Manual Fagus 7

FAGUS

Manual since version 1.00 Copyright Cubus AG, Zurich

Table of Contents

Vol. A Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A1

A 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A1

A 1.1 What is FAGUS ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Basic Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Analysis Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 1.2 Program Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross section management / Cross section elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 1.3 Analysis Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 1.3.1 Bending with axial force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 1.3.2 Shear force and torsion for reinforced concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 1.3.3 Elastic shear flow analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 1.3.4 Limitations of beam theory and cross section considerations . . . . . . . . . . . . . . . . . . . . . . A 1.4 Checks to be performed by User . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 1.5 Limited Guarantee (excerpt from Cubus Licence Agreement) . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 1.6 Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A1 A1 A1 A2 A2 A2 A2 A4 A4 A6 A7 A7 A7 A8 A8

A 2 Starting the Program  Brief Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A9

A 2.1 Starting the Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CubusExplorer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 2.2 Brief Description of the Most Important Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A9 A9 A9

Vol. B Working with FAGUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B1

B 1 Overview and Management of Cross Section Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B 1

B 1.1 The Program Window of FAGUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 1.2 The Menu List of FAGUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 1.2.1 The menu 'File' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 1.2.2 The menu 'Options' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 1.2.3 The menu 'Display' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 1.2.4 The menu 'Window' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 1.2.5 The menu 'Help' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 1.3 Management of Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dialogue 'Cross Section Management' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 1.4 The Layer Buttons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B1 B2 B2 B3 B3 B4 B4 B4 B4 B6

FAGUS‐7

i

Table of Contents

B 2 Input of Cross Section Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.1 The Tab sheet 'Geometry' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.1.1 General information on the input of cross section elements . . . . . . . . . . . . . . . . . . . . . . . B 2.1.2 Drawn partial cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.1.3 Openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.1.4 Thinwalled partial cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.1.5 Parametrised cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.1.6 Standard rolledsteel sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.1.7 Introduction of existing cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.1.8 Axial points and handles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.1.9 PT constraint lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.1.10 Result points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard result points (only in layer button): . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.1.11 Result combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.1.12 Cross section results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.1.13 Checking the cross section input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.1.14 Legends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.2 The Tab sheet 'VTModel' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.2.1 Shear walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input of shear wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of shear wall forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical description of shear wall forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simplified shear wall input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Torsion values from basic cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weight factors for wall groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional input help for compact cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stirrups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.3 The Tab sheet 'Reinforcement' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.3.1 Reinforcement groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.3.2 Auxiliary points to graphically construct the reinforcement . . . . . . . . . . . . . . . . . . . . . . . . B 2.3.3 Point, line and circular reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.3.4 Tendons / Strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Initial strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slope of tendons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Longterm losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.4 The Tab sheet 'Variants' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 2.4.1 Properties of the variant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tab sheet 'Properties': . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tab sheet 'Member' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tab sheet 'VT Model' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tab sheet 'Cross Section Values' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii

B7 B7 B7 B8 B8 B9 B9 B11 B12 B12 B12 B13 B13 B14 B15 B15 B15 B16 B16 B17 B18 B19 B19 B19 B19 B20 B20 B20 B22 B22 B22 B22 B25 B25 B26 B27 B27 B30 B30 B30 B31 B32 B33

FAGUS‐7

Table of Contents

B 3 Design and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B34

B 3.1 Loading of Cross Section, Section Force Reference System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 3.2 The Tab sheet 'Reinforcement Design' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Automatic adjustment of the required reinforcement areas . . . . . . . . . . . . . . . . . . . . . . . Interactive selection of diameter and adjustment of the layer . . . . . . . . . . . . . . . . . . . . . . Numerical legend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 3.2.1 General remarks on the dimensioning of the axial reinforcement . . . . . . . . . . . . . . . . . . . B 3.2.2 General remarks on shear force and torsional dimensioning . . . . . . . . . . . . . . . . . . . . . . . Check the loading of the concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial reinforcement due to force in shear wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 3.3 The Tab sheet 'Analyses' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 3.3.1 Ultimate load/ Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 3.3.2 Stress analysis with forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crack width verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 3.3.3 Stress analysis with strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 3.3.4 Elastic shear flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 3.3.5 MN interaction diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 3.3.6 General interaction diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 3.3.7 Momentcurvature diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 3.3.8 Momentstiffness diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 3.3.9 Analysis of second order effects with axial load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 3.4 Batch Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 3.4.1 Management of batch analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B 3.4.2 Batch analysis/create task list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B34 B36 B37 B37 B37 B38 B38 B39 B40 B42 B42 B43 B44 B45 B46 B47 B48 B49 B50 B51 B53 B54 B54

Vol. C Special Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C1

C 1 Coordinate System and Sign Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 2 Section properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C1 C2

C 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 2.1.1 Area, moments of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 2.1.2 Torsional Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 2.1.3 Torsionskonstante Ix mittels FEAnalyse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C2 C2 C3 C3 C5

C 3 Analyses with ThinWalled Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C6

C 3.1 Shear Stresses due to Shear Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 3.2 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 3.2.1 Open Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shear Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Torsional Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Warping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 3.2.2 Closed Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C6 C9 C9 C9 C9 C10 C10

C 4 Shear and Torsion for Reinforced Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C11

C 4.1 National Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 4.1.1 EN 199211:2004 Eurocode 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 4.1.2 SIA 262 Swisscode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 4.1.3 EHE08 Spanish Code: Instrucción de Hormigón Estructural . . . . . . . . . . . . . . . . . . . . . C 4.2 Older Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 4.2.1 Design according to SIA 162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 4.2.2 Design according to DIN 1045 (07/1988) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 4.3 Design for torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 4.3.1 Überlagerung Querkraft und Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 4.3.2 Überlagerung Querkraft und Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 4.4 Combination of Shear Wall Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C11 C11 C13 C13 C14 C14 C14 C15 C15 C16 C17

FAGUS‐7

iii

Table of Contents

C 5 Overview of Parametrised Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C18

Vol. D Timber Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D1

D 1 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D1

D 2 Remarks for the analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D1

D 2.1 Material Specifics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D1

D 3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D2

D 3.1 Stress Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D 3.2 Efficiency, Capacity Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D 3.2.1 Bending and axial normal force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D 3.2.2 Shear and torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D2 D2 D2 D4

D 4 Composite Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D4

D 5 Calculations directly from STATIK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D4

Vol. E Introductory examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

E1

E 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.1 Presentation Conventions for the Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.1.1 Load Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.2 Starting FAGUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.3 Opening a Cross Section Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.4 Example 1: Cross Section with Simple Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.4.1 Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.4.2 The program window of FAGUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.4.3 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.4.4 Input of the cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.4.5 Inputting the opening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.4.6 Input of the reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.4.7 Tab sheet Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.4.8 Documentation of the cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensioning the cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Select content of figures with the layer buttons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enter figure in the print list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Creation of a text legend with print entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Print preview and printing with the CubusViewer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.5 Example 2: Composite Cross Section with Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.5.1 Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.5.2 Definition of the required materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.5.3 Input of the concrete slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.5.4 Input of the rolled steel section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.5.5 Definition of the variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.5.6 Input of the axis point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.5.7 Anchor points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1.5.8 Introduction of result points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

E1 E1 E1 E2 E2 E4 E4 E5 E5 E6 E7 E8 E11 E12 E12 E13 E13 E13 E14 E16 E16 E16 E17 E17 E18 E19 E19 E20

iv

FAGUS‐7

Table of Contents

E 2 Beispiele Analysemodul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

E21

E 2.1 Datei: 'S1' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 2.1.1 Träger, Bewehrungsbemessung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 2.1.2 Träger R2, Grenzwerte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 2.1.3 ITräger, Bemessung für kombinierte Beanspruchung . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 2.1.4 Kreisstütze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 2.1.5 Stützenquerschnitt, Steifigkeitsdiagramme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 2.1.6 Spannungsnachweise vorgespannter Träger, Schiefe Biegung . . . . . . . . . . . . . . . . . . . . . E 2.1.7 Interaktionsdiagramme für einbetoniertes Walzprofil . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 2.1.8 Spannungsanalyse für vorgespannten Brückenträger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 2.1.9 Brückenträger mit inaktiven Querschnittsteilen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 2.1.10 Dünnwandige Querschnitte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 2.2 Datei 'S2': Verbundquerschnitt mit Belastungsstufen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 2.3 Datei 'S3': Verbundquerschnitt unter Langzeitbelastung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

E21 E21 E21 E22 E22 E22 E22 E23 E23 E23 E24 E25 E26

Vol. F Fire Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

F1

F 1 Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

F1

F 1.1 Theoretic Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 1.2 Register 'Thermal Analysis' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 1.2.1 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal values for concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal values for steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special (Userdefined) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 1.2.2 Temperaturetime curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 1.2.3 Thermal bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tab sheet 'Thermal action' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tab sheet 'Encasement' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visibility and graphic representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 1.2.4 Analysis Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FEMesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 1.2.5 Run thermal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Check Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Run thermal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reset thermal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 1.2.6 Result Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 1.3 Validationexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 1.3.1 DIN EN 199112/NA:2010:12 Appendix CC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 1.3.2 EN 199212 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reinforced concrete column 30 x 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slab crosssection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

F 2 Mechanical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 2.1 Influence of temperature on material behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 2.1.1 Strain/Stress relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 2.1.2 Thermal strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 2.2.1 Basics of the solution process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 2.2.2 Using the program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 2.3 Utilisation / Capacity / Failure time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 2.4 Stress analysis with given forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 2.5 Stress analysis with given strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 2.6 Momentcurvaturediagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 2.7 Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

FAGUS‐7

F1 F3 F3 F4 F4 F4 F4 F6 F6 F7 F7 F7 F8 F8 F9 F9 F9 F9 F9 F11 F11 F11 F11 F11

F13 F13 F13 F15 F16 F16 F16 F17 F17 F17 F18 F18

v

Table of Contents

Vol. G Baustoffe, Analyseparameter . . . . . . . . . . . . . . . . . . . . . . . . . . .

G1

G 1 Baustoffe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

G1

G 1.1 BaustoffDialoge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G 1.2 Eindimensionale SpannungsDehnungsbeziehungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G 1.2.1 BetonDruckspannungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G 1.2.2 BetonZugspannungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G 1.2.3 Mitwirkung des Betons zwischen den Rissen mittels Verbundbeiwert . . . . . . . . . . . . . . . G 1.2.4 Betonstahl, Baustahl und Spannstahl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

G 2 Analyseparameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G 2.1 Der AnalyseparameterDialog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G 2.1.1 Registerblatt Grenzdehnungen und Grenzspannungen . . . . . . . . . . . . . . . . . . . . . . . . . . . G 2.1.2 Registerblatt Widerstandsbeiwerte / Teilsicherheitsbeiwerte: . . . . . . . . . . . . . . . . . . . . . . G 2.1.3 Registerblatt BetonDiagramm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G 2.1.4 Registerblatt Vorspannung: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G 2.1.5 Registerblatt Bewehrung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G 2.1.6 Weitere Werte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G 2.1.7 Zusätzliche Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G 2.1.8 Echoprint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

G1 G2 G2 G3 G4 G4

G6 G6 G6 G8 G8 G8 G8 G9 G9 G9

FAGUS‐7

A 1 Overview

Vol. A Introduction FAGUS‐7

A 1 Overview A 1.1 What is FAGUS ? FAGUS is a program for analysing the properties of cross sections, which can be used either alone or together with other Cubus programs. This chapter provides a general overview of its capabilities, the concept underlying its use and the way it works in conjunction with the other Cubus programs. In Part B an indepth description is given of each program component. Parts A and B together with the Introductory Example should be sufficient to allow one to use the program. The other sections deal with some specific aspects. These chapters can be consulted at any time using the program's Help System. They are available as PDF files and thus, if necessary, they can be printed individually.

The Basic Module EAx EJy

 E



The Basic Module is part of the STATIK packet and serves to analyse arbitrarilyshaped cross section geometries and calculate the cross section properties (Ax, Iy, Iz etc.). Simpler cross sections can be analysed directly in the corresponding structural member (beam) dialogue as socalled parametrised cross sections by supplying their dimensions. A simplified input is also possible for rolled steel sections. For the analysis of cross sections of arbitrary geometry the objectoriented Graphics Editor, which should be well known from the other Cubus programs, is used. The cross sections may then have an arbitrary shape described by polygonal or circular boundary. All similar types of cross sections are stored in a socalled Cross Section Library. A cross section library is in fact a Windows folder containing FAGUS cross sections. It may be an analysis folder, which is created directly by FAGUS (or in the CubusExplorer for FAGUS), or a project folder from another Cubus program, e.g. STATIK, which also works with FAGUS cross sections. The cross sections stored in various libraries can be imported and exported.

The Analysis Module The Analysis Module, which has to be purchased as an additional option, includes the 

Interactive analysis and dimensioning of reinforced and prestressed concrete cross sections (for bending about one or two axes, together with axial and shear forces and torsional moments).



Dimensioning and verification of whole frame structures (= postprocessing with an automatic import of section forces from STATIK)



Dimensioning and verification of timber cross sections

The computational core of FAGUS is also used in other Cubus programs (STATIK, CEDRUS, LARIX, PYRUS). Therefore, with the aid of interactive FAGUS the corresponding cross section results can be reproduced in detail interactively.

FAGUS‐7

A-1

Vol. A Introduction

A 1.2 Program Input With the help of figure A1 it will be attempted to explain graphically some technical terms and concepts.

cross section analysis M N

 c



Material management Corresponding to the two abovementioned program modules the material management consists of two parts. If only cross section properties (and possibly masses) have to be calculated, it is sufficient if the parameters on the left side are defined. If the cross section is analysed using FAGUS additional material parameters have to be known. This is achieved by assignment to a Material Class. The modulus of elasticity therefore is defined twice. The value on the left side determines the corresponding section forces or deformation analysis of the associated FE program (STATIK, CEDRUS) and can be changed at any time for special analyses without affecting the cross section analysis.

Cross section management / Cross section elements For a valid cross section that can be used in STATIK at least one complete boundary or a socalled thinwalled section has to be input. Parametrised cross sections or steel sections provide simplified input aids, by extending these elements, if necessary. If reinforced concrete analyses have to be carried out, the reinforcement also has to be known. A cross section can be dimensioned if a minimum of one reinforcement layer (line or point) is present on the tension side. In order to be able to switch quickly between different reinforcement layouts, it is best to group together the individual reinforcement elements to form Reinforcement Groups. Besides bending with axial force, with FAGUS the effect of shear force and torsion can also be investigated. For an elastic consideration of shear flow the cross section must consist of thinwalled elements. In the case of reinforced concrete analyses, as ”structural modelling elements” socalled shear walls are used, which have to be input in addition to the existing cross section boundary.

Cross section variants Cross section variants are needed for STATIK construction states. If the cross section changes during construction, this can be achieved with the aid of cross section variants by activating/deactivating individual cross section elements. A new cross section always has the variant "Standard".

General cross section attributes Each cross sectional element is input graphically as a point, line, polygon, etc., and is assigned additional attributes (e.g. material for point reinforcement). Properties, which apply to the whole cross section, are called variant properties.

Analysis parameters All other settings that cannot be included in the cross sectional or material properties are contained in the analysis parameters. These, e.g., are code provisions such as limiting strains, minimum reinforcement contents, etc. In all Cubus programs there is an identical program module for the management of the materials and analysis parameters. The description can be found in a separate document.

A-2

FAGUS‐7

A 1 Overview

FAGUS Basic Module

Analysis Module

Material management Materials

Material classes Component

E

G

ρ

concete concrete

general columns

36 38

15 13

2.5 2.5

reinf. steel

general

210

80

8.0

ID

Type

C CC R

Material class C40/50

fcd 24

Ecm

fctm

36

3.5

The lists can be extended by the user.

Cross section management Cross sections consisting of geometry and material

Analysis parameters

Partial cross sections

Predefined lists for different codes

circumscribed:

steel section

polygon

Name

Partial factors

 s, adm t u p n I

max  c 20 o/oo 1.5

ULS

reinforcement

 p

1.15

150 N  mm2

SLS thinwalled:

 s

The list can be extended by the user.

point

line

circular

untensioned or prestressed

shear walls

Section forces / strains

 cross section variants (arbitrary combinations of partial cross sections and reinforcement of a cross section)

N

M

reinforcement design t u p t u O / s i s y l a n A

static cross section properties mass per unit length of member

ultimate load analyses stress analyses interaction diagrams moment/curvature diagrams moment/stiffness diagrams shear flow distributions

Frame analysis program STATIK

STATIK Post−processing

Fig. A−1 Overview of cross section program FAGUS

FAGUS‐7

A-3

Vol. A Introduction

A 1.3 Analysis Modules A 1.3.1 Bending with axial force The figure A2 shows various possibilities of the Analysis Module: Top left is the simplest type of analysis, the stress analysis for a given strain plane, shown schematically. If the strain plane is known, a cross section integration suffices (summation of all stresses) to obtain the section forces. The assumptions made there also apply to all subsequent analyses listed below: 

Cross sections remain plane (Bernoulli hypothesis), i.e. linear strain distribution over the cross section (the strain plane).



Fully effective bonding between concrete and reinforcing steel. Exception: prestressed elements can be introduced with the option 'without bonding'.



Zero concrete tensile strength, i.e. cracked concrete tensile zone (= State II). (For special analyses a small amount of concrete tensile strength can be taken into account.)

Regarding computational effort, the executed stress analysis for given forces top right requires some extra computational effort. The associated strain plane, corresponding to the given section forces, can only be determined after many iterations. The procedure is as follows: 

Assumption of a strain plane



Determination of the internal forces and moments (integration over cross section)



Comparison of external and internal forces. If the outofbalance loads are too great, the iteration is repeated with an improved strain plane.

In the determination of the ultimate load or the strength utilisation (efficiency) of the given section, forces are increased continuously until a specified limit state is reached, i.e. until the edge strains defined by the analysis parameters are reached. The analysis is carried out as a repeated stress analysis and is therefore more timeconsuming. The program permits, among other things, simply increasing individual section force components. In reinforcement design the area of reinforcement is determined such that a particular loading can just be resisted. The dimensioning is based on the ultimate load analysis. For the existing reinforcement an efficiency factor is determined. If this is insufficient the area of reinforcement is increased until the required value is reached. In the case of beam cross sections, primarily the tension side is dimensioned, whereas in the case of column cross sections the complete reinforcement. Individual reinforcement layers can be defined as "constant". In this case their area remains unchanged during the dimensioning process. The diagrams shown in the lower part represent the graphical evaluations of the analyses described above: In the case of the simple MyN interaction diagrams all section forces are displayed graphically for a number of strain planes corresponding to a limit state. This analysis is performed for the reinforcement already present in the cross section or for specific reinforcement contents. Each point in the diagram can also be reproduced with the analyses carried out above. The general section forceinteraction diagram is determined using a number of ultimate load/efficiency analyses. Of the three components, one component must be chosen to be constant, so that the corresponding (2dimensional) diagram can be constructed. However, several curves can be displayed simultaneously in the same diagram (e.g. for N=100 kN, 200 kN, etc.). The last two diagrams represent graphical evaluations of stress analyses. For a particular axial force the bending moment is increased continuously and the result (stiffness orcurvature) is displayed graphically.

A-4

FAGUS‐7

A 1 Overview

Stress analysis for given force

Stress analysis for given strain plane 

 (y,z)



 (y,z)

(y,z)

?



M

N

N

 (y,z)

?

M





Method of solution: Method of solution:

Iteration:

Integration of the stresses over the cross section

given external section forces

Reinforcement design

Ultimate load / Efficiency

 ..u

N



?

M

variation of the strains until internal section forces =



.N



.M

Nd

?

Md

As=? Method of solution:

Method of solution:

Section forces varied until the limit state is reached

As varied until efficiency =1

(repeated stress analysis)

My-N interaction diagram

General interaction diagram

N

One value from N, My, Mz kept constant e.g.

M

z N=constant (given value)

M y M y

Method of solution:

Method of solution:

Section forces determined for all possible limit states

Repeated ultimate load analysis (with 3rd component constant)

Moment-stiffness diagram

Moment-curvature diagram My

EI y

 y

My Method of solution:

Method of solution:

Repeated stress analysis with N=constant

Repeated stress analysis with N=constant

Fig. A− 2 Overview of analysis types

FAGUS‐7

A-5

Vol. A Introduction

A 1.3.2 Shear force and torsion for reinforced concrete For the analysis and dimensioning in the case of shear force and torsion, the polygonal boundary is not enough to automatically generate from it a suitable reinforced concrete model. The user therefore also has to define a socalled shear wall model, as shown in the figure below.

Fig. A− 3 Shear wall model for bridge beam

Here it is assumed that a shear wall can only resist forces in the direction of the plane of the wall. The sum of all inplane forces must be in equilibrium with the external loads. Thus, at least three shear walls are needed that do not intersect at a point to be able to resist an arbitrary combination of shear force and torsion. If one has more than three shear walls the problem is statically indeterminate and the distribution is carried out on the basis of the existing stiffnesses. In the literature on earthquake design for buildings it is, e.g., proposed to distribute the forces according to the moment of inertia of each wall. Such an analysis can also be carried out using FAGUS . Vz T

Centre of stiffness: y s



z s



Vy

I iy �y i)  (

 I iy I iz �z i)  (  I iz

Fig. A− 4 Walls in building plan view

If the inplane forces are known, the required reinforcement content can be determined from the corresponding provisions of the code. compression flange

In FAGUS z is defined by the shear wall geometry (= user input), i.e. there is no automatic import from the bending analysis!

z tension flange

A sl2

A sl2

diagonal strut

V M N shear reinforcement

Fig. A− 5 Reinforced concrete truss model

Further information on this topic can be found in the following sections :

A-6



Ch. B 2.2 : Tabsheet 'Shear Walls' General description of shear wall input



Ch. B 2.4 : Tabsheet 'Variants' > 'VT Models' Control possibilities open / closed cross sections, etc)



Ch. C 4

: Further explanations on determining shear wall forces. Dimensioning requirements of different codes.

FAGUS‐7

A 1 Overview

A 1.3.3 Elastic shear flow analysis

V

A quite different approach is available for cross sections that are made up of thinwalled elements. An elastic shear flow analysis is sometimes needed for the analysis of steel cross sections, but can also be used for comparison purposes for the type of wall loading described above. In addition to the flow diagram shown on the left, summed shear stresses can be output for each wall. These inplane forces can then, in special cases, be compared with the reinforced concrete shear wall forces.

A 1.3.4 Limitations of beam theory and cross section considerations In principle, it should be mentioned that for a cross section analysis it is required that the assumptions of the beam theory apply. However, a pure consideration of the cross section, strictly speaking, is only permissible in the B regions (Bending, Beam, Bernoulli) shown below with the following conditions: 

uniformly distributed loading



uniformly or gradually changing thickness



cross sections remain plane

In the other regions (nodes, geometrical Discontinuities, etc.), often additional considerations (truss models, stress fields) are often necessary. concentrated point load

corner of frame

D

B

D

B

support

B

D

beam with opening

gantry crane  corbel

B

D

B

D

Fig. A−6 Comparison of Beam and Discontinuity regions

A 1.4 Checks to be performed by User A computational model is always a simplification of reality. Thus, it is important to interpret the results and to obtain an overview by varying the model parameters. Not all analysis quantities have the same influence on the final result! Besides questions of modelling, there are a number of error sources, beginning with the data input, followed by numerical problems. Finally, there may be programming errors, which despite taking every care in the development work can never be completely excluded. The user is responsible for the interpretation and application of the results. It is required that the results are randomly checked using a simple calculation by hand together with plausibility considerations.

FAGUS‐7

A-7

Vol. A Introduction

A 1.5 Limited Guarantee (excerpt from Cubus Licence Agreement) Limited Guarantee  Cubus guarantees for a period of six months from the first date of de-

livery that the software basically works according to the accompanying computer manual and that the hardware module distributed with the software is free from material and workmanship defects. Satisfying customer demands  the complete liability of Cubus is covered by the choice

of either (a) refunding the whole of the payment price or (b) the repair or replacement of the software or hardware, which does not fulfil the Cubus guarantee, provided they are returned to Cubus. This limited guarantee does not apply if the malfunctioning of the software or hardware is due to an accident, misuse or inappropriate application. No other guarantee  Cubus excludes every additional guarantee regarding the software,

the accompanying manuals and the hardware module supplied. No liability (without restriction) regarding consequential damage  neither Cubus nor

its suppliers are liable to pay compensation for any damage (including damage due to a loss of profit, business interruption, loss of business information or data or any other financial loss), arising from the use of the Cubus product or the inability to use this product. In any case the liability of Cubus is limited to the purchase price the user actually paid for the product.

A 1.6 Codes In accordance with the standard verifications carried out today, various limit states (ultimate load/serviceability) have to be investigated. FAGUS is appropriate for determining the cross section behaviour, i.e. the section forces have to be known! In the interactive mode of operation the section forces have to be input manually (design level!) and each analysis has to be performed individually. Thus this mode mainly serves preliminary dimensioning or the verification of certain results in other programs. If FAGUS is started from CEDRUS, STATIK or LARIX they produce the envelopes of all combinations that have to be investigated and the safety factors are automatically set correctly according to the particular verifications that are required. For some codes, both materials properties and analysis parameters are tabulated. Under the term 'analysis parameters' all values are included that influence the analysis in one way or another, e.g. limiting strains, partial safety factors, etc. Of course, not all codes can be modelled on a reference code simply by means of parameter selection. Especially for shear force and torsion each country has its own code provisions. Standardisation within Europe has been achieved through Eurocode EC2 (Edition:2004). Within the framework of code selection it is shown whether the selected code is based on an "original" implementation or uses a reference code that can be simulated by means of parameter adjustment. It may be that the selected code only carries out the analyses for bending and normal force. Although many technical terms were unified in the new Eurocodes, even in German speaking countries certain differences exist, e.g. SIA262 :  s = “Widerstandsbeiwert für Betonstahl und Spannstahl” (Resistance factor for ordinary reinforcing steel and prestressing steel) EC2,DIN 10451 :  s = “Teilsicherheitsbeiwert für Betonstahl” (Partial safety factor for ordinary reinforcing steel) As a basis for the program development, among others the following documents were used

A-8



EN 199211: Planung von Stahlbeton und Spannbetontragwerken (Eurocode EN 199211: Design of Concrete Structures)



SIA Normen 260,261,262, ...: Schweiz. Ingenieur und ArchitektenVerein (Swiss Codes SIA260 SIA267,etc.)



DINNorm 10451: Tragwerke aus Beton und Stahlbeton (German Code DIN 10451)

FAGUS‐7

A 2 Starting the Program - Brief Description

A 2 Starting the Program  Brief Description A 2.1 Starting the Program FAGUS can be started in two ways: 

By direct execution of the program file FAGUS7.EXE, e.g. using the Windows Start Menu (with the standard installation: [Start] > Programs > Cubus > FAGUS7) FAGUS7 is displayed with an empty window and the following menu bar:

This method of starting is recommended above all if one wants to continue with one of the recently modified cross section libraries (these are listed in the File menu). 

Using the CubusExplorer: The CubusExplorer is an independent program for managing the analyses of the different Cubus programs and is also called using the Windows Start Menu (with the standard installation: [Start] > Programs > Cubus > CubusExplorer)

CubusExplorer Jede Installation von CubusProgrammen beinhaltet den CubusExplorer. Es ist das zentrale Modul bei der Verwendung der Programme und dient folgenden Zwecken: 

Start der einzelnen Programme



Verwaltung der Berechnungsdaten



Verwaltung der Installation über den CubusManager, der vom CubusExplorer aus aufgerufen wird

Start CubusManager zur Verwaltung der Installation

Start der Anwendungsprogramme der aktuellen Generation

Verwaltung der Berechnungsdaten

Start von Programmen früherer Generationen

A 2.2 Brief Description of the Most Important Points 

FAGUS‐7

Working with the mouse: In Windows one generally works with the left mouse button. Clicking on or selecting a symbol on the screen involves moving the mouse pointer over the symbol and then pressing for a short time on the left mouse button. The right mouse button is only for displaying a context menu on the screen in a particular situation.

A-9

Vol. A Introduction

Online Help:

Many of the dialogue windows that appear during input have a Help button. The corresponding link brings the user directly to the corresponding page in the PDF manual. (Depending on the zoom factor the page may have to be moved up or down a little. ) By pressing the key, while the mouse pointer is situated over a button, the FAGUS manual is opened ( during the input of a graphics element opens the manual of the Graphics Editor.) Input sequence for the cross section input:

For the first input of a cross section one proceeds as follows: 1) New cross section



Select button for new cross section; input desired name



Work through the Control tabsheet sequentially from left to right (geometry, shear walls, reinforcement)



Select desired cross section elements. The input sequence is basically free. However it is recommended to work through the tabsheets and the buttons sequentially from left to right.



As soon as the corresponding button is clicked with the left mouse button, the properties dialogue is opened and all numerical attributes can be input. At the same time all currently permissible drawing tools are activated on the left side of the screen.

2) Tabsheet selection 3) Cross section element

4) Select desired attributes

5a) 'Create'

A-10

FAGUS‐7

A 2 Starting the Program - Brief Description



5b) Graphics Editor Tools

6) Input boundary

y 4, z 4

y 5, z 5 y 7, z 7

y 2, z2 y 3, z 3

y 8, z8

y 1, z1

The geometrical input procedure is now started using the button 'Create' or using the corresponding input symbol on the left side of the screen. The button 'Create' starts the polygon input. Alternative tools of the Graphics Editor are available on the left side of the screen, whereby instead of the button 'Create' one of the symbols is selected. Then the individual coordinate pairs of the bounding line are input either with the keyboard or with the mouse (the hint on the mouse pointer indicates what the program now expects and how the input is closed). At the end of the manual there is an example with a complete cross section input ("key by key"). A complete description of all possibilities of the Graphics Editor is available with the Help function (grabbing, grid points, relative, orthogonal input ...) A cross section boundary may consist of individual lines, of a closed polygon or of different polygon sections. With [OK] the dialogue ”Modify cross section” is closed and from the input the program tries to form a single closed polygon. Each tabsheet has a Check function with which the input thus far can be checked.

Subsequent changes / objectoriented procedure

Existing input can be modified by clicking on the corresponding object and then the context menu is activated using the right mouse button. If, e.g., parameters are to be verified or changed, the properties dialogue must be called. After the selected input fields have been modified, the changes are made effective by using the button 'Apply' . (Apply is only active if something was actually modified). The number of “affected” objects is shown in brackets. Input of shear walls

If a reinforced concrete cross section is subjected to shear forces or torsional action, shear walls have to be input. The tabsheet “Shear Walls” is available for this purpose. A single shear wall is input geometrically as a line. For complicated (noncontiguous) cross sections there is the Help function “Generate shear walls automatically”, which attempts to analyse an existing outline geometry and suggest the most feasible solution. Possibly, this suggestion still has to be modified manually. Input of reinforcement

Reinforced concrete cross sections should contain at least two reinforcement layers (with an initial reinforcement content), to allow them to be analysed or dimensioned. The geometrical positioning is performed very quickly and accurately, if beforehand stirrups have been input. Here too there is a function, which, based on the existing shear walls and the cross section geometry, makes a fairly reasonable suggestion. Afterwards the cross section is ready for dimensioning or analysis. These are described in Part B of this manual

FAGUS‐7

A-11

Vol. A Introduction

A-12

FAGUS‐7

B 1 Overview and Management of Cross Section Data

Vol. B Working with FAGUS FAGUS‐7

B 1 Overview and Management of Cross Section Data B 1.1 The Program Window of FAGUS Each purchased license of FAGUS can only run on one computer at a time, but several cross section libraries (analysis windows) can be open at the same time. After opening a cross section library, the FAGUS window looks as follows:

Menu

Management of cross sections

Graphics Editor functions and tools:

Drawing tools

Control Tab sheet Drawing area

Selection Part of drawing (Zoom...)

Undo/Redo Layer switches Input options

Coordinates input fields

Exaggerated scale

Status line

In general, the first time a cross section is input the individual Tab sheets should be worked through from left to right. As mentioned previously, the computational core “FGx.DLL” is also used by other Cubus programs. In this way it is ensured that the cross section analysis can be carried out with all these programs. The interactive program FAGUS serves to create cross sections and to analyse them manually.

FAGUS‐7

B-1

Vol. B Working with FAGUS

B 1.2 The Menu List of FAGUS B 1.2.1 The menu 'File' CubusExplorer: The CubusExplorer is a program for managing the analysis data. It looks

like the normal Windows Explorer, but only “Cubus Folders” are shown. The CubusExplorer program contains its own Help menu with a detailed description. New: With this command a new FAGUS cross section library is created. The command starts

the CubusExplorer in the limited functionality of creating a new analysis folder (for details see there). Save: The current state of the active cross section library is saved under the current analysis

name. There is an automatic save when starting an analysis or exiting the program (possibly after a request for confirmation). Save as ..: The current state of the active cross section library is saved under a new name. Close: The current state of the active cross section library is closed. However, FAGUS itself

is not exited. Documents: In the dialogue 'Documents' additional documents together with the analysis

data can be saved. The insertion (pasting) of ones own documents (“PDF”, “Word”) is carried out using “Drag&Drop” in the shown dialogue area. Some of the documentation of the examples supplied with the program by Cubus are also saved in this area. Print: Calls the submenu on the left which offers the following possibilities: Print immediately: Direct print of the current working area. A dialogue window appears

for setting certain parameters or defining any desired part of a figure. Print entry: The content of the current working area is entered in an output list to be prin-

ted later. A dialogue window appears for setting certain parameters or to select any desired part of a figure. The print preview program is called CubusViewer and is described in a separate document. Print preview: Starts the CubusViewer to view and print the documents entered in the list.



These print functions may also be activated using the buttons shown on the left Import: With this function, depending on the source, the geometry or cross section data

from external files can be imported. The submenu shown on the left appears. 

FAGUSx..y: With this command data from older FAGUS analyses can be imported. (If

with this project STATIK data has also be saved, the import should be carried out using STATIK , so that both STATIK data and cross section data can be imported) 

DXF: Serves to import geometrical data from a DXF file.

Export: Serves to export the graphics in the drawing area in different graphics formats. (It

can also be called directly using the right mouse button) Recovery: This function serves to reconstruct partially damaged data structures.



Parallel to the binary data, the input data can also be saved in a text file. With the Recovery command the input from this “Backup“ can be read in. Additional information such as labelling and dimensioning, however, are not included. If necessary, all cross sections in a cross section library can be recovered immediately using the commandRecovery > Cross Section Library as with the method described above.



The exchange of binary data created by the various program versions basically proceeds only ”upwards", i.e. a new program can read old data but not vice versa ! 1 .. 2 .. (Lists with the last cross section libraries to be used): As usual in Windows pro-

grams , the File menu provides a list to select the most recently used cross section libraries. This is the quickest way to continue working with the cross section libraries just used.

B-2

FAGUS‐7


End: Closes all analyses that may still be open and then closes FAGUS.

B 1.2.2 The menu 'Options' General: Starts the dialogue 'General Settings', which among other things manages the

parameters: 

Code



Descriptions of objects and parts of structures, authors, any commentary, etc.

Reinforcement/Checks:



Reinforcement: Serves to manage the reinforcement diameters suggested by the program.



Checks: Before allowing a cross section to be analysed, several checks are carried out. Certain tolerances and checks to be performed can be changed or omitted here for special requirements

Dialogue settings:

This function serves to manage the default dialogue values. On closing a dialogue window, the values used by the user are saved as default values for the next session. This only applies, however, to the current cross section library. In the case of new analyses an installation default value applies, which can be overwritten as follows (for each user, identified by Login Name): 

Save as default value (with a corresponding open dialogue) or using the key combination .



If the current dialogue needs to be overwritten by the user settings previously saved, this can be achieved by Inputting default value or using the key combination .

The above two commands are confirmed in the confirmation dialogue. Materials: The dialogue 'Materials' manages all materials used in an analysis. Each of these

materials has a specific name for the analysis, which can be assigned to the individual elements of a cross section. In the case of a bridge, e.g., with two types of concrete and one type of reinforcing steel, these three materials are defined at the start of an input using provisional input data. Later, e.g., a concrete class can be changed and all elements with this reference are adjusted automatically, i.e. no modifications are necessary for the individual cross sections. Analysis parameters: The analysis parameters include all additional analysis parameters

required for the FAGUS analysis. For the analysis parameters and materials there is a description that is valid for each application. Minimise dialogues: Dialogues often cover a large part of the screen and hamper the in-

troduction of new objects. If this menu option is activated the dialogues are reduced to their headings when introducing new objects and after completing the action they are displayed once more in full size. If one moves over the heading with the mouse, the whole dialogue is also shown. Automatic save: Using this function the time interval for an automatic save can be set. Language: Serves to change the program language. For a complete recreation of all dia-

logues it is best to exit the program and make a restart. Existing print entries remain in their original language.

B 1.2.3 The menu 'Display' Units/Number of decimal places: This command opens a dialogue window for the selec-

tion of units and the number of decimal places. The settings made here are saved userspecific (not analysisspecific).

FAGUS‐7

B-3


Colours/ Line types: Line thicknesses, line types, colours and fill can be set in this dialogue

separately for the screen, black and white and coloured printers. The settings made here are saved userspecific. Font size: This allows the font size to be changed in the graphics input. This setting, how-

ever, has no influence on the font size in the printed output. For this we have the setting in the Print Entry dialogue or in the CubusViewer.

B 1.2.4 The menu 'Window' This menu is only important if several cross sections (=windows) are currently in use. It allows arranging the windows according to various criteria.

B 1.2.5 The menu 'Help' This provides access to the various Help documents of FAGUS. In addition, the dialogue window 'About FAGUS' can be called, which gives information on current program versions, etc. Several other examples can also be found under the menu item 'Help'.

B 1.3 Management of Cross Sections At the top of the program window there are various aids for managing a cross section library. The list field serves to select the cross section to be modified. New cross sections can be created by clicking on the the button on the left. The name of the new cross section can be input in a separate dialogue window. Cross sections can also be duplicated. The button on the left opens a window in which the name of the duplicate can be input. The name of a cross section can be changed by clicking on this button. By clicking on this button existing cross sections can be deleted. It should be noted that cross sections deleted in this way are no longer available. 

If FAGUS is started from another program, it is not possible to delete or rename cross sections, materials or analysis parameters. This has to be done in the original program.

Dialogue 'Cross Section Management' The button on the left opens the dialogue 'Cross Section Management', which can be used to copy cross sections between different cross section libraries. This cross section management can be started both in STATIK and in FAGUS.

B-4

FAGUS‐7


The dialogue consists of a window with two halves. On the left side the cross section library of STATIK or FAGUS that is currently in use can be displayed. At first the right side is empty. By clicking on this button the CubusExplorer is started and the user can then select any other cross section library, whose contents are shown on the right side. With this button you can delete one or more selected cross sections. With this button you can copy individual cross sections selected on the right side into the current project. With this button all cross sections can be copied from the right side to the left side. If a cross section with the same name already exists, the cross section to be copied is renamed in the process.

FAGUS‐7

B-5


B 1.4 The Layer Buttons On the right of the Application window layer buttons are grouped individually. Normally the same symbol is used for the mouse pointer as in the corresponding Control Tab sheet. The hint given by the mouse pointer indicates which elements are to be found in this layer. Cross section layer:

The first group of layer buttons contains the graphics objects input in the Tab sheet 'Geometry'. In the following, two special layer buttons are discussed: Direction of drawing the polygons: In the case of tapered members (e.g. haunched

beams) the cross sections lying within the member have to be interpolated from the member's end cross sections. This is only possible if the two cross sections actually match each other. Thus it is sometimes necessary to specify the drawing direction and especially the starting point of the polygon. The first polygon side is marked on this layer using an arrow (Change: 1: Select polygon, 2. 3. From context menu select 'First polygon side'). Standard result points: FAGUS generates for each cross section a number of standard res-

ult points which can be used in STATIK to determine cross section results. These points denoted by A, TA, BA (for plane frames) and also AR, AL, TR, TL, BR and BR (for space frames) can be displayed with this layer button. It should be noted that these points are only displayed if the cross section has been checked successfully. If necessary, the button 'Check' must be activated to show these points again. Reinforcement layer: For each input reinforcement group in the Tab sheet 'Reinforce-

ment' this group contains a button. The group Output contains four buttons for dimensioning lines, labelling the cross section elements, details regarding the labelling and a button for scaled text display. Scaled text signifies that the text size parallel to the screen excerpt/zoom factor is changed. (For general information on text size see menu > 'Display' > 'Font Size') Shear walls: Top left this group has a button to display the shear walls and the stirrups.

Moreover, there are three Help buttons that can be used to display individual active components (Vy, Vz, T) (corresponding to the input shear wall attributes).

Showing the centre of gravity and the centre of shear: The group on the left contains the

centre of gravity(S), the principal axes and, if present, the centre of shear (M) of the cross section. In the Tab sheets 'Variant', 'Reinforcement' and 'Analysis' the graph is shown dependently of the cross section variant. In the other cases (and if no variant is selected) the values of the basic cross section are shown. Results: With the first four buttons the result for bending with normal force can be influ-

enced: strain plane, stress diagram, inner lever arm The second four buttons are for management of the shear wall results: group button, shear forces, shear reinforcement, stress in stirrup

User: In the layer group 'User' one can draw freely, i.e. the inputs made here have no influ-

ence on the analysis. Normally, the User layer is used to include dimensioning lines and additional labelling. For the corresponding tools to be available on the left side this layer has first to be 'activated'. (Click on User layer with left mouse button> activate). If necessary, new layers can also be created or layers no longer needed can be deleted.

B-6

FAGUS‐7

B 2 Input of Cross Section Data

B 2 Input of Cross Section Data B 2.1 The Tab sheet 'Geometry' In this Tab sheet drawn (polygon input) or thin walled partial sections are defined, as also result points :

s s n g o i n i t c n e e p s l O a i t r a p n w a r D

s n o i t c e s l a i t r a p d e l l a w n i h T

s n s o n i o t i c t e c s e s s l s e o e r t c S d d e l e s l i r o t e R m a r a P

s n o i t c e s s s o r c g n i t s i x e t r e s n I

t n i o p s i x A

) t n i o p ( e l d n a H

s e n i l t n i a r t s n o C T P

s t n i o p t l u s e R

s n o i t a n i b m o c t l u s e R

s t l u s e r n o i t c e s s s o r C

k d c n e e h g C e L

s t c e j b o e l b a t c e l e S

B 2.1.1 General information on the input of cross section elements A cross section element consists of the geometry, material and possibly some other attributes. After selecting the corresponding button a Properties dialogue opens and at the same time all relevant drawing tools of the Graphics Editor are activated. First the desired properties must be selected and then the geometry can be input with the aid of the Graphics Editor. The dialogues for the different cross section elements are very similar and thus we only need to describe them once in the case of the drawn partial section:

Identifier: In the first Tab sheet there is always an input field for the name of the cross sec-

tion element. The identifier is automatically suggested by the program. For each cross section a selected name may only be used once. Material: Most cross section elements contain a list field for assigning the desired material.

If this is already in the list, then use of the button on the left leads directly to the material management. Fibre Direction: Only active for timber cross sections. Active/Inactive: Only statically active elements can resist stresses. An inactive element,

however, still counts in regard to the selfweight of the cross section.

FAGUS‐7

B-7


Options: Depending on the cross section element it has various settings, including Checks: For a correct analysis the individual cross section elements should not overlap. If

necessary, this check can be switched off using the corresponding buttons for the current cross section. Example of reinforcement check (default values set for the whole project: Menu > 'Options' > 'Reinforcement/Checks'):

B 2.1.2 Drawn partial cross sections Using this button partial cross sections enclosed by a polygon can be input. Active/Inactive: For the determination of cross section values and for the analyses only the

active cross section parts are used. For each partial section this can be set in the corresponding Tab sheet using the option 'Active/Inactive'. In the case of the bridge beam shown below the circumference described by the formwork plan may be used for determining the selfweight. All other calculations, however, have to be carried out taking into account the contributing widths and other code provisions. Statically effective ?

Fig. B−1 Bridge cross section

Contributing widths:

FAGUS does not have any rules regarding contributing widths! All statically active cross section elements are taken into account in the determination of the cross section values and in the cross section analyses. In the case of complicated cross sections there is sometimes the need to be able to switch in a simple way between different models. This can be done with the help of additionally introduced cross section elements as follows: inactive partial cross section “placed” over the original cross section

Fig. B− 2 Modelling contributing widths with the help of inactive cross section elements

If an inactive cross section element is placed over an active cross section element with the same material properties, then these (doubly defined) zones are considered as openings, both for the determination of the cross section values and for the reinforced concrete analyses. The only exception to this rule is the determination of torsion constants, which is carried out using the original cross section !

B 2.1.3 Openings Openings are input using the button on the left. Openings are also of arbitrary polygonal shape, but with no material. They have to be completely within an existing circumference.

B-8

FAGUS‐7


B 2.1.4 Thinwalled partial cross sections With this button a thinwalled cross section element can be input. If the cross section only consists of thinwalled elements, which are also connected at the nodal points, the following cross section values and graphics may be required: •

Centre of shear

•

Warping constant (open cross sections)

•

Shear flow diagram due to Vy, Vz and T

A thinwalled cross section element is defined by its axis and the corresponding width, whereby the width should be much smaller than the length, for the corresponding conditions to apply.

admissible

inadmissible

Fig. B− 3 Examples of thin−walled cross sections

For typical analyses (ultimate load, stress verification) thinwalled cross section elements are considered as linear reinforcement. When determining the cross section resistance (with nonlinear material behaviour) a possible buckling of individual cross sections at high loading is not taken into account by the program !

B 2.1.5 Parametrised cross sections With this button the dialogue to define a parametrised (i.e. typical) cross section is opened. Instead of polygonal cross section circumferences, here one can specify directly the desired cross section dimensions and the program generates from them a complete cross section.

FAGUS‐7

B-9


The default values for the different parametrised cross sections are contained in several Tab sheets: 

concrete (= unreinforced cross sections )



reinforced concrete (including VT model  to check program's default values)



steel (standard steel sections)



timber (various common timber cross sections)



thinwalled sections (in addition to the standard steel sections)



user (= folder with userdefined cross sections)



old (= needed for compatibility reasons for old projects)

The most important dialogue functions:

The selected cross section is displayed in the right upper half of the dialogue, whereby the display can be still changed using settings. Drawing:



Scale: The cross section is shown to scale taking into account the specified parameters.



Schematically: The cross section is shown schematically, on the basis of the program's default values (not taking into account the selected parameters). The use of this type of display is necessary if different labellings overlap and thus some values can no longer be read.

Dimensioning :



Identifier: The dimensioning lines are labelled with the variable identifier. (Thereby the text of the underlying input fields is visible)



Numerically: The dimensioning lines are labelled with the numerical input values.

The Tab sheet Reinforcement Input contains a table with 4 columns, whereby each time two values can be edited and the other two are given by these values. (For the columns  and s max to be active, firstly n  0 must be set)

Modification of existing parametrised cross sections:

After 'introducing' the cross section, individual parameters can subsequently be changed as follows: 

Double click on Dimension Line/Reinforcement directly in the normal FAGUS interface (or with Dimension Line select, , 'Properties').



The button 'Parametrised Cross Sections' in the Control Tab sheet brings you back to the Input dialogue.

Convert the parametrised cross section into general FAGUS cross sections:

Parametrised cross sections cannot be extended with partial cross sections or reinforcement. They can, however, be changed into the normal standard cross sections and then modified as desired If the button on the left is active (Tab sheet 'Geometry'), it is shown whether it is a case of a parametrised cross section or not. Using 'Convert' it is no longer parametrised. Note, a converted cross section cannot be reconverted back into a parametrised cross section !

B-10

FAGUS‐7


B 2.1.6 Standard rolledsteel sections Standard rolledsteel sections can be selected in STATIK directly by name and do not have to be available as a FAGUS cross section. If a steel section is part of a composite cross section, then if it has to be modified or analysed using FAGUS, a FAGUS cross section has to be created: This button starts the corresponding input. After the desired steel section has been selected from the list provided, it still has to be placed in the drawing area by inputting its coordinates. The point of insertion is shown in the drawing area by means of the colour red. It can be placed at any one of the 9 given locations. After insertion, the steel section can be changed using all the tools of the Graphics Editor (rotate, creating mirror image, duplicate); Some steel sections can be rotated in the dialogue (0,90,180.20) or duplicated, as illustrated in the case of the metal section shown below.

Modifying:

Sometimes the steel sections are not used in their original shape, but are changed by cutting, as required. For this purpose, in this Tab sheet the following modification steps can be carried out.

No modification: The rolledsteel profile is introduced as a complete unit and retains its

tabulated cross section values. Converted to drawn cross section: Instead of the steel section, a drawn partial section of

approximately the same geometry is introduced. The cross section values determined from such a partial section may differ slightly from the tabulated cross section values, since now only the polygonal circumference is known. Cut horizontally: In addition to ”Converting to a drawn cross section" the steel section is cut into two partial sections with the parameters ”Cut position dz" und ”Cut loss". The cut position gives the distance of the axis of the cut from the axis of symmetry of the steel section. This may involve a possible loss of section (width of the section centred on the cut position). Generate additional concrete: Depends on type of cross section (see drawing). Alternative: Input with two partial cross sections (rolled steel profile plus concrete quadrilateral).

FAGUS‐7

B-11


B 2.1.7 Introduction of existing cross sections With the button on the left an existing cross section from the current library can be introduced into the cross section under consideration (analogous to the steel section). Placing: In the preview shown below the red 'insertion point' may be placed at one of the

given 9 locations.

B 2.1.8 Axial points and handles Both axial points and handles may be necessary, above all within STATIK applications. Detailed descriptions may be found in the manual of STATIK. With the aid of an axial point, a reference point for the loading can be specified. If there is no axial point (=standard case), then the cross section is loaded at the centre of gravity with the given section forces (see also Ch. B 3.1). With the aid of a handle the cross section can be positioned relative to the STATIK member's axis. For purely FAGUS applications the handle is of no significance.

B 2.1.9 PT constraint lines PT constraint lines are an aid for positioning tendons in connection with the Prestressing Module of STATIK. (The yeccentricity of the prestressing member's point in the vertical section is defined by the horizontal projection of the corresponding auxiliary line)

‐ez ey

Prestressing member Auxiliary line

Fig. B− 4 y-eccentricity by means of auxiliary line in cross section

B-12

FAGUS‐7


B 2.1.10 Result points The echoprint of each FAGUS analysis contains a series of standard results. If additional results are required, e.g. stress at a particular point, then a result point must be input. The values at the result points appear both during interactive work with FAGUS and within the postprocessing program for STATIK. Thus a result point also serves to control the amount of output: 

FAGUS interactive: For additional labelling of the cross section or for the tabular outputs.



FAGUS as postprocessing program: In the corresponding tables and as graphics directly on the STATIK structure.

A result point consists basically of two types (strain, stress, reinforcement area, etc.), position and a reference element. The Properties dialogue looks as follows: ID: Identification: name of result point Reference element: partial section, which

provides the value of the result. Results type: strain, stress, area (depends on reference element) Visibility: For which analyses the value

should be presented

The results point takes the desired value from its reference element (e.g. area of reinforcement of a point reinforcement). When a result point is first introduced, it can be placed straightaway on the desired element and the program then automatically takes the underlying element as the reference element. Both the given results type and the assignment of the reference element are independent of the Tab sheet that was used for the input ('Geometry', 'VTModel', 'Reinforcement'). For stress and strain results it is checked whether the coordinates of the reference point lie within the corresponding reference element. If this is not the case an error message is given. This check can be switched off in the Tab sheet Options. Kontrolle der Eingabe:

Die gemachten Eingaben sind in der QuerschnittsLegende im Register 'Varianten' aufgeführt. In der Tabelle 'Resultatpunkte' wird jeder vom Benutzer eingegebene Resultatpunkt in der folgenden Art und Weise ausgewiesen:  (R1( 3.,4,2.5))  Längsspanungen in der Bewehrung "R1" an der Stelle ( 3..4,2.5)

QS2 QS1 RP RP



ID: In connection with a STATIK reinforced concrete analysis it is recommended to use for a particular result the same ID in all cross sections. (In the results tables the individual columns are labelled with the results IDs of the first cross section. Subsequent cross sections provide results in the corresponding column with the same ID)

With the button on the left the “Snap Mode” is activated in the Graphics Editor . By clicking on a partial cross section (e.g. the circumference or reinforcement) the reference cross section for the result point can be redefined (in the initial input this button is not yet active).

Standard result points (only in layer button): In order to provide (linearelastic) edge stresses immediately in STATIK, the following automatically introduced standard result points can be made visible by means of the button on the left.

FAGUS‐7

B-13


TL

TA

TR

AL

A

AR

BL

BA

BR

The point (A) corresponds to the elastic centre of gravity of the cross section or (if defined) the axis point. The other points are given by the rectangle enclosing the cross section and the axis intersection points going out from (A).

B 2.1.11 Result combinations With the help of ”Result Combinations" values can be defined from different result points or values from cross section results using a formula. The following example should provide some explanations:

Under result combination "ASW1" on the parametrised cross section "RR" the following formula can be found: ASWMIN(SWZ) : ASW(SWZ) \ ASW(SWY) + ASW(SWT) With this formula the results in the three shear walls SWZ, SWY and SWT, which are independent of one another, can be combined in the following way: Combined stirrups area ASW1 := Min. stirrups area (SWZ) ALTERNATIV

Asw (SWZ) ODER

Asw (SWY)

PLUS

Asw (SWT)

B-14

FAGUS‐7


Hint: The terms Alternativ , Plus , Oder (in the formula abbreviated to the characters ':', '+', und '\') are taken from the determination of the limit state in STATIK and have the same meaning. (Corresponding description in STATIK, Dimensioning Limit State Values). Take care: The program permits the combining of result points with different physical meanings, for example one can combine stresses and strains. This possibility is hardly likely to lead to meaningful results. The present program version currently still uses a very simple ”Parser" (breakdown of the formula into individual terms), which is used above all for internal purposes. It is planned to extend both the description and the functionality in future versions. In addition to the functions treated below there are for example the following possibilities: RES(...): 'Request of information from another another result point, cross section results or result combination 

The list of result combinations is evaluated in alphabetical order. This has to be kept in mind when using another result combination in the expression of a result combination (no recur- sion). QRT (...): square root of the expression given in parenthesis SQR(...): square of the expression given in parenthesis

The desired formulas can either be input directly using text or using input aids. With the help of the shown selectability button one can switch between these modes. The input aids located at the bottom of the dialogue insert the corresponding expressions directly into the formula: [ASW] (stirrup reinforcement): After grabbing the desired shear wall the expression ”ASW(Shear Wall Name )" )" is inserted. [ASWmin] (minimum stirrup reinforcement): After grabbing the desired shear wall the expression ”ASWmin(Shear Wall Name )" )" is inserted. [ASL] (axial reinforcement):Permits areas of several reinforcement elements to be added

together (e.g. upper chord, lower chord, etc.). After grabbing the desired reinforcement the expression ”ASL(Reinforcement Name )" )" is inserted [oder] : inserts the character ”\" for an Oder (=or) combination. [plus] : inserts the character ”+" ”+" for a Plus combination. combination. [alt.] : inserts the character ”:" for for an Alternative combination. combination. The expressions are inserted from left to right. With the button on the left the whole of the formula or the previously selected text can be deleted.

B 2.1.12 Cross section results With the aid of cross section results the output values listed below can be defined and placed in the graphics. In contrast to the result points, which are dependent on position, the cross section results refer to the whole cross section.

B 2.1.13 Checking the cross section input The current input is checked using the button shown on the left. If an input error is found, a dialogue appears with a description of the error and the faulty cross section element blinks in the Graphics area. It is normally not really necessary to use the button, since this is carried out by the program program automatically before an an analysis.

B 2.1.14 Legends Each Tab sheet has a 'Legend' button as shown on the left. The number and size of tables depends on the current open Tab sheet (e.g. only reinforcement output in output Tab sheet, etc.). A complete cross section output is obtained with this button in the Tab sheet sheet 'Variant'.

FAGUS‐7

B-15


B 2.2 The Tab sheet 'VTModel'

s l l a w r a e h S

t u p n i s p u r r i t S


s k c e h C

d n e g e L

f s t o y c t e i j l b i b o a e t c h t e l e S

B 2.2.1 Shear walls A static model for shear force and torsion loading is defined with the aid of shear walls. A shear wall is input as a line in the Graphics area , whereby the start and end points denote the centre of action of the forces in the chords and the shear wall width as well as other properties are assigned as attributes. Type A: Wall A: Wall forces only in walldirections

Type B: Solid (compact) cross sections

polygonal circumference

Shear wall

z

centre of compression chord Shear wall

V z b w

centre of tension chord polygonal circumference

Fig. B− 5 FAGUS FAGUS cross section and torsion model

In Fig B5 three typical cross sections are shown. In Model Type A in response to shear force and torsion a shear wall can only resist forces acting in the direction of the shear wall. For the other two cross sections model Type B is available. 

Type A: Wall forces only in the direction of the walls

For each cross section numerous shear walls can be defined, whereby in general at least three shear walls must be input, which in addition may not intersect at any point, so that torsional forces can also be resisted. All shear walls act completely independently of each other, i.e. there is no compatibility at the edges of two touching shear walls. Therefore the shear walls of open cross sections do not need to be connected to each other. 

Type B: Solid cross sections

In order to reduce the input work somewhat, one can stipulate that for solid cross sections in each shear wall four small internal shear walls are used to resist the torsion internally. 

Type Automatic (program default):

Depending on the cross section input the program selects Type A or B and supplements the input if necessary. For torsion one can request that the calculation is performed with values taken from the basic cross section (polygonal circumference). circumference). (see also “Torsion values from basic cross section” page B19). B19 ). The cross section legend then shows the selected type. In the case shown above the program would react as follows: Bridge girder cross section: Type A is chosen. However, the closed cross section is detected as such. Circular cross section:  Type B. The shear force resistance is represented with the two shear walls. For the torsion the basic circular cross sections is responsible. Girder:  Type B: The program inserts a small shear wall in the horizontal direction and for torsion calculations the basic cross section values are used. This kind of input should only be applied if the vertical direction is the dominant action.

B B-16

FAGUS‐7


The VTModell type can be selected under Tab sheet “Variants”, icon “Properties” (see also chap. B 2.3). For a bending analysis a cross section does not need a VTModell. The inplane forces in the individual shear walls are determined, based on the model defined by the user. Then the section area of stirrups (and all other quantities to be verified) can be determined from the corresponding code provisions. These are usually based on a strut model, as illustrated below. diagonal strut

compression chord

V

z

M N tension chord

shear reinforcement

Fig. B−6 General reinforced concrete strut model



If only bending with normal force has to be investigated the cross section does not require any shear walls.

Input of shear wall The dialogue for the input of individual shear walls looks as shown below: If the character '@' appears in the identifier, the subsequent part is assumed to be the group name. Shear walls with the same group Id belong together and thus have a greater weighting (E.g.: S1@G1 = shear wall 'S1' in group group 'G1') Width of shear wall Options

Assignment of stirrup to shear wall Automatic generation generation of shear walls

Tab sheet 'Properties' 'Properties' 

Shear wall width(s): Even if this is different at the start and end of the member, the

computational model assumes a constant width. However, the automatically generated stirrup reinforcement may may be adjusted for a tapering form of cross section. 

Length of inner lever arm:

Normally this value is taken from the shear wall length. It can also, however, be input explicitly. 

Assignment of stirrups to shear walls:

This assignment is not absolutely necessary for dimensioning purposes. Without the input of stirrups, dimensioning is carried out with the standard material and the required stirrup area is given in the units of mm2/m. If the assignment command is present, however, the stirrup cross section can be adjusted at the end of the dimensioning process. If, e.g., an ultimate load analysis has to be carried out for shear force and torsion the existing stirrup cross section must be known.

FAGUS‐7

B-17


Tab sheet 'Options' 

Weight factors for Type A:

For each shear wall an individual weight factor can be given 

Type B: Active components / weight factors:

Each component can have its own factor.

Calculation of shear wall forces With the standard setting the program automatically specifies how the shear force components have to be distributed between the individual shear walls. The individual fields in the dialogue shown on the right are made invisible and the weight factor for each shear wall is proportional to the moment of inertia. These initial settings, however, can be changed in the program using the Tab sheet: Variants > Properties > VT Model > Shear and Torsion Model

V y

T

V z

Weight factors for type A The determination of the individual shear wall forces is based on purely equilibrium considerations and the calculation is carried out according to the deformation method. In the general case, compared to the three action quantities (Vy,Vz, and T) there are n unknown internal forces. The stiffness s of each shear wall is proportional to its moment of inertia. s

Spring with stiffness s

� h  Weightfactor b�12

3

Result check: The existing forces in the shear walls must be in equilibrium with the external forces. Si d i

i

V y

B-18

T

V z

V y   Si  sin i V z   Si  cos i T   S i  d i S i = Section force of i−th shear wall Sign in text output and results: A positive shear wall force points from the start point in the direction of the end point of the shear wall.  i = Inclination of i−th shear wall (with respect to z−axis or y−axis) d i = distance of i−th shear wall from reference point

FAGUS‐7


Graphical description of shear wall forces For positiv input values Vy, Vz und T the shear wall forces are displayed as follows (The layer can be set on/off in lower part of the layer list on the right hand side of the screen): z

T V y

y

V z

V y

T

x

V z Fig. B−7 Sign of shear wall forces

Simplified shear wall input Sometimes and in simple cases only the results in the main direction are needed. In the CEDRUS building module there are lots of rectangular cross sections with main acting forces in local zdirection. To ensure equilibrium for the other two compents the program introduces automatically a wall in the orthogonal direction if this component is active. Internal model Vz

Input Shear wall activ for Vy, Vz, T

Main action

Vy

T

Secondary actions

With the input shown above it is clear that the three internal shears wall are belonging together and therefore can be combined as shown in the table below (Extreme value out of “Vy+T" or “Vz+T").

Torsion values from basic cross section The torsion values are calculated from the basic (polygonal) cross section in the following case:  Tab sheet 'VTModel' > 'Typ Auto:'  One or two (crossing) shear walls have been defined The legend of the shear wall data shows following message: Torsional model Typ B:

Ak.. uk.. tef.. values from entire cross section

Weight factors for wall groups Pilar cross sections from CEDRUS6 act as a group. In this case the weight factor is calculated with the moment of inertia of the whole group. To achieve the same effect in FAGUS the name of shear wall has to end with the name of the group, separated with '@' e.g. .

FAGUS‐7

B-19


S4 S5

S1@G1 S2@G1

S6

S3@G1

Fig. B− 8 Grouped walls S1 S2 and S3 (Pilar G1)

Additional remarks

F d S

F d   F d

V z



An elastic shear flow analysis also delivers resulting wall forces and could be used to compare the “FAGUS shear wall solution” with the elastic solution. (Requirement for an elastic analysis: All elements have to be connected)



Further, it is necessary to point out the classical problem of the shear connection of the flange to the web as shown on the left: For a vertical shear force Vz FAGUS only gives values for the vertical web stirrups. The horizontal web stirrups area could be obtained, e.g., as follows: • Express method: Knowing that the shear flow is uniformly transferred to both sides, a further analysis is dis- pensed with and half the web stirrups reinforcement is simply used for both flanges (= standard method for simple cases) • Exact investigation: Firstly, ∆F is determined by comparing the axial forces in the flanges (known from bending analysis) and then a design of the splices is carried out by means of a truss model.

Automatic generation of shear walls The individual shear walls can either be input individually or generated automatically using the function shown on the left for the whole cross section. Based on the current polygonal circumference it is attempted to suggest a suitable model including stirrups. This can subsequently be adjusted selectively.

Additional input help for compact cross sections Above all, for compact cross sections the input of the shear wall model can be relatively timeconsuming. Besides the two shear force directions a ring must also be defined to resist the torsion. Therefore in the Tab sheet 'Variants' > Properties > VT Model there is the possibility of selecting the torsion model of type B, which introduces an appropriate model with four inner shear walls for each shear wall with an existing torsion attribute. As a last variant, the program still provides the possibility of importing the geometric data for the torsion analysis directly from the cross section circumference. For this purpose one proceeds as follows:  Select Tab sheet 'VT Model' > 'Type Auto: automatic'  Input one or two intersecting shear walls Advantage: If the basic cross section is circular or ringshaped, the values Ak, u, tef can be calculated in accordance with the code and accurately. In the case of a general convex circumference this corresponds to a good approximation.

Stirrups As mentioned above, for the cross section dimensioning it suffices to have a valid shear wall to work with. The input of stirrups increases the quality of the analysis in that, e.g., the axial reinforcement is more accurately positioned and the output can be documented more clearly. FAGUS, therefore, permits the input of stirrups as graphics elements.

B-20

FAGUS‐7


If there is a distribution between shear wall and stirrups, the stirrups areas and materials are taken from the corresponding stirrups attributes. The stirrup geometry is not input directly, but a polygon has to be input which then serves as a reference for the stirrup. The stirrup can be generated with the aid of the geometry parameters shown below. If the polygon is closed (start point=end point), the stirrup is also closed. Subsequent changes are also carried out using this reference polygon.

Input polygon as 'formwork edge' for the stirrup: if this is given in the anticlockwise direction, the direction should be set to 'inside'.

Automatic generation of stirrups As with the shear walls this dialogue also provides the possibility of an automatic generation of stirrups.

FAGUS‐7

B-21


B 2.3 The Tab sheet 'Reinforcement' This Tab sheet serves to input untensioned and prestressed axial reinforcement.

List field to select the reinforcement group

p p p u u u o r o r o r g g g t e f n t o e e s l e i m e e D t r c r e o p f o n r i e r P w e N

t w n o e h m s e o c t r o s f t n n i i o e p r y r a i l i x u A

- - t i l e n i e x c u r m o f a n i n e e e r r w o t f e s b t n i e c o n p a y t s r i a D

t n e m e c r o f n i e r e n i L

t n e m e c r o f n i e r t n i o P

t n e m e c r o f n i e r r a l u c r i C

s n o d n e T


p u r r i t s o t t n e m e c r o f n i e r t s u j d A

k c e h C

t n e m e c r o f n i e r r o f d n e g e L

y t i l i b a t c e l e S

B 2.3.1 Reinforcement groups In order to be able to manage many layers of reinforcement more easily, several layers can be grouped together to form a reinforcement group. Example: beam cross section with two renforcement layouts (sections within span and at column support). When listing cross section variants and in the case of analyses taken directly from the program STATIK the desired group can then be selected. The above shown section of the Reinforcement Tab sheet contains different ways of managing reinforcement groups.

B 2.3.2 Auxiliary points to graphically construct the reinforcement In the input of reinforcement one normally does it with reference to the edge of the cross section. With this button auxiliary points can be generated, which can then be referenced ('grabbed') in the input process. These auxiliary points are not of course part of the reinforcement input ! In the input field shown on the left the desired distances of the auxiliary points to the reinforcement diagram can be input. By clicking on the arrow the input can be moved upwards or downwards in steps. If necessary, it is also possible to make a numerical input directly in the corresponding numerical input field.

B 2.3.3 Point, line and circular reinforcement FAGUS cross sections may have have as many reinforcement elements as desired. The numerical properties for the three types of reinforcement are similar. point

line

B-22

circular



Point reinforcement is defined by inputting a material, an area and a pair of coordinates y,z. For the analysis, the given reinforcement area is concentrated in the given point, i.e. the element does not have a moment of inertia).



Line and circular reinforcement are “infinitesimally thin”. Regarding distribution along the length for the line and circular (i.e ringshaped) reinforcement there are the following possibilities:

FAGUS‐7


Linearly distributed reinforcement: The input reinforcement is distributed uniformly

along the given line Single members: In the cross section integration the reinforcment area along the line is

concentrated in the given line. In some circumstances this can have a considerable influence on the result (e.g. in the case of PYRUS analyses with circular cross sections or square supports with skew reinforcement. In the dimensioning, the area in the individual points is increased uniformly and continuously; as result a diameter may be obtained with a number of places after the decimal point. Below the Properties dialogue is shown for line reinforcement. In the Tab sheet 'Attributes' all the settings may be found which are used for the analyses. The Tab sheet 'Dimensioning' contains the information for controlling the dimensioning. If the cross section is dimensioned (adjusting the reinforcement according to Ch. B 3.2), then the reinforcement areas defined in the Tab sheet 'Attributes' (or the number and diaimeter) are automatically adjusted, i.e. the user input is overwritten! Line Single members with end members without end members Input of total area

Automatic generation of axial reinforcement:

Input of total area using  diameter  distance  number

Geometrical positioning with reference to the stirrup (see below)

For every automatically generated stirrup two layers of axial reinforcement are generated

Automatic modification of geometry: The repositioning is carried out either by explicitly clicking on the button on the left or automatically after dimensioning with interactive selection of the axial reinforcement (see Ch. B 3.2). The three possibilities provided above have the following significance: 

none: the geometrical position of this reinforcement layer is not changed



to the stirrup: the nearest stirrup is sought and the position is moved towards and normal to the stirrup (path A)



to the corner of the stirrup: the input reinforcement is lengthened, so that the end bar lies in the corner (path B)

A B

FAGUS‐7

B-23


Dimensioning strategies / Tab sheet 'Design': Reinforcement attributes for dimensioning

The following dimensioning rules are possible at the present time: 

Constant area: The area of this reinforcement layer is not changed during the

dimensioning. 

Dimension reinforcement for tension and compression: The reinforcement in the

layers is increased proportional to its starting value, irrespective of whether it is under tension or compression. 

Dimension reinforcement in tension zone only

This option is meaningful above all for beam cross sections. The tension and compression zones within the cross section are determined on the basis of the current loading before the actual dimensioning iterations on the homogeneous cross section. If additional compression reinforcement is needed to achieve the required cross sectional resistance, then for this purpose all existing layers in the compression zone are used (Exception: Layers with the attribute ”Area remains constant”). 

Default dimensioning according to cross section type

This option serves as a default value for new projects. The dimensioning rule is thereby modified depending on the cross section type. Starting value of the reinforcement area:

In the case of beam cross sections the default reinforcement starting value is considered to be the minimum reinforcement. The area given by the program is therefore never smaller than the area already input. In the case of column cross sections, the starting values for the reinforcement areas to be dimensioned are scaled corresponding to the given minimum reinforcement content. In the analysis of the reinforcement content, the layers declared to have a constant area are also taken into account. If several layers are dimensioned at the same time, for each layer to be dimensioned the reinforcement area is increased proportional to the starting value. Thus at the same time with this area a proportionality factor is defined. Result of the dimensioning

Input

1 cm2 2 cm2

8 cm2 16 cm2

Fig. B− 9 Reinforcement dimensioning: Adjustment according to starting values



B-24

If the reinforcement should also be taken into account in the analysis of the static cross section values, this has to be specified explicitly. (Tab sheet 'Variants' > Properties> Models).

FAGUS‐7


B 2.3.4 Tendons / Strips Tendons are represented by point reinforcement and can either be input as corresponding reinforcement elements in the cross section (if one only works with FAGUS) or more conveniently with the Prestressing Module of STATIK, which “places” them in the reinforced concrete analyses automatically in the correct place in the cross section. In FAGUS this input of a tendon corresponds practically to the input of a point reinforcement. Additionally, however, it is still necessary to specify a value for the initial strain and information on the bonding action with the surrounding concrete. In order to be able to input adhesive strip reinforcement with initial strain, this dialogue is also used for the input of strips (lamella). The selected prestressing system (strands, wires, rods) is of no importance in the following considerations. Below an excerpt from the corresponding dialogue is shown:

Input of initial strain

Only as aid for the input

Allowance for longterm losses

Slope of tendon can be taken into account by inputting the values dz and dy

Initial strain  p As a starting value for the analysis, FAGUS requires the initial strain in the tendon  p on the strainfree cross section (εx, κ y, κ z = 0). For beams constructed using the pretensioning method this value can be input immediately:



 p 

 p E p

with  p = steel stress, E p = elastic modulus of prestressing steel

 p  p

FAGUS‐7



In all other cases the initial strain has to be determined from the steel stress or from the forces measured at the prestressing jacks and depends on the friction and anchor losses, as well as on the prestressing conditions (elastic shortening). In EC2 there is the following formula:

B-25


P  P o   P c   P    P sl

= prestressing force at the end of the tendon immediately after prestressing

P o

 P c

= prestress loss due to elastic deformation (loss of compression)

 P   P sl

= prestress loss due to friction = prestress loss due to slipping (pullin) of wedges at anchorage = prestressing force for the analysis of the initial strain

P

The strain relation is as follows:  p   o  

where  o is the “existing” strain in the prestressing steel in section x after prestressing and  corresponds to the strains in the neighbouring concrete fibres due to elastic deformation during prestressing.  o 



P o   P    P sl A p  E p

 

P

  c E c

 c = concrete stress at the height of the tendon under consideration E c = modulus of elasticity of the concrete : strains due to elastic deformation during stressing

The contribution of  is usually small. Thus in many cases one can dispense with a very exact analysis or otherwise other related factors like “rebound of the formwork", “sequence of stressing the different tendons" have to be clarified in detail. For the direct input of prestressed cross section elements in FAGUS two separate input fields for  o and  are provided. 

Sign (algebraic) of : If the concrete is in compression,  has a positive sign. “Prestressing steel strains increase due to loss of compression".

If the tendons are defined using STATIK, the concrete stress is calculated there with the loading given there under G1 at the corresponding height of each tendon. The corresponding input field is to be found in the dialogue 'Reinforced Concrete Analyses' and often for this purpose simply the selfweight and the effect of the prestressing are used, i.e. it is assumed that during prestressing the beam separates (i.e. lifts up) from the formwork. A modification can be effected using the parameter “PREFAC" (see below and in the STATIK manual) Starting from the forces measurable at the ends the procedure is illustrated by means of the following comparison:

Bonding It has to be shown, whether the tendon is effectively bonded to the surrounding concrete. Since for all analyses for the cross section it is assumed that “plane sections remain plane", with bonding the additional strains in the prestressing steel due to external loading of the cross section are the same size as the strains in the neighbouring concrete fibres, i.e. in the failure state this leads to a considerable increase in stress in the prestressing steel. By contrast, the unbonded tendons retain their initial strain input by the user (and thus the prestressing force) independent of the current strain plane.

B-26

FAGUS‐7


Prestressing with cable in sheath “Posttensioning"

Pretensioning method: “Pretensioning"

Measured force at jack: P o

Before cutting the wires the cross section is unstrained !

 p 

P o A p� E p

 o 

P o A p� E p

Strain change in prestressing steel when cutting the wires (the cross section is now strained:

Beam shortening due to prestressing:

 P o  

 

A c� E c

  o   p  

,

 P o

A c� E c

  p   o  

P o : prestress force in tendon (tension positive)

,

A c , A p : concrete and steel cross sections, resp. sheath

P o

P o

P o

P o

Fig. B−10 Definition of the initial strain in two simple cases

Slope of tendons In the case of sloping tendons, P is split into individual components, e.g. longitudinal component: P x( d)  P ( d)  dx

(the area is also adjusted: A x( d)  A( d)  dx

{ dx, dy, dz } = unit directional vectors of tendon in the tendon coordinate system

The vertical component P z has a role in dimensioning the stirrups and is taken into account in the determination of the inplane forces. With a favourable action of P z in general only the forces at time t  need be used, which however is only possible when also using STATIK (see below) P z ( d)  P ( d)  dz

Longterm losses Interactive analysis in FAGUS

Only taking into account the cross section the losses cannot be calculated by the program. But with a userspecified global loss factor P oo P o certain effects can be estimated. Whether the analysis should be carried out at time t  0 or time t   can be specified in the analaysis parameters. This higherlevel control also applies to the analyses described below.

FAGUS‐7

B-27


STATIK (postprocessing/reinforced concrete analyses) For verifications at time t   the prestress losses due to creep, shrinkage and relaxation Directly from

are considered as follows: P   P   P (t)

 P (t) = loss of prestress force due to creep, shrinkage and relaxation

The corresponding strain relation is:     o   cs   cc   sr

= final rate of shrinkage = creep strain at height of considered tendon

 cs  cc

 cc  

g  p)  c ( E c

 = creep coefficient  c = concrete stress at height of considered tendon due to permanent

loads (in STATIK defined by G1 and G2) and prestressing = strain due to relaxation of prestressing steel

 sr

Creep and shrinkage: In FAGUS the losses due to creep and shrinkage are calculated using

a formula proposed by CEB/FIP, which is also to be found in a similar form in EC2:  cs   cc 

 cs  

1

 c( p) E c

E

g  p)  c ( E c

  p p  (1  2 )

= k  ( cs   cc)

 cs = final rate of creep (input by user)  = creep coefficient (input by user) g  p)  c ( = concrete stress at height of tendon due to G1 and G2 p) = initial concrete stress at height of tendon due to prestressing  c ( E c, E p = elastic modulus of concrete, elastic modulus of prestressing steel k = “correction factor": given specially by program for manual checking

(see below) Relaxation: The relaxation of the prestressing steel depends on the prestressing system

used and from the level of prestressing. In the various codes, the curves are usually presented graphically, e.g. for the time period up to 1000 [h]. FAGUS uses the curves shown below from SIA 262, which can be modified using the commands given at the end of this section.

[o/o] 12

 P o  P o

A

10 8

B

6

C 4.5

4 2

2.5 1.0 60

70

80

[o/o]

 P o f pk

Fig. B−11 Relaxation losses after 1000 [h]

For checking purposes the loss values are presented in a table for each section:

B-28

FAGUS‐7


P o

Simple example ('STATIK' > 'Help' > 'Examples', Materials acc. to SIA262 ) Concrete cross section: A c  0.16 m2, C20/25, E c  30.4 kN  m2,   2 Prestress: A p  400 mm2, Y1860, f pk  1860 N  mm2 , E p  195 kN  m2, 0.75  f pk  1395 N  mm2 , P o   558 kN ,  o  7.15 o oo

P o

Concrete stress:  ( P o)   3.49 N  mm2 Shortening:     c E c  0.114 o oo Correction factor: k  0.969 Losses (for initial parameters see table below): Shrinkage : k   cs o  0.969  0.27.15 Creep : k   cc o  0.969  2  0.1147.15 Relaxation : k   P o  P o  0.969  0.105

 2.7 o o  3.09 o o  10.2 o o

Tendons: Initial strains and long−term losses Parameters used (final values): creep φ=2 , shrinkage εcs=0.2 [‰] , relaxation (0.75 f pk) = 10.5 [%] Member Elem Distance Id Initial strains Long−term losses εo ∆ε k Shrinkage Creep Relaxation [m]

SL_1_0

[‰]

1

0

SG1

7.2

[‰]

0.1

[−]

0.97

[%]

2.7

[%]

[%]

3.1

10.2

Total [%]

16.0



With which initial strains an analysis is carried out depends on the corresponding settings in the dialogue 'Analysis Parameters' > 'Prestressing': Under the section 'STATIK Reinforced Concrete Analysis with Prestressing' the options field 'Prestressing with longterm losses' may be found, which determines whether an analysis is carried out at time t=0 or at t   .



Definition of losses in [o/o]: k   k   k   Shrinkage:  cs  100 , Creep:  cc  100 , Relaxation:  sr  100 o o o Changing the default values:

The default values set by the program can be changed following the instructions given below. In the STATIK manual under “Reinforced Concrete Analyses", “additional parameters" are given. The parameters currently possible relate to (the values given below are the program's default values): 

SHRINK=0.2

Final value for  cs in [o/oo] 

CREEP=2

Final creep coefficient  for the analysis of prestressing losses. For the actual FAGUS analysis, however, the value defined in the Analysis parameters is used ! 

RELAX=4.5

Relaxation loss in [o/o] after1000h under a stress of 0.8  f tk. The curve C in B11 is scaled. 

RCLASS=C

Input of a relaxation class ('A','B','C') corresponding to B11. If the relaxation value is input directly (with RELAX=xx) , this command has no effect ! 

PREFAC=1

Multiplicator for the shortening part : When using this command the calculated values  are multiplied with this factor and as confirmation the corresponding output column is labelled f �  . Further information may be found in the STATIK manual.

FAGUS‐7

B-29


B 2.4 The Tab sheet 'Variants' A member ,e.g. , can be built up in a particular construction stage of the cross section variant "A" and in another stage of the variant "B". If one speaks of a “cross section” , in reality a particular cross section variant is meant. The (standard) variant contains all input cross section elements. Variants derived from them are obtained by deactivation of certain partial sections (for more details see Tab sheet "Attributes") The Tab sheet contains the following buttons:

List field to choose variant

t n a i r a v w e n e t e l e D

s e i t r e p o r P

s t n a i r a v k c e h C

d d s n t e n o e a i g l t e o c e L p r s e t s n s i o k r c c e h C

Check the variant: Manual checking of the whole cross section is automatically carried out

when switching to another cross section or on leaving the program. Any objects with errors in them are shown with a blinking animation. Legend with all cross section data: Using the button on the left the legend of the whole

cross section variant is created. The corresponding function in the previous Tab sheet only creates a reduced output (with the cross section elements, that were input in the corresponding Tab sheet) Checking interpolated cross sections: With STATIK structures having varying cross sec-

tions (tapering) both sections must be 'affine' to each other (same number of cross section elements, same number of circumference points for each partial section, etc.) This is best achieved by duplicating and changing the coordinates of a particular cross section. The corresponding button opens a new dialogue, with which two arbitrary cross sections can be checked visually (two list fields at the top of the dialogue). A slider is located on the right of it , with which the desired position of the interpolated cross section can be set. 

In this dialogue it is allowed to combine arbitrary cross sections from the available library. An error message is shown at the top of the dialogue if the two cross sections do not match..



It is also possible to use a 3D representation (drawing function on the left edge)

B 2.4.1 Properties of the variant This button opens a Properties dialogue with the following Tab sheets:

Tab sheet 'Properties': Identifier: serves to change the name of the variant Cross section elements:

All cross section elements (and reinforcement groups) are listed here and using the button can be set to active or inactive.

B-30

FAGUS‐7


Tab sheet 'Member' Cross section type:

The input of a cross section type is important above all for dimensioning. One the one hand, this affects the way in which the reinforcement area should be increased in a dimensioning iteration, and, on the other, depending on the code to take into account minimum reinforcement rules (axial and stirrup reinforcement). 

Beams

In the case of a beam cross section, firstly the reinforcement on the tension side is increased. Only if the compression side is governing, is it necessary (possibly) to increase the reinforcement area on the compression side. 

Columns

The reinforcement of column cross sections is uniformly changed in the dimensioning. The dimensioning begins with a userdefined or codedefined minimum reinforcement content (can be set in Analysis Parameters). 

Wall

For dimensioning purposes a wall is treated like a column, but no stirrup reinforcement is considered. 

Platte

Für Plattenquerschnitte (bestehend aus einem rechteckigen Stahlbetonquerschnitt und beliebig vielen Bewehrungen) wird ein Querkraftnachweis durchgeführt, auch wenn kein Stahlbetonschubwandmodell vorliegt. Bei den parametrisierten Querschnitten wird dazu ein entsprechender Querschnitt angeboten. Slope of haunching:

Normally in FAGUS it is assumed that the cross section plane is normal to the axis of the member (the axial stresses act normal to the considered cross section ) and that the member is prismatic. Above all, in connection with the input of the section forces from STATIK it is sometimes required in the case of members with different end sections, to consider the influence of sloping flange forces. In a postprocessing analysis the angle  shown below is taken into accounft as an approximation of the geometry of the participating cross sections. For individual analyses these can also be input directly as a cross section attribute. The cross section analysis is then influenced as follows: 

Bending with axial force

The area of the axial reinforcement A s is reduced in the analysis according to the angle  of the reinforcement rod as follows: A s red  A s cos 

 is calculated for each reinforcement layer from the corresponding distance to the

edge. 

Shear force V z

The modification of the shear force in the local zdirection due to haunching may be clearly seen in the following figure. F c

 o M

V z u

z

N

1 F t

 u

Fig. B−12 Section forces on haunched cross section

M 1  M  N � z u M  u  tan  o)  N � tan  o V  z 1 �(tan

FAGUS‐7

B-31




Hint:  In the current version only haunching in the zdirection is taken into acccount.  In the case of large slope angles the results have to be compared with other analysis methods (e.g. truss models).

Tab sheet 'VT Model' In this Tab sheet the analysis model for shear force and torsion can be controlled. Slope of stirrups:

In the current program version the same stirrup slope applies to all shear walls of the cross section. Shear and torsion model: Type Auto: Automatic Distribution

The program itself tries to find a meaningful model. Depending on the number of input shear walls, internally the program switches to one of the models given below:

A k u t ef



Open/plateshaped cross sections: If at least three shear walls have been input and these do not intersect at a point, it is assumed that we are dealing with a cross section of type A . The weighting factors are specified to be proportional to the moments of inertia of the individual walls.



Compact cross sections (one or two shear walls) If only one shear wall was input, automatically a second wall is introduced orthogonal to it in order to deal with the secondary direction (= simplified input for simple cases). Additional information on this may be found in Ch. C 4.2. If only one shear wall was input or the two shear walls for Vy and Vz intersect, the values for torsion are taken from the basic cross section. In the cross section legend a corresponding remark may be found. For circular cross sections, the verifications required by the code can be carried out correctly. For elliptical or convex cross sections this is the best approximation.

Falls mindestens drei Schubwände eingegeben wurden, wird die Torsionbeanspruchung durch Aufteilung in Scheibenkräfte behandelt. Sind alle eingegebenen Stahlbetonschubwände miteinander verbunden, so werden die Scheibenkräfte infolge V,T aus einer elastischen Schubflussberechnung entnommen (gemäss Kapitel C3: Dünnwandige Elemente). In der Querschnittslegende wird dieses Modell wie folgt angezeigt:

Für die so modellierten Querschnitte können die Schubspannungen am Querschnitt dargestellt werden (gemäss Kapitel LEERER MERKER). Bei nicht zusammenhängen Strukturen wird der unten besprochene Typ A verwendet. Type A: Distribution into inplane forces

This is the standard model for open cross sections. Inplane forces can only be resisted in the direction of the wall. The distribution of the inplane forces is carried out on the basis of equilibrium considerations together with any weighting factors. If shear walls with the attribute “Torsion” form a closed ring, this is taken into account by the program (constant shear flow). The requirement is that each shear wall is in perfect contact with the adjacent one (coordinate input using grabbing function) and only the shear walls that form part of the ring exhibit the torsion attribute. In this case there results around the circumference a constant shear flow due to torsion. The shear force S i in shear wall i due to torsion (apart from any assigned weighting factors) is: z S i  T i 2� A ef 

B-32

The current program version can only deal with a ring. In practical cases the shear walls of the outermost ring should be set as active for torsion. More refined (elastic) estimates, ho-

FAGUS‐7


T

wever, can be made using thinwalled elements

Fig. B−13 Girder with 4 webs: Active shear walls specified for torsion

Type B: Individual full cross sections

This is the standard model for simple (compact) cross sections. The shear force is resisted in the given wall direction. For shear walls that can resist torsion internally the model shown below is formed. Input

Torsion model

Application

Standard weighting factors: If the cross section consists of several shear walls specified to be active for torsion then the distribution of the torsional moment is normally based on the areas. Materials for shear dimensioning: These are either taken automatically from the basic cross section or they can be explicitly overwritten here. The same materials are used for all shear walls.

Tab sheet 'Cross Section Values' All functions discussed here are mainly important in relation to STATIK.  Determination of the static cross section values / reinforcement taken into account: Normally, in determining the cross section values it is recommended not to take the influence of the reinforcement into account. The default program value can, if necessary, be overwritten here (e.g. in the case of very high reinforcement contents).  Check cross section and export it (special function seldom required): If the corresponding check box is not activated, this cross section is not visible in the corresponding list in STATIK. 

Reference material:

In the case of composite cross sections one can specify here, for which material the cross section values should be output. If no information is provided this is the material with the smallest modulus of elasticity E.  Overwriting the cross section values: Since the values listed here G� I x, G� A y, G� A z are only approximately calculated in FAGUS, they can be overwritten by the user. STATIK then uses this userdefined value.  Additional masses: This value is input together with the cross section and is taken into account in the determination of the selfweight of the member (= purely a “convenience function”).

FAGUS‐7

B-33


B 3 Design and Analysis B 3.1 Loading of Cross Section, Section Force Reference System Normally, the section forces act at the centre of gravity of the cross section, which is determined by the program. In the Graphics Area this can be shown using the layer button on the right side or presented in numerical form in the table with the cross section values.

Fig. B−14 Example of a simple echo print of the loading

If an axis point is input, then in the interactive use of FAGUS for M,N loading one can switch between the centre of gravity and the axis point. For shear force and torsion loading the centre of shear is also available as a reference point. Below the section force input table the following choice is available: Cross section without axis point

M, N: centre of gravity V,T : centre of gravity or centre of shear

Cross section with axis point

M, N: axis point or centre of gravity V,T : axis point or centre of shear

N A

M S

N

Fig. B−15 Section force transformation axis point − centre of gravity

Transformation of the section forces

Before the actual analysis, firstly the section forces are transformed in the centre of gravity. In the following circumstances a second modification is carried out:

B-34



Haunching (see Fig. B12)



Prestressing (see slope of tendons, B 2.3.4)



Consider shear force and torsion as internal axial force (Analysis Parameters)

FAGUS‐7

B 3 Design and Analysis

In the echo print of the input section forces then for each input line at most the following three lines may be found: 

Input section forces (with respect to axis point, centre of shear)



Staticallyequivalent section forces with respect to centre of gravity



Modified section forces (due to haunching, prestressing, internal axial forces)

Fig. B−16 Echo print including transformed section forces

Postprocessing Postprocessing analyses In the case of postprocessing analyses directly from the STATIK program for M,N loading the axis point is always used if one exists. The same applies for the V,T loading. If the coordinates of these two points are not the same the additional torsion must be taken into account. Here, however, it is still possible to overrule this convention by inputting a corresponding parameter. STATIK > Tab sheet 'analysis' > 'special analysis' > 'RC analysis' > edit field: 'additional parameter') parameter') > command “SHEARCENTER=ACTIVE” “SHEARCENTER=ACTIVE” Vz

Vy S

T

S

M Fig. B−17 L 17 L−cross section with centre of gravity S and centre of shear M

FAGUS‐7



In the building model, the axis of the member in the case shown above is introduced at the centre of gravity S. If in this storey there is a sufficient number of walls, the vertical member member does not receive any large torsional forces from the frame analysis. By letting the shear forces act directly at the centre of shear, due to the eccentricity SM no additional torsional torsional moments result .



If the structure is calculated as a plane frame model or if the member is taged with “uniaxial bending” the eccentricity SM is not considered either.

B-35


B 3.2 The Tab sheet 'Reinforcement Design' With the Tab sheet 'Reinforcement Design' an individual cross section can be dimensioned interactively for bending and axial axial force, as well as for shear force and torsion. In the case of combined loading the following analyses are carried out: 

Dimensioning of the axial reinforcement for bending with axial force



Determination of the additional axial reinforcement due to shear force and torsion



Stirrup reinforcement

The result of the dimensioning is the required reinforcement areas presented in graphical and in tabular form and the input reinforcement is adjusted. Below the main operating elements are shown. One or more loadings can be input, whereby the section forces are input at the dimensioning level. If several input lines are active, the dimensioning is peformed successively for all loadings. The result of each dimensioning step is used as starting value for the next dimensioning step.

List field to select cross section variant

Management of member diameter

Delete line

New input line Loadings: By selecting a particular line this can be activated for the analysis Current analysis parameters Various dimensioning parameters Start of dimensioning Design

Automatic adjustment of the reinforcement areas

Interactive reinforcement adjustment If stirrups present: geometrical adjustment Efficiency check with new reinforcement configuration: (error message for value > 1.02)

Graphical output

Graphical output

Table with results

The adjustment of the reinforcement areas at the end of dimensiong is carried out according to the above scheme. The path shown on the left (black lightening button) correponds to the procedure used in earlier FAGUS versions. The method shown in the middle with the interactive selection of the diameter is described below.

B B-36

FAGUS‐7


Automatic adjustment of the required reinforcement areas The dimensioning provides the required reinforcement area, which corresponds to an efficiency factor = 1. The efficiency depends among other things on the selected analysis parameters and also, of course, on the given loading. At the end of the dimensioning dimensioning the reinforcement areas input by the user are adjusted. In the case of line reinforcement layers, the output form depends on how the reinforcement layer was originally input: 

For distributed line reinforcement layers the result is output in the form (A s = mm2).



For point reinforcement, the required reinforcement area is converted into a (fictitious) diameter,e.g. 4  12.34 .

If, with the adjusted cross section, subsequently an ultimate load analysis is carried out with the same section forces and analysis parameters, the cross section should have an efficiency factor of exactly1.

Interactive selection of diameter and adjustment of the layer Firstly, the required reinforcemement areas are determined as described above. Then follows an interactive selection of the desired axial reinforcement. In a special dialogue for each reinforcement element various possible reinforcement diameters are shown (sorted according to the divergence from the required result). If a valid VT model exists as well as a shear wall stirrup assignment, stirrup diameters/spacings can be selected. Now follows the automatic geometry adjustment: firstly, all stirrup diameters are adjusted (concrete cover remains the same). Then for each axial reinforcement layer the critical stirrup is sought and the reinforcement is adjusted according to the input attributes (Ch. B 2.3.3). 2.3.3). Checking the new reinforcement configuration: After the final specifying, the program

carries out once again a dimensioning with the latest selected reinforcement configuration. For various reasons (larger diameters, second layer necessary) the inner lever arm may have changed in comparison with the original dimensioning. If, with this selection, the admissible efficiency is exceeded, one is required to carry out the dimensioning again (the new dimensioning now starts with the new geometry positions).

Numerical legend With the button on the left the result of the dimensioning is displayed directly in tabular form.

Option: Uniaxial bending Selecting this check box has the following effects:



FAGUS‐7



Bending with normal force:  the loading Mz is set to zero.  the neutral axis is always horizontal. In the case of nonsymmetrical cross sections the internal forces are only in equilibrium with the external section forces about the yaxis!



Shear force and torsion:  T = 0, Vy = 0  Vz is distributed to all existing (active in relation to Vz ) shear walls in accordance with the given weighting. For a nonsymmetrical arrangement of the shear walls (in relation to the zaxis) the inplane forces are only in equilibrium in the zdirection !

In the case of reinforced concrete analyses directly from STATIK for plane systems all all cross sections are dimensioned automatically under "uniaxial bending".

B-37


B 3.2.1 General remarks on the dimensioning of the axial reinforcement With the dimensioning of the axial reinforcement, all the reinforcement areas designated by the user are changed until the required cross section resistance is attained for the given loading. Dimensioning can only be carried out if the cross section already possesses a (minimum) reinforcement. This initial reinforcement serves, among other things, to define the geomety of each layer. For beam cross sections it is recommended to input at least one line reinforcement at the top and the bottom. For column cross sections (or biaxial loading) there should be reinforcement along all four sides. Dimensioning control in the case of several layers:

Each reinforcement layer contains information regarding adjustment of the reinforcement area (Ch. B 2.3). In the case of beam cross sections, the given initial value of reinforcement is considered as a minimum reinforcement. Thus the area provided by the program is never smaller than the area already input. For column cross sections the initial values of the reinforcement area to be dimensioned are scaled in proportion to the given minimum reinforcement content (see Analysis Parameters). When calculating the reinforcement content, also the layers with the areas declared to be constant are taken into account. If the total area is too large, then the minimum reinforcement condition prescribed for the columns cannot be fulfilled.  Circular line reinforcement is dimensioned for tension and compression.  Prestressing reinforcement areas normally remain constant. 

General procedure: To be able to interpret the dimensioning results more easily even in the case of complicated cross sections, it pays to carry out the first dimensioning only with the above recommended second and fourth layers. (Additional reinforcement, if present, may be declared as remaining constant) Finally, one should decide upon a particular reinforcement layout and by means of analyses (efficiency, crack widths) that the required verification can be achieved.

B 3.2.2 General remarks on shear force and torsional dimensioning VT dimensioning is only carried out if a valid shear wall model exists. As the result of dimensioning , among others the following quantities are available  Shear wall forces (for details on analysis see C 4.1)  In the graphical presentation a positive shear wall force points from the starting point to the end point of the shear wall Efficiency of concrete compression diagonals (column with the heading eff c)  Required stirrup area (column with the heading a sw) Stirrup dimensioning for different codes is described in Ch. C. 



Required additional axial reinforcement for each shear wall side: (heading A sl2 )

In compliance with the common reinforced concrete codes, for the dimensioning of the stirrups the classical truss model shown below is used ( ,  from EC 2): z�( cot   cot ) F c z

F w Stirrup tension force



V  S i Concrete compression force



F s

Fig. B−18 Truss geometry

B-38

FAGUS‐7


The corresponding descriptions in the codes are usually based on a single web, i.e. the critical inplane force in this case is the same as the corresponding shear force S i  V sd

Therefore, in order to be able to compare the following formulas with the notation used in the codes, in this section the term V sd is used for the inplane loading. Assuming a freely selectable slope of the concrete compression diagonal and stirrups results in the following tensile force to be resisted per unit length F w. F w 

V sd z� sin �( cot   cot )

Resisting shear force with stirrups alone

If this force has to be resisted by the stirrups alone, i.e. without the additional shear force resistance of the concrete, we have (for vertical stirrups with a spacing of s w), the usual formula: A sw 

V sd s w z� f ywd� cot 

In the program the value a sw , which depends on the stirrups spacing, is output: A a sw  s sw  sw

V sd z� f ywd�cot 

2

(unit: mm m

or selected by user)

i.e. for a specific stirrup area the program output value has to be multiplied by the desired stirrups spacing. V sd z f ywd

 

dimensioning value of the inplane force: is given by the FAGUS shear wall model inner lever arm defined by the length of the corresponding shear wall. dimensioning value of the elastic limit of the stirrup reinforcing steel: from the assi gned stirrups or fromthe standard default value: > Variant Properties > Models slope of the concrete compression diagonal: Analysis Parameters > Further Values. slope of the stirrups/Shear Force Reinforcement: > Variant Properties > Models

Minimum reinforcement

Besides the reinforcement required to satisfy the statics in the codes, there are quite different rules about minimum reinforcement contents. For details refer to Ch. C.

Check the loading of the concrete An upper limit for each loading is given by the maximum admissible concrete stress in the compression diagonals. Usually this condition is given as the maximum admissible shear force, e.g. in EC2 as V Rd, max: V sd  V Rd2

The determination of V Rd2 depends on the code and is described in Part C. For practical reasons in FAGUS the efficiency factor is always output. eff c 

FAGUS‐7

V Rd, max  1.00 V d

B-39


Axial reinforcement due to force in shear wall 

Basically, the shear force analysis is completely separate from the dimensioning for bending. Regarding output two settings are possible in the Analysis Parameters :



Standard Method A sl2 is listed in the numerical output for each shear wall. It is the user's responsibility to calculate this value for the corresponding axial reinforcement layers.



V + T as internal axial forces In this case the axial reinforcement includes the influence of V + T.

Standard method

The additional required axial reinforcement due to the force in the shear wall S i for each flange is: Si

F s  0.5�Si�( cot   cot )

A sl2



A sl2 

 s� F s f y

This value is output in the shear wall table together with the required stirrup reinforcement. In the case of the standard method, the axial reinforcment table only includes the reinforcement area due to M,N. The additionaly required value A sl2 has to be added “manually”. Shear force and torsion as internal axial force

If the corresponding check box in the analysis parameters is not ticked, then the influence of shear force and torsion on the axial forces is integrated directly in the bending analysis by introducing an additional ”internal" tensile axial force. The changed axial loading is determined as follows: F si  Si�( cot   cot ) (sum of all tensile axial forces in the shear wall) * N  N   F si M y*  M y   F si �  z si * M z  M z   F si �  y si F si : normal force contribution of the ith shear wall  y si ,  z si : distance from middle of shear wall to centre of gravity of cross section.

Fig. B−19 Part of the dialogue for the analysis parameters

The individual shear wall forces are determined using the same rules that were described in the previous sections. A prerequisite for this method is that for the VT loading there is a valid cross section input. Otherwise or if the cross section has no shear walls, the total shear force is assumed to act at the centre of the cross section.



N (V y,V z, T )  V y2V z2 cot �(   cot ) 



T �u ef � cot  2� A ef

Since with this model all the reinforcement of the cross section is taken into account, it must be ensured that the reinforcement lying outside the webs is correctly “connected” (key word “Shear Connection” e.g. slab  web). This is already implied in this analysis model !

Overview table for reinforced concrte analyses directly from STATIK

In the case of dimensioning directly from the program STATIK, an overview table with the title ”Standard Results" is created together with some graphics, in which the sum of all required reinforcement areas is presented.

B-40

FAGUS‐7


The curve A results from the analysis for bending with normal force. If there are several reinforcement layers, the reinforcement areas for each half of the beam is summarized and written on the corresponding half. Curve B still contains the A sl2 portion of all shear walls z �cot  2

 x Asw1 (stirrups)

1

2

tensile flange forces:

initial value determined in point 1 and horizontally increased in arrow direction to point 2.

contribution of shear force 1

support fans

2

B A

shear force from bending analysis (with normal force)

A sections with fan attribute

B

Curve A: As( M , N ) Curve B: As( M , N , V )  As( M , N )   A sl2 For column cross sections the total amount of reinforcement is output in a single column and in the case of unsymmetrical conditions the greater value of A sl2 is used for the summation (earthquake analyses, etc.).  A sl2  Maximum(  A sl2 top,  A sl2 bottom) As total( M , N , V )  Astotal( M , N )  2�  A sl2 Adjustments in support area 12

If a section member has the attribute ”fans”, the results are adjusted in the way described above (no additional axial reinforcement due to shear and torsion over the support cross section). 

These adjustments are only made in the table ”Standard Results" and the corresponding graphics. Additional convention: cross sections without shear walls Sometimes in simple cases one may also want some information on the ”amount of offset”, without having to define extra shear walls in the cross sections. If the cross section has no shear walls F s is calculated directly from V z , i.e. F s  0.5�V z�( cot   cot ). Comparison of the two methods

If one only considers the tension side of the cross section (reinforcement), then with the aproach “internal axial tensile force” in general one obtains similar results to those of the standard method. The advantage of this method, however, is that the influence of V+T can also be included in the analysis types Ultimate Load and Stress Verification. The resulting decrease of normal force on the compression side of the cross section due to V and T have to be examined more closely (sometimes the shear force influence in the case of column cross sections is deliberately neglected). Dimensioning directly from STATIK for the two models:

Depending on the setting in the check box, the first curve is determined using one of the above methods and the second curve by addition or subtraction of the corresponding part  A sl2.

curve determined using V+T as internal tensile axial stress (includes the offset) curve determined using “standard method“

FAGUS‐7

B-41


B 3.3 The Tab sheet 'Analyses' All cross section analyses are carried out in this Tab sheet.

List field for selection of desired cross section variant

n d m o n a i t c a r s g a a r i s i e y d t l n a n i n o i N a t /  s c a r M s e e r t d t n n S i l a a y r c e n n e e i g c i f f E

 m d m t n a n a e r a r g w g m a a o i o l i d d f m r s e a s d r u e e n t n a a h f S f ) v i r t n i u s  a r c t t n s e ( m s i o s y m l a n a s s e r t S

t r o p e r d n a s i s y l a n a l a i r e s e t a r e n e G

If several section force lines were input, each line is analysed individually and the results are the limit state values of all output analyses (e.g. stresses min/max). Each analysis is carried out with a particular cross section variant and taking into account the specified analysis parameters.

B 3.3.1 Ultimate load/ Efficiency With the aid of this analysis the cross section efficiency can be determined, which is defined as follows: eff  S

R d

eff = efficiency. ( In the Eurocode unfortunately at the moment there is still no official term for this value) S = input applied stress R d = dimensioning value of strength The required verification is deemed to be satisfactory if eff  1.00 . Depending on the selected analysis parameters (ULS, SLS) both the ultimate load and the serviceability are verified. 

Comparison with earlier versions of FAGUS: In earlier versions of FAGUS an ultimate load factor  was determined. The relationship between eff and  is:

eff  1 

The advantage of an efficiency factor is that it is always in the range of 0 .. 1 if the verification check is fulfilled. This is very convenient for showing the results in graphical diagrams. The capacity factor can be very high for small action forces and the value is even infinite in a zero point of a beam.

B-42

FAGUS‐7


Additional condition:

At the top of the table there is a line labelled 'constant' and alongside it above each column

additional condition

see dimensioning

there are corresponding check boxes. If some of these check boxes are ticked, in the resistance analysis only the 'free' section forces are increased. In the case shown below with two components the component Y is left untouched ( remains at the input value) and only the component X is increased. The loading path for the resistance analyses may be seen in the diagram shown below. Y

Y d) ( � X d ,

Y d

X d

X

Fig. B− 20 Loading path with constant component Y



Application: This possibility is sometimes needed for column cross sections ( N constant, M y and M z variable)



Efficiency of shear force and torsion If we have a valid shear wall model and there are corresponding stirrup cross sections, this factor is also calculated for V+T. The influence on the axial reinforcement is considered using the model “Shear force and torsion as internal tensile axial force”, i.e. the analysis parameters are automatically adjusted. (For details see the chapter on dimensioning)



Several actions If several lines are activated at the same time the ultimate load factor and the corresponding results are output.

B 3.3.2 Stress analysis with forces In the case of the stress analysis, the section forces input by the user are applied to the cross section and the result is all the required force and strain quanties displayed graphically and numerically. The input dialogue is similar to that for the ultimate load analysis. There is no error message saying whether the calculated stresses fulfil certain conditions. The obtained result without further evaluation is output graphically and numerically. If the given section forces are too large, a corresponding error message is given. In this case it is recommended to carry out first an ultimate load analysis. If a valid shear wall model exists and the stirrup areas are known, the stresses in the stirrups are also output (analysis with the same truss model as for the dimensioning). Load steps: If several lines are active and this check box is ticked, the given section forces are applied in loading steps (summed!). Otherwise it is a question of independent analyses and the result includes the Min/Max values, which, e.g., are required for fatigue verifications.

FAGUS‐7

B-43




If, within the load steps, individual partial sections also have to be activated, this can only be accomplished by means of a series of analyses.

Crack width verification As part of the stress verification the crack widths and crack spacings are calculated according to the codes nach DIN 10451 and EN199211:2004. For all other codes a verification according to EN1992 is carried out (material values taken from the national codes, but the procedure is as described below) The crack width according to EN1992 is given by (the procedure is similar to DIN)  sm   cm) w k  s r , max (

where

( sm   cm) 

f ct, eff  p ,eff

(1   e  p, eff )

E s

 0.6  s

E s

= steel stress

 s k 1

 s  kt

 0.4 (for longterm loading)

f ct, eff = average value of the effective concrete tensile strength f ct, eff  f ctm or smaller, if cracking is expected before 28 days the corresponding reduction factor can be input in the Tab sheet Analysis Parameters

 p, eff = reinforcement content (in the zone ”A c, eff " according to 7.3.2) = ratio

 e

E s

E cm

The maximum crack spacing is: s r , max  k3 c  k1 k 2 k 4   p, eff with the following values k 1 k 2 k 2

 0.8  0.5  1.0

for reinforcement bars with good bonding (for pure bending) (for pure tension, lies between k 2  (1  2)2 1

From the national appendix k 3 k 4

 3.4  0.425

Reinforcement: The reinforcement considered in the analysis is defined by the hatched area in Fig. 7.1 (EN1992). In the case of sloping reinforcement in FAGUS a line parallel to the neutral axis is assumed as the boundary of the zone at a distance h c, eff from the edge of the tension side. According to the code the distance is: h  d) ; ( h  x)3 ; h2 ] h c, eff  Minimum [ 2.5(

If several different diameters are encountered, the following conversion is peformed to find the equivalent diameter:

 

B-44

n 1 21  n 2  22 n 1 1  n 2  2

FAGUS‐7


If the cross section also contains prestressing layers, which act compositely with the rest of the concrete cross section, then they may be included in the ”active" reinforcement: A s   21 A p  p, eff  A c, eff

 1 = modified ratio of the composite strength of prestressing steel and ordinary steel reinforcement. The default value for  1 can be overruled in the analysis parameter dialogue (Tab sheet 'Additional values'). With a value  1 = 1 the calculation is executed with fol-

lowing assumptions:



 s  p

(7.5)

1  

 = ratio of bond strength of prestressing and reinforcing steel, according to Table 6.2 in 6.8.2. (default value =0.6)  s = largest bar diameter of reinforcing steel  p = equivalent diameter of tendon according to 6.8.2

for bundles (Default): for single 7 wire strands for single 3 wire strands:

 p =1.6 A p  p =1.75  wire  p =1.2  wire

 ( wire = wire diameter)

B 3.3.3 Stress analysis with strains With the stress analysis, with a given strain plane the three components εx, χy, χz are given and from these the corresponding section forces N, My, Mz, as well as other results required by the user are determined (stresses at given positions in the cross section, cross section stiffnesses, etc.) For the program this is a simple task, since the required section forces can be directly determined by a single cross section integration. All other analysis types use the procedure described here, but may require several iterations to find the required result. The cross section integration comprises the following steps: •

determination of ε(y,z) in the integration points of the cross section

( y, z)   x  z   y  y   z •

determination of the corresponding stress using the material law selected by the user

 ( y, z)  f (( y, z)) •

summation over all "fibres" of the cross section

       

N x 

( y, z) dA

M y

( y, z) z dA

M z

( y, z) y dA

The polygonal or circular circumference of the cross section can be integrated exactly by means of numerical integration along the boundary of the cross section. Together with other results the output of the secant stiffnesses can be requested. These are defined as follows:

FAGUS‐7

B-45


EA  N  x

M y

EI y  

y

M

EI z   z z

For the determination of the secant stiffnesses, therefore, neither an Emodulus nor a static cross section value is used, but they are always calculated from the existing loading actions (N,M) and strains (, ). If individual components are equal to zero, instead of the secant stiffness the corresponding tangent stiffness is output. The following analysis parameters are taken into account in the analysis: • type of stressstrain diagram • creep coefficient • resistance factor (normally these are equal to 1) All other parameters have no influence on this analysis. Consideration of the limiting strains, for example, would be meaningless in this case, since the cross section deformations are input by the user. After the completion of an analysis the results are displayed on the screen with the aid of the diagrams shown below. With a ”print entry” the graphics and the numerical results can be included in the print list and inspected there. −0.10 o/oo

−3.41 N/mm2 −12.82 N/mm2

−220.56 kN

γ c=1 γ s=1

0.54 o/oo

105.58 N/mm2 10.56 kN

Fig. B− 21 Stress− strain diagrams

In the strain plane the maximum edge strains are labelled; in the stress figure the edge stresses and the stresses in the extreme reinforcement layers. In the diagram with the couple, the point of action of the tensile and compressive forces are shown with an arrow, whereby the compressive force includes both the part due to the concrete and the contribution of the compression reinforcement, if present. The diagrams should be helpful when checking “manually”. Within FAGUS, however, the shown quantities are never used in the analysis , i.e. at the end of the iterations for the manual calculation the "inner lever arm" with the tensile and compressive forces is “backcalculated”. The couple is always drawn in the direction of the neutral axis. However, in some circumstances (see examples) the neutral axis is not parallel to the resulting moment vector ({My}+{Mz}). Thus, in the case of an equilibrium check this has to be taken into account.

B 3.3.4 Elastic shear flow This analysis is available only if the cross sections meets one of the following conditions:  A: Thinwalled cross section: Cross section consists solely of thinwalled elements (building one well connected structure) 

B: Reinforced concrete cross section with well connected shear walls (as shown below): Cross section consists of various (polygonally enclosed) partial cross sections

and a shear wall model which fullfills the same conditions as mentioned under point A above. (VT-model must be set on „Automatic“) 

For type “B” the properties of the variant have to be set to "Automatic" otherwise a message like "The shear flow for the present cross section cannot be displayed" will appear.

The corresponding analysis is started with the button shown on the left and the result is displayed graphically or tabularly. Normal and shear stresses are available. In the analysis the cross section is assumed to be homogenous (not cracked) and the normal stresses are calculated with the usual formula shown below (The shear flow theory is explained in chap. C 3.):

B-46

FAGUS‐7


M y M � z  z � y I y I z

 ( y, z)  N  A

If the cross section is defined according to type „B“ the normal stresses are calculated with the cross sectional values of the „basic“ (polygonally enclosed) cross section. The graphic diagram is done along the shear all axis

Fig. B− 22 Normal stresses due to M,N

Integrated shear wall force = Summation of distributed shear stresses. Shear stresses or shear flow

Fig. B− 23 Shear stresses due to V,T

With the button shown on the left an equivalent stress   f (  , ) is shown: If the cross section is made out of steel the „von Mises“ criterion will be shown . For concrete cross sections the results are evaluated according to the Rankine's formula. In the numerical table both values can be found. Von Mises shape modification hypothesis:  v   2  3�  2    2  4�  2 2 Reference point for the section forces:

Rankine:  R 

Depending on the state of the button for the cross section, the cross section is loaded either at the centre of gravity or at the centre of shear . The second possibility should help above all in working through examples from the text book. Computed value: One can select shear stresses (e.g. [N/mm2]) orshear flow ([kN/m]) for output. Several loadings:

The dialogue allows the input of all six components but only one input line can be active. Output settings: The program scales the graphical output automatically. If the scaling has

to be adjusted, this can be done with the button shown on the left or using the corresponding dialogue.

B 3.3.5 MN interaction diagram With the MyN interaction diagram, all possible limit states of MyN combinations for one or more reinforcement contents can be displayed graphically. The limit state is defined in the Tab sheet Analysis Parameters. Because these, as in all other analyses, can be defined freely, it is, e.g., possible to create an interaction diagram to check for admissible tensile steel stresses (serviceability).

FAGUS‐7

B-47


The program creates an interaction diagram by determining the section forces for a series of limit states. For each point of the diagram the cross section has to be integrated once. In the case of nonsymmetrical cross sections, with the strain planes for χy≠0 and χz=0 there may be moments Mz not equal to zero. These are not shown in the diagram. Nevertheless, the presented NMy interaction curve gives a correct polygonal line in the space NMyMz diagram. An NMy diagram (with Mz = 0) can be obtain using the function "General Interaction Diagram".

N

M

In the creation of a diagram a reinforcement content can be input. The existing reinforcement area is then scaled accordingly, i.e. the reinforcement areas of different input reinforcement are adjusted until the required reinforcement content is reached. Here the input made in the Tab sheet 'Dimensioning' is taken into account in the reinforcement input. Areas of reinforcement layers with the attribute "Reinforcement area remains constant" are not changed. If prestressing tendons are present, the user input "with/without bonding" is also taken into account. If the cross section doesn't contain any reinforcement (e.g. steel sections), then aninteraction diagram is created with the existing cross section (independent of input reinforcement). If no reinforcement content is input (incomplete or empty input line), the diagram is created for the reinforcement already existing in the cross section. With several values (separated by blanks), for each input value a curve is created. The shown section force combinations are always with reference to the centre of gravity of the cross section. Reinforcement content :

For all required reinforcement contents  a momentnormal force interaction diagram is displayed. The individual values have to be separated by blanks, e.g. ”0.1 0.3 0.5". If the input field is empty, a diagram with the reinforcement already existing in the cross section is created.

B 3.3.6 General interaction diagram With the general section force interaction diagram two components out of N, My and Mz can be selected, for which the corresponding graphics will be created. The third component is constant, whereby a value not equal to zero can also be input. It is also possible to input several values (separated by blanks), and then in the same diagram the corresponding curves are displayed M

z N=konst.

M

y

The analyses are then carried out with the reinforcement already existing in the cross section, i.e. it is not possible to specify the reinforcement content as in the case of the MyN interaction diagram. The program then determines the individual corner points of the diagram with the help each time of an ultimate load analysis. The numerical expenditure for the creation of the diagram is thus considerably more than in the case of the MyN interaction diagram. The user, however, can select the number of points to be calculated and thus influence the required accuracy suited to his particular needs. The input values for the component held constant must lie within admissible ranges, otherwise a diagram is not created. Admissible means that solutions are possible for the required ultimate load analyses with a constant third component. For specified values of the component :

Here one inputs which component is constant (out of N x, M y and M z) and the required values are then input in the corresponding input field separated by blanks (e.g. ”100 200 500"). Number of calculated points :

In order to save computational time the number of calculated points can be influenced using the slider. For larger numbers the curves are more rounded, but the analysis takes longer.

B-48

FAGUS‐7


B 3.3.7 Momentcurvature diagram To better visualize the deformation behaviour of the cross section the momentcurvature diagram shown below can be created. It serves, for example, to evaluate the cross section ductility needed for earthquake analyses. The analyses are carried out with the reinforcement already existing in the cross section. As parameter the required axial force is input, whereby several values can be input (separated by blanks). For each value a curve is created. The program then determines the individual points of the curve with the aid of a stress verification.

M y

N

] m N = k −100.0 [ y M

−200.0 −500.0 kN

∆ &− 500.0

0 0 5

o _

 y

0 0 4

_

0 0 3

&− 200.0

o

&− 100.0

o

_

0 0 2

0 0 1

ρ

& = ε = 20.0 o/oo o = Zugrand: 5.0 o/oo

ρ = f ct = 0 N/mm _ = + ε y = 2.19 o/oo ∆ = − εy = −2.19 o/oo

ρ ρ

0

2

4

6

8

10

12

Fig. B− 24 Moment −curvature diagram

14

16

18

χ [km−1]

In the figure, the following points are identified by means of signatures: 

reaching the concrete's tensile strength (if input)



reaching the elastic limit in the extreme steel layers on the compression side



reaching the elastic limit in the extreme steel layers on the tension side



reaching the limiting “compression strain" at the edge on the compression side defined in the analysis parameters



reaching the limiting strain in the steel at the edge on the tension side defined in the analysis parameters



reaching the “last” calculated steel strain at the edge on the tension side

Depending on the type of cross section and the input axial force it is possible that not all of the points listed above are reached and the sequence is also not always the same. Bending about :

Selection of axis. The analysis is carried out either for bending about the y axis or the z axis.

FAGUS‐7

B-49


Loading : :

As many values for N x can be input as required. The individual values have to be separated by blanks, e.g. ”100 200 500".

B 3.3.8 Momentstiffness diagram For frame analyses , for which the assumed cross section stiffnesses have a big influence on the result, among other things the diagram on the left is required. It shows the relationship between the moment M y and the secant stiffness EI sec sec for one or more given normal/axial forces.

E I y

M E I sec sec  

If the cross section contains tendons, in general the moment My already differs from zero for a curvature =0 , i.e. the curve begins at E at E I sec sec  . For cross sections with prestrained tendons, therefore, no diagram is created. To deal with such cases it is recommended to include the prestressing force in the analysis as an external loading and create a diagram with a corresponding axial force (without prestrained reinforcement layers).

M y y

By means of symbols on the curve it is indicated when the extreme steel layer begins to yield or when the limiting strains are reached.

M

To compare different cross sections, it may be useful to output directly an (ideal) secant Emodulus. This is defined as follows:

arc tan EI sec sec

E sec  M



I  

I = moment of inertia of the cross section according to Ch.C Ch.C 2.1.1 The program performs the analysis for a single point of the curve following the stress analysis described above. Thus individual values can be verified ”manually", provided the same analysis parameters are used. The adopted concrete material law is very important for the stiffness of a cross section. Also of importance are the input resistance factors and a concrete creep coefficient, if present. Based on numerical values, the differences between the two most common stressstrain curves for concrete are documented: 

S

Care is necessary in the case of unsymmetrical cross sections! The relationship relationship between M and  is plotted with reference to the elasticallycalculated centre of gravity according to Ch. A 2 (without the reinforcement part, if this was specified in the variant properties). properties). The centre of gravity calculated using the nonlinear materials, does not necessarily have to be identical with the first value. For N  0 and M  0





  0 ( N , M and  with reference to S)

Axis points: In the diagrams no axis points are taken into account, i.e. the presented curve represents the relationship to the centre of gravity S.

Bending about :

Selection of axis. The analysis analysis is either performed for bending about about the y or the z axis. Loading : :

Any number of values of N x can be input. The individual values must be separated by blanks, e.g. ”100 200 500". Value of the ordinate : Both stiffnesses and the virtual ”elastic moduli" can be output.

B B-50

FAGUS‐7


B 3.3.9 Analysis of second order effects with axial load With this analysis it is possible to do a simple check of a column according to 2nd order theory. As input following parameters have to be given: e1 Imperfection

e0

l k

l k : Buckling length e o : Eccentricity due to imperfection e 1 : Eccentricity due due to an an end moment (calculated with e 1 = M/N) The procedure described below is suitable for isolated members with a constant normal force, with a constant cross section and a well defined buckling length l length l k as well.

e2

Background / Theorie based on: SIA 262: 2013 : 4.3.7 Druckglieder EN 199211:20 199211:2004 04 : 5.8.8 Method Method based based on nominal nominal curvature curvature Iteration with one degree of freedom: The maximum normal force is determined with an iteration procedure where the maximum deflection e deflection e 2 is the only unknown. e 2 is calculated based on the standard stiffness or curvature procedures described in the previous chapters ("cracked" cross section assumption, concrete normally without tension). In the current version buckling around the yaxis is considered only. (positive moment, tension side at the bottom of the cross section). One should use only symmetric cross sections. l k2 e  e 0  e 1  e 2 e 2   � c c   2  y

The calculation of the curvature is carried out with the real MNcombination and the applied values can be checked and compared with a normal cross section stress analysis, provided that the same analysis parameters are used. This is more accurate than certain methods where simplified simplified techniques are used to calculate the curvature. Graphic results

The loading path until failure is shown in the standard MNdiagram. There are two possible failure situations: 1 Not very slender: Cross section strain limit is reached (last point on MyNCurve) 1 2 Slender column: = stability problem 2

Last point is inside the the My Ndiagram.

Fig. B− 25 My −N−diagram with loading path

Elevated temperatures (fire resistance)

If a thermal analysis has been previously carried out and the corresponding check box has

FAGUS‐7

B-51


been activated, the loading paths paths for all intermediate points in time time are shown. (These points are defined in the thermal analysis of the cross section. See Fig. F7 /“ Output times“). Curvature

t n e m o M

Moment

t f a r k l a m r o N

Fig. B− 26 Temperature dependend moment moment −curvate diagram and corresponding My −N−Diagram for various output times

Numerical results

With this button all results are shown in a table. If a thermal analysis has been previously carried out it is also possible to determine the failure time. Comparison between FAGUS and PYRUS

Both programs use an iterative algorithm but in FAGUS there is only one unknown  the deflection in the middle of the column. PYRUS is a general FEprogram and uses several internal members (with additional integration integration points) to represent represent the variable stiffness more accurately. The static system has six degrees of freedom per node. In general FAGUS FAGUS results are a little bit more conservative than PYRUS results but a FAGUS analysis has the advantage of its simplicity. In more complex situations (variable cross section, loads in different directions etc.) only PYRUS can deliver the correct result. 

Remark regarding elevated temperatures: The FAGUS predication seems to be less accurate but not much experience is available at the time

Column with normal force

Deflection

e 2



B B-52

Stiffness EJ

In FAGUS: Assumption that the stiffness is constant along the column

The maximum horizontal deflection shown in PYRUS corresponds with e 2.

FAGUS‐7


B 3.4 Batch Analyses The functions presented here should help if frequently repeated analyses with reduced input are carried out. ”Batch Analysis" refers to the stepwise, automatic working though of analyses and cross section dimensionings. The following tasks can be solved: 

Automatic repeat of unaltered types of analysis on different cross sections.



Investigation of construction states with active/inactive partial sections and with the accompanying loading history.



Creation of internal stress states.



Activation and deactivating of partial cross sections.

A Batch Analysis is created in a special dialogue window, which is opened in the Tab sheet 'Analyses' using the button on the left. Each Batch Analysis consists of a number of command lines, which are then executed step bystep. With the excerpt from the corresponding dialogue shown below the following three steps are executed: In the first line the cross section 'QS' is loaded, i.e. made available for the subsequent analyses. Then in the second line an ultimate load analysis with My=50 kNm is carried out and with the command in the third line a print entry is made, so that the results can then be inspected in the CubusViewer. Print entry

Active/Inactive, strains

Load cross section Analyses Management of Batch Analysis

List with all tasks

Interactive creation of task list

New command lines are created with one of the buttons placed at the top right corner. If in the input window a line is already selected, then the new command line is inserted directly after it. If no command line is selected, new lines are added to the end of the list. Changes to existing entries can be modified with the help of the context menu (right mouse button).

FAGUS‐7

B-53


Creation of output lists based on analyses already carried out

Every dimensioning or analysis already carried out is plotted for each cross section. The button shown on the left opens a dialogue window with a corresponding list. Activating a single check box causes the corresponding analysis to be included in a Batch Analysis with the name "autogenerated". These can then be used directly or duplicated and modified as desired. Creation of an output list with a Text Editor

Parallel to the interactive creation of the task list, the program creates a text file [File name = name of Batch Analysis + ”.macro.txt"], which can be modified using a Text Editor (e.g. Notepad). The button on the left starts a Standard Editor with the current Batch Analysis: The syntax used within the text file is more or less selfexplanatory. For a specific task it is best to create a small Test Task interactively and the subsequent modifications can then be carried out by Duplication/Modifying the corresponding lines. Syntax check: When read in, only the lines with a valid syntax are accepted, while the rest is assumed to be comments.

B 3.4.1 Management of batch analyses In each cross section library as many different Batch Analyses can be defined and managed with the functions shown on the left:  Creation of a new Batch Analysis. The identifier is input in a separate dialogue  Change the identifier of a Batch Analysis  Delete a Batch Analysis  Save  Select an existing Batch Analysis

B 3.4.2 Batch analysis/create task list To check the analysis process and select various parameters the following (described) buttons are available: The sequence of the lines can be changed with the help of this button. The selected line can be moved up or down. Likewise, individual lines can be moved with the mouse to the desired position. Load (select) a cross section (including variant, if present). Creation of a print entry for the current state (text or graphics output). Selection of analysis parameters: Generally, the analysis parameters are selected by the program on the basis of the analysis type, e.g. Stress analyses, etc. > !Serviceability Dimensioning > !Ultimate load With this command the automatic selection can be overwritten. For all subsequent analyses the last parameters to be input are used. Creation of comment lines: This possibility is only used for documenting the Batch Analysis (useful in creating long input lists with the aid of a Text Editor).

B-54

FAGUS‐7


Stop command to halt the analysis: The following command will no longer be carried out, which can be of help in setting up large analysis processes. For the actual analyses the same symbols and buttons are used as in the Tab sheets already described for the normal interactive procedure. Load history, construction states, initial strains

With the functions available to the Batch Analysis it is possible to load just an andividual part of the cross section, as well as to investigate temperature effects (possibly only on specified parts of the cross section). This is normally carried out by specifying a "load history", with which a cross section is initially loaded and then the desired cross section elements are activated and loaded in sequence. In addition to the command "Stress Analysis with Forces" the three buttons shown below are needed. They serve to "set Active/Inactive" for the desired cross section elements (upper row of buttons), as well as input initial strains (lower row of buttons). Within the load history individual analysis parameters can also be modified (e.g. creep coefficient). The ma jority of the existing parameters (e.g. limit strains, resistance factors) should not normally be changed during a load history. Mixing with other analysis types (e.g ultimate load) is not possible in the case of a stepwise loaded cross section. Activating partial sections: In a special dialogue window the whole of the cross sction is shown and by clicking on the corresponding check boxes or with ”graphical selection" of the desired cross section elements it is communicated to the program which element should be activated. Initial state: After ”loading" a new cross section all elements are active ! Deactivating of partial cross sections: (analogue to activating).

Definition of strains: The selection of the desired cross section elements is carried out in the same way as when activating partial sections. In addition, the desired strain state has to be defined that should act in the designated cross section elements. This is done by inputting two arbitrary points with the coordinates yi, zi and their corresponding strainsεi. These two points define the ”dip" of the corresponding strain plane. Hints: 1) By inputting a strain plane an equilibrium iteration is not carried out automatically ! If, e.g., one wants to define a nonlinear temperature distribution over the cross section, several strain instructions (inputs) may be necessary and only then should the analysis be performed on the complete definition. 2) Stress analyses with ”zero section forces” are admissible ! Even if no external section quantities exist, by means of such an input line an equilibrium iteration is started, in which the (by introducing strains possibly modified) internal equilibrium is then restored again. 3) With repeated use of the command "Stress Analysis with Forces" these are summed, i.e. for each section the load increment has to be defined each time. Beton (C25/30)

An example is available under '> Help > Examples > Ex_A'

HEA300

FAGUS‐7

B-55


B-56

FAGUS‐7

C 1 Coordinate System and Sign Conventions

Vol. C Special Topics FAGUS‐7

C 1 Coordinate System and Sign Conventions The geometry of the cross section as well as the position of the reinforcement are described in a right angled, right hand y q, z q coordinate system, which is designated the input coordinate system. For the sake of compatibility with the program STATIK the xaxis lies parallel to the member axis, i.e. normal to the cross section and points in the figure below towards the reader. The position of the cross section with respect to the input coordinate system is arbitrary. The principal axes system differs from the input coordinate system in the most general case by displacements y s and z s, as well as rotation by an angle  (range of values:  45°    45° , signs according to the right hand rule, i.e. in the diagram below it represents a “positive" rotation about the xaxis). These values are calculated by the program and output together with the other structural cross section values. Unless specifically stated the member axis passes through the centroid of the cross section. If a so called 'axis point' is defined, the member axis passes through this axis point and all quantities related to the member (local member loads, section forces) act in the axis point. z q

z h

z

y q , z q

Input coordinate system of the cross section (= coordinate system of the FAGUS Graphics Editor) y , z Local coordinate system of the cross section or of the member (section forces, strains, loads) y h , z h Principal axes system of the cross section (= internal reference system for all STATIK analyses)

y h z s

S

 y

y s Fig. 1



y q

S

Centroid

Cross section coordinate systems

Remark: For an angle   0 the STATIK analyses are carried out automatically internally in the principal axes system. From the point of view of the user however only the coordinate system y, z is of interest (in the centroid or in the axis point, if one is defined). z

M z

start of member

M y

V y

y

S

N

tension positive

T V z Fig. 2

FAGUS‐7

x

Member coordinate system and section forces

C-1

Vol. C Special Topics

C 2 Section properties C 2.1 Introduction In the case of frame analysis one assumes the classical assumptions of beam theory. Elastic material behaviour is presupposed so that one can obtain a linear relationship between the individual reference quantities. In the plane of the cross section the well known relationships between the kinematic quantities and the section forces are as follows: du dx

 EN , � A x

d 2 w  M y , E� I y dx2

d 2 v  dx2

z  EM , � I z

dr x  GT � I dx x

u, v, w : displacements in the corresponding coordinate directions x,y,z r x : rotation about the member's axis N , M , T : axial/normal force, bending and torsional moments E, G : Young's (or E)modulus, shear modulus (for composite construction: E  E ref ,G  G ref ) A x : area of cross section I y, I z : moment of inertia (second moment of area) about the corresponding

axes If the influence of shear on the deformations is also considered, there are in addition the following two relationships:  y 

V y , G� A y

 z 

V z G� A z

 y ,  z : shear strains V y , V z : shear forces A y, A z : reduced cross section areas: A y   y� A x , A z   z� A x (see alsoC 3.1) FAGUS assumes for arbitrary cross sections:  y   z  1

The two material parameters E andG are defined in FAGUS by the selection of a particular material. For analyses with STATIK, FAGUS produces directly the stiffness values listed in the denominator ( E� A x, E� I y etc.), which can be overwritten there for each member or even individually. When determining I x, as well as A y and A z , certain assumptions have to be made. The program therefore allows the overwriting of these three values directly as part of the cross section/variant properties.

C 2.1.1 Area, moments of inertia The area and the moments of inertia are calculated by evaluating the integral given below with respect to the centroid and they are then output. In the case of cross sections with different materials (composite sections) these represent ideal values, related to a reference material (concrete), which is independently selected by the program for each cross section. Each cross section then contributes its contribution according to the weighting factor wi. In FAGUS this is calculated from the input elastic moduli E of the Materials. For the subsection i:

C-2

Ei E ref

Weighting:

w i 

Area:

A xi  w i

dA

Moment of inertia:

I yi  wi

z 2 dA

 

FAGUS‐7

C 2 Section properties

Moment of inertia:

   

y 2 dA

I zi  w i

y dA

Centroid:

y si

  

z dA

, z si

dA

dA

For the complete cross all subsections are summed up: Area:

A x   A xi

Moment of inertia:

I y   I yi ,

Centroid:

y s 

 y si A xi , A x

I z   I zi z s 

 z si A xi A x

To calculate the stiffnesses of the cross section these (ideal) cross section values are then multiplied by E ref . Point and line reinforcement can be considered, if desired. In the normal case however it is recommendable to omit this contribution, so that the centroid and the cross section values do not change with changes in the reinforcement. (changes in: Tabsheet > 'Variants' > 'Properties of variant' > 'Cross section values' Documentation: The moments of inertia are output by the program with respect to the principle axes system y h z , h of the cross section (cf.Fig. 1). For cross sections with   0 in Iz *, Iyz * an extra column entitled “... rel. to input coordinates” the values Iy *, are output y z , with respect to the local coordinate system of the cross section.

Reference material By default FAGUS selects the material with the lowest weighting factor as reference. Dadurch sind die Gewichte der Teilquerschntte immer 1  w i. Für einen StahlBetonVerbundträger wird der Beton damit standardmässig zum Referenzbaustoff. In einigen Tabellenwerken finden sich aber Querschnittswerte mit Stahl als Referenzbaustoff. This can be changed in: Tabsheet > 'Variants' > 'Properties of variant'> 'Cross section values' Die StabBiegesteifigkeiten werden in STATIK wie folgt bestimmt: Biegesteifigkeit : EJ y  E ref  I y( E ref ) Dazu ist In Bezug auf die Schubsteifigkeiten folgendes zu vermerken: A z variiert zwar linear mit E ref , der verwendete Schubmodul ändert sich aber nicht im glei-

chen Mass, da die Querdehnungszahlen (insbesondere von Beton und Stahl) der beteiligten Baustoff unterschiedlich sind. Schubsteifigkeit allgemein : GA z  G ref  A z( E ref ) Schubsteifigkeit im Speziellen : G Beton  A z( E Beton)  G Stahl  A z( E Stahl)

C 2.1.2 Torsional Constant 1

2

The relationship between torsional moment and the derivation of the member rotation about its axis is described by the following relationship: dr x  GT � I dx x Here I x denotes the torsional constant, torsional resistance or warping resistance. Small re-

inforced concrete cross sections are usually “compact" and the torsional constant is calculated on the basis of St. Venant's theory (membrane analogy). For arbitrary compact cross sections, in general the exact value can only be found by solving the differential equation (or using an FE program). FAGUS uses at present an approximation, in which use is made of well known formulas for many common cross section shapes.

FAGUS‐7

C-3


Cross section without openings:

Ix Model No. 1 2 3 4 5 3

Type of cross section rectangle circle ∼ thin-walled (t < 0.1 l) compact steel section

Torsional constant for STATIK Ix=1 / 3bh3 (1− 0.63h/b+0.052 ....) Ix=0.5 πr4 Ix=1 / 3 l t3 Ix=0.5 πr4 Ix=from steelwork tables

approximation!! approximation!!

Remarks on Ix Model No. 3: For polygonal subsections with a large perimeter (in comparison to area) l andt can be estimated by the program from these two values. If however we are dealing with a thinwalled section, it is better to work with the corresponding input elements (see below).

4

Remarks on Ix Model No. 4: The radius r is obtained for a circle of equal area

5

For arbitrary polygonal cross sections and without further information the program automatically assigns each subsection to one of the above cases, whereby the dimensions of the cross section needed in the above formulas are estimated partly from the area and the length of the outline. The value of Ix determined by FAGUS has to be checked and if necessary replaced by an improved value supplied by the user, especially if the size of the torsional constant greatly influences the rest of the analysis. For a cross section consisting of several subsections, the parts of the indivial subsections are determined separately and added together. Only “outlined" cross section parts contribute to Ix (i.e. no reinforcement)

6

For cross sections with large openings, the torsional constant is calculated using Bredt's formula:

7

I x Bredt  4�

A o2

 dst

A o2 u t

 4�

u = circumference, t = (assumed constant) wall thickness of the equivalent box section Cross Section with one or more Openings:

Ix Model No. 6 7

Cross section type Axopeni ng < 30 % Ax total Axopeni ng > 30 % Ax total

Torsional constant for STATIK Ix = Σ Ixtotal − Σ Ixopening Ix = IxBredt

approximation approximation

The parameters Ao , u, t are estimated by the program from the individual polygonal areas and the length of the outline. The model number used is given in the tabular output of the cross section values of the individual subsections (1..7). 8

Thinwalled cross sections

9

St. Venant's torsional constant for open, thinwalled cross sections is given by the summation of all wall elements. If the wall elements form a box section with one or more openings, the torsional constant is obtained basically from Bredt's formula for the “external" circumference. More on this can be found in Ch. C 3. In summarising it can be said that a cross section built up of thinwalled wall elements gives correct values for Ix! If therefore the torsional constant of a multiple cell bridge beam has to be known exactly, it is recommended to model it as a cross section variant with wall elements. Thinwalled cross sections:

C-4

FAGUS‐7

C 2 Section properties

Ix Model No. 8 9

Cross section type

Torsional constant for STATIK-5

open cross section box section

Ix = IxSaint−Venant (see Ch. C 3) Ix = IxBredt + ( IxSaint−Venant ) (see Ch. C 3)

In the tabular output of the cross section values of the individual subsections the model number used is given (1..9).

C 2.1.3 Torsionskonstante Ix mittels FEAnalyse Bei Querschnitten, die aus einem zusammenhängenden polygonal umrandeten Teilquerschnitt (mit optionalen Aussparungen) bestehen, kann die Torsionskonstante nach St. Venant aus einer FEAnalyse bestimmt werden. Die entsprechende Schaltfläche befindet sich im Register 'Varianten' > 'Varianteneigenschaften'. Die numerische Berechnung basiert auf folgender Theorie: Mit der Einführung einer Spannungsfunktion  wird das elastische Torsionsverhalten eines homogenen Gebietes mit folgender Gleichung beschrieben:

 2   2    y 2   z 2  2G   0  = Drehwinkel pro Längeneinheit

G = Schubmodul

wobei folgende Randbedingungen gelten:   0 Aussenrand: Aussparungen:  konstant entlang einer bestimmten Aussparung

Mit Hilfe der Spannungsfunktion  können die Schubspannungskomponenten in einem beliebigen Punkt angegeben werden:  xy 

   z

 xz  

   y

Die Schiebung ist  xy 

 xy G

Das Torsionsmoment ergibt sich aus der Integration über die gesamte Querschnittsfläche A: T  2



 dA

A

  T GI x

FAGUS‐7

für = G = 1



I x  T

C-5


C 3 Analyses with ThinWalled Elements Bei einem dünnwandigen Querschnitt ist die Dicke jedes Elementes im Vergleich zur Länge klein und die Variation der Normalspannungen über die Dicke des Elementes ist deshalb vernachlässigbar.

C 3.1 Shear Stresses due to Shear Force Basic Theory

Shear stresses are determined on the basis of equilibrium considerations in an infinitesimal element and depend on the normal stress changes along the axis of the beam x. With the conventional assumptions structural theory provides for a rectangular cross section the following relation ("Bisquit" formula): ( z) 

z

V S�( z) b � I

V =shear force,

S= stat. moment, I = moment of inertia

  d 



z

  x

y b

dx Fig. 3

Shear stresses due to shear force

For the cross section shown above one can assume that the shear stresses are parallel to the shear force and uniformly distributed over the width of the section b. Insert: Reduced area of cross section when considering deformation due to shear

For some statical analyses one wishes to take into account the influence of shear force in the analysis of the strain energy. Analogous to Hooke's law the shear strain ( z) in in a particular section (see above) is: ( z) 

( z) , G

G = shear modulus

For the whole cross section one assumes an average shear strain  m. The corresponding term in the work equation (virtual work per unit length of member dx) is: V �V =  m�V GA *

where  m  V * G� A

 m depends onG, V and thereduced area of cross section A *. With the help of the above formulas one can then find for A * the following definition:



 m�V =  m�  dA =



V �V S 2 dA GI 2 b 2





1  1 S 2 dA A * I 2 b 2

For space frame structures two shear force directions have to be considered. In STATIK/FAGUS the corresponding reduced section areas are designated by A y and A z.

C-6

FAGUS‐7

C 3 Analyses with Thin-Walled Elements

ThinWalled Cross Sections

In the case of general thinwalled cross sections one proceeds in a similar fashion: It is assumed that the shear stresses are uniformly distributed over the width of the wall and that their magnitude is also obtained from the difference between the normal stresses on the “cut" section. For nonsymmetrical cross sections there is in addition the new definition of the shear centre with the following properties: S

M



Each load passing through the shear centre stresses the beam according to classical bending theory with bending without rotation.



If the load does not pass through the shear centre, it can be split up into two partial loads: a) a load V  through the shear centre, for which one can perform a stress analysis according to classical bending theory and b) a torsional moment T  V � e M e( M = distance between shear centre and centroid), which stresses the section shown on the left for bending in the flanges.

V

V  e M

For a member or section only loaded by a shear force, therefore, the location of the shear centre has to be known in order to be able to calculate the resulting shear stresses. The procedure is explained by means of the following numerical example: Firstly, we assume a distribution of normal stresses according to classical bending theory (1) and calculate from them the corresponding shear stresses (2). The position of the shear centre is given by the condition that the moment of the forces S 1 andS 2 about this point vanish. Then we introduce a torsional moment V � e M , which also produces shear stresses (3) (for analysis see next chapter). The resulting shear stresses are then given by the sum of these two parts (2+3). 

In the interactive FAGUS mode one can choose which of distributions (2) or (4) should be presented. Amongst other things this should help to check more simply the text book examples.

Normal stresses (1)

Geometry

Shear stresses (2)

(3)

(4) S1

S1 S1

e s

S

M

h

V 

+

S2

S2

=

T

V

S1

e M

S1

S1

b

Section width: b  0.40 h  1.00 Height: Centroid: e S  0.0889 Distance MS: e M   0.23 Fig. 4

FAGUS‐7

V   1.00 S 1  0.14117 S 2  1.00

T   0.23 S 1  0.23 S 2  0.00

V  1.00 S 1  0.0889 S 2  1.00

Numerical example for U− section, shear flow and section forces

C-7


By integrating the shear flow one obtains for each wall element a resulting force in the section. The following check shows that the internal section forces (4) are in equilibrium with the given external forces. V  S 2 = 1.00 (O.K.) T  h�S 1  e s�S2 V z

 1.00�0.0889  0.0889�1.00  0.00 (O.K.)

Only with open cross sections can the shear stresses be directly determined using the “Bisquit" formula. Closed cross sections have to be imagined to be cut and by introducing appropriate forces the compatibility at the boundaries can be restored, as shown below. The output is automatically carried out by the program and the user does not need to bother about it (For checking the graphical input see Ch. C 3.2.2).

V y a

t

a

 S

1.0

S

S

 G�t � l

l

Fig. 5

C-8

Determination of shear flow for closed cross sections

FAGUS‐7

C 3 Analyses with Thin-Walled Elements

C 3.2 Torsion C 3.2.1 Open Cross Sections Shear Stresses For the shear stresses of open thinwalled cross sections FAGUS automatically assumes a distribution corresponding to warping torsion theory. By comparing the next two models their most important features are described:



T s

 Fig. 6

Shear stresses according to St.Venant

Normal stresses

Shear stresses





S2 T w Resultant flange forces



S1



Conditions:

  dA  0 My=0:   � z dA  0 Mz=0:   � y dA  0 N=0:

Fig. 7

Normal and and shear stresses for warping torsion

Depending on the geometry and statics in reality a mixture of the two models will be taken. Only on the basis of the cross section geometry alone, however, nothing can be said about the ratio of T s (St. Venant) toT w (warping torsion): T  T s  T w

 assumption for thinwalled cross sections in FAGUS:T  T w

If all thinwalled wall elements intersect in a point, the cross section is free of warping and only St. Venant's torsion exists.

Torsional Constant The wellknown torsional constant used for statical analyses is based on St.Venant's torsion and as usual is calculated using the following formula: I x St.Venant   1 l�t3 3

FAGUS‐7

l, t : length, thickness of thinwalled element

C-9


Warping Constant For all open cross sections, besides the shear centre also the warping constant (warping resistance) is output: s

 l

( s) I   s s

0



 2 dA

( s) 



( s) ds

For more on this see, e.g.:  Chwalla, E.: "Einführung in die Baustatik", StahlbauVerlag  Sattler, K.: "Lehrbuch der Statik", Springer Verlag, 1969

C 3.2.2 Closed Cross Sections For closed cross sections it is characteristic that under torsional loading for every hollow cell a constant shear flow is obtained. The torsional constant of a single hollow cell is calculated using Bredt's formula: I x Bredt  4�

T

A o2

 dst

For multiple cell subsections the torsional constant and shear flow are given by the compatibility and equilibrium conditions. This is performed by FAGUS automatically by solving the corresponding system of equations. For the corresponding statical section property I x of the total cross section the default value is the sum of the following two parts: I x  I x St.Venant  I x Bredt

In the case of thinwalled cross sections the first term, however, can usually be neglected (for definition see open cross sections). The Bredt part determined by the program depends on whether in fact in the cross section input one or more closed hollow cells are defined (connection produced by “snapping" the end points!). The topology recognised by the program can be checked in the numerical output. For the cross section shown below, in the legend of the table with the cross section geometry, e.g., there should be the following remark: “The thin-walled wall elements form a box section with 3 cells"

Otherwise it can be seen straightaway which node is “leaking" from the shear flow diagram of a test analysis.

0.6

0 4 . 9 2

0.5

0.5

0.5

0.5 0.6

26.00

Fig. 8

26.00

26.00

Comparison of the two torsion parts by means of a numerical ex ample (from K. Sattler Vol. I/8, units [cm])

I x St.Venant  16 cm2, I x Bredt  56900 cm2

C-10

FAGUS‐7

C 4 Shear and Torsion for Reinforced Concrete

C 4 Shear and Torsion for Reinforced Concrete C 4.1 National Codes

C 4.1.1 EN 199211:2004 Eurocode 2 Shear reinforcement area

The stirrup area per unit length is (6.13) z



V sd A sw  s w f ywd� z�( cot   cot )� sin 



A sw s w 

= crosssectional area of shear reinforcement = spacing of stirrups = inclination of stirrups  ( = 90 = vertical stirrups) = inclination of concrete compression strut



f ywd

= design value of strength of shear reinforcement at elastic limit f y f ywd  

s

z



= inner lever arm (= 0.9 d in simple cases)

6.2.3 says: "z should correspond to the bending moment in the element under consideration" meaning that for a simple beam z should be taken at midspan for the shear force checks at both supports. Therefore in FAGUS z cannot be directly taken from a flexure resistance. z has to be given by the user either with the shear wall height or with an explict value. Checking stresses in concrete

As a check on possible failure of the concrete compression strut Eq. 6.14 is used, i.e. the shear stresses have to fulfil the following condition: V Rd, max   cw b w� z�  1 � f cd �( cot   cot )(1  cot 2 )

 1 = The value of  1 may be found in its National Annex. 

Recommended value:  1 =   from National Annex or 



f

ck  0.6  250

( f ck in N/mm2 )

If the design stress of the shear reinforcement is below 80% of the characteristic yield stress f yk,  1 may be taken as:  1 = 0.6  1  0.9 

f ck 200

for f ck  60 N  mm2

(6.10aN)

 0.5 for f ck  60 N  mm2

(6.10bN)

 cw = is a coefficient taking account of the state of the stress in the compression chord

For nonprestressed structures:

FAGUS‐7

C-11


 cw = 1.0

Prestressed structures:  cw =(1 

 cp ) f cd

for

(6.11aN)  cw =1.25 (6.11bN)

 cw =2.5�(1 

(6.11cN) 

0   cp  0.25 f cd

for 0.25 f cd   cp  0.5 f cd  cp ) f cd

for 0.5 f cd   cp  1.00 f cd

The values  1 and  cw can be found in the legend and can be overwritten.

The value of  w, min for beams may be found in the National Annex. The recommended value is given as  w, min  0.08 f ck  f yk

(9.5N)

Summary of input parameters:

Analysis parameters (> 'Options' > 'Analysis parameters') θ : in tabsheet > 'Miscellaneous' > ...  w, min : in tabsheet > 'Reinforcement' > ... Cross section input α : In tabsheet 'Variants' > 'Properties' > 'VTModell' > 'Inclination ..' z, b w : Given as shear wall attributes f ywd : Given through selected material class: (1) Default values in 'Variants' > 'Properties' > 'VTModell' > .. (2) As stirrup attributes

6.2.2 Members not requiring design shear reinforcement Der Durchstanzwiderstand ohne Schubbewehrung beträgt: V Rd, c  [C Rd, c  k   100   l  f ck

(6.2.a)

1 3

 k1   cp] b w  d

Minimum V Rd, c  [ v min  k1   cp] b w  d C Rd, c  0.18  c



k  1  200  d  2.0

 l 

A sl  0.02 b w d

 min  0.035  k 32  f ck f ck

C-12

:

12

charakteristische Zylinderdruckfestigkeit

FAGUS‐7


C 4.1.2 SIA 262

Swisscode Stirrup

z





The stirrup area per unit length is: V Rd, s A sw  s w f sd� z�sin �( cot   cot )

(262.38)

Check of concrete stresses:

V Rd, c  b w  z  k c  f cd  ( cos   cot  sin ) sin 

FAGUS uses k c  0.6

(262.39) (262.4.2.1.7) ).

The efficiency eff c is defined as eff c 

V Sd V Rd, c

Minimum reinforcement a sw, min  0.2%� b w , b w  40 cm

C 4.1.3 EHE08

Spanish Code: Instrucción de Hormigón Estructural EHE08 is based on Eurocode with following national adaptions:

z





0.5

 cot   2.0

Check of concrete stresses: see Eurocode Shear force resistance

V u2  V cu  V su V cu = Contribution of concrete

In general cases V cu is not easy to evaluate as it is dependend on the longitudinal reinforcement (e.g. dependent on ,  and d). In practical cases there are three different solutions:  V cu is set to zero (This option is on the safe side and used in most European countries)  V cu is calculated according the old EH91code  V cu has to be calculated manually and given explicitely for each shear wall In the current program version, only option 2 is available, which can be can be used in a pre-design phase. Option 1 und 3 will be available in future versions. V su = contribution of stirrups : V su  A sw� f ywd� z � sin �( cot   cot ) Stirrup area per unit length V sd  V cu A sw s w  f � z�( cot   cot )� sin  ywd

FAGUS‐7

C-13


C 4.2 Older Codes C 4.2.1 Design according to SIA 162 Design of stirrup reinforcement

z

The stirrups area per unit length is determined using: 



Checking stresses in concrete A sw is only output if the stresses in the concrete compression diagonals

fulfil the following condition:  R  f c , red �(cos   cot �sin )� sin 

(f c , red after 3 24 211)

f c , f cw , min red  �sin  �

 = 0.4

Check at design level   Rd   R  Sd 

V Sd   Rd b w� z

Minimum reinforcement

Explicit information is missing in SIA 162, i.e. a numerical verification does not have to be given. Following current practice, however, adequate stirrups reinforcement is provided for constructional reasons.

C 4.2.2 Design according to DIN 1045 (07/1988) z



  45°

The inclination of the compression diagonals following Section 17.5.4 is generally ussumed to be 45 degrees, i.e. the default program value should not be changed. Depending on the level of loading the design value of shear force (or shear stress τ) may not be reduced. One distinguishes between: Shear Region 1 (region with minimum reinforcement: 

  012)

The design value of the action is:  nom  0.4�  o

This value corresponds to a minimum shear reinforcement of 

 o  0.4�  s

 ( s = permissible steel stress =

fy , µ = reinforcement content) 1.75

Normally, the minimum reinforcement is determined by the program for the material quality used. For special analyses this value can be overwritten by the user. Shear Region 2 (reduced shear cover:  012 0.4�     o s



    02)

2

  o  0.4�  0 02

i.e. for the design the reduced shear force is V  � b w� z Shear Region 3 (full shear cover:  02

C-14

   03 )

FAGUS‐7


In this region the whole shear force has to be resisted by the shear reinforcement alone. A sw V s w  z �  s � sin �( cot   cot )

V ( = Q = shear force at service level)

Apart from Shear Region 3 no reinforcement areas are output. The limits of the above shear regions can be found in Table 13 of the code. a sw, min 

85� 1� b 0  s

C 4.3 Design for torsion Torsion models Type A

In the case of open cross sections as well as with (automatically recognised) closed hollow boxes the given torsional force corresponding to the method described at the beginning of this chapter is distributed to the individual shear walls. A single shear wall no longer “knows", where its internal force comes from. The design of the shear reinforcement, therefore, proceeds according to the rules discussed in the previous section.

V

T

Torsion model Type B: Compact cross sections

If the shear walls have to be designed for torsion as “full sections" (enclosed shear flow),then the additional axial reinforcement due to a torsional moment T is determined using the following formula: A sl 

t ef t ef A ef

u ef = A ef =  = f yd =

T �u ef f yd 2 � A ef �

�cot 

circumference of coacting cross section area with torsion coacting cross section area with torsion (= width• height of shear wall) slope of concrete compression diagonal  ( = 45° according to DIN 1045) design value of strength at the yield limit of axial reinforcement

The shear stress due to torsion is: * sw

A

 

T 2 � A ef �t ef

t ef = coacting (equivalent) wall thickness with torsion d o , d o  min( b, h) 8

according to SIA :

t ef 

according to DIN:

t ef only indirectly specified, in FAGUS limited to d o6

according to EC2:

t ef  A u =

A 2�( b  h)

The required stirrups area is * A sw 

T � tan  f ywd 2 � A ef �

f ywd = design value of strength at the elastic limit of stirrups reinforcement

A sl2 A sw A sl2

To be able to output these values together with the shear force results in the same column, two values are output (applies only to compact cross sections!): * A sw  2 � A sw : total area of stirrups element (both top and bottom sides of beam)

C 4.3.1 Überlagerung Querkraft und Torsion Die Betonausnutzung wird für alle Normen mit folgender Formel nachgewiesen:

FAGUS‐7

C-15


T ed V Ed   1.0 T Rd, max V Rd, max

EN 199211 (6.29)

Nach DIN 10451 dürfen die beiden Terme für kompakte Querschnitte noch quadriert werden. Es ist ein entsprechende Hinweis in der Legende der entsprechenden Tabelle zu finden und als Betonausnutzung wird das Maximum aus folgenden drei Termen gebildet: T ed 2 V Ed 2     T Rd, max V Rd, max

effc = Maximum { 

,

T ed V Ed , ) T Rd, max V Rd, max

C 4.3.2 Überlagerung Querkraft und Torsion Die Betonausnutzung wird für alle Normen mit folgender Formel nachgewiesen: T ed V Ed   1.0 T Rd, max V Rd, max

EN 199211 (6.29)

Nach DIN 10451 dürfen die beiden Terme für kompakte Querschnitte noch quadriert werden. Es ist ein entsprechende Hinweis in der Legende der entsprechenden Tabelle zu finden und als Betonausnutzung wird das Maximum aus folgenden drei Termen gebildet: T ed 2 V Ed 2     T Rd, max V Rd, max

effc = Maximum { 

C-16

,

T ed V Ed , ) T Rd, max V Rd, max

FAGUS‐7


C 4.4 Combination of Shear Wall Results With compac cross sections, e.g. also for the circular section shown below, several shear walls are necessary in order to obtain a complete model for resisting a combined action (Vy, Vz, T). In this case each shear wall provides its special result, but the cross section, however, may contain only one stirrup, i.e. for the final stirrups area we have to be able to combine different results (also considering minimum reinforcement rules). This is done by means of results combinations. A general description of the results combinations can be found in the program's help system in the corresponding dialogue window. Below are some examples for the calculation of Asw1 (=total stirrups reinforcement per web): Formulas used for the calculation of Asw1

Cross section

Distribution of load among the shear walls

Asw1:=

Vy

Vz

AswMin(SWZ) OR Asw(SWZ)+Asw(SWT) OR Asw(SWY)+Asw(SWT)

Asw1

SWY

T o r s i o n m o d e l :

T

Z S W -

SWT

B

SWY AswMin(SWZ) OR Asw(SWZ[Vz])+Asw(SWZ[T])

SWY Asw1

S W Z

S W Z

B

SWY2 AswMin(SWZ) OR Asw(SWZ[Vz])+Asw(SWZ[T])

SWY2 Asw1

S W Z

S W Z

SWY1

B SWY1 SWT1 .. SWT8

Asw(SWY[Vy])+2*Asw(SWT1[T]) OR Asw(SWZ[Vz])+2*Asw(SWT1[T])

Asw1

SW Y

S W Z

A Conventions: AswMin(SWY) = minimum reinforcement of corresponding shear wall SWY Asw(SWY[Vy]) = statically required stirrup reinforcement in the shear wall SWY shear wall model for Vy Fig. 9

FAGUS‐7

Combination of different shear walls for some simple cross sections

C-17


C 5 Overview of Parametrised Cross Sections As previously mentioned with the help of parametrised cross sections a complete cross section input can be quickly generated. A further advantage of this input method is that different cross section attributes, based on the types given below, can be automatically set by the program “correctly". In a first group of PCS the “solid" cross sections given in Fig. 10 are summarised. They are unreinforced and can be of any material.

S-R

S-RV

S-L

S-T

S-Z

Fig. 10

S-TV

S-U

S-TV2

S-I

S-UV

S-O

S-OH

S-RH2

S-C

S-IV

S-RH

Parametrised cross sections of type “Solid” (S=Solid)

A second group contains a series of thinwalled PCS. They are also unreinforced and may be of any arbitrary material.

T-T

T-T2

Fig. 11

T-I

T-I3

T-I2

T-I4

T-HO

T-C

T-C2

T-Z

Parametricised cross section of type “Thin−walled” (T=thin−walled)

The parametricised cross sections shown in Fig. 12 contain axial reinforcement, a complete shear wall model, as well as a results combination Asw1 (for explanation see Part B), which represents the required main stirrup reinforcement according to the diagram shown below. This simplification is necessary, above all for tabular output in connection with STATIK/ FAGUS analyses, so that uniform tables for the complete structure can be output. In the detailed output in a FAGUS manual operation details are given on each cross section element.

Asw 1 G-R

Asw 1 G-T

G-TV

Asw 1 G-U Fig. 12

Asw 1

Asw 1 G-I

Asw 1 G-UV

Asw 1 G-IV

Asw 1 G-RH

Asw 1

Asw 1 G-B1

G-B2

Parametricised cross sections of type “RC beam” (G=Girder)

The cross sections shown above are dimensioned for all six section quantities (N,My,Mz,Vy,Vz,Tx). It is assumed however that the main loading direction is in the local z direction (cf. reinforcement layout) and if the shear centre is not at the centroid, no conversion for the horizontal loading Vy is carried out. As column and wall cross sections the following parametricised cross sections are available:

C-18

FAGUS‐7

C 5 Overview of Parametrised Cross Sections

C-RP Fig. 13

W-R Fig. 14

C-RL

C-O

C-RH

C-OH

Parametricised cross sections of type “RC columns” (C=Column)

W-L

W-Z

W-U

Parametricised cross sections of type “RC walls” (W=Wall)

To be able also to design circular column cross sections for (as a rule small) shear force and torsion, (to meet the wish of a user) a conservative shear wall model was defined. In the case of large shear forces however the assumptions made have to be checked in each case with the recommendations in the technical iterature.

SLAB

FAGUS‐7



Linienförmige Bewehrungen mit diskreten Bewehrungsstäben : In FAGUS kann nun auch bei parametrisierten Querschnitten zwischen 'linienförmig verteilter Bewehrung' und 'Einzelstäben' umgeschaltet werden. Um eine ”Kollision" in den Bewehrungseckpunkten zu vermeiden (z.B. CRH), werden die vertikalen Linien in diesem Fall ohne Endstäbe gene- riert. Dies führt aber bei den oben dargestellten Wandelementen selten zum gewünschten Resultat. Falls man tatsächlich das Modell 'Einzelstäbe' verwenden möchte, sollten solche Querschnitte mit Vorteil in normale FAGUSQuerschnitte umgewandelt werden.



Damit die Stützenkreisquerschnitte ebenfalls auf (in der Regel kleine) Querkraft und Tor- sionsbeanspruchungen bemessen werden können, wurde hierfür entsprechend einem Benutzerwunsch ein konservatives Schubwandmodell definiert. Bei grossen Querkraftbe- anspruchungen sind die getroffenen Annahmen aber in jedem Fall mit den Empfehlungen der Fachliteratur zu überprüfen.

C-19


C-20

FAGUS‐7

D 1 Introduction and Overview

Vol. D Timber Cross Sections FAGUS‐7

D 1 Introduction and Overview If a cross section is built up solely by different types of wood FAGUS uses a special procedure according to the requirements of this material. Only three analysis types are available in this case  Stress analysis  Ultimate load / Efficiency / Capacity check  General interaction diagram and the following steps are needed: 1. Check material list and select desired timber material classes (Menu: Options > Materials) 2. Define the cross section geometry  with a parametrised cross section (tabsheet “Timber")  with one or more polygonal enclosed partial cross section 3. Select tabsheet 'Analysis' and define action forces and optionally buckling lengths 4. Create results: The flash button starts the calculation and shows a graph with the results. The efficency factor should be below 1.00 to fullfill the safety requirements.

D 2 Remarks for the analysis D 2.1 Material Specifics 

For this type of analysis a linearelastic relationship is assumed. Timber is an orthotropic material and the national codes require different allowable stresses due to bending and axial forces.

f .. k f .. d

Example C24 (“Coniferous tree") acc. SIA 265: char. value f m, k  24 N  mm2 

Parallel to the grain: f c,0, d   12 N  mm2, f t,0, d  8 N  mm2 , f m, d  14 N  mm2 Perpendicular to the grain: f c,90, d   1.8 N  mm 2, f t,90, d  0.1 N  mm2 In a timber cross section the stress in each grain has to be checked. The conversion in an arbitray direction is done in a similar way described below for the modulus of elasticity. In contrast to the (concrete) analysis methodes descriped in the previous chapters, for timber cross sections design values for stresses are used to define a limit state instead of limit strains. Conversion between characteristic and design values

Each national annex has its specific characteristics: The design values according SIA 265 (1) are defined as follows: R d 

 M  t  w  M R k

In table 6 (solid timber) and table 7 (glued laminated timber) design values are given for the most frequent cases (Duration of load:  t  1 ; moisture class  w  1 = 1). In other cases the term  t�  w can directly be given in Menu 'Options' > 'Analysis parameters'

FAGUS‐7

D-1

Vol. D Timber Cross Sections

Design values according to DIN 1052 und EC 5: f m, d 

kmod� f m, k  M

In the German Code DIN 1052 characteristic values are predefined. The partial factor  M is given by  M  1.3. k mod is a factor taking into account the effect of the duration of load and moisture. The program assumption have to be checked by the user and eventually adapted! Grain direction

Each partial cross section can have its own fibre direction . The modulus of elasticity E (and the cross sectional properties) is calculated with E  

E 0� E90 E0 sin   E 90 cos 2  2

For E 0 and E90 mean values are used: E 0  E0, mean , E90  E90, mean (SIA 265, 2.3.1.1)

D 3 Analysis D 3.1 Stress Analysis

M

In the case of a stress analysis, the section forces input by the user are applied to the cross section and the result is all the required force and strain quanties displayed graphically and numerically. The results are also shown if the action is too big. The material is assumed to behave elastically unconditionally but in the table “Max. strains and stresses” there is a message indicating this fact. Additional stresses at any point in the cross section are available by means of 'Results points'.

D 3.2 Efficiency, Capacity Check The action forces given by user are augmented until an allowable stress is reached somewhere in the cross section.

D 3.2.1 Bending and axial normal force The general rules defined in the national codes require a distinction between stresses  t due to an axial normal forces and stresses  m due to bending moments. The check is done in the follwing way:  t  m  1 f t, d f m, d Combined bending and axial tension

SIA 265

D-2

 t,0, d  m, y, d  m, z, d   1 f t,0, d f m, y, d f m, z, d

FAGUS‐7

D 3 Analysis

In DIN 1052 and EC5 a factor k m is introduced:  t,0, d  m, y, d    k m� m ,z, d  1 f t,0, d f m, y, d f m, z ,d

EC5 (6.16)

  t,0, d   k m� m, y ,d  m, z, d  1 f t,0, d f m, y, d f m, z, d

EC5 (6.17)

For solid timber, glued laminated timber and laminated veneer lumber:  for rectangular sections : k m = 0,7  for other crosssections : k m = 1,0 For other wood based structural products:  for all cross sections : k m = 1,0 Combined bending and axial compression  c,0, d 2  m, y, d  m, z, d ( )   1 SIA 265 f c,0, d f m, y, d f m ,z, d

DIN 1052, EC5  c,0, d 2  m, y, d  ( )   k m m, z, d  1 f c,0, d f m, y, d f m, z, d  m, y, d  m, z, d  c,0, d 2  1 ( )  k m f c,0, d f m, y, d f m, z, d

EC5 (6.18) EC5 (6.19)

Stability The stability verfication is performed in an similar way. The “allowable” stresses are reduced by the two factors k c and k crit . The following formulas are used in SIA, DIN and EC5. k( c stands for k c ,y and k c, z , k crit = k m in SIA) k c 



1

EC5 (6.25)

2 k  k 2   rel

2 k  0.5�[1   c�(  rel  0.3)   rel ]

Laterial torsional buckling: k crit k crit  1 k crit  1.56  0.75  rel, m 2 k crit  1  rel , m

EC5 (6.26)

for  rel, m  0.75 for 0.75   rel, m  1.4 for 1.4   rel ,m

EC5 (6.32)

In the case where only a moment M y exists about the strong axis y, the stresses should satisfy the following expression:  m, d  k crit f m ,d

In the case where a combination of moment M y about the strong axis y and compressive force N c exists, the stresses should satisfy the following expression :  m, y 2  c 1 ( )  k crit f m, d k c, zt f c,0, d

EC5 (6.33)

SIA 265 (37) requires:   c,0, d  m ,y, d  1 k c ,y� f c,0, d k m� f m ,y, d

Example for combined bending and axial compression according SIA 265: cm, C24 (SIA265), l ky  l kz  l D  400 cm Geometry: 25 cm x 25 Action forces: N   200 kN , My  15 kNm, Mz  7.6 kNm Result from FAGUS analysis: 1.00

FAGUS‐7

D-3


Some intermediate results:  rel, m  0.07



ah b

 400�25  0.07 25  0.28  k m  1

Jy = 7.22    400 55.42   rel  0.98  k c  0.704 A 7.22  200 kN  c, d    3.2 N  mm2 625 cm 2  c,0, d 3.2  0.378  Compression:  0.704�12 k c f c,0, d i



Bending stresses:  m, y, d 

M y M  5.76 N  mm2,  m, z, d  z  2.918 N  mm2 W y W z

 m, y, d  m, z, d   5.76  2.918 f m ,y, d f m, z, d 14 14  c,0, d  m, y, d  m, z, d   1 Combination: f c,0, d f m, y, d f m, z, d

Bending check:

 0.6198

D 3.2.2 Shear and torsion In the actual version this analysis is for rectangular cross sections available only.

D 4 Composite Cross Sections 

The explanations of the previous chapter are related to pure timber cross sections. Basically FAGUS allows to build up a cross section with various material types. In this case FAGUS assumes an ideal interconnection between the different types of material (assumption of a strain plain).

f c.. k f c.. d

Load factor / Efficiency / Limit state:

If the cross section contains a second material other than timber, the usual stress/strain limits defined in analysis parameters and the nonlinear material behaviour of these materials are also taken into account. But as the stresses can no longer be divided into stress due to axial force and stress due to bending a second rule is introduced: The “allowable” stresses are established in a preevaluation of the cross section. If the cross section is stressed under pure bending: f d : f m, d For pure axial tension and compression the following values are used: f d  : f t, d (tension) and f d  : f c, d compresssion

 cu f t.. d

For all intermediate combinations the values are interpolated. 

It should be emphasized that this approach cannot be found in any national code but the developers feel that it could serve as a calculation tool under some circumstances.

D 5 Calculations directly from STATIK For an automatic execution directly from STATIK the same input procedure as for steel members is required (see manual of STATIK). The results (efficiency factors) can be viewed in numerical or graphical from. The required steps in STATIK are:  Define geometry of timber structure and loadings as usual Check given buckling lengths and distances between lateral buckling supports (member attributes > last tabsheet)  Ask for the desired analysis: Tabsheet 'Calculation' > 'Special analysis' > 'Timber ..'  View desired results in the tabsheet 'Results'

D-4

FAGUS‐7

D 5 Calculations directly from STATIK

The output of the ULS verification holds three colum groups. On the left hand side the results for bending and axial force can be found. Shear and Torsion is in the middle of the page (if the cross section allows such results) and the overall results (M,N,V,T) are on the right hand side of the chart.

FAGUS‐7

D-5


D-6

FAGUS‐7

Vol. E Introductory examples FAGUS‐7

E 1 Introduction Two short sections dealing with starting the program and opening a new cross section library are followed by some examples, which will help you to become familiar with important aspects of the program. The examples are ordered according to increasing specialisation. It is essential to work through these introductory examples yourself, before you attemp real cases. You will then be acquainted with the program's logic and for further details of program operation you can consult FAGUS's Help System, whose intensive use will be well repaid. It can be started in different ways:  By clicking on the menu “Help“ in the program's menu bar. A list of all available Help documents is given, which you can enter immediately.  By pressing the key you obtain specific help on the action you are about to carry out (e.g. when inputting a point) or on the input element that the mouse is currently pointing to.  Many of the dialogue windows, which appear during input, have their own Help button for information on the corresponding dialogue. A prerequisite is that one has some basic knowledge of the use of the Windows Operating System. This includes the manipulation of windows (moving, increasing and decreasing their size, etc.), of the Start menu, the Task bar, the Clipboard and of the Windows Explorer. 

As in Windows in general one always works with the left mouse button. To click or select a symbol on the screen means: move the mouse pointer onto the symbol and then press briefly on the left mouse button. The right mouse button is only used to bring up a context menu (see later) in a particular situation on the screen.

E 1.1 Presentation Conventions for the Examples For all examples of application the following presentation conventions apply:  All actions to be carried out are indented and marked as follows:  Description of action       

Everything in bold print in the description of an action has to be typed in exactly as given except for the following exceptions. Special keys are denoted by <..> (e.g.=<>, , , etc.) The mouse buttons are abbreviated to and (left and right) Buttons on the screen are shown in square brackets (e.g. [Cancel]) Words printed in italics Click , Select , ... are previously clearly defined user actions An entry to be made in a menu is given in inverted commas, separated in the case of multiple selection by “>“ (e.g. >Options>Language>English>)

The texts printed in this way within this chapter represent insertions with useful additional information on the current topic. They do not imply any executed actions in connection with the examples.

E 1.1.1 Load Examples The examples described are stored in your program installation and can be fetched using the Help System in order to view and modify them. This also applies especially to examples which are not developed from the beginning, but serve as a starting point for modifications and further developing them.

FAGUS‐7

E-1

Vol. E Introductory examples

When working with a downloaded example not the original, but an automatically created copy is modified in the TEMP folder of Windows. Thus the original data cannot be lost and one can load an example as often as one desires. Downloading an example is carried out as follows:  Start FAGUS directly, as described in Example 1 and from the menu bar of the program se-

lect: >Help>Load Examples> Name of an Example>

E 1.2 Starting FAGUS FAGUS can be started in two ways: 1) By direct execution of the program file FAGUS.EXE, e.g. using the Start menu of Windows (for a standard installation: [Start] > Programs > Cubus > FAGUS) FAGUS responds with an empty window and a menu bar:

This way of entering the program is recommended above all if you want to continue working with one of the recently modified cross section libraries (these are listed in the file Menu). 2) Using the CubusExplorer: the CubusExplorer is an independent program for the management of analyses with the different Cubus programs and is also opened using the Start menu of Windows (for the standard installation: [Start] > Programs > Cubus > CubusExplorer)

E 1.3 Opening a Cross Section Library By cross section library is meant a folder, in which FAGUS cross sections are kept. This may be an analysis folder, which has been created directly by FAGUS (or in CubusExplorer for FAGUS) or an analysis folder of other Cubus programs, e.g. STATIK, which also works with FAGUS cross sections. Specific FAGUS analysis folders are used to carry out independent cross section analyses with other programs. They may however be built up as a genuine library, since copying cross sections from one library to another (e.g. in a STATIK analysis) is simple and possible at any time. We now want to create a FAGUS cross section library. To do this start the CubusExplorer using the Windows Start menu, as described in the previous chapter. If this is the first time for you to do this, then it may appear as in the following graphic shown on the left. Click on the “+” beside “My Computer” and you will obtain the presentation on the right. The CubusExplorer is very similar to the Windows Explorer. The difference is above all that in the left window only those directories are shown that you want and these are normally those that contain analysis data from Cubus programs. For a detailed description of the CubusExplorer see its Help menu. For our example we want to create a folder called ”FGData”on a harddisk (here D:), in which we then want to create our first cross section library. Basically you are completely free when creating a folder structure to manage your projects. You can also at any time reorganize and rename or move folders.

E-2

FAGUS‐7

First click on the symbol of the desired harddisk and then on the symbol shown to create or make folders visible.

Create new folder

The window that then appears shows the folder structure on the chosen harddisk. You can now select one of the folders or subfolders shown and [Select] with the button in the CubusExplorer to display them. We want however to create a new folder and first select the object (harddisk or folder), where the new folder should be placed ‐ in our case therefore the harddisk symbol D:. To create a new folder click on the adjacent button. This causes a new folder to appear at the desired level, which we rename as ”FGData” and introduce with the button [Select] in CubusExplorer. Back in CubusExplorer you should check that in the display filter shown below the button for FAGUS cross section libraries has been activated (only the analysis folders for the activated programs are shown in the filter).

Now select the newly created folder and click on the buttonshown on the left to create a new cross section library. Directly below the button there appears a row of buttons for all the installed Cubus programs. Select the symbol for FAGUS cross section libraries, and then in the middle part of the CubusExplorer a new entry with the standard name “Fagus Cross Sections” appears. Since the name of the new cross section library has already been chosen (i.e. highlighted), you can rename it by typing directly ”Example 1”. Thus you have created the desired cross section library and you can start FAGUS with the button [Modify] bottom right in the CubusExplorer. We do not want to do this here but in the first example and therefore we now exit the CubusExplorer.

FAGUS‐7

E-3


E 1.4 Example 1: Cross Section with Simple Outline E 1.4.1 Task

180

Measurements in cm

72

Materials: concrete and reinforcing steel as specified

6 6

3 3

7 2

As = 26 cm2 6 3

zq

7 5

8

7 2

8

5 * ∅ 22

12

yq

30

120

Aim: To become acquainted with the most important functions of the basic module by inputting and documenting a simple cross section (with reinforcement optional)

 Start the CubusExplorer and select a cross section library (as created in Ch. E 1.3)  Click on the button[Edit] (bottom right of CubusExplorer window)

If FAGUS is started with a cross section library that is still empty, then a first cross section is started immediately. If this is not the case then do it manually:  Click on this button, if the dialogue 'New Cross Section' is still displayed, to create a new

cross section The following dialogue appears to input the cross section name:

 Overwrite 'QS' with Ex1 and select [OK]

E-4

FAGUS‐7

E 1.4.2 The program window of FAGUS The FAGUS window, which you now have before you, responds with a program window that looks as follows: Menu

Management of cross sections

Graphics Editor functions and tools:

Drawing tools

Control tabsheet Drawing area

Selection Part of drawing (Zoom...)

Undo/Redo Layer switches Input options

Coordinates input fields

Exaggerated scale

Status line

A cross section library consists in general of several cross sections, which are created, deleted, copied, etc. using the cross section management ( on the button of the cross section management explains their function). At any time one cross section of the library is always active. It is shown in the list field of the cross section management ('Ex1') and designated as the current cross section. The Control tabsheet is a kind of menu for editing the active cross section. Its tabsheets are activated by clicking on the tab. The Control tabsheets 'Reinforcement design' and 'Analyses' are only available in the licensed Analysis Module.

E 1.4.3 Units The units of mass for cross section input and output quantities can be freely selected at any time within the framework of existing choices. To each quantity that can be set the number of decimal places for the presentation of the numerical results can be elected.

FAGUS‐7

E-5


 Call the Settings dialogue using the menu 'Presentation' > 'Units/Decimal Places' and make

the settings as desired.

E 1.4.4 Input of the cross section The cross section input is done in the 'Geometry' tabsheet:

s s n g o i n i t c n e e p s l O a i t r a p n w a r D

s n o i t c e s l a i t r a p d e l l a w n i h T

s n o i t c e s s s o r c d e s i r t e m a r a P

s n o i t c e s l e e t S

s n o i t c e s s s o r c g n i t s i x e t r e s n I

t n i o p s i x A

) t n i o p ( e l d n a H

s e n i l t n i a r t s n o C T P


s n o i t a n i b m o c t l u s e R

s t l u s e r n o i t c e s s s o r C

k d c n e e h g C e L

s t c e j b o e l b a t c e l e S

Usually in reinforced concrete practice one will work with rectangular, T and circular cross sections. FAGUS offers a quick input of these and a series of other section shapes with parametricised cross sections, some of which are provided with reinforcement and shear walls. The Structure tabsheet has a button for inputting parametricised cross sections (see above). Cross sections, for which there is no parametricised type, have to be constructed. They can be built up of one or more subsections with their own materials, whereby a particular subsection can be defined by 

the input of the polygonal outline, which can contain one or more openings that have to lie completely within the corresponding subsection.



taking a section from the existing rolled steel section library



the input of a thinwalled subsection (line or polygon with details of wall thickness)

Our example is constructed by inputting an outline polygon with one opening:  Click on this button in the Geometry tabsheet to create a new subsection outline

The dialogue shown appears, in which you can name the subsection and select the desired material.  Select in the list field under Material the existing material Concrete

The list field for Materials contains all currently defined materials. With the button to the right of the list field the existing materials can be modified and also redefined. We will deal with this point in the example 'Composite Cross Section'. Now the geometry of the output has to be input. The tools for this purpose are available on the left side of the FAGUS window. These are lines, rectangles, polygons and circles. You can click on the appropriate tool and with [Introduce] in the dialogue you can automatically start the polygon tool.

E-6

FAGUS‐7

 Select this button to start the polygon tool.

With this (or any other drawing tool) you enter the mode to edit the geometry, which is confirmed by the appearance of the dialogue 'Edit Geometry' on the left. As long as this dialogue is shown, one is still in the edit mode for an outline or an opening and with the drawing tools supplied in the Graphics Editor you can draw an arbitrary outline shape. On exiting the edit mode with [OK] the program tries to find a valid cross section outline from the drawing elements, which encloses a contiguous area without holes. This is possible in the following cases: 1) The only drawing element is a closed polygon. 2) There are several lines and/or polygon segments connected together, such that visually they represent a single valid outline. 3) As drawing elements only quadrilaterals and a closed polygon were used; with several such elements they have to enclose a contiguous area without holes. They may intersect. With [Introduce] we start the polygon tool and input the outline as a closed polygon. This is done by inputting its corner point coordinates. The starting point and direction (clockwise or anticlockwise) of the polygon are not prescribed by the program.  Type in 60 <> 0 <> 60 <> 0 <> 30 <> 57 <> 0 <> 33 <> 180 <> 0 <> 0 <> 33 <>

An before an input coordinate pair stands for relative input, i.e. the coordinates refer to the point last input, the relative point

Select the first point again to close the polygon

Insertion: Point input by grabbing Point Input by Grabbing (= Select) is done as follows: bring the mouse pointer as close as possible to the point to be grabbed, until at the top of the mouse pointer a small circle is shown, which on clicking on the point nearest to the crosshairs grabs it.



As well as typing in the absolute and relative coordinates there are also other methods of point input. Consult for this action as also for the polygon input and modification the Help System in the Graphics Editor.

You can modify the outline later, by selecting it (=clicking) and then with the call its Context menu and select 'Edit Geometry' there. You are once again in the above mentioned Edit mode.  Click on this button in the dialogue 'Edit Geometry', to finish the editing of this part of the

cross section  Click on the button shown (Graphics Editor tool on left side of screen), in order to centre

the drawing on the screen

E 1.4.5 Inputting the opening  Click on this button in the Structure tabsheet to input an opening

FAGUS‐7

E-7


An opening is the same as creating the outline of a part of a cross section, except that it does not have any material attribute. It is also input in the same way. As an alternative to the above, where we started the polygon input using [Introduce], we now want to use the Rectangle tool of the Graphics Editor.  Click on the Rectangle tool (variants of the rectangle input are shown by clicking on the

small arrow lower right on the button) Now the two corner points of the opening have to be input.  Type in 36 <> 27 <> 72 <> 36 <>  Click on this button in the dialogue 'Edit Geometry',to end the input of the opening

The input of the cross section is now complete.  Click on this button to check the cross section

The cross section is now shown correctly, and we want to input the required reinforcement. Pass over this section, if you think you will never have to work with RC cross sections.

E 1.4.6 Input of the reinforcement 

Normally reinforcement has no influence on the cross section values and therefore doesn't need to be input, if no analyses are planned. If, however, you want to take into account its influence, you have to do this with the property settings of the variants (see later in the example).

 Change to the tabsheet 'Reinforcement'

List field to select the reinforcement group

p p p u u u o r o r o r g g g t e f n t o e e s l e i m e e D t r c e r o p f o n r i e r P w e N

t w n o e h m s e o c t r o s f t n n i i e o r p y r a i l i x u A

- - t i l e n i e x c u r o m f a n i n e e e r r w o t f e s b t n i e c o n p a y t s r i a D

t n e m e c r o f n i e r e t n n i i o L P t n e m e c r o f n i e r

t n e m e c r o f n i e r

s n o d n e T

r a l u c r i C


p u r r i t s o t t n e m e c r o f n i e r t s u j d A

k c e h C

t n e m e c r o f n i e r r o f d n e g e L

y t i l i b a t c e l e S

First we introduce point reinforcement. The bars have a spacing to the boundary in all directions of 80 mm, so it is easier to use an auxiliary polygon:  Click on the button for Auxiliary lines, to activate this function  Set to 80mm. You can now see a polygon at a distance of 8cm from the boundary and the

opening, whose corner points can be grabbed. 

The input of the desired 5 bottom bars could also  and more quickly  be input as line reinforcement. But firstly it would not be possible to modify the bars individually afterwards, and secondly we want here to work with both kinds.

 Click on the button for Point reinforcement

E-8

FAGUS‐7

 Select in the dialogue 'Point Reinforcement' that appears the given material for reinforce-

ment (if already set)  Select the option 'Members given'

and in the corresponding list field a diameter of 22  Click on [Create] and with the grab the lower left corner point of the auxiliary poly-

gon.

We introduce the remaining 5 bars by duplicating:  Select the point reinforcement just introduced  Press the and select from the context menu 'Duplicate'

 Set the dialogue as shown. (At the start point of the line (='duplication ruler'), along which one has to fill, we already have an object, which is why the control field 'Duplication at start point' is inactive)  Select arrow button to select the following points:

1) the reference point of the objects to be duplicated, i.e. the point reinforcement to be duplicated here. (The reference point is the point of the object / objects, which has to be placed along the subsequently to be defined duplication ruler) 2) the start point of the duplication ruler, i.e. here the same point. 3) the end point of the duplication ruler, i.e. the lower right point of the auxiliary polygon 1), 2)

3)

 Shows in the preview ([Preview] button is active) the desired duplications, and so select

[OK] The duplication just performed is a typical example of objectoriented working. The Graphics Editor of FAGUS functions without exception according to this principle, which is generally formulated in the following insertion.

FAGUS‐7

E-9


Insertion: Object-Oriented Method of Working You select one or more desired objects, then press the right mouse button and a context menu will appear with the functions, which are possible with all the selected objects. 

If you do not see a function in the context menu, it may be that you have also selected objects which do not work with these functions!

The selection of objects therefore plays an important rôle. Besides clicking with the mouse or using a window there are a number of other very useful selection methods, or information on selection, which are essential to know. Therefore please read also chapter “The Graphics Editor”.

A typical case of objectoriented working is also changing the properties of objects.  Selektieren Sie die fünf Punktbewehrungen, indem Sie ein Fenster über diese aufziehen.

E-10

FAGUS‐7

 Holen Sie mit der das Kontextmenu zu den selektierten Objekten undwählen Sie daraus 'Eigenschaften' . Diese Menuzeile ist nur aktiv, wenn Sie ausschliesslich gleichartige Objekte selektiert haben! Sie könnten jetzt beliebige Attribute verändern und diese an-

schliessend den selektieren Objekten zuweisen. 

Die Schaltfläche [Anwenden] ist allerdings nur aktiv, wenn Objekte selektiert sind und etwas in den Dialogeinstellungen verändert wurde. In Klammern hinter 'Anwenden' steht immer die Anzahl der selektierten Objekte, auf welche sich das Anwenden der geänderten Eigenschaften auswirken wird.

The line reinforcement is still missing:  Set the auxiliary polygon spacing to 60mm.  Click on the button for line reinforcement  Select in the dialogue that appears the given material for the reinforcement (should al-

ready be set)  Select the option 'Area given' and define the desired value As = 26 cm2

 Click on [Create]  Select as Start point of the line the upper left corner point of the auxiliary polygon and as

End point the upper right corner point  This button enables you to check the cross section

E 1.4.7 Tab sheet Variants  Change to the Variants tabsheet

List field to choose variant

t n a i r a v w e n e t e l e D

s e i t r e p o r P

s t n a i r a v k c e h C

d d s n t e n o e a i g l t e o c e L p r s e t s n s i o k r c c e h C

Example of a cross section with a variant section:

Two input subsections = variant 'Standard'

A cross section can have several variants, which can differ with respect to the choice of subsections and reinforcement as well as in other properties. Usually, as also in our example, one does not work with several variants. Thus a variant ' Standard' is always predefined, which always includes all input subsections and reinforcement. However, what is always of interest in this tabsheet are the variant's properties  including those of the standard variant  and the button to provide a tabular documentation of the cross section.

Variant 'Trough'

 Click on it to show the Properties dialogue of the current variant 'Standard'.

The first tabsheet of the dialogue serves to define the subsections and reinforcement that belong to a variant. Since the standard variant always includes all these elements, nothing can be changed here. However, a further tabsheet includes the settings for cross section values exported to STATIK and should be considered:

FAGUS‐7

E-11


In this way the influence of reinforcement on the cross section values is considered. Be careful: may change the position of centroid!

These cross section stiffnesses cannot always be calculated correctly by the program and so they can be overwritten. STATIK caters for acceleration forces, which act on the cross section masses. The selfweight is also input in this way. Thus an additional mass can be input, which does not come from an input subsection (can also be negative).

The other tabsheets in the dialogue are only of interest in connection with cross section analyses and are described later.

E 1.4.8 Documentation of the cross section For our example we want here to carry out the following steps in the documentation of the cross section: 

Adding dimensioning to the cross section as in the current task



Set the desired figure content with the help of the layer button



Enter a figure of the structure in the print list



Create a text legend with all cross section data and cross section values and enter it in the print list



Call the CubusViewer to preview, to set possible presentation parameters and print

To document a cross section one changes to the Variants tabsheet and selects the variant to be documented, if more than one exists.

Dimensioning the cross section On the right side of the FAGUS window is the Layer bar. In layers, graphics objects of the same type are placed, as in CAD systems. By clicking on the layer buttons you can blend in or blend out the corresponding layers. Details on this are given below. If you want to add to figures to be printed information like dimensioning lines or extra labels, this is done using the layer group “User", in which the button for a first user layer is already given. You can also distribute these additions to the figures in different layers, by creating as many other user layers as desired, using the context menu of the group header “User". It may sometimes happen that points which one wants to input have to be constructed with the aid of auxiliary points and auxiliary lines. Such constructions are likewise carried out in user layers. Points constructed here can be grabbed in the input of cross section ob jects. To be able to draw in a user layer, this first has to be made the active layer (not to be confused with making it visible):  Press on the button of the existing user layer and then in the context menu on the

line 'Active' Now all drawing tools of the Graphics Editor have been activated.

E-12

FAGUS‐7

 Click on the dimensioning tool  Specify the direction of the dimensioning line using Select , e.g. of the left cross section

edge (1) (5)

 Click on a position through which the dimensioning line should pass (2)

(1) (2)

 Click now sequentially on the points that should be included on this dimensioning line

(4)

(3),(4),(5)  Close the dimensioning line using (3)

 Proceed in the same way for the dimensioning lines in the example.

Also try out here the other drawing tools and delete the corresponding 'creations' again with [Undo] (see following insertion). Insertion: Undo/Redo If you have made an incorrect input or even more than one in succession, this is not a problem: With the Undo function (on the left side of the window) as many changes to graphics objects as you like (provided the button is active) can be reversed stepwise. Key combination: + With the Redo function changes, which were made using the Undo function, can be reversed stepwise. Key combination: +



Creating other user layers is done in the context menu for the group button 'User'

Select content of figures with the layer buttons As mentioned above, the contents of figures can be activated or deactivated using the corresponding layer button as desired. Try out the settings...  Wählen Sie zum Schluss die Einstellungen, wie nebenan abgebildet.

Enter figure in the print list The content of the graphics area can be sent directly to the printer at any time (left button) or, with the right button, to the output list for printing later. If you click on one of these buttons a dialogue appears, which among other things gives you the choice, whether a figure should be created with the complete contents of the graphics area or only with a part of it. For details on the dialogue for print entry use its help button. Here we want to enter a figure of the whole structure:  Click on the button for a print entry and then on [OK] in the unchanged print entry dialogue

Creation of a text legend with print entry  Click on the button to create a text legend (top right in the Variants tabsheet)

FAGUS‐7

E-13


This, as all text output, is created in a separate window, which looks as follows: Presentation of the numerical output with a pressed preview key

The presentation of individual tables can be activated or deactivated (applies also to the print entry)

 Click on the button for the print entry  Close the text window ([x] button top right corner)

Print preview and printing with the CubusViewer Up to now we have created two print entries, one with a figure of the cross section and one with the text legend, and now we want to call the CubusViewer, in order to see how printing is prepared and how one prints.  Click on the button for the CubusViewer

The CubusViewer appears in a separate window and should look as in the following figure, in which you can also see its most important functions. A detailed description of the CubusViewer can be found in its help menu. Editing the entries in the window with the list is done using the menu 'Print Entries' on the menu bar or using the context menu for the print entries shown below.  Select the entry for the figure and change the scale to 1 : 20  Print the two entries and then close the CubusViewer

E-14

FAGUS‐7

In the menu 'File' you can among other things select the printer

Activate / deactivate entries (Presentation and Printing) Page setup (size, page header, ...) Zoom functions for preview For quick editing of a long list of entries, or of large files:  deactivate filling  deactivate preview  present picture with less content

Window with list of print entries (supports moving lines with (drag & drop)

Preview window

FAGUS‐7

E-15


E 1.5 Example 2: Composite Cross Section with Variants E 1.5.1 Task 150

Measurements in cm

zq

8

Concrete yq

4

slab R.S. section  Materials:

 Variants:

IPE240 S275

Id

Class

E‐Modulus

C (concrete) C2 (concrete longterm) S (structural steel)

C25/30 C25/30 S275

32.0 kN/mm2 12.0 kN/mm2 210 kN/mm2

Name

Formed plate in transverse direction (statically i neffective for this cross section)

Active subsections

construction state rolled steel section longterm R.S.section, slab (concrete longterm)  centroid at (0,0)  what is an anchor point?  result points: PLMU PRMO

Aim: to familiarise yourself with the following special aspects:  editing the (project) materials  several subsections with different materials  introduce subsections from the cross section library  cross section variants  input of centroid, anchor point and result points

E 1.5.2 Definition of the required materials To each part of a FAGUS cross section a material chosen from a material list maintained by the user is assigned. Such a material has a freely selectable name, possibly a material class and parameters, which e.g. are required for the determination of the cross section stiffnesses. In this way all required properties of a material are summarised under a name. In the case of the other material properties, which are not defined by means of the material classes in the codes, it is a question of 

the E and G moduli, which are used by STATIK for the determination of the required cross section stiffnesses; the E moduli can, for subsections with different materials, also have an influence of the position of the centroid of the cross section



the specific mass required by STATIK (e.g. for selfweight)

The material management is called from the main menu of FAGUS.  Select from the menu: 'Options' > 'Materials'

The corresponding dialogue shows a list of already defined or given materials, which can be modified at will and extended. Below the buttons of the dialogue are explained:

E-16

FAGUS‐7

Definition of new materials Changing properties of the selected material Deleting the selected material In this list we now want to give the existing material the required properties and define the additionally required concrete with a reduced Emodulus.  Select the line with the material 'Concrete'  Click on the button of the properties dialogue for the selected material:  Set the dialogue as follows, and then press [OK]:

 With this button you create a new material 

When creating a new material the dialogue is initialised with the properties of the previously selected material

 Select the same class, but change the Young's modulus to E = 12 kN/mm2  Ensure that a material with the name 'Structural Steel' needed for the current task exists; if

necessary create one  Confirm the changes in the material list with [OK]

E 1.5.3 Input of the concrete slab  Introduce a new cross section called 'Ex2'  Input as subsection 'Slab' the rectangular section of the slab with the material Concrete for

the given task

E 1.5.4 Input of the rolled steel section  Click on the adjacent button in the Geometry tabsheet, to introduce a rolled steel section

as subsection

FAGUS‐7

E-17


 Select IPE and 240 in the dialogue shown

The chosen section is now shown, together with the possible insertion points, which you can use to place the section. Here the most suitable point is at the top in the middle.  Click on the insertion point top middle (becomes red)

Depending on the type, sections can be introduced in a rotated position. The corresponding rotation angles are shown in the dialogue as buttons, and the section is presented according to the selected rotation.  Ensure that the rotation is set to [0].  Select as project material 'Structural Steel'.

The steel section can now be introduced.  Click on [Create]  Input the point where the selected insertion point of the section should be: 0 <> 0 <>

and close the rolled steel section dialogue

E 1.5.5 Definition of the variants Besides the standard variant, which includes all input subsections with the assigned materials, for the given task we need two additional variants of the cross section: 

the steel section by itself



the steel section and slab, the latter however with the material 'Concrete longterm'

 Change to the tabsheet 'Variants'  Create a new variant and set the dialogue as shown:

E-18

FAGUS‐7

 Create a second variant and activate both Slab and Steel section, then press [OK]

 Select the subsection 'Slab' and call its properties dialogue using the context menu ()  Change the material to 'Concrete longterm' and then select [Apply (1)] This setting applies only to the currently set variant 'Longterm', i.e. the material of a subsection can vary in the variants.

E 1.5.6 Input of the axis point If an axis point is input, it applies to all variants. The input of an axis point is only necessary in special cases. Its existence has the following consequences: 

STATIK: not the centroid of the cross section or the cross section variant, but the axis point lies on the axis of the member (see STATIK Manual). Since in STATIK the position of a member axis cannot be changed in a construction state, only the input axis point can guarantee that each cross section variant lies in the correct position with respect to the member axis. The use of cross sections with axis points is possible in STATIK only for the licensed option 'Specialities'.



FAGUS: The loading of the cross section required in the Analysis module refers not to the centroid but to the axis point.

In our example an axis point is to be input at (0,0).  Change again to the Geometry tabsheet and click on this button to introduce an axis point  Click on [Introduce] in the Axis Point dialogue  Introduce the axis point by typing its coordinates: 0 <> 0 <>

E 1.5.7 Anchor points Anchor points are only meaningful in connection with STATIK and serve there to be able to place cross sections easily with respect to input member lines (see STATIK manual). There can be predefined anchor points, which are defined at the same time as the given result points (see nect section). Alternatively, in FAGUS with the adjacent button an anchor point can be defined anywhere.

FAGUS‐7

E-19


E 1.5.8 Introduction of result points Result points are intended for use in STATIK. STATIK offers the possibility, in a homogeneous cross section, to output calculated stresses (e.g. stresses at the boundaries) and strains in certain cross section points, the socalled results points. These results points have an identifier (ID), with which the results in STATIK are called. FAGUS supplies predefined result points according to the following scheme, so that these do not have to be defined in the standard case: TL

AT

AL

AA

TR

Lines through centroid or axis points parallel to the input axes

AR Bounding rectangle parallel to input axes

BL

BA

BR Centroid or axis point, respectively

If one wants result in other points, these can be defined accordingly.

PLMU

For our example we want to introduce the two result points as shown:

PRMO

 Click on this button in the Geometry tabsheet in order to introduce a result point and set

the dialogue as shown:

Click on it and select the outline of the subsection, to which the result point should belong.



Da es Fälle gibt, bei denen es nicht eindeutig klar ist, zu welchem Querschnittsteil ein Resultatpunkt gehört (z.B. auf der Grenze zwischen zwei Teilquerschnitten), muss der zum Resultatpunkt gehörende Querschnittsteil grundsätzlich eingegeben werden.

 Klicken Sie auf [Einführen] im Resultatpunkt‐Dialog  Führen Sie den Resultatpunkt durch Tippen seiner Koordinaten ein: 0 <> 4 <>  Geben Sie den zweiten, zum Profil gehörenden Resultatpunkt auf die gleiche Weise ein

E-20

FAGUS‐7

E 2 Beispiele Analysemodul

E 2 Beispiele Analysemodul Die folgenden Programmbeispiele sind unter der Funktion 'Serienrechnung' in der Beispielsammlung mit folgenden Schritten verfügbar: 1) Neue Berechnung starten (mit CubusExplorer oder mit FAGUS) 2) Beispielsammlung laden (Menu 'Hilfe' > 'Beispiele laden' > 'Bsp Analysemodul') 3) Gewünschte Norm wählen (Menu 'Einstellungen' > 'Norm') 4) Funktion 'Serienrechnung' im Register 'Analyse' wählen 5) Aus 'Datei' die gewünschte Serie ('S1' .. 'S3') wählen und Berechnung starten 6) Dokumentation mit CubusViewer betrachten und auf Wunsch ausdrucken

E 2.1 Datei: 'S1' In 'S1' sind eine paar kleinere, voneinander unabhängige Beispiele zusammengestellt, wobei die unten dargestellte Angaben als Übersicht dienen sollen. Die genauen geometrischen Daten und gewählten Baustoffklassen (normabhängig) sind direkt aus der (mit dem CubusViewer zu druckenden) Dokumentation zu entnehmen.

E 2.1.1 Träger, Bewehrungsbemessung Querschnitt “R1"

Aufgabe: Bemessung für

Myd = 75 kNm Vyd = 84 kN T = 10 kNm 4 0 . 0 0

3 2 . 0 0

Gesucht:

Längsbewehrung: As1 = ? Bügelbewehrung: Asw (Vy) = ? Asw (T) = ?, Asw(Vy+T) = ? 30.00

E 2.1.2 Träger R2, Grenzwerte Querschnitt “R2"

Aufgabe: Bemessung für

N kN

1 3 5 . 0 0

1 2 3 . 0 0

25.00

FAGUS‐7

0. 0. 0.170 0.222 0.643 0.279 0.127

My kNm 193. 198. 18. 178. 205. 17. 214.

Gesucht: As1 = ?, As2 = ?

E-21


E 2.1.3 ITräger, Bemessung für kombinierte Beanspruchung Querschnitt “I1"

Schnittkräfte: N My Mz -100.0 80.0 10.0 0.0 100.0 0.0 -200.0 -300.0 -40.0

Vy 100.0 200.0 90.0

Vz 100.0 130.0 85.0

T 80.0 90.0 50.0

Gesucht:

erforderliche Längs und Bügelbewehrungen

E 2.1.4 Kreisstütze Querschnitt “C1"

Schnittkräfte: N My Mz -7200.0 2000.0 0.0

Vy 0.0

Vz 200.0

T 200.0

Gesucht:

erforderliche Längs und Bügelbewehrungen

E 2.1.5 Stützenquerschnitt, Steifigkeitsdiagramme Querschnitt “S1" 0.75

0 . 5 0

0.60

Momenten-Steifigkeitsdiagramm: 0.30

0.60

0.75

0 . 1 0

Randabstand der Bewehrung = 6 cm

1 . 0 0

My [kNm] 0 . 5 0

E 2.1.6 Spannungsnachweise vorgespannter Träger, Schiefe Biegung Querschnitt “T1"

Schnittkräfte:

My = 120.0 kNm Mz =12.0 kNm

Gesucht:

Randspannungen, Neutralachse, Grafik

E-22

FAGUS‐7


E 2.1.7 Interaktionsdiagramme für einbetoniertes Walzprofil Querschnitt “W1"

N-My-Diagramm, N-My-Mz-Diagramm: N= −2000.00 −3000.00 −4000.00 −5000.00 kN ] 0 m 0 N 2 k [ z M

HEA300

− 2000.00 − 3000.00 − 4000.00

0 0 1

3 9 . 0 0

− 5000.00

0 0 0 1 −

0 0 2 −

40.00

− 400

− 200

0

My [kNm] 200

400

E 2.1.8 Spannungsanalyse für vorgespannten Brückenträger Beanspruchung: My = -80 MNm

Querschnitt “B1" 1575.00

2 0 0 . 0 0

E 2.1.9 Brückenträger mit inaktiven Querschnittsteilen Querschnitt “B2"

Beanspruchung: My = -80 MNm 1575.000

statisch inaktiv

statisch inaktiv

2 0 0 . 0 0

Die Konsolen sind statisch inaktiv > gleiche Resultate wie Querschnitt “B1" oben

FAGUS‐7

E-23


E 2.1.10 Dünnwandige Querschnitte Querschnitt “D1"

Querschnitt “D2"

Querschnitt “D3"

M M

M

S S S

Drei Beispiele aus: Gruttman, F. Wagner W.: “Ein Weggrössenverfahren zur Berechnung von Querschnittsschubspannungen in dünnwandigen Querschnitten", Der Bauingenieur, Okt. 2001

Querschnitt “D4"

Querschnitt “D5"

S M

M

Brückenquerschnitt

S

Aus Thürlimann, B.: “Torsion" Vorlesungsunterlagen ETH Zürich

Querschnitt “D6"

Querschnitt “D7"

M S M S

Zwei Beispiele aus Aus Sattler K..: “Lehrbuch der Statik", Springer Verlag, Band I/B, 1969

E-24

FAGUS‐7


E 2.2 Datei 'S2': Verbundquerschnitt mit Belastungsstufen Der links dargestellte Verbundträger (Walzprofil HEA300, Betonplatte 100 x 15 cm) wird in folgenden drei Schritten belastet:

Betonplatte CS1 HEA300

Schritt Belastung auf

Beanspruchung

A B C

M y1  150 kNm M y2  150 kNm (zusätzlich)  x   0.2 o oo (Betonplatte CS1 allein)

Stahlträger allein Stahlträger + Betonplatte Stahlträger + Betonplatte

Untenstehend ist die unter 'S2' verfügbare Belastungsgeschichte abgebildet. Da nach dem “Laden" standardmässig alle Teilquerschnitte einer Querschnittsvariante aktiv sind, muss in Schritt Nr. 1 zuerst die Betonplatte inaktiv gesetzt werden. Beanspruchung / Belastungsgeschichte − Berechnung mit Laststufen, (in−)aktiven Teil−QS, Dehnungen

(1):

Nr.

Aufgabe,Analyseparameter

N [kN]

My [kNm]

Mz [kNm]

0 1 2

Querschnitt laden “V1” Inaktiv setzen “CS1” A:Gebrauchstauglichkeit

0

150.0

0

3 4

Aktiv setzen “CS1” A:Gebrauchstauglichkeit

0

150.0

0

5 6

Dehnungen (1) A:Gebrauchstauglichkeit

0

0

0

yq1 = 0 cm zq1 = 20.00 cm  1 = −0.2 o/oo yq2 = 0 cm zq2 = −20.00 cm  2 = −0.2 o/oo −−> CS1

In der entsprechenden Eingabe wurde nach jedem Belastungsschritt ein Druckbefehl eingefügt, sodass in den Druckeinträgen auch die Zwischenstadien erscheinen. Für die Norm “Swisscode SIA 262" mit den Baustoffklassen “C25/30" und “S275" sollten folgende Spannungsbilder ersichtlich sein:  ( M y1)

 ( M y1

 M y2)

 ( M y1

5.7 119.1

119.1 [ N  mm 2]



FAGUS‐7

6.1

118.5

193.4

 M y2   x)

144.7

201 [ N  mm 2]

[ mm 2] N

Desaktivieren von Querschnittselementen zu einem späteren Zeitpunkt: Falls einzelne Querschnittselemente, die von Null verschiedene Spannungen aufweisen, inaktiv gesetzt werden sollen, so muss im Anschluss an diesen Schritt eine Spannungs- analyse (evtl. ohne Kräfte) durchgeführt werden, damit das interne Querschnittsgleichge- wicht wieder hergestellt wird.

E-25


E 2.3 Datei 'S3': Verbundquerschnitt unter Langzeitbelastung Im Beispiel 'S3" soll der Verbundquerschnitt “V1" für eine Langzeitbelastung (Ständige Lasten,   2) mit anschliessender Kurzzeitbelastung (Verkehrslasten,   0) analysiert werden. Die vorgegebene Belastungssequenz besteht aus folgenden Schritten: 0 1 2 3 4

Querschnitt V1 “laden" Wahl Analyseparameter Spannungsanalyse Wahl Analyseparameter Spannungsanalyse

“Phi=2" M y  50 kNm

“Phi=0"

M y  50 kNm (zusätzlich)

Für die Norm “SIA 262" mit den Baustoffklassen “C25/30" und “S275" ist am Schluss der Berechnung folgendes SpannungsDehnungsbild ersichtlich (Mit Betondiagramm Typ 4) N [ mm2] 

[ o oo] 0.1

0.2

3.3

51.7

Zum Vergleich: Dehnungen und Spannungen mit konstanter Kriechzahl für beide Schritte:   0

 o oo]  [

N [ mm 2] 

0.1

0.2



o oo]  [

4.0

48.7

2 N [ mm2] 

0.1

0.3

2.7

54.9

Bemerkungen: Wie früher entsprechen die dargestellten Dehnungen den elastischen, spannungserzeugenden Dehnungen (Sprung bei Unterkante Platte für   0 ). Natürlich könnten die beiden Berechnungen von 'S2' und 'S3' auch aneinandergehängt werden. Bei einer umfangreichen Berechnung mit mehreren Schritten muss vom Benutzer sichergestellt sein, dass die Schnittkräfte auch tatsächlich aufgebracht werden können, andernfalls meldet sich das Programm mit “Kein Gleichgewicht gefunden".

E-26

FAGUS‐7

F 1 Thermal Analysis

Vol. F Fire Analyses FAGUS‐7

F 1 Thermal Analysis In this chapter the transient thermal analysis will get explained more closely. The mechanical crosssection analysis under elevated temperature will get examined in chapter F2. A transient thermal analysis can be done for all FAGUS cross sections. Limitations exist only in reinforcement input: As temperature in each reinforcement point has to be set, only point reinforcements are allowed.

F 1.1 Theoretic Basics The transient thermal flow through a solid body can be described by Fourier's law of thermal conduction as follows:



       2   2  c p  x 2 t  c p  y 2



with    c p t 

Temperature Heat conductivity Density Specific heat capacity Time Laplaceoperator

[°C] mK] [W m3] [kg kgK] [J [s]



Prerequisites for the validity of this description are: 1. The material properties  ,  , c p are only dependent on temperature 2. Isotropic material behaviour is assumed 3. There is no heat source or heat sink inside the solid body

  i

20°C

 a Fig. F −1

Integration region  with a thermal influence at the outer bound  a and an opening with the bound  i

Fourier's law of thermal conduction is a so called initialboundary value problem in the form of a parabolic (partial) differential equation. Fig. F1 shows a twodimensional integration region  with an outer bound  a and the inner bound  i of an opening.

FAGUS‐7

F-1

Vol. F Fire Analyses

The initial condition is the initial temperature state of  at the timet  0 . The boundary conditions result from the thermal influences at the bounds  a and  i and are mathemati. cally depicted as the net heat flux h net normal to  . The heat transfer occurs here through . . two parts, one part h net, c through convection and one part h net, r through radiation: .

h net

.

 h net, c    c ( g   m) 

.

h net, r

  m  f  [( r  273) 4  ( m  273)4]

with  c  g  m  r   m  f 

Heat transfer coefficient Gastemperature around the element Surfacetemperature of the element Effective radiation temperature of the fire Configuration factor Emissivity of the element Emissivity of the fire StephanBoltzmannConstant (  5.68  10 8)

m 2] [W [°C] [°C] [°C]    4 m 2K [W ]

For simplification these assumptions are taken: 1. 2. 3. 4. 5.



The configuration factor is   1 by default. The effective radiation temperature  r of the fire equates the gas temperature g . The progress of the gas temperature  g is given by a temperature time curve. The emissivity of the fire is  f  1. The heat transfer due to “inner radiation” inside of openings in  is neglected, so that at the bounds  i adiabiatic boundary conditions (No heat transfer with the environment) should be chosen.

The configuration factor  describes the radiation exchange between two surfaces. The mathematical definition of  can be found in appendix G of EN 199112.  is defined as part of the radiated heat from a surface, that is caught by another surface. The geometric position of the origin of fire in relation to the element as well as shadow effects of the cross section have an influence on the size of . The condition applies   1 and with the assumption   1 one remains on the safe side.

For the numeric solution of the initialboundary value problem the BackwardEuler method as well as theMethod of Finite Elements is used. The differential equation is turned into a weak formulation with the Galerkin method and the region  is discretised with triangular elements. The interpolation of the temperature field inside the elements is quadratic. 

F-2

The chosen element formulation is achieving the adiabatic boundary condition approximately, i.e. if along a bound  no boundary condition is set, no transfer of heat is taking place.

FAGUS‐7


F 1.2 Register 'Thermal Analysis' The complete input, calculation and output is done in the tab sheet 'Thermal Analysis': Check Input

Analysis Options Temperaturetime curves

Run thermal analysis Temperature along priLegend ciple axis

Material Properties Thermal bounds Reset thermal analysis Temperature at result points Output Settings

Fig. F − 2 Buttons for the thermal Analysis

To enable a thermal temperature calculation, the type 'Transient' in the first selection list has to be chosen first. After that the other buttons in the register are active. 

Each input in this register is either valid globally for the entire project or for the current cross section.

F 1.2.1 Material Properties The specific material properties concerning thermal analysis are added in this window. The data given here applies for the entire project. Should differing thermal properties be needed for the same material (e.g. for comparisons), then a new material has to be added.

Every material defined in the project is available. The values defined in this area are start values for calculating the heat conductivity, density and specific heat capacity according to E 1.1 For the mechanical analysis under elevated temperature additional specifications may be required for some material types (Example Concrete: Type of aggregate: siliceous, calcareous). The expression 'Automatic' means that the type is chosen by the program.

Various graphic illustrations depending on the temperature.

Fig. F − 3 Input of thermal material properties.



FAGUS‐7

A numeric summary of the thermal material properties can be found in the legend (F 1.2.6)

F-3


Thermal values for concrete For the description of its thermal behaviour concrete is seen as homogenous. The temperaturesensitive values  c,  c and c p are calculated according to EN 199212. The moisture content u as well as density  c,20° C are to be defined under normal conditions. EN 199212 gives a lower limit  c,u and a upper limit  c, o for heat conductivity. With the input of a interpolation factor f  a linear interpolation between these two values is possible:  c()  [  c, o()   c,u()]  f    c,u()

0  f   1

The energy consumption for evaporation is taken into consideration. Depending on the moisture content u the specific heat capacity c p between100°C and200° C is adjusted accordingly. The thermal material types for mechanical analysis under elevated temperatures are:





Siliceous aggregates,



Calcareous aggregates,



High strength Class 1,



High strength Class 2,



High strength Class 3

If 'Automatic' is selected the most suitable class for high strength concrete is chosen. For standard concrete the least suitable type is chosen.

Thermal values for steel The thermal properties  a,  a and c a for construction steel, reinforcement steel and prestressing steel are temperature dependent according to EN 199212. Types for rebars: 

Hot rolled reinforcing steel,



Cold worked reinforcing steel

Prestressing steel:





Class A (Wires and strands),



Class B (Wires and strands)



Tempered prestressing steel

With 'Automatic' selected the least suitable type is chosen.

Special (Userdefined) For userdefined materials the program uses type 'X' . In this case only constant values for  x,  x and c x can be assigned. Nichtlineare, temperaturabhängige Beziehungen wie oben beschrieben können nicht definiert werden.

F 1.2.2 Temperaturetime curves The button 'Temperaturetime curves' opens a dialog for editing temperaturetime curves (Fig. F4). Die Temperaturzeitkurven sind Bestandteil der Einwirkungsspezifikation an den thermischen Rändern.

F-4

FAGUS‐7


Four standard temperaturetime curves are already defined, which correspond with the EN 199112 normcurves, by default: 1. Standard temperature time This curve corresponds to the standard ISO 834 description. It is mainly used for fire resistance classifications and for structural design. The mathematical representation of this curve is  g [°C]  20  345  log10(8  t [min]  1).

2. Normal conditions 20° C At all bounds which are not under fire exposure, a normal gas temperature of 20° C can be defined with this condition. 3. External fire curve This curve corresponds to a lowered unit temperaturetime curve and is used for examining elements outside of the fire compartment, for example balustrades. The mathematical representation of this curve is  g [°C]  20  660  (1  0.687  e 0.32t [min] 0.313  e 3.8t [min])

4. Hydrocarbon curve A hydrocarbon fire reaches a noticeably higher temperature in a shorter time than a standard temperaturetime fire. In standard buildings it is not used very often but if required the evolution of the gas temperature can be described with this curve.  g [°C]  20  1080  (1  0.325  e 0.167 t [min] 0.675  e 2.5t [min]) Userdefined temperaturetime curve

To ensure standard testing methods in a laboratory, the four temperaturetime curves above show a constant increase in temperature. In natural fire scenarios the gas curve has development phases as well as cooling phases. For describing the natural fire curve there are several approaches, for example the parameterized temperaturetime curve according to EN 199112. With this button a new temperaturetime curve is generated, which has a constant temperature of 20° C. Copies the selected temperaturetime curve. Deletes the selected temperaturetime curve. 

The button 'delete curve' is only active if the temperaturetime curve is not being used, this means it is not assigned to a thermal boundary. The standard time curves can not be deleted.

With the button 'New curve from clipboard' a new temperaturetime curve is created and it takes the numerical values from the clipboard. Per row two numbers have to be entered, separated by a tabulator. t1

1

 n

until t n

FAGUS‐7

F-5


Fig. F − 4 Temperature−time curves

New number pairs (t i,  i) can be added to a temperaturetime curve by adding a time t i  0 [min] with a corresponding temperature i [°C]in the two fields in the upper part of the dialog and pressing 'Add/update timetemperature value'. The new timetemperature value is added into the sorted list and the graphic diagram is updated accordingly. Existing number pairs can be edited by clicking on the table on the left hand side of the dialog. The program moves the selected values t i and i in the input fields. A selected row can be deleted with this button.



The temperature value of the temperaturetime curve „TZK_“ for t  n.

 t n is constant

F 1.2.3 Thermal bounds The effect of a fire situation is achieved through the input of thermal boundaries. A thermal boundary is a line object, which optionally can also include a (thin) insulation. If a cross section is heated up uniformly from all sides, the desired properties can be given to the whole cross section with the button shown to the left. Should a cross section consist of several independent partcross sections or recesses, one point of the desired boundary has to be selected. With this button different thermal boundaries can be introduced on some parts of the cross section only. The input is created with a single thermal boundary line (holding attributes and start and endpoint).

Tab sheet 'Thermal action' The requested temperaturetime curve, the heat transfer coefficient  c and the emissivity of the element  m can be defined in the tab sheet 'Thermal action' . If no temperaturetime curve has been chosen (represented with '') then at this boundary no heat transfer happens and adiabatic boundary conditions are assumed. The heat transfer coefficient is normally  c  25 WmK for bounds facing the fire and  c  4 WmK for bounds facing away from the fire. More information can be found in EN 199112.

F-6

FAGUS‐7


Fig. F − 5 Thermal Bounds: Tab sheets Thermal action and Encasement

According to EN 199212  m  0.7. (  m  0.8 could also be used, given that no other information concerning the material properties have been made) 

The emissivity of the fire  f and the configuration factor  are both assumed as 1.0 ( F 1.1 ). Other values could be taken into account by giving the product of   m  f . instead of  m .

Tab sheet 'Encasement' In this tab sheet a cross section encasement with the thickness d p can be defined. The coating can consist either of standard FAGUS material (e.g. steel casing) or a fire protection material. All the materials from Steeldoc 02/06 are available. In addition the thermal values  p ,  p and c p can be entered manually. 

Box casings can not be generated this way. They have to be created manually through in- serting several cross section parts.

Visibility and graphic representation The visibility of thermal bounds is managed through the layers 'Elevated Temperature'. Bounds without encasements are orange, while bounds with encasements are brown and slightly thicker. (Fig. F6)

Fig. F −6 Representation of the thermal bounds

F 1.2.4 Analysis Options In Fig. F7 the dialog 'Analysis options' is shown

FAGUS‐7

F-7


Initial temperature of the cross section at time t=0. With the standard value of 20 °C no thermal expansions occur. Thermal analysis ends at given time. If the ultimate limit is beyond this point a new thermal analysis with a higer value has to be done.

To keep the results well organised the output is available only at certain time steps. The field 'additional' times allows to complement this definition. For all times defined here a mechanical analysis can be made

Fig. F −7 Input of the analysis options

Time The time step parameters can either be set automatically (Default) or set with constant time steps t. The automatic time step parameters are set by the program to keep calculation time as short as possible. 

A calculation with a constant time step is an alternative if the standard calculation was not successful. (In this respect time steps should not be too big)



To obtain the temperature in a certain point of the cross section a result point of type 'longitudinal stress' has to be introduced in the tab sheet 'Geometry'.

FEMesh The mesh width h of the FEmesh has an influence on the precision of the calculated results of the cross section temperatures and the same FEmesh is used for the mechanical analysis (s. F 2.2.1). A very narrow FEmesh entails longer calculation times. If the mesh width h is not manually restricted with the maximal value h max an optimised mesh width is chosen in respect to calculation time and precision. In the areas of bigger temperature gradients a narrower mash should be used. An option 'Refine mesh at bounds' exists to generate a narrower FEmesh at the thermal bounds. If the cross section has rebars with large diameters the option 'Mesh point reinf.' should be activated. The rebars are then taken into mesh consideration and the thermal properties of reinforcement steel are taken into account for the numerical calculation. If the cross section consists of several subsections, it is important that the adjoining boundaries fit together well. Therefore the check described in Chap. B 2.1.1 (checking overlapping elements) may not be switched off.

F-8

FAGUS‐7


F 1.2.5 Run thermal analysis Check Input Once all the input has been entered for the calculation, the cross section is examined for the calculation. The geometric input as well as the definition of the thermal bounds are controlled here. Should the cross section have several variants, the cross section bounds of the current variant is used for examining the geometric position of the thermal bounds. If the examination is successful the FEmesh will be generated. The visibility of the FEMesh can be controlled with the layer button 'FEMesh' in the layergroup 'Temperature'. The encasement mesh at the thermal bounds is not shown.

Run thermal analysis After a successful examination of the input the calculation can be started with the button to the left. If an examination has not been done yet, it is done automatically. If the calculation was successful numerical and graphical results are ready and the cross section is now ready for a mechanical analysis.

Reset thermal analysis The thermal calculations can be reset with the button shown on the left. This erases all saved calculation results.

F 1.2.6 Result Output The button 'Legend thermal analysis' creates a tabular summary of all inputs as well as the calculated temperature values in the result points. This button shows a graphic with a timeline of the temperature at the result points. For control purposes the defined temperature time curves of the bounds are also shown. For the Isoline representation a certain time has to be chosen. This is done by the drop down menu shown to the left. (The different layers are controlled by the layer menu 'elevated temperture'.) Window 'Output Settings'

The temperature of each FEnode is shown (Normally these are a lot of values that overlap, this means that one should work in a highly zoomed in mode )

Fig. F − 8 Controlling the output

FAGUS‐7

F-9


Scaling the colour gradient

The colour density of the isoareadepiction is controlled with the field 'Scaling'. The colour scaling always runs from blue to red. If the setting 'Automatic scaling' is active the colour blue is always the lowest and the colour red is the highest temperature. Should the automatic scaling not be active, then the colours red and blue are assigned according to the inputs “Min” and “Max” for all times. This makes comparing isoline representations easier. This button shows the temperature profile along the main axis graphically.

F-10

FAGUS‐7


F 1.3 Validationexamples F 1.3.1 DIN EN 199112/NA:2010:12 Appendix CC The national appendix CC of DIN EN 199112/NA:201012 includes several examples for validating and checking a computer program. The documentation and calculations are available in FAGUS under > Help > Load Examples > Elevated Temperatures ...

F 1.3.2 EN 199212 Appendix A Two additional reference examples can be found in EN 199212 Appendix A with these thermal properties: Heat transfer coefficient Moisture content Emission factor Heat conductivity Exposure 

  25Wm 2 K u  1.5%   0.7 f   0 (lower bound)

ETK

The calculations are included in FAGUS as an introductory example.

Reinforced concrete column 30 x 30 Fig. A.13 in EN 199212 appendix A shows the temperature profile of a column exposed to fire on all four sides with a quadratic crosssection of 30  30 cm. In the FAGUS introductory example the crosssection “QS_STUETZE” has the suitable input. Due to double symmetry only ¼ is modeled. Fig. F9 shows, that the isolines calculated by FAGUS are almost identical with the ones provided by the reference document.

Fig. F − 9 Calculated Cross− section−temperatures and EN 1992−1− 2, Dpc.13

Slab crosssection Fig. A.2 in EN 199212 appendix A shows the temperture profile of a slab cross section with a thickness of h  20 cm exposed to fire on one side. The properties are provided in the cross section “QS_Platte”. The comparison in intervals of x  5 cm and x  10 cm proves a good accordance with the reference document. (s. Fig. F10 and F11).

FAGUS‐7

F-11


Fig. F −10 EN 1992−1− 2 Appendix A, Dep. 2

Fig. F −11 Temperature curve in panel cross− section with x=5 cm and x=10 cm

F-12

FAGUS‐7

F 2 Mechanical Analysis

F 2 Mechanical Analysis F 2.1 Influence of temperature on material behaviour

F 2.1.1 Strain/Stress relations The temperature dependent mechanical material behaviour is described as depicted down below for different concrete classes. This curve shows the reduction of compressive strength in relation to temperature: k()  f c, f ck(20°) k 1

Normal concrete K = calcareous Q = siliceous

2

Q

K

3

High strenght concrete 1 = C55/67 .. 2 = C70/85 .. 3 = C90/105

Fig. F −12 Reducation factors k according to EN 1992−1− 2

The automatic classification for high performance concrete is done according to section 6.1 (5) of the EN norm: 

Class 1: C 55/67, C60/75



Class 2: C 70/85, C80/95



Class 3: ab C90/105

For userdefined materials and 'Automatic' classification the program uses the given material strength to determine an appropriate class. The compressive strength reduction from class 2 to class 3 can be quite large under certain circumstances and has to be taken into consideration when comparing results from different concrete classes. (A highgrade concrete can give worse results under certain conditions)

FAGUS‐7

F-13


For a quick check a table with the minimal and maximal values for concrete and steel can be found in the standardresults of FAGUS. Additional results at certain points in the cross sections are accessible through result points.

Of course the mechanical analysis needs the whole    diagram: The implementions are based on the Norm EN 199212: 2004. For concrete the strain stressrelation in Fig.3.1. provides a graphical and analytical description. The two essential input parameters, the compressive strength reduction and the strain at the peak of the curve can be found in table 3.1. In case of tension the concrete is assumed to cracked. 2 m m / N n i n o i s n e T

Elongation [o/oo]

Fig. F −13 Concrete diagram for differing temperatures

The steel diagrams are described in the EN Norm in Fig.3.3. or table 3.2 (Reinforcing steel, structural steel) and table 3.3. (Prestressed steel). Below is the Strain/Stress diagramm for reinforcing steel.

F-14

FAGUS‐7


2 m m / N n i n o i s n e T

0..400

500

Temperature in Celsius 600

700

Elongation[o/oo]

Fig. F −14 Temperaturedependent Strain/Stress−diagramm for B500B

F 2.1.2 Thermal strain The thermal expansions are temperature dependent and are calculated according to sections 3.3.1 (concrete) and 3.4 (steel) of the EN norm.

Fig. F −15 Thermal expansion according to EN 1992−1−1 for concrete and steel (right side)

For illustrative purposes the strain distribution of a centrally loaded cross section with fire exposure on all four sides is shown below. +



Fig. F −16 Strain distribution of a centrally loaded cross section

FAGUS‐7

F-15


The total strains  tot consist of the thermal expansion  and the elastic strains  as follows:  tot     

The stress can then be calculated with the before mentioned    diagramms .

   f (  )

F 2.2 Analysis

F 2.2.1 Basics of the solution process At elevated temperatures the mechanical behaviour at each point of the crosssection is different. The cross section integration is done via a finite element discretisation. The same mesh size as in the thermal analysis is used. Depending on the degree of discretisation the results of the FEmodeled cross section at time t=0 can differ from „normal“ FAGUS results. A “cold” cross section can be integrated exactly whereas the “hot” FEapproach implies a certain approximation. The numerical effort naturally grows with a growing number of FEelements. The calculated temperature in the rebars is crucial for the mechanical results. Because the temperature gradient close to the edge of the cross section is big, a correct placement plays a decisive role. As mentioned before all rebars have to be entered as point reinforcements.

F 2.2.2 Using the program After a thermal analysis has been done, the tab sheet 'Analysis' displays a check button 'Fire exposure' which allows the following analysis under elevated temperatures: 

Utilization / Load capacity / Failure time



Stress analysis with given forces



Stress analysis with given strain



Moment curvature diagram



Analysis of second order effect under axial load

For analysis under elevated temperatures the analysis parameter are given by the programm and cannot be changed.

F-16

FAGUS‐7


F 2.3 Utilisation / Capacity / Failure time With this analysis the utilisation or load factor is determined for each result time and represented in a table. In the last row of this table the failure time is shown should it be within the specified calculation time. Result time as well as calculation time can be specified during the thermal analysis (see Chapter E.1.2.4 Calculation options).

Depending on the amount of result steps and number of FEelements the numerical effort can be significant.

F 2.4 Stress analysis with given forces For a given time and for given section forces this analysis shows: 

A contour plot (isolines) of strain and stresses (The “classic” images with the strain and stress diagrams, the graph of the inner lever arm are not significant for such an analysis and are not shown.)



Numerical results in the result points



Cross section stiffness



Maximum strain and stresses

F 2.5 Stress analysis with given strain With this analysis the elongations εx, χy, and χz are entered directly. It is the only calculation type, whose results do not have to be determined by iterations and will presumably only be used in special cases (For example for the verification of a cross section integration).

FAGUS‐7

F-17


F 2.6 Momentcurvaturediagram The momentcurvaturediagram gives a graphical depiction. Depending on amount of points in time and FEelements the calculation time can be considerable. In the input field for the normal force only the first value is used (For analysis under normal temperature several normal forces can be input divided by a space character). On demand curves for all result times are created.

F 2.7 Checklist To determine the failure time of a crosssection the following steps are necessary:  Enter cross section as usual. The reinforcement should be defined as a point reinforcement.  Switch to tab sheet 'Thermal analysis'  Choose calculation type 'transient'  Adjust calculation time and result time points if needed in the dialog 'Calculation options'  Define thermal bounds  Check cross section. Following the check the FEmesh is shown. In general a discretization with around 400 elements should be enough  Carry out thermal analysis  Switch to the tab sheet 'Analysis'  Activate check box  Choose desired analysis, enter forces and carry out calculation

F-18

FAGUS‐7


Several examples can be found in the menu 'Help' (For example validation examples according to EN) and under 'File' > 'Documents' the related explanations can be found.

FAGUS‐7

F-19


F-20

FAGUS‐7

G 1 Baustoffe

Vol. G Baustoffe, Analyseparameter FAGUS‐6

G 1 Baustoffe G 1.1 BaustoffDialoge Zu Beginn einer neuen Berechnung sollten alle Baustoffe und ihre für die Berechnung relevanten Eigenschaften definiert werden (Menu > Einstellungen > Baustoffe). Der Umfang der Baustofftabelle ist einen allen CubusProgrammen identisch. Je nach Anwendung werden nicht alle Werte für die Berechnung benötigt::

ID: Die BaustoffZuweisung zu einzelnen Strukturelementen (Bauteil, Querschnitt, Be-

wehrung) findet über die “ID” statt. Wird z.B. in der oben dargestellten ersten Zeile die Betonklassse gewechselt, so passen sich alle Elemente mit der Referenz “C” automatisch der neuen Definition an. Die Bezeichnung ist bewusst sehr kurz gehalten, damit die numerischen Ausgaben nicht mit langen Texten belastet werden. Der Benutzer kann eigene BaustoffeID's erzeugen, allerdings nur durch Zusatz einer einstelligen Zahl (Index). Der erste Buchstabe wird verwendet um den Baustofftyp festzulegen. 

Beton

C (Concrete)



Betonstahl

R (Reinforcment)



Spannstahl

P (Posttensioning)



Baustahl

S (Steel)



Holz

W (Wood)



Aluminium

U



Faserverbund F (FRP)



Mauerwerk

M (Masonry)



Spezial

X

Typ: Der Baustofftyp bestimmt u.a. das eindimensionale Materialverhalten (Kap. G 1.2) und

kann nur bei der erstmaligen Eingabe gewählt frei gewählt werden. (Gewisse Strukturelemente lassen nur bestimmte Baustofftypen zu. Eine Bewehrung kann z.B. nicht vom Typ 'Beton' sein) Bauteil: Frei wählbare Bezeichnung

E: Für Schnittkraft und Durchbiegungsberechnungen (CEDRUS, STATIK) wird der in dieser

Kolonne aufgeführte Wert verwendet. Bei der Initialisierung wird E vom Vorgabewert aus der Baustoffklasse übernommen und kann dann vom Benutzer verändert werden.

FAGUS‐6

G-1

Vol. G Baustoffe, Analyseparameter



E ist ein Ausgangswert für die Steifigkeitsbestimmung. In den verschiedenen Programmen lassen sich diese noch durch Einführung eines entsprechenden Faktors zusätzlich an- E0 passen z.B. CEDRUS6 pro Zone: E  f E �

G:  : : :

Schubmodul: In STATIK6 für die die Steifigkeiten GIx, verwendet. Für die Bestimmung von Massen und EigengewichtLastfällen verwendet. Temperaturausdehnungskoeffizient Querdehnungszahl (CEDRUS)

Farbe/Farbnuance : Für Bildschirmdarstellung und gedruckte Ausgabe verwendet Klassen:

Die Werte der vom Programm vorgegebenen Klassen können nicht verändert werden, aber die Tabelle kann mit eigenen Baustoffklassen ergänzt werden. Die Klasseninformationen sind vor allem im Zusammenhang mit der Querschnittsanalyse und Bemessung relevant. Umfang und Bezeichnungen sind normabhängig. Normalerweise sind charakteristische Werte einzutragen.: Der Bemessungswert X d einer einzelnen Baustoffeigenschaft ergibt sich aus x X d   k

M

x k = charakteristische Baustofffestigkeit  M = Teilsicherheitsbeiwert

Bezogen auf die verschiedenen Baustoffe sind für X k folgende Werte einzusetzen: f ck, f yk, f pk

Charakt. Werte der Beton, Betonstahl und Spannstahlfestigkeiten

und für  M  c,  s,  p,  a

Teilsicherheitsbeiwerte für Beton, Bewehrungs, Spann u. Baustahl.

Dieser Teilsicherheitsbeiwerte sind bei den Analyseparametern definiert. Beim Baustoff 'Beton' kommt in der Regel noch ein Abminderungsbeiwert  für die Berücksichtigung der Dauerstandsfestigkeit hinzu.   f

f cd   ck c Für die SIA262 wird in dieser Tabelle direkt der Wert für  fc  f ck eingegeben  (  1). f cd 

 fc  f ck  c

G 1.2 Eindimensionale SpannungsDehnungsbeziehungen In diesem Kapitel werden die für die Querschnittsanalysen verwendeten eindimensionalen SpannungsDehnungsbeziehungen beschrieben. Die Form der    Diagramme oft berechnungsabhängig kann deshalb via Analyseparameter gesteutert werden.

G 1.2.1 BetonDruckspannungen Bei Biegung mit Längskraft wird die einaxiale SpannungsDehnungsbeziehung in jedem Punkt des Querschnittes durch eines der untenstehend abgebildeten Diagramme beschrieben. Für die Berechnung des Querschnittswiderstandes und für Bemessungsaufgaben wird von den meisten Normen das Diagramm Nr. 2 vorgeschlagen, wobei für den ersten Kurvenabschnitt eine quadratische Parabel verwendet wird, welche durch die beiden Parameter  2.0   2.00 o oo und f c definiert wird. Der tangentielle EModul bei Belastungsbeginn beträgt somit E co  1000� f c .

G-2

FAGUS‐6

G 1 Baustoffe



 f c

f c

E co



f c

f c

0.4 f c E c







 cu

E c

E cm

 2.0 o oo  cu

1: Bilinear

2: Quadratische Parabel



  c1 d  c2 d

 c1  c1u 3: EC2, DIN10451

4: SIA 262

Fig. G−1 Spannungs−Dehnungs−Diagramme für Betondruckspannungen

Für nichtlineare Berechnungen wird in EC2 und DIN10451 das Diagramm Typ 3 verlangt, welches durch untenstehende Gleichung beschrieben wird:

 c  f c 

k    2 1  ( k  2) 

wobei: 

=  c / c 1 (beide negativ einsetzen)

 c1

=  0.0022 (Stauchung beim Erreichen des Höchstwertes der Betondruckspan nung f c)

k

= 1.1 . E c,nom .  c1 /f c (f c negativ)

E c,nom = entweder Mittelwert E cm (Tabelle 3.2 EC2) oder entsprechender Bemessungs wertE cd des Elastizitätsmodules

Die Norm SIA 262 benützt für den aufsteigenden Ast einen ähnlichen Ansatz

 c f cd



k     2 E  mit k   cd und   c c1 d 1  ( k   2)  400 f cd

wobei nach dem Erreichen von f cd ein konstantes Plateau bis zum Bruch verwendet wird und die Bruchstauchung auf  c2 d  3 [ o oo] beschränkt wird. 

Hinweis Vorzeichen: Analog zu den Stablängskräften sind die Zugspannungen in FAGUS6 auch positiv. Die oben und im Programm gewählte Darstellung des Spannungsdehnungsdiagrammes für Beton entspricht den üblichen Konventionen.

G 1.2.2 BetonZugspannungen Im Normalfall wird der Beton unter Zug als gerissen angenommen (Zustand II), d.h. er übernimmt keine Zugspannungen. Für spezielle Untersuchungen, z.B. wenn die Steifigkeit des Querschnittes eine wesentliche Rolle spielt, kann aber auch ein Diagramm Nr. 1.. 3 gewählt werden: 

 



f ct 0



f ct

1

 

f ct)  ct  f ( r ,



f ct

2

f ct 3

Fig. G− 2 Spannungs−Dehnungs−Diagramme für Betonzugspannungen

FAGUS‐6

G-3


Während der Beton beim Diagramm Nr. 1 nach dem Erreichen von f ct keine Spannungen mehr übernimmt, bleiben diese beim Diagramm Nr. 3 auf dem Niveau  = =f ct konstant.f ct kann als Parameter bei der Materialeingabe eingegeben bzw. verändert werden. Der Verlauf der Kurven entspricht jeweils dem für die Druckspannungen gewählten Typ (Punktspiegelung bezüglich Nullpunkt). Ein etwas wirklichkeitsgetreueres Materialverhalten wird mit Nr. 2 realisiert, indem die Grösse der Betonspannung abhängig gemacht wird von der aktuellen Randzugdehnung. Randzugdehnung. Für Diagramm Nr. 2 wurde folgender Ansatz gewählt:  r 2 )) 0.2%

0   c c  f ct  (1  ( wobei:  ct

: Betonzugspannung Betonzugspannung

f ct

: eingegebene Betonzugfestigkeit Betonzugfestigkeit

 r

: aktuelle Randdehnung auf der Zugseite des Querschnittes

0.2 % ≈ Dehnung bei der Streckgrenze eines üblichen Betonstahles (S500) Mit diesem Modell beträgt die Betonzugfestigkeit zu Beginn der Belastung  =f ct und nimmt mit gesteigerten Krümmungen quadratisch ab. Ab dem Erreichen der Fliessspannung am Zugrand (bzw. bei  r   0.2 %) ist keine Betonzugfestigkeit mehr vorhanden.

G 1.2.3 Mitwirkung des Betons zwischen den Rissen mittels Verbundbeiwert A





In den Querschnittsanalysen wird im Normalfall der gerissene Querschnitt (Schnitt A) betrachtet. Gewisse Korrekturmöglichkeiten wurden im vorangegangenen Abschnitt besprochen. Ein anderer Ansatz für die Berücksichtigung der Mitwirkung des Betons zwischen den Rissen wird in der Norm SIA E 166 “Klebebewehrungen" beschrieben. Vorgehen: Für die Ermittlung der Dehnungsebene werden mittlere Dehnungen ( ) verwendet, für die Gleichgewichtsbeziehungen Gleichgewichtsbeziehungen wird aber der Schnitt im Riss betrachtet. Das Verhältnis zwischen mittleren und maximalen Werten ( ") wird durch den Verbundbeiwert  beschrieben.    

Falls das entsprechende Kontrollfeld ("Verbundbeiwert berücksichtigen") bei den Analyseparametern aktiviert ist, so wird die QuerschnittsAnalyse mit Berücksichtigung dieses als konstant angenommenen Faktors durchgeführt. Im Allgemeinen ist  abhängig davon, ob das Grundmaterial gerissen ist (Beton) und ob es sich um eine innere Stabbewehrung oder eine äussere Klebebewehrung handelt. Strukturen aus Metallen oder Holz sind im Allgemeinen ungerissen und man nimmt an, dass hier die Klebebewehrung an jeder Stelle die gleiche Dehnung wie das Grundmaterial aufweist. In der gegenwärtigen Programmversion werden folgende nicht veränderbare Werte verwendet: Verbundbeiwert der inneren Stabbewehrung :  s = 0.7 Verbundbeiwert der äusseren Klebebewehrung: Klebebewehrung:  l = 0.9 Für Bewehrungs und Vorspannstahl Vorspannstahl wird  s verwendet. Für Faserverbundwerkstoffe und Baustahl, welche als “Spannglied/Lamellen" eingegeben wurden, wird  l eingesetzt. Das entsprechende Querschnittselement muss zudem unter Zugbeanspruchung Zugbeanspruchung sein, damit  berücksichtigt wird. Bei linienförmigen Bewehrungen gilt dies nur für den entsprechenden Abschnitt.

G 1.2.4 Betonstahl, Baustahl und Spannstahl Für Betonstahl, Baustahl und Spannstahl werden die unten dargestellten Diagramme verwendet.

G G-4

FAGUS‐6

G 1 Baustoffe



 f pk f y

f p0.1 k E T

E s

E s

 uk Betonstahl

Baustahl

Spannstahl

Fig. G− 3 Spannungs−Dehnungs−Diagramme für Stahl



Falls eigene Baustoffklassen definiert werden, so ist darauf zu achten, dass Kennwerte eingegeben werden. Diese werden dann entsprechend den in den Analyseparametern definierten Widerstandsbeiwerten/Teilsicherheitsfaktoren reduziert. Die Reduktion erfolgt normabhängig entsprechend entsprechend der nebenstehenden nebenstehenden Figur. Ob mit oder ohne Verfestigung gerechnet werden soll, kann ebenfalls bei den Analyseparametern Analyseparametern eingegeben werden.

SIA162

f .. k k f .. d d

 

Neue Normen

f .. k k

Bei den meisten Normen ist für die Zug und Druckfestigkeit der gleiche Wert anzunehmen. Für spezielle Untersuchungen (z.B. British Standard BS5400) können aber auch unterschiedliche Werte gewählt werden. Falls zugleich fy <
f .. d d



f tk  f y E T     y uk

In den Materialtabellen von FAGUS6 werden auch für den Spannstahl für alle Normen gewisse Vorschläge gemacht. Bei der Wahl eines Spannstahles ist aber unbedingt darauf zu achten, dass die vom Programm vorgegebenen Werte mit den Herstellerangaben übereinstimmen, d.h. in der Regel sind diese anzupassen. Die initiale Vorspannkraft wird durch Eingabe einer Vordehnung angegeben. angegeben. Weitere Erklärungen dazu sind z.B. im Handbuch von FAGUS zu finden.

FAGUS‐6

G-5


G 2 Analyseparameter G 2.1 Der AnalyseparameterDialog Als Analyseparameter werden alle Parameter bezeichnet. welche das Verhalten einer Querschnittsanalyse beeinflussen (und nicht bereits in der Querschnittsgeometrie oder den Baustoffparametern enthalten sind). Es können mehrere Sätze von Analyseparametern unterhalten werden, wobei standardmässig je eine Definitionen Definitionen für den GZG (Gebrauchstauglichkeit) (Gebrauchstauglichkeit) und für den GZT (Tragsicherheit) (Tragsiche rheit) zur Verfügung stehen. Der Dialog ist in mehrere Registerblätter aufgeteilt:



Beispiel: Ein MNInteraktionsdiagramm kann wahlweise für den “Grenzzustand der Ge- brauchstauglichkeit“ wie auch für den “Grenzzustand der Tragsicherheit" erzeugt werden, indem der entsprechende Analyseparametername gewählt wird. Vorgabewerte: Standardmässig werden für die verschiedenen Analysen folgende Zuord-

nungen verwendet: AP

Analysetyp

GZG

Spannungsanalyse, MomentenKrümmungsdiagramme

GZT

Traglastanalyse, Bemessung, Interaktionsdiagramme Interaktionsdiagramme Der Benutzer kann die bestehenden Programmvorgaben Programmvorgaben verändern sowie eigene Definitionen hinzufügen.

G 2.1.1 Registerblatt Grenzdehnungen und Grenzspannungen Die Grenztragfähigkeit eines Querschnittes gilt als erreicht, wenn die Dehnung der äussersten Querschnittsfaser auf der Druckseite oder der äussersten Bewehrungslage auf der Zugseite einen bestimmten Grenzwert erreicht hat. Für zentrischen Druck und für Biegung sind die Grenzdehnungen unterschiedlich, wie dies in untenstehender Figur dargestellt wird: 

Für die Parameter  cu c . c,  cu b .b,  su fehlen einheitliche Bezeichnungen. Es wurde deshalb eine Definition gewählt, welche für alle unterstützten Normen einigermassen akzeptabel sein sollte. Dabei steht der erste Index c für “concrete" bzw. s für “Stahl", “Stahl", u für ultimate (grenz) Bedingung und der Buchstabe nach dem Punkt für c=zentrisch bzw b=Biegung .

Die fünf Dehnungsbereiche werden mit folgenden Begriffen charakterisiert: Bereich 1: Mittiger Zug und Zugkraft mit kleiner Ausmitte Bereich 2: Biegung (mit Längskraft) unter Ausnutzung der Bewehrung Bereich 3: Biegung (mit L.Kraft) unter Ausnutzung der Bewehrung und des Betons Bereich 4: Biegung (mit Längskraft) unter Ausnutzung des Betons Bereich 5: Längskraft innerhalb Kernbereich des Querschnittes, zentrischer Druck

G G-6

FAGUS‐6

G 2 Analyseparameter

 cu. c  cu. b

3

2

1

d

h

4 5

  su

 sy

Zug

Druck

 sy = Stahlstreckgrenze

Fig. G− 4 Grenzdehnungsebenen für GZT

Im Grenzzustand befinden sich die Dehnungen am Querschnittsrand also auf folgender, geschlossener Figur:  2  Dehnungen am oberen Rand

Zug

Biegung

1

1

  cu. b

 1  Dehnungen am unteren Rand Druck Biegung

5 1

 2   cu. c

3,4 2

  cu. b

1

 h d

1

 h d

2

 su. c

( . c   cu. b)   cu. b  su

Fig. G− 5 Dehnung am Querschnittsrand im Grenzzustand

Der Querschnittswiderstand wird somit nie rein statisch aus den Rechenwerten der Baustofffestigkeiten ermittelt, sondern es wird immer ein Verzerrungszustand gesucht, bei dem die Dehnungen mindestens an einem Ort des Querschnittes gerade den zulässigen Grenzwert erreichen. Für eine exakte Bestimmung des plastischen Momentes (mit vollständiger Plastifizierung des Querschnittes) müssten unendlich grosse Randdehnungen vorgegeben werden können. Folgende Tabelle gibt Auskunft über die FAGUS6 Voreinstellungen für einige Normen: Norm

 cu.c

 cu.b

 su

SIA 262 DIN1045−1 EC2 OeNorm B4200 OeNorm B4700

mittiger Druck −2.0 o/oo (bzw. −3.0) −2.0 o/oo −2.0 o/oo −2.0 o/oo −2.0 o/oo

Biegung −3.0 o/oo −3.5 o/oo −3.5 o/oo (*) −2.0 o/oo −3.5 o/oo

Stahlgrenzdehnung (Zug) 5.0 o/oo 5.0 o/oo 20 o/oo (falls fy=ftk, sonst 10 o/oo) 4 o/oo 10 o/oo

(*) abhängig von Betonqualität Bei einer Berechnung mit schiefer Biegung werden diese Bedingungen entsprechend der aktuellen Lage der Neutralachse kontrolliert. Querschnitt ohne Bewehrung: Falls auf der Zugseite keine schlaffe Bewehrung angetroffen wird, gilt der eingegebene maximale Wert  su (bzw. die eingegeben maximale Stahlspannung) am Querschnittsrand.

FAGUS‐6

G-7


Nachweise für den GZG

Für die Gebrauchstauglichkeit sind oft Rissweiten (*) oder maximale Stahlspannungen in Abhängigkeit vom Durchmesser und Abstand nachzuweisen. Anstelle von Grenzdehnungen können auch zulässige Stahlspannungen eingegeben werden. Von gewissen Anwendungen (CEDRUS6) wird dieser Wert aber “von aussen” entsprechenden den vorgefundenen Eingabedaten gesetzt. (*) Rissweiten sind als Teil von Spannungsnachweisen verfügbar.

G 2.1.2 Registerblatt Widerstandsbeiwerte / Teilsicherheitsbeiwerte: Die Baustoffkennwerte (Festigkeiten, etc.) werden durch die Angabe einer Baustoffklasse im Rahmen der Querschnittseingabe jedem Querschnittselement fix zugewiesen. Das zu  s , ) .. können jeverwendende     Diagramm und die Teilsicherheitsbeiwerte ( c , doch für jede Analyse unterschiedlich festgelegt werden.

G 2.1.3 Registerblatt BetonDiagramm Hier kann die gewünschte    Beziehung sowohl im Druckbereich sowie für allfällig zu berücksichtigende Zugspannungen gewählt werden. Eine Beschreibung dazu befindet sich in Kap. G 1.2.

G 2.1.4 Registerblatt Vorspannung: Bei der Vorspannung kann zwischen zwei    Beziehungen umgeschaltet werden. Ein weiterer Punkt betrifft die Berücksichtigung von Langzeitverlusten, wobei diese Eingabe nur im Zusammenhang mit dem VorspannModul von STATIK6 verwendet wird.

G 2.1.5 Registerblatt Bewehrung In diesem Register werden verschiedene Bewehrungsgehalte für die Bemessung der Längs und Bügelbewehrung vorgegeben:  Minimaler Längsbewehrungsgehalt für Stützen Für Stützenquerschnitte existieren in allen Normen Vorschriften über minimale Bewehrungsgehalte. Bei der Bemessung werden die eingegebenen Bewehrungen so skaliert, dass der Minimalwert auch für geringe Beanspruchungen nicht unterschritten wird.  Maximaler Längsbewehrungsgehalt (alle Querschnittstypen) Dient als Abbruchkriterium (und somit als Schutz gegen zu lange Rechenzeiten) im Rahmen der Bemessungsiterationen.  Minimaler Bügelbewehrungsgehalt Im Rahmen der Bügelbemessung wird eine statisch erforderliche Bewehrung bestimmt (Kap. LEERER MERKER). Mit Hilfe von Resultatkombinationen kann dieser Wert anschliessend mit dem vom Benutzer gegebenen bzw. nach Norm erforderlichen Minimalwert verglichen werden (vgl. Kap. C 4.4)

G-8

FAGUS‐6

Manual Fagus 7

Recommend Documents