitude-differences in the offdiagonal stiffness terms. A kinematic constraint relating the displacements at the two points eliminates the ill conditioning problem. The kinematic constraint is modeled as a rigid link between the points connecting the stiff component.

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Consider two grid points A and B, as shown by Figure 6-1 with a rigid link connecting the two points. z

B

l

y A x Figure 6-1 Rigid Connection of Two Grid Points

The rotations of the two points are θ A and θ B , respectively; and the displacements are u A and u B . The displacement and rotation of B can then be expressed in terms of the displacement at A as u B = u A + θ A × r AB

(Eq. 6-1)

where r AB is the vector from point A to point B and, since the rotation of all points on the rigid link are the same:

θ A = θ B

(Eq. 6-2)

. These vector equations can be written in scalar form as u1B = u1A - r yu6A + r z u5A u2B = u2A - r z u4A + r xu6A u3B = u3A - r xu5A +r yu4A u4B = u4A

(Eq. 6-3)

u5B = u5A u6B = u6A

where r x , r y , r z are the components of r AB in the x-, y-, and z -directions. As a numerical example, let the coordinates of points A and B be given as A => (1, 2, 3) B => (5, 3, 1)

which are identified as grid points 363 and 132, respectively. The components of r AB can then be found to be r x = 4; r y = 1; and r z = 2 The constraint (7-34) can then be written as in the form of (7-19) as follows

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1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

' $ $ $ $ $ $ 0 – 1 0 0 0 –2 1 $ $ 0 0 – 1 0 2 0 – 4 $ 0 0 0 – 1 –1 4 0 $ % 0 0 0 0 –1 0 0 $ $ 0 0 0 0 0 –1 0 $ 1 0 0 0 0 0 – 1 $ $ $ $ $ $ $ "

u 1 A (

$

u 2 A $

$

u 3 A $

$

u 4 A $

$

u A $

$

u 6 A $

& = 0

(Eq. 6-4)

u 1 B $

$

u 2 B $

$

u 3 B $

$

u 4 B $

$

u 5 B $

$

u 6 B #

There are six constraint equation, therefore six MPC data statements are required. Looking at the first equation, there are four non zero coefficients. The m-set degree of freedom must have a non zero coefficient so the displacements at B will be arbitrarily chosen to be members of the m-set. Letting the Grid ID of A and B be 383 and 132, respectively, the MPC entities defining the constraint equation are: 1 MPC

2 101

+ M1 MPC

101

+ M2 MPC

101

+M3

3

4

5

6

7

8

132

1

1.

363

1

-1.

363

5

-2.

363

6

1.

132

2

1.

363

2

-1.

363

4

2.

363

6

-4.

132

3

1.

363

3

-1.

363

4

-1.

363

5

4.

MPC

101

132

4

1.

363

4

-1.

MPC

101

132

5

1.

363

5

-1.

MPC

101

132

6

1.

363

6

-1.

9

10

+ M1 +M2 +M3

where the MPC set identification MPCSID = 101 has been used for each MPC data statement. The user will note that nine lines of input are required to define the simple kinematic constraint defined by the rigid link. The specification of even simple rigid link connecting two points is a laborious task. Since all of the geometric data required to define the constraint equations is included on the GRID it would seem logical to let the program generate the associated MPC set programmatically. NASTRAN does this by including a library of constraint elements. These elements defined the topology of the rigid element and generate the constraint equations. The example described above could have been defined by using a single RBAR constraint element The rigid and constraint elements are described in the chapter on elements

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6.11

Global Analysis Procedures Single Point Constraints - SPC

Single Point Constraints - SPC Displacement boundary conditions are used to specify degrees having known values. Generally the value of these degrees of freedom will be zero but this need not be the case. The specification of displacement boundary conditions is the most common use of SPC. It is used, for example to define the “built-in”, or fully constrained, conditions at one end of a cantilever beam. The statement “built-in” means the displacements associated with the fixed degree of freedom are not unknowns. They are known and the fact they are zero for a beam results in a unique solution for the cantilever beam under a certain loading condition. Single point constraints and multi-point constraints are also used to specify symmetric boundary conditions. The use of symmetry conditions, if present, can lead to a great reduction in the resources required to obtain a solution. For example, a square plate with symmetric boundary conditions and isotropic material can be modeled using one eighth of the structural model. Symmetry boundary conditions are then used to represent the rest of the plate. Without getting into the details of how to model symmetry conditions at this time it should be rather obvious that the size of the structural model is greatly reduced. The specification of displacement boundary conditions are applied to degrees of freedom in the s-set. Degrees of freedom specified on SPC entities have an explicit or implicit constraint value and are partitioned into the s-set. SPC thus defines the partition of the n-set as follows:

{ un } =

{ u s }

(Eq. 6-5)

{ u f }

where { u s } is prescribed as:

{ u s } = { Y s }

(Eq. 6-6)

Where { Y s } are known values specified on SPC Bulk Data statements.

6.11.1

Reduction to the f -Set

The equations associated with the n-set are reduced to the f -set by partitioning the n-set equations,(Eq. 6-12), using (Eq. 6-5) which gives:

' [ K ss ] [ K sf ] $ { u s } % [ K fs ] [ K ff ] $ { u f } "

( ' $ $ { P s } & = % ˜ } $ $ { P f # "

( $ & $ #

(Eq. 6-7)

The { u s }set is prescribed as indicated by (Eq. 6-6) so that the reduced set of equations for the free or f -set becomes

[ K ff ] { u f } = { P f }

(Eq. 6-8)

{ P f } = { P f } – [ K fs ] { Y s }

(Eq. 6-9)

where

6.11.2

Single Point Constraint Forces

The internal forces are given by

184 Nastran Primer


' $ { c s } % $ { c f } "

( $ [ K ss ] [ K sf ] & = [ K fs ] [ K ff ] $ #

' $ { Y s } % $ { u f } "

( ' $ $ { P s } & – % $ $ { P f } # "

( $ & $ #

(Eq. 6-10)

The forces of reaction associated with { u s} degrees of freedom are found from the first of equations (Eq. 6-10) to be:

{ c s } = [ K ss ] { Y s } + [ K sf ] { u f } – { P s }

(Eq. 6-11)

where { c f } = 0 since there are no constraints on the f -set. The single point constraint forces can be recovered by the Case Control Directive. SPCFORCE =
6.11.3

SPC Data Entity

Single Point Constraint (SPC) and is defined by SPC, SPC1, and SPCADD data statements as shown in Bulk Data Image 6-1 1

2

3

4

5

6

7

SPC

SID

GID1

C1

VALUE1

GID2

C2

SPCADD

SID

SID1

SID2

SID3

SID4

-etc.-

8

9

10

10

VALUE2

*Alternate Forms 1 SPC1

2

3

4

5

6

7

8

9

GID3

GID4

GID5

GID6

SID

C

GID1

GID2

GID7

GID8

GID9

-etc.-

SID

C

GID1

THRU

or SPC1

GID2

Bulk Data Image 6-1 Single Point Constraint Specification

where Field

Description

SID

SPC set identification number (integer, > 0)

(GIDi, Ci)

Pair defining external degree of freedom number. GID is the grid or scalar point identification number (integer, > 0) and C is the degree of freedom code. C may be a packed integer number 0 < C < 6 to specify all degrees of freedom at G that have the value given by D.

VALUEi

Value of the prescribed displacement (real).

SID1, SID2, ...

SPC Set identification numbers

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6.11.4

Selecting SPC Sets in Case Control

Several sets of single point constraints can be defined in Bulk Data where each set i s given a unique set identification number. The SPC set that is to be used for a given Subcase is then specified by the Case Control directive SPC =

where SID is the integer number assigned to a SPC set in Bulk Data. SPC sets can be combined into a single logical set by SPCADD. The SPC sets specified by set identification numbers SID1, SID2, ... ,etc. are combined and are given a unique SID on the SPCADD, which is then selected by the SPC Case Control directive.

6.11.5

Specifying s-set Degrees of Freedom and Value

The SPC Bulk Data perform two independent functions. These are 1.Purging nonexistent degrees of freedom. 2.Setting displacement boundary conditions. The form of the SPC data statement supports these two functions by 1.Defining degrees of freedom to be included in the s-set. 2.Specifying a value for each degree of freedom either implicitly or explicitly. The SPC data statement is used to explicitly set the degree(s) of freedom defined by the pair (GID, C) to the value, VALUE, where GID

is a grid or scalar point number.

C

is a packed set of one to six degrees of freedom codes at point GID.

All of the C degrees of freedom at grid point GID then are set equal to the value VALUE The constraint value of a degree of freedom specified by SPC1 is implicitly zero. The degrees of freedom specified by the packed code C are set equal to zero at all the grid points listed on the remainder of the SPC1 entity. SPC1 can be continued whereas SPC cannot. A set of points can be defined using the second form of SPC1 using of the literal string THRU. The total set of degrees of freedom included in the s-set is taken to be the union of all degrees of freedom defined on SPC-type Bulk Data and those defined as permanent single point constraints (PSPC) on GRID Bulk Data.

6.11.6

Purging Degrees of Freedom

It is frequently necessary to purge degrees of freedom from the solution set. Typically these are degrees of freedom defined for a GRID point but which are not connected by the finite element model. Examples would include out-of-plane behavior for plane problems, the rotational degrees of freedom for membranes and solids, and the normal rotation for plates and shells. Purging can be accomplished by simply including all unconnected degrees of freedom in the s-set. Suppose, for example, that only unconnected degrees of freedom are included in the s-set. Then, since there is no stiffness associated with these degrees of freedom, all of the matrix partitions in (7-41) associated with the s-set are null. The reduced set of equations is then given by (7-39) where { P f } independent of the prescribed values { Y s } , since [ K fs ] is null.

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6.11.7

AUTOSPC - Automatic Purging

In large analysis problems the task identifying degrees of freedom which are unconnected in the system stiffness matrix can be difficult. NASTRAN thus includes the capability of automatically detecting singular degrees of freedom which can then be removed under user control. The process, which takes place after the unconstrained stiffness matrix [ K gg ] has been generated, is controlled by the AUTOSPC parameter. Both Legacy and MSC NASTRAN and MSC include a capability for detecting essential singularities but the procedure differs as does the form of the AUTOSPC parameter. The procedures described below do not deal with the finding linear dependences that are due to rigid body motion or one or more components of the structure. These can only be determined during the matrix solve operation in static analysis. 6.11.7.1

Legacy NASTRAN

Legacy NASTRAN find the rank of the 3x3 matrices associated with displacement and rotational degrees of freedom on the diagonal of the [ K gg ] . The AUTOSPC parameter then controls the action taken if singularities that are not removed by existing SPC or MPC Bulk Data. The AUTOSPC parameter has the following form 1 PARAM

2 AUTOSPC

3 value

where value is an integer number whose default value is zero. If singularities are found and no AUTOSPC parameter having a non zero value is present then a list of singularities is printed in the output file and program execution proceeds and will fail in the matrix solve phase due to one or more singular columns, The best advice is to include 1 PARAM

2 AUTOSPC

3 1

in the Bulk Data section.

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6.11.7.2

MSC NASTRAN

The mathematical procedure for detecting potential singularities looks at the eigenvalues of the 3x3 matrices for displacement and rotational degrees of freedom on the diagonal of [ K gg ] . In essence the inclusion of a PARAM as shown below with the name AUTOSPC and a value of YES will cause NASTRAN to remove unconnected degrees of freedom. The default value of the parameter is “YES” in the MSC solution sequences so no user action is required to remove essential singularities 1

2

PARAM

6.11.8

3

AUTOSPC

YES

Example: Specification of Single Point Constraints

Consider the cantilever beam in Figure 6-1. y u y 2 = 2 in. EI

1

2 x

L

Figure 6-1 Enforced Displacement of a Beam

The beam is built-in at the left end can deflect only in the xy- plane. An enforced displacement u y2 = 2 in. is to be specified at the right end of the beam. The constraints are specified by the following SPC:

SPC1

1

1345

1

2

SPC1

2

26

1

SPC

3

2

2

.02

SPCADD

4

1

2

3

where: 1.The SPC1having SID = 1 retains only displacements in the xy-plane, 2.The SPC1having SID = 2 constrains u x and θ z at Grid 1, 3.The SPC having SID = 3 specifies the enforced displacement at Grid 2, 4.The SPCADD having SID = 4 combines the previous sets of constraints The combined set of constraints would be selected by a SPC = 4 Case Control directive. It should be noted that the out-of-plane displacements could also be removed by using a PSPC having a value of 1345 at all Grid points.

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Global Analysis Procedures Static Condensation -OMIT and ASET

6.12

Static Condensation -OMIT and ASET Static condensation is a basic transformation used for retaining a boundary set in substructures and as an effective transformation for dynamics. Static condensation in substructures is used to represent the boundary displacements of a component in terms of displacements internal to the component. The internal displacements are partitioned into the oset and removed from the analysis set. The intent in dynamics is entirely different. In dynamics, static condensation results in a transformation that is useful for reducing the number of dynamic degrees of freedom by redistributing mass and stiffness coefficients to a smaller number of degrees of freedom. The mathematical procedure is the same for both applications: the f -set is partitioned into the o-set which is to be removed from the analysis; and, the a-set which is retained. In normal modes analysis the a-set is the solution set in which eigenvalues and eigenvectors are extracted as described a a later chapter. In statics, the a-set may be further partitioned by SUPORT as described in the next section.

6.12.1

Reduction to the a-Set

The first step in static condensation is to partition the f -set into two complementary sets: the o-set that will be omitted from the analysis, and the a-set hat will be retained in the analysis. Partitioning { u f } gives:

' $ { uo } { u f } = % $ { ua } "

( $ & $ #

(Eq. 6-1)

The reduced stiffness matrix associated with the { u f } set is written in partitioned form as

' [ K oo ] [ K oa ] $ { u o } % [ K ao ] [ K aa ] $ { u a } "

( ' $ $ { P o } & = % $ $ { P a } # "

( $ & $ #

(Eq. 6-2)

The transformation between the { uo } and { ua } displacement sets, which is used in the reduction process, is determined by solving for { u o } from the first of the two matrix equations in (Eq. 6-2):

{ u o } = [ G oa ] { ua }

(Eq. 6-3)

where T

[ Goa ] = – [ K oo ] [ K oa ]

(Eq. 6-4)

T

where the term – [ K oo ] { P o } which is associated with the loads on the o-set will be added during the recovery of the o-set displacements as shown by(Eq. 6-10). The {u f } set can be expressed entirely in terms of { ua}by substituting (Eq. 6-3) into (Eq. 6-1) giving:

{ u f } =

[ G oa ] [ I aa ]

{ ua }

(Eq. 6-5)

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or

{ u f } = [ G fa ] { ua }

(Eq. 6-6)

where [ I aa ] is an identity matrix of a-size. The reduced set of equations for the a-set is then given by:

[ K aa ] { u a } = { P a }

(Eq. 6-7)

where T

[ K aa ] = [ G fa ] [ K ff ] [ G fa ]

(Eq. 6-8)

and T

{ P a } = [ G fa ] { P f } 6.12.2

(Eq. 6-9)

Recovery of Omitted Degrees of Freedom

After the displacements in the a-set are determined, the o-set is recovered by augmenting the displacements calculated by means of(Eq. 6-3) with the displacements produced by the loads { P o } . The o-set displacements are given by: o

{ u o } = [ G oa ] { u a } + { uo }

(Eq. 6-10)

where o

– 1

{ u o } = [ K oo ] { P o } 6.12.3

(Eq. 6-11)

Physical Interpretation of [Goa]

The transformation matrix [ G oa ] (Eq. 6-4), that relates the omitted degrees of freedom to those retained in the analysis has a useful physical interpretation. The columns of are static displacement modes of the structure. That is each column represents the static displacement of the entire o-set for a unit displacement of the a-set degree of freedom associated with that column, with all other a-set degrees of freedom set equal to zero.

6.12.4

Why Use Static Condensation?

The purpose in going through this rather straight forward derivation is to show you how simple the idea of static condensation really is. It is a partitioning operation that is exact for statics that leads to a smaller solution set. The partitioning is defined by defining either the oset or the a-set. The question is why it might be useful to define a reduced set of analysis degrees-of-freedom. In your mind's eye, think about a large structure, such as a large commercial airliner. Let's say that your task is to analyze the wing due to a variety of flight loads. How do you support the wing? You could choose to fix the points where the wing connects to the fuselage. But that support condition doesn't properly represent the elastic compliance of the fuselage structure.

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The use of a model of both the fuselage and wing will have the correct compliance but may result in a very large analysis model. However, if static condensation was used on the fuselage model to reduce the stiffness to the boundary displacements between the two structures, the compliance would be correct and the model sized reduced. This is the basic idea in the using components in NASTRAN. After completing the stress analysis of a component, such as a wing, the next task might be to determine the ten lowest modes and associated natural frequencies. Here the problem is one of problem size and computer resources. The static stress analysis model for the wing might have 500,000 to 1,000,000 degrees-of-freedom. Based on the fact that more than one normal mode is required, one of several eigenvalue routines such as Givens, Modified Givens or Lanczos might be employed. Assuming the model will be used for design trade-off studies a reduced solution set size is desirable. The use a static condensation to reduce the size of the analysis set is one option. The transformation represented by (Eq. 6-3) might be of interest. It describes how a “dependent” set of displacements, the o-set, responds, statically, to displacements in the a-set, where the transformation is based solely on the stiffness of the system. As a matter of fact, think about the a-set as a displacement vector which is all zeros except for a single component which has a unit value. This little thought experiment tells us that the columns of the transformation [ G oa ] are the “static displacement modes” of the o-set relative to the a-set. Guyan, [1], recognized that the "static displacement modes", if properly chosen, could be used in dynamics to reduce the number of dynamic degrees-of-freedom. It was such a good idea that the technique is now a standard tool in the dynamic analyst’s bag of tricks and is called Guyan reduction. Guyan reduction can lead to accurate dynamic response provided the o-set is properly selected as described in the chapter on normal modes.

6.12.5

The OMIT and ASET Data Entities

Static condensation is specified by the inclusion of either OMIT or ASET entities, described below by Bulk Data Image 6-1 and Bulk Data Image 6-2, respectively:. 1 OMIT

2 G

3

4

5

6

7

8

9

C

G

C

G

C

G

C

G3

G4

G5

G6

G7

10

*Alternate Forms

OMIT1

C

G1

G2

+OM1

G8

G8

-etc.-

C

GI

THRU

+ OM1

or OMIT1

GJ

where G j

Grid or scalar point identification number (integer, > 0)

C

Degree of freedom code, an integer if G is a geometric point or zero or blank if G is a scalar point. Bulk Data Image 6-1 Matrix Reduction - OMIT

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OMIT is used to specify degrees of freedom to be partitioned into the removed o-set and the ASET specifies degrees of freedom to be retained in the a-set. Either OMIT or ASET can be used. 1 ASET

2

3

G

4

5

6

7

8

9

C

G

C

G

C

G

C

G3

G4

G5

G6

G7

10

*Alternate Forms

ASET1

C

G1

G2

+A1

G8

G8

-etc.-

C

GI

THRU

+ A1

or ASET1

GJ

Bulk Data Image 6-2 Matrix Partitioning - ASET

6.12.6

Specifying Degrees of Freedom

There are alternate forms of the OMIT or ASET called OMIT1 or ASET1s. The OMIT and ASET data statements allow the user to specify a degree of freedom by the external degree of freedom number, which consists of the pair (G, C) where G

is the grid or scalar point number

and C

is one or more degree of freedom codes at point G.

Degrees of freedom specified on OMIT are partitioned into the o-set while those specified on ASET data statements are partitioned into the a-set. Note that neither the OMIT nor ASET data statements can be continued. The OMIT1 and ASET1 allow the user to define degrees of freedom to be partitioned into the o- or a-sets, respectively, by specifying the degrees of freedom using the degree of freedom codes, C, followed by all the Grid points at which the degree(s) of freedom are to be omitted or retained. This form of the OMIT1 and ASET1 data statements can have continuations. Finally, both the OMIT1 and ASET1 data a statements have an alternative form that allows the user to specify an inclusive set of grid and scalar points by use of the literal string THRU in the field separating the first and last grid or scalar point.

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Global Analysis Procedures Support for Free Bodies - SUPORT

6.12.7

Example: Static Condensation

Consider the finite element model shown inFigure 6-2, in which we wish to place all degrees of freedom associated with the interior grid points in the o-set. z

21

22

16 11 6 1

2

17

12

7

23

13

8

3

18

19

14 9

4

24

25

y

20

15

10

5

x Figure 6-2 Elimination of Internal Degrees of Freedom using OMIT

The structure is that of a plate, and it is assumed that the rotation about the normal has been purged by use of single point constraints. The following Bulk Data then reduce the degrees of freedom in the analysis set to those associated with the boundary. 1

2

3

OMIT1

12345

7

+ O1

18

19

4 8

5 9

6 12

7 13

8 14

9 17

10 + O1

where the number in the second field specifies the degrees of freedom that are to be included in the o-set for each of the grid points listed. Conversely, the following ASET1 data statements could be used 1

2

3

4

5

ASET1

12345

1

THRU

5

ASET1

12345

6

10

11

ASET1

12345

21

THRU

25

6

15

7

16

8

9

10

20

For this example the o-set is smaller than the a-set, and it makes sense to use the OMIT type data statement. One could visualize the case where the converse would be true and then the ASET data statement should be used.

6.13

Support for Free Bodies - SUPORT An elastic body that is capable of undergoing rigid body motion is termed a free body. Free elastic bodies are capable of motion that produces no internal forces in the body. The stiffness matrix for a free body is singular and the structure must be supported to remove the rigid body motion for static analysis. The required constraints could be specified by single point or multi point constraints, or by means of free body supports. The advantage in using free body supports is that the rigid body characteristics are calculated and the supports are checked for sufficiency.

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The singularity can be removed by restraining sufficient degrees of freedom { ur }to eliminate rigid body motion of the structure without introducing redundant reactions. The set of forces associated with the r -set thus must be statically determinate.

6.13.1

Reduction to the l-Set

The stiffness equation for the a-set(Eq. 6-7) can be written in partitioned form as ˜ ] [ K ] [ K rr rl

{ ur }

[ K lr ] [ K ll ]

{ u l }

=

{ P r } { P l }

' { C r } ( & " 0 #

+%

(Eq. 6-12)

where{ur }is a set of displacements that if constrained would remove rigid body motion; { ul } is the set left over after all constraint and partition operations and is the solution set for static analysis; and, { C r } are constraint forces associated with the r -set and where the a-set is partitioned as follows

{ ua } =

{ ur }

(Eq. 6-13)

{ ul }

The r -set is specified on the SUPORT data statement described by Bulk Data Image 6-3. If the displacements of the r -set and are set equal to zero we have, from the second of (Eq. 6-12),

[ K ll ] { u l } = { P l }

(Eq. 6-14)

The stiffness matrix [ K ll ] is non singular so that (Eq. 6-14) can be solved for the {ul } set.

6.13.2

Rigid Body Transformation Matrix

Although the SUPORTed degrees of freedom, { ur }, are set equal to zero in reducing to the solution set, {ul }, this operation differs from that associated with the use of SPCs in a very fundamental way. The forces of constraint associated with the r -set are equal to zero. Thus, from the first of (Eq. 6-12) we find

[ C r ] = 0 = – { P r } + [ K rl ] { ul }

(Eq. 6-15)

The forces applied to the SUPORTed degrees of freedom, { P r }, can now be expressed in terms of the loads applied to the solution set, { P l }, by noting that { ul }can be found from (Eq. 6-14) as – 1

{ ul } = [ K ll ] { P l }

(Eq. 6-16)

{ P r } = – [ G rl ] { P l }

(Eq. 6-17)

so that

where [ G rl ] is called the rigid body transformation matrix and is given by – 1

[ G rl ] = – [ K rl ] [ K ll ]

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(Eq. 6-18)


It is left as an exercise to show that, in the absence of external loads, the motion of the f-set is related to motion of the r -set through the following relation T

{ ul } = [ G lr ] { u r } = [ G rl ] { u r } 6.13.3

(Eq. 6-19)

Rigid Body Stiffness Matrix

The SUPORTed degrees of freedom, { u r } , define the rigid body motion of the structure. Therefore, the reduced stiffness matrix for the r -set should be null. The reduced matrix is found by noting that rigid body motion of the { ul } is related to r -set displacements by(Eq. 6-19) so we can write (Eq. 6-13) for rigid body motion as

{ ua } = [ G ar ] { u r }

(Eq. 6-20)

where the rigid body transformation matrix is

[ G ar ] =

[ I rr ] [ G lr ]

(Eq. 6-21)

Then, the reduced stiffness matrix [ K rr ] is given by T

[ K rr ] = [ G ar ] [ K aa ] [ G ar ]

(Eq. 6-22)

Representing [ K aa] in partitioned form, the substitution of (Eq. 6-21) into (Eq. 6-22) then gives

[ K rr ] = [ K ˜rr ] + [ K rl ] [ G lr ]

(Eq. 6-23)

where [ G rl ] is given by (Eq. 6-18). The reduced matrix [ K rr ] should be completely null at this point. NASTRAN calculates the rigid body error ratio K ˜ K

rr ε = ------------

(Eq. 6-24)

rr

and the strain energy associated with giving the r -set degrees of freedom unit displacements or rotations where the notation K rr means the Euclidean norm of the matrix. These quantities are automatically printed after they are calculated. The rigid body error ratio and the strain energy should be zero, neglecting round-off error, if a compatible set of statically determinate supports have been chosen by the user. These quantities may be nonzero for any of the following reasons. 1.Round off error accumulation. 2.The { u r } set is over-determined leading to redundant supports (high strain energy). 3.The { u r } set is under-specified leading to a singular reduced stiffness matrix (high rigid body error ratio). 4.The multipoint constraints are incompatible (high strain energy and high rigid body error ratio). 5.There are too many single point constraints (high strain energy and high rigid body error ratio).

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Global Analysis Procedures Flexibility to Stiffness Transformation

6. [ K rr ] is null (unit value for rigid body error but low strain energy). This is an acceptable condition and may occur when generalized dynamic reduction is used. NASTRAN can not determine the modeling error. The user must review the prescribed constraints to determine possible changes in the finite element model. The structure could be restrained against rigid body motion by single point constraints, but there are advantages to using the SUPORT option. These are 1.The [ G rl ] -matrix is used to calculate the rigid body mass matrix for dynamic analysis and static analysis with inertia relief. 2.The NASTRAN program automatically calculates and prints diagnostic data concerning sufficiency of constraints.

6.13.4

SUPORT Bulk Data

The SUPORT data statement is used to specify the degrees of freedom { ur }that will remove rigid body motion.SUPORT is shown by Bulk Data Image 6-3

1 SUPORT

2 GID1

3

4

C1

GID2

5 C2

6 GID3

7 C3

8 GID4

9

10

C4

where GIDi

Grid or scalar point identification (integer, > 0)

Ci

Component number integer 0 < C < 6 or blank if G is a scalar point. Bulk Data Image 6-3 Specification of Statically Determinate Supports.

6.13.5

Specifying Degrees of Freedom

The user specifies the degrees of freedom that will restrain rigid body motion by the external degrees of freedom numbers, which consist of the pair (G, C) where G is the ID of a grid or scalar point and C is a packed set of one or more degree od freedom codes.

6.14

Flexibility to Stiffness Transformation The stiffness equation for an element was found to be of the form

[ K ee ] { ue } = { P e }

(Eq. 6-25)

There is a complementary approach based on flexibility whereby we find a set of relations in the form

[ F ] { P } = { u }

(Eq. 6-26)

where [ F ] is called the flexibility matrix. The stiffness coefficients k ij are the forces at degrees of freedom, i, due to unit displacement of degree of freedom, j, holding all other displacements equal to zero. It is tempting to use an analogous approach to define the flexibility coefficients, however a force cannot be specified for an unconstrained element. Since there are no support forces the equilibrium conditions for the element would not be satisfied so that flexibility coefficients cannot be defined for an unsupported element.

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As an alternative, consider an element supported in such a manner that the support forces can be expressed in terms of the forces at the unsupported degrees of freedom. Structures that are supported in this manner are called statically determinate because the support forces can be evaluated solely by static equilibrium considerations. The flexibility equations for a statically determinate element are

[ F 11 ] [ F 12 ]

{ P 1 }

[ F 12 ] [ F 22 ]

{ P 2 }

=

{ u1 }

(Eq. 6-27)

{ u2 }

The flexibility coefficients are then seen to be the displacements of degree of freedom i due to a unit force at and in the direction of degree of freedom j with all other forces equal to zero. It is easy to show that the flexibility matrices for the simply supported and cantilever beam which are shown by Figure 6-3 and Figure 6-4 are given by (Eq. 6-28) and (Eq. 6-29), respectively. y, u 1 u3

1

P 4, u 4 x

P 2, u 2

' ( $ u2 $ l 2 – 1 % & = --------$ u4 $ 6 EI –1 2 " #

' $ P 2 % $ P 4 "

( $ & $ #

(Eq. 6-28)

Figure 6-3 Flexibility equation for simple supported beam

y, u 1 u2

1

u 3, P 3

2

u 4, P 4

' ( ' ( $ u3 $ l 2 l 2 3 l $ P 3 $ % & = --------% & u $ 4 $ 6 EI 3 l 6 $ P 4 $ " # " #

x

(Eq. 6-29)

Figure 6-4 Flexibility equation for cantilever beam

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The flexibility coefficients are much easier to obtain by a laboratory test than the stiffness coefficients. All that is required for the flexibility approach is mounting of the structure in a statically determinate manner. Loads are then applied at each of the selected node points, and the displacements at all node point degrees of freedom are measured using appropriate instrumentation. These measurements are then scaled by the magnitude of the applied load to give the flexibility coefficients While the analysis problem can be formulated using a flexibility approach, the stiffness approach has found almost universal acceptance and is the only approach implemented in NASTRAN. A method is therefore required to transform element flexibility coefficients into an equivalent stiffness matrix. We proceed by first making the observation that 1.The stiffness matrix is singular because the node point displacements include rigid body motion. 2.The flexibility matrix is formed for a structural element that is statically determinate. These two statements are really equivalent and the transformation of the flexibility coefficients to a set of stiffness coefficients for the elements must include the rigid body degrees of freedom. Let us consider the set of all grid point displacements for the element as the union of two subsets; the set of degrees of freedom { ud }and {ui}where {ud }

are a set of degree of freedoms that would eliminate rigid body motion if constrained.

and

{ ui }

are unconstrained degrees of freedom that may displace relative to the d -set

The flexibility equations relate the forces and displacements of the i-set so that

[ F ii ] { P i } = { ui }

(Eq. 6-30)

The stiffness equations relate the forces and displacements for both the d- and i-sets. The stiffness equations can thus be represented in partitioned form as

[ K ii ] [ K id ]

{ ui }

[ K di ] [ K dd ]

{ u d }

=

{ P i } { P d }

(Eq. 6-31)

Setting the set of displacements { ud } equal to zero then gives

[ K ii ] { ui } = { P i }

(Eq. 6-32)

A comparison of (Eq. 6-32) with (Eq. 6-30) shows that the [ K ii ] partition of the stiffness matrix is given by – 1

[ K ii ] = [ F ii ]

198 Nastran Primer

(Eq. 6-33)


In order to determine the other partitions of the stiffness matrix we must include information about the relation between the { ui } and { ud } displacement sets, since all the other partitions involve the { ud } set. In the absence of forces { P i } , the element may displace as a rigid body so that a relation between the { ui } and the { ud } degrees of freedom must exist in the form r

{ ui } = [ S id ] { u d }

(Eq. 6-34)

If the element is subjected to a set of forces { P i } and the { ud } set is not equal to zero, then the total displacement of the { ui } set is the sum of the displacements given by f

r

{ u i } = { ui } + { ui }

(Eq. 6-35)

f

r

where { ui } are the displacements due to structural flexibility and { ui } represent rigid body motion relative to the d -set. The substitution of equations (Eq. 6-30) and (Eq. 6-34) into(Eq. 6-35) gives

{ ui } = [ F ii ] { P i } + [ S id ] { u d }

(Eq. 6-36)

which can be put into the form – 1

– 1

[ F ii ] { ui } – [ F ii ] [ S id ] { u d } = { P i }

(Eq. 6-37)

Now, comparing the first of (Eq. 6-31) with (Eq. 6-37) we see that – 1

[ K id ] = – [ F ii ] [ S id ]

(Eq. 6-38)

and, since the stiffness matrix and flexibility matrices are symmetric, we have – 1

T

[ K di ] = – [ S id ] [ F ii ]

(Eq. 6-39)

The final part of the flexibility to stiffness transformation is the determination of the partition, [ K dd ] , which relates the { u d } displacements to the forces, { P d } . This can be done by noting that the flexibility relations for an element are obtained by specifying that { u d } are constrained so that the element is statically determinate. This means that the forces, { P d } , can be determined from{ P i } by a relation of the form

{ P d } = [ R di ] { P i }

(Eq. 6-40)

The substitution of(Eq. 6-37) into (Eq. 6-40) then gives – 1

– 1

{ P d } = [ R di ] [ F ii ] { u i } – [ Rdi ] [ F ii ] [ S id ] { ud }

(Eq. 6-41)

The [ Rdi ] and [ S id ] matrices are related. The relation is implicit if we note that [ K di ] given by (7-81) as a result of symmetry and is also obtained from (Eq. 6-41) as – 1

[ K di ] = [ Rdi ] [ F ii ]

(Eq. 6-42)

A comparison of these two expressions shows that

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[ R di ] = – [ S id ]

T

(Eq. 6-43)

where we have used (Eq. 6-39) to relate [ R di ] and [ S id ] . A more direct way of showing that the relation holds is to note that the virtual work of the external forces must vanish if we allow only rigid body motion. T

T

{ P i } { δ u i } + { P d } { δ u d } = 0

(Eq. 6-44)

Then we have from (Eq. 6-40)) T

T

{ P d } = { P i } [ R di ]

T

(Eq. 6-45)

and, from (Eq. 6-34),

{ δ ui } = [ S id ] { δ u d }

(Eq. 6-46)

The substitution of (Eq. 6-45)and (Eq. 6-46) into (Eq. 6-44)then gives T

[ R di ] = – [ S id ] Finally, the coefficients of { u d } are the [ K dd ] partition and are given by – 1

T

[ K dd ] = [ S id ] [ F ii ] [ S id ]

(Eq. 6-47)

The element stiffness matrix can thus be formed from the element flexibility, [ F ] , and either [ S ] or [ R ] as follows

[ K ] =

[ K ii ] [ K id ] T

[ K id ] [ K dd ]

– 1

– 1

[ F ]

=

– [ F ] [ S ] – 1

T

– [ S ] [ F ]

(Eq. 6-48)

T

[ S ] [ F ] [ S ]

The stiffness equations for an element can be generated using the flexibility and rigid body transformation matrices specified using GENEL Bulk Data as described in the next section.

6.14.1

GENEL Bulk Data

The GENEL contains four logical sections that 1.Specify flexible degrees of freedom that comprise the i-set. These degrees of freedom must be defined using GRID or SPOINT Bulk Data. 2.Specify the restrained degrees of freedom. 3.List the coefficients of the flexibility matrix. 4.List the coefficients of the rigid body transformation matrix. *Define degrees of freedom in the i-set 1

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2

GENEL

EID

+1

GI4

3

4 GI1

CI4

etc.

5 CI1

6 GI2

7 CI2

8 GI3

9 CI3

10 +1 +2


*Define degrees of freedom in the d -set +2

UD

+3

GD4

GD1 CD4

CD1

GD2

CD2

GD3

CD3

etc.

+3 +4

*Define elements of flexibility matrix by columns +4

Z

Z11

Z21

Z31

etc.

Z22

Z32

Z42

+5

+5

etc.

Z33

Z43

Z53

etc.

Z44

Z54

etc.

+6

+6

etc.

+7

*Define Rigid Body Transformation Matrix by Rows +7

S

S11

S12

S13

+8

etc.

S33

S32

etc.

S21

etc.

S22

S23

+B

where Field

Description

EID

Unique element number (integer, > 0).

GIk ,CIk

Pairs defining the degrees of freedom that comprise the i-set of displacements for the element (integer).

UD

Characters in field two, which defines the beginning of the list of d -set degrees of freedom.

GDk , CDk

Pairs defining the degrees of freedom that comprise the d -set.

Z

Character in field two that defines the beginning of flexibility coefficients.

Zmn

Elements of the flexibility matrix by columns starting with the diagonal in the order of the list for the i-set.

S

Character in field two, which defines the beginning of rigid body coefficients.

Smn

Elements of the rigid body transformation matrix by rows in the order of the list for the d -set.

Bulk Data Image 6-4 Flexibility to Stiffness Transformation - GENEL

6.14.2

Example:Beam Stiffness Using GENEL

The stiffness properties for the beam element can be defined by using either (Eq. 6-28) or(Eq. 6-29) together with the appropriate rigid body transformation matrix. We therefore consider the problem of determining the response of a beam supported as a cantilever using the flexibility coefficients for the simple supported beam as given by (Eq. 6-28). Let the beam properties be given as EI = 1 L = 6

Then the flexibility coefficients (Eq. 6-26) become

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6

Global Analysis Procedures References

[ F ] =

2 – 1 –1 2

The rigid body transformation matrix is obtained by nothing that

( u 3 – u1 ) u 2 = u 3 = -------------------- L so that [ S id ] is given by 1 [ S id ] = --- –1 1 L –1 1

We will define the degrees of freedom connected by the beam by SPOINT rather than by Grid points using the displacement numbers defined on Figure 6-4. The bulk data required to define the beam element and the constraints and apply a unit force at the tip are:

1

2

3

5

SPOINT

1

GENEL

2

+G1

UD

1

+G2

Z

2.

+G3

S

-.1667

SPC

100

1

0.

SPC

100

2

0.

SLOAD

100

3

6.14.3

THRU

4

6

7

8

9

10

4 4

+G1 3

+G2

-1.

2.

+G3

1.

-.1667

1.

1.

Direct Specification of Element Stiffness

The GENEL can also be used in an alternative form for specifying the stiffness coefficients associated with the partitioned equations

' $ { F i } % $ { F d } "

( [ K ii ] – [ K ii ] [ S id ] $ & = T T $ – [ S id ] [ K ii ] [ S id ] [ K ii ] [ S id ] #

' ( $ { ui } $ % & $ { u d } $ " #

(6-49)

The only change to the GENEL data statement is the replacement of the character ‘ Z’ by ‘ K’ on continuation statement ‘+4’ shown in Bulk Data Image 6-4. The coefficients are then taken to be the elements of the stiffness partition [ K ii ] rather than the flexibility matrix. The GENEL element thus provides the user with a means of defining elements of the global stiffness matrix directly without using a finite element model. The coefficients can be defined either in terms of stiffness or flexibility.

6.15

References [1] R. Guyan, “Reduction of stiffness and mass matrices”, AIAA J., 3(2), p 380, 1965

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Global Analysis Procedures Problems

6.16

Problems 7.1

The Lagrange multiplier approach to the incorporation of linear constraint equation involves the definition of a set of workless constraint forces { λ m } . Using the fact that the constraint forces produce no virtual work show that the augmented stiffness equations can be written as

T ' { P } ( [ K gg ] [ G mg ] ' { u } ( % & = % g & " O # [ G mg ] [ 0 ] " { λ m } #

7.2

Given the augmented set of equations from problem 7.1 show that { λ m } can be found as

{ λ m } = [ G ] [ K ] – 1 [ G ] T mg gg mg 7.3

– 1

/ [ G ] [ K ] { P } 0 mg gg g , +

Suppose that a rigid link is to connect node points 1 and 2 as shown below where there is no restraint to rotation θ y2,

u y 2

u x 2

2 y

θ z 2

u y 1

1

u x 1

Rigid Element

θ z 1

x

z

Assuming that the nodes are located the basic coordinate system at X 1 = (1.,1.,0.) and X 2 = (3.,5.,0.), write the set of constraint relations which define the motion of point 2 in terms of the displacements at point 1. 7.4

Given the system of axial members shown below show that the augmented set of equations for the constraint condition u2 - u3 = 0 are as follows

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6

Global Analysis Procedures Problems

2 k o – k o 1 ' u2 (

' P $ $ $ 2 –k o 2 k o – 1 % u3 & = % P 3 $ $ $ " 0 1 –1 0 " λ # where

AE k o = ------ L

1

2

1

2

3

4

P 3

P 2

L

L

3

( $ & $ #

L

AE = constant

7.5

Show that the Lagrange multiplier for problem 7.4 is λ = 1/2(P 2-P 3 ). What is the physical interpretation of λ?

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7.6

Using the stiffness to flexibility relation (Eq. 7-90) show that the correct element stiffness matrix for the beam is obtained when starting from the flexibility for either the simply supported beam (Eq. 7-70) or the cantilever beam (Eq. 7-71).

7.7

The stiffness matrix for the beam element is singular because the node point displacements include rigid body motion. Show that the reduced stiffness matrix associated with including the transverse displacements at each end of the beam in the r -set is null.

7.8

Verify that (Eq. 7-82) relates the rigid body displacements of the f -set to rigid body motion of the r -set displacement.

7 7.1

Accuracy and Performance of Finite Elements

Element Performance MacNeal[1] describes a simple test called the patch test, first proposed by Irons [2], for studying the performance of finite elements. The idea behind the test is that under simple loading conditions the finite element should be capable of representing exact behavior. For example, if a two dimensional region is subject to a uniform boundary traction, an arbitrary element should represent the correct value of strain and therefore stress. MacNeal [3] shows that the patch test imposes the following set of sufficient conditions on the element formulation: 1.Strains and stress computed from nodal point displacements are correct to the desired order 2.The integral relating the generalized forces to stresses

{ F i } =

* [ Bi ] { σ }d Ω

(Eq. 7-1)

Ω

is exact for the desired order of { σ}. 3.Equilibrium at internal nodes is satisfied because either a) the elements are conforming, or b) the nonconforming part of the strain-displacement matrix of each element produces zero grid point forces, or c) the shape functions associated with interior nodes satisfy the interelement traction boundary conditions Failure to satisfy these three conditions is due to interpolation failure, integration failure or equilibrium failure.

7.2

Interpolation Failure Interpolation failure is due to the incorrect interpolation of the node point generalized coordinates. This type of failure is one of the consequences of parametric mapping in isoparametric elements. Interpolation failure also causes a phenomenon called locking that is caused by excessive stiffness for a particular deformation mode of the element. In hindsight locking was responsible for the poor performance of early lower-order elements and gave impetus to developing higher-order elements.

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7

Accuracy and Performance of Finite Elements Effect of Element Shape

The locking phenomenon is still not well understood by finite element developers and certainly is somewhat of a mystery to the users, especially those who expect analysis fidelity to be equivalent to geometric fidelity. MacNeal suggests that interpolation failure must be present for locking to exist; but, this contention is still the subject of future research on the subject. Locking is caused by a number of element parameters including loading, element geometry, part geometry and material properties. Incompressible materials are an obvious problem since the bulk modulus becomes infinite for an isotropic material when Poisson’s ratio becomes 0.5. This can be overcome by using a reformulated element or by taking Poisson’s ratio close to, but not equal to 0.5.

7.3

Effect of Element Shape Element testing is a vital part of due diligence that must be performed by both the developer and the user. After-the-fact testing of the original NASTRAN membrane element, the QUAD2, showed that it gave good results only for rectangular geometry and then only for a restricted range of element aspect ratio. For non rectangular regions, and for aspect ratios greater than 2:1 for rectangular regions, the element produced spectacularly poor results. Due to a concern about element performance in general and the elements included in MSC.Nastran in particular, MacNeal and Harder [4] proposed a set of standard tests for element performance that included all of the parameters identified in the proceeding section. The loading should induce all behavior modes in the element including inplane extension, bending in two plane and twist of two dimensional elements, for example. Elements have standard shapes. A quadrilateral element formulation is based on a bisquare in parameter space and preforms best when the Jacobian of transformation is a unit matrix, or at least a diagonal matrix. The test geometry must therefore include all types of geometric anomalies such as those shown by Figure 7-1for a quadrilateral element.

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Aspect Ratio

a ⁄ b

b a

δ

Skew

Taper b

d

(2 Directions)

d --b

h --a

h

Warp

a Figure 7-1 Types of Geometric Distortion of a Quadrilateral

In addition to element shape the part geometry also affects the accuracy of the finite element model and of course dictates the type of element to be employed. Curved shells can only be approximated by flat surfaces; and, shells with one and two radii of curvature are chosen for the tests. Even the boundary geometry of a flat assemblage can have on effect which could be tested using a skewed plate, a swept wing and a rhombic plate. The test problems described below are called the MacNeal-Harder tests which are summarized by Table 7-1 Table 7-1 Summary of Tests Elements appropriate for the Test Test problem

Plate Membrane

Beam

Patch test

Plate Bending Shell

Solid

X

X

X

Straight slender cantilever beam

X

X

X

X

Curved Beam

X

X

X

X

Twisted Beam

X

X

X

X

X

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Table 7-1 Summary of Tests Elements appropriate for the Test Test problem

Plate Membrane

Beam

Plate Bending Shell

Rectangular Plate

X

X

Scordelis-Lo Cylindrical Section

X

X

Spherical Shell

X

Solid X X

X

X

X

Thick-walled cylinder

7.3.1

X

Two Dimensional Patch Test

The geometry for the two dimensional patch test is shown by Figure 7-2 where the node points are defined by Table 7-2. y b

4

3

a

2

1

x Figure 7-2 Two Dimensional Patch Test Geometry

The dimensions of the rectangular region and material properties are a = 0.12, b = 0.24, t = 0.001; E = 1.0 x106; ν = 0.25. Table 7-2 Vertices for 2D Patch

7.3.1.1

Node

x

y

1

0.04

0.02

2

0.18

0.03

3

0.16

0.08

4

0.08

0.08

Membrane Boundary Conditions

The loads are applied using the following displacement boundary conditions – 3

u = 10 ( x + y ⁄ 2 ) – 3

v = 10 ( y + x ⁄ 2 )

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Accuracy and Performance of Finite Elements Effect of Element Shape 7.3.1.2

Membrane Theoretical Solution

The membrane strain and stress components are – 3

ε xx = ε yy = γ xy = 10 σ xx = σ yy = 1333 τ xy = 400 7.3.1.3

Bending Boundary Conditions

The loads are applied by the following boundary displacements – 3

10 2 2 w = ----------- ( x + xy + y ) 2 – 3 w θ x = ∂ = 10 ( x + y ⁄ 2 ) ∂ y

∂w = – 10 y + x 2 θ y = – ( ⁄ ) ∂ x – 3

7.3.1.4

Bending Theoretical Solution

The bending moments and surface stresses are – 7

m x = m y = 1.111 x 10 – 7

m xy = 10

σ xx = σ yy = − + 0.667 τ = − + 0.200 xy

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7.3.2

Three Dimensional Patch Test

The geometry for the two dimensional patch test is shown by Figure 7-1 where the node points are defined by Table 7-1.

y

4 8

3

7 1

2 5

6 x

z Figure 7-1 Three Dimensional Patch Test Geometry

The region is a unit cube; E = 1.0 x106; ν = 0.25. Table 7-1 Vertex Locations for 3D Patch Node

x

y

z

1

00.249

0.342

0.192

2

0.826

0.288

0.288

3

0.850

0.649

0.263

4

0.273

0.750

0.230

5

0.320

0.186

0.643

6

0.677

0.305

0.683

7

0.788

0.693

0.644

8

0.165

0.745

0.702

The region is subject to the following displacement boundary conditions – 3

u = 10 ( 2 x + y + z ) ⁄ 2 – 3

v = 10 ( x + 2 y + z ) ⁄ 2 – 3

w = 10 ( x + y + 2 z ) ⁄ 2

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The strain and stress components for the theoretical solution are – 3

ε xx = ε yy = ε zz = 10

– 3

γ xy = γ yz = γ zx = 10

σ xx = σ yy = σ zz = 2000 τ xy = τ yz = τ zx = 400 7.3.3

Patch Test Results

The results of the patch test for MSC.Nastran elements is presented by Table 7-2 Table 7-2 Patch Test Results Element

Error in Stress, percent Constant stress Loading

Constant Curvature Loading

QUAD2

0.0

30.7

QUAD4

0.0

0.0

QUAD8

18.0

51.6

HEXA(8)

0.0

N/A

HEXA(20)

0.0

N/A

where the HEXA(8) and the HEXA(20) are results for the HEXA element with 8 and 20 nodes, respectively. The QUAD2 element was one of the original elements in MSC.Nastran. Due to its poor performance it was replaced by the QUAD4 element.

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7.3.4

MacNeal Beam Tests

Three types of beams were used by MacNeal: the straight beam, the curved beams and the twisted beam shown by Figure 7-2 through Figure 7-4 respectively with exact results for four loading conditions shown by Table 7-3.

(a)

45 °

45 °

(b)

45 °

(c) Figure 7-2 MacNeal’s Slender Straight Beam.

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90 °

FIXED Figure 7-3 MacNeal’s Curved Beam.

FIXED END Figure 7-4 MacNeal’s Twisted Beam.

Table 7-3 Theoretical Solutions for Beam Problems Tip load direction

Displacement in load direction Straight beam

Curved beam

Twisted beam

Extension

3.0x10 -5

In-plane shear

0.1081

0.08734

0.005424

Out-of-plane shear

0.4321

0.5022

0.001754

Twist

0.03206

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7.3.4.1

Results for Slender Straight Beam

The results for the slender straight beam test for MSC.Nastran two and three dimensional elements by Table 7-1. Table 7-1 Results for Straight Beam Tip Load Direction

Normalized tip displacement in load direction

QUAD2

QUAD4

QUAD8

HEXA8

HEXA20

(a) Rectangular elements

Extension

0.992

0.995

0.999

0.988

0.994

In-plane shear

0.032

0.904

0.987

0.981

0.970

Out-of-plane shear

0.971

0.986

0.991

0.981

0.961

Twist

0.566

0.041

0.950

0.910

0.904

Extension

0.992

0.996

0.999

0.989

0.994

In-plane shear

0.016

0.071

0.946

0.069

0.886

Out-of-plane shear

0.963

0.968

0.998

0.051

0.920

Twist

0.616

0.951

0.943

0.906

0.904

(b) Trapezoidal elements

(c) Parallelogram elements

Extension

0.992

0.996

0.999

0.989

0.994

In-plane shear

0.014

0.080

0.995

0.080

0.967

Out-of-plane shear

0.963

0.977

0.985

0.055

0.941

Twist

0.615

0.945

0.965

0.910

0.904

7.3.4.2

Results for Curved Beam

The results for the curved beam are shown by Table 7-1. Table 7-1 Results for Curved Beam Tip Loading Direction

Normalized tip displacement in load direction QUAD2

QUAD4

QUAD8

HEXA8

HEXA20

In-plane (vertical)

0.025

0.833

1.007

0.880

0.875

Out-of-plane shear

0.594

0.951

0.97

0.849

0.946

7.3.4.3

Results for Twisted Beam

The results for the twisted beam are shown by Table 7-1. Table 7-1 Results for Twisted Beam Tip Loading Direction

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Normalized tip displacement in load direction QUAD2

QUAD4

QUAD8

HEXA8

HEXA20

In-plane (vertical)

100.4

0.993

0.998

0.983

0.991

Out-of-plane shear

228.9

0.985

0.998

0.977

0.995


7.3.5

Rectangular Plate Tests

The transversely loaded rectangular plate is shown by Figure 7-1

sym

a

sym

b Figure 7-1 Rectangular Plate

The plate dimensions and material properties are; a = 2, b = 2 or 10, E = 1.7472e7, ν = 0.3, t = 0.0001 for plates and 0.001 for solids. The plate is subject to two symmetric loading conditions and two sets of symmetric boundary conditions. The loads are a uniform normal pressure or a central concentrated force. The boundary conditions are: all edges simply supported or all edges clamped. Only one quarter of the plate is modeled using symmetry conditions, and the mesh density can be varied to study convergence. The theoretical solutions for all cases of loads, boundary conditions and aspect ratios are presented by Table 7-2 Table 7-2 Theoretical Center Displacement Displacement at Center of Plate Boundary Supports

Aspect Ratio b/a Uniform Pressure

Concentrated Force

Simple Support

1.0

4.062

11.60

Simple Support

5.0

12.97

16.96

Clamped

1.0

1.26

5.60

Clamped

5.0

2.56

7.23

The results for the simple support cases with uniform load and clamped cases with concentrated load are shown by Table 7-3 throughTable 7-6. Table 7-3 Simple Supports, Uniform Load, a/b = 1 Number of Elements per Edge Element Type 2

4

6

8

QUAD2

0.971

0.995

0.998

0.999

QUAD4

0.981

1.004

1.003

1.002

QUAD8

0.927

0.996

0.999

1.000

HEXA(8)

0.989

0.998

0.999

1.000

HEXA(20)

0.023

0.738

0.967

0.991

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Table 7-4 Simple Supports, Uniform Load, a/b = 5 Number of Elements per Edge Element Type 2

4

6

8

QUAD2

0.773

0.968

0.993

0.998

QUAD4

1.052

0.991

0.997

0.998

QUAD8

1.223

1.003

1.000

1.000

HEXA(8)

0.955

0.978

0.990

0.995

HEXA(20)

0.028

0.693

1.066

1.026

Table 7-5 Clamped Edges, Concentrated Load, a/b = 1 Number of Elements per Edge Element Type 2

4

6

8

QUAD2

0.979

1.008

1.006

1.005

QUAD4

0.934

1.010

1.012

1.010

QUAD8

1.076

0.969

0.992

0.997

HEXA(8)

0.885

0.972

0.988

0.994

HEXA(20)

0.002

0.072

0.552

0.821

Table 7-6 Clamped Edges, Concentrated Load, a/b = 5 Number of Elements per Edge Element Type 2

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4

6

8

QUAD2

0.333

0.512

0.638

0.723

QUAD4

0.519

0.863

0.940

0.972

QUAD8

0.542

0.754

0.932

0.975

HEXA(8)

0.321

0.850

0.927

0.957

HEXA(20)

0.001

0.041

0.220

0.374


7.3.6

Cylindrical Shell Tests

The Scordelis-Lo roof [5] is used to test the ability of elements to represent a shell with one radius of curvature. The shell is shown by Figure 7-2

z y x

S Y M

M Y S

E E R F

40 °

L

R

Figure 7-2 Scordelis-Lo Roof

The dimensions and material properties of the roof are; R = 25, L = 50, t = 0.25, E = 4.32e8, ν= 0.0. The shell is loaded by a uniform load of 90 units of force per unit area in the negative Z direction where the units are consistent among all physical quantities. The boundary conditions on the curved edges are u x = u z = 0. The shell is modeled using an N by N mesh of the shaded area. Since the loads, geometry, material are symmetric, only one quarter of the structure is modeled using appropriate symmetry conditions. The theoretical solution for the vertical displacement at the center of the straight edge is given by [5] to be 0.3086. MacNeal[4] suggest using a slightly different value, 0.3024. The results of as a function of mesh refinement is presented by Table 7-7. Table 7-7 Results for Scordelis-Lo Roof Number of Elements per Edge Element Type 2

4

6

8

10

QUAD2

0.784

0.665

0.781

0.854

0.987

QUAD4

1.376

1.050

1.018

1.008

1.004

QUAD8

1.021

0.984

1.002

0.997

0.996

HEXA(8)

1.320

1.028

1.012

1.005

HEXA(20)

0.092

0.258

0.589

0.812

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7.3.7

Pinched Sphere

The spherical shell us used to test the ability of elements to represent shell structures having two radii of curvature. The shell is shown by Figure 7-3. z

18 °

F R E E

S Y M

M Y S

y F = 2.0 (on quadrant)

E F R E

x

F = 2.0

(on quadrant)

Figure 7-3 Spherical Shell with Opening

The dimensions and material properties of the shell are; R = 10.0, t = 0.04, E = 6.825e7, ν = 0.3. The shell is open on the top as shown. The boundaries are free with equal and opposite concentrated loads applied to quadrants as shown. The geometry and loads are symmetric so only one quarter of the shell is modeled by an N by N mesh using appropriate symmetry conditions . There is no closed form solution for the test case. MacNeal [4] recommends a value of 0.0940 for the radial displacement at the point of load application. The normalize results are presented as a function of mesh refinement by Table 7-8. Table 7-8 Spherical Shell with a Central hole Number of Elements per Edge Element Type 2

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4

6

8

10

12

QUAD2

0.928

0.990

0.990

0.986

0.984

0.982

QUAD4

0.972

1.024

1.013

1.005

1.001

0.998

Accuracy and Performance of Finite Elements Effect of Element Shape Table 7-8 Spherical Shell with a Central hole Number of Elements per Edge Element Type 2

QUAD8

4

0.025

6

0.121

8

0.494

0.823

10

12

0.955

0.992

HEXA(8)

0.039

0.730

0.955

HEXA(20)

0.001

0.021

0.097

7.3.8

Incompressible Material

A thick cylindrical shell shown by Figure 7-4 is used to test the effect of Poisson’s ratio.

Radius 0 0 .

0 0 0 5 . . 3 3

0 2 . 5

0 2 . 4

S Y M

9

5 7 . 6

E E R F

p

10 °

SYM Figure 7-4 Thick Walled Cylinder

The Cylinder has an inner radius, Ri = 3.0, an outer radius, Ro = 9.0, the elastic modulus is E = 1000 and Poisson’s ratio is varied. The cylinder is loaded with a unit internal pressure. Plane strain is assumed and the 5 by 1 mesh shown is used for ν = 0.49, 0.499 and 0.4999. The theoretical result for the radial displacement is given by 2

2

( 1 + ν ) pR i R o

u r = --------------------------- ------ + ( 1 – 2 ν ) r 2 2 E ( Ro – Ri ) r

(Eq. 7-1)

The radial displacement at the inner surface is presented by Table 7-9. Table 7-9 Radial Displacement at Inner Surface Poisson’s Ratio

Radial Displacement

0.49

5.0399e-3

0.499

5.0602e-3

0.4999

5.0623e-3

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Accuracy and Performance of Finite Elements Summary of Test Results

The results for the MSC.Nastran elements are shown by Table 7-10. Table 7-10 Test Results for Radial Displacement Poisson’s Ratio Element Type 0.49

7.4

0.499

0.4999

QUAD2

0.643

0.156

0.018

QUAD4

0.846

0.359

0.053

QUAD8

1.0000

0.997

0.967

HEXA(8)

0.986

0.986

0.986

HEXA(20)

0.999

0.986

0.879

Summary of Test Results The tests used in this study are by no means complete and should be augmented by others presented in the literature several of which are described by Table 7-11 Table 7-11 Additional Tests for Finite Elements Test Name

Description

Pinched Ring

The pinched ring test presented by Sze[7] can be used to test an assemblage elements whose stiffness is coupled by the structural geometry

Swept Plate

The swept plate test presented by Cook [16] can be used to test an assemblage of elements having variable trapezoidal geometry

Short Beam

The short beam test was proposed by Cheung [10] and is used to test an assemblage of elements having both skewed and trapezoidal geometry.

Short Beam for Three Dimensional Elements

The short beam test generalized to test three dimensional stress elements by Cheung [11]]

Circular Plate

The circular plate has been used by several authors, including[11] to test the effect of meshing strategies in best representing circular plates

Skew Plate

The skew plate test presented by [17]can be used to test an assemblage of skewed elements and to compare the results to theory [15]

NAFEMS

The NAFEMS organization was established in Great Briton to develop standard tests for elements [13].

These tests for static loading can be used to determine the relative performance of elements. These tests must be augmented by tests for other types of analyses including normal modes, transient dynamics, buckling and nonlinear analyses. There are many factors that affect the ability of the model to simulate the behavior of a virtual part or assemblage. As we rely more heavily on simulation in design as an alternative to prototype test it becomes more and more important to use a part of the analysis budget to design and verify the virtual prototype. The results of the tests presented in this Chapter are encouraging. They are restricted to only those elements included in NASTRAN since those are the elements we have at our disposal. The QUAD2 was included to show how poorly early elements performed and to provide a baseline for showing the improvements in elements that resulted from this type of testing.

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Accuracy and Performance of Finite Elements Summary of Test Results

In [4] the elements are graded based on the error in achieving the exact or target result. The test results are summarized for two and three dimensional elements by Table 7-12 and Table 7-13, respectively using the following grading scale: Grade

Rule

A

2% ≥ error

B

10% ≥ error < 2%

C

20% ≥ error < 10%

D

50% ≥ error < 20%

F

error > 50%

Table 7-12 Summary of Results for Shell Elements Element Loading Test In-plane

Patch test

Out of plane

X

Patch test

X

Element Shape

Element Type QUAD2

QUAD4

QUAD8

Irregular

A

A

C

Irregular

D

A

D

Straight beam, extension

X

All

A

A

A

Straight beam, bending

X

Regular

F

B

A


X

Irregular

F

F

B


X

Regular

B

A

A


X

Irregular

B

B

A

All

D

B

B

Regular

F

C

A

X

Regular

D

B

B

X

Regular

F

A

A

X

Regular

C

B

B

Straight beam, twist Curved beam

X

Curved beam Twisted beam

X

Rectangular plate, N = 4 Scordelis-Lo

X

X

Regular

D

B

A

Spherical Shell

X

X

Regular

A

A

C

Incompressible Material

X

F

F

B

9

2

1

Regular

( ν = 0.4999) Number of Failures (D’s and F’s)

Table 7-13 Summary of Results for Solid Elements Test

Element Shape

Element Type Hexa(8)

Hexa(20)

Patch test

Irregular

A

A

Straight beam, extension

All

A

A


Regular

A

B

Straight beam, bending, inplane

Irregular

F

B

Straight beam, bending, out of plane

Irregular

F

B

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Accuracy and Performance of Finite Elements Problems

Table 7-13 Summary of Results for Solid Elements Test

Element Shape

Element Type Hexa(8)

Hexa(20)

Straight beam, twist

All

B

B

Curved beam, in plane loading

Regular

C

C

Curved beam, out of plane loading Regular

C

B

Twisted beam

Regular

B

B

Rectangular plate, N = 4

Regular

B

F

Scordelis-Lo, N = 4

Regular

B

F

Spherical Shell, N = 8

Regular

D

F

A

C

3

3

Incompressible Material ( ν = 0.4999) Number of Failures (D’s and F’s)

Regular

The QUAD4 element, performs well for all tests except the curved beam with out of plane loading. The failure of this test is due to locking. The QUAD8 performs well for all tests except for the patch test. The HEXA(8) passes the patch test but performs badly in bending. However, it handles incompressible materials well. The HEXA(20) does well for beam bending but performs badly for plate and shell problems. Elements performance is continuously evaluated by the element designers, and new or revised formulation are implemented as appropriate to improve element performance. Certainly, the implementation of p-element technology into MSC/NASTAN is evidence of the commitment of program developers to provide state of the art performance. There are some exciting developments in the area of hybrid formulations, such as the Pian-Sumihara membrane element[6], and significant contributions by Sze[7],[8], To [9] and Cheung [10], [11] and [14], which will undoubtedly be evaluated and perhaps find their way into commercial programs in the future.

7.5

Problems [1] Verify the results for the two dimensional patch test using the geometric and material parameters given in Sec. 8.3.1 for the membrane and bending displacements given. Normalize the results using the exact values. [2] For the 2-D patch, use the geometry given in Sec. 8.3.1 with material properties, 6 E = 30 ×10 ps i and ν = 0.3 and determine the stress components normaized by the exact values. [3] Verify the results for the three dimensional patch test using the geometric and material parameters given in Sec. 8.3.2 for the Hexa element with 8 nodes. Normalize the results using the exact values. [4] Repeat Problem 3 using the Hexa element with 20 nodes. [5] Repeat Problem 3 using the Tetra element with 4 nodes. [6] Repeat Problem 3 using the Tetra element with 10 nodes. [7] Verify the results for the straight beam test uning the geometry shown in Figure 7-2 for an extensional load, normalize the results using the exact values.

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Accuracy and Performance of Finite Elements References

[8] Repeat Problem 7 for a unit inplane shear at the free end [9] Repeat Problem 7 for a unit out of plane shear at the free end. [10] Verify the results for the rectangular plate tests given by Table 7-3 using the Quad4 element [11] Repeat Problem 10 using the Quad8 with 8 nodes [12] Repeat Problem 10 using the Tria3 element. [13] Repeat Problem 10 using the Tria6 with 6 nodes. [14] Verify the results for Scordelis-Lo given by Table 7-7 for the Hexa with 20 nodes.

7.6

References [1] R.H. MacNeal, Finite Elements: Their Design and Performance, Marcel Decker, Inc., New York, NY, p. 179, 1994. [2] B.W. Irons, “Numerical integration applied to finite element methods”, Conf. on Use of Digital Computers in Structural Engineering , Univ. of Newcastle, 1966. [3] R.H. MacNeal, Finite Elements: Their Design and Performance, Marcel Decker, Inc., New York, NY, pp. 193-194, 1994. [4] R.H. MacNeal and R.L Harder, “A proposed set of standard problems to test finite element accuracy”, Finite Elements in Analysis and Design, 1, 3-20, 1993. [5] A.C. Scordelis and K.S. Lo, “Computer analysis of shells”, J. Amer. Concrete Inst. 61, 539-561. 1969. [6] T.H.H. Pian and K. Sumihara, “Rational approach for assumed stress finite elements,”. Int. J. Num. Meth. Engr.,20, 1685-95, 1984. [7] K.Y. Sze, “Hybrid plane quadrilateral element with corner rotations,”. Journal of Structural Engineering , 119(9), 2552-2572, 1993. [8] K.Y Sze, “Hybrid hexahedral element for solids, plates, shells and beams by selective scaling”, Inter. J. Num. Meth. in Engr ., 36, 677-693, 1993. [9] C.W.S. To, “Hybrid strain-based three node flat triangular laminated composite shell elements”, Finite elements in Analysis and Design, 28(3), 177-208, 1998. [10] Wanji Chen and Y.K. Cheung, “A new approach for the hybrid element method”,. Inter. J. Num. Meth. in Engr ., 24, 1697-1709, 1987. [11] Y.K. Cheung and Chen Wanji, “Isoparametric hybrid hexahedral elements for three dimensional stress analysis,”. Inter. J. Num. Meth. in Engr ., 26, 677-693, 1988. [12] Y.F. Dong, C.C. Wu and J.A. Teixeira de Freitas, “The hybrid model for MindlinReissner plates based on a stress optimization condition,”, Comp. and Struct ., 46(5) 877-897, 1993. [13] anon., The Finite Element Primer , National Agency for Finite Elements, Department of Trade and Industry, Glasgow, pp 85-115, 1986. [14] Y.K. Cheung and Chen Wanji, “Hybrid quadrilateral element based on Mindlin/Reissner plate theory,”. Comp. and Struct., 32(2), 327-329. [15] L.S.D. Morley, Skew Plates and Structure s, Pergamon Press, Oxford, 1963. [16] R.D. Cook, “Improved two-dimensional finite element”, J. Struct. Div, ASCE , ST9, 18511863, 1974.

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Accuracy and Performance of Finite Elements References

[17] R.L. Taylor and F. Auriccho, Linked interpolation for Reissner-Mindlin plate elements: Part II - A simple triangle”, Inter. J. Num. Meth. in Engr ., 36,3057-3066, 1993.

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8 8.1

Rigid and Constraint Elements

Rigid and Constraint Elements The rigid and constraint elements automate the generation of linear constraint equations. Linear constraint equations can always be defined using the MPC data statement as described in Chapter 6. However, when the constraint equations are associated with rigid body motion or other specific types of constraints such as weighted averages or spline fits to Grid point degrees of freedom the generation of the constraint equations can be automated. NASTRAN incorporates a number of rigid and constraint elements that generate linear constraint equations between degrees of freedom for: • Rigid Elements • Elastic Spline Fit • Weighted Average

The NASTRAN constraint elements are summarized by Table 8-1 where the element names such as RROD are the mnemonics for the associated data statements. The NASTRAN rigid elements represent kinematic constraints between degrees of freedom based on rigid body motion between connected degrees of freedom. They include the RROD, RBAR, RTRPLT, RBE1 and RBE2. The other two elements, the RSPLINE and the RBE3, are more correctly termed constraint elements because the relationship between degrees of freedom is based on assumptions other than rigid body motion. The RSPLINE assumes an cubic spline fit and the RBE3 assumes a weighted average between specified degrees of freedom. Constraint Element Name

Description

m

RROD

An extensional constraint in the direction of a line segment between two grid points.

1

RBAR

A rigid link between two grid points

1≤m≤6

RTRPLT

A rigid link between three grid points

1 ≤ m ≤ 12

RBE1

A rigid connection between an arbitrary number of grid points where both independent and dependent degrees of m ≥ 1 freedom are specified by the user.

RBE2

A rigid connection between an arbitrary number of grid points where independent degrees of freedom are at a specified reference point.

m≥1

RBE3

A weighted average constraint

1≤m≤6

RSPLINE

A spline element

m≥1

Table 8-1 Rigid and Constraint Elements

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Number of equations

8

Rigid and Constraint Elements RROD - Extensional Constraint

The specification of constraint elements requires an understanding of the form and function of the linear constraint equations defined in NASTRAN. As was noted in Chapter 7, a set of m linear constraints is incorporated into the solution set by eliminating m degrees of freedom. The MPC operation partitions the g -set into two subsets, the m-set and the n-set that are related by a set of m linear constraint equations. Using the constraint equations, the m-set is related to the n-set and is removed from the analysis. The purpose of the NASTRAN constraint elements is to provide an efficient means of generating constraint equations in the form of (Eq. 6-2). The data statements associated constraint element must therefore provide a mechanism for specifying the set membership of the connected degrees of freedom. Furthermore, since the constraint elements generate MPCs, all of the rules associated with MPCs must be followed including: 1.A degree of freedom specified in the m-set cannot be included in any of the other removed sets. This means that it cannot specified on SPC, OMIT or SUPORT data statements. Doing so will result in a fatal error message. 2.A degree of freedom can be specified in the m-set only once. A fatal error will be issued if the same degree of freedom is designated as dependent on two constraint elements. 3.The degrees of freedom in the n-set for all rigid elements except the RROD must be sufficient to represent rigid body motion.

8.2

RROD - Extensional Constraint The RROD constraint element is shown by Figure 8-1 and is defined by the RROD Bulk Data shown on Bulk Data Image 8-1. z

GB

GA

y

x Figure 8-1 RROD Constraint Element

1 RROD

2 EID

3 GA

4 GB

5

6

CMA

CMB

7

8

where GA,GB

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are grid point identification numbers (Integer >0)

9

10

Rigid and Constraint Elements RBAR - Rigid Link Connecting Two Grids

CMA, CMB

component number of one translational degree of freedom at either end A or at end B that is to be included in the m-set. One of the two fields contains the integer code for the m-set degree of freedom while the other field is blank.

Bulk Data Image 8-1 Extensional Constraint Element - RROD

The extensional constraint condition implies that the relative displacement in the direction of the line segment joining the connected geometric grid points must be zero. Thus we have

{ d } A = { d } B

(Eq. 8-1)

where { d } is the displacement in the direction of the line segment from grid point A to grid point B. Expressing { d } in terms of displacement components at A and B gives u xA α x + u yA α y + u zA α z = u xB α x + u yB α y + u zB α z

(Eq. 8-2)

where (αx, αy, αz) are direction cosines between the directed line segment and the coordinate axes. The RROD thus specifies a single equation which can be cast in the form of the MPC, Multipoint Constraints - MPC (p. 175), provided that the dependent degree of freedom, i.e. the m-set degree of freedom, is specified. This specification is made by identifying one of the three displacement degrees of freedom in (Eq. 8-2) as an m-set degree of freedom and entering the associated degree of freedom code in field 5 or field 6 of the RROD depending on whether the m-set degree of freedom is at end A or B, respectively. Finally, with reference to (Eq. 8-2), it is apparent that the direction cosine associated with the m-set degree of freedom must be non zero. This implies that the m-set degree of freedom cannot be perpendicular to the line segment joining A and B.

8.3

RBAR - Rigid Link Connecting Two Grids The RBAR element provides the capability of modeling the rigid link shown by Figure 6-1 by using the single data statement shown on Bulk Data Image 8-2 rather than a set of MPC data statements necessary to specify the associated constraint equations (Eq. 6-3). 1

2

RBAR

EID

3 GA

4 GB

5

6

7

8

CNA

CNB

CMA

CMB

9

10

where CNA, CNB

are codes for n-set degrees of freedom at ends A and B, respectively (integer > 0 or blank)

CMA, CMB

are codes for m-set degrees of freedom at ends A and B, respectively (integer > 0 or blank) Bulk Data Image 8-2 RBAR Data Statement

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Rigid and Constraint Elements RTRPLT - Rigid Triangular Constraint Element

The RBAR element defines a rigid connection for up to six degrees of freedom at the connected grid points. The constraint equations conditions for rigid body motion can then be expressed by (Eq. 6-3). The RBAR, and the other rigid elements, allow the user to specify which of the constraint equations is to be generated. The first requirement is to specify exactly six degrees of freedom for the rigid element that represent rigid body motion using the CN fields. Then, the specific constraint equation(s) to be generated are specified by the CM fields which serves the dual purpose of assigning the associated degrees of freedom to the m-set. The genius of the formulation of the rigid elements is that the CM fields allows the user to specify which of the six constraint equations are to be generated. If the RBAR is to generate only constraint equations associated with displacements then the CM field would include only displacement degree of freedom codes and no constraints would be generated for the rotational degrees of freedom.

8.3.1

RBAR Example: Rigid Link Connecting Two Points

Suppose that two grid points, 363 and 132, are connected by a member which is extremely stiff compared to other elements in the structure. The correct modeling procedure in this case is to define a set of constraint equations using a RBAR rather than using an extremely stiff elastic beam element. The set of MPC Bulk Data required to model this rigid body constraint equations is given in Defining the Constraint Equation (p. 181). The RBAR can be used as an alternative in which case the rigid link is defined by the following single RBAR statement. 1 RBAR

2 153

3

4

363

132

5

6

7

8

9

10

123456

where the six degrees of freedom at point 363 have been included in the n-set implying that the six degrees of freedom at point 132 are in the m-set in agreement with the MPC Specification.

8.4

RTRPLT - Rigid Triangular Constraint Element A constraint element which connects three geometric grid points is shown by Figure 8-2 and is described using the RTRPLT data statement shown on Bulk Data Image 8-3. z

GC

GA

x

y

GB

Figure 8-2 Triangular Constraint Element

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Rigid and Constraint Elements Rigid Constraint Element - RBE1 and RBE2

1

2

RTRPLT EID CMA

3

4

GA

GB

CMB

CMC

5 GC

6

7

8

CNA

CNB

CNC

9

10

where CNA, CNB, CNC

are codes for n-set degrees of freedom at points A, B, and C, respectively (integer > 0 or blank)

CMA, CMB, CMC

are codes for m-set degrees of freedom at A, B, and C, respectively. (Integer > 0 or blank)

Bulk Data Image 8-3 RTRPLT Data Statement

The six rigid body degrees of freedom for the rigid element are represented by the n-set degrees of freedom which must be specified at the connected grid points by the CN A, CNB, and CNC fields. By default, if the CM fields are left blank, all of the remaining degrees of freedom are includes in the m-set and twelve constraint equations will be generated. As in the case of the RBAR, the CM fields can be used to generate specific constraint equations at the grid points and include the associate degrees of freedom in the m-set.

8.5

Rigid Constraint Element - RBE1 and RBE2 The RBE1 and RBE2 constraint elements are generalizations of the RBAR and RTRPLT elements that allow connection of an arbitrary number of geometric grid points. Six degrees of freedom representing rigid body motion must be assigned to the n-set. RBE1 and RBE2 differ only in way the n-set is specified. The form of the RBE1 and RBE2 rigid elements is shown on Bulk Data Image 8-4. 1 RBE1

2

3

4

5

6

7

8

EID

GN1

CN1

GN2

CN2

GN3

CN3

GN4

CN4

GN5

CN5

GN6

CN6

GM1

CM1

GM2

CM2

GM3

CM3

GM4

CM4

-etc.-

3

4

5

6

7

GM1

GM2

GM3

“UM”

1 RBE2

2 EID

GN

CM

GM6

GM7

etc.

9

10

8

9

10

GM4

GM5

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Rigid and Constraint Elements Rigid Constraint Element - RBE1 and RBE2

where the fields of the RBE1 are Field

Description

GNi

are grid points at which n-set degrees of freedom

GNi

are grid points at which n-set degrees of freedom

UM

is a literal string which terminates the specification of the n-set

GMi

are grid points at which m-set degrees of freedom are specified

CMi

are codes for degrees of freedom which are to be included in the m-set.

and the fields of the RBE2 are Field

Description

GN

Is the reference point whose six degrees of freedom represent rigid body motion.

CM

Degree of freedom codes for constraint equations to be generated at the GMi grid points

GMi

Grid points at which m-set degrees of freedom are specified and for which constraint equations are generated.

Bulk Data Image 8-4 RBE1 and RBE2

The RBE1 and RBE2 perform the same function but have the following differences 1.The RBE1 allows the user to define the six degrees of freedom in the n-set that are capable of representing rigid body motion by specifying exactly six grid point degrees of freedom. These degrees of freedom may be at one or more Grid points. If the n-set is defined at three or less grid points then the first continuation is not required. The end of the n-set is signified by the literal string ‘UM’ on a continuation statement. The dependent degrees of freedom are then specified after the UM field by pairs (GM i, CMi) where GMi is the grid point number and CMi are degree of freedom code numbers. 2.The RBE2 takes the six degrees of freedom at the grid point defined in the GN-field to be in the n-set. The CM-field then contains the code for up to six degrees of freedom at the grid points GM i that are included in the m-set and for which constraint equations are generated

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Rigid and Constraint Elements Elastic Constraint Element - RSPLINE

8.5.1

Example: Rigid Inclusion

It may be necessary to model a rigid inclusion as a structure such as a tightly fitting bolt in a bolt hole. The grid points on the circumference of the bolt hole shown by Figure 8-3 can thus have no relative motion. For this example it assumed that the elastic continuum is modeled using solid elements so that constraint equations are to be generated for only the displacement degrees of freedom.

5

4

6

100

1

3

2

Figure 8-3 Rigid Circular Inclusion.

The equations that specify the desired constraint condition can be defined by using either the RBE1 or RBE2 rigid element, both of which are shown below 1

2

3

4

5

6

7

8

RBE1

156

100

123456

+001

UM

1

123

2

123

3

123

4

123

5

123

6

123

100

123

1

2

3

4

+002

9

10 0001 0002

or RBE2

157

+003

6

5

0003

In both cases the six degrees of freedom at the center of the hole are included in the n-set while the three displacements degrees of freedom at each point around the circumference are included in the m-set.

8.6

Elastic Constraint Element - RSPLINE The elements described in the previous sections are rigid elements. For those elements the constraint equations result from the kinematic condition that there is no relative motion between specified points on the rigid body. The RSPLINE element described in this section and the RBE3 element described in the next section define relationships between degrees of freedom based on assumptions other than rigidity. The RSPLINE uses a cubic spline fit and the RBE3 uses a weighted average to relate degrees of freedom. In either case, since the

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concept of rigidity is not employed, it is no longer appropriate to specify six degrees of freedom which are capable of representing rigid body motion in the n-set. On the contrary, for both of these elements an arbitrary number of degrees of freedom can be put in the n-set. However, there must be at least six. The elastic interpolation element is shown by Figure 8-4 and is defined by the RSPLINE Bulk Data shown on Bulk Data Image 8-5 1

2

RSPLINE

3

4

5

EID

D/L

G1

G2

C4

G5

C5

-etc.-

6 C2

7 G3

8 C3

9

10

G4

where D/L

is the ratio of elastic tube diameter to total tube length. (Real > 0. Default = 0.1)

Gi

are grid point numbers. (Integer > 0)

Ci

are degree of freedom codes. (Integer > 0 or blank) Bulk Data Image 8-5 RSPLINE Data Statement

The elastic interpolation element connects two end points (identified by solid circles) and user specified interior points (identified as solid triangle). z

G

G1 y

x Figure 8-4 Elastic Interpolation Element - RSPLINE

The RSPLINE cubic spline represents a curve and has the property of matching the displacement and slope at the end points and at the user-specified 'primary' interior points. The displacements at user-designated 'secondary' interior points (indicated by the open squares) can then be determined by using the equation for the elastic curve which matches displacement at the primary designated points.

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The values of the displacement at the secondary points are thus dependent on the displacements of the primary points that determine the equation of the spline. The displacement degrees of freedom at the secondary points are thus included in the m-set while the degrees of freedom associated with the end points and primary interior points are independent and are included in the n-set. Consider the case shown by Figure 8-5, where we wish to connect the grid points which lie along the line y = constant using an elastic spline. y

θ z 3 θ z 1

4’

3’

2’

1’

θ z 5

u y 3

5’

u y 1

1

u y 5 2

3

4

5

x Figure 8-5 Elastic Spline Fit Passing Through Displaced Points

The elastic curve is to be determined by the displacement u y and rotation θ, at end points 1 and 5 and a primary interior point 3. The displacements at secondary points 2 and 4 are then to be evaluated from the equation of the elastic curve which passes through points 1, 3, and 5. The RSPLINE constraint element in effect uses the elastic beam equation to provide a relation between specified and interpolated degrees of freedom. The RSPLINE data statement allows the user to define the elastic interpolation function as follows 1.The first and last grid points are taken to be the ends of the spline. All of the degrees of freedom at these points are members of the n-set. 2.An arbitrary number of interior points can be defined which may be either primary points that define the elastic curve such as point 3 in Figure 8-5 or secondary points at which interpolated values are to be determined such as points 2 and 4. The interior degrees of freedom are defined by a pair of fields such as (G 2, C2) where G2 is the grid point number. The presence or absence of the associated degree of freedom code field then determines whether the points are primary or secondary as follows. Value of C-field

Meaning

Blank

The associated grid point is primary and the degrees of freedom at the grid point are used to determine the equation for the elastic curve and are thus to be included in the n-set.

Integer > 0

A secondary grid point. The degrees of freedom associated with the packed degree of freedom code are to be evaluated from the equation for the elastic curve and are thus to be included in the m-set. All other degrees of freedom at the grid point (if any) are to be used to define elastic behavior and are thus in the n-set.

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Rigid and Constraint Elements Weighted Average Constraint Element - RBE3

3.The ratio D/L defines the ratio of bending "stiffness" to torsional "stiffness" for the elastic tube which connects the independent (primary) degrees of freedom. A value for this parameter can be specified or the default value D/L = 0.1 will be taken if the field is blank. The value is of interest only if the RSPLINE connects grid points which do not lie on a straight line. In that case bending and torsional moments are coupled and some difference in the behavior of the RSPLINE element will be observed if D/L is changed. The default is recommended unless the user has a specific reason to use another value. The RSPLINE was developed to provide a way of connecting regions of different mesh density. However, Ransom [1] presented the results of a study done in conjunction with development of interface elements that show that the results for even simple test cases do not give good results. We therefore do not recommend the RSPLINE for transitioning.

8.7

Weighted Average Constraint Element - RBE3 The RBE3 constraint element,Bulk Data Image 8-6, provides a means of specifying that the value of selected degrees of freedom is to be a weighted average of displacement of the other degrees of freedom 1 RBE3

2

3

4

5

GREF

CREF

Wi

Ci

G1i

G2i

W j

C j

G1 j

G2 j

Wk

Ck

G1k

GM1

CM1

GM2

CM2

GM3

CM3

GM4

CM4

GM5

CM5

-etc.-

EID G3i

6

7

8

9

10

-etc.“UM”

Bulk Data Image 8-6 RBE3 Data Statement

The form of the RBE3 constraint element allows the user to define the motion of a reference point by 1.Specifying the reference point using the fields • GREF - the reference grid point • CREF - the degrees of freedom at the reference grid point.

2.Specifying weighting factors, Wi. 3.Specifying degrees of freedoms associated with the weighting factors by first specifying the degree of freedom code, C i, followed by the grid point identification numbers of which the degree of freedom, C i, are to be given the weight W i. 4.Specifying up to six degrees of freedom which are to be placed in the m-set. These (the m- degrees of freedom may be specified as follows

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a.

The default m-set includes the degrees of freedom specified at the reference point.

b.

The m-set can be explicitly defined by placing the literal string UM in field two of a continuation statement after the definition of the weighting factors and associated degrees of freedom. The literal string (UM) is then followed by pairs (GM, CM) which specify one or more degrees of freedom (CM) at the grid point (GM) that are to be included in the m-set.

Rigid and Constraint Elements Weighted Average Constraint Element - RBE3

A set of linear equations is then formed by taking the displacement of the reference point to be the weighted average of the set of independent displacements. The m-set is then taken to be the reference set by default or those degrees of freedom explicitly defined as the m-set. If the m-set is explicitly defined then

8.7.1

a.

The total number of degrees of freedom in the m-set must be the same as the number of degrees of freedom defined at the reference point.

b.

The degrees of freedom must be a subset of those at the reference and the weighted grid points.

c.

The coefficient matrix of the m-set, Rmm, must be nonsingular.

Example -"Beaming" Loads and Masses

The user will often have to "transfer" loads and masses from a reference node, which might not be part of the structural model, to nodes which are part of the model, Consider the cross-section of a cylindrical shell as shown by Figure 8-6 where the total mass of the adjacent structure and the associated center of gravity is at node 100 which lies on the axis of the cylinder, This mass and inertia must be distributed to the structural nodes 1 through 6. The following RBE3 element will properly distribute the mass equally to the six nodes on the shell. l z 5

6

4 100 60o

1

3 2

x

y

Figure 8-6 Circular Cross-Section 1

2

RBE3

74

+RB1

3

3

4

4

5

100

123456

5

6

6 1.

7 123

8 1

9 2

10 +RB1

The mass could be defined by means of a CONM2 mass element that specifies a concentrated mass at Grid point 100. Note that degrees of freedom 1 through 6 of point 100 are dependent while all components at nodes 1 through 6 are independent. The displacements of node 100 represent the weighted average motions at nodes 1 through 6.

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Rigid and Constraint Elements References

We might also consider the case where the mass is to be distributed to the nodes on the cylinder but where 2/3 of the total mass is to be equally distributed to nodes 2 through 4 and that 1/3 of the total mass is to be equally distributed to points 1, 5, and 6. The associated RBE3 bulk data is then 1

2

RBE3

75

+EF

4

3

1.

4

5

6

7

100

123456

2.

123

123

1

5

6

8 2

9 3

10 +EF

Applied static or dynamic loads can be distributed to structural degrees of freedom by means of the same technique. For example, if loads are defined at point 100 then the first RBE3 which was defined in this section would equally distribute the loads to points 1 through 6. The degrees of freedom at the reference point are generally included in the m-set. In some modeling situations the degrees of freedom at the reference point must be included in the n-set. In these cases the user can explicitly define the degrees of freedom in the m-set. For example, the following RBE3 element will evenly distribute loads at the reference point to the structure: 1

2

RBE3

74

+EF

3

+HI

UM

3

4

5

100

123

4

5

6

6

123

6 1.0

7 123

8 1

9 2

10 +EF +HI

In this case the first two statements are similar to the first example discussed above except that here the RBE3 element is connected to only the three translational degrees of freedom of the reference grid point. The last data statement has been added to define components 123 of node 6 as dependent. Now components 123 of node 100 are included in the n-set and can thus be included in a removed data set or if not removed will be included in the m-set. Any of the digits 1-6 may be used in the C REF field. Omission of any of these digits merely indicates that the RBE3 element is not connected to that degree of freedom. The recommended values for the C fields are the codes 1-3. While the codes 4-6 may be used in these fields, determination of proper weighting factors is much more difficult as the ratios between translation and rotational weighting factors are related to the dimensions of the structure which is connected by the RBE3 element.

8.8

References [1] J.A. Ransom, private communication, October, 1999.

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9

Material Properties

The constitutive equations as well as other material properties are described in this chapter. Constitutive relations may be of the following types in NASTRAN: 1. Isotropic temperature-dependent material. 2. Anisotropic temperature-dependent material for two- and three-dimensional elements. Temperature-independent properties are defined using MATi Bulk Data statement. Temperature dependence of properties is then prescribed by MATTi Bulk Data, which define pointers to appropriate material tables on which the temperature dependence is defined. The NASTRAN Bulk Data data statement used to specify material properties are summarized in Table 9-1. Table 9-1 Material Specifications for Structural Analysis Data Name

Description

MAT1

Linear, temperature-independent, isotropic material

MAT2

Linear, temperature-independent, anisotropic material for two-dimensional elements

MAT8

Linear, temperature-independent, orthotrophic material for two-dimensional elements

MAT9

Linear, temperature-independent, anisotropic material for three-dimensional elements

MATT1

Specifies table references for temperature-dependent material properties in conjunction with MAT1

MATT2

Specifies temperature-dependent material properties in conjunction with MAT2

MATT9

Specifies temperature-dependent material properties in conjunction with MAT9

The allowable material relations for the recommended NASTRAN elements are summarized in Table 9-2. Table 9-2 Allowable Material Relations for Structural Elements in NASTRAN. Element Type

Material Type MAT1

239 Nastran Primer

BAR

X

BEAM

X

BEND

X

HEXA

X

MAT2

MAT8

MAT9

X

9

Material Properties Isotropic Material

Table 9-2 Allowable Material Relations for Structural Elements in NASTRAN. Element Type

Material Type MAT1

9.1

MAT2

MAT8

PENTA

X

QUAD4

X

X

X

QUAD8

X

X

X

ROD

X

SHEAR

X

TETRA

X

TRIA3

X

X

X

TRIA6

X

X

X

TUBE

X

MAT9

X

X

Isotropic Material Material properties for a linear, temperature-independent isotropic material are specified on a MAT1 data statement, which is shown in Bulk Data Image 9-1. 1

2

MAT1 MID ST

3

4

5

E

G

NU

SC

SS

MCSID

6

RHO

7

A

8

TREF

9

10

GE

where Field

Description

MID

Material identification number (integer, > 0)

E

Modulus of elasticity (real, > 0. or blank)

G

Shear modulus (real, > 0. or blank)

NU

Poisson's ratio (- 1.0< NU < 0.5, real or blank)

RHO

Mass density (real)

A LPHA

Coefficient of thermal expansion (real)

TREF

Reference temperature for thermal expansion (real)

Ge

Structural element damping coefficient (real)

ST, SC, SS

Stress limits for tension, compression, and shear, respectively, (real). Used to compute margins of safety in certain elements and have no effect on computational procedures.

MATCSID

Coordinate System with respect to which material properties are defined (integer)

Bulk Data Image 9-1 MAT1 - Isotropic Material Property Definition

Features of the MAT1 data statement are: 1. The MID field must be unique among all MAT i-type data statements.

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2. The value of RHO is used to calculate the structural mass matrix for all elements with a defined volume. (The weight density may be specified in weight units and is converted to mass by including the WTMASS parameter having a value 1/ g where g is the acceleration of gravity (see Appendix B for parameters). 3. If E and NU, or G and NU, are both blank, they will be set equal to zero. If two of the three coefficients are defined, the remaining coefficient is calculated using the isotropic relation (Eq. 9-4)).

9.1.1

One-Dimensional Elements

The constitutive relations for the one-dimensional element is of the form

σ xx = E ( ε xx – α∆ T )

(Eq. 9-1)

where E is the modulus of elasticity, α is the coefficient of thermal expansion, and ∆ T = T – T RE F . The only elastic constant that needs to be specified for the ROD and BAR elements is the modulus E . The shear modulus and Poisson's ratio may be left blank.

9.1.2

Two-Dimensional Elements - Plane Stress

The isotropic constitutive relations for the plane stress formulation are of the form:

{ σ } = [ E ] ( { ε } – { α }∆ T )

(Eq. 9-2)

where

{ σ } ≡ σ xx σ yy τ xy { ε } ≡ ε xx ε yy γ xy {α } ≡ α α 0

T

T

T

(Eq. 9-3) E E ------------------- ν ------------------- 0 2 2 ( 1 – ν ) ( 1 – ν ) [ E ] = E E ------------------- 0 ν ------------------2 2 ( 1 – ν ) ( 1 – ν ) 0 0 G

and where E

Modulus of Elasticity

ν

Poisson's ratio

G

Shear modulus

a

Coefficient of thermal expansion

∆T

Temperature change, T- T REF

For an isotropic material, only two of the three material coefficients are independent since E G = -------------------2(1 + ν)

(Eq. 9-4)

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9.1.3

Two-Dimensional Elements - Plane Strain

The isotropic constitutive relations for the plane strain formulation are of the form

{ σ } = [ E ′ ] ( { ε } – { α′ }∆ T )

(Eq. 9-5)

where: 0 ( 1 – ν ) ν E ( 1 – ν ) 0 [ E ′ ] = -------------------------------------- ν ( 1 + ν ) ( 1 – 2 ν ) 1 – 2 ν ) (-------------------0

0

2

and

{ α′ } = ( 1 + ν ) α α 0

T

The material properties for plane strain can be specified by using the MAT1 Bulk Data data statement if E , ν, and α are replaced by E' , ν', and α' where: E E ′ = ------------------2 ( 1 – ν )

(Eq. 9-6)

ν ν′ = ---------------( 1 – ν ) α′ = ( 1 + ν )α 9.1.4

Three- Dimensional Elements

The constitutive relations for the three-dimensional case are:

{ σ } = [ E ] ( { ε } – { α }∆ T )

(Eq. 9-7)

where:

{ σ } = σ xx σ yy σ zz τ xy τ yz τ zx { ε } = ε xx ε yy ε zz γ xy γ yz γ zx {α} = α α α 0 0 0

T

T

(Eq. 9-8)

T

and

E [ E ] = ------------------2 ( 1 – ν )

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1

ν

ν ν

1

ν ν

ν

1

0

0

0

0

0

0

0

0

0

0 0 0 1 – ν ) (---------------2

0 0 0

0 0 0

0

0

0

1 – ν ) (----------------

0

0

0

(---------------1 – ν )

2

2

(Eq. 9-9)

Material Properties Anisotropic Material for Surface Elements

9.2

Anisotropic Material for Surface Elements The material coefficients for a linear temperature-independent anisotropic material are shown in Bulk Data Image 9-2 1

MAT2

2

3

4

5

6

7

8

9

MID

G11

G12

G13

G22

G23

G33

RHO

A1

A2

A3

TREF

Ge

ST

SC

SS

10

MCSID

where Field

Description

MID

Material ID number (integer). Must be unique over all material IDs.

G11, G12, G13, G22, etc.

Diagonal and Upper triangular coefficients of elasticity matrix as defined by Eq. 10-11 (real).

ALPHA1, ALPHA2, ALPHA12

Coefficients of thermal expansion (real).

MATCID

Coordinate ID for coordinate system used for material coefficients (integer). Bulk Data Image 9-2 MAT2 Anisotropic Material

The Stress strain equation is given by (Eq. 9-2) and the components of stress and strain are the same as those presented by (Eq. 9-3). The thermal strain vector and elastic matric must be modified for non isotropic behavior as shown:

{ α } ≡ α1 α 2 α 12

T

G 11 G 12 G 13

(Eq. 9-10)

[ E ] = G 12 G 22 G 23 G 13 G 23 G 33

where: 1.[ E ] is a symmetric matrix so that only the upper half is specified by MAT2. 2.The 1 and 2 directions are relative to the MATCID coordinate system. 3.The α12 term can exist as a result of a coordinate transformation.

9.2.1

Orthotropic Material Properties

The orthotropic relations for two-dimensional behavior are given by:

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Material Properties Anisotropic Material for Surface Elements

' $ σ11 $ % σ22 $ $ σ12 "

/' ( G 11 G 12 0 - $ ε 11 $ -$ $ & = G 12 G22 0 - % ε 22 -$ $ 0 0 G33 - $ γ 12 $ +" #

0 ( $ ' α1 (. $ $ $. & – ∆ T % α2 &. $ $ $. $ " 0 #. , #

(Eq. 9-11)

The orthotropic material coefficients are defined relative to a specific set of axes which are called the material axes. The elastic coefficients for an orthotropic material are related to engineering material coefficients as: E 1 G 11 = -----------------------------( 1 – ν 12 ν 21 )

ν 12 E 2 ν 21 E 1 G 12 = -----------------------------= -----------------------------( 1 – ν 12 ν 21 ) ( 1 – ν12 ν 21 )

(Eq. 9-12)

E 2 G 22 = -----------------------------( 1 – ν 12 ν 21 )

where

9.2.2

E 1

Elastic modulus in the direction of the 1- material axis.

E 2

Elastic modulus in the direction of the 2- material axis.

ν 12

Poisson's ratio for transverse strain in the 2-direction when stress is applied in the 1-direction.

ν 21

Poisson's ratio for transverse strain in the 1-direction when stress is applied in the 2-direction.

G33

Shear modulus

α1

Coefficient of thermal expansion in the 1-direction

α2

Thermal expansion coefficient in the 2-direction

Example: Orthotropic Material

The properties of an E-glass epoxy lamina are E x = 7.8 x 106 psi

E y = 2.6 x 10 6 psi

νxy = 0.25

G33 = 1.25 x 10 6 psi

α1 = 3.5 x 10-6 in. /in. -oF

α2 = 11.4 x 10-6 in./in. -oF

The stress limits for a composite material are generally vastly different in the directions of the material axes since one axis is aligned parallel to the fiber direction and one axis is transverse to the fiber direction. Typical values for the lamina strengths in strength of E-glass are X T = X c = 150 x 10 3 psi Y T = 4x103 psi Y c = 20 x 103 psi S s = 6 x 103 psi

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Material Properties Orthotropic Material for Surface Elements

where: ( X T, X c)

Tensile and compressive strengths in the fiber direction

(Y T, Y c)

Tensile and compressive strengths transverse to fiber direction

S S

Shear strength

For the purpose of this example, we will assume that the longitudinal stress is of interest, and we will therefore take the following stress limits: S T = 150 x 103 psi S C = 150 x 103 psi S S = 6 x 103 psi

The elastic constants are calculated by first noting that the missing Poisson's ratio can be calculated using the second of Eq. 10-15 to give:

ν xy E y - = 0.083 ν yx = ------------ E x

We then find: G11 = 7.965 x 10 6 psi G12 = 0.664 x 10 6 psi G22 = 2.655 x 10 6 psi

These material properties can be defined on a MAT2 data statement as follows 1

9.3

2

3

4

5

MAT2

101

7.965+6

0.664+6

0.0

+M1

3.5-6

11.4-6

0.0

72.0

6

7

2.655+6

8

9

10

0.0

1.25+6 0.0

+M1

1.5+5

6.0+3

Orthotropic Material for Surface Elements The calculation of the coefficients of the elastic matrix using (Eq. 9-12) can be eliminated if we use the MAT8 data statement as shown by Bulk Data Image 9-3, which allows us to specify engineering material constants 1

MAT8

2

3

4

5

6

7

8

9

MID

E1

E2

ν12

G12

G1z

G2z

RHO

Α1

Α2

TREF

Xt

Xc

Yt

Yc

Ss

Ge

F12

STRN

10

Bulk Data Image 9-3 MAT8 Data Statement

The MAT8 material data statement is used in conjunction with the PCOMP property data statement, for thin shell elements to define the lamina properties with respect to a set of principal material axes ( X 1 , X 2). The elastic constants with respect to the material coordinates are then defined by(Eq. 9-10) and are transformed by using the lamina orientation angle θ, which is defined on PCOMP statement, by using appropriate transformation equations (see Ref [1], for example).

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Material Properties Anisotropic Material for Solid Elements

The material coefficients that are specified by MAT8 Bulk Data are described as follows

9.4

E l, E 2

Modulus of Elasticity along and normal to the fiber direction, respectively (real not ≠ 0)

ν12

Poisson's ratio for loading in the 1-direction. The poisson's ratio ν21 is related to ν12 by the equation ν12E2 = ν21E1. (real)

G12

In-plane shear modulus. (real ≠ 0.)

G1z,G2z,

Transverse shear modulae (real or blank)

RHO

Mass Density (real or blank)

α1,α2

Coefficients of thermal expansion in 1 - and 2 - directions. (real or blank)

T REF

Reference temperature for calculation of thermal expansion

X T, X C

Allowable stress in tension and compression, respectively, in the fiber direction. (real or blank)

Y T, Y C

Allowable stress in tension and compression, respectively, normal to fiber direction. (real or blank)

S S

Allowable Shear stress (real or blank)

Ge

Structural damping coefficient. (real or blank)

F 12

Interaction term in tensor polynomial for Tsai-Wu. (real or blank)

STRN

If maximum strain failure theory is chosen by the FT field of the associated PCOMP data statement, then the value of this field indicates whether the allowables are stress or strain allowables. STRN =1. (default) for strain, STRN = 0. for stress.

Anisotropic Material for Solid Elements The material coefficients for a linear temperature-independent three dimensional material are defined using the MAT9 data statement shown on Bulk Data Image 9-4 The only elements that can reference this material formulation are the TETRA, PENTA and HEXA three-dimensional isoparametric elements. 1

MAT9

2

3

4

5

6

7

8

9

MID

C11

C12

C13

C14

C15

C16

C22

C23

C24

C25

C26

C33

C34

C35

C36

C44

C45

C46

C55

C56

C66

RHO

α1

α2

α3

α4

α5

α6

TREF

Ce

10

where: Field

Description

MID

The material ID (integer). Must be unique among all material type data statements.

C11,C12, …, C66

Coefficients of upper half of symmetrical elastic matrix (real)

RHO

Mass density per unit volume. Bulk Data Image 9-4 MAT9 Data Statement

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Material Properties Anisotropic Material for Solid Elements

Field

Description

α1,α2, …, α6

Coefficients of thermal expansion (real).

TREF

Reference temperature for calculation of thermal expansion (real). Bulk Data Image 9-4 MAT9 Data Statement

The general form of the elastic relationship for three-dimensional elements in a three-dimensional stress state is given by (Eq. 9-8) and the coefficients of stress and strain by the (Eq. 9-9). The thermal strain vector is:

{α} =

T

(Eq. 9-13)

α1 α2 α3 α 4 α5 α 6

and where the elasticity matrix, [ E ], is a symmetric 6 by 6 matrix whose coefficients are C ij. The form of the MAT9 data statement thus allows the user to define an anisotropic material having twenty-one independent material coefficients. The inclusion of the six components of the thermal expansion vector allows the definition of transformed material coefficients.

9.4.1

Orthotropic Constants for Solid Elements

Engineering materials having orthotropic properties are finding increased application in the design of structural systems. An orthotropic material is completely defined by nine independent elastic constants. The most common elastic constants are the following 1.Elastic modulae, E 1 , E 2 , E 3 in three orthogonal directions 2.Poisson's ratios, νij, for transverse strain in the j-directions due to stress in i-direction. 3.Shear moduli, G12, G23, G31 in the 1-2, 2-3, and 3-1 planes, respectively. The compliance matrix, [S ], which is the inverse of the elastic matrix is given by:

– 1

[ S ] = [ E ]

ν 21 ν 31 1 ------ – ------- – -------- 0 E 1 E 2 E 3

0

0

ν 12 1 ν 32 – ------- ------ – -------0 E 1 E 2 E 3

0

0

0

0

ν 13 ν 23 1 – ------- – -------- -----= E 1 E 2 E 3

0

(Eq. 9-14) 1 --------G 12

0

0

0

0

0

0

0

1 0 --------G 23

0

0

0

0

0

0 0 1 --------G 31

The compliance matrix is symmetric so that the following symmetry relations must hold:

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Material Properties Transformation of Elastic Constants

ν 12 E 2 = ν 21 E 1 ν 23 E 3 = ν 32 E 2

(Eq. 9-15)

ν 31 E 1 = ν 13 E 3 The nonzero stiffness coefficients, C ij, are found by inverting the compliance matrix (Eq. 9-14) and are: 1 – ν23 ν 32 ν 12 + ν 32 ν 13 ν 13 + ν 12 ν23 ------------------------- ------------------------------ -----------------------------CE 2 E 3 CE 1 E 3 CE 1 E 3

ν 12 + ν 32 ν 13 1 – ν 31 ν 13 ν 23 + ν 21 ν13 -----------------------------CE 1 E 3

------------------------- -----------------------------CE 1 E 3 CE 1 E 2

[ E ] = ν 13 + ν 12 ν 23 ν 23 + ν 21 ν 13 1 – ν 12 ν 21 ------------------------------ -----------------------------CE 1 E 3 CE 1 E 2

------------------------CE 1 E 2

0

0

0

0

0

0

0

0

0

G 12 0

0

0

0

0

0

0

0

0 G 23 0

0

0

0

0

(Eq. 9-16)

0 G 31

where: 1 – ν12 ν21 – ν 23 ν 32 – ν 31 ν 13 – 2 ν 21 ν 32 ν13 C = ------------------------------------------------------------------------------------------------------- E 1 E 2 E 3

9.5

(Eq. 9-17)

Transformation of Elastic Constants The elastic constants for non-isotropic materials must be defined with respect to a specific material coordinate system. The user then specifies both a material set identification number and the coordinate identification number associated with the material coefficients on a PSOLID data statement as shown by Bulk Data Image 6-10. The ability to specify coordinate systems using the CORD-type data statement and the reference system used on the PSOLID relieves the analyst from the burden of transforming elastic constants from one coordinate system to another.

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Material Properties Temperature- Dependent Materials

9.6

Temperature- Dependent Materials Any of the temperature-independent material property fields specified on the MAT1, MAT2, or MAT9 data statements can be made temperature dependent by means of MATTi data statements, as shown in Bulk Data Image 9-5. 1

2

MATT1 MID T(ST) MATT2 MID T(A1) MATT8 MID

3

4

T(E)

T(G)

T(SC)

T(SS)

T(G11)

T(G12)

T(A2)

T(A3)

T(E1)

T(E2)

T(A1)

T(A2)

T(Ge)

T(F12)

5

T(NU)

6

7

8

T(Ge)

T(G23) T(G33) T(RHO) T(ST)

T(SC)

T(SS)

T( NU12) T(G12)

T(G2z)

T(G2z)

T(RHO)

T(Xt)

T(Xc)

T(Yt)

T(YC)

T(Ss)

T(C11)

T(C12)

T(C13)

T(C14)

T(C15)

T(C16)

T(C22)

T(C23)

T(C24)

T(C25)

T(C26)

T(C33)

T(C34)

T(C35)

T(C36)

T(C44)

T(C45)

T(C46)

T(C55)

T(C56)

T(C66)

T(RHO) T(A1)

T(A2)

T(A3)

T(A4)

T(A5)

T(A6)

MATT9 MID

10

T(Ge)

T(RHO) T(A)

T(G13) T(G22)

9

T(Ge)

where: Field

Description

MID

The ID of a MAT-type data statement whose coefficients are to be temperature-dependent (integer > 0).

T(E), T(G),T(NU),etc.

The ID of a TABLEM-type tables that specifies the temperature dependence of the associated field of the MAT1that is indicated in parentheses.

T(G11), …, T(G33),etc.

The ID of a TABLEM-type tables that specifies the temperature dependence of the associated field of the MAT2 that is indicated in parentheses.

T(E1), …, T(F12),etc.

The ID of a TABLEM-type table that specifies the temperature dependence of the associated field of the MAT8 that is indicated in parentheses.

T(C11), …,T(C66), etc.

The ID of a TABLEM-type table that specifies the temperature dependence of the associated field of the MAT9 that is indicated in parentheses.

Bulk Data Image 9-5 MATTi Data Statements

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Material Properties Material Property Table

The form of all MATTi data statements is the same. The MID of the MATTi statement must match the MID on the corresponding MAT-type statement whose properties are to be made temperature dependent. The temperature-dependence of a specific property is then defined by reference to an appropriate table in the field of the MATTi statement associated with the temperature-dependent property. Blank or zero entries in the table reference field of MATTi are taken to mean that the property on the associated MATi data statement is temperature independent.

9.7

Material Property Table The temperature dependence of material properties is defined by means of material property tables. Each of the four types of TABLEMi data statements that are shown in Bulk Data Image 9-6 uses a different algorithm evaluate the function.Tables TABLEM1 through TABLEM3 use linear interpolation between table entries. TABLEM4 uses a power series expansion. The algorithms are defined in Table 9-3 1

2

TABLEM1 TID T1 TABLEM2 TID T1 TABLEM3 TID T1 TABLEM4 TID A0

3

4

5

6

7

8

9

XAXIS

YAXIS

P1

T2

P2

T3

P3

-etc.-

ENDT

P1

T2

P2

T3

P3

-etc.-

ENDT

Ta

T b

P1

T2

P2

T3

P3

-etc.-

ENDT

Ta

T b

TL

TU

A1

A2

A3

A4

A5

-etc.-

ENDT

10

Ta

where Field

Meaning

Table identification number, Integer > 0

TID XAXIS,YAXIS

Specifies the type of interpolation to be applied in interpolating temperature values. Blank or one of the character strings, ‘LINEAR’ or ‘LOG’. Linear interpolation is used if the field is blank.

(Ti, Pi)

Pairs of temperature and property value, real.

Ta, T b, TL,TU

Table parameters whose use depends on the specific table algorithm, real.

Ai

Coefficients of power series expansion used only by TABLEM4, real.

ENDT

A required character string, ‘ENDT’, which signifies the end of table entries. Bulk Data Image 9-6 Temperature- Dependent Material Property Tables

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Table 9-3 Algorithms Used for Material Tables Table Type

Look-up Algorithm

TABLEM1

P = P T ( T )

TABLEM2

P = P T ( T – T 1 )

TABLEM3

T – T 1 P = P re f P T / ----------------0 + T 2 , N

TABLEM4

P = P re f

1

T – T 10 i A i / --------------+ T 2 ,

i=0

Features of the TABLEMi data statements are: 1. At least two entries must exist for tables TABLEMi (i = 1,2,3). 2. At least one entry must exist for TABLEM4. 3. Tables must be terminated with the alphabetic string 'ENDT' in the field following the last entry. 4. The table temperatures must define either an ascending or descending sequence. TABLEM1 provides a simple tabular look-up. The pairs ( T , i P ) i are specified where T is the temperature and P is the associated material property. The table look-up evaluates the property P at temperature T by using linear interpolation within the table and linear extrapolation outside the table based on the last two points at the end of the table. A discontinuous material property may be defined at a specific temperature, as indicated in Figure 9-1 Figure 9-1 Discontinuity of Property Value

P 4 P 3

P 2 P 1 T 1

T 2

T 3

Discontinuities may not be defined at the endpoints of the table and the average value of the property is taken at the jump. For example, at T = T 2 P 3 + P 2 P ( T 2 ) = -----------------2

(Eq. 9-18)

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TABLEM2 is similar to TABLEM1 except that tabulated values are used to scale the basic property value Pref defined on the MATi data statement which is referenced on the MATTi data statement. The inclusion of a reference temperature, T a, is an added convenience since many material properties are defined relative to room temperature. TABLEM3 data statement is similar to the TABLEM2 data statement, except that the temperature difference is normalized with respect to the parameter T b. TABLEM4 allows the analyst to define the temperature-dependent property using a power series in the normalized independent variable T = (T - T a )/T b. As in TABLEM2 and TABLEM3, the reference value of the, property, P ref , is that which is on the appropriate field of the MATi data statement. The coefficients of a power series expansion are entered on the continuation statement. The first coefficient A0 must appear in the second field of the continuation statement, followed by A1 , A2 ,...,A N . If one of the coefficients is zero, then a zero entry must be included in the corresponding field. The parameters T L and T U are lower and upper temperature limits, respectively so that T is bounded as:

/ T L if T < T L 0 . + T U if T > T U ,

T = -

9.7.1

Example: Temperature- Dependent Elasticity

Consider the temperature-dependent modulus of elasticity that is shown in Figure 9-2 The isotropic room temperature properties are E = 10x 10 6 psi

ν = 0.3 α = 7 x 10-6 in. /in. -oF γ = 0. 1 lbl /in.3 T ref = 72o

– 6

E × 10 psi

10 8 6 4 2

72

250

400

Figure 9-2 Temperature- Dependent Elastic Modulus

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T (oF)

Material Properties Modeling Composite Materials

The following data statements specify this a temperature dependent modulus of Elasticity, E: 1

2

3

MAT1

1

1.E7

MATT1

1

37

4

5

.3

6

.1

7

7.E-6

8

9

72.

TABLEM1 37 +TM1

10

+TM1

72.

1.0E7

250.

0.8E7

400.

2.0E6

ENDT

where 1.The MATT1 data statement has the same material identification number, MID, as the MAT1 data statement. 2.The TID of the temperature table is entered in the field of MATT1 which corresponds to the modulus of elasticity, E , and points to a TABLEM1 data statement whose identification number is 37. 3.The table must be terminated by the alphabetic string 'ENDT' in the field following the last entry. The temperature-dependent modulus could have been defined using a TABLEM2 data statement: 1

2

3

TABLEM2 37

72.

+TM2

1.

0.

4

5

6

7

8

9

10

+TM2 178.

.8

328.

.2

ENDT

where the table temperatures have been reduced by the value of the reference temperature and the property entries have been normalized with respect to the value on the MAT1 data statement.

9.8

Modeling Composite Materials Composite materials are attractive for design of structures since materials having attractive attributes including high strength-to weight ratio and low coefficients of thermal expansion can be designed and manufactured. The mechanics of composite materials is well covered in the literature (Ref. [1] through [4]) and will only summarized here.

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Material Properties Modeling Composite Composite Materials

9.8.1

Lamina Pro Properties

A laminated composite material is manufactured by bonding lamina in some stacking sequence as shown by Figure 9-3.

Figure 9-3 Laminate Construction

The individual lamina have orthotropic properties defined relative to a set of lamina material axes as shown by Figure 9-4. 3 2

1 Figure 9-4 Lamina Material Axes

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The orthotropic lamina strain-stress relations relative to the lamina coordinate system are given by(Eq. 9-10) and (Eq. 9-12). These lamina equations must be transformed to the Laminate material coordinate system, system, which for a plane Lamina is a rotation about the normal through an angle, θ, as shown by Figure 9-5. y

2 1

+θ x

Transformation rmation of Lamina Axes Figure 9-5 Transfo The transformation of the stress strain relations is relatively straight forward since the stress components components are a second order tensor and the engineering strain components can be transformed to tensor strain components as follow:

{ ε }tensor = [ T ] { ε' }tensor

(Eq. (Eq. 9-19) 9-19)

{ σ } = [ T ] { σ' } The engineering strains can be transformed to tensor components as follows:

' $ ε 11 $ % ε 22 $ $ γ 12 "

( ' $ 1 0 0 $ ε 11 $ $ = 0 1 0 % ε 22 & $ $ 0 0 2 $ ε 12 $ # En gr · "

( $ $ = [ R ] { ε }tensor & $ $ #Tensor

(Eq. (Eq. 9-20) 9-20)

Jones[1] shows that the transformed material matrix and thermal strain vector are then given by: T

[ G ] = [ T ] [ G' ] [ T ] T

(Eq. (Eq. 9-21) 9-21)

{ α } = [ T ] [ R ] { α' } where [T ] is the transform from the prime to the unprime coordinate system. The Lamina stress strain relationship referred to the material coordinate system is then:

{ σ } = [ G ] { ε } + ∆ T { α } 9.8.2

(Eq. (Eq. 9-22) 9-22)

Laminate Pr Properties

The laminate is assumed to be constructed by bonding individual lamina. The bond between lamina lamina is to be infinitesimally infinitesimally thin and and have no shear shear deformations. deformations. The strains and curvatures of the laminate are then defined by plate bending theory as described in Chapter 2. The stress coefficients are not continuous through the thickness. However, the variation of strain is a function of the strain at the reference surface and the curvatures as follows:

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' $ ε 11 $ % ε 22 $ $ γ 12 "

( ' o ( ' $ $ ε 11 $ $ χ 11 $ $ o $ $ & = % ε 22 & + z % χ 22 $ $ $ $ $ $ γ o $ $ χ 12 # " 12 # "

( $ $ & $ $ #

(Eq. (Eq. 9-23) 9-23)

where the reference surface strains and curvatures are given by Eqs 2-53 and 2-54. The stress strain relation for the k th lamina is then found by substituting (Eq. 9-23) into (Eq. 9-22):

' $ σ11 $ % σ22 $ $ τ 12 "

( G11 G 12 G 13 $ $ & = G21 G22 G23 $ G31 G 32 G 33 $ #k

/' o - $ ε 11 -$ - % εo - $ 22 -$ o k + " γ 12

( ' $ $ χ 11 $ $ & + z % χ 22 $ $ $ $ χ 12 # "

( ' (0 $ $ α1 $ . $ $ $. & + ∆ T % α2 & . $ $ $. $ $ α3 $ . # " #k ,

(Eq. (Eq. 9-24) 9-24)

The laminate stacking order is assume as to be as shown by Figure 9-6. 1 1 -2

z0

2 z1

MIDDLE SURFACE

z2 z k

z k – 1

z k

z N – 1

z N

N

LAYER LAYER NUMBER NUMBE R Figure 9-6 Laminate Stacking Order

The stress and moment resultants described in Chapter 2 are then found by piecewise integration over the thickness to be:

' $ N x $ { N } = % N y $ $ N xy "

( $ $ & = $ $ #

' $ M x $ { M } = % M y $ $ M xy "

( $ $ & = $ $ #

z / z k 0 k . 1 [ G ]k -- * { ε }d z + * { χ } z d z .. k = 1 z + z k – 1 , k – 1

(Eq. (Eq. 9-25) 9-25)

z / z k 0 k 2 . 1 [ G ]k -- * { ε } z d z + * { κ } z d z .. k = 1 z + z k – 1 , k – 1

(Eq. (Eq. 9-26) 9-26)

N

N

Since the reference surface strains and curvatures are not functions of z , the stress and moment resultants can be written as:

256 Nastran Primer

Material Properties Modeling Composite Composite Materials o

{ N } = [ A ] { ε } + [ B ] { χ }

(Eq. (Eq. 9-27) 9-27)

o

{ M } = [ B ] { ε } + [ D ] { χ } where the elastic stiffness coefficients are:

/ N 0 A ij = - 1 ( G ij )k ( z k – z k – 1 ). . + k = 1 , / N 0 1 2 2 B ij = -- - 1 ( G ij ) k ( z k – z k – 1 ). . 2+ k = 1 ,

(Eq. (Eq. 9-28) 9-28)

/ N 0 1 3 3 D ij = -- - 1 ( G ij ) k ( z k – z k – 1 ). . 3+ k = 1 , The elastic matrix, [ A] and [ D extensional and bending stiffness, D], are called the extensional respectively and the matrix [ B] is the coupling matrix. Since the coupling matrix is an even function it will be null for a symmetric laminate. laminate.

9.8. 9. 8.3 3

Defi De fini ning ng a Com Compo posi site te La Lami mina nate te Pr Prop opert erties ies

The laminate is defined by the order and orientation of each lamina with respect to a material coordinate system. The PCOMP data statement defines the properties of an N -ply -ply laminate as shown by Bulk Data Image 9-7.

Properties of n-ply laminated composite material 1

2

3

4

5

6

Z0

NSM

S b

MID1

T1

θ1

SOUT1 MID2

MID3

T3

θ3

SOUT3 -etc.-

PCOMP PID

FT

7

8

9

Tref

Ge

LAM

T2

θ2

SOUT2

10 10

Properties of n-ply laminated composite material where all plies are of the same material and equal thickness 1

2

PCOM COMP1 PID

θ1

3

4

5

Z0

NSM

S b

θ2

θ3

-etc.-

6 FT

7

8

MID

TPLY PLY

9

10

LAM

Properties of n-ply laminated composite material where all plies are of the same material 1

2

PCOM COMP2 PID T1

3

4

5

6

Z0

NSM

S b

FT

θ1

T2

θ2

-etc.-

7 MID

8

9

10

LAM

Nastran Primer 257

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where Field

Description

PI D

Property identification number. (0 < integer < 1000000)

z0

Distance from the reference plane to the bottom surface, real. The default is -0.5*T where T is the element thickness.

NSM

Non structural mass, mass, real > 0.

S b

Allowable interlaminar shear shear stress, required if FT is also specified. Real. Failure theory. One of the following character strings: “HILL” - Hill Theory

FT

“HOFF” - Hoffman theory “TSAI” - Tsai-Wu theory “STRN” - Maximum strain theory

Tref

Reference temperature, temperature, real or blank. Default is 0.0.

Ge

Damping Coefficient, real or blank

LAM

Symmetric laminate option, either the character string, SYM, or blank. If LAM = SYM then only plies on one side side of middle surface are specified. If an odd number of plies is desired then the center ply thickness is one half the actual thickness. If blank all plies must be be specified.

MIDi

MID of a lamina starting with the bottom lamina. Must refer to a MAT1, MAT2 or MAT8 data statement. Integer> 0 or blank.

Ti

Ply thickness, starting with the bottom ply. Real

θi

Orientation angle of the longitudinal axis of each ply with the element material axis, starting with the bottom ply. Real or blank A character string requesting lamina stress and strain output. SOUT = YES - print stress and strain for the lamina

SOUTi

SOUT = NO - do not print stress and strain output Default is NO Bulk Data Image 9-7 PCOMP Layered Composite

9.8.3.1

Remarks

1.The minimum number of plies is one. 2.One of the fields, MID i, Ti, and θi must exist for each ply. Those not specified default to the last specified value for the field. 3.The reference non uniform structural damping coefficient, Ge, and temperature specified on the material data statements is not used. The damping coefficient and reference temperature specified specified on PCOMP is used for all plies. 4.The ELFORCE and STRESS Case Control directives must be present if ply stress or failure indices are desired. 5.PCOMP is processed by the Input File Processor and to generate PSHELL and MAT2 data statements. Theses generated statements are printed if an ECHO = SORT request is included in Case Control.

258 Nastran Primer


6.PCOMP is also used for post processing ply stresses so that the equivalent PSHELL and MAT2 will not result in ply stresses. 7.If the specified value of z0 is not one half the thickness of the laminate and PARAM,NOCOMPS,-1 is specified then the homogeneous element stresses are incorrect while the lamina stress and force and strains are correct. For correct homogeneous homogeneous stresses use ZOFFS on the shell connection statement.

9.8 9. 8.4

Lam amin ina a Fa Fail ilu ure Cri rite teri ria a

The general topic of failure of fiber-reinforced orthotropic lamina is presented in detail by Jones[1]and Jones[1]and Gurdal [2]. The failure criteria incorporated incorporated in the composite modeling modeling capability of NASTRAN is summarized in the following sections. 9.8.4.1

Maximum Strain Theory

The maximum strain criterion states that failure occurs when one of the following inequalities is violated: X

t

ε 1 < ε 1 = ------t E 1

c

X

ε 1 < – ε1 = – ------c E 1

s

Y t t ,ε 2 < ε 2 = ------ for ε 1 ε , 2>0 E 2 Y c ,ε2 < – ε 2 = – ------t for ε 1 ε , 2<0 E 2

(Eq. (Eq. 9-1) 9-1)

S G 12

γ 12 < γ 12 = ---------

where S is is the Ultimate in-plane shear strength, X and and Y are are ultimate strengths along and transverse to the fiber directions, respectively respectively;; and, t and and c sub and superscripts refer to tension and compressive values, respectively. 9.8.4.2

Tsai-Hill Theory

The maximum strain criteria fail to represent the effect of interactions of strain components on failure. The Von Mises criteria accounts for these interactions an various modifications are used to represent the failure surface. Those used most frequently are due to Tsai-Hill, Tsai-WU and Hoffman [5]and [6]]. Hill Theory[3] extends von Mises’ failure criterion for anisotropic materials with equal strength in tension and compression for a three dimensional stress state. The modified theory for plane stress is generally termed the Tsai-Hill theory which has the following failure criterion: 1 2 1 2 1 1 2 ------ σ 11 + ----- σ 22 – ------ σ 11 σ + ----- τ 12 < 1 22 2 2 2 2 X t Y t X t S 9.8.4.3

(Eq. (Eq. 9-1) 9-1)

Hoffman Theory

Hoffman[4] generalized the Tsai-Hill theory to allow for different strengths in tension and compression. The Hoffman Criterion is: 1 2 1 2 1 1 2 10 1 1 ----------- σ 11 + ---------- σ 22 – ----------- σ 11 σ + / --1--- – ----σ 1 + /+ ---- – -----0, σ 2 + -----2 τ 12 < 1 22 + , X t X c Y t Y c X t X c X t X c Y t Y c S

(Eq. (Eq. 9-1) 9-1)

Nastran Primer 259

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Material Properties Reference

9.8.4.4

Tsai-Wu Theory

The Tsai-Wu theory also allows for different strengths in tension and compression. The Tsai-Wu criterion is: 1 2 1 2 1 10 ----------- σ 11 + ---------- σ 22 – -------------------------- σ 11 σ + / --1--- – ----σ 22 + X t X c Y t Y c X t X c, 1 X X Y Y

(Eq. (Eq. 9-1) 9-1)

t c t c

1 2 1 1 + / ---- – -----0 σ 2 + ----- τ 12 < 1 2 + Y Y , t c S

9.9

Reference [1] [1] Jone Jones, s, R. M., M., Mechanics of Composite Composite Materials, McGraw-Hill, 1975, pp. 147-164. Optimization of Laminated Laminated Composite Composite [2] Z. Gurdal, Gurdal, R.T. R.T. Haftka Haftka and and P. P. Hajela, Hajela, Design and Optimization Materials, Wiley Interscience, 1999, pp. 231-245. Proceedings of the [3] R. Hill, Hill, “A theory theory of the yielding yielding and flow flow of anisotrop anisotropic ic materials materials”, ”, Proceedings Royal Society, A(193), 1948, 281-297. Composite [4] O. Hoffman, Hoffman, “The “The brittle brittle strength strength of orthotro orthotropic pic materials materials”, ”, Journal of Composite s, 1(2), 1967, 200-29. Material s,

[5] S.W. S.W. Tsai, Tsai, “Strength “Strength theories of filamentary filamentary composites”, composites”, In Fundamental Aspects Aspects of Fiber Reinforced Reinforced Plastic Composites Composites, R.T. Schwartz and H.S. Schwartz (eds.), Wiley Interscience, New York, 1968, pp. 3-11. [6] E.M. Wu, Wu, “Streng “Strength th and fractu fracture re of composi composites” tes”.. In Composite Materials Vol. Vol. 5, pp. p p. 191247. L.J. Brountma and R.H. Krock (eds.) Academic Press, New York, 1974.

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262 Nastran Primer


10

External Loads

External applied loads may result from a variety of effects, including: 1. Concentrated forces and moments at Grid points 2. Applied element surface forces 3. Distributed element body forces 4. Enforced Grid point displacement 5. Enforced element deformation 6. Element thermal fields The total load applied to each element is the superposition of each type of loading so that the total element load, { P e}, is given by:

{ P e } = { P ex } + { P es } + { P eg } + { P ei }

(Eq. 10-1)

where { P eg} are the concentrated loads applied to the grid points and where { P ex}, { P eS }, and { P ei}, are the work equivalent Grid point forces associated with body forces, surface tractions and initial strains, respectively. The data statements for specifying loads are summarized by Table 10-1. Table 10-1 External Static Loads Load Type

FORCE MOMENT FORCE1 MOMENT1

Reference

Concentrated static force or moment vector at a grid point, defined using components.

Bulk Data Image 1-1

Concentrated static force or moment vector having a magnitude and a direction specified by the line connecting two grid points.

Bulk Data Image 1-2

MOMENT2

Concentrated static force or moment having a magnitude Bulk Data and a direction determined by the cross product of two Image 1-3 vectors.

SLOAD

Load at scalar point

Bulk Data Image 1-4

PLOAD1*

Distributed load along the length of BAR, BEAM, and BEND elements

Bulk Data Image 1-5

PLOAD2

Uniform normal pressure on two-dimensional elements

Bulk Data Image 1-6

PLOAD4**

Surface traction for TETRA, HEXA, PENTA, QUAD4, Bulk Data QUAD8, TRIA3, and TRIA6 elements Image 1-7

GRAV

Gravity load specified by defining gravity vector in any Bulk Data defined coordinate system Image 1-8

RFORCE

Load due to centrifugal force field

FORCE2 and

263 Nastran Primer

Description

Bulk Data Image 1-9

10

External Loads Concentrated External Forces at Grid Points

Table 10-1 External Static Loads

SPCD

Load due to enforced grid point displacement

Bulk Data Image 1-9

DEFORM

Enforced axial deformation for one-dimensional elements

Bulk Data Image 1-12

TEMP

Temperature at a grid point


TEMPD

Default grid point temperature


TEMPRB

One-dimensional element temperature field


TEMPP1

Plate element temperature field


* MSC Nastran only **Restricted to surface elements in Legacy NASTRAN These data statements are used to specify concentrated forces and moments at the grid points, scalar forces, distributed body forces due to gravity and centrifugal acceleration fields, enforced grid point displacements, enforced element deformations, and temperature fields. These various types of structural loading are described in subsequent sections of this chapter.

10.1

Concentrated External Forces at Grid Points Concentrated force and moment vectors can be defined at geometric grid points in the following alternative ways: 1. Specification of the Grid point G at which the vector acts and the vector components, as shown in Figure 1-1a. 2. Specification of the Grid point G at which the vector acts and the sense of the vector using two geometric grid points, as shown in Figure 1-1 b.

264 Nastran Primer

External Loads Force Vector Defined by Components

3.Specification of the geometric grid point G at which the vector acts with the vector direction specified as the cross product of two vectors, as shown in Figure 1-1. z

z F, M

F, M F z , M z

G

G

G2 N

F , M y y

G1 F x , M x

x

x

y

y

b) Directed Line Segment Between Two Grid Points

a) Vector Components

z F, M

G

G2 G4

a x

b y

G1

G3 c) Vector Direction Using Cross-Product 4.

Figure 1-1 Definition of Concentrated Force and Moments

10.2

Force Vector Defined by Components The force or moment vector can be defined by specifying the magnitude, A, and the components of a free vector (N 1, N2, N3) that acts in the direction of the force by means of the FORCE and MOMENT data statements shown in Bulk Data Image 1-1 . 1

2

3

4

5

6

7

8

SID

G

CID

A

N1

N2

N3

MOMENT SID

G

CID

A

N1

N2

N3

FORCE

9

10

Nastran Primer 265

10

External Loads Force Direction Defined by Two Points

where Field

Description

SID

Load set identification number. Load sets must be selected using a LOAD = SID statement in Case Control. (integer, > 0)

G

Grid point at which load is to be applied. (integer, > 0)

A

Magnitudes of force or moment. (real ≠ 0.) Components of a free vector defined with respect to the CID coordinate system. (real)

N1, N2, N3 CID

Coordinate identification number. (integer)

Bulk Data Image 1-1 Static Force and Moment Using Vector Components.

The vector components ( N 1 , N 2, N 3) define a vector

{ N } = n1 i1 + n 2 i 2 + n 3 i 3 where the unit vectors are taken in the sense of the displacement degrees of freedom for the CID coordinate system, as shown by Fig. 7-4. The resulting vector is then given by

{ F } = A { N }

(Eq. 10-2)

so that the magnitude of the force is given by

{ F } = A { N }

(Eq. 10-3)

The magnitude of the force (or moment) is thus equal to the magnitude A specified on the FORCE or MOMENT data statement only if { N } is a unit vector.

10.3

Force Direction Defined by Two Points The force or moment vector can be defined by specifying a magnitude A and a direction determined by the directed line segment between two grid points by means of the FORCE1 and MOMENT1 data statements shown in Bulk Data Image 1-2. 1

2

3

4

5

6

SID

G

A

G1

G2

MOMENT1 SID

G

A

G1

G2

FORCE1

7

8

9

10

where Field

266 Nastran Primer

Description

SID

Load set identification number. (integer, > 0)

G

Grid point at which the load is to be applied. (integer, > 0)

A

Magnitude of the force or moment. (real)

G1, G2

Two non coincident grid points which define the sense of the load vector as the unit vector, { N }, along the directed line segment from G1 to G2. (integers, > 0, G1 ≠ G2)

External Loads Vector Normal to a Surface Bulk Data Image 1-2 Direction of Force Defined by Two Points

The force or moment is defined as

{ F } = A { N }

(Eq. 10-4)

where { N } is a unit vector in the direction of the line segment drawn between geometric grid points G1, G2

10.4

Vector Normal to a Surface The force or moment vector can be defined by specifying a magnitude A and a direction determined by the vector product of two vectors by using the FORCE2 and MOMENT2 data statements shown in Bulk Data Image 1-3 1

2

3

4

5

6

7

8

SID

G

A

G1

G2

G3

G4

MOMENT2 SID

G

A

G1

G2

G3

G4

FORCE2

9

10

where Field

Description

SID

Load set identification number. (integer, > 0)

G

Grid point at which the load is to be applied. (integer, > 0)

A

Magnitude of the force or moment. (real)

G1, G2, G3, G4

Pairs of grid points that define the vectors a and b , respectively as shown on Fig.11-1c. G 1 ≠ G2, G3 ≠ G4. (integers, > 0)

Bulk Data Image 1-3 Direction of Force Defined by Vector Cross Product

The force or moment is defined as { F } = A { N } where { N } is a unit vector defined by the cross product:

a× b

N = ---------------

(Eq. 10-5)

a× b and where

a is the vector from grid point G 1 to G2 and

b is the vector from grid point G 3 to G4 The force acts at grid point G, which may be the same as points G 1 and/or G3. This form of force specification is useful when defining a force vector that is perpendicular to the surface of an element, such as that shown in Figure 1-2, where a force is to be defined at grid point one in the direction of the outward normal.

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External Loads Load Applied to Scalar Point - SLOAD

The force can be most conveniently specified by means of the FORCE2 data statement as shown below. 1 FORCE2

2

3

101

4

1

5

10.

6

1

7

2

8

1

9

10

3

where a is defined from point one to two and b is defined from one to three. Since three points define a plane, it follows that the cross product defines a vector normal to the plane as desired. z F = 10

3 b

1 a

2 x y Figure 1-2 Force Acting Normal to a Surface.

10.5

Load Applied to Scalar Point - SLOAD A static load can be applied to a scalar degree of freedom by the SLOAD data statement shown in Bulk Data Image 1-4. 1

2

SLOAD SID

3 S

4 F

5 S

6 F

7 S

8 F

where Field

Description

SID

Load identification number. (integer, >0)

S

Scalar point identification number. (integer, >0)

F

Magnitude of the force. (real)

Bulk Data Image 1-4 Specification of Force at Scalar Points

268 Nastran Primer

9

10

External Loads Distributed Load on Beam - PLOAD1

10.6

Distributed Load on Beam - PLOAD1 The PLOAD1 shown by Bulk Data Image 1-5 is used to specify distributed and/or concentrated loads along the BAR, BEAM, and BEND element. PLOAD1 generates workequivalent forces at the grid points associated with the element. 1

2

3

PLOAD1 SID

EID

4 TYPE

5 SCALE X1

6

7 P1

8 X2

9

10

P2

where Field

Description

SID

The set identification number assigned to the load. Must be selected using a LOAD = Case Control statement (integer>0)

EID

The identification number of the element to be loaded (integer>0)

TYPE

Defines the type of load as described below.

SCALE

Distance scaling as described below

(Xi, Pi)

Load position and value as described below

Bulk Data Image 1-5 Distributed Loads on Beams-Bending Elements - PLOAD1

10.6.1

Load Type and Direction

The type of load, i.e., force or moment, and the vector sense of the load with respect to either local element or basic coordinates is specified by an appropriate character strings in the TYPE field. The allowable character strings are: TYPE Field

Description

FX, FY, FZ

Force in the Basic X, Y , or Z directions, respectively

MX, MY, MZ

Moment about the Basic X, Y, or Z axes, respectively

FXE, FYE, FZE

Force in the local. x, y, or z directions, respectively

MXE, MYE, MZE

Moment about the local x, y, or z axes, respectively.

10.6.2

Load and SCALE Fields

PLOAD1 can be used to define a concentrated force or moment at a point along the length of the element; or, a distributed force or moment. The load is described by: 1. Specifying the region over which the load is to be distributed using the X1 and X2 fields where X1 defines the start of the loaded region and X 2 defines its end. If a concentrated load at X1 is desired then X 2 must either be blank or equal to X 1. 2. Specifying whether the length is defined as the actual element length or is scaled by either actual length or projected length, using the following character strings in the SCALE field. 3.Specifying the load intensity using the P1, P2 fields where • A concentrated load is defined by a real value for P 1 with P2 blank • A uniform load between X 1 and X2 is defined by setting P 2 = P1

Nastran Primer 269

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External Loads Distributed Load on Beam - PLOAD1

• A linearly varying load is defined between X 1 an X2 if P2 ≠ P1. SCALE Field

Description

LE

Distance is actual length along the neutral axis. If X 2 > X1 the load is a force or moment per unit length.

FR

Distances are normalized with respect to total element length. If X2 > X1 the load is a force or moment per unit length.

LEPR

Same as LE except loads are defined in terms of projected lengths of the neutral axis on the Basic coordinate axes.

FRPR

Same as FR except loads are defined in terms of normalized projected length of the neutral axis on the Basic coordinate axes.

10.6.2.1

Example - Load Distribution on Beam Elements

Consider the specification of the loads applied to the structural system as shown by Figure 1-3 where the loads are as follows: 1. A uniform load, pa, having a magnitude 20 lb./in. applied to element 2 in the negative x-direction in the Basic coordinate system. 2. A concentrated load P b applied to element 2 at a distance of 10 inches from grid point 2 (taken to be the A-end of the element) having a magnitude of 30 lb. 3. A concentrated force P c of 20 lb. and a concentrated moment M d of 60 in.-lb. at the midpoint of element 1.

y

2 l 1

1 0 i n

P c = 20 lb M d = 60 in-lb

P a = 20 lb

2

1 2 l 1 1 /

P b = 30 lb

1

x

3 Figure 1-3 Distributed and Concentrated Forces Along Beam-Type Elements

Assuming that the element z-axes coincide with the Basic Z-axis, these loads could be described by using the following PLOAD1 data statements. 1

2

3

4

5

6

7

8 1.

-20.

10

PLOAD1 101

2

FX

FRPR

0.

-20.

PLOAD1 101

2

FY

LE

10.

-30.

0002

PLOAD1 101

1

FYE

FR

0.5

20.

0003

PLOAD1 101

1

MZE

FR

0.5

60.

0004

where statements 0001 thru 0004 define P a, P b, P c, and M d , respectively.

270 Nastran Primer

9

0001

External Loads Uniform Normal Pressure - PLOAD2

10.7

Uniform Normal Pressure - PLOAD2 The PLOAD2 data statement shown by Bulk Data Image 1-6 specifies a uniform normal pressure on the surface of the two dimensional shell elements including the QUAD4, QUAD8, TRIA3, and TRIA6 1

2

PLOAD2 SID

3

4

5

6

PRESS

EID1

EID2

EID3

PRESS

EID1

THRU

EIDn

7

8

9

10

etc.

or PLOAD2 SID where Field

Description

SID

Load set identification number, (integer, > 0)

PRESS

Magnitude of the pressure. (real)

ElDi

Element identification numbers. All elements referenced must exist and must be two-dimensional. When using the alternate form of EID1 THRU EIDn, all elements must exist. (Integer)

Bulk Data Image 1-6 Pload2 - Normal Pressure for Bending Elements

10.7.1

Sign Convention for Positive Pressure

PLOAD2 generates grid point forces in the element coordinate system. The positive sense of the pressure is determined by the grid number sequence specified on the associated element connection data statement. The sense of positive pressure is found by applying the right hand rule to the grid point numbering sequence for the element vertices. A counterclockwise sequence defines positive pressure in the negative element z-direction. Since the element grid point numbering sequence defines the sense of positive pressure it is important that a consistent numbering sequence be used.

Nastran Primer 271

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10.8

External Loads Nonuniform Surface Tractions - PLOAD4

Nonuniform Surface Tractions - PLOAD4 A nonuniform surface traction can be applied to the surface of the two dimensional shell elements or any face of three dimensional TETRA, HEXA or PENTA elements by the PLOAD4 data statement shown by Bulk Data Image 1-7. 1

2

3

PLOAD4 SID

EID

P1

P2

N1

N2

N3

CID

4

5

6

7

8

9

P3

P4

G1

G3 or G4

P3

P4

THRU

EID2

10

or, alternatively for two dimensional elements PLOAD4 SID CID

EID1

P1

P2

N1

N2

N3

where: Field

Description

SID

The identification number for the load. Must be selected sing a LOAD = Case Control statement.

EID

Identification number of element to be loaded.

P1, P2, P3, and P4

Value of surface traction at vertices of loaded face. (Real)

G1, G3

Grid point IDs used to specify loaded face of three dimensional elements as described below.

THRU, EIDn

Defines set of two dimensional elements to be loaded.

CID

ID of coordinate system used to define vector direction of traction. (Integer)

N1, N2, N3

Components of traction direction. (Real)

Bulk Data Image 1-7 PLOAD4 - Nonuniform Surface Traction

The alternate form can be used to define a surface traction over a set of TRIA3, QUAD4, TRIA6, and QUAD8 elements.

10.8.1

Direction of Surface Traction

The direction of the surface traction can be defined by specifying the three real components of a vector ( N 1 , N 2 , N 3) with respect to the CID coordinate system on the optional continuation statement. If the continuation is not present then the traction is taken to be normal to the surface using the following sign convention. 1. For two dimensional elements positive pressure is the direction determined by applying the right hand rule to the node number sequence as described in the previous section on PLOAD2. 2. For three-dimensional elements positive pressure is in the negative direction of the normal to the face.

10.8.2

Specifying Load Intensity

The fields P 1 , P 2 , P 3 , P 4 are interpreted as

272 Nastran Primer

External Loads Nonuniform Surface Tractions - PLOAD4

1. The values of the surface traction at the ordered set of connected grid points G 1, G2, G3, and G4 as defined on the connection statement for a two dimensional element where P4 is blank if the element is triangular. 2. The values of surface traction at the ordered set of connected vertex grid points on the face of a three dimensional element. The face is identified by specifying two grid points, G1 and G3, which define a diagonal on the face as described below. The field P1 is then taken to be the traction at G 1 and P2, P3, (and P4) are tractions at the other vertices in a sequence determined by applying the right hand rule to the outward normal to the face.

10.8.3

Specifying the Loaded Face of Solid Elements

The loaded face of a solid element is specified using the G1 and G3 fields as described below: 1.For a HEXA element, G1 and G3 are diagonally opposite grid point numbers on a face. 2.For quadrilateral faces of a PENTA, G1 and G3 are diagonally opposite grid point numbers on a face. 3.For triangular faces of a PENTA, G 1 is a grid point number on the face and G3 is blank. 4.For TETRA elements, G1 is a grid point number on the face and G3 is the grid point number not on the face. 10.8.3.1

Example -Surface Traction on Three Dimensional Element

Consider the specification of the surface traction applied to the eight-node HEXA element shown by Figure 1-4. 3 psi 5 psi z

4 psi

2

3

1 4 EI D = 1

6 5 x

7

y

8 Figure 1-4 Surface Traction on Three Dimensional Element

The magnitude on the surface traction is taken to be 5 psi at node 1 3 psi at node 2 4 psi at node 3

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10

External Loads Gravity Loads - GRAV

0 psi at node 4 and varies linearly along the edge of the element. A surface traction of this magnitude can be specified in the direction of the Basic X-axis by means of the following PLOAD4 data statement. 1

2

PLOAD4 101 +P1

0

3

4

5

1

5.

0.

1.

0.

0.

6 4.

7 3.

8 1

9 3

10 +P1

where the face is identified by grid points 1 and three in fields 8 and 9 and P 1, P2, P3, and P4 are associated with grid points 1, 4, 3, and 2, respectively.

10.9

Gravity Loads - GRAV A uniform gravity load can be applied to the structure by specifying an acceleration vector using the GRAV data statement, which is shown in Bulk Data Image 1-8. 1 GRAV

2

3

SID

CID

4 A

5 N1

6 N2

7 N3

8

9

10

MB

where Field

Description

SID

Load set identification number. The SID cannot be the same as any other static load sets. Load sets in Bulk Data can be combined using the LOAD data statement (see section 11.11). (integer > 0)

CID

Set identification of the coordinate system in which the vector components N1, N2, and N3 are defined. (integer > 0) Scale factor such that the desired acceleration is given by

A

{a} = A{N} where {N} need not be a unit vector. (real ≠ 0.)

N1, N2, N3

Components of a vector, {N}, taken in the sense of the displacement degrees of freedom shown by Fig. 7-4 for the coordinate type specified by CID. (real) Used for PART superements to specify which coordinate system to used in an assembled model.

MB

MB = 0, The coordinate systems in the Main Bulk Data section are considered to be stationary with respect to the assembly Basic coordinate system (default) MB = -1, the CID coordinate system is defined in the main bulk data section

Bulk Data Image 1-8 GRAV - Specification of Gravity Vector

The acceleration vector is transformed to the basic coordinate system and expanded to a set of vectors acting on the g -set. Letting { a g } be the set of node point accelerations, the gravity loads are calculated using the structural mass matrix as:

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External Loads Rotation About an Axis- RFORCE

{ P g } = [ M gg ] { a g }

(Eq. 10-1)

where [ M gg ] is the system mass matrix. The inclusion of the GRAV loads in the input implies that the mass matrix must be generated. The user must therefore insure that the mass for all elements is defined as described in Specification of Element Inertia Properties (p. 320) of the text. Sets of gravity loads may be specified by giving each set an identification number that is unique among all load set identifications. The gravity load to be used as execution time is then specified by the Case Control Directive LOAD = SID

It is important to note that NASTRAN does not allow GRAV data statements to have the same set number as the applied external loads described in previous sections. If both gravity and applied external loads are desired they must be combined using LOAD data statement, which is described in Combined Loading - LOAD (p. 277).

10.10 Rotation About an Axis- RFORCE The force field due to angular velocity and acceleration about a grid point, is shown by Figure 1-5. z

Ω Go r i

R o

i Ri

y

x

Figure 1-5 Angular Rotation Vector for Centrifugal Forces

The RFORCE data statement is shown in Bulk Data Image 1-9. 1

2

RFORCE SID RACC

3 G0

4 CID

5 A

6 R 1

7 R 2

8 R 3

9

10

METHOD

MB

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External Loads Rotation About an Axis- RFORCE

where Field

Description

SID

Load set identification number. The SID cannot be the same as any other static load sets. Load sets in Bulk Data can be combined using the LOAD data statement (see section 11.11). (integer, > 0)

G0

ID of a Grid point lying on the axis of rotation. ( integer ≥ 0

CID

Set identification of the coordinate system in which the vector components N1, N2, and N3 are defined. (integer, > 0)

A

The magnitude of the angular velocity about the axis of rotation in revolutions per unit time, real.

R 1, R 2, R 3

Components of the rotation vector, R ,taken in the sense of the displacement degrees of freedom shown by Figure 6-4 for the coordinate type specified by CID. (real)

METHOD

Method used to compute centrifugal force vector. (integer = 1 or 2, or blank, default = 1)

RACC

The magnitude of the angular acceleration about the axis of rotation in revolutions per unit time squared, real. Used for PART superements to specify which coordinate system to used in an assembled model.

MB

MB = 0, The coordinate systems in the Main Bulk Data section are considered to be stationary with respect to the assembly Basic coordinate system (default) MB = -1, the CID coordinate system is defined in the main bulk data section

Bulk Data Image 1-9 RFORCE - Specification of Angular Velocity Vector

The structure is assumed to rotate such that point Go lies on the axis of rotation. If the G o field is left blank, then the spin axis passes through the origin of the Basic coordinate system. The angular rotation vector is defined as R = R1 i 1 + R2 i 2 + R3 i 3

(Eq. 10-2)

The angular velocity vector in radians per unit time is

ω = 2 π AR

(Eq. 10-3)

and, the angular acceleration vector in radians per unit time squared is

α = 2 π ( RA CC ) R

(Eq. 10-4)

The acceleration at Grid point G i in the structure is then given by a i = α × r i + ω × ( ω × r i )

where we see from Figure 1-5 that

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(Eq. 10-5)

External Loads Combined Loading - LOAD

R i = r i + R 0

(Eq. 10-6)

Applying Newton’s second law of motion, the force at grid point G i is then: F i = m i a i

(Eq. 10-7)

Denoting the set of accelerations associated with all geometric grid point degrees of freedom as {a g }, the centrifugal forces are then calculated to be

{ P g } = – [ M gg ] { a g }

(10-8)

10.10.1 Remarks 1.The axis of rotation passes thorough the origin of the Basic Coordinate system if GID = 0 2.If CID = 0 then the axis of rotation vector is defined relative to the Basic Coordinate system 3.The inclusion of centrifugal forces in the analysis implies that the system mass matrix must be calculated. The user must therefore insure that the mass for all structural elements as described in Section 15.4 of the text is defined. The user is also cautioned that the use of consistent mass may lead to erroneous centrifugal forces if the default for METHOD is taken. If METHOD = 2 then the off-diagonal mass terms which are associated with the consistent mass formulation are correctly incorporated unless the model contains offset CONM2 data statements. 4.If angular acceleration is included METHOD = 2 ia always used. 5.The axis of rotation must coincide with the axis of symmetry for Cyclic Symmetry. 6.For superelement analysis the reference grid point must be in the residual structure.

10.11 Combined Loading - LOAD The loads defined in the previous sections are identified by means of a set number. The load to be used is specified by means of the following Case Control Directive LOAD = SID

Combinations of loads may be required and can be specified by means of the LOAD data statement shown by Bulk Data Image 1-10. The LOAD data statement is used to combine sets of static loads as well as to combine RFORCE and GRAV loads with other static loads.

1 LOAD

2

3

SID

S

S4

L4

4 S1

5 L1

6 S2

7 L2

8 S3

9

10

L3

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External Loads Enforced Displacements - SPCD

where

Set identification of the combined load set. The magnitude of entire load is given by N

SID L = S 0

1 S i ( LS ID i ) i=1

Si

(i = 0, 1, 2, N) are scale factors. (real)

LSIDi

(i = 1, 2, N) are LOAD set identification numbers of any load set which can be specified by a LOAD =< SID> Case Control Directive.

Bulk Data Image 1-10 LOAD - Combined Load Sets.

The static load vector specified by the set of loads used at execution time can be displayed by the Case Control Directive. OLOAD =

where is a set of grid and scalar points defined using a SET Case Control Directive.

10.12

Enforced Displacements - SPCD The SPCD data statement shown by Bulk Data Image 1-11 provides the user with the capability of defining a constraint that will be treated as load associated with a degree of freedom in the s-set. The use of the SPCD statement avoids the decomposition of the stiffness matrix when only the magnitudes of enforced displacements are changed. 1

2

SPCD

SID

3 G1

4 C1

5 D1

6 G2

7 C2

8

9

10

D2

where: Field

Description

SID

Load set identification number. The SID cannot be the same as any other static load sets. Load sets in Bulk Data can be combined using the LOAD data statement (see section 11.12, (integer, > 0)

G, C

Degree(s) of freedom associated with the enforced displacement set where G is the grid point number and C is the degree of freedom code(s). (Integers > 0)

D

Value of enforced displacement. (Real)

Bulk Data Image 1-11 SPCD - Displacement Boundary Conditions in Load Set

The SPCD is similar to the SPC data statement described in Chapter 6 except that the set identification associated with the SPCD refers to a load set and is specified using a LOAD = Case Control statement. The following rules must be followed in using the load-set enforced displacement.

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External Loads Enforced Deformation - DEFORM

1. The degrees of freedom specified on the SPCD data statement must also be defined to be members of the s-set by SPC or SPC1 data statements. 2. The value D replaces the value of the constraint defined by the SPC or SPC1 statements only if the appropriate load set is selected in Case Control. 3. The LOAD combination capability can not be used to select SPCD sets. 4. At least one external load specification (FORCE, MOMENT, etc.) must be present in the input, however, the associated load may be zero. The SPCD assists the analyst in performing rigid body checks for large structural models. Using SPCD, displacement boundary conditions can be specified as loading conditions so that rigid body checks can be performed in a single solution with only one matrix decomposition.

10.13

Enforced Deformation - DEFORM An enforced axial deformation can be specified for the ROD-type and BAR-type elements by means of the DEFORM data statement shown by Bulk Data Image 1-12 1 DEFORM

2

3

SID

EID

4 D

5 EID

6 D

7 EID

8

9

10

D

where

SID

Load set identification. (integer, > 0)

EID

Element number of a ROD, TUBE, CONROD, BAR, or BEAM element. (integer, > 0)

D

Enforced axial deformation, plus for elongation. (real)

Bulk Data Image 1-12 DEFORM - Enforced Axial Displacement

The set of enforced displacements that is to be used is specified by means of the Case Control Directive DEFORM = SID

where SID is the identification number of a DEFORM set in Bulk Data.

10.14 Thermal Loading Element and grid point temperatures are used for the determination of: 1. Thermal loading 2. Temperature-dependent material properties 3. Stress recovery The analyst selects a specific set of temperature data and specifies whether the temperatures are to be used to determine temperature-dependent material properties or to define equivalent thermal loads by the TEMPERATURE Case Control Directive, which has the format

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External Loads Thermal Loading

/ MATERIAL0 TEMPERATURE- LOAD . = SID . + BOTH , where BOTH is the default option. For example, TEMP(LOAD) = 30

specifies that temperature set 30 is to be used to calculate thermal loads and TEMP(MATERIAL) = 20

specifies that temperature set 20 is to be used to calculate temperature-dependent material properties. These specifications are mutually exclusive; i.e. temperature-dependent material properties will not be calculated if LOAD is specified, and thermal loads will not be calculated if MATERIAL is specified. Both thermal loads and material-dependent properties are specified by either of the equivalent forms TEMP(BOTH) = SID

or TEMP = SID

If the thermal effects are specified, then all elements must have a temperature field defined either directly by an element temperature field or indirectly by grid point temperatures. If both are given then precedence is always given to element temperatures. The equivalent concentrated thermal loads are calculated as follows

{ P ei } =

* [ be ]

T

[ E e ] { e ei } d Ω

(Eq. 10-9)

Ω

where Ω is the region of the element and the initial strains, { eei},due to thermal effects and are

{ e ei } = { α }∆ T

(Eq. 10-10)

where {α} is the set of thermal coefficients defined on an appropriate MATi data statement, as described in Chapter 9. If grid point temperatures are used to calculate the thermal strains, then the element temperature is assumed to be the average of the temperature of the connected node points for the lineal elements. In this case the thermal strains for an element are constant over the element and are given by:

{ e ei } = { α } T av e

(Eq. 10-11)

The effective thermal loads are then given by T

{ P ei } = T a ve { α }

T

* [ E ][ bc ] d Ω

(Eq. 10-12)

Ω

The temperature at interior points of the isoparametric elements is found by interpolation of the temperatures at the corner nodes.

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External Loads Grid Point Temperatures - TEMP and TEMPD

10.15 Grid Point Temperatures - TEMP and TEMPD The grid point temperature can be specified by means of the TEMP and TEMPD data statements, which are shown in Bulk Data Image 1-13. 1 TEMP

2 SID

TEMPD SID1

3

4

5

6

7

8

G1

T1

G2

T2

G3

T3

T1

SID2

T2

SID3

T3

SID4

9

10

T4

where: Field

Description

SID

Identification number of the temperature set.

G,T

Pairs that define the temperature T (real) at grid point G (integer).

SID, T

Pairs on the TEMPD statement defining default values for grid point temperatures which have not been specified on a TEMP statement for temperature set SID.

Bulk Data Image 1-13 Specification of Grid Point Temperatures

The TEMP statement specifies a scalar value of temperature at grid points, and the TEMPD data statement specifies a default temperature for all grid points whose temperatures are not defined by a TEMP statement. The grid point temperatures may be calculated in a separate execution of NASTRAN by specifying the HEAT approach in Executive Control. This feature is especially attractive since the grid point temperatures are calculated using basically the same finite element model as that used for the structural analysis.

10.16 . Thermal Field for Axial Elements - TEMPRB The temperature is assumed to vary linearly over the length of an axial element. If only axial thermal strains are of interest then it may be reasonable to define grid point temperatures using the TEMP data statement described in the previous section. For elements that exhibit bending behavior the situation is not quite so simple since the temperature gradients through the thickness produce thermal moments. The effective thermal gradients in the bending elements are calculated using the following relations: 1 T y = ---- zT ( y, z ) d A I y

*

A

1 T z = ---- yT ( y, z ) d A I z

(Eq. 10-13)

*

A

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External Loads Thermal Field for Surface Elements

where T(y,z) is the temperature distribution over the plane normal to the neutral axis, and I y and Iz are the area moments of inertia of the cross-section about the element y- and z-axes, respectively. In this case the temperature field is defined using the TEMPRB data statement, which is shown in Bulk Data Image 1-14. 1

2

3

4

5

6

7

8

9

EID1

Ta

T b

Tya

Tyb

Tza

Tzb

TCa

TDa

TEa

TFa

TC b

TD b

TE b

TF b

EID2

EID3

EID4

EID5

EID6

EID7

-etc.-

EID j

THRU

EIDK

TEMPRB SID

10

Alternate form of second continuation

EID2

THRU

EIDi

where Field

Description

SID

Temperature set identification. (integer, > 0)

ElDi

Unique element identification numbers of elements with specified temperature field. (integer, > 0)

Ta, T b

Average temperature at ends a and b, respectively. (real)

Tya, Tyb, Tza, Tzb

Effective thermal gradient for calculation of thermal moment in a bending element. (real)

TCi, TDi, TEi, TFi

Temperatures at points C, D, E, and F at end i (i = a, b) at which stresses are to be recovered in a bending element. (real)

Bulk Data Image 1-14 TEMPRB - Temperature Field for Axial Elements

The bending stresses in a BEAM-type element that result from the mechanical loads are modified by the element temperatures if at least one of the temperatures TC a, TDa, etc. are different from zero. The thermal stresses at end a are then given by:

∆σ c = – E α ( TC a – T ya CY a – T za CZ a – T a ) ∆σ d = – E α ( TD a – T ya DY a – T za DZ a – T a ) ∆σ e = – E α ( TE a – T ya EY a – T za EZ a – T a )

(Eq. 10-14)

∆σ f = – E α ( TF a – T ya FY a – T za FZ a – T a ) where the material properties are defined using MAT1 data statement and where the distances to the four points C, D, E and F on the cross-section are specified on an appropriate property statement. The thermal stress at end b of the element are given by a set of equations that are similar in form to (Eq. 10-14).

10.17 Thermal Field for Surface Elements For two-dimensional elements that have bending behavior, a thermal moment will result if the temperature distribution through the thickness is an odd function of the thickness relative to the plate neutral surface. The user may specify the effect of the temperature variation through the thickness by specifying the effective gradient:

282 Nastran Primer


1 T ′ = --- zT ( z ) d z I

*

(Eq. 10-15)

t

on a TEMPP1 data statement as shown by Bulk Data Image 1-15 1

2

TEMPP1 SID

EID2

3

4

5

6

7

EID1

TBAR

TPRIME T1

T2

EID3

EID4

EID5

EID6

EID7

8

9

10

-etc.-

alternative form for second continuation

EID2

“THRU EIDI ”

EIDj

THRU

EIDk

1

THRU

30

THRU

61

10

where

SID

Load set identification. (integer, > 0)

ElDi

Element identification numbers. (integer, > 0)

T

Average temperature of the element. (real)

T1, T2

Temperatures at the upper and lower surfaces used only for stress recovery. (real)

T’

Effective thermal gradient. (real)

Bulk Data Image 1-15 TEMPP1 - Thermal Gradients for Plate

Nastran Primer 283

10

284 Nastran Primer


11

Static Analysis

Solution 1 and 101 are used for linear static analysis based on the finite element method in Legacy and MSC NASTRAN, respectively. These solutions supports all of the modern NASTRAN features including: • Modern element library • Constraint and rigid elements • Automatic purging of unconnected degrees of freedom • Distributed surface tractions on two and three dimensional elements • Inertia relief

Other solution sequences for static analysis include nonlinear strain displacement relations or nonlinear material behavior and are therefore beyond the scope of the present text.

11.1

System Matrices The system matrices can include the mass matrix, [ M gg ] , as well as the stiffness matrix, [ K gg ] and the load vector [ P g ] . These matrices can be • Generated from a finite element model as described in previous chapters • Specified directly using, Direct Matrix I nput (DMIG) • Specified using mass elements described in this chapter • Specified element stiffness using GENEL.

The system matrices associated with the finite element model can be defined and calculated by NASTRAN by • Defining degrees of freedom using GRID and SPOINT data statements as described in Chapter 6. • Defining structural element connectivity and properties using the recommended elements described in Chapter 5. • Defining material properties using MAT-type data statements described in Chapter 10. • Defining inertia properties as described in this chapter.

11.1.1

Direct Specification of System Matrices

The stiffness and mass matrices associated with g -set of displacements are of the following form:

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11

Static Analysis System Matrices

1

2

[ K gg ] = [ K gg ] + [ K gg ] 1

2

(Eq. 11-1)

[ M gg ] = [ M gg ] + [ M gg ] 1

2

{ P g } = { P g } + { P g } where the subscript, g , refers to the all inclusive set of displacements defined on Grid and 1 1 SPOINT data statements; and, where [ K gg ] and [ M gg ] are the stiffness and mass matrices 2 2 generated using the finite element model and where [ K gg ] and [ M gg ] are stiffness and mass coefficients which can be defined directly by using the DMIG data statement shown by Bulk Data Image 11-1. 1

2

3

4

5

6

7

8

9

10

*Header statement DMIG

NAME 0

FORM

TIN

TOUT

0001

*Column definition statement

NAME Gcol

Ccol

Gr1

Cr1

REr1,col IMr1,col 0002

+001

Gr2

Cr2

REr2,col IMr2,col Gr3

Cr3

REr3,clo IMr3,col 0003

+002

Gr4

Cr4

REr4,col IMr4,col etc.

DMIG

0004

where: Field

Description

NAME

The user-defined character tag for the matrix data block. (Unique Character string, 1 to 8 characters)

FORM

Form code for matrix data block as defined below. (integer > 0)

TIN

Type of matrix element defined in DMIG as described below. (integer > 0)

TOUT

Type of matrix to be generated as described below. (integer > 0)

Gcol, Ccol

Column number defined by grid number and a single degree of freedom code. (Integers)

Gri, Cri

Row number defined by grid number and a single degree of freedom code, (Integers)

REr1,col, IMr3,col

Real and imaginary components of matrix coefficient associated with row (Gri, Cri) in column (Gcol, Ccol). (Real or double precision)

Bulk Data Image 11-1 DMIG - Direct Matrix Input

The matrix data block is associated with an internal name by the ‘NAME’ field which is a character string that must begin with an alphabetic character. The matrix is then specified by • A header data statement having a zero, (0), in field 3 and where

FORM

Matrix form defined by Table 11-2. (integer >0)

TIN

Type of input defined by Table 11-3. (integer >0)

TOUT

Type of output defined by Table 11-3 (integer >0)

• A DMIG statement having the same NAME as the header for each non-null column

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Static Analysis System Matrices Table 11-2 Matrix FORM Codes FORM CODE

MATRIX FORM

1

Square but not Symmetric

2

General Rectangular

3

Diagonal (N = 1, M = no. of rows)

4

Upper Triangular Factor

5

Lower Triangular Factor

6

Symmetric

7

Row (M = no. of Columns, N = 1)

8

Unit (M = no. of rows, N = 1)

Table 11-3 Matrix Type Codess TYPE CODE

NUMERICAL REPRESENTATION OF MATRIX

0

Pick best for machine configuration

1

Real, single precision

2

Real, double precision

3

Complex, single precision

4

Complex, double precision

Each nonzero coefficient in the column must be specified by identifying the row associated with the element followed by the value of the matrix element. DMIG has the general capability of defining complex matrix data blocks so that the row designation is followed by two fields, the first, RE ij, is the real part and the second, IM ij, is the imaginary part. Since only real matrices are used in static and normal modes analysis the second field must be left blank. The more general capability for defining complex matrices is described in Bulk Data Image 11-1. 2

2

The matrices defined on DMIG Bulk Data can be selected as K gg or M gg type input by the following Case Control Directives 2

• M2GG= for [ M gg ] input 2

• K2GG= for [ K gg ] input

where is the NAME assigned to the matrix data block on the DMIG statement. Consider the specification of the stiffness matrix for the cantilever beam shown by Figure 11-1, for example z 2 EI = 1 1

2

3 x

l = 1

l = 1

Figure 11-1 Cantilever Beam Defined by DMIG

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Static Analysis System Matrices

The stiffness matrix for the beam was determined for a bending in the x-y plane in Chapter 5, (Eq. 4-26). It is not difficult to show that the stiffness matrix associated with bending in the x-z plane is: 6 – 3 l –6 – 3 l 2

2

2 EI –3 l 2 l 3 l l [ k ee ] = --------3

(Eq. 11-2)

–6 3 l 6 3 l

l

2

2

– 3 l l

3 l 2 l

Substituting the values for l and EI gives 1 – 3 – 6 – 3 [ k ee ] = –3 2 3 1 –6 3 6 3 –3 1 3 2 Using the techniques described in Chapter 6 to assemble the system matrices and apply the displacement constraints then leads to the following 4 x 4 matrix for the solution set displacements 12 [ K ll ] = 0 – 6 – 3

0 –6 – 3 4 3 1 3 6 3 1 3 2

where for the purpose of this example the g -set includes the following physical degrees of freedom:

{ u g } = u z 1 θ y 1 u z 2 θ y 2 u z 3 θ y 3

T

and the l -set is:

{ u l } = u z 2 θ y 2 u z 3 θ y 3

T

The stiffness matrix could then be defined by the following data statements. 1

288 Nastran Primer

2

3

4

5

6

7

8

9

10

GRID

2

0

1.

0.

0.

GRID

3

0

2.

0.

0.

0001

DMIG

BEAM 0

6

1

0

0002

DMIG

BEAM 2

3

2

3

12.

0003

+003

3

-6.

3

5

-3.

0004

DMIG

BEAM 2

5

2

5

4.

0005

+005

3

3

3.

3

5

1.

0006

DMIG

BEAM 3

3

3

3

6.

0007

+007

3

3.

3

5

0008

Static Analysis Constraint and Static Condensation

1 DMIG

2

3

BEAM 3

4

5

5

6

3

7

5

8

9

2.

10

0009

where NAME = BEAM. The first DMIG, statement 0002 in field 10, is the header specifying a symmetric matrix, (FORM = 6), whose elements are real single precision, (TIN = 1), whose output type will be appropriate for the type of computer (TOUT = 0). 1. The specification of symmetric matrix on DMIG implies that a given off-diagonal element can be input either above or below the diagonal, but not both. Thus only the diagonal and either upper or lower off-diagonal elements are to be entered for a symmetric matrix. 2. The external column and row codes represent the non-null degrees of freedom in the displacement set. 3. The stiffness matrix would be selected by the following Case Control Directive K2GG = BEAM

11.2

Constraint and Static Condensation The stiffness matrix associated with the set of all grid and scalar degrees of freedom can be modified by means of various constraint and partitioning operations. These include 1. Specification of multipoint constraints Multipoint Constraints - MPC (p. 175)) 2. Specification of single point constraints Single Point Constraints - SPC (p. 184) 3. Static Condensation Static Condensation -OMIT and ASET (p. 189) 4. Support of free bodies Support for Free Bodies - SUPORT (p. 193) 5. Specification of matrix elements using Flexibility to Stiffness Transformation (p. 196) The set of displacements remaining after these operations is called the l -set which is the solution set for static analysis.

11.3

Static Loads The set of static loads that is applied to the grid and scalar points can result from a variety of sources, including 1. Concentrated forces and moments applied directly to the grid points Concentrated External Forces at Grid Points (p. 264) 2. Concentrated loads applied to scalar points Load Applied to Scalar Point - SLOAD (p. 268) 3. Distributed pressure load on one-, two-, and three-dimensional elements Uniform Normal Pressure - PLOAD2 (p. 271) 4.Gravity loads Gravity Loads - GRAV (p. 274) 5. Centripetal acceleration Rotation About an Axis- RFORCE (p. 275) 6. Enforced boundary displacement Enforced Displacements - SPCD (p. 278) 7. Enforced axial deformation Enforced Deformation - DEFORM (p. 279) 8. Temperature field Grid Point Temperatures - TEMP and TEMPD (p. 281)

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Static Analysis Inertia Relief

The set of node point forces from all these effects is reduced to the l -set of displacements by using the transformation defined by the constraints and partitioning specifications. The resulting set of equations is of the form

[ K ll ] { ul } = { P l }

(Eq. 11-3)

This set of equations is solved by first decomposing the stiffness matrix into its lower and upper triangular factors. A forward-backward substitution is then performed for all load subcases which have the same set of constraints.

11.4

Inertia Relief A free body may be subject to a set of external forces which are not in static equilibrium and which therefore produce rigid body accelerations. If the rate of change of the external forces is sufficiently small compared to the lowest natural frequency, we may consider the external forces to be in equilibrium with inertia forces. The analyst may choose to apply a set of forces that represents a non equilibrium set. The analyst may be interested in the effect of the inertia forces. In this case, a condition must be defined in which the effects of the inertia forces are not present, hence the term "inertia relief." The general procedure for removing the effects of inertia is 1. Select a set of determinate support points, e.g., the r -set, which is defined using a SUPORT data statement 2. Determine the rigid body acceleration of the r -set from the loads applied to the a-set of degrees of freedom 3. Calculate the rigid body accelerations of the l -set of degrees of freedom 4. Subtract the inertia forces that result from accelerations of the f -set from the external load, { P l }, and solve for the set of displacements obtained by constraining the r -set. The forces of reaction of the r -set of degrees of freedom will be zero. The equations of motion that include inertia relief are obtained by noting that the a-set is partitioned into the l - and r -sets where the degrees of freedom in the r -set are just sufficient to remove rigid body motion. The l - and r -sets are then related by the rigid body transformation matrix defined by (7-63). The reduced mass matrix which is obtained by applying the same set of transformations that were used to reduce the stiffness is then obtained by partitioning the a-set of equations into the l - and r -sets as:

[ M aa ] =

[ M l l ] [ M lr ] [ M rl ] [ M rr ]

(Eq. 11-4)

˜

The reduced mass matrix is obtained by means of the transformation: T

[ M rr ] = [ G ar ] [ M aa ] [ G ar ]

(Eq. 11-5)

Where the transformation matrix relates the a- and r -sets and is expressed in terms of the rigid body transformation matrix (Eq. 6-21) as:

{ ua } =

290 Nastran Primer

[ G lr ] [ I rr ]

{ ur } = [ G ar ] { ur }

(Eq. 11-6)

Static Analysis Data Recovery

The reduced mass matrix then becomes: T

T

T

[ M rr ] = [ M rr ] + [ M lr ] [ G lr ] + [ G lr ] [ M lr ] + [ G lr ] [ M ll ] [ G lr ]

(Eq. 11-7)

˜

The set of forces associated with the r -set is reduced in a similar way to give: T

{ P l } = [ G lr ] { P a }

(Eq. 11-8)

The equations of motion for the supported r -set can be written as:

[ M rr ] { u··r } = { P r }

(Eq. 11-9)

The accelerations of the r -set can now be determined as: – 1

{ u··r } = – [ M rr ] { P r }

(Eq. 11-10)

and by using (Eq. 6-21), the accelerations of the l -set become:

{ u··l } = [ G lr ] { u··r }

(Eq. 11-11)

The inertia forces associated with the l -set can now be determined in terms of the partitions of the matrix as:

{ ql } = – [ M ll ] { u··l } – [ M lr ] { u··r }

(Eq. 11-12)

where { q l } are the inertia forces. These forces can be expressed by using (Eq. 11-10) and (Eq. 11-11) as: – 1

{ q l } = – ( [ M ll ] [ G lr ] + [ M lr ] ) [ M rr ] { P r }

(Eq. 11-13)

The inertia forces are then added to the set of applied loads when inertia relief effects are desired.

11.5

Data Recovery The displacement sets associated with the various constraint and static condensation operations are recovered to obtain the g -set. The details concerning this recovery process are described in Chapter 6. Once the displacements have been recovered, the element forces, stresses, forces of constraint, and strain energy are selectively calculated based on user-supplied Case Control Directives that are described in Chapter 1.

11.6

Input Specifications 11.6.1

Executive Control Section

The Executive Control section is described in Chapter 1. The user specifies static analysis by means of the SOL statement. A minimum Executive Control deck includes SOL

101

CEND

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Static Analysis Input Specifications

11.6.2

Case Control Section

The Case Control section for static analysis and static analysis with inertia relief are similar. The Case Control directives that apply to both are: 1.Output MUST be requested in Case Control. The following are appropriate for Static Analysis

Directive

Action

DISP

Output a set of displacement as specified by a preceding SET directive or the set ALL to write all displacements on the selected file (the .out file by default or the PUNCH file.)

ESE

Calculate and write element strain energy to the selected file (the .out file by default or the PUNCH file.)

GPFORCE

Calculate and write grid point force balance to the selected file (the .out file by default or the PUNCH file.)

MPCFORCE

Calculate and write MPC forces at the grid points to the selected file (the .out file by default or the PUNCH file.)

OLOAD

Write the input forces at the grid points to the selected file (the .out file by default or the PUNCH file.)

SPCFORCE

Calculate and write the output to the selected file (the out file by default or the PUNCH file.

STRAIN

Calculate the element strain for the set of elements specified by a preceding SET directive. The element stresses and dispalcements are written to the .out file and also to an external file called .in where is the name of the input file with no extension. The strain values at the grid points are the volumeweighted average of the stress components for all elements which connect at a grid point.

STRESS

Calculate the element stress for the set of elements specified by a preceding SET directive. The element stresses and dispalcements are written to the .out file and also to an external file called .in where is the name of the input file with no extension. The stress values at the grid points are the volumeweighted average of the stress components for all elements which connect at a grid point.

2.A separate subcase must be defined for each unique combination of constraints and static loads 3. A static load must be defined above the subcase level with a LOAD, TEMPERATURE (LOAD), DEFORM, or SPC Case Control Directive. Loads defined within a subcase supersede the load defined above the subcase level. If loading results from SPC sets, then at least one specified displacement degree of freedom must be nonzero. 4. An SPC set must be specified for each subcase unless the body is a properly supported free body, or all constraints are specified on GRID data statements, scalar connections (i.e., springs to ground) or with the general element.

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Static Analysis Solution Sequence Output

5. Loading conditions associated with same set of constraints should be contiguous subcases. 6. The REPCASE directive may be used to allow multiple data recovery specifications for the same displacement set. 7. A single set of direct input matrices may be selected with K2GG, M2GG and P2G Case Control directives above the subcase level. For inertia relief, a SUPORT Bulk Data entity must be selected to remove at least one rigid body degree of freedom. Constraints can be used to remove some, but not all, of the rigid body degrees of freedom. If the SUPORT data statement is present then the mass matrix is required. An error condition exists if the mass matrix is null.

11.6.3

Parameters

The parameters which are useful for static analysis are described in Appendix B. They include • GRDPNT -Turns on the weight generator. • WTMASS -Conversion factor applied to mass matrix. • AUTOSPC - Automatically purges unconnected degrees of freedom. • COUPMASS -Optionally calculates consistent rather than lumped mass

11.7

Solution Sequence Output Although most of the solution sequence output which describes the structural behavior is optional and is requested in the Case Control section, some of the output is automatic or under control of DIAG requests in the Executive Control section. The printer output is designed for 132 characters per line, with the lines per page controlled by the LINE directive in Case Control. The default is LINE = 50. Optional titles defined by TITLE, SUBTITLE, and LABEL Case Control Directives are printed at the top of each page. Titling directives may be defined at the subcase level. The pages are automatically dated and numbered, and the NASTRAN generation date is specified. All of the output from data recovery and plot modules is optional, and its selection is controlled by appropriate Case Control directives. The details of making printer selections in Case Control are described in Section 1.11 for printer and punch file output. Detailed information on the force and stress output available for each element type is described in Chapter 6. A few printer output items are under the control of PARAM statements. The use of PARAM statements is described in Appendix B. The DIAG Executive Control directive is used to control the printing of the compiled DMAP (DIAG 14) and executive tables such as the File Allocation Table (FIAT) and the Operation sequence array (OSCAR). The use of the DIAG statement is described of the Chapter 1.

11.7.1

Automatic Output

The first part of the output for a NASTRAN run is prepared during the execution of the Preface, prior to the execution of the DMAP sequence. The following output is either automatically or optionally provided during the execution of the Preface: 1. NASTRAN title page - automatic. 2. Executive Control echo - automatic.

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Static Analysis Fatal Errors

3. Case Control echo - automatic. 4. Unsorted Bulk Data echo - optional, selected in Case Control using ECHO = UNSORT. 5. Sorted Bulk Data echo - automatic, unless suppressed using ECHO = NONE in Case Control. The Grid Point Singularity Table is automatically output if essential singularities are identified in the stiffness matrix. This table lists singular degrees of freedom in the displacement coordinate system. These singular degrees of freedom are automatically purged unless inhibited by the user by an AUTOSPC as described in Grid Point Singularity Processing (p. 295).

11.7.2

Output of System Response Variables

NASTRAN is designed to solve large problems which may include several tens of thousand variables and can produce a very large amount of data. The data base for large problems may be several gigabytes. Data recovery and printout is therefore performed under user control. The output options provided as a part of the static solution include the following Case Control directives 1. Displacements at selected GRID points - DISP 2. Nonzero components of applied loads - OLOAD 3. Nonzero components of single point constraint forces - SPCF 4. Grid point force balance of selected points - GPFO 5. Element forces in selected elements - ELFO 6. Stress in selected elements - STRESS 7. Strain energy in selected elements - ESE

11.8

Fatal Errors There are two classes of fatal errors in NASTRAN: 1. Those that occur during the execution of a DMAP module 2. Those which are associated with a logical inconsistencies during execution of the DMAP instructions The first type of fatal error is identified with a message in the printout and is associated with a message number. The message numbers are assigned in groups as follows 1-1000

Errors found in the form of data deck (termed Preface Errors)

1001-200 0

Errors found by the executive DMAP modules

2001-899 9

Errors found by the functional modules

For example, the message *** USER FATAL ERROR 2025, UNDEFINED COORDINATE SYSTEM 102

means that the user has made an error. A reference has been made to coordinate system 102 but that coordinate system has not been defined on CORD-type data statement.

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Static Analysis Grid Point Singularity Processing

All user and system errors are tabulated by message number in Ref. [2] In most cases the description of the errors in the reference is more informative than the terse message which is printed in the output. The other class of error is associated with logical errors in the DMAP Solution Sequence which sets the values of certain variable parameters to cause a jump to an error condition in the DMAP sequence. The contents of the variable parameter table, which includes the values of all variable parameters, is then printed and NASTRAN terminates normally. In these cases the fact that the variable parameter table has been printed is the clue which tells us that an error condition has been found. The variable parameters which control this action, and the associated logical error are described by Table 11-4. Parameter Name

Meaning

NOL

There are no independent degrees of freedom in the f-set

NOMGG

The mass matrix has not been generated and is required

NOKGG

The stiffness matrix has not been generated

SING

Matrix decomposition of a singular matrix has been attempted

NOGPDT

No grid or scalar points have been defined

TEST

An incompatibility exists between a SOL subset and the Case Control looping directives Table 11-4 Variable Parameters Used for Error Exit in Statics

The diagnostic messages may be either user or program diagnostics that contain warnings, information, or indication of fatal errors. These messages are usually identified by number and are documented in the NASTRAN User’s Manual. These diagnostic messages are identified by three asterisks that precede the message. The messages are self-explanatory and are thus not included in this text. In addition to the numbered diagnostic messages there are a few self-explanatory messages, such as the time required for matrix decomposition and the grid point singularity table if potential singularities are found.

11.9

Grid Point Singularity Processing The assembled [ K gg ] stiffness matrix will be singular since it contains rigid body motion and is therefore rank deficient. Hopefully proper use of SPC, MPC and SUPORT will remove of rigid body motion. But even then it may happen that the run fails during the solve operation due to rank deficiency. This could be caused by a mechanism in the model but is more than likely caused by the presence of one or more zero terms on the diagonal. The objective of grid point singularity processing is to detect and to allow the user to cause the singular degree of freedom to be removed. The details of the process used by Legacy NASTRAN and the MSC and NX versions to detect singularities is different. In both cases the 3 by 3 submatrices of the translational and rotational degrees of freedom are formed and each submatrix is tested for rank. Those degrees of freedom that are identified as potentially singular are identified. Those that are not removed by SPC, MPC or other means are flagged as being singular and unless removed will lead to failure during the solve operation. The user control of this process for Legacy NASTRAN and MSC Nastran are described in following sections.

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11.9.1

Legacy NASTRAN

If singularities are detected, the resulting action is controlled by an integer parameter called AUTOSPC. (A character parameter of the same name is used by MSC NASTRAN as described below.) The action taken is then dependent of the value of AUTOSPS as described by Table 11-5: Table 11-5 AUTOSPC Actions Value

Action

0 (Default)

Write a table of singular degrees of freedom and terminate

>0

Write a list of SPC1 Bulk Data that will be added to the current sset and proceed

<0

Write a list of SPC1 Bulk Data that will be added to the current sset, and, in addition, write the spc1 to the punch file before proceeding

11.9.2

MSC and NX NASTRAN

The most common modeling situation that leads to singular degrees of freedom occurs when the elements chosen to represent the structure do not connect all of the structural degrees of freedom at a grid point. The criteria used to identify singular degrees of freedom are thus biased toward that situation. The user should carefully review the list of singularities which are identified and removed from the analysis set. The local grid point singularities are identified by examining the stiffness matrix, [ K gg ] , by the following procedure. 1. Calculate the principal stiffness, K i , and principal direction of stiffness, { d i } , for each 3 x 3 diagonal partitions of [ K gg ] . 2. Normalize the three principal stiffness as follows K i Ri = ----------- K ma x

if

K ma x > 0 i = 1, 2, 3

R i = 0

if

K ma x = 0

(Eq. 11-14)

3. Compare the ratio, Ri , to a small positive number, EPZERO, which is defined by a parameter If Ri < EPZERO then a potential singularity exists and can be removed by including the degree of freedom having a direction closest to the principal direction, { d i } , in the s-set. The list of potentially singular degrees of freedom is then compared with the displacement set membership as defined by MPCs and SPCs. A potentially singular degree of freedom which is not included in the m- or s-sets is then classified as a true singularity. Singular degrees of freedom are automatically purged by appending them to the user-defined s-set unless the AUTOSPC parameter is explicitly set to NO by a PARAM statement.

11.9.2.1

User Control

The user can exercise control over the identification of potential singularities as well as their removal by means of the following parameters.

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AUTOSPC Parameter YES

Automatic purge i.e. append identified singularities to the s-set. (Default)

NO

No purge

EPZERO Singularity test value 10.-8

(Default)

PRGPST Controls the printout of singularity messages YES

Print messages (Default)

NO

No messages

SPCGEN Controls SPC generation >0

Generate SPC Bulk Data on the system PUNCH file for all user defined and internally generated SPCs

0

No data statements generated (Default)

11.9.2.2

Automatic Output

The grid point singularity table is automatically printed if the associated control parameter PRGPST is YES. An example of the table which could result due to the definition of a planar bending problem is as follows: GRID POINT SINGULARITY TABLE POINT ID

TYPE

Failed Direction

Stiffness Ratio

Old

New

USET

USET

1

G

1

0.

L

S

1

G

2

0.

L

S

1

G

4

0.

L

S

1

G

6

0.

L

S

2

G

1

0.

L

S

2

G

2

0.

L

S

2

G

4

0.

L

S

2

G

6

0.

L

S

3

G

1

0.

L

S

3

G

2

0.

L

S

3

G

4

0.

L

S

3

G

6

0.

L

S

This table contains one line for each degree of freedom which is not elastically connected and which has not been explicitly purged by SPC-type input. Degrees of freedom which have been purged by the action of PARAM, AUTOSPC, YES are identified by an asterisk following the last "New Uset" entry. The list can be lengthy if, for example, AUTOSPC is used to purge unconnected degree of freedom for a solid element model.

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11.9.3

Displacement Set Membership

It is often necessary for the analyst to know the displacement set membership of each degree of freedom and the internal equation number associated with each degree of freedom in a specific set. For example, it is inconsistent and improper to define a degree of freedom in more than one of the removed displacement sets. That is, with reference to Table 6-6, a degree of freedom cannot be defined in more than one of the m-, s-, o-, or r -sets. If the analyst attempts to do so the following error message will be printed ***USER FATAL MESSAGE 2101A, GRID POINT 21 COMPONENT 3 ILLEGALLY DEFINED IN SETS M AND O

The user control for printing the Set Membership Tables for Legacy and MCSC and NX NASTRAN is decrined ine the following sections.

11.9.3.1

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Legacy NASTRAN


The print of the set memebership tables is controles by DIAG 21 and DIAG 22 as descrined in the follwing table:

DISP

Action

21

Print a table whoserows are degrees of freedom and whose columns are: Internal dof numbers External GRID and dof code SAUTO - S-set dof generated by GPSP SB - SPC Bulk Data SG - Permanent SPC L - Set of solution set dof A - Retained set associated with ASET/OMIT F - Free set N - MPC Independent set G - Set of all dof O - Omitted set S - Union of all SPC sets M - Dependent set on MPC

22

Contents of various displacement sets

The DIAG 21 format produces a very long table since there is one line per degree of freedom. However, it is also relates the internal dof number to GRID dof.

11.9.3.2

MSC and NX NASTRAN

The set membership tables are printed where the user can optionally select the type and number of tables to be printed by means of the parameters USETPRT and USETSEL as follows: Parameter Name

USETPRT

USETSEL

Parameter Value

Action

0 (Default)

Print a table showing membership of each degree of freedom

1

Print selected displacement sets specified by USETSEL parameter

>1

Both of the above

1

M-set

2

S-Set

4

O-set

8

R-set

16

G-set

32

N-set

64

F-Set

128

A-Set

256

L-Set

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Static Analysis Grid Point Weight Generator

Parameter Name

Parameter Value

Action

512

SG-Set (PSPC on Grid statements)

1024

SB-Set (SPC statements)

647

Union of SB, M, O, and A sets

The USETSEL values define a bit position associated with a specific set. The above values are an abbreviated set which are appropriate for static analysis. For the complete set of values see the Quick Reference Guide.

11.10 Grid Point Weight Generator Weight and balance information can be calculated in any of the solution sequences for structural analysis. The calculation is requested by the inclusion of a GRDPNT parameter. The value of the parameter defines the reference GRID point. A value of zero (0) is used to specify the origin of basic coordinates as the reference point. A blank field for the value is not allowed. The grid point weight generator calculates the mass, center of gravity and mass moments of inertia for the finite element model of the structure. These data are calculated from the system mass matrix by a rigid body transformation about a reference point that is specified on the GRDPNT parameter.

11.10.1

Rigid Body Transformation Matrix

Let { u0 } represent the six degrees of freedom associated with the reference point. The rigid body transformation matrix,[ D g 0 ] ,that relates the displacement of the set of all grid point displacements to displacements of the reference point:

{ u g } = [ D g 0 ] { u 0 }

(Eq. 11-1)

where [ D g 0 ] is a g -by-6 matrix.

11.10.2

Rigid Body Mass Matrix

The rigid body mass matrix is calculated with respect to the reference point by the transformation: T

[ M 0 ] = [ D g 0 ] [ M gg ] [ D g 0 ]

(Eq. 11-2)

which results in a 6-by-6 rigid body mass matrix. The mass matrix may have directional properties because of scalar mass effects.

11.10.3

Principal Mass Axes

The set of grid point displacements is partitioned into translational (subscript t ) and rotational degrees of freedom (subscript r ) so that the mass matrix can be partitioned as:

[ M 0 ] =

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[ M tt ] [ M tr ] T

[ M tr ] [ M rr ]

(Eq. 11-3)

Static Analysis Example Problems

If the [ M t t ] matrix is not diagonal (i.e. if the elements of the off-diagonal terms are larger than some preassigned small value) then the principal masses and associated principal axes of the translational mass matrix are calculated. The transformation, [ S ] , from basic coordinates to principal directions is defined in terms of the normalized eigenvectors as:

[ S ] = { e 1 } { e 2 } { e 3 }

(Eq. 11-4)

Each of the partitions of [ M 0 ] is then transformed to the principal axes of [ M tt ] as follows: T

[ M tr ] = [ S ] [ M tr ] [ S ] T [ M rr ] = [ S ] [ M rr ] [ S ]

(Eq. 11-5)

T

[ M tt ] = [ S ] [ M tt [ S ] ] 11.10.4

Centroid

The position of the center of gravity may be different when taken with respect to different mass directions since the diagonal elements of the translational mass elements may be different if scalar masses are specified (see Specification of Element Inertia Properties (p. 320)). The location of the center of mass is therefore calculated with respect to the reference point for each mass direction.

11.10.5

Moments of Inertia

The moments of inertia with respect to the principal mass axes at the center of gravity are determined using the parallel axis theorem for each of the mass systems. The resulting inertia tensor is then labeled [ I ( [ S ] ) ] . The set of principal directions, [ Q ] , is then found so that the principal moments of inertia are given by: T

[ I ( [ Q ] ) ] = [ Q ] [ I [ ( S ) ] ] [ Q ]

(Eq. 11-6)

11.11 Example Problems General purpose finite element programs such as NASTRAN are designed to solve large problems represented by several thousand degrees of freedom. However, it is always a good idea to build up a base of experience and confidence in our computational tool by exploring all of its features and capabilities for the solution of relatively simple problems before moving to the complex world of real-life structures. We will, therefore, use NASTRAN to solve several problems for which solutions are known including: 1. A uniformly loaded cantilever beam 2. A stringer stiffened panel These problems will allow us to use- several of the elements and modeling features together with data recovery features in NASTRAN including • Use of BAR, BEAM, QUAD4 and HEXA elements • Use of single point constraints (SPC)

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Static Analysis Cantilever Beam with Uniform Load

• Use of PLOAD1, PLOAD2 and PLOAD4 • Grid point weight generator • Printing displacement set membership tables • Data recovery including • Node point displacements • Element forces and stresses • Element strains and curvatures • Single point Constraint forces

While no one example problem will incorporate all of these features, each will be included in at least one of the examples which are described in the following sections.

11.12 Cantilever Beam with Uniform Load The structural element for this example is shown by Figure 11-1.

y’

z

y

30 ° z‘

p = p o

1 h

2 L

4 3 b

Figure 11-1 Beam Element

Where the beam has the following geometric and elastic properties E = 30 x 106 psi

ν= 0.3 L = 10 in. h = 0.5 in.

γ = 0.281 Ibf /in.3 b = 0.25 in.

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x


and transverse shear coefficients, K y = K z = 0.833. The uniform applied loading, po = 10 lb./in., acts in a plane rotated 60 degrees from the Basic X-Z plane as shown. The element y-z coordinates of the cross section are chosen so that they coincide with the Basic Y-Z coordinates. The load is then defined either in terms of its components in the Basic Y and Z directions or in terms of components in the element y-z directions. For the purpose of this example the loads will be described as components referred to the Basic Y-Z coordinates. The loads are modeled using three load subcases using suitable SUBCASE directives a. Uniform load of magnitude posinθ in Basic Y -Direction using PLOAD1 with basic coordinate option (i.e., TYPE = FY). b. Uniform load of magnitude pocosθ in basic Z-direction using PLOAD1 with basic coordinate option (i.e., TYPE = FZ). c. Combination of two previous subcases using the LOAD data statement that combines the results of the two previous subcases to obtain response to actual load vector.

11.12.1

Element Properties

With reference toFigure 11-1, (Eq. 5-3) through (Eq. 5-5) are used to calculate the moments of inertia with respect to the principal axes ( y, z ). The principal moments of inertia are: I yy = 0.00260 in4 I zz = 0.00651 in4 I yz = 0 A = 0.125 in2

The orientation of element axes is described using the components of the orientation vector on the CBAR data statement. Taking the components as: {V } = {0., 1., 0.} will align the element y-axis with the Basic Y -axis. The area properties associated with the principal axes are then specified using the PBAR data statement. The CBAR and PBAR data statements for this case are: 1

2

3

4

5

6

7

CBAR

1

269

1

2

0.

PBAR

269

30

.125

6.51-4

2.604-4

+001

-.125

-.25

.125

-.25

.125

+002

.833

.833

8

1.

9

10

0.

0001 0002

.25

-.125

.25

0003 0004

where the stress recovery points at the corners and the transverse shear factors are specified on continuations.

11.12.2

Distributed Line Load

The distributed line load for this case is specified by two PLOAD1 as described above. The PLOAD1 together with a CBARAO to recover element force and stress output at 10 intermediate points along the length of the Bar for this case are: 1

2

3

4

5

6

7

8

9

PLOAD1

100

1

FY

FR

0.

5.

1.

5.

PLOAD1

200

1

FZ

FR

0.

8.6603

1.

86603.

10

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1

2

CBARAO

11.12.3

1

3

FR

4

9

5

6

.1

7

8

9

10

.1

The Input File

We will now create the complete NASTRAN input file that • Requests static analysis, solution 101 • Requests SPC set 23 which will define geometric boundary conditions at X = 0. • Purges unconnected degrees of freedom using AUTOSPC • Selects static load set 101 which defines a uniform load by means of PLOAD1 Bulk Data • Requests displacements, SPC forces, element forces, and stresses at corner points as output • Generates weight and balance information using GRDPNT parameter

The NASTRAN input is then 1

2

3

4

5

6

7

9

SOL 101 $ REQUESTS STATIC SOLUTION SEQUENCE

0001

TIME 100 $ SETS SOLUTION TIME TO 100 MINUTES

0002

CEND

0003

TITLE=UNIFORMLY LOADED CANTILEVER BEAM

0004

SUBTITLE= PRINCIPAL ELEMENT AXES

0005

SPC = 23

0006

DISP = ALL

0007

ELFO = ALL

0008

STRESS = ALL

0009

OLOAD = ALL

0010

SPCFORCE = ALL

0011

SUBCASE 1

0012

LOAD = 100

0013

SUBCASE 2

0014

LOAD = 200

0015

SUBCASE 3

0016

LOAD = 300

0017

BEGIN BULK

0018

PARAM

GRDPNT

0

0019

PARAM

AUTOSPC YES

0020

PARAM

USETSEL

0021

1746

GRDSET

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14

0022

GRID

1

0

10.

0.

0.

0023

GRID

2

0

0.

0.

0.

0024

CBAR

1

229

2

1

0.

1.

0.

0025


1

2

3

PBAR

229

30

+028

-.125

-.25

4

.125

5

6

6.5104-4

2.604-3

-.25

.125

7

8

9

0026 .25

-.125

.25

+029

0027 0028

MAT1

30

3.+7

+025

1.+4

1.+4

.5+4

SPC

23

2

2356

0.

PLOAD1 100

1

FY

FR

0.

5.

1.

5.

0032

PLOAD1 200

1

FZ

FR

0.

8.6603

1.

8.6603

0033

CBARAO 1

FR

9

.1

.1

LOAD

1.

1.

100

1.

300

.3

0029 0030 0031

0034 200

0035

ENDDAT A

0036

With reference to the statements numbers in field 10: 0001

SOL requests solution 101, Static analysis

0002

TIME specifies two minutes of cpu time for

0003

CEND terminates Executive Control

0004 - 0005

TITLE and SUBTITLE are optional titling directives

0006

SPC Specifies that SPC set 23 is to be used for all subcases

0007 thru 0011

Output requests for all subcases

0012 thru 0017

Load subcases

0018

BULK DATA terminates Case Control

0019

GRDPNT parameter requests calculation of weight and balance data with reference to the origin of the Basic coordinate system

0020

AUTOSPC set equal to YES will result in the automatic removal of singular degrees of freedom. This is the default value for the parameter so its presence is optional

0021

USETSEL will display the SB, SG, S, G, F and A sets

0022

GRDSET specifies default value for PSPC = 14 if the PSPC field of a Grid is left blank. This value will remove u 1 and u4 at the point.

0023 -0024

GRID points 1 and 2

0025

CBAR connects Grids 1 and 2 with a BAR bending element and aligns elemental y axis with the Basic Y axis. Points to PBAR with PID = 229

0026

PBAR specifies I yy = 2.604 x 10-4 in4, I zz = 6.5104 x 10 -4 in4, relative to element principal axes and points to material set 30

0027

PBAR continuation specifies the four stress recovery points, as shown by Fig. 12-2 in element coordinates

0028

Specifies transverse shear flexibility coefficients

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0029

MAT1 specifies an isotropic material with E = 30 x 10 6 psi and ν= 0.3

0030

Stress limits in tension, compression and shear

0031

SPC Specifies constraints to physical degrees of freedom at Grid point 1 (u x2 and θx2 are not connected since A = 0 and J = 0 on PBAR data statement). All unconnected degrees of freedom will be purged using AUTOSPC.

0032

PLOAD1 specifies uniform pressure load of 5 lb./in. in the Basic Y -direction along length of the BAR

0033

PLOAD1 specifies uniform pressure load of 8.6603 lb./in. in the Basic Z -direction along length of the BAR

0034

CBARAO requests calculation of element forces and stresses at 9 intermediate points along the length of the Bar

0035

LOAD combines the loads for subcases 1 and 2 for subcase 3

0036

ENDDATA terminates BULK DATA

11.12.4

Running NASTRAN

Running NASTRAN depends on the installation and the it’s dialect. NASA NASTRAN processes the external files differently than MSC NASTRAN. This means that a bash shell that defines file allocation is required for NASA NASTRAN. These details are taken care of using ASSIGN statement in MSC NASTRAN. • If using NASA NASTRAN make sure the script file in the BIN directory is PATHed and the enter script.sh barPA

• Create a script file in the PATHed directory. The script file, which might be named runit.bat contains the line of text: c:\msc\bin\nastran %1

The program could then be executed by entering: runit barPA.dat

Where it is assumed that the input file is named barPA.dat. In either case the results will be written in the directory in which the data file exists.

11.12.5

The Output File

The contents of the output file include echoes of the Executive Control and Case Control sections and the sorted bulk data which is not described here. The calculated output components are described below.

11.12.5.1

Grid Point Weight Table

The weight and balance information requested by the inclusion of the GRDPNT parameter. The units of the coefficients of the mass and mass moment of inertia matrices are in weight units because the weight per unit volume was specified on MAT1. The weight of the structure is the same in the three coordinate directions and is equal to 0.35375 lb. O U T P U T

F R O M

306 Nastran Primer

G R I D P O I N T W E I G H T REFERENCE POINT = 0 M O

G E N E R A T O R

Static Analysis Cantilever Beam with Uniform Load * * * * * *

3.512500E-01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00

0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 3.512500E-01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 3.512500E-01 0.000000E+00 -1.756250E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -1.756250E+00 0.000000E+00 1.756250E+01 1.756250E+00 0.000000E+00 0.000000E+00 0.000000E+00 S * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 *

DIRECTION MASS AXIS SYSTEM (S) MASS X 3.512500E-01 Y 3.512500E-01 Z 3.512500E-01 * 0.000000E+00 * 0.000000E+00 * 0.000000E+00 * 0.000000E+00 * * * * *

1.000000E+00 0.000000E+00 0.000000E+00

X-C.G. Y-C.G. 0.000000E+00 0.000000E+00 5.000000E+00 0.000000E+00 5.000000E+00 0.000000E+00 I(S) 0.000000E+00 0.000000E+00 * 8.781250E+00 0.000000E+00 * 0.000000E+00 8.781250E+00 * I(Q) * 8.781250E+00 * 8.781250E+00 * Q 0.000000E+00 0.000000E+00 * 1.000000E+00 0.000000E+00 * 0.000000E+00 1.000000E+00 *

0.000000E+00 1.756250E+00 0.000000E+00 0.000000E+00 0.000000E+00 1.756250E+01

* * * * * *

Z-C.G. 0.000000E+00 0.000000E+00 0.000000E+00

Figure 11-1 Grid Point Weight Table

11.12.5.2

Displacement Set Membership Tables

The displacement set membership for each degree of freedom can be printed using the USETPRT and USETSEL parameters. USETPRT controls whether the USET table is to be printed. Setting USETPRT to 12 requests two type of tables using external grid point identification numbers. The first type of table contains a row for each degree of freedom in the model and identifies the set membership for each degree of freedom. The headings for the columns in this table are described by Table 11-1. If the USETPRT parameter requests row tables this means that the degrees of freedom for specific sets are to be printed. Setting the USETSEL value equal to 1746 requests the S, SB, SG, G, A, and F sets. Table 11-1 Column Heading of Uset Table Column Heading

Description

EXT. GP. DOF

The external degree of freedom code. The first number is the grid or scalar point number, and the second is the degree of freedom code at the grid or scalar point.

INT DOF

The internal number for the degree of freedom

INT GP

The internal point number and type: G - Grid point S - Scalar point E - Extra point

EXT. GP. DOF

The external degree of freedom code. The first number is the grid or scalar point number, and the second is the degree of freedom code at the grid or scalar point.

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Table 11-1 Column Heading of Uset Table Column Heading

Description

SB

Single point constraints specified on SPC data statements

SG

Single point constraints specified on GRID Bulk Data data statements

L

Degrees of freedom in the l -set

A

Degrees of freedom in the a-set

F

Degrees of freedom in the f -set

N

Degrees of freedom in the n-set

G

Degrees of freedom in the g -set

R

Degrees of freedom in the r -set

O

Degrees of freedom in the o-set

S

Degrees of freedom specified in SG and SB sets

M

Degrees of freedom in the m-set

E

Degrees of freedom in the e-set

The information contained in these tables can be extremely valuable, especially when the user is attempting to find the cause of a singular stiffness or mass matrix.

11.12.5.3

Selected Output

The displacements, forces of constraint, grid point loads, forces in the BAR element, and the stress in the BAR element are shown below. The GRID point-oriented output such as displacements, grid point loads and SPC forces are sorted by GRID point number. The components are then printed under the headings T1, T2, T3 for displacement or force components and Rl, R2, R3 for rotation or moment components. The components are interpreted with respect to the displacement coordinate system which is associated with grid point displacements. Displacements - Subcase ‘ POINT ID. TYPE T1 1 G 2

T2

T3

0.0 G

0.0 0.0

R1

0.0 3.201971E-01

R2

R3

0.0 0.0

0.0 0.0

0.0 0.0

4.266677E-02

Displacements - Subcase 2 POINT ID.

TYPE 1 2

G G

T1 0.0 0.0

T2 0.0 0.0

T3 0.0 1.389341E-01

R1 0.0 0.0

R2 0.0 -1.847649E-02

R3 0.0 0.0

Displacements - Subcase 3 POINT ID. 1 2

TYPE G G

T1 0.0 0.0

T2 0.0 3.201971E-01

T3 0.0 1.389341E-01

R1 0.0 0.0

R2 0.0 -1.847649E-02

R3 0.0 4.266677E-02

Load Vector - Subcase 1 POINT ID.

TYPE

1 2

G G

T1 0.0 0.0

T2 2.500000E+01 2.500000E+01

T3 0.0 0.0

R1 0.0 0.0

R2 0.0 0.0

R3 4.166667E+01 -4.166667E+01

Load Vector - Subcase 2 POINT ID.

TYPE

308 Nastran Primer

T1

T2

T3

R1

R2

R3

Static Analysis Cantilever Beam with Uniform Load 1 2

G G

0.0 0.0

0.0 0.0

4.330150E+01 4.330150E+01

0.0 0.0

-7.216917E+01 7.216917E+01

0.0 0.0

Load Vector - Subcase 3 POINT ID. 1 2

TYPE G G

T1

T2

0.0 0.0

2.500000E+01 2.500000E+01

T3 4.330150E+01 4.330150E+01

R1 0.0 0.0

R2

R3

-7.216917E+01 7.216917E+01

4.166667E+01 -4.166667E+01

R2

R3

Forces of Single- Point Constraint - Subcase 1 POINT ID. 1 G

TYPE 0.0

T1

T2 -5.000000E+01

T3 0.0

R1 0.0

0.0

-2.500000E+02

Forces of Single- Point Constraint - Subcase 2 POINT ID. 1

TYPE G

T1

T2

0.0

0.0

T3 -8.660300E+01

R1 0.0

R2

R3

4.330150E+02

0.0

Forces of Single- Point Constraint - Subcase 3 POINT ID.

TYPE 1

G

T1 0.0

T2 -5.000000E+01

T3 -8.660300E+01

R1 0.0

R2 4.330150E+02

R3 -2.500000E+02

Force Distribution in BAR Elements - Subcase 1 0

ELEMENT STATION ID. (PCT) 1 0.000 1 0.100 1 0.200 1 0.300 1 0.400 1 0.500 1 0.600 1 0.700 1 0.800 1 0.900 1 1.000

BEND-MOMENT PLANE 1 PLANE 2 2.500000E+02 0.0 2.025000E+02 0.0 1.600000E+02 0.0 1.225000E+02 0.0 8.999999E+01 0.0 6.250000E+01 0.0 3.999998E+01 0.0 2.250000E+01 0.0 9.999990E+00 0.0 2.499978E+00 0.0 -3.814697E-06 0.0

SHEAR FORCE PLANE 1 PLANE 2 5.000000E+01 0.0 4.500000E+01 0.0 4.000000E+01 0.0 3.500000E+01 0.0 3.000000E+01 0.0 2.500000E+01 0.0 2.000000E+01 0.0 1.500000E+01 0.0 1.000000E+01 0.0 5.000000E+00 0.0 0.0 0.0

AXIAL FORCE 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

TORQUE 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0


ELEMENT ID. 1 1 1 1 1 1 1 1 1 1 1

STATION (PCT) 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000

BEND-MOMENT PLANE 1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

PLANE 2 4.330150E+02 3.507422E+02 2.771296E+02 2.121774E+02 1.558854E+02 1.082538E+02 6.928239E+01 3.897137E+01 1.732059E+01 4.330115E+00 7.629395E-06

SHEAR FORCE PLANE 1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

PLANE 2 8.660300E+01 7.794270E+01 6.928240E+01 6.062210E+01 5.196180E+01 4.330150E+01 3.464120E+01 2.598090E+01 1.732060E+01 8.660301E+00 0.0

AXIAL FORCE 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

TORQUE 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0


ELEMENT

STATION

ID.

(PCT) 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000

1 1 1 1 1 1 1 1 1 1 1

BEND-MOMENT PLANE 1 2.500000E+02 2.025000E+02 1.600000E+02 1.225000E+02 8.999999E+01 6.250000E+01 3.999998E+01 2.250000E+01 9.999990E+00 2.499978E+00 -3.814697E-06

PLANE 2 4.330150E+02 3.507422E+02 2.771296E+02 2.121774E+02 1.558854E+02 1.082538E+02 6.928239E+01 3.897137E+01 1.732059E+01 4.330115E+00 7.629395E-06

SHEAR FORCE PLANE 1 5.000000E+01 4.500000E+01 4.000000E+01 3.500000E+01 3.000000E+01 2.500000E+01 2.000000E+01 1.500000E+01 1.000000E+01 5.000000E+00 0.0

PLANE 2 8.660300E+01 7.794270E+01 6.928240E+01 6.062210E+01 5.196180E+01 4.330150E+01 3.464120E+01 2.598090E+01 1.732060E+01 8.660301E+00 0.0

AXIAL FORCE 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

TORQUE 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

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Stress Distribution in BAR Element - Subcase 1 S T R E S S D I S T R I B U T I O N I N B A R E L E M E N T S ( C B A R ) 0 ELEMENT STATION SXC SXD SXE SXF AXIAL ID. (PCT) 1 0.000 4.800012E+04 4.800012E+04 -4.800012E+04 -4.800012E+04 0.0 1 0.100 3.888010E+04 3.888010E+04 -3.888010E+04 -3.888010E+04 0.0 1 0.200 3.072008E+04 3.072008E+04 -3.072008E+04 -3.072008E+04 0.0 1 0.300 2.352006E+04 2.352006E+04 -2.352006E+04 -2.352006E+04 0.0 1 0.400 1.728004E+04 1.728004E+04 -1.728004E+04 -1.728004E+04 0.0 1 0.500 1.200003E+04 1.200003E+04 -1.200003E+04 -1.200003E+04 0.0 1 0.600 7.680017E+03 7.680017E+03 -7.680017E+03 -7.680017E+03 0.0 1 0.700 4.320011E+03 4.320011E+03 -4.320011E+03 -4.320011E+03 0.0 1 0.800 1.920003E+03 1.920003E+03 -1.920003E+03 -1.920003E+03 0.0 1 0.900 4.799969E+02 4.799969E+02 -4.799969E+02 -4.799969E+02 0.0 1 1.000 -7.324237E-04 -7.324237E-04 7.324237E-04 7.324237E-04 0.0

S-MAX 4.800012E+04 3.888010E+04 3.072008E+04 2.352006E+04 1.728004E+04 1.200003E+04 7.680017E+03 4.320011E+03 1.920003E+03 4.799969E+02 7.324237E-04

S-MIN -4.800012E+04 -3.888010E+04 -3.072008E+04 -2.352006E+04 -1.728004E+04 -1.200003E+04 -7.680017E+03 -4.320011E+03 -1.920003E+03 -4.799969E+02 -7.324237E-04

Stress Distribution in BAR Element - Subcase 2 0


SXC

SXD

4.157210E+04 3.367340E+04 2.660614E+04 2.037033E+04 1.496596E+04 1.039303E+04 6.651535E+03 3.741491E+03 1.662883E+03 4.157177E+02 7.324687E-04

SXE

-4.157210E+04 -3.367340E+04 -2.660614E+04 -2.037033E+04 -1.496596E+04 -1.039303E+04 -6.651535E+03 -3.741491E+03 -1.662883E+03 -4.157177E+02 -7.324687E-04

-4.157210E+04 -3.367340E+04 -2.660614E+04 -2.037033E+04 -1.496596E+04 -1.039303E+04 -6.651535E+03 -3.741491E+03 -1.662883E+03 -4.157177E+02 -7.324687E-04

SXF 4.157210E+04 3.367340E+04 2.660614E+04 2.037033E+04 1.496596E+04 1.039303E+04 6.651535E+03 3.741491E+03 1.662883E+03 4.157177E+02 7.324687E-04

AXIAL 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0



Stress Distribution in BAR Element - Subcase 3 0


SXC

SXD

SXE

8.957222E+04 6.428020E+03 7.255350E+04 5.206699E+03 5.732622E+04 4.113934E+03 4.389039E+04 3.149729E+03 3.224600E+04 2.314086E+03 2.239305E+04 1.607005E+03 1.433155E+04 1.028482E+03 8.061503E+03 5.785198E+02 3.582886E+03 2.571199E+02 8.957146E+02 6.427924E+01 4.499452E-08 -1.464892E-03

11.12.6

-8.957222E+04 -7.255350E+04 -5.732622E+04 -4.389039E+04 -3.224600E+04 -2.239305E+04 -1.433155E+04 -8.061503E+03 -3.582886E+03 -8.957146E+02 -4.499452E-08

SXF -6.428020E+03 -5.206699E+03 -4.113934E+03 -3.149729E+03 -2.314086E+03 -1.607005E+03 -1.028482E+03 -5.785198E+02 -2.571199E+02 -6.427924E+01 1.464892E-03

AXIAL 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0



Discussion of Results

The element forces are defined using the displacement sign convention rather than the equilibrium convention because PLOAD1 and CBARAO are present in the input file. The headings under Force Distribution in BAR Elements are: Element ID

The identification number of the element

Station PCT

The normalized length along the Bar

Moment

310 Nastran Primer

Plane 1 is Mz, the moment about the element z -axis Plane 2 is My, the moment about the element y-axis

Shear Force

Plane 1 is Vy, the transverse shear in the element y direction

Force

Fx, The force in the element x -direction

Axial Torque

Mx, the moment about the element x-axis

Plane 2 is Vz, the transverse shear in the element y direction


The headings under Stress Distribution are: Element ID

The identification number of the element

SXC

Bending stress at the four stress recovery points identified as points C, D, E, and F

SXD SXE SXF AXIAL

Axial stress, P/A

S-MAX

The maximum combined stress at the four stress recovery points

S-MIN

The minimum combined stress at the four stress recovery points

M.S

The margin of safety for stress based on stress limit defined by MAT1 Bulk DATA for the element

The displacement at the end of the Bar element from beam theory with no correction for transverse shear deflection is given by: 4

p 0 L δ = ----------8 EI

(Eq. 11-1)

where p0 is the magnitude of the uniform line load, L is the length of the Bar, E is the elastic modulus and I is the area moment of inertia of the cross section. Substituting the values used for this problem we find that δ y = 0.032 for p0Y = 5.0 lb/in and δ z = 0.01358 in. for p0Z = 8.6603 lb./in. You should recall that the stiffness for the Bar element was obtained by using the shape functions for a beam loaded with end loads. The PLOAD1 generates work equivalent loads which lead to the exact solution for a uniformly distributed load using only one element. Now looking at the transverse shear distribution we see that it varies linearly from a value of zero at the free end to p0 L at the fixed end which agrees exactly with equilibrium considerations. Similarly the moment at any cross section is given by: 1 2 M ( x ) = --- p 0 ( L – x ) 2

(Eq. 11-2)

Looking at the moment distribution, for loading in the Y -direction for example, it can be verified that the moment distribution agrees exactly with that predicted by Eq. 12-23. It is left for an exercise to verify that the results for other loadings supported by PLOAD1 result in exact agreement with beam theory for a single element.

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Static Analysis Simply-Supported Rib-Stiffened Plate

11.13 Simply-Supported Rib-Stiffened Plate The next example, which is a stiffened flat plate shown by Figure 11-1, will allow us to explore the use of the NASTRAN shell elements for modeling. 1.0 in.

1.0 in.

0.5 in.

0.5 in. z

0.4 in.

15 in. x

15 in. 15 in.

15 in.

y 6 E = 10 × 10 psi ν = 0.3 3 γ = 0.1 lb/in

Figure 11-1 Eccentrically Stiffened Plate

With reference to Fig. 12-4, we will consider the following solutions a) A simply supported flat plate subject to uniform pressure load without stiffeners using appropriate symmetry conditions to model 1/4 of the plate b)A simply supported plate with Bar stiffeners where the middle of the plate is used as the reference surface and where the stiffener is modeled with an offset Bar element.

11.14 Problems 6

[1] Given a uniform cantilever beam having and elastic modulus, E = 30 ×10 ps i .a cross section moment of inertia, I = 1 ⁄ 12 in4,a length of 10 in. and a load at the free end of 300 lb. Model the beam using DMIG and determine the displacement and slope at the free end normalized using the exact solution. [2] Repeat Problem 1 using the GENEL element [3] Given the rectangular plate described in Sec. 8.3.5, for the case where a ⁄ b = 1 and a meshing factor of 4, remove all displacements at the interior node points using static condensation. Determine the displacement at the center normalized using the exact value. [4] Model the simply supported rectangular plate having an aspect ratio, a ⁄ b = 5 , using Hexa elements with 8 nodes for a mesh density of 4 by 4 elements. Load the top of the plate with a unit uniform pressure using Pload4, Determine the normalized center displacement. [5] Repeat problem 4 using Hexa 20 elements. Determine the normalized center displacement and verify that the total load is correct. [6] Repeat problem 4 but load the plate having a weight density of 1 lb per sq. in. using the Grav load. Determine the normalized center displacement. [7] Determine the converged center displacement for the stiffened plate described in Sect. 12.11.2 [8] Determine the normalized tip displacement of the swept plate described by Cook (Ref. [16], Chapter 7) using a 5 by 5 mesh of Quad4 elements.

312 Nastran Primer

Static Analysis References

11.15 References [1] R.J. Roark, Formulas for Stress and Strain, Third Edition, McGraw-Hill, New York, 1954, p. 198. [2] Anon, NASTRAN Users Manual, NASA SP-222, Vol. I , 1986. Sec. 6

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314 Nastran Primer

Static Analysis References

12


The general problem of Structural Dynamic analysis is addressed by several authors including Bathe [1], Zienkiewicz [2]and Hughes [3] and is beyond the scope of the present text. We will restrict our attention in this chapter to the real eigenvalue problem. In succeeding sections we will present the equations of motion for dynamic systems and define the eigenvalue problem. We will then describe the important properties of normal modes. Finally, the various eigenvalue extraction techniques which are available in NASTRAN will be described.

12.1

Dynamic Motion The motion of structural dynamic systems is described by a set of equations expresses the balance between external applied loads, the internal forces and the inertial forces. For a structural system these forces are: the inertial forces, { f i } , the damping or dissipative forces, { f d } , the internal elastic forces, { f e } , and the external time-dependent forces, { P } . Using Newton’s second law of motion then leads to

{ f i ( t ) } + { f d ( t ) } + { f e ( t ) } = { P ( t ) }

(Eq. 12-1)

where the internal forces can be expressed in terms of displacements and derivatives of displacements as follows: ··

{ f i ( t ) } = [ M ] { u } ·

{ f d ( t ) } = [ B ] { u }

(Eq. 12-2)

{ f e ( t ) } = [ K ] { u } using these definitions for internal forces we can then write (Eq. 12-1) as: ··

·

[ M ] { u } + [ B ] { u } + [ K ] { u } = { P ( t ) }

(Eq. 12-3)

where the stiffness matrix, [ K ] , is the same as that for the static case. We also note that (Eq. 12-3) reduces to the static equilibrium equation when the time derivatives of ·· displacements, { u· } and { u } , are zero. The additional system matrices associated with a dynamic system are [ M ] , the system mass matrix and [ B ] , the system damping matrix. The solution of (Eq. 12-3) is beyond the scope of this text. However it is worthwhile to at least describe the solution methods which can be used in order to provide motivation for the topic of normal modes and frequencies. In our discussion we will assume that the dynamic model is based on the structural model used for a static analysis. This implies that the number of degrees of freedom can be large so that cost, measured in terms of time and computer resources, will be a factor in solving the set of dynamic equations.

315 Nastran Primer

12

Normal Modes Analysis The Eigenvalue Problem

Again, just thinking the problem through for a moment, we recognize that if solution time is a factor (and it is) then we should be able to reduce the solution time by reducing the number of degrees of freedom. The question is: How can the system of equations be reduced from some large number which was used for static analysis to a fairly small number which is appropriate for dynamic analysis. Let's be a little more specific. Without worrying at the present time about how [ M ] and [ B ] are defined, let's assume that MPC and SPC and OMIT operations have been applied so that the solution set is reduced to the a-set, which is the solution set for normal modes analysis. The differential equation, (Eq. 12-3) is then written using standard NASTRAN set notation as:

[ M aa ] { u··a } + [ Baa ] { u· a } + [ K aa ] { ua } = { P a }

(Eq. 12-4)

Now suppose we can find a transformation, [ G ah ] ,which relates { ua } to a greatly reduced set of degrees of freedom { uh } . That is:

{ ua } = [ G ah ] { uh }

(Eq. 12-5)

We can then use this transformation to reduce(Eq. 12-4)to the h-set resulting in:

[ M hh ] { u··h } + [ Bhh ] { u· h } + [ K hh ] { uh } = { P h }

(Eq. 12-6)

where: T

[ M hh ] = [ G ah ] [ M aa ] [ G ah ] T

[ Bhh ] = [ G ah ] [ B aa ] [ G ah ] T

(Eq. 12-7)

[ K hh ] = [ G ah ] [ K aa ] [ G ah ] T

{ P hh } = [ G ah ] { P a } There isn't any question about being able to define the transformation(Eq. 12-5). The real question is: what transformations would be useful and would retain the essence of the dynamic DNA, as it were. A particularly attractive one would be one for which the reduced system matrices [ M hh ] , [ Bhh ] , and [ M hh ] , are diagonal. The resulting set of equations, (Eq. 12-6), would be completely decoupled and their solution would be reduced to quadrature using the convolution integral which is described by Meirovich [4] and others. We will see in the next section that the eigenvectors for the real eigenvalue problem transform the mass and stiffness matrices to diagonal forms. The study of and solution for eigenvectors is thus of great importance in transforming large sets of coupled dynamic problems to a small set of uncoupled dynamic problems because of their orthonormal properties and the fact they form a basis in a-dimensional solution space. The calculation of the eigenvalues and eigenvectors is also of importance in its own right, of course, because they define the undamped free vibrational modes and frequencies of the system.

12.2

The Eigenvalue Problem The eigenvalue problem is associated with the solutions of the undamped (i.e., [ B ] = 0 ) unforced (i.e.,{ P } = 0 ) system of equations. We have then, from (Eq. 12-3):

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··

[ M ] { u } + [ K ] { u } = 0

(Eq. 12-8)

Assuming separation of variables and harmonic motion gives the following:

{ u ( ( x, y, z ), t ) } = { φ ( x, y, z ) } e

i ω t n

(Eq. 12-9)

where { φ ( x, y, z ) } is a vector of displacement amplitudes which defines the spacial distribution of the displacement vector { u } . The substitution of { u } and { u··} into (Eq. 12-8) then gives the following equation which relates { φ } and ωn: 2

2

[ K ] { φ } – ωn [ M ] { φ } = ( { K } – ωn [ M ] ) { φ } = 0

(Eq. 12-10)

From (Eq. 12-10) we obtain the eigenvalue problem:

( [ K aa ] – λ [ M aa ] ) { φa } = 0

(Eq. 12-11)

where the subscript a indicates the eigenvalue problem is associated with a-set degrees of freedom and where: 2

λ = ωn

(Eq. 12-12)

The eigenvalue problem, equation (Eq. 12-11) has non trivial solutions (i.e., a non null values of { φ } ) only if the determinant of the coefficient matrix is zero; i.e. det ( [ K ] – λ [ M ] ) = 0

(Eq. 12-13)

This determinant, which is called the characteristic equation, is a polynomial in terms of powers of λ where the highest power is the same as the order of matrices [ M ] and [ K ] . The values of λ which satisfy the characteristic equation are called the eigenvalues; and, since an nth order polynomial has n roots there are exactly n eigenvalues. Each one of the roots, λi (i = 1,2,...,n), then satisfies (Eq. 12-11) so that we have:

( [ K ] – λi [ M ] ) { φi } = 0

(Eq. 12-14)

where{ φ i }is the vector of displacement amplitudes, i.e., the eigenvector, associated with the eigenvalue λi. The procedure for determining eigenvalues and the associated eigenvector is conceptually quite straightforward. The eigenvalues are obtained by solving for the roots of the characteristic polynomial (Eq. 12-13). An eigenvalue is then substituted into (Eq. 12-14) and the resulting set of algebraic equations is solved for the associated eigenvector. As an example, consider a system with [ K ] and [ M ] given by:

[ K ] =

6 – 2 –2 4

[ M ] = 2 0 0 1

(Eq. 12-15)

the characteristic equation (Eq. 12-13) then becomes: = 0 de t ( 6 – 2 λ ) – 2 –2 ( 4 – λ )

(Eq. 12-16)

which has two distinct real roots:, λ1 = 2 and λ2 = 5. Nastran Primer 317

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The substitution of these two roots and the given [ K ] and [ M ] matrices into (Eq. 12-14) then leads to the following eigenvectors:

fo r

' ( { φ 1 } = α1 % 1 & " 1 #

λ1 = 2

(Eq. 12-17) fo r

' ( { φ 2 } = α2 % 2 & " – 1 #

λ2 = 5

where the coefficients, αi, indicate that we can only determine the relative values for the amplitudes of the eigenvectors. We can generalize this last statement to say that if { φ i } is an eigenvector, i.e., it satisfies (Eq. 12-13), then α i { φ i } is also an eigenvector. Now that we know how to calculate eigenvalues and eigenvectors we can consider some of their important properties and in the process see that the eigenvectors do have the special property to which we alluded in the last section, i.e., they transform the mass and stiffness matrices to diagonal form. Let's suppose that we have two distinct eigenvalues and their associated eigenvectors (λi,{ φ i }) and (λ j, { φ j } ). Each of their pairs satisfy (Eq. 12-14) so that we have:

( [ K ] – λi [ M ] ) { φi } = 0

(Eq. 12-18)

( [ K ] – λ j [ M ] ) { φ j } = 0 T

T

The premultiplication of the first of (Eq. 12-18) by { φ j } and the second by { φ i } followed by the transposition of the second equation then leads to: T

{ φ j } ( [ K ] – λi [ M ] ) { φi } = 0 T

{ φ j } ( [ K ] – λ j [ M ] )

(Eq. 12-19)

= 0

where we have used the fact that [ K ] and [ M ] are symmetric matrices. The subtraction of the second of (Eq. 12-19) from the first leads to: T

( λ j – λ i ) { φ j } [ M ] { φi } = 0

(Eq. 12-20)

Since λi and λ j are distinct, it follows that: T

{ φ j } [ M ] { φ j } = 0

i≠j

(Eq. 12-21)

This relation can be generalized as:

' M ii T { φ j } [ M ] { φi } = % " 0

fo r

i = j

fo r

i≠j

(Eq. 12-22)

Where M ii is the modal mass. This relation shows that the real eigenvectors are orthonormal with respect to the mass matrix. It is a simple matter to show that the eigenvectors are also orthonormal with respect to the stiffness matrix:

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Normal Modes Analysis Standard Form of Eigenvalue Problem

' K ii T { a j } [ K ] { ai } = % "0

fo r

i = j

fo r

i≠j

(Eq. 12-23)

and that the modal mass, M ii and modal stiffness K ii are related to the eigenvalue as follows: K

ii λ i = --------

(Eq. 12-24)

M ii

Returning briefly to the transformation (Eq. 12-5), let us suppose that we have found h eigenvectors each of which has a-rows. We can then define the transformation [ G ah ] to be a rectangular array of the h eigenvectors, arranged in order of ascending eigenvalue as follows:

[ G ah ] = { a 1 } { a2 } { a 3 } … { ah }

(Eq. 12-25)

Since each of the eigenvectors is orthonormal with respect to the mass and stiffness matrices, it follows that the use of the transformation (Eq. 12-5) will result in modal mass and modal stiffness matrices which are diagonal. The use of real eigenvectors for the transformation from physical set of degrees of freedom, { ua } , to modal h-set, { uh } , coordinates results in the modal formulation of dynamic algorithms in NASTRAN for transient and frequency response using the modal formulation. We noted earlier in this section that if { φ i } is an eigenvector which satisfies (Eq. 12-11) then α i { φ i } , where αi is a scale factor for the it h mode is also a eigenvector since it satisfies (Eq. 12-11) The value of αi depends on the method used to normalize the eigenvector. Two popular methods incorporated in NASTRAN are MASS normalization where the eigenvectors are scaled to make M ii = 1; and, MAX normalization where the scale factor is the largest element of the eigenvector so that the largest element of the scaled eigenvector is equal to one.

12.3

Standard Form of Eigenvalue Problem Several of the eigenvalue algorithms included in NASTRAN require that the eigenvalue problem be transformed to the standard form of the eigenvalue problem which is:

( [ J ] – λ [ I ] ) { w } = 0

(Eq. 12-26)

where [ J ] is real and symmetric and [ I ] is the identity matrix. The transformation must be a similarity transformation so that the symmetry of [ K ] and [ M ] is preserved and [ K ] and [ J ] have the same eigenvalues with the same multiplicity, as shown by Strang [5]. The required transformation is obtained by first finding the Choleski factors, [ C ] ,[6] of the mass matrix, where:

[ M ] = [ C ] [ C ]

T

(Eq. 12-27)

The substitution of (Eq. 12-27) into (Eq. 12-11) then gives: T

( – λ [ C ] [ C ] + [ K ] ) { a } = 0

(Eq. 12-28)

A set of transformed variables is then defined as: T

{ w } = [ C ] { a }

(Eq. 12-29)

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Normal Modes Analysis Specification of Element Inertia Properties

The substitution of (Eq. 12-29) into (Eq. 12-28) and subsequent premultiplication – 1 by [ C ] then leads to the form of equations (Eq. 12-26) where: – 1

– T

[ J ] = [ C ] [ K ] [ C ]

(Eq. 12-30)

The transformation, (Eq. 12-30), is called a similarity transformation. This type of transformation preserves the character of the matrix [ K ] so that if it is real and symmetric [ J ] will also have those characteristics. It is important to note that the Choleski factors will be singular if [ M ] is singular. This implies that an eigenvalue extraction algorithm which requires transformations to the standard form using the algorithm described above will fail during the inversion of the Choleski factor. The diagnostic message from NASTRAN will simply state that ‘MAA is singular’, where ‘MAA’ refers to the reduced mass matrix for the a-set. Special procedures are used in NASTRAN to remove these singularities.

12.4

Specification of Element Inertia Properties Element inertia properties can be generated using both structural mass and nonstructural mass or by special the mass elements. The mass matrix will be calculated: 1. For all dynamic solution sequences, including normal modes and frequencies. 2. If the structural weight is requested by including the parameter GRDPNT in any of the static solutions. 3. If external forces are to be generated using GRAV and RFORCE load specifications.

12.4.1

Structural Mass

Structural mass is associated with the volume of the element and is defined by means of the mass density field, RHO, on a MAT-type data statements. The value of RHO is used to calculate the mass matrix for all elements for which a volume can be calculated.

12.4.2

Nonstructural Mass

Non structural mass allows the analyst to account for the inertia effects of components that are non-structural in nature, such as hydraulic lines and wiring in an aircraft for example. Non structural mass is associated with an element’s length or area rather than on its volume. Non-structural mass is defined for one and two dimensional elements, by the NSM field on the associated property data statement. The NSM will be interpreted as mass per unit length for lineal elements and mass per unit volume for two dimensional elements. By its very nature, NSM is meant to augment the structural mass. Thus, if both structural mass, RHO, and non structural mass, NSM, are specified, the element mass matrix will be calculated using the sum of structural and nonstructural mass values.

12.4.3

Consistent and Lumped Formulations

The masses associated with the structural finite elements can be calculated by two methods using either a lumped or a consistent mass formulations. The lumped formulation is generally the default unless the user requests the consistent formulation by means of the COUPMASS parameter data statement which is described in a later section.

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In the lumped mass method the total mass for the element is calculated using the RHO and NSM values as appropriate for the element. The mass at each grid point is then calculated by dividing the total element mass by the number of element vertices. This mass fraction is then associated with each of the translational degrees of freedom for the element. The lumped mass formulation does not generate inertia terms for rotational degrees of freedom. This means that special care must be taken when using eigenvalue algorithms using the standard form of the eigenvalue problem as described in the preceding sections since the associated mass matrix in this case will be singular.

12.4.4

Mass Elements

Mass can be assigned to scalar and grid point degrees of freedom by means of the CMASSi, CONM1 and CONM2 described in the following sections.

12.4.5

Scalar Mass - CMASS

A scalar mass, which is associated with a SPOINT or a single degree of freedom at a Grid point is specified by one of the CMASS-type data statements shown by Bulk Data Image 12-1. 1

2

3

4

CMASS1

EID

PID

PMASS

PID

MASS

CMASS2

EID

CMASS3

5

6

7

G1

C1

G2

C2

MASS

G1

C1

G2

C2

EID

PID

S1

S2

PMASS

PID

MASS

CMASS4

EID

MASS

S1

S2

8

9

10

Where: Field

Description

EID

Element identification number, integer >0. Must be unique among all elements in the Bulk Data section.

PID

Identification number of a PMASS data statement, integer >0.

G1, C1

The grid point number and degree of freedom code defining a s ingle degree of freedom to which mass is to assigned, integers; C 1 is blank or zero if G1 refers to a SCALAR point.

G2, C2

The grid point number and degree of freedom code defining a s ingle degree of freedom to which mass is to assigned, integers; C 1 is blank or zero if G1 refers to a SCALAR point.

S1

The identification number of a SCALAR point to which mass is to assigned, integer > 0.

S2

The identification number of a SCALAR point to which mass is to assigned, integer > 0.

MASS

Mass coefficient to be assigned to the degree of freedom, real > 0.

Bulk Data Image 12-1 Scalar Mass Elements

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The form of the scalar mass element allows the element to be connected to two degrees of freedom which might be more appropriate for modeling passive electrical circuits than structural systems. The MASS value is placed on the diagonal of the mass matrix associated with the first of the two connected degrees of freedom and the off diagonal term associated with the second. This allows the CMASS to be used to specify an inductance in an electrical circuit element which is analogous to mass. When used to define the concentrated mass associated with a single degree of freedom the second connected point must be left blank.

12.4.6

Concentrated Grid Point Mass - CONM1

The CONM1 data statement specifies the mass matrix coefficients relating all six degrees of freedom at a grid point using the following CONM1 shown by Bulk Data Image 12-2. 1

2

3

4

5

6

7

8

9

10

CONM1 EID

G

CID

M11

M21

M22

M31

M32

0001

+001

M33

M41

M42

M43

M44

M15

M25

M35

0002

+002

M45

M55

M16

M26

M36

M46

M56

M66

Where: Field

Description

EID

Identification for the element, integer > 0. The element ID must be unique among all elements in Bulk Data.

G

The identification number of the Grid point for which the mass coefficients are specified.

CID

Identification number of the reference coordinate system in which the mass coefficients are defined, integer =>0.

Mij

The coefficients of the diagonal terms of the system mass matrix associated with Grid point G, real.

Bulk Data Image 12-2 Grid Point Mass Matrix - CONM1

12.4.7

Offset Concentrated Grid Point Mass - CONM2

An offset concentrated mass at a grid point is specifies using the CONM2 as shown by Bulk Data Image 12-3: 1

2

3

4

5

6

7

CONM2 EID

G

CID

M

X1

X2

+001

I21

i22

i31

I32

I33

I11

8

9

X3

10 0001

Where: Field

322 Nastran Primer

Description

EID

Identification for the element, integer > 0. The element ID must be unique among all elements in Bulk Data.

G

The identification number of the Grid point for which the mass coefficients are generated, integer > 0.

CID

Identification number of the reference coordinate system in which the offset, Xi, and moment of inertia tensor, I ij, are defined, integer =>0.

Normal Modes Analysis Real Eigenvalue Extraction Techniques

Field

M Xi

Description

The offset concentrated mass, real. Components of the vector offset of the mass from the grid point defined with reference to the displacement coordinate system at the grid point if CID => 0. If CID = - 1 the offsets are taken to be the

difference between the grid point location and the position defined by the offset coordinates. Iij

The coefficients of the mass moment of inertia tensor defined with reference to the CID coordinate system.

Bulk Data Image 12-3 Concentrated Mass - CONM2

The form of the mass matrix generated at grid point G for the CONM2 specification is as follows:

I 22 I 33 I 12 I 23 I 31

2

2

* ρ ( x2 + x3 )d V 2 2 = * ρ ( x 3 + x 1 ) d V 2 2 = * ρ ( x 1 + x 2 ) d V = * ρ x 1 x 2 d V = * ρ x 2 x 3 d V = * ρ x 3 x 1 d V

I 11 =

M 0 0

[ M ] =

0 0 0 0 M 0 0 0 0 0 0 M 0 0 0 0 0 0 I 11 – I 12 – I 13

(Eq. 12-31)

(Eq. 12-32)

0 0 0 – I 21 I 22 – I 23 0 0 0 – I 31 – I 32 I 33 where the Grid point mass, M , is: M =

12.5

* ρ d V

(Eq. 12-33)

Real Eigenvalue Extraction Techniques The eigenvalue problem is defined for the a-set degrees of freedom. The eigenvalue problem in NASTRAN set notation is:

( – λ [ M aa ] + K aa ) { φa } = 0

(Eq. 12-34)

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Normal Modes Analysis Real Eigenvalue Extraction Techniques

In the “early days” of NASTRAN eigenvalue extraction was an expensive operation. Since those days great strides in Linear Algebra have led to efficient methods that have made most of the methods in the following table obsolete. Modern methods are based on the Lanczos method. All algorithms are maintained in the program to support “dusty decks”, I suspect. That said the correct algorithm is the “LANC” method for Legacy NASTRAN and the use of EIGRL in NASTRAN derivatives for the Lanczos method, Table 1-1 Eigen Solution Methods in NASTRAN Method Name

Description

INV

Inverse power with shifts

GIV

Givens method of triadiagonalization

LANC

Lanczos method

FEER**

Fast Eigenvalue Extraction

SINV*

Inverse power with enhancements

MGIV*

Modified Givens

HOU*

Housholder’s method of triadiagonalization

MHOU*

Modified Householder

AGIV*

Automatic selection of Givens or modified Givens methods

AHOU*

Automatic selection of Givens or modified Givens methods

Note: Those options identified by a single asterisk are only available in MSC and NX NASTRAN. The FEER method, identified by double astericks, is only available in Legacy NASTRAN An algorithm together with its associated parameters is specified by an appropriate EIGR or EIGRL data specification. EIGRL(p. 331) is used to specify the Lanczos method and the EIGR is used to specify all other real eigenvalue extraction methods available in NASTRAN. Inverse power is a technique that finds one eigenvalue and an associated eigenvector at a time. The Givens and Householder algorithms use transformation techniques to find all the eigenvalues by transforming the eigenvalue problem to a diagonal form. The associated eigenvectors can then be selectively evaluated. The Lanczos method is a subspace algorithm. The idea is to represent a subspace of the eigenvectors using a linearly independent set of approximate transformation vectors which are determined using a truncated inverse power solution and which span the subspace. The matrices are then transformed to a subspace in which the eigenvalues and eigenvectors are determined using Givens, House-holder or the Jacoby method. As a last step the subspace eigenvectors are transformed to the solution set. The relative efficiency of any of the methods is dependent on the matrix topology, the order of the system matrices, and the number of eigenvalues and eigenvectors to be extracted. Numerical studies have shown that the Lanczos technique is the most efficient of the eigenvalue solvers for determining a large number of eigenvectors for large problem size. The Givens method requires that the eigenvalue problem be cast in standard form as defined by (Eq. 12-26). Since [ J ] in this form must be real and symmetric and since [ J ] is obtained by the similarity transformation, (Eq. 12-30), this implies that the mass matrix must be positive definite. Thus, the Givens method cannot be used for the buckling problem. The following general observations can be made about the choice of eigenvalue extraction technique 1.Lanczos is the recommended procedure for one-to-many eigenvalues

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2.The inverse power method is obsolete. But, if used, is best suited for obtaining only a few eigenvectors for large systems with sparse matrices. 3.The tridiagonal methods can for those cases where several eigenvectors are required but the Lanczos method is to be recommended in their place,.

12.6

The Inverse Power Method The following observations can be made about the inverse iteration with shifts procedure, where one or more shift points λ0 are introduce in the frequency range that is specified on the frequency range of interest. 1.The stiffness matrix can be singular, provided λ0 is nonzero, so rigid body modes offer no special difficulty. Tests of this algorithm suggest that repeated rigid body modes are evaluated with no difficulty. 2.The shift point, λ0, may be changed at any stage in the solution to improve convergence or to improve accuracy and convergence rate after several roots have been extracted. A change in starting point requires that an additional matrix decomposition be performed. 3.The shift point can be specified so that eigenvalues can be obtained within a specified frequency band rather than obtaining the smallest eigenvalue.

12.6.1

Specification of EIGR for Inverse Power with Shifts

The format for the EIGR data statement for the inverse power with shifts is: 1

2

3

4

EIGR

Method

SID

F1

+001

NORM

G

C

5 F2

6 Ne

7

8

9

Nd

10 0001

Where Field

Description

Method

The value of this field is the character string, INV, for inverse power or SINV for enhanced INV

SID

The set identification number, integer >0

f min, f max

Minimum and maximum frequencies defining the frequency range of interest, real > 0. Cycles per unit time. ( 0. ≤ f mi n < f ma x )

Ne

The number of estimated roots in the frequency range of interest, integer > 0.

Nd

The number of eigenvalues desired. Integer >0 or blank. If blank the N d = 3N e.

NORM

Method of normalizing the eigenvector. One of the character strings, MAX, MASS or POINT

G, C

The grid point number and degree of freedom code defining the degree of freedom to be used for POINT normalization, integers.

Bulk Data Image 12-4 Inverse Power Method - INV and SINV

The data specified on EIGR are used to control the inverse iteration method in the following manner: Nastran Primer 325

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1.The maximum and minimum frequencies of interest are converted to maximum and minimum values of the eigenvalue λmin and λmax where

λ = ( 2 π f )

2

2.The number of roots estimated in the range, N e, is used to determine the number of starting points, N s, which is chosen to satisfy the inequality N e N s – 1 < ------ ≤ N e 6

The starting points are then distributed, as shown in Figure 12-1 where the increment in the eigenvalue is found from:

∆λ = λ ma x – λ mi n and the position of the nth starting point is given by:

( n – 1 ⁄ 2 ) λ ns = λmi n + ----------------------- ∆λ N s

The algorithm assumes that approximately six good eigenvalues and associated eigenvectors can be found from a single starting point. Thus, if the estimate number of roots is six or less, the algorithm will use only one starting point, which is located in the middle of the eigenvalue range of interest. The eigenvalue analysis is terminated after N d roots are found. If the user does not specify N d , the algorithm will terminate after finding three times the number of roots estimated.

∆λ

∆λ

λ s1

∆λ

∆λ -------

------ N s

--------2 N s

--------2 N s

N s

λ s2

λ s3

λ mi n

λ ma x Figure 12-1 Distribution of Starting Points for Inverse Power

12.6.2

Summary for the Inverse Iteration Method

A summary of the eigenvalues found by the inverse power method is printed automatically. To interpret this summary, the user must have some understanding of the inverse iteration algorithm. First of all, there is a difference between the starting point and the shift point. The algorithm starts the eigenvalue search at a starting point and then may shift to another point (indeed the same concept as a starting point) in order to speed convergence. If, for example, the ratio of the first two shifted eigenvalues relative to the current starting point is approximately equal to unity, the rate of convergence will be very slow. Since the value of the closest eigenvalue and the time required to converge from both the current starting point and the shifted point can be estimated, a decision to either shift or continue to iterate can be made.

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The algorithms will continue to search for eigenvalues relative to the current starting point until an eigenvalue is found in the range of the next starting point, where a certain overlap between range of starting points is allowed. The algorithm finds successive eigenvalues, the first of which is the closest to the starting point. The eigenvalues then increase in absolute magnitude from the starting point. This means that one cannot be certain that the lowest root has been found in a specified range unless all the roots in the range have been found. The summary of the parameters associated with the algorithm and reasons for termination are as follows. 1.Number of eigenvalues extracted 2.Number of starting points used 3.Number of starting points moves 4.Number of matrix decompositions 5.Number of iterations 6.Reason for termination 1.Two consecutive singularities encountered while performing matrix decompositions. If two successive singularities in ( [ K ] – λ0 [ M ] ) are found, this implies that there may be something abnormal about the problem. Check the input to insure that the mass matrix has been specified. 2.Four shift points used while tracking a single root. If four shifts are required to track a single root, this implies that the first root is a long way from the first starting point and that the user did not exercise proper care in specifying the frequency range of interest. A preliminary estimate of the lowest root should be made with changes to the frequency range, and perhaps, the number of roots estimated in the range. 3.All eigenvalues found in the frequency range specified. All the roots in the range have been found. An excellent reason for terminating. 4.Three times the number of estimated eigenvalues have been found. If the user has not specified the number of roots desired, the algorithm will terminate after finding 3 Ne roots. Generally speaking, this is not a good reason for terminating. 5.All the eigenvalues that exist in the problem have been found. A valid reason for termination, but one that the user should not receive except for those cases where there are a very few degrees of freedom. If the user desires all the roots, Givens method should be specified since it is more economical than the inverse method if several roots and associated vectors are desired. 6.The number of roots desired have been found. same comment as reason 4 7.One or more eigenvalues have been found outside the specified frequency range.

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If one or more eigenvalues eigenvalues have been found outside the range, then either all eigenvalues in the range have been found or there are no roots in the range. A good reason for terminating, especially if only one root is desired. 8.Insufficient time to find another root. A bad reason for terminating. Re-execute Re-execute the problem taking advantage of current results, if possible, by restricting the range of interest. 9.Unable to converge. A bad reason for termination because because the algorithm can be shown to converge if [ M aa] and [ K K aa] are symmetric. Since the algorithm uses system matrices generated by the finite element model, there should be no convergence problem unless the user has modified the [ M M aa] and/or [ K K aa] matrices by some nonstandard technique. 7. Largest off-diagonal modal mass term and the numbers of the eigenvector pairs that fail the criteria.

12.6.3 12. 6.3

Findin Fin ding g Low Lowest est Eig Eigenv envalu aluee wit with h Inv Invers ersee Met Method hod

The inverse power method is frequently used to determine the lowest eigenvalue, i.e, the lowest natural frequency for a normal mode solution. If one has no knowledge of the value for the eigenvalue then one is on the horns of a dilemma because 1.The specification of a large frequency range for normal modes may lead to the determination of "several eigenvalue-eigenvector eigenvalue-eigenvector pairs at great cost. 2.Even if all roots are found in the range there is always a possibility of having a root below f min if f min > 0. There isn't any magic prescription in Inverse Power that in essence says "find the lowest root". All we can do is give some rather general guidelines which may help reduce the solution costs and which will guarantee that the lowest root is found. The suggested technique for finding the lowest eigenvalue is to specify a small frequency range using f min and f max which is so small that the root will be outside the range. Then specify that N e = 1, which will generate one starting point in the middle of the range, and that N d = 1 which will cause the algorithm to terminate after the extraction of one eigenvalue and associated eigenvector. eigenvector. If we were successful in choosing the frequency range correctly then the root is outside the range and is closest to the starting point, which is at the center of the specified frequency range. In order to guarantee that there are no roots below that calculated you can request that parameters which which are calculated calculated within the the eigenvalue extraction module be printed by using DIAG 16 in Executive Control. While this data includes much which will either be incomprehensible incomprehensible or useless it does include the number of roots below the current starting point λ0. If this number is zero then we can be assured that the one root which was calculated is indeed the lowest. A description of the data which is printed by DIAG 16 in Solution 1, ot Solution 103 is included as part of the example problem in a later section.

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12.7

The Lanczos Method 12.7.1 12. 7.1

Descri Des cripti ption on of the Lan Lanczo czoss Pro Proced cedure ure

The Lanczos procedure is described in detail by Nour-Omid[8] who notes that it is closely related to inverse power which was described in the previous section. In this section we summarize the description of the Lanczos procedure presented in [8], which should be consulted for more detailed information. In the inverse power with shifts procedure, given a pair of matrices, [ A 0 ] = ( [ K ] – λ0 [ M ] ) and [ M ] , and a starting vector, { r } , after n iterations inverse power generates a sequence of vectors: n / 0 2 – 1 – 1 – 1 + { r } ,[ A 0 ] [ M ] { r } ,( [ A 0 ] [ M ] ) { r } … ,( [ A0 ] [ M ] ) { r } ,

(Eq. (Eq. 12-35) 12-35)

The sequence converges as n → ∞ to the eigenvector eigenvector closest to the shift point. In the Lanczos procedure each vector in the sequence is used to obtain the best approximation to the eigenvectors. Instead of tossing the intermediate information away, all of the vectors in the sequence are used to obtain a Rayleigh-Ritz approximation approximation using the GramSchmidt orthonormization procedure which is described by Bathe [7]. In describing the algorithm, assume the first n Lanczos vectors have been found,

/ { q } ,{ q } ,{ q } … ,{ q } 0 n , 1 2 3 +

(Eq. (Eq. 12-36) 12-36)

and that we want to construct the (n+1) st vector. Each of the preceding vectors has been normalized with respect to the mass matrix and therefore satisfies the condition that T

{ φ i } [ M ] { φ j } = δij

(Eq. (Eq. 12-37) 12-37)

wher wheree δ ij is the the Kro Krone neke kerr del delta ta..

– 1

n

To calculate the next Lanczos vector, { s n } = ( [ A0 ] [ M ] ) { r } , must be normalized with respect to all of the preceding { q n } . By definition – 1

{ s n } = ( [ A 0 ] [ M ] ) { s n – 1 } .

(Eq. 12-38) 12-38)

Since s n – 1 is already [ M ]-orthogonal to the previous ( n - 2) Lanczos vectors it follows that M ]-orthogonal n

{ s n – 1 } =

1 yi { qi }

(Eq. (Eq. 12-39) 12-39)

i=1

where y i is the component of the Lanczos vector in the { s n – 1 } direction. The substitution 0f (Eq. 12-39) into (Eq. 12-38) then gives n

{ s n } =

– 1

1 yi [ A0 ]

[ M ] { qi }

i=1

(Eq. (Eq. 12-40) 12-40)

n – 1

– 1

{ s n } = y n [ A 0 ] [ M ] { qn } +

– 1

1 yi [ A0 ]

[ M ] { qi }

i=1

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– 1

Noting that each each vector, [ A0 ] [ M ] { q i } , can be written written as the sum of the first n Lanczos vectors so that n – 1

– 1

1 yi { qi }

{ s n } = y n [ A 0 ] [ M ] { q n } +

(Eq. (Eq. 12-41) 12-41)

i=1

where the last term of (Eq. 12-41) does not affect the determination of the next Lanczos vector since it will disappear in the [ M ]-normalization ]-normalization of the { sn} against all preceding Lanczos vectors. The next Lanczos vector is determines by first evaluating – 1

{ z n } = [ A 0 ] [ M ] { qn }

(Eq. (Eq. 12-42) 12-42)

Which is then [ M ]-normalized ]-normalized against all previous eigenvectors. Since { z n } contains components of all previous Lanczos vectors it can be expressed as

{ z n } = { z n } + αn { qn } + βn { qn – 1 } + γ n { qn – 2 } + …

(Eq. (Eq. 12-43) 12-43)

where { z n } is the component of { z n } which is orthogonal to all previous Lanczos vectors. Nour-Omid shows shows that all coefficients coefficients in this expansion expansion except except αn and βn vanish where T

αn = { q n } [ M ] { z n } T

(Eq. (Eq. 12-44) 12-44)

β n = { z n – 1 } [ M ] { qn } Solving for { z n } by substituting (Eq. 12-43) into (Eq. 12-42) gives – 1

{ z n } = [ A0 ] [ M ] { qn } – α n { qn } – β n { q n – 1 }

(Eq. (Eq. 12-45) 12-45)

The Lanczos vector the then obtained by normalizing { z n } 1

{ q n + 1 } = ---------------------------------------- { z n } T { z n } [ M ] { z n }

(Eq. (Eq. 12-46) 12-46)

After m Lanczos vectors have been generated they can be arranged in a rectangular array of a-sized vectors, [ Lam]. The eigenvector in the standard form of the eigenvalue problem, (Eq. 12-26) is approximated by using the following transformation using the Lanczos vectors

{ w a } = [ L am ] { ξm }

(Eq. (Eq. 12-47) 12-47)

The substitution of (Eq. 12-47) into (Eq. 12-29) and the application of the [ M ]-orthonormal ]-orthonormal properties of the the Lanczos vectors vectors then gives: gives: T

– 1

[ Lam ] [ M ] ( [ A 0 ] [ M ] – [ Θmm ] ) [ L am ] { ξm } = { 0 }

(Eq. (Eq. 12-48) 12-48)

where [ Θmm ] is a diagonal matrix of eigenvalues. eigenvalues. Because of the form of (Eq. 12-45), (Eq. 12-48) reduces to the tridiagonal eigenvalue problem

( [ T mm ] – [ Θmm ] ) { ξm } = { 0 }

330 Nastran Primer

(Eq. (Eq. 12-49) 12-49)


where – 1

T

[ T mm ] = [ L am ] [ M ] [ A 0 ] [ M ] [ Lam ]

(Eq. (Eq. 12-50) 12-50)

At each iteration of the Lanczos procedure the eigenvalue problem is transformed to a tridiagonal form in the a subspace. All of the eigenvalues of the reduced problem can now be economically determined using Givens method, which is described in the next section. The implementation of the algorithm requires that sufficient Lanczos vectors be generated to produce sufficient eigenvalues is a given frequency range. The use of the Strum sequence check[9] which requires a decomposition of [ A0] at the upper frequency of interest. The number of eigenvalues is then equal to the number negative terms on the factor diagonal. In practice the number of Lanczos vectors will be some multiple, greater than one, of the number of roots at an below the upper frequency range. The algorithm also must determine whether the roots in the subspace have converged by iteratively increasing the number of Lanczos vectors and applying a convergence criterion. The procedural steps define the Lanczos method. The actual implementation details, which are not given here, are the key to rendering the method effective and reliable.

12.7.2 12. 7.2

The Spe Specif cifica icatio tion n of the Lan Lanczo czoss Pro Proced cedure ure - EIGR EIGRL L

Not available in Legacy NASTRAN. NASTRAN. Use the “FEER” option in in EIGR ia an alternative. alternative. The Lanczos method is specified by the EIGRL described by Bulk Data Image 12-5 1 EIGRL

2 SID

3 V1

4 V2

5 Nd

6

7

8

MSGLVL MAXSET SHFSCL

9

10

NORM

option_1=value_1 option_1=value_1 option_2=value_2 option_2=value_2

where. Field

Description

SID

Set identification number, integer > 0.

V1, V2

Frequency range of interest for vibration analysis; eigenvalue range for buckling, real or blank.

Nd

Number of roots desired, desired, integer > 0 or or blank.

MSGLVL

Diagnostic message level, integer 0 - no messages (default) 1 - Prints eigenvalues accepted at each shift >1 - increases level of diagnostic output

MAXSET

Number of vectors in block or set, integer.

SHFSCH

Estimate of the frequency of the first flexible mode, real or blank. A reasonable estimate will improve performance when rigid body modes are present.

NORM

The method of normalizing normalizing the eigenvectors, eigenvectors, Character Character string MAX - Normalize largest component to one, default for vibration MASS - Normalize so that modal mass is one, default for buckling

Bulk Data Image 12-5 Specification of Lanczos Method - EIGRL

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Normal Modes Analysis Tridiagonal Methods

Remarks

1.The real eigenvalue method must be selected by the Case Control directive METHOD =

2.The units of V1 and V2 are cycles per unit time for vibration analysis and are multiples of the load for buckling. 3.MAX normalization is used for buckling analysis 4.The eigenvalues are found in increasing magnitude. magnitude. The number and type of eigenvalue is determine by the frequency range and the number of roots entries according to Table 1-2 Type of Roots Table 1-2 Number and Type V1

V2

Nd

Action

V1

V2

Nd

Lowest Nd or all in the range, whichever is smaller

V1

V2

blank

V1

blank

Nd

, ) Lowest Nd in the range ( V 1 ∞

V1

blank

blank

, ) Lowest root in the range ( V 1 ∞

blank

blank

Nd

Lowest Nd in the range ( – ∞ ,∞ )

blank

blank

blank

blank

V2

Nd

blank

V2

blank

All in the range

Lowest root Lowest Nd roots below V 2 All below V2

5.In vibration analysis the negative frequency range will be searched if V 1 < 0.0. If V1 is blank all roots less than 0.0 are calculated. Small negative roots generally indicate rigid body modes. Significant negative values indicates a modeling problem 6.The eigenvalues are sorted in increasing value for output. An eigenvector is found for each eigenvector 7.Special control parameters are available when using parallel implementation on selected computes. See the NASTRAN Quick Reference Guide for further details.

12.8

Tridiago gon nal Me Meth thod odss NASTRAN incorporates two basic basic algorithms, Givens (GIV) and and Householder Householder as well as as modified Givens (MGIV) and modified Householder (MHOU) tridiagonal methods. Of these only GIV is available in Legacy NASTRAN. Wilkinson [12] notes that for a number of years it seemed that the Givens process was likely to be the most efficient method of reducing a matrix to tridiagonal form. Householder suggested that the reduction would be more efficient if Hermetian orthogonal orthogonal matrices were used instead of plane rotations used by Givens. Wilkinson Wilkinson notes that the two methods are essentially the same but that Householders method is more efficient. Both of these methods are based on the standard form of the eigenvalue problem given by (Eq. 12-26) The difference between the two basic methods and their modifications is that the traditional methods uses the Choleski factors of the mass matrix as shown by (Eq. 12-27)while the modified methods reformulate the eigenvalue problem by defining the eigenvalue, λ, as:

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1

λ = ----- – λ 0 Λ′

(Eq. (Eq. 12-51) 12-51)

where λ0 is a shift point. The substitution of ((Eq. 12-51)into ((Eq. 12-11) then gives:

[ – Λ′ ( [ K aa ] + λ0 [ M aa ] ) + [ M aa ] ] { φa } = 0

(Eq. (Eq. 12-52) 12-52)

The modified eigenvalue problem is thus reduced to standard form by finding the Choleski factors of ( [ K aa ] + λ0 [ M aa ] ) which will be non singular even if the a-set contains massless degrees of freedom. A user-friendly feature is available for removing null columns which exist in both [ K aa ] and [ M a a ] that is controlled by the ASING parameter which is described in a later section. The tridiagonal methods employ the following three steps to determine the eigenvalues and eigenvectors: eigenvectors: 1.Use the Givens or Householder Householder method to reduce ( [ K aa ] + λ 0 [ M aa ] ) to tridiagonal form 2.Use QR iteration to determine all of the eigenvalues 3.Use inverse iteration to determine eigenvectors in a specified frequency range.

12.8.1 12. 8.1

EIGR EIG R for the Tri Triadi adiago agonal naliza izatio tion n Met Method hodss

The EIGR data statement for the Givens methods, GIV, MGIV, AGIV and the Householder methods, HOU, MHOU and AHOU are shown by Bulk Data Image 12-6. 1

2

3

4

EIGR

SID

Method f 1

+E‘

NORM

G

5

6

f 2

7

8

9

Nd

10 +E1

C

Where: Field

Description

SID

The set identification number that is selected by a METHOD = SID Case Control directive, integer > 0

Method

The name of the requested algorithm, Character string: GIV - Givens method MGIV - Modified Givens method AGIV - Automatic choice of GIV or MGIV HOU - Householder method MHOU - Modified Householder method AHOU - Automatic choice of Householder method

f 1 , f 2

The frequency range over which eigenvectors are to be calculated, real, f 2 > f 1.

Nd

The number of eigenvectors to be calculated starting with the lowest, integer.

NORM

The method used for eigenvector normalization: normalization: MASS - Mass normalization MAX - Max normalization POINT - Normalize with respect to displacement degree of freedom

G, C

Grid point number and degree of freedom code used if NORM = POINT

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Normal Modes Analysis Dynamic Degrees Degrees of Freedom

Bulk Data Image 12-6 Specification of Tridiagonal Eigenvalue Extraction

The data specified on EIGR is the same for all of the methods except for the second field which specifies the specific method. Both of the Householder and Givens methods find all of the eigenvalues in the a-set simultaneously. simultaneously. The recovery of eigenvectors is then controlled by specifying: • A frequency range ( f 1 , f 2) where f 1, is the lowest and f 2 is the highest frequency, in cycles per unit time. • The number of lowest frequencies for which eigenvectors are to be recovered - N d

The Nd field is an integer number which defines the number of eigenvectors starting with that associated with the lowest eigenvalue for either the transformations transformations methods. If present the Nd specification takes precedence over the frequency range for the transformation methods. The NORM field contains one of the literal strings MASS, MAX, or POINT which defines how the eigenvectors are to be normalized. If POINT is specified then the eigenvectors are normalized with respect to the displacement displacement associated with the degree of freedom defined by the pair (G, (G, C) where G is a grid point identification number and C is a degree of freedom code. The entry in field 3 contains one of the literal strings GIV, MGIV or AGIV to request one of the Givens methods; or HOU, MHOU or AHOU to request one of the Householder methods. In either case the SID field is an integer set identification number. The particular eigenvalue algorithm which is to be incorporated in the analysis is then specified by the following Case Control Directive. METHOD =

12.9 12 .9

Dyn ynam amic ic Deg egrree eess of of Fre Freeedo dom m The workflow in perfomning a design simulation rarely starts and ends with a static analysis. Linear and non-linear Dynamic analyses may be required to verify the design integrity. It is accepted practice, especially using Automated Multilevel Substructure analysis, [16], that is included in Legaacy Nastran to validate each structure component component by performing a component mode anaysis. This has several advantages: 1.Validates the existance of six rigid body modes. 2.Identifies mechnisms if present. Mechanisms will lead to failure in statics unless they are identified and removed. 3.Identifies normal modes. Are there suffient to span the space? Performing a static simulation can be deceiving: only on beam element is required to find the static solution. But the beam must be subdivied into many elements to represent, let’s say the 4th mode. In that case perhaps 10 spacial points are required to adeqently represent a mode so 40 element would be require to obtain good resoumtion og the 4th. This same ideal is also true in buckling analyses.

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12.10 Reducing Dynamic Degrees of Freedom On the other hand, NASTRAN can be used to model an almost unlimited variety of modern structures. It is not uncommon to use several thousand grid points, with tens of thousands of degrees of freedom in the static analysis of ships, aircraft, automobiles or nuclear reactors to name a few. Realistic procedures for reducing the number of degrees of freedom prior to performing dynamic analyses must therefore be formulated. A number of techniques for reducing the dynamic degrees of freedom have been presented in the literature including the Guyan Reduction[12], the use of static deflection shapes and Component Mode Synthesis (CMS)[14]. CMS is, perhaps, more appropriate for reducing the size of substructures and is not covered here. It is a technique that is supported in the Automated Multilevel Substructure (AMSS) procedure that is included in Legacy NASTRAN. See Reference [16].

12.10.1

Guyan Reduction

Static condensation is associated with the OMIT feature in NASTRAN which is described in Chapter 6 where it is shown that the { uo } and { ua } displacements sets are related by

( { u o } = [ G oa ] { ua } )

(Eq. 12-53)

– 1

[ G oa ] = – [ K oo ] [ K oa ] The { u f } set is then related to the { ua } set by the transformation

( { u f } = [ G fa ] { ua } ) [ G fa ] =

(Eq. 12-54)

[ G oa ] [ I aa ]

If we use the static condensation for the dynamic equations we would run into certain difficulties. As an illustration consider the following equations for normal modes analysis partitioned into the a- and o-sets:

/ 0 - – ω 2 [ M o o ] [ M oa ] + [ K oo ] [ K oa ] . - n ˜ ] ˜ ]. [ M a o ] [ M [ K ao ] [ K aa aa , +

{ uo } { ua }

' ( = % 0 & " 0 #

(Eq. 12-55)

If we attempt to solve for { uo } using the first matrix equation of (Eq. 12-55), as was done in the static case, we find that the frequency appears in the transformation matrix which relates { uo } and { u a } . An iterative procedure would therefore be required to find the transformation associated with each natural frequency. An approximate technique which is independent of frequency is therefore desirable. The approximate approach, called the Guyan Reduction, uses the static condensation transformation (Eq. 12-53) for dynamics so that the reduced mass matrix becomes: T

[ M aa ] = [ G fa ] [ M ff ] [ G fa ] ˜ ] + [ M ] [ G ] + [ G ] T [ M ] + [ G ]T [ M ] [ G ] [ M aa ] = [ M aa ao oa oa oa oa oo oa

(Eq. 12-56)

˜ ] and the If the mass coefficients associated with { uo } are null then [ M aa ] = [ M aa transformation (Eq. 12-53) is exact.

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Normal Modes Analysis Removing Matrix Singularities

In order to assess the approximation which is associated with the use of the static relation (Eq. 12-53) in dynamics it is reasonable to consider the consequences of using (Eq. 12-55) to define a relation between the o- and a-sets. From the upper partition we find that: – 1

2

2

{ u o } = – ( ω [ M oo ] + [ K oo ] ) ( – ω [ M oa ] + [ K oa ] ) { u a }

(Eq. 12-57)

The substitution of (Eq. 12-57) into the lower partition of (Eq. 12-55) then gives the following reduced equation: 2

[ – ω [ M aa ] + [ K aa ]

(Eq. 12-58)

2

– 1

2

2

+ ( – ω [ M ao ] + [ K au ] ) ( – ω [ M oo ] + [ K oo ] ) ( – ω [ M oa ] + [ K oa ] ) ] { u a } = 0 – 1

2

The inverse term, ( – ω [ M oo ] + [ K oo ] ) as follows: 2

– 1

( – ω [ M oo ] + [ K oo ] )

– 1

= [ K oo ]

, can be represented using a series of expansion

2

– 1

– 1

+ ω [ K oo ] [ M oo ] [ K oo ]

+…

(Eq. 12-59)

The substitution of (Eq. 12-59) into (Eq. 12-57) then leads to the same reduced stiffness matrix and reduced mass matrix as those obtained by using static condensation when only terms through ω2 are retained. The exact solution of the reduced problem is thus seen to involve an iterated procedure. Such a procedure is not automated in the normal modes analysis solution algorithm so that some means of selecting the OMIT degrees of freedom to minimize the frequency-dependent error in the reduced mass matrix is required. Unfortunately, there is no automated procedure for selecting those degrees of freedom to be included in OMIT. There are some guidelines, based on experience, which suggest retaining 1.Degrees of freedom with large inertia. 2.Degrees of freedom with large accelerations (i.e., large relative displacements for normal mode analysis). 3.Approximately 3.5 times as many well-chosen degrees of freedom as the number of accurate modes desired in the solution. 4.Translational displacement degrees of freedom for bending problems. These suggestions almost presuppose that the user knows the solution to the eigenvalue problem in order to make an intelligent choice of retained physical degrees of freedom in the a-set. While Guyan Reduction (i.e., the use of the OMIT feature) is inexpensive and is therefore popular, it requires user prescience to obtain accurate solutions.

12.11 Removing Matrix Singularities All of the eigenvalue extraction techniques perform a decomposition of the stiffness or mass matrix or a linear combination of the two. Any degree of freedom which is unconnected in both matrices is undefined, and its presence in the analysis set will lead to a singular matrix that, by definition, has no inverse. A singularity check is made in the matrix decomposition modules in NASTRAN so that a fatal error will be issued and program execution will be terminated.

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The unconnected degrees of freedom in the stiffness matrix can be purged either explicitly by single point constraints or implicitly by means of the AUTOSPC parameter (See Chapter 11 and Appendix B). A singular mass matrix is then the analyst's real concern, especially in the GIV or HOU methods. The mass matrix associated with the a-set can be made non singular by using static condensation to remove all massless degrees of freedom from the analysis set. This procedure leads to a positive definite mass matrix which is required for the GIV and HOU methods and while a positive definite mass matrix is not required by the MGIV or MHOU methods the algorithm is generally faster if massless degrees of freedom are removed by static condensation (OMIT). However, the use of static condensation leads to matrices with greater density and increased active columns so that the INV generally runs slower if static condensation is performed on massless degrees of freedom. MSC and NX NASTRAN provides an user option that can be used to detect singularities in the stiffness and mass matrices and to then perform matrix operations which are appropriate for the eigensolution method to remove the singularity. This action is controlled by the integer ASING parameter which provides two options for dealing with the singularity. The parameter values and the associated actions are described below.

12.11.1

Detect Singularities and Exit - ASING = -1

ASING is set to - 1 all degrees of freedom with null columns in both the stiffness and mass matrix are identified and a fatal error exit is taken. This is the proper option to use to assure that no modeling errors exist prior to performing an expensive eigensolution.

12.11.2

Remove Massless Degrees of Freedom ASING = 0

If ASING is set to zero then all degrees of freedom with null columns are identified and placed in the w-set where:

{ ua } =

{ uw }

(Eq. 12-60)

{ u x }

The a-set mass and stiffness matrices are then partitioned so that

/ ˜ - – ω2 [ M xx ] [ M xw ] + [ K xx ] [ K xw ] [ M wx ] [ M ww ] [ K wx ] [ K ww ] +

0 {u } {0} x . . {u } = {0} w ,

(Eq. 12-61)

where [ M xx ] , [ M wx ] and [ M xw ] , are null by definition. All zero diagonal terms in [ K ww ] are replaced with unit diagonal terms so that we have from the second of (Eq. 12-61)

[ K wx ] { u x } + [ K ww ] { uw } = { 0 }

(Eq. 12-62)

The inverse of [ K ww ] now exists because former null columns have been replaced with unit diagonals so that we can solve for { uw } by using(Eq. 12-62) to obtain

{ u w } = [ G wx ] { u x }

(Eq. 12-63)

where – 1

[ G wx ] = – [ K ww ] [ K wx ]

(Eq. 12-64)

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Normal Modes Analysis Input File for Normal Modes Analysis

The reduced set of equations for the x-set (i.e. the solution set for eigensolution) is then found to be 2

( – ω [ M xx ] + [ K xx ] ) { u x } = { 0 }

(Eq. 12-65)

where T

˜ ] + [ K ] [ K ] [ K xx ] = [ K xx wx wx

(Eq. 12-66)

This formulation, which is associated with ASING = 0, is appropriate for the GIV, HOU, MGIV and MHOU methods. It will prevent fatal errors which are due to a singular mass matrix in the GIV and HOU methods if the singularity is caused by null columns in [ M ww ] . However, mechanisms cannot be detected. That type of singularity does not necessarily prevent eigensolutions but can cause poor numerical stability. The MGIV, AGIV MHOU and AHOU methods do not suffer from numerical instability due to singular or nearly singular mass matrices so that they is more reliable than the GIV and HOU methods, respectively. The removal of the massless degrees of freedom by static reduction will reduce solution costs and will also eliminate the calculation of the eigenvalues for high frequency "noise". The static reduction of massless degrees of freedom does not introduce any of the approximations which are associated with the Guyan Reduction. The expediency of replacing the null columns of the stiffness matrix by unit diagonals is proper because the associated columns of [ G wx ] are null. The same result could be achieved by first purging the unconnected degrees of freedom o [ K gg ] followed by an OMIT of massless degrees of freedom, but at a higher cost. If the INV method is specified then there is no mass reduction. However, all degrees of freedom having null columns in both mass and stiffness matrices are removed before performing the eigensolution.

12.12 Input File for Normal Modes Analysis 12.12.1

Executive Control Section

The executive control section must specify that the normal modes and frequency analysis, solution 103, is to be executed by means of the following Executive Control Directive.

SOL 103 in MSC/NASTRAN or

SOL 3 in Legacy NASTRAN

12.12.2

Case Control Section

The following items relate to the selection of Bulk Data and subcase definitions for normal modes. 1.A METHOD = SID Case Control Directive must be included to select an EIGR Bulk Data statement defining an eigenvalue extraction method. 2.An SPC set must be selected unless the model is a free body or if the constraints are specified as permanent constraints on a GRID card or by means of general elements.

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Normal Modes Analysis Parameters

3.Multiple subcases may be used only for output requests. A single subcase is sufficient if the same output requests are appropriate for all modes. The use of multiple subcases is discussed in Chapter One. All of the output Case Control Directives that are associated with static analysis can be specified for Normal Modes analysis.

12.13 Parameters The following optional parameters may be used in normal modes analysis. (See Appendix B of the Primer for the form of the PARAM Bulk Data card and additional discussion of use of parameters) 1.AUTOSPC

Purges unconnected degrees of freedom YES - Remove singular DOF (default) NO - Terminate if singular DOF are found

2.COUPMASS

Causes the generation of consistent rather than lumped mass. -1 Lumped formulation (default) >0 Consistent formulation

3.WTMASS

Factor which multiplies all terms in the mass matrix (real)

Other parameters of interest are described in Appendix B.

12.14 Example Problems We will consider several example problems whose purpose is to show how to employ the new capability which we have described in this chapter. We will thus use relatively simple structural topologies to solve several eigenvalue problems including 1.A Cantilever beam 2.A simply supported rectangular plate

12.14.1

Cantilever Beam

The cantilever beam considered for this example is that presented by Fig. 12-2 where it was noted that the weight density of the beam is 0.281 lb./in3. Since this value will be specified on the Mat1 data statement we will need to include a WTMASS parameter to change from weight to mass units. In defining the finite element model we recognize that the mesh refinement is dictated by the number of mode shapes that to be adequately represented. We will therefore suppose that the model is to be capable of representing the fourth symmetric mode, and that approximately 10 equally spaced increments are required to represent a mode. The number of grid points is thus 41 for the entire beam. The boundary conditions for a fully fixed boundary are to be imposed at x=0 and the beam is free at x = L. The problem will be solved using the Inverse power method to solve for a lumped mass formulation and Givens method to solve for the consistent mass formulation. The results for the two formulations will then be compared with the theoretical solutions for natural frequencies. Common Model Components

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Normal Modes Analysis Example Problems

The finite element model for the static problem described in Chapter 12 is not adequate since it does not have sufficient Grid points to represent the modes of interest. A more refined model that can serve as the basis for both the inverse power and Givens methods. In preparing this model: 1.Use the area properties described in Chapter 12 2.Use the material properties specified in Chapter 12 including the weight density 3.Include the WTMASS parameter having a value of 1/ g . Lanczos

Lanczos is to be run for the lumped mass formulation. As we noted earlier, the COUPMASS parameter controls the type of formulation. Its default value is -1 which results in the lumped formulation. Selected Output

The output data for this example problem is shown below where we note 1.The INPUT DATA DECK ECHO, which was requested by ECHO = BOTH in Case Control prints the input Bulk Data deck exactly as it is read by NASTRAN. In this case we have used free-field input together with generation and repeat features 2.REAL EIGENVALUES. The table of eigenvalues is automatically printed for normal modes. The eigenvalues are sorted in increasing value in the 3rd column and are converted to radians and cycles/second in columns 4 and 5. The generalized mass is only determinable for calculated eigenvectors (three in this case) and is equal to 1. because we requested MASS normalization. The generalized stiffness for this case is seen to be equal to the eigenvalue for modes for which modes were calculated. 3.REAL EIGENVECTOR - The displacements in the a-set for the four modes is printed as a result of SVECTOR = ALL Theoretical Results

The exact solution for the cantilever beam are presented by several authors including Meirovich [13] who shows that the first three natural frequencies for a uniform cantilever beam are:

ω1 = ( 1.875 ) ω2 = ( 4.694 ) ω1 = ( 7.855 )

2

EI ------------4 ρ AL

2

EI ------------4 ρ AL

2

EI ------------4 ρ AL

(Eq. 12-67)

where I is either I yy or I zz since the section is rectangular.

12.14.2

Simply Supported Plate

The structural model is a square plate with simply supported edges. The finite element model, which is included in the inp directory of the NASA NASTRAN download from the GitHub site uses a 10x20 mesh that represents one-half of the plate reducing the number of GRID points by one-half. Symmetric boundary conditions are used on the midline (x = 0) as described below.

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Normal Modes Analysis Problems

Only bending modes are desired so in-plane displacements and rotations about the normal are constrained. The FEER method is used for eigen extraction. Both structural mass and non structural mass are used to define mass matrix. Propertites

The properties of the square plate are shown by the following table: Table 1-3 Properties of Square Plate Property

Value

l - Plate length

20 in.

w - Plate width

20 in.

t - Plate thickness

1 in.

E - Elastic Modulus

30 x 10 6 lbf/in2

ν − Poisson ratio

0.3

ρ - Mass density

Structural mass = 200 lb f sec2/in4 Nonstructural mass = 6.039 lb f sec2/in4

Boundary Conditions

The boundary conditions are: 1.Symmetry conditions along x = 0: θx = 0, uy 2.Simple support along y = 0: u z = 0, θx = 0 3.Simple support along x = 10: u z = 0, θy= 0 4.Simple support along y = 20: u z = 0, θx = 0 Eigenvalue Data

1.Method = FEER 2.Center point for FEER: 0.87 3.Number of desired roots: 3 Elements

CQUAD1 elements are used. Results

The following table lists the NASTRAN and theoretical natural frequencies (cps), where the theoretical values are defined in Ref. [15] Table 1-4 Comparison of Plate Natural Frequencies (cps) Mode 1

.9069

.9056

2

2.2672

2.2634

3

4.5345

4.5329

12.15 Problems [1] Determine the first three elastic bending modes and frequencies for a free-free beam 6 bending in the x-y plane where the elastic modulus is, E = 30 ×10 ps i , the weight density, γ = 0.1 lb/in3, the cross section moment of inertia, I = 0.1 in4, and the length, L = 10 in.

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Normal Modes Analysis References

[2] Determine the first four bending modes and frequencies of the simply supported plate described in Section 12.14.2. Plot the modes and normalize the frequencies using the exact values. [3] Using 30 elements along the length, use static condensation to remove the displacement and rotation degrees of freedom at all even numbered node points. Compare the frequencies to those found in problem 1. [4] Using the model for problem 1, compare results for Givens, Lanczos or FEER, and Inverse Power methods. [5] Using the model for the square plate example, Section 12.14.2, change the element from CQUAD1 to CQUAD4, making sure to also use the correct property type, and use the Lanczos method for eigenvalue extraction.

12.16 References [1] K-J. Bathe, Finite Element Procedures, Prentice Hall, Englewood Cliffs, N. J., 1996. [2] O.C. Zienkiewicz and R.L Taylor, The Finite Element Method, Fourth Edition, Volume 2: Solid and Fluid Mechanics Dynamics and Non-linearity, McGraw-Hill, Berkshire England, 1991. [3] T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Analysis , Prentice Hall, Englewood Cliffs, NJ, 1987. [4] L. Meirovich, Elements of Vibration Analysis, McGraw-Hill, New York, 1972, pp. 65-70. [5] G. Strang, Linear Algebra and its Applications, Academic Press, New York, 1976, p 221. [6] ibid, p. 241. [7] K-J. Bathe, Finite Element Procedures, Prentice Hall, Englewood Cliffs, N. J., 1996, pp. 906-908. [8] B. Nour-Omid, “The lanczos algorithm for the solution of large generalized eigenproblems”, in the book by T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Analysis, Prentice Hall, Englewood Cliffs, NJ, 1987., pp. 582-629 [9] K-J Bathe and E.L. Wilson, Numerical Methods in Finite Element Analysis, Prentice Hall, Englewood Cliffs, N.J., 1976, pp 505-506 [10] ibid, pp. 465-471. [11] NASTRAN Theoretical Manual , NASA SP-221, Section 5.5.3. [12] R. Guyan, “Reduction of Stiffness and Mass Matrices,” AIAA J., 3,(2), 1965, pp. 380. [13] L. Meirovich, Elements of Vibration Analysis, McGraw-Hill, New York, 1972, p. 212. [14] R.R. Craig, “A Review of Time-Domain and Frequency Domain Component Mode Synthesis Methods”, Int. J. Analytical and Experimental Modal Analysis, Vol. 2, No. 2, 1987 [15] W.F Stokley, “Vibration of Systems Having Distributed Mass and Elasticity”, Chap. 7, Shock and Vibration Handbook , C.M. Harris and C.E. Crede, Editors, McGraw-Hill, 1961. [16] Anon., “ NASTRAN USER’S MANUAL”, NASA SP-222, VOL. I, Sec. 1.10.2.

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Build Legacy NASTRAN NASA has published the source code for NASA NASTRAN on their GitHub site. (Googling for “NASA NASTRAN” should get you there). The complete download includes the source code, the DMAP for all rigid formats, the NASA NASTRAN manuals and the verifications problems. It’s quite a treasure trove for software archeologists to rummage around in!

13.1 Before You Start NASTRAN is a fairly complicated program. You really need some understanding of the architecture to get started. I am going to assume that is the case. In what follows I will outline the workflow for getting Legacy NASTRAN running. You can then judge what to do next.

13.1.1Go Windows I have lots of experience with Compac FORTRAN Visual Studio 6.5 which is a wonderful development environment. The compiler has all of the right stuff to compile and link Legacy NASTRAN and the debugger is unequalled in my humble opinion. However, the current FORTRAN compilers that support Windows are too dear for my budget so I continue to rely on my old faithful. If you have access to either Portland Group or Intel compilers just make sure they support Cray pointers. With that as a background and having access to a suitable FORTRAN installation you will have no problem with building an executable for Legacy NASTRAN. You will find some compile errors, perhaps the same as those noted for gfortran in the next section, but they are easy to overcome. The best part is that the compiler will support varaible formats a capability in which gfortran failed. So read through the next section, fire up your Visual Studio FORTRAN and copy all the source files to whatever project you choose, perhaps nastran. Then build to find compiler errors, fix them, build again and you should have it. You will still need to create a script file; or, perhaps create a FORTRAN front end to serve the same funtion. That is what I did since I have more confidence in my FORTRAN skills than my Powershell skills. Either way you can use the script file in the NASA NASTRAN installation directory as a guide.

13.1.2Go Linux You need a FORTRAN compiler. I suggest using the GNU toolchain including the C, C++ and gfortran compilers. You don’t need the C compilers right now but you will if you want to add new code. If you have a Linux system installed all you need to do is install the GNU compilers. If you don’t know how to do that you probably shouldn’t be thinking about giving Legacy NASTRAN new life. If your computer has windows installed you will need to install MinGW and MSYS. It’s not hard to do, just do some Googling, it’s amazing what you can find on the web. I am going to assume that you have done this and have installed the GNU tool chain as I described above.

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Build Legacy NASTRAN Compiling and linking.

You are almost there. Now download Legacy NASTRAN from the NASA GitHub site into your Linux (MinGW) home directory. The path will be something like: c:/mingw/msys/1.0/. If you haven’t already done so open a command shell in Windows, change directory to c:/mingw/msys and type “msys’ to open a bash shell and then cd to the nastran directory (I am assuming you have downloaded to that directory).

13.2 Compiling and linking. If you haven’t used a make file you need to do a little research on the subject. Make is a utility that controls compiling and linking the source code. Once you have created the file called makefile all you need to do is enter make in the bash shell. But I am getting ahead of myself.

13.2.1Work directory Create a work directory as a subdirectory in the nastran directory and, in the work directory, create the following subdirectories:

Table 13-1 Sandbox Directories Directory Name

Description

sb

sandbox for the executable

include

include file

bin

bash files

test

test files

Right now all these directories are empty but we will start working on that by:

1. copying all *.f files from nastran/mds to work/sb 2. copying all *.f files from nastran/mis to work/sb We still need to create a bash script for executing nastran and copy a model file from the nastran/nid demonstration directory to the test directory but we have some serious work ahead of us before we get that far. Once we have a successfully linked nastran program we will get to those details.

13.2.2Build NASTRAN Firstly. you will need a makefile. Copy the following and save it as “makefile” in work/sb: # Declaration of variables FC = gfortran FFLAGS = -w -fno-range-check -fno-automatic -fcray-pointer -g # file names EXEC=nast SOURCES = $(wildcard *.f) OBJECTS = $(SOURCES:.f=.o)

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# Main target $(EXEC):$(OBJECTS) $(FC) $(OBJECTS) -g -o $(EXEC) # To obtain the object files %.o:%.f $(FC) $(FFLAGS) -g -c $< -o $@ # To remove generated files clean: rm -f $(EXEC) $(OBJECTS)

Then, create a new directory “nastprog” in the work directory and move nasthelp.f and nastplot.f there. These are utility programs that are not a intrinsic part of nastran. Then open a bash shell and cd to sb and enter make in the command window. The following routines will fail to compile.

Table 13-2 sb Compile failures Routine

Problem

conmsg.f

Error in time and date intrinsic

ffread.f

Error in open statement at line 376

mapfns.f

Error in calling LOC at line 138. Change %LOC to LOC

nsinfo.f

Error in open at line 68

pexit.f

Error at line 45. LINK is external

rfopen.f

Need to change open at line 81

sgino.f

Need to change open statements.

sofut,f

Error in calling rename. It is external

tdate.f

Need to replace time and date intrinsic

After making these changes the routines will compile but fail to link. The link errors are:

1.Both errtrc.f and dummy.f produce the symbol “errtrc”. Remove the symbol from dummy.f 2. remove endsys.f. It is no longer used. 3. move chkfil.f. to nastprog directory. It is a utility program. 4. remove shtrmd.f and shhmgd.f Nastran should now link and create the executable called run.exe. Now the fun begins.

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13.2.3Create the nastran bash file In order to execute run.exe you need to create a bash script. You can use the script f ile in the download as a guide or you can copy the following script file and save in it the work/test directory. #!/bin/bash cwd=$(pwd) echo The current work Directory is - $cwd progname=”$1” ft05=”$progname.nid” ft06=”$progname.out” nptp=”$progname.nptp” echo Input file is: $ft05 sof1=”none” sof2=”none” rfdir=”/home/harry/nasanastran/rf” nasthome=”/home/harry/nasanastran/work” tmp=”$nasthome/test/tmp” projectdir=”$nasthome/test” nasexec=”$nasthome/sb/run” mkdir -p $tmp nasscr=$tmp dbmem=”14000000” ocmem=”14000000” echo ==== NASTRAN is beginning execution of $progname ==== NPTPNM=$nptp;export NPTPNM PLTNM=”$progname.plt”;export PLTNM DICTNM=”none”;export DICTNM PUNCHNM=”$progname.pnh” ;export PUNCHNM OUT11=”$progname.out11” ;export OUT11 IN12=”$progname.in12” ;export IN12 FTN11=”none”;export FTN11 FTN12=”none”;export FTN12 DIRCTY=$nasscr ;export DIRCTY lognm=”$progname.log” LOGNM=$lognm;export LOGNM OPTPNM=”none”;export OPTPNM RFDIR=$rfdir ;export RFDIR FTN13=”none”;export FTN13 SOF1=$sof1;export SOF1 SOF2=$sof2 ;export SOF2 FTN14=”none”;export FTN14 FTN17=”none”;export FTN17 FTN18=”none”;export FTN18 FTN19=”none”;export FTN19 FTN20=”none”;export FTN20 FTN15=”none”;export FTN15 FTN16=”none”;export FTB16 FTN21=”none”;export FTN21 FTN22=”none”;export FTN22 FTN23=”none”;export FTN23 DBMEM=$dbmem;export DBMEM OCMEM=$ocmem ;export OCMEM FT05=$ft05;export FT05 346 Nastran Primer


FT06=$ft06;export FT06 CWD=$cwd;export CWD PROJ=$projectdir;export PROJ #printenv $nasexec #gdb $nasexec echo ===== NASTRAN has completed problem $progname ===== You will need to edit the paths for rfdir and nasthome. You can also change the extensions for ft05, the input filr and ft06, the output file. Many nastran users us “bdf” or “dat” for the input file and “f06” for the output file extensions.

13.2.4Additional Modifications Additional changes are required to eliminate run time errors associated with file I/O. Firstly the size of the character variable associated with environment variables and an array called dsnames must be changed. to do this make modifications to files in the sb directory as follows:

1.In nastrn.f change the lengths of the character variables “value” and “dsnames” from 44 to 80, and add an integer parameter “len_dsnames” with a value of 80. 2. In nastrn.f, change “do 5 i=44,1,-1” to “do 5 i=len_dsnames,1,-1”. 3. In gino.f and ginio.f, change the size of the dsnames variable from 44 to 80. 4. In xsem00.f, add the following in the definition section: !

Set variables integer error_id integer nin, nout character(80) proj,ft05,ft06,output,infile

5. In xsem00.f, add the following equivalence, “equivalence (xx(3),nin). 6. In xsem00.f, add the following after “ call btstrp”: !---------------------------------------------------------------------! hgs 12/06/2104 - The NASA delivery uses redirected input and out put ! so these files are not explicitly opened. I need ! to change the stdin and stdout to allow the use of ! GDB in the script file. Therefore the following mod! ifications explicitly open the stdin and stdout files ! using the FTN5 and FTN6 ENV set by the script. ! c c open inout and output files c call getenv(‘PROJ’,proj) call getenv(‘FT05’,ft05) call getenv(‘FT06’,ft06) Nastran Primer 347

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output = trim(proj)//’/’//trim(ft06) error_id = 0 101 continue ifile = nout open(nout,file=output,form=’formatted’, 1status=’unknown’,iostat=ierr,err=102) go to 103 102 continue error_id = -2 go to 104 103 continue ifile = nin infile = trim(proj)//’/’//trim(ft05) open(nin,file=infile,form=’formatted’ 1 ,status=’unknown’, iostat =ierr,err=106) go to 105 106 continue error_id = -1 104 continue c c open error c write(nout,*) ‘Error in opening file =’,ifile,’ IOSTAT = ‘,ierr select case(error_id) case(-1) write(nout,’(a)’) ‘File name: ‘,infile case(-2) write(nout,’(a)’) ‘File name: ‘,output end select call pexit ! close down and exit with ierror return 105 continue c I notice that there are tab characters in the above code fragment so some editing might be required.

7. In btstrp.f, at line 180 change the variable “mach” to 6. 8. run make in the sb directory. Hopefully there are no errors. If there are rather than trying to sort them out you can got to “www.mycosim.com” for help.

13.2.5Run a test file Copy the file d01081a.nid to the test directory. Then change to the test directory and enter “nast.sh d01081a”. Assuming no link errors the problem will apparently run to completion, and it almost makes it to the end. However, there is a problem in the OFP module that leads to an error. It turns out that OFP used a viable format write that gfortran won’t handle. So, bummer. If you had used another compiler such as as visual studio fortran you won’t have this problem and your run should go to s successful completion.

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Don’t give up on gfortran, this is the only limitation I have found but we need a workaround. That is to replace the current ofpprt.f and add a routine forwrt.f. I include those routines here, but suggest that might make more sense to go to my github site and download a modified nastran installation that I call “classicnastran”.

13.2.5.1

OFPPRT

SUBROUTINE OFPPNT (OUT,NWDS,FMT) C CWKBD LOGICAL DEBUG CWKBR INTEGER OUT(NWDS), FMT(300) INTEGER OUT(NWDS) CHARACTER*1 FMT(1200) CWKBI COMMON /MACHIN/ MACHX COMMON /SYSTEM/ SYSBUF, L CWKBD DATA DEBUG / .FALSE. / C CWKBD IF (DEBUG) WRITE (L,10) (FMT(K),K=1,32) 10 FORMAT (‘ FMT=’,32A4) CWKBR 5/95 IF ( MACHX.EQ.2 .OR. MACHX.EQ.5 ) IF ( MACHX.EQ.2 .OR. MACHX.EQ.5 .OR. MACHX .EQ. 21 ) * WRITE (L,FMT,IOSTAT=IOSXX) (OUT(K),K=1,NWDS) CWKBR 5/95 IF ( MACHX.NE.2 .AND. MACHX.NE.5 ) IF ( MACHX.NE.2 .AND. MACHX.NE.5 .AND. MACHX .NE. 21 ) * CALL FORWRT (FMT, OUT, NWDS) RETURN END

13.2.5.2

FORWRT SUBROUTINE FORWRT ( FORM, INDATA, NWDS )

C******************************************************************** C EXPECTED TYPES OF FORMAT CODES ARE AS FOLLOWS C NH-----NENN.N NDNN.N NX C NFNN.N NINN NGNN.N NAN C NPENN.N NPFNN.N NPN(----) NP,ENN.N C NP,FNN.N NP,N(----) C SPECIAL CHARACTERS: /(), C ICHAR = CURRENT CHARACTER NUMBER BEING PROCESSED IN “FORM” C ICOL = CURRENT CHARACTER COLUMN POSITION WITHIN THE LINE C NCNT = NUMBER OF VALUES OF IDATA AND DATA THAT HAVE BEEN PROCESSE C******************************************************************** CHARACTER*1 FORM(1000) CHARACTER*1 SLASH , BLANK CHARACTER*1 LPAREN, RPAREN, PERIOD, COMMA, NUMBER(10) CHARACTER*1 H, E, D, X, F, I, G, A, P CHARACTER*2 PFACT CHARACTER*4 CDATA(200) CHARACTER*132 LINE CHARACTER*132 TFORM INTEGER*4 INDATA(NWDS), IDATA(200) REAL*4 DATA(200)

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REAL*8 DDATA(100) COMMON /SYSTEM/ ISYSBF, IWR EQUIVALENCE (IDATA, DATA, DDATA, CDATA ) DATA H/’H’/, E/’E’/, D/’D’/, X/’X’/, F/’F’/ DATA I/’I’/, G/’G’/, A/’A’/, P/’P’/ DATA LPAREN /’(‘/, RPAREN/’)’/, PERIOD/’.’/ DATA COMMA /’,’/, SLASH /’/’/, BLANK /’ ‘/ DATA NUMBER /’0’,’1’,’2’,’3’,’4’,’5’,’6’,’7’,’8’,’9’/ IF ( NWDS .LE. 200 ) GO TO 2 PRINT *,’ LIMIT OF WORDS REACHED IN FORWRT, LIMIT=200’ CALL PEXIT 2 DO 3 KB = 1, NWDS IDATA( KB ) = INDATA( KB ) 3 CONTINUE ILOOP = 0 ICHAR = 1 NCNT = 1 ICOL = 1 LINE = BLANK PFACT = BLANK ICYCLE= 0 5 IF ( FORM(ICHAR) .EQ. LPAREN ) GO TO 75 ICHAR = ICHAR + 1 IF ( ICHAR .LE. 1000 ) GO TO 5 GO TO 7702 70 IF ( ICHAR .GT. 1000 ) GO TO 7702 IF ( FORM(ICHAR) .EQ. BLANK ) GO TO 75 IF ( FORM(ICHAR) .EQ. SLASH ) GO TO 100 IF ( FORM(ICHAR) .GE. NUMBER(1) .AND. & FORM(ICHAR) .LE. NUMBER(10) ) GO TO 200 IF ( FORM(ICHAR) .EQ. A ) GO TO 300 IF ( FORM(ICHAR) .EQ. I ) GO TO 400 IF ( FORM(ICHAR) .EQ. H ) GO TO 500 IF ( FORM(ICHAR) .EQ. X ) GO TO 600 IF ( FORM(ICHAR) .EQ. P ) GO TO 700 IF ( FORM(ICHAR) .EQ. F ) GO TO 800 IF ( FORM(ICHAR) .EQ. G ) GO TO 800 IF ( FORM(ICHAR) .EQ. D ) GO TO 800 IF ( FORM(ICHAR) .EQ. E ) GO TO 800 IF ( FORM(ICHAR) .EQ. LPAREN ) GO TO 1000 IF ( FORM(ICHAR) .EQ. RPAREN ) GO TO 1100 IF ( FORM(ICHAR) .NE. COMMA ) GO TO 7702 IF ( ICYCLE .EQ. 0 ) PFACT = BLANK 75 ICHAR = ICHAR + 1 GO TO 70 C PROCESS SLASH 100 CONTINUE IF ( LINE .NE. BLANK ) WRITE ( IWR,900 ) LINE 900 FORMAT(A132) IF ( LINE .EQ. BLANK ) WRITE ( IWR,901 ) 901 FORMAT(/) LINE = BLANK IF ( ICYCLE .EQ. 0 ) PFACT = BLANK ICOL = 1

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GO TO 75 C GET MULTIPLIER FOR FIELD CONVERSION 200 CALL FORNUM ( FORM, ICHAR, IMULT ) GO TO 70 C PROCESS ALPHA FIELD--FORMAT(NNANNN) (NN=IMULT,NNN=IFIELD) 300 ICHAR = ICHAR + 1 IF ( NCNT .GT. NWDS ) GO TO 1200 CALL FORNUM ( FORM, ICHAR, IFIELD ) ILEFT = NWDS - NCNT + 1 IF ( ILEFT .LT. IMULT ) IMULT = ILEFT IF ( IMULT .EQ. 0 ) IMULT = 1 WRITE ( TFORM, 902 ) IMULT, IFIELD 902 FORMAT(‘(‘,I2,’A’,I2,’)’) I1 = ICOL LENGTH = IMULT*IFIELD NEND = NCNT + IMULT - 1 LAST = ICOL + LENGTH - 1 WRITE( LINE(ICOL:LAST), TFORM ) (CDATA(KK),KK=NCNT,NEND) ICOL = ICOL + LENGTH NCNT = NCNT + IMULT IMULT = 1 GO TO 70 C PROCESS INTEGER FIELD -- FORMAT(NNINNN) (NN=IMULT,NNN=IFIELD) 400 ICHAR = ICHAR + 1 IF ( NCNT .GT. NWDS ) GO TO 1200 CALL FORNUM ( FORM, ICHAR, IFIELD ) IF ( IMULT .EQ. 0 ) IMULT = 1 WRITE ( TFORM, 903 ) IMULT, IFIELD 903 FORMAT(‘(‘,I2,’I’,I2,’)’) I1 = ICOL LENGTH = IMULT*IFIELD NEND = NCNT + IMULT - 1 LAST = ICOL + LENGTH - 1 WRITE( LINE(ICOL:LAST), TFORM ) (IDATA(KK),KK=NCNT,NEND) ICOL = ICOL + LENGTH NCNT = NCNT + IMULT IMULT = 1 GO TO 70 C PROCESS HOLERITH FIELD -- FORMAT(NNH----) (NN=IMULT) 500 LAST = ICOL + IMULT - 1 ICHAR = ICHAR + 1 LCHAR = ICHAR + IMULT - 1 WRITE ( LINE(ICOL:LAST), 904 ) (FORM(KK),KK=ICHAR,LCHAR) 904 FORMAT(133A1) ICOL = ICOL + IMULT ICHAR = LCHAR IMULT = 1 GO TO 75 C PROCESS X FIELD -- FORMAT(NNX) (NN=IMULT) 600 WRITE ( TFORM, 905 ) IMULT 905 FORMAT(‘(‘,I2,’X’,’)’) LAST = ICOL + IMULT - 1 WRITE( LINE(ICOL:LAST), TFORM ) ICOL = ICOL + IMULT

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IMULT = 1 GO TO 75 C PROCESS P FACTOR FOR FLOATING FORMAT 700 WRITE ( PFACT,904 ) FORM(ICHAR-1), FORM(ICHAR) IF ( NCNT .GT. NWDS ) GO TO 1200 710 IF ( FORM( ICHAR+1 ) .NE. BLANK .AND. FORM( ICHAR+1 ) .NE. & COMMA ) GO TO 75 ICHAR = ICHAR + 1 IF ( ICHAR .GT. 1000 ) GO TO 7702 GO TO 710 C PROCESS FLOATING FIELD -- FORMAT(NPNNXNNN.NNNN) WHERE C (NP = PFACT, NN=IMULT, NNN=IFIELD, NNNN=IDEC) 800 ITYPE = ICHAR IF ( NCNT .GT. NWDS ) GO TO 1200 ICHAR = ICHAR + 1 CALL FORNUM ( FORM, ICHAR, IFIELD ) 810 IF ( FORM( ICHAR ) .EQ. PERIOD ) GO TO 820 ICHAR = ICHAR + 1 GO TO 810 820 ICHAR = ICHAR + 1 CALL FORNUM ( FORM, ICHAR, IDEC ) IF ( IMULT .EQ. 0 ) IMULT = 1 WRITE ( TFORM, 906 ) PFACT, IMULT, FORM(ITYPE),IFIELD, IDEC 906 FORMAT(‘(‘,A2,I2,A1,I2,’.’,I2,’)’) I1 = ICOL LENGTH = IMULT*IFIELD NEND = NCNT + IMULT - 1 LAST = ICOL + LENGTH - 1 IF ( FORM(ITYPE) .EQ. D ) & WRITE( LINE(ICOL:LAST), TFORM ) (DDATA(KK),KK=NCNT,NEND) IF ( FORM(ITYPE) .NE. D ) & WRITE( LINE(ICOL:LAST), TFORM ) (DATA(KK),KK=NCNT,NEND) ICOL = ICOL + LENGTH NCNT = NCNT + IMULT IMULT = 1 GO TO 70 C PROCESS LEFT PAREN (NOT THE FIRST LEFT PAREN BUT ONE FOR A GROUP) C IMULT HAS THE MULTIPLIER TO BE APPLIED TO THE GROUP 1000 ICYCLE = IMULT-1 ICSAVE = ICHAR+1 ILOOP = 1 IMULT = 1 GO TO 75 C PROCESS RIGHT PAREN ( CHECK IF IT IS THE LAST OF THE FORMAT) C IF IT IS PART OF A GROUP, THEN ICYCLE WILL BE NON-ZERO 1100 IF ( ICYCLE .GT. 0 ) GO TO 1110 IF ( ILOOP .NE. 0 ) GO TO 1120 IF ( NCNT .GT. NWDS ) GO TO 1200 C NO GROUP, THEREFORE MUST RE CYCLE THROUGH FORMAT C UNTIL LIST IS SATISFIED WRITE ( IWR,900 ) LINE ICHAR = 2 LINE = BLANK

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PFACT = BLANK ICOL = 1 GO TO 70 C GROUP BEING PROCESSED, DECREMENT COUNT AND RESET ICHAR TO BEGINNING C OF THE GROUP 1110 ICYCLE = ICYCLE - 1 ICHAR = ICSAVE GO TO 70 C FINISHED WITH LOOP, CONTINUE WITH FORMAT 1120 ILOOP = 0 ICYCLE = 0 GO TO 75 1200 WRITE ( IWR,900 ) LINE 7000 CONTINUE RETURN 7702 WRITE( IWR, 9901 ) ICHAR, FORM 9901 FORMAT(///’ SUBROUTINE FORWRT UNABLE TO DECIPHER THE FOLLOWING’ & ,’ FORMAT AT CHARACTER ‘,I4,/,’ FORMAT GIVEN WAS THE FOLLOWING:’ & ,/,(1X,131A1)) END

13.3 Taking Stock If all has gone well you now have a working version of the NASTRAN installation that you downloaded from the NASA NASTRAN GitHub site. It has been a struggle at times but congratulations if you persevered. Now, where to we go from here? There are lots of changes, the most pressing is an improved element library. The next is to replace or augment the current solvers for statics and eigenvalue extraction with those using sparse logic. There are many open source codes out there. Then there are utilities to be upgraded such as replacing the current MPC module with one that finds the best m-set rather than requiring the user to do so. And speaking of MPC equations, additional constraint generators are required including contact and glue connections. But the JEWEL that comes with Legacy NASTRAN is the substructure capability called Automated Multilevel SubStructures (AMSS). It truly is a great start to a new paradigm for changing the workflow in creating models of assemblies.

13.3.1Next Step I have a lot of experience with NASTRAN development starting with an installation delivered in 1995. In the two years after the 1993 release that is on the NASA GitHub, the maintenance team at COSMIC made some important changes. Those changes are included in the download for classicalnastran that is discussed on “www.mycosim.com”. I suggest starting there. It’s a much improved baseline for future developments.

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A

Use of Parameters

Parameters may be specified by the user by means of a PARAM-card in Bulk Data. The PARAM-card is as shown by Card Image A-1. 1

2

3

4

PARAM

N

V1

V2

5

6

7

8

9

10

where: N

Parameter name

V1, V2

Parameter value depending on the parameter type as follows: Type

V1

V2

Integer

Integer

Blank

Real, single-precision

Real

Blank

Literal

Literal

Blank

Real, double-precision

D.P.

Blank

Complex, single-precision

Real

Real

Complex, double-precision

D.P.

D.P.

Bulk Data Image A-1 The PARAM Bulk Data Statement

The allowable PARAMeters that might be useful for rigid formats 1, 101, 3, and 103 are summarized below. ASING (MSC Only)

Automatic elimination of matrix singularities for normal modes analysis (Sec. Chapter 12)

AUTOSPC (MSC)

Automatic purge of unconnected degrees of freedom Chapters 6 and 12). associated PARAMeters include EPPRT EPZERO PRGPST SPCGEN USETPRT USETSEL

AUTOSPC (Legacy NASTRAN))

Automaticically detect and, if found, remove under user control. Integer 0 - Detect, create and print the Grid Point Singularity Table. (GPST). (Will probaly result in failure in the solve operation.) 1 - Process GPST and create SPC1s to remove the singular degrees of freedom -1 - Same as = 1 but write the SPC1s to the PUNCH file

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A

Use of Parameters

COUPMASS

A positive value (default is -1) requests the coupled rather than the lumped mass formulation.

GRDPNT

A positive value (default is -1) requests the weight and balance information described in Sec. 12.10 to be calculated and displayed using the grid point number, defined as the value of the PARAMeter as the reference point.

IRES

A positive value (default is -1) requests that the residual for the aand o-sets be displayed.

MAXRATIO

The presence of this parameter will cause the conditioning numbers for each degree of freedom to be compared to the value of the 5

PARAMeter (default is 10 ). If they are greater than MAXRATIO the matrix is considered to include a mechanism. The conditioning numbers and the external degree of freedom codes are printed and execution is terminated. WTMASS

356 Nastran Primer

The mass matrix is multiplied by the value of this parameter (default is 1.) prior to dynamic analysis. The use of this PARAMeter allows the analyst to define the density and nonstructural mass in terms of weight units which are then multiplied by a value of 1/g which is specified as the WTMASS PARAMeter value.

Nastran Primer

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