Pushover Analysis using ETABS and SAP2000

Pushover Analysis Using ETABS and SAP2000

June 18-19, Manila, Philippines For

Association of Structural Engineers Philippines

By

Naveed Anwar Asian Center for Engineering Computations and Software Asian Institute of Technology In Association with

Computers and Structures Inc., Berkeley, California, USA

Pushover Analysis Using ETABS (and SAP2000)

June 22-23, CEBU, Philippines

By

Naveed Anwar Asian Center for Engineering Computations and Software Asian Institute of Technology In Association with

Computers and Structures Inc., Berkeley, California, USA

Acknowledgements

Pushover Analysis, ACECOMS, AIT

• Some of the material presented in these notes is based on following sources: – Class notes by Prof. Worsak Kanok-Nukulchai – Seminar notes from Computers and Structures Incorporated, USA – Notes from various workshops conducted by Naveed Anwar – SAP2000 User and Technical Manuals – ETABS User and Technical Manuals – ATC40, Applied Technology Council, USA – FEMA-273, Federal Emergency Management Agency, USA


Objectives • Introduce the basic Modeling and Analysis Concepts • To provide an understanding of Static Nonlinear Pushover Analysis for Seismic Performance • To demonstrate the application of Pushover Analysis for buildings using ETABS and SAP2000 and to provide a comparison

The Questions


• Why use Pushover Analysis • What is Pushover Analysis • How to carryout Pushover Analysis • What to do before Pushover Analysis • What to do after Pushover Analysis


Modeling and Analysis


Summary • • • • •

The Purpose of Analysis The Significance of Modeling Analysis Types Linearity and Non-Linearity Static and Dynamic Analysis

Structural System – Analysis Model STRUCTURE

RESPONSES


EXCITATION Loads Vibrations Settlements Thermal Changes

Displacements Strains Stress Stress Resultants

pv

Structural Model

Analysis of Structures  xx  yy  zz    pvx  0 x y z Pushover Analysis, ACECOMS, AIT

pv

Real Structure is governed by “Partial Differential Equations” of various order Direct solution is only possible for: • Simple geometry • Simple Boundary • Simple Loading.


The Need for Modeling A - Real Structure cannot be Analyzed: It can only be “Load Tested” to determine response B - We can only analyze a “Model” of the Structure C - We therefore need tools to Model the Structure and to Analyze the Model

Finite Element Method: The Analysis Tool

• Finite Element Analysis (FEA) – “A discretized solution to a continuum problem using FEM”


• Finite Element Method (FEM) – “A numerical procedure for solving (partial) differential equations associated with field problems, with an accuracy acceptable to engineers”


Continuum to Discrete Model

pv 3D-CONTINUM MODEL

CONTINUOUS MODEL OF STRUCTURE

(Governed by partial (Governed by either differential equations) partial or total differential equations)

DISCRETE MODEL OF STRUCTURE

(Governed by algebraic equations)

From Classical to FEM Solution Equilibrium

Actual Structure


 xx  yy  zz    pvx  0 x y z “Partial Differential Equations”

FEM

Assumptions

Classical

Structural Model

Kr  R

Stress-Strain Law Compatibility



t

_

_

“Algebraic Equations” _

 dV   p u dV   p u ds t v

t s

v

(Principle of Virtual Work)

K = Stiffness r = Response R = Loads

Simplified Structural System Deformations (D)

Loads (F)


Fv

D

K

F

F=KD

The Analysis System STRUCTURE

RESPONSES

EXCITATION


pv

• Static • Dynamic

• Elastic • Inelastic

• Linear • Nonlinear

Eight types of equilibrium equations are possible!

The Equilibrium Equations 1. Linear-Static

Elastic

Ku  F

2. Linear-Dynamic

Elastic


Mu(t )  Cu(t )  Ku(t )  F (t )

3. Nonlinear - Static Elastic OR Inelastic Ku  FNL  F

4. Nonlinear-Dynamic Inelastic

Elastic OR

Mu(t )  Cu(t )  Ku(t )  F (t ) NL  F (t )


Basic Analysis Types Excitation

Structure

Response

Basic Analysis Type

Static

Elastic

Linear

Linear-Elastic-Static Analysis

Static

Elastic

Nonlinear

Nonlinear-Elastic-Static Analysis

Static

Inelastic

Linear

Linear-Inelastic-Static Analysis

Static

Inelastic

Nonlinear

Nonlinear-Inelastic-Static Analysis

Dynamic

Elastic

Linear

Linear-Elastic-Dynamic Analysis

Dynamic

Elastic

Nonlinear

Nonlinear-Elastic-Dynamic Analysis

Dynamic

Inelastic

Linear

Linear-Inelastic-Dynamic Analysis

Dynamic

Inelastic

Nonlinear

Nonlinear-Inelastic-Dynamic Analysis

Some More Solution Types


• Non-linear Analysis – – – – –

P-Delta Analysis Buckling Analysis Static Pushover Analysis Fast Non-Linear Analysis (FNA) Large Displacement Analysis

• Dynamic Analysis – Free Vibration and Modal Analysis – Response Spectrum Analysis – Steady State Dynamic Analysis

Analysis Type


The type of Analysis to be carried out depends on the Structural System – The Type of Excitation (Loads) – The Type Structure (Material and Geometry) – The Type Response

Static Vs Dynamic • Static Excitation


– When the Excitation (Load) does not vary rapidly with Time – When the Load can be assumed to be applied “Slowly”

• Dynamic Excitation – When the Excitation varies rapidly with Time – When the “Inertial Force” becomes significant

• Most Real Excitation are Dynamic but are considered“Quasi Static” • Most Dynamic Excitation can be converted to “Equivalent Static Loads”

Elastic Vs Inelastic • Elastic Material – Follows the same path during loading and unloading and returns to initial state of deformation, stress, strain etc. after removal of load/ excitation


• Inelastic Material – Does not follow the same path during loading and unloading and may not returns to initial state of deformation, stress, strain etc. after removal of load/ excitation

• Most materials exhibit both, elastic and inelastic behavior depending upon level of loading.

Linear Vs Nonlinear • Linearity – The response is directly proportional to excitation – (Deflection doubles if load is doubled)


• Non-Linearity – The response is not directly proportional to excitation – (deflection may become 4 times if load is doubled)

• Non-linear response may be produced by: – Geometric Effects (Geometric non-linearity) – Material Effects (Material non-linearity) – Both

Linear-Elastic

Action

Action

Elasticity and Linearity

Deformation

Action

Action


Deformation

Linear-Inelastic

Nonlinear-Elastic Deformation

Nonlinear-Inelastic Deformation

Linear and Nonlinear Linear, Static and Dynamic

Ku  F Pushover Analysis, ACECOMS, AIT

F

FNL

(t )  Cu (t )  Ku(t )  F (t ) Mu Ku = F Ku - FNL = F

Nonlinear, Static and Dynamic

Ku  FNL  F u

Non Linear Equilibrium

Mu(t )  Cu(t )  Ku(t )  F (t ) NL  F (t )

Basic Concepts for Analysis


• • • •

DOF (Degree of Freedom) Stiffness Static Analysis Process Dynamic Analysis Procedures


The Seven Degrees of Freedom • The General Beam Element may have 7 degrees of freedom • The seventh degree is Warping • Warping is out-of plane distortion of the beam crosssection

ry uy y

u x rx x z uz rz wz

Each section on a beam member can have seven Degrees Of Freedom (DOF) with respect to its local axis.


§ § § § § § §

DOF Picture uz The  AxialComplete deformation  Axial strain  Axial stress ux  Shear deformation  Shear strain  Shear stress uy  Shear deformation  Shear strain  Shear stress rz  Torsion  Shear strain  Shear stress r y  Curvature  Axial strain  Axial stress rx  Curvature  Axial strain  Axial stress wz Warping  Axial strain  Axial stress


What is Stiffness ? • In structural terms, stiffness may be defined as “Resistance to Deformation” • So for each type of deformation, there is a corresponding stiffness • Stiffness can be considered or evaluated at various levels • Stiffness is also the “constant” in the ActionDeformation Relationship

For Linear Response

uF Ku  F F K u

The Structure Stiffness Material Stiffness

Cross-section Geometry


Section Stiffness

Member Geometry Member Stiffness

Structure Geometry Structure Stiffness

The Matrices in FEM Global Nodal Deformations T-Matrix Global-Local Cords.

Element Nodal Deformations Pushover Analysis, ACECOMS, AIT

N-Matrix Shape Functions

Deformation in Element Space B-Matrix Strain-Deforrmation

Strain In Element Space D-Matrix Stress-Strain

Stress in Element Space


Linear-Static Analysis Process • • • • • •

Generate Stiffness Matrix for each Element Form Global Stiffness Matrix Form Load Vector Modify for boundary conditions Solve for unknown Displacements Compute element actions/ stresses from end displacements

Methods of Dynamic Analysis • For Both Linear and Non-Linear Systems


– Step-by-Step Integration – Use of Mode Superposition with Eigen or LoadDependent Ritz Vector for Fast Nonlinear Analysis (FNA)

• For Linear Systems Only – Transformation of frequency domain and FFT Method – Response Spectrum Method – CQC - SRSS


Step by Step Solution Method • Form Effective Stiffness Matrix • Solve Set of Dynamic Equilibrium Equations for Displacement at Each Time Step • For Non-Linear Problems Calculate Member Forces for Each Time Step and Iterate for Equilibrium – Brute Force Method


Mode Superposition Method • Generate Orthogonal Dependent Vectors and Frequencies • Form Uncoupled Modal Equations and Solve Using Exact Method for Each Time Increment • Recover Nodal Displacement as a Function of Time • Calculate Member Forces as a Function of Time


Load Dependent Ritz Vector • Approximately Three Times Faster than the Calculation of Exact Eigen Vectors • Results in Improved Accuracy using a Smaller Number of LDR Vector • Computer Storage Requirements are Reduced • Can be Used for Non-Linear analysis to Capture Local Static Response


Fast Non-Linear Analysis • Evaluate LDR Vectors with Non-Linear Elements Removed and Dummy Elements Added for Stability • Solve All Modal Equations with Non-Linear Forces on the Right Hand Side • Use Exact Integration within Each Time Step • Force and Energy Equilibrium are Satisfied at Each Time Step by Iteration • The FNA Method is Designed for Static and Dynamic Analysis of Non-Linear Structures with a Limited Number of Pre-Defined Non-Linear Elements


Pushover Analysis • • • • •

One Dimensional Static Loads No Energy Dissipation Inertia Forces Not Considered Defined One Failure Mode Higher Mode Effects Neglected


The Modal Analysis


The Modal Analysis • The modal analysis determines the inherent natural frequencies of vibration • Each natural frequency is related to a time period and a mode shape • Time Period is the time it takes to complete one cycle of vibration • The Mode Shape is normalized deformation pattern • The number of Modes is typically equal to the number of Degrees of Freedom • The Time Period and Mode Shapes are inherent properties of the structure and do not depend on the applied loads

Free Vibration Analysis • Definition – Natural vibration of a structure released from initial condition and subjected to no external load or damping


• Main governing equation -Eigenvalue Problem     M  u   c u   K  ut  Pt  t  t

• Solution gives – Natural Frequencies – Associated mode shapes – An insight into the dynamic behavior and response of the structure


The Modal Analysis • The Modal Analysis should be run before applying loads any other analysis to check the model and to understand the response of the structure • Modal analysis is precursor to most types of analysis including Response Spectrum, Time History, Push-over analysis etc. • Modal analysis is a useful tool even if full Dynamic Analysis is not performed • Modal analysis easy to run and is a fun to watch the animations

Application of Modal Analysis


• The Time Period and Mode Shapes, together with animation immediately exhibit the strengths and weaknesses of the structure • Modal analysis can be used to check the accuracy of the structural model – The Time Period should be within reasonable range, (Ex: 0.1 x number of stories seconds) – The disconnected members are identified – Local modes are identified that may need suppression

Application of Modal Analysis


• The symmetry of the structure can be determined – For doubly symmetrical buildings, generally the first two modes are translational and third mode is rotational – If first mode is rotational, the structural is unsymmetrical

• The resonance with the applied loads or excitation can be avoided – The natural frequency of the structure should not be close to excitation frequency

Eccentric and Concentric Response


Unsymmetrical Mass and Stiffness

Symmetrical Mass and Stiffness

Mode-1

Mode-2

Mode-3


Modes and Pushover • Generally the deformation pattern corresponding to the First Mode is used as the basis for analysis • This is acceptable for structures with time period less than or equal to 1 second • For more flexible structures, higher mode contribution may become significant


Special Analysis Problems


Base Isolation

Isolators

Building Impact


Building Impact Analysis

Dampers Friction device


Concentrated damper Nonlinear element

Gaps and Joints Gap Element


Bridge Deck ABUTMENT

Tension only element

Hinges


PLASTIC HINGES

2 Rotational DOF

Degrading Stiffness?

Dampers


Mechanical Damper F= f(u,v,umax)

F= ku

F= CvN Mathematical Model


Linear Viscous Damping • Does not Exist in Normal Structures and Foundations • 5 or 10 Percent modal Damping Values are Often Used to Justify Energy Dissipation Due to Non-Linear Effects • If Energy Dissipation Devices are Used Then 1 Percent Modal Damping should be Used for the Elastic Part of the Structure

Uplift


FRAME WITH UPLIFTING ALLOWED

Uplifting Allowed


Structural Modeling

Structure Types • Cable Structures • Cable Nets • Cable Stayed


• Bar Structures • 2D/3D Trusses • 2D/3D Frames, Grids

• Surface Structures • Plate, Shell • In-Plane, Plane Stress

• Solid Structures

Global Modeling of Structural Geometry


(a) Real Structure

(b) Solid Model

(c) 3D Plate-Frame

(d) 3D Fram e

(f) Grid-Plate

(e) 2D Fram e Fig. 1 Various Ways to Model a Real Struture

Some Sample Finite Elements


Truss and Beam Elements (1D,2D,3D)

Plane Stress, Plane Strain, Axisymmetric, Plate and Shell Elements (2D,3D)

Brick Elements


Model Creation Tools • • • • • • • • •

Defining Individual Nodes and Elements Using Graphical Modeling Tools Using Numerical Generation Using Mathematical Generation Using Copy and Replication Using Subdivision and Meshing Using Geometric Extrusions Using Parametric Structures

Graphic Object Modeling


• Use basic Geometric Entities to create FE Models • Simple Graphic Objects – – – –

Point Object Line Object Area Object Brick Object

Represents Node Represents 1D Elements Represents 2D Elements Represents 3D Elements

• Graphic Objects can be used to represent geometry, boundary and loads • SAP2000, ETABS and SAFE use the concept of Graphic Objects


Modeling Objects and Finite Elements • Structural Members are representation of actual structural components • Finite Elements are discretized representation of Structural Members • The concept of Graphic Objects can be used to represent both, the Structural Members as well as Finite Elements • In ETABS, the Graphic Objects representing the Structural Members are automatically divided into Finite Elements for analysis and then back to structural members for result interpretation


Design Methods and Concepts

From Loads to Stresses


Applied Loads

Building Analysis

Member Actions

Cross-section Actions

Material Stress/Strain

From Strains to Response

The Response and Design Material Response

Section Response

Member Response

Building Response

Load Capacity

Three Design Approaches • Working Stress Design – Stress is primary concern and objective

• Ultimate Strength Design Pushover Analysis, ACECOMS, AIT

– Strain is primary concern

• Performance Based Design – Deformation is primary concern

From Serviceability to Performance


Allowable material, control on deformation limits for design loads Material failure criteria, section capacity for factored loads Ductility considerations, deformation capacity, load capacity at large deformations. Extraordinary load considerations

Serviceability Design

Strength Design

Performance Design



• Satisfying one design level does not ensure that other design levels will be satisfied – Serviceability design only ensures that deflections and vibrations etc. for service loads are within limits but says nothing about strength – Strength design ensures that a certain factor of safety against overload is available within a member or a cross-section but says nothing about what happens if load exceeds design level – Performance design ensures that structure as a whole reaches a specified demand level. Performance design can include, both service and strength design levels


A – Serviceability B – Cracking Limit C – Strength Limit D – Failure Limit

P

D

C B

Load


• The entire response of structure or a member can be determined, in an integrated manner from the ActionDeformation Curve

P

Δ A

Deformation

D

Cross-section Reponses • Stresses – Tension – Compression – Shear > Tension-Compression


• Strains – Normal strain – Shear Strain

• Deformations – – – –

Rotation Shortening Shearing Twisting

Determining Cross-section Response Material Stress-Strain Curves Cross-section Dimensions

Performance


Given P value

Given Moments

Given Axial Load

P-M Curve

M-M Curve

Moment-Curvature Curves

•Moment for Given Curvature •Curvature for Given Moment •Yield Moment •Stiffness •Ductility

•Moment for Given Load •Load for Given Moment •Capacity Ratio

•Mx for Given My •My for Given Mx •Capacity Ratio

Strength

Capacity Interaction Surface

Given Moment Direction


Capacity Interaction Surface P

My

Mx


P-M and M-M Interaction Curves


The Moment Curvature Curve


Cross-section Stresses


Original Cross-sections

Plain concrete shape

Compact Built-up steel section

Reinforced concrete section

Composite section

Compact Hot-rolled steel shape

Reinforced concrete, composite section


Sections After Strengthening


Strength and Performance • In Strength Design, every member and every cross-section must satisfy strength equation • Even if all members and sections are designed for strength, the structure may not perform well in case of overload • In Performance Based Design, only a few members on the critical load path need to perform well for the structure to perform well • Therefore for strengthening of structures, we may only need to strengthen members or section in the critical load path

Members on Critical Load Path


• In Performance Based Design, only a few members on the critical load path need to perform well for the structure to perform well

• Therefore for strengthening of structures, we may only need to strengthen members or section in the critical load path

What Effects Serviceability? • Anything that reduces cracking – The presence of appropriate amount of reinforcement at appropriate locations


• Anything that increases stiffness – Reasonable sizes and proportions of member cross-sections

• Anything that reduces Creep/ Shrinkage – Presence of compressive reinforcement

• Anything that improves Durability – High strength concrete – Proper cver and protection of rebars

What Effects Strength? • The basic Material Strength


– Concrete crushing strength – Reinforcement yield strength

• The Cross-section Dimensions • The amount of Rebars • The framing conditions


What Effects Performance? • Performance is generally of concern for lateral loads such as earthquake and wind • The main factor that effects performance is the Ductility of the members on the critical load path • In frame structures, the design of the joints between columns and beams is critical • The performance of shear walls if great importance for lateral load demands


• Ductility can be defined as the “ratio of deformation and a given stage to the maximum deformation capacity” • Normally ductility is measured from the deformation at design strength to the maximum deformation at failure

Load

Ductility – Definition and Usage

Yield/ Design Strength

Dy

Du

Deformation Ductility = Dy / Du

What Effects Ductility! • The most important factor effecting ductility of reinforced concrete cross-section is the confinement of concrete


– Amount of confinement steel – Shape of confinement steel

• Other factors include: – – – – –

Presence of Axial Load Stress-strain curve of rebars Amount of rebars in tension Amount of rebars in compression The shape of cross-section

Action – Deformation Curves


• Relationship between action and corresponding deformation • These relationships can be obtained at several levels – – – –

The Structural Level: Load - Deflection The Member Level: Moment - Rotation The Cross-section Level: Moment - Curvature The Material Level : Stress-Strain

• The Action-Deformation curves show the entire response of the structure, member, cross-section or material

How to Get Action-Deformation Curves • By actual measurements – Apply load, measure deflection – Apply load, measure stress and strain


• By computations – Use material models, cross-section dimensions to get Moment-Curvature Curves

• By combination of measurement and computations – Calibrate computation models with actual measurements – Some parameters obtained by measurement and some by computations


The Moment Curvature Curve

The Moment-Curvature Curve


• Probably the most important action-deformation curve for beams, columns, shear walls and consequently for building structures • Significant information can be obtained from Moment Curvature Curve to compute: – – – – – – – –

Yield Point Failure Point Ductility Stiffness Crack Width Rotation Deflection Strain


What is Curvature • In geometry, it is rate of change of rotation • In structural behavior, Curvature is related to Moment • For a cross-section undergoing flexural deformation, it can computed as the ratio of the strain to the depth of neutral axis

e C

Curvature = e / C (radian / unit length)


How to Read M-Phi Curve

Outputs from M-Phi Curve


2 -Failure Point

1 -Yield Point

y 3 - Ductility  u

Outputs from M-Phi Curve


4 - Stiffness of the Section at given M and Phi

M  EI M EI   5 - Slope of the section at given Moment b

M    dx EI a

Outputs from M-Phi Curve 6 - Deflection of the section at given Moment

M  D    x dx EI  a Pushover Analysis, ACECOMS, AIT

b

7 - Strain at given Moment

  c

c = distance from the NA to the point where strain is required

Outputs from M-Phi Curve 8 - Crack Width at given crack spacing

Specified Crack Spacing = X

W  s X Pushover Analysis, ACECOMS, AIT

W  yX



NA

y

Rebar Centroid

s

9 - Crack Spacing at given crack width W X

s

W X y

W

Outputs from M-Phi Curve - Summary Plot M-Phi Curve

EI 

Determine curvature at known moment




Determine Flexural Stiffness (EI) b

  a

M D   EI a b

M

 x dx 

Determine Deflection

  c

M dx EI Determine Slope

X 

Determine Strain

W

W  s X

s Determine Crack Spacing/Width

Outputs from M-Phi Curve - Example For M=600 Phi = 0.00006 From M-Phi Diagram

P=160 K

L/2 24 in

EI 

M 

36 in 15 ft


EI=600x12/0.00006 EI=1.2E8 k-in^2

Slope at Mid Span M=600 k-ft b

 

M dx EI

a =600x7.5x144/1.2E8 =0.0054 rad

Outputs from M-Phi Curve - Example Deflection at Mid Span

M  D    x dx EI  a Pushover Analysis, ACECOMS, AIT

b

=600x7.5x144x15x12/(6x1.2E8) =0.162 in


Strain in Steel

  c M = 600 k-ft, y=16

=0.00006x16 =0.00096



NA

y

Rebar Centroid

s

W

Outputs from M-Phi Curve - Example Crack Width Assuming crack spacing of 18 in


W  s X




NA

=0.00096 x 18 =0.01728 in

Crack Spacing Assuming crack width of 0.02 in

X

W s

=0.02/0.00096 =20.8 in

y

Rebar Centroid

s

W

M-Phi Curve and Ductility


• • • •

Effect of Axial Load Effect of Compression Steel Effect of Confinement Model Effect of Confinement Shape


Axial Load and Ductility

12#8 bars


Compression Steel and Ductility

a)

b)

8#8 8#8bars bars

2#8 bars

8#8 bars

c)

4#8 bars

8#8 bars

d)

8#8 bars

8#8 bars

Confinement Model and Ductility Effect of Concrete Confinement Model on Ductility of Cross-Section 350

300


Moment (kip-ft)

250

200

Whitney Rectangle Mander Circular Confined

150

Mander Pipe Filled 100

50

0 0

0.001

0.002

0.003

0.004

0.005

0.006

Curvature (rad/in)

a)

8#8 bars Whitney Rectangle (both)

b)

c)

8#8 bars 8#8 bars Whitney Rectangle (outside) Whitney Rectangle (outside) Mander Circular Confined (inside) Mander Pipe Filled (inside)

Confinement Steel and Ductility Effect of Confinement Steel Spacing on Ductility 160

140


Moment (kip-ft)

120

100

Spacing = 3in 80

Spacing = 6 in 60

Spacing = 12 in

40

20 0 -0.0005

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

-20

Curvature (in/rad) a)

8#6 bars Mander’s Rectangular Confined


Confinement Shape and Ductility

a)

8#6 bars Mander’s Rectangular Confined

b)

a)

8#6 bars Mander’s Circular Confined

8#6 bars Whitney Rectangle


Introducing

Pushover Analysis


The Pushover Analysis • An alternate method of analysis for carrying out the Performance Based Design • Pushover analysis is carried out after the Linear Analysis has been done and Serviceability and Strength design has been completed • Pushover analysis is most suitable for determining the performance, specially for lateral loads such as Earthquake or even wind

Why Pushover Analysis


• Buildings do not respond as linearly elastic systems during strong ground shaking • Improve Understanding of Building Behavior – More accurate prediction of global displacement – More realistic prediction of earthquake demand on individual components and elements – More reliable identification of “bad actors”

• Reduce Impact and Cost of Seismic Retrofit – Less conservative acceptance criteria – Less extensive construction

• Advance the State of the Practice

Performance Based Design - Basics • Design is based not on Ultimate Strength but rather on Expected Performance


– Basic Ultimate Strength does not tell us what will be performance of the structure at Ultimate Capacity

• Performance Based Design Levels – – – – –

Fully Operational Operational Life Safe Near Collapse Collapse


Pushover Spectrum


Pushover Demand Curves


Earthquake Push on Building


The Pushover Curve


Pushover Capacity Curves


Demand Vs Capacity

Non-linearity in Pushover • Material nonlinearity at discrete, user-defined hinges in frame/line elements. 1. Material nonlinearity in the link elements.


• Gap (compression only), hook (tension only), uniaxial plasticity base isolators (biaxial plasticity and biaxial friction/pendulum)..

2. Geometric nonlinearity in all elements. • Only P-delta effects • P-delta effects plus large displacements

3. Staged (sequential) construction. • Members can be added or removed in a sequence of stages during each analysis case.

Important Considerations


• • • •

Nonlinear analysis takes time and patience Each nonlinear problem is different Start simple and build up gradually. Run linear static loads and modal analysis first • Add hinges gradually beginning with the areas where you expect the most nonlinearity. • Perform initial analyses without geometric non-linearity. Add P-delta effects, and large deformations, much later.


Important Considerations • Mathematically, static nonlinear analysis does not always guarantee a unique solution. • Small changes in properties or loading can cause large changes in nonlinear response. • It is Important to consider many different loading cases, and sensitivity studies on the effect of varying the properties of the structure • Nonlinear analysis takes time and patience. Don’t Rush it or Push to Hard


Procedure for Pushover Analysis • Create a model just like for any other analysis. • Define the static load cases, if any, needed for use in the static nonlinear analysis (Define > Static Load Cases). • Define any other static and dynamic analysis cases that may be needed for steel or concrete design of frame elements.


Procedure for Pushover Analysis • Define hinge properties, if any (Define > Frame Nonlinear Hinge Properties). • Assign hinge properties, if any, to frame/line elements (Assign > Frame/Line > Frame Nonlinear Hinges). • Define nonlinear link properties, if any (Define > Link Properties).


Procedure for Pushover Analysis • Assign link properties, if any, to frame/line elements (Assign > Frame/Line > Link Properties). • Run the basic linear and dynamic analyses (Analyze > Run). • Perform concrete design/steel design so that reinforcing steel/ section is determined for concrete/steel hinge if properties are based on default values to be computed by the program.


Procedure for Pushover Analysis • For staged construction, define groups that represent the various completed stages of construction. • Define the static nonlinear load cases (Define > Static Nonlinear/Pushover Cases). • Run the static nonlinear analysis (Analyze > Run Static Nonlinear Analysis).


Procedure for Pushover Analysis • Review the static nonlinear results (Display > Show Static Pushover Curve), (Display > Show Deformed Shape), (Display > Show Member Forces/Stress Diagram), and (File > Print Tables > Analysis Output). • Perform any design checks that utilize static nonlinear cases. • Revise the model as necessary and repeat.


Summary • We have to think in terms of “Displacements” and not in terms of loads, stresses or strains • The main idea is to compare expected displacements or required displacements with the ability of the structure to reach those displacements without failing OR indicating that it will not reach those displacements

Performance Based Design and

Pushover Analysis Technical Background By:

Iqbal Suharwardy, PhD, S.E Director Development Computers and Structures Inc., Berkeley, USA

Performance Check for Structures • Purpose


– How will a structure perform when subjected to a given level of earthquake? • Definition of Structural Performance • Definition of Earthquake Level • Determination of performance level

Performance Check for Structures • Process – Guidelines for Seismic Rehabilitation of Buildings:


• ATC-40 • ATC-33 (FEMA 273 and 274)

– SEAOC Vision 2000 Framework


Types of Performance Checks • Linear Static Analysis • Linear Dynamic Analysis • Non Linear Static Analysis (Pushover Analysis) • Non Linear Dynamic Analysis

Performance Check Using Pushover

Force Measure


Expected Performance Point for given Earthquake

Performance Limits (IO, LS, CP)

Deformation Measure

Steps in Performance Check


• • • •

Construct Pushover Curve Select Earthquake Level to check Select Performance Level to check Select acceptance criteria for each Performance Level • Verify Acceptance – ATC-40 Method – ATC-33 Method

Constructing Pushover Curve • Define Structural Model – Elements – Strength-Deformation properties


• Define Loads – Gravity – Lateral Load Patterns

• Select Control Displacements or Drifts • Perform Pushover Analysis

Pushover Modeling (Elements)


• Types – Truss – Yielding and Buckling – 3D Beam – Major direction Flexural and Shear Hinging – 3D Column – P-M-M Interaction and shear Hinging – Panel Zone – Shear Yielding – In-Fill Panel – Shear Failure – Shear Wall – P-M-Shear Interaction! – Spring – for foundation modeling

Pushover Modeling (Properties) Force - Deformation Relationship

Force


C

B D A

Deformation

E

Pushover Modeling (Beam Element)

Three Dimensional Beam Element Pushover Analysis, ACECOMS, AIT

Span Loads

Shear Hinge

Flexible connection

Plastic Hinge

Rigid Zone

Pushover Modeling (Column Element)

Three Dimensional Column Element Pushover Analysis, ACECOMS, AIT

Shear Hinge

Plastic Hinge

Rigid Zone

Pushover Modeling


• Types of Deformation Properties – – – – –

Axial Moment only P-M : Uniaxial P-M Interaction P-M-M : Biaxial P-M Interaction Shear

Pushover Modeling (Loads) • Start with Gravity Loads – Dead Load – Some Portion of Live Load


• Select Lateral Load Patterns – – – –

Uniform Code Static Lateral Load Distribution First Mode Combination of Modes

Pushover Analysis (Control) • Force Controlled Analysis • Deformation Controlled Analysis – Roof Displacement – Generalized Displacement Definitions Pushover Analysis, ACECOMS, AIT

• Story Drift

• Limit of Analysis – Instability – Loss of Gravity Load Carry Capacity – Excessive Distortions

Pushover Analysis (Solution Schemes) • Event by Event Strategies – Manual


• Newton-Raphson Type Strategies – Constant Stiffness iteration – Tangent Stiffness iteration

• Problem of Degradation of Strength • Ritz Method (Reduced Space) Strategies


Use of Pushover Curve (ATC-40) • • • • •

Construct Capacity Spectrum Estimation of Equivalent Damping Determine Demand Spectrum Determine Performance Point Verify Acceptance

Use of Pushover Curve (ATC-40)

Spectral Acceleration


Capacity Spectrum

Spectral Displacement

Use of Pushover Curve (ATC-40) Response Spectrum (5% Damping)



2.5CA Cv/T

Time Period

Use of Pushover Curve (ATC-40) Reduced Spectrum (Equivalent Damping)



2.5CA/Bs

Cv /(T BL)

Time Period




Performance Point

Spectral Displacement


Force Measure


Expected Performance Point for given Earthquake

Performance Limits (IO, LS, CP)

Deformation Measure

Use of Pushover Curve (FEMA-273)


• • • •

Displacement Coefficient Method Estimate Target Displacement Verify Acceptance Estimation of Target Displacement – – – –

Estimate effective elastic stiffness , Ke Estimate post yield stiffness, Ks Estimate effective fundamental period, Te Calculate target roof displacement

Use of Pushover Curve (FEMA-273) • Estimation of Target Displacement


– – – –

Co, Relates spectral to roof displacement C1, Modifier for inelastic displacement C2, Modifier for hysteresis loop shape C3, Modifier for second order effects

SAP2000/ETABS Pushover Options


• Full 3D implementation • Single Model for – – – – – –

Linear Static Analysis Linear Response Spectrum Analysis Linear Time History Analysis Non Linear Time History Analysis Non Linear Static Pushover Analysis Steel and Concrete Design



• Generally Follows ATC-40 and FEMA-273 • Available Pushover Element Types – – – – – – –

Truss – Yielding and Buckling 3D Beam – Major direction Flexural and Shear Hinging 3D Column – P-M-M Interaction and shear Hinging Shell, Solids, etc (Considered Linear) Panel Zone – (later) Shear Wall – (Later) Non-Linear Spring – (Later)

SAP2000/ETABS Pushover Options Force - Deformation Relationship

Force


C

B D A

Deformation

E


Three Dimensional Beam Element Pushover Analysis, ACECOMS, AIT

Span Loads

Shear Hinge

Flexible connection

Plastic Hinge

Rigid Zone

SAP2000/ETABS Pushover Options • Strength – Deformation and P-M-M curves can be calculated by program for:


– Steel beams (FEMA-273) – Steel columns (FEMA-273) – Shear Hinges in EBF Links (FEMA-273) – Concrete Beams (ATC-40) – Concrete Columns (ATC-40) – Shear hinge in Coupling Beams (ATC-40)

SAP2000/ETABS Pushover Options • Gravity Load Analysis


– Nodal Loads – Element Loads – Load Controlled Analysis

• Pushover Analysis – Starts from Gravity loads – Nodal Load Patterns (User, Modal, Mass) – Multi-Step Displacement or Drift Controlled



• Available Results for each step of Loading – – – – – – –

Base Shear Element Forces Section Forces Joint Displacement Drifts Element hinge Deformations Limit Points reached



• Pushover Curve Post-Processing (ACT-40) – – – – – –

Conversion to Capacity Spectrum Calculation of Effective Period (per step) Calculation of Effective Damping (per step) Calculation of Demand Spectrum (per step) Location of Performance Point Limit Points (acceptable criteria) reached

SAP2000/ETABS Pushover Options • Visual Display for Each Step – Deformed Shape – Member Force Diagrams – Hinge Locations and Stages


• Graphs – – – – –

Base Shear VS Roof Displacement Capacity Curves Demand Curves Demand Spectra at different Damping Effective Period Lines


Examples

Example 1 P=100 Kip Gravity Load m=3.6

W36x120

10 ft


Lateral Push to 0.5ft Disp

Default M3 Pushover Hinge


Base Shear Vs Displacement


Capacity Spectrum

Example 2 P=Unit Load

Axial Force, P (Kips)

24"x24" Conc Col

12 ft


Desired Behavior

User P Hinge

2100 1700 1000

0.1

0.6

0.8

Measured Axial Displacement at Joint 2 (in)

Find Column E • Determine Column E to give Appropriate Initial Stiffness:

PL E AD

= (1700 *12*12)/(24*24*0.1) = 4250 Ksi



Column

Desired Behavior

2100 1700 1000

0.1

0.6

0.8


Find Column Deflection


Column

PL D AE

= [(2100-1700) *12*12)]/(24*24*4250) = 0.0235 in


• Determine Elastic Column Lengthening when loading from 1700 to 2100 K:

Desired Behavior

2100 1700 1000

0.1

0.6

0.8




Column D 

PL AE

= [(2100-1000) *12*12)]/(24*24*4250) = 0.0647 in


• Determine Elastic Column Lengthening when loading from 2100 to 1000 K:

Desired Behavior

2100 1700 1000

0.1

0.6

0.8



Column

PL D AE

= 1000 *12*12)/(24*24*4250) = 0.0588 in

Desired Behavior



• Determine Elastic Column Lengthening when loading from 1000 to 0 K: 2100 1700 1000

0.1

0.6

0.8


Find Hinge Properties Hinge Properties

B

D

E

2100 1700 1000

0.8

0.7412

0.4765 0.5412

A 0.0


1000


C

2100 1700

Desired Behavior

B = 0.1 - 0.1 = 0 C = 0.6 - 0.1 - 0.0235 = 0.4765 D = 0.6 - 0.1 - 0.0235 + 0.0647 = 0.5412 E = 0.8 - 0.1 - 0.0235 + 0.0647 = 0.7412

0.1

0.6

0.8



Hinge Properties


Pushover Curve

0.8 kip/ft

W14x90

Push 0 19 2x 0 W1 x1 W8

W1 2x 19 0 W8 x1 0

W14x90


Example 3 1.2 kip/ft 0.8 kip/ft

W24x55

Example 3 M3

M3


PMM

M3 V M3 M3 MR

MR

PMM

MR

P

PMM

M3

P Legend P = Axial Hinge MR = Moment Release M3 = Moment Hinge V2 = Shear Hinge PMM = PMM Hinge

PMM MR


With W12x190 Brace


With W8x10 Brace


Conversion to ADRS Spectra ATC-40

Response Spectrum Conversion


• Acceleration-Displacement Response Spectra (ADRS) • Every Point on a Response Spectrum curve has a unique – – – –

Spectral Acceleration, Sa Spectral Velocity, Sv Spectral Displacement, Sd Time, T

Response Spectrum Conversion • For Each value or Sai and Ti determine the value of Sdi using the equation 2


Ti S di  S ai g 2 4

• Spectral Acceleration and Displacement at period Ti are given by

2 S ai g  Sv Ti

Ti S di  Sv 2


Capacity Spectrum Conversion • Capacity Spectrum from Capacity or Pushover Curve • Point by Point conversion to first mode spectral coordinates • Vi and D roof on capacity curves are converted to corresponding Sai and Sdi on capacity spectrum using: Vi S ai  W

1

S di 

D roof

PF   1

1, roof




Moment Hinge Properties Using M-Fi Curve

Procedure


• Plot M-Fi curve for cross-section • Estimate EI value from M-Fi Curve using the following equation M  EI M EI  

• Calculate Rotations from Curvature using: b

M    dx EI a

• Reinforced Concrete Beam-Column CrossSection • 24”x24” • Reinforced with 12 #9 bars • Length is 12 ft

24"

24"


Example


Example

370

0.00028

Example


M EI   • So EI = 370/0.00028 = 1321428.6 b M M   Ip    dx EI EI a • So  = 0.00336 rad • Find  for other Moment Values and input in Hinge Property

Considerations


• Keep moment Constant over hinge length when integrating or integrate over the whole member length with actual moment diagram

• Only one value of EI at Yield is sufficient • Ip = h/2


Comparisons of SAP2000 and ETABS

SAP2000 vs ETABS •

SAP2000 – General Purpose FEA Software


– Classic Finite Element Software – Steel, and Concrete Frame Element Design – Shear Wall Design Not Supported – Fewer Automated Meshing Options – Does not Support Composite Design

•

ETABS – Specialized FEA Software for Building analysis and design – Fully Object based Modeling and Design – Steel, concrete, composite Frame Element design – Supports Shear wall design

– Full and practical auto meshing options – Supports Composite Design

SAP2000 vs ETABS •

SAP2000 – General output related to nodes and elements is reported

•

ETABS – Floor wise representation of results such as story drift, floor mass participation, story shear, etc.


– General Report (text files)

– Professional Report – Powerful load cases, combinations, envelopes, multiple case, etc. – Cables, Dampers, and NL Links and Hinges

– Relatively less ability to handle load combinations – Only Nonlinear links and Hinges

SAP2000 vs ETABS • SAP2000 – Supports Solid Elements


– Relatively low versatility for defining and editing grid systems

• ETABS – Does not support solid elements – Powerful grid system definition and editing


ETABS Pushover


ETABS Pushover


ETABS Pushover


SAP2000 Pushover


SAP2000 Pushover


SAP2000 Pushover


SAP2000 Pushover


SAP2000 Pushover


1 Use Load Patterns Steps to compute the Displacement (Displacement not Monitored) Divide the Specified Displacement into Steps and apply loads to attain that displacement Monitor which DOF at what level/story Save Positive Results only 2 After a member fails redistribute loads locally around failed members or reanalyze structure using a new stiffness matrix 3 Which Pattern Loads to apply and what is the scaling factor for each loading case included in the load factor

ETABS Pushover 4 Consider P-Delta effects and Large Displacements due to gravity loads caused by each step of lateral loading

1

2

4

3

5

5 For Construction Sequence analysis. Specify which Pushover case to be applied to which stage of construction or strengthening.


SAP Pushover

1 Weather to start from unstressed condition or if more than one Pushover cases are defined then may be start the later pushover case from the final state of the pervious case

2 When the load type in 3 is set to Loads this becomes irrelevant and if the Load Type in 3 is set to Acceleration then to find modal masses, select the analysis case from which the modal masses may be

3 Specify if Loads or Accelerations needs to be applied and what is the scale factor for each load case

4 Load Application Use full load application without monitoring the displacement or use the displacement control. Also specify the DOF to be Monitored and the Joint at which the DOF is to be monitored Results Saved Save Results at only final stage of Loading or after each step. Specify Max and Min number of steps Staged Construction For Construction Sequence analysis. Specify which Pushover case to be applied to which stage of construction or strengthening Nonlinear Parameters Those explained in 2 and 4 on previous slide

SAP/ETABS Pushover Output

2 3 4


1

1 V=Base Shear D=Displacement Sa=Spectral Acceleration Sd=Spectral Disp Teff=Effective Fundamental Period Beff=Effective Viscous Damping

5

2

3 Demand Curves plotted for these Damping Ratios 4 Grey Lines are the Constant Period Lines drawing for period specified here

5 If there is additional viscous damping provided in the structure, perhaps by viscous dampers that are not specifically included in the model The Structural Behavior Types A, B and C default to the values defined for those structural behavior types in Section 8.2.2.1.1 of ATC-40 . The User Defined Kappa

Pushover Analysis using ETABS and SAP2000

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