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 ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1
The Basic Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1
The Analysis Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1
A 1.2 Program Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2
Material management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2
Cross section management / Cross section elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2
Analysis parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2
A 1.3 Analysis Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-4
A 1.3.1 Bending with axial force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-4
A 1.3.2 Shear force and torsion for reinforced concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-6
A 1.3.3 Elastic shear flow analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-7
A 1.3.4 Limitations of beam theory and cross section considerations . . . . . . . . . . . . . . . . . . . . . . A-7
A 1.4 Checks to be performed by User . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-7
A 1.5 Limited Guarantee (excerpt from Cubus Licence Agreement) . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-8
A 1.6 Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-8

A 2 Starting the Program - Brief Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-9

A 2.1 Starting the Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-9
CubusExplorer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-9
A 2.2 Brief Description of the Most Important Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
B 1.2 The Menu List of FAGUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
B 1.2.1 The menu 'File' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
B 1.2.2 The menu 'Options' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-3
B 1.2.3 The menu 'Display' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-3
B 1.2.4 The menu 'Window' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-4
B 1.2.5 The menu 'Help' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-4
B 1.3 Management of Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-4
Dialogue 'Cross Section Management' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-4
B 1.4 The Layer Buttons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-6

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B 2 Input of Cross Section Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-7

B 2.1 The Tab sheet 'Geometry' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-7
B 2.1.1 General information on the input of cross section elements . . . . . . . . . . . . . . . . . . . . . . . B-7
B 2.1.2 Drawn partial cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-8
B 2.1.3 Openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-8
B 2.1.4 Thin-walled partial cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-9
B 2.1.5 Parametrised cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-9
B 2.1.6 Standard rolled-steel sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-11
B 2.1.7 Introduction of existing cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-12
B 2.1.8 Axial points and handles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-12
B 2.1.9 PT constraint lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-12
B 2.1.10 Result points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-13
Standard result points (only in layer button): . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-13
B 2.1.11 Result combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-14
B 2.1.12 Cross section results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-15
B 2.1.13 Checking the cross section input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-15
B 2.1.14 Legends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-15
B 2.2 The Tab sheet 'VT-Model' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-16
B 2.2.1 Shear walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-16
Input of shear wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-17
Calculation of shear wall forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-18
Graphical description of shear wall forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-19
Simplified shear wall input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-19
Torsion values from basic cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-19
Weight factors for wall groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-19
Additional remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-20
Additional input help for compact cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-20
Stirrups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-20
B 2.3 The Tab sheet 'Reinforcement' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-22
B 2.3.1 Reinforcement groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-22
B 2.3.2 Auxiliary points to graphically construct the reinforcement . . . . . . . . . . . . . . . . . . . . . . . . B-22
B 2.3.3 Point, line and circular reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-22
B 2.3.4 Tendons / Strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-25
Initial strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-25
Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-26
Slope of tendons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-27
Long-term losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-27
B 2.4 The Tab sheet 'Variants' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-30
B 2.4.1 Properties of the variant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-30
Tab sheet 'Properties': . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-30
Tab sheet 'Member' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-31
Tab sheet 'VT Model' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-32
Tab sheet 'Cross Section Values' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-33

ii FAGUS7
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B 3 Design and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-34

B 3.1 Loading of Cross Section, Section Force Reference System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-34
B 3.2 The Tab sheet 'Reinforcement Design' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-36
Automatic adjustment of the required reinforcement areas . . . . . . . . . . . . . . . . . . . . . . . B-37
Interactive selection of diameter and adjustment of the layer . . . . . . . . . . . . . . . . . . . . . . B-37
Numerical legend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-37
B 3.2.1 General remarks on the dimensioning of the axial reinforcement . . . . . . . . . . . . . . . . . . . B-38
B 3.2.2 General remarks on shear force and torsional dimensioning . . . . . . . . . . . . . . . . . . . . . . . B-38
Check the loading of the concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-39
Axial reinforcement due to force in shear wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-40
B 3.3 The Tab sheet 'Analyses' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-42
B 3.3.1 Ultimate load/ Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-42
B 3.3.2 Stress analysis with forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-43
Crack width verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-44
B 3.3.3 Stress analysis with strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-45
B 3.3.4 Elastic shear flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-46
B 3.3.5 M-N interaction diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-47
B 3.3.6 General interaction diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-48
B 3.3.7 Moment-curvature diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-49
B 3.3.8 Moment-stiffness diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-50
B 3.3.9 Analysis of second order effects with axial load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-51
B 3.4 Batch Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-53
B 3.4.1 Management of batch analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-54
B 3.4.2 Batch analysis/create task list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-54

Vol. C Special Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-1

C 1 Coordinate System and Sign Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-1
C 2 Section properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-2
C 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-2
C 2.1.1 Area, moments of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-2
Reference material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-3
C 2.1.2 Torsional Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-3
C 2.1.3 Torsionskonstante Ix mittels FE-Analyse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-5
C 3 Analyses with Thin-Walled Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-6
C 3.1 Shear Stresses due to Shear Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-6
C 3.2 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-9
C 3.2.1 Open Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-9
Shear Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-9
Torsional Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-9
Warping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-10
C 3.2.2 Closed Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-10
C 4 Shear and Torsion for Reinforced Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-11
C 4.1 National Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-11
C 4.1.1 EN 1992-1-1:2004 Eurocode 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-11
C 4.1.2 SIA 262 Swisscode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-13
C 4.1.3 EHE-08 Spanish Code: Instruccin de Hormign Estructural . . . . . . . . . . . . . . . . . . . . . C-13
C 4.2 Older Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-14
C 4.2.1 Design according to SIA 162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-14
C 4.2.2 Design according to DIN 1045 (07/1988) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-14
C 4.3 Design for torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-15
C 4.3.1 berlagerung Querkraft und Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-15
C 4.3.2 berlagerung Querkraft und Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-16
C 4.4 Combination of Shear Wall Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-17

FAGUS7 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-2
D 3.2 Efficiency, Capacity Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2
D 3.2.1 Bending and axial normal force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2
D 3.2.2 Shear and torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
E 1.1 Presentation Conventions for the Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-1
E 1.1.1 Load Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-1
E 1.2 Starting FAGUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-2
E 1.3 Opening a Cross Section Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-2
E 1.4 Example 1: Cross Section with Simple Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-4
E 1.4.1 Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-4
E 1.4.2 The program window of FAGUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-5
E 1.4.3 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-5
E 1.4.4 Input of the cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-6
E 1.4.5 Inputting the opening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-7
E 1.4.6 Input of the reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-8
E 1.4.7 Tab sheet Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-11
E 1.4.8 Documentation of the cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-12
Dimensioning the cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-12
Select content of figures with the layer buttons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-13
Enter figure in the print list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-13
Creation of a text legend with print entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-13
Print preview and printing with the CubusViewer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-14
E 1.5 Example 2: Composite Cross Section with Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-16
E 1.5.1 Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-16
E 1.5.2 Definition of the required materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-16
E 1.5.3 Input of the concrete slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-17
E 1.5.4 Input of the rolled steel section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-17
E 1.5.5 Definition of the variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-18
E 1.5.6 Input of the axis point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-19
E 1.5.7 Anchor points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-19
E 1.5.8 Introduction of result points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-20

iv FAGUS7
 Table of Contents

E 2 Beispiele Analysemodul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-21

E 2.1 Datei: 'S1' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-21
E 2.1.1 Trger, Bewehrungsbemessung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-21
E 2.1.2 Trger R2, Grenzwerte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-21
E 2.1.3 I-Trger, Bemessung fr kombinierte Beanspruchung . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-22
E 2.1.4 Kreissttze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-22
E 2.1.5 Sttzenquerschnitt, Steifigkeitsdiagramme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-22
E 2.1.6 Spannungsnachweise vorgespannter Trger, Schiefe Biegung . . . . . . . . . . . . . . . . . . . . . E-22
E 2.1.7 Interaktionsdiagramme fr einbetoniertes Walzprofil . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-23
E 2.1.8 Spannungsanalyse fr vorgespannten Brckentrger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-23
E 2.1.9 Brckentrger mit inaktiven Querschnittsteilen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-23
E 2.1.10 Dnnwandige Querschnitte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-24
E 2.2 Datei 'S2': Verbundquerschnitt mit Belastungsstufen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-25
E 2.3 Datei 'S3': Verbundquerschnitt unter Langzeitbelastung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-26

Vol. F Fire Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-1

F 1 Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-1
F 1.1 Theoretic Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-1
F 1.2 Register 'Thermal Analysis' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-3
F 1.2.1 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-3
Thermal values for concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-4
Thermal values for steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-4
Special (User-defined) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-4
F 1.2.2 Temperature-time curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-4
F 1.2.3 Thermal bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-6
Tab sheet 'Thermal action' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-6
Tab sheet 'Encasement' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-7
Visibility and graphic representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-7
F 1.2.4 Analysis Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-7
Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-8
FE-Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-8
F 1.2.5 Run thermal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-9
Check Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-9
Run thermal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-9
Reset thermal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-9
F 1.2.6 Result Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-9
F 1.3 Validationexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-11
F 1.3.1 DIN EN 1991-1-2/NA:2010:12 Appendix CC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-11
F 1.3.2 EN 1992-1-2 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-11
Reinforced concrete column 30 x 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-11
Slab cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-11
F 2 Mechanical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-13
F 2.1 Influence of temperature on material behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-13
F 2.1.1 Strain/Stress relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-13
F 2.1.2 Thermal strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-15
F 2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-16
F 2.2.1 Basics of the solution process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-16
F 2.2.2 Using the program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-16
F 2.3 Utilisation / Capacity / Failure time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-17
F 2.4 Stress analysis with given forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-17
F 2.5 Stress analysis with given strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-17
F 2.6 Moment-curvature-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-18
F 2.7 Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-18

FAGUS7 v
 Table of Contents

Vol. G Baustoffe, Analyseparameter . . . . . . . . . . . . . . . . . . . . . . . . . . . G-1

G 1 Baustoffe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-1
G 1.1 Baustoff-Dialoge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-1
G 1.2 Eindimensionale Spannungs-Dehnungsbeziehungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-2
G 1.2.1 Beton-Druckspannungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-2
G 1.2.2 Beton-Zugspannungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-3
G 1.2.3 Mitwirkung des Betons zwischen den Rissen mittels Verbundbeiwert . . . . . . . . . . . . . . . G-4
G 1.2.4 Betonstahl, Baustahl und Spannstahl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-4
G 2 Analyseparameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-6
G 2.1 Der Analyseparameter-Dialog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-6
G 2.1.1 Registerblatt Grenzdehnungen und Grenzspannungen . . . . . . . . . . . . . . . . . . . . . . . . . . . G-6
G 2.1.2 Registerblatt Widerstandsbeiwerte / Teilsicherheitsbeiwerte: . . . . . . . . . . . . . . . . . . . . . . G-8
G 2.1.3 Registerblatt Beton-Diagramm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-8
G 2.1.4 Registerblatt Vorspannung: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-8
G 2.1.5 Registerblatt Bewehrung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-8
G 2.1.6 Weitere Werte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-9
G 2.1.7 Zustzliche Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-9
G 2.1.8 Echoprint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-9

vi FAGUS7
 A 1 Overview

Vol. A Introduction
FAGUS7

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 over
view 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
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
s is also possible for rolled steel sections. For the analysis of cross sections of arbitrary geo
metry the object-oriented Graphics Editor, which should be well known from the other
E 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
S Interactive analysis and dimensioning of reinforced and prestressed concrete cross sec
tions (for bending about one or two axes, together with axial and shear forces and tor
sional moments).
S Dimensioning and verification of whole frame structures (= post-processing with an au
tomatic import of section forces from STATIK)
S 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 sec
tion results can be reproduced in detail interactively.

FAGUS7 A-1
Vol. A Introduction

A 1.2 Program Input

cross section analysis With the help of figure A-1 it will be attempted to explain graphically some technical terms
and concepts.
M
N
Material management
sc 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 calcu
lated, it is sufficient if the parameters on the left side are defined. If the cross section is ana
lysed 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 con
crete analyses have to be carried out, the reinforcement also has to be known. A cross sec
tion 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 Rein
forcement 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 as
signed 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 ma
terials and analysis parameters. The description can be found in a separate document.

A-2 FAGUS7
 A 1 Overview

FAGUS
Basic Module Analysis Module

Material management

Materials Material classes

ID Type Component E G Material class fcd Ecm fctm

C concete general 36 15 2.5 C40/50 -24 36 3.5

CC concrete columns 38 13 2.5
R reinf. steel general 210 80 8.0
The lists can be extended by the user.

Cross section management

Cross sections consisting of geometry and material Analysis parameters

Predefined lists for different codes
Partial cross sections
circumscribed: polygon steel section Name Partial factors
ss,adm max gc gs gp
ULS 20 o/oo 1.5 1.15
Input

SLS 150
Nmm 2
thin-walled:
The list can be extended by the user.

reinforcement point line circular

untensioned or
prestressed

shear walls Section forces / strains

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

reinforcement design
static cross section properties ultimate load analyses
Analysis / Output

mass per unit length of member stress analyses

interaction diagrams
moment/curvature diagrams
moment/stiffness diagrams
shear flow distributions

Frame analysis program STATIK STATIK Postprocessing

Fig. A1 Overview of cross section program FAGUS

FAGUS7 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:
S Cross sections remain plane (Bernoulli hypothesis), i.e. linear strain distribution over the
cross section (the strain plane).
S Fully effective bonding between concrete and reinforcing steel.
Exception: prestressed elements can be introduced with the option 'without bonding'.
S 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 re
quires 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 fol
lows:
S Assumption of a strain plane
S Determination of the internal forces and moments (integration over cross section)
S 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 per
mits, 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 graph
ically 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 con
tents. 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. How
ever, 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 ax
ial force the bending moment is increased continuously and the result (stiffness or curva
ture) is displayed graphically.

A-4 FAGUS7
 A 1 Overview

Stress analysis for given strain plane Stress analysis for given force

(y,z)  (y,z)
(y,z)  (y,z)

? x
x M M ?

N N

Method of solution:
Method of solution: Iteration: variation of the strains until internal section forces =
Integration of the stresses over the cross section given external section forces

Ultimate load / Efficiency Reinforcement design

..u 

?
N M  .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 M
z
kept constant e.g.

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

EI My
y

My xy

Method of solution: Method of solution:

Repeated stress analysis with N=constant Repeated stress analysis with N=constant

Fig. A2 Overview of analysis types

FAGUS7 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. A3 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 arbit
rary combination of shear force and torsion.
If one has more than three shear walls the problem is statically indeterminate and the distri
bution 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 Centre of stiffness:
S
(I iyyi)
ys +
T S
I iy
Vy S
(I izz i)
zs +
S
I iz

Fig. A4 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
In FAGUS z is defined by the diagonal strut
flange
shear wall geometry (= user V
input), i.e. there is no auto
z M
matic import from the ben
ding analysis! N
tension
flange shear reinforcement
A sl2 A sl2

Fig. A5 Reinforced concrete truss model

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

S Ch. B 2.2 : Tabsheet 'Shear Walls'
General description of shear wall input
S Ch. B 2.4 : Tabsheet 'Variants' > 'V-T Models'
Control possibilities open / closed cross sections, etc)
S Ch. C 4 : Further explanations on determining shear wall forces.
Dimensioning requirements of different codes.

A-6 FAGUS7
 A 1 Overview

A 1.3.3 Elastic shear flow analysis

A quite different approach is available for cross sections that are made up of thin-walled el
ements. 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 de

V scribed 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 be
low with the following conditions:
S uniformly distributed loading
S uniformly or gradually changing thickness
S cross sections remain plane
In the other regions (nodes, geometrical Discontinuities, etc.), often additional considera
tions (truss models, stress fields) are often necessary.

corner of frame concentrated point load support

D B
D B
D

B beam with opening

gantry crane - corbel
B D B

Fig. A6 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 inter
pretation and application of the results. It is required that the results are randomly checked
using a simple calculation by hand together with plausibility considerations.

FAGUS7 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 work
manship 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 re
turned to Cubus. This limited guarantee does not apply if the malfunctioning of the soft
ware 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 prod
uct.

A 1.6 Codes

In accordance with the standard verifications carried out today, various limit states (ulti
mate 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 speak
ing countries certain differences exist, e.g.
SIA262 : g s = Widerstandsbeiwert fr Betonstahl und Spannstahl
(Resistance factor for ordinary reinforcing steel and prestressing steel)
EC2,DIN 1045-1 : g s = Teilsicherheitsbeiwert fr Betonstahl
(Partial safety factor for ordinary reinforcing steel)
As a basis for the program development, among others the following documents were
used
S EN 1992-1-1: Planung von Stahlbeton- und Spannbetontragwerken (Eurocode EN
1992-1-1: Design of Concrete Structures)
S SIA Normen 260,261,262, ...: Schweiz. Ingenieur- und Architekten-Verein (Swiss Codes
SIA260 -SIA267,etc.)
S DIN-Norm 1045-1: Tragwerke aus Beton- und Stahlbeton (German Code DIN 1045-1)

A-8 FAGUS7
 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:

S 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 dis
played 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).
S 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:
S Start der einzelnen Programme
S Verwaltung der Berechnungsdaten
S Verwaltung der Installation ber den CubusManager, der vom CubusExplorer aus auf
gerufen wird

Start CubusManager
zur Verwaltung der
Installation

Start der
Anwendungspro Verwaltung
gramme der aktu der Berech
ellen Generation nungsdaten

Start von Pro

grammen frherer
Generationen

A 2.2 Brief Description of the Most Important Points

. 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 pres
sing 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.

FAGUS7 A-9
Vol. A Introduction

Online Help:
Many of the dialogue windows that appear during input have a Help button. The corres
ponding link brings the user directly to the corresponding page in the PDF manual. (De
pending on the zoom factor the page may have to be moved up or down a little. )

By pressing the <F1> key, while the mouse pointer is situated over a button, the FAGUS
manual is opened (<F1> 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:
S Select button for new cross section; input desired name
1) New cross section S Work through the Control tabsheet sequentially from left to right (geometry, shear
walls, reinforcement)

2) Tabsheet selection

3) Cross section element

S 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.
S 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.

4) Select desired attributes

5a) 'Create'

A-10 FAGUS7
 A 2 Starting the Program - Brief Description

S 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.
5b) Graphics Editor Alternative tools of the Graphics Editor are available on the left side of the screen, whe
Tools reby 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 ...)

6) Input boundary A cross section boundary may consist of individual lines, of a closed polygon or of diffe
rent polygon sections. With [OK] the dialogue Modify cross section is closed and from
y 5, z 5 y 4, z 4
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.
y 7, z 7 y 2, z 2
Subsequent changes / object-oriented procedure
y 3, z 3
Existing input can be modified by clicking on the corresponding object and then the con
text 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
y 1, z 1 been modified, the changes are made effective by using the button 'Apply'. (Apply is only
y 8, z 8 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 sugges
tion 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 geo
metrical 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

FAGUS7 A-11
Vol. A Introduction

A-12 FAGUS7
 B 1 Overview and Management of Cross Section Data

Vol. B Working with FAGUS

FAGUS7

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 Exaggerated Status line

input fields scale

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 ana
lyse them manually.

FAGUS7 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 (possi
bly 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 ex
amples 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 sep
arate 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.
S 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)
S 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 command Recovery > 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 FAGUS7
 B 1 Overview and Management of Cross Section Data

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

Reinforcement/Checks:
S Reinforcement: Serves to manage the reinforcement diameters suggested by the pro
gram.
S 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 spe
cial 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 ap
plies, however, to the current cross section library. In the case of new analyses an installa
tion default value applies, which can be overwritten as follows (for each user, identified by
Login Name):
S Save as default value (with a corresponding open dialogue) or using the key combina
tion <Ctrl><F9>.
S 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
<Ctrl><F10> .
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 ele
ments 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 auto
matically, 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 de
scription 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 dis
played once more in full size. If one moves over the heading with the mouse, the whole dia
logue 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-spe
cific (not analysis-specific).

FAGUS7 B-3
Vol. B Working with FAGUS

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 al
lows 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 ver
sions, 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 lib
rary.
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 sec
tions, 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 manage
ment can be started both in STATIK and in FAGUS.

B-4 FAGUS7
 B 1 Overview and Management of Cross Section Data

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 re
named in the process.

FAGUS7 B-5
Vol. B Working with FAGUS

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 'Geo
metry'. 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 mem
ber'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. <RMB> 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 de
noted 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 dis
played 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 gen
eral 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 com
ponents (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 ad
ditional labelling.
For the corresponding tools to be available on the left side this layer has first to be 'activ
ated'. (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 FAGUS7
 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 :

Legend
Openings

Check
Handle (point)

Result combinations
Parametrised cross sections

Insert existing cross sections

PT Constraint lines
Axis point

Result points
Drawn partial sections

Thin walled partial sections

Cross section results

Rolled Steel sections

Selectable objects
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 attrib
utes. 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 proper
ties must be selected and then the geometry can be input with the aid of the Graphics Ed
itor.

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 sec
tion 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.

FAGUS7 B-7
Vol. B Working with FAGUS

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 corres
ponding 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. B1 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 some
times 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. B2 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 ana
lyses. The only exception to this rule is the determination of torsion constants, which is car
ried 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 FAGUS7
 B 2 Input of Cross Section Data

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 follow
ing 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 condi
tions to apply.

admissible inadmissible

Fig. B3 Examples of thinwalled 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.

FAGUS7 B-9
Vol. B Working with FAGUS

The default values for the different parametrised cross sections are contained in several Tab
sheets:
S concrete (= unreinforced cross sections )
S reinforced concrete (including V-T model to check program's default values)
S steel (standard steel sections)
S timber (various common timber cross sections)
S thin-walled sections (in addition to the standard steel sections)
S user (= folder with user-defined cross sections)
S 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:
S Scale: The cross section is shown to scale taking into account the specified parameters.
S 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:
S Identifier: The dimensioning lines are labelled with the variable identifier. (Thereby the
text of the underlying input fields is visible)
S 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 f
and s max to be active, firstly n u 0 must be set)

Modification of existing parametrised cross sections:

After 'introducing' the cross section, individual parameters can subsequently be changed
as follows:
S Double click on Dimension Line/Reinforcement directly in the normal FAGUS interface
(or with Dimension Line select, <RMB>, 'Properties').
S 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 reinforce
ment. 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 FAGUS7
 B 2 Input of Cross Section Data

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 sec
tion, 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 selec
ted from the list provided, it still has to be placed in the drawing area by inputting its co
ordinates.
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 illus
trated 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 cut
ting, 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 sec
tion. This may involve a possible loss of section (width of the section centred on the cut pos
ition).
Generate additional concrete: Depends on type of cross section (see drawing). Alterna
tive: Input with two partial cross sections (rolled steel profile plus concrete quadrilateral).

FAGUS7 B-11
Vol. B Working with FAGUS

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 intro
duced 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. De
tailed 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. B4 y-eccentricity by means of auxiliary line in cross section

B-12 FAGUS7
 B 2 Input of Cross Section Data

B 2.1.10 Result points

The echoprint of each FAGUS analysis contains a series of standard results. If additional res
ults 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:
S FAGUS interactive: For additional labelling of the cross section or for the tabular out
puts.
S FAGUS as post-processing program: In the corresponding tables and as graphics di
rectly on the STATIK structure.
A result point consists basically of two types (strain, stress, reinforcement area, etc.), posi
tion 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 reinforce
ment 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 under
lying 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' aufge
fhrt. In der Tabelle 'Resultatpunkte' wird jeder vom Benutzer eingegebene Resultatpunkt
in der folgenden Art und Weise ausgewiesen:
s(R1( 3.,4,2.5)) Lngsspanungen in der Bewehrung "R1" an der Stelle ( 3..4,2.5)
QS2
QS1
RP . ID: In connection with a STATIK reinforced concrete analysis it is recommended to use for
RP
a particular result the same ID in all cross sections. (In the results tables the individual co
lumns 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 sec
tion 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 auto
matically introduced standard result points can be made visible by means of the button on
the left.

FAGUS7 B-13
Vol. B Working with FAGUS

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 for
mula 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 inde
pendent 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 FAGUS7
 B 2 Input of Cross Section Data

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 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 ex
pression 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.
[alt.] : inserts the character :" for an Alternative 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 automatically before 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
'Variant'.

FAGUS7 B-15
Vol. B Working with FAGUS

B 2.2 The Tab sheet 'VT-Model'

Legend
Checks
Shear walls

Stirrups input

the objects
Selectability of
Result points

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 forces only in wall-directions Type B: Solid (compact) cross sections
centre of
com
pression
chord
z Shear wall
Shear wall polygonal Vz
circumfe bw
rence centre of
tension
chord
polygonal circumference
Fig. B5 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.
S 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. There
fore the shear walls of open cross sections do not need to be connected to each other.
S Type B: Solid cross sections
In order to reduce the input work somewhat, one can stipulate that for solid cross sec
tions in each shear wall four small internal shear walls are used to resist the torsion in
ternally.
S 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). (see also Torsion
values from basic cross section page 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 de
tected 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-16 FAGUS7
 B 2 Input of Cross Section Data

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 de
fined 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. B6 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 subse

quent 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 'G1')

Width of shear wall

Options

Assignment of stirrup to shear wall

Automatic generation of shear walls

Tab sheet 'Properties'

S 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 gene
rated stirrup reinforcement may be adjusted for a tapering form of cross section.
S Length of inner lever arm:
Normally this value is taken from the shear wall length. It can also, however, be input
explicitly.
S Assignment of stirrups to shear walls:
This assignment is not absolutely necessary for dimensioning purposes. Without the in
put 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, how
ever, 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 ex
isting stirrup cross section must be known.

FAGUS7 B-17
 Vol. B Working with FAGUS

Tab sheet 'Options'

S Weight factors for Type A:
For each shear wall an individual weight factor can be given
S 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: Vari
ants > Properties > V-T Model > Shear and Torsion Model

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 un
Vy
T known internal forces.

Vz The stiffness s of each shear wall is proportional to its moment of inertia.

s + Weightfactor

bh
3

12
Spring with stiffness s
Result check: The existing forces in the shear walls must be in equilibrium with the external
forces.

Si
V y + S
Si @ sin di
di T
V z + S
Si @ cos di
di T + S
S i @ d i
Vy
Vz
S i = Section force of ith 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.
d i = Inclination of ith shear wall (with respect to zaxis or yaxis)
d i = distance of ith shear wall from reference point

B-18 FAGUS7
 B 2 Input of Cross Section Data

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

y
T Vz Vy T
Vy x
Vz

Fig. B7 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 CE
DRUS 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 intro
duces automatically a wall in the orthogonal direction if this component is active.

Input Internal model

Shear wall activ for Vz Vy T

Vy, Vz, T Main action Secondary actions

With the input shown above it is clear that the three internal shears wall are belonging to
gether 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:
S Tab sheet 'VT-Model' > 'Typ Auto:'
S 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.
<shearwall name @ groupname>
.

FAGUS7 B-19
 Vol. B Working with FAGUS

S4 S1@G1

S5
S2@G1

S6 S3@G1

Fig. B8 Grouped walls S1 S2 and S3 (Pilar G1)

Additional remarks
. An elastic shear flow analysis also delivers resulting wall forces and could be used to com
pare 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
Fd flange to the web as shown on the left: For a vertical shear force Vz FAGUS only gives values
S for the vertical web stirrups. The horizontal web stirrups area could be obtained, e.g., as fol
lows:

F d ) DF d 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 (= stan
dard method for simple cases)
Vz
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 us
ing the function shown on the left for the whole cross section. Based on the current poly
gonal 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 se
lecting 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:
S Select Tab sheet 'V-T Model' > 'Type Auto: automatic'
S 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 cir
cumference 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 FAGUS7
 B 2 Input of Cross Section Data

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 pa
rameters 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 direc
tion should be set to 'inside'.

Automatic generation of stirrups

As with the shear walls this dialogue also provides the possibility of an automatic genera
tion of stirrups.

FAGUS7 B-21
Vol. B Working with FAGUS

B 2.3 The Tab sheet 'Reinforcement'

This Tab sheet serves to input untensioned and prestressed axial reinforcement.

Point reinforcement
Circular reinforcement
reinforcement

Line reinforcement
New reinforcement group
Delete group
Properties of group

ment
Auxiliary points to show

Distance between auxili

ary points for reinforce

Result points
Tendons

Adjust reinforcement to stirrup

Legend for reinforcement

Check

Selectability
List field to select the
reinforcement group

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 sec
tion 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 con
tains 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 rein
forcement input !
In the input field shown on the left the desired distances of the auxiliary points to the rein
forcement 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 nu
merical properties for the three types of reinforcement are similar.
point S 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 circular S 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 fol
lowing possibilities:

B-22 FAGUS7
 B 2 Input of Cross Section Data

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 influ
ence 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 num
ber 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 dimen
sioned (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 adjus
ted, i.e. the user input is overwritten!

Line
Single members
with end members
without end members

Automatic generation of axial rein Input of total area

forcement:
Input of total area using
- diameter
- distance
- number

Geometrical positioning with refe

For every automatically generated rence to the stirrup (see below)
stirrup two layers of axial re
inforcement are generated

Automatic modification of geometry:

The re-positioning is carried out either by explicitly clicking on the button on the left or au
tomatically after dimensioning with interactive selection of the axial reinforcement (see Ch.
B 3.2).
The three possibilities provided above have the following significance:
S none: the geometrical position of this reinforcement layer is not changed
S to the stirrup: the nearest stirrup is sought and the position is moved towards and nor
mal to the stirrup (path A)
S to the corner of the stirrup: the input reinforcement is lengthened, so that the end bar
lies in the corner (path B)

A
B

FAGUS7 B-23
Vol. B Working with FAGUS

Dimensioning strategies / Tab sheet 'Design':

Reinforcement attributes for dimensioning

The following dimensioning rules are possible at the present time:

S Constant area: The area of this reinforcement layer is not changed during the
dimensioning.
S 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.
S Dimension reinforcement in tension zone only
This option is meaningful above all for beam cross sections. The tension and com
pression zones within the cross section are determined on the basis of the current loa
ding before the actual dimensioning iterations on the homogeneous cross section.
If additional compression reinforcement is needed to achieve the required cross sectio
nal resistance, then for this purpose all existing layers in the compression zone are used
(Exception: Layers with the attribute Area remains constant).
S 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 re
inforcement area is increased proportional to the starting value. Thus at the same time with
this area a proportionality factor is defined.

Input Result of the dimensioning

1 cm2 8 cm2
2 cm2 16 cm2

Fig. B9 Reinforcement dimensioning: Adjustment according to starting values

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

B-24 FAGUS7
 B 2 Input of Cross Section Data

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 con
veniently 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 reinforce
ment. 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 import
ance 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
s strain-free cross section (x, y, z = 0). For beams constructed using the pretensioning
method this value can be input immediately:

sp
p + with s p = steel stress, E p = elastic modulus of prestressing steel
Ep
sp
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,
p
as well as on the prestressing conditions (elastic shortening).
In EC2 there is the following formula:

FAGUS7 B-25
Vol. B Working with FAGUS

P + P o * DP c * DP m * DP sl

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

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

DP m = prestress loss due to friction

DP sl = prestress loss due to slipping (pull-in) of wedges at anchorage

P = prestressing force for the analysis of the initial strain

The strain relation is as follows:

p + o ) D

where o is the existing strain in the prestressing steel in section x after prestressing and
D corresponds to the strains in the neighbouring concrete fibres due to elastic deforma
tion during prestressing.
Po * DPm * DPsl
o +
D A p @ Ep
* sc
D +
Ec
P
s c = concrete stress at the height of the tendon under consideration
E c = modulus of elasticity of the concrete

D: strains due to elastic deformation during stressing

The contribution of D 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 D are provided.
. Sign (algebraic) of D:
If the concrete is in compression, D 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 correspond
ing 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 as
sumed that during prestressing the beam separates (i.e. lifts up) from the formwork. A mod
ification can be effected using the parameter PREFAC" (see below and in the STATIK man
ual)

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 con
trast, the unbonded tendons retain their initial strain input by the user (and thus the pre
stressing force) independent of the current strain plane.

B-26 FAGUS7
 B 2 Input of Cross Section Data

Pretensioning method: Prestressing with cable in sheath

Pre-tensioning" Post-tensioning"
Before cutting the wires the cross section is Measured force at jack: P o
unstrained !
Po
p +
Po o +
ApE p A pE p

Strain change in prestressing steel

when cutting the wires (the cross sec Beam shortening due to prest
tion is now strained: ressing:

* Po
* Po D +
D + A cEc
A cEc
o + p ) D, p + o * D,

P o : prestress force in tendon A c
,
A p : concrete and steel cross

(tension positive) sections, resp.
sheath

Po Po Po Po

Fig. B10 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 com
ponent:

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 + Rneed 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 ooPo certain effects can be estimated.
Whether the analysis should be carried out at time t + 0 or time t + Rcan be specified
in the analaysis parameters. This higher-level control also applies to the analyses described
below.

FAGUS7 B-27
Vol. B Working with FAGUS

Directly from STATIK (postprocessing/reinforced concrete analyses)

For verifications at time t + R the prestress losses due to creep, shrinkage and relaxation
are considered as follows:

P R + P * DP(t)

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

The corresponding strain relation is:

R + o * cs * cc * sr

cs = final rate of shrinkage

cc = creep strain at height of considered tendon
sc
(g ) p)
cc +

Ec
= creep coefficient
s c = concrete stress at height of considered tendon due to permanent
loads (in STATIK defined by G1 and G2) and prestressing

sr = strain due to relaxation of prestressing steel

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:

s c
(g)p)
csR )
Ec
cs ) cc + s c(p) Ep
= k @ ( csR ) cc)
1* Ec @ sp @ (1 ) 2 )
csR = final rate of creep (input by user)
= creep coefficient (input by user)
s c
(g ) p) = concrete stress at height of tendon due to G1 and G2
s c
(p) = initial concrete stress at height of tendon due to prestressing
E c, Ep = 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 presen
ted 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] Ds Po
s Po A
12

10

8 B

6
C
4 4.5
2 2.5
1.0
s Po
60 70 80 [o/o] f pk
Fig. B11 Relaxation losses after 1000 [h]

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

B-28 FAGUS7
 B 2 Input of Cross Section Data

Simple example ('STATIK' > 'Help' > 'Examples', Materials acc. to SIA262 )
Po Po

Concrete cross section: A c + 0.16
m2, C20/25, E c + 30.4
kNm2, + 2
Prestress: A p + 400
mm2, Y1860, f pk + 1860
Nmm2 , E p + 195
kNm2 ,

0.75 @ f pk + 1395
Nmm 2 , P o + * 558
kN, o + 7.15
ooo

Concrete stress: s(P o) + * 3.49
Nmm 2
Shortening: D + * s c
E c + 0.114
ooo
Correction factor: k + 0.969

Losses (for initial parameters see table below):

Shrinkage : k @ cs o + 0.969 @ 0.27.15 2.7
oo
Creep : k @ cc o + 0.969 @ 2 @ 0.1147.15 3.09
oo
Relaxation : k @ Ds Pos Po + 0.969 @ 0.105 10.2
oo

Tendons: Initial strains and longterm losses

Parameters used (final values): creep =2 , shrinkage cs=0.2 [] , relaxation (0.75 fpk) = 10.5 [%]
Member Elem Distance Id Initial strains Longterm losses
o k Shrinkage Creep Relaxation Total
[m] [] [] [] [%] [%] [%] [%]
SL_1_0 1 0 SG1 7.2 0.1 0.97 2.7 3.1 10.2 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 + R.

. 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 be
low. 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 de
fault values):
S SHRINK=0.2
Final value for cs in [o/oo]
S 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 !
S 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.
S 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 !
S PREFAC=1
Multiplicator for the shortening part D:
When using this command the calculated values D are multiplied with this factor and
as confirmation the corresponding output column is labelled fD . Further information
may be found in the STATIK manual.

FAGUS7 B-29
Vol. B Working with FAGUS

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 par
ticular 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 sec
tions (for more details see Tab sheet "Attributes")

The Tab sheet contains the following buttons:

Legend

Check interpolated
Check variants

cross sections
Delete new variant

Properties
List field to choose
variant

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 corres
ponding 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 FAGUS7
 B 2 Input of Cross Section Data

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 rein
forcement rules (axial and stirrup reinforcement).
S Beams
In the case of a beam cross section, firstly the reinforcement on the tension side is incre
ased. Only if the compression side is governing, is it necessary (possibly) to increase the
reinforcement area on the compression side.
S 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).
S Wall
For dimensioning purposes a wall is treated like a column, but no stirrup reinforcement
is considered.
S Platte
Fr Plattenquerschnitte (bestehend aus einem rechteckigen Stahlbetonquerschnitt
und beliebig vielen Bewehrungen) wird ein Querkraftnachweis durchgefhrt, auch
wenn kein Stahlbetonschubwandmodell vorliegt. Bei den parametrisierten Qu
erschnitten 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 mem
ber 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 y shown below
is taken into accounft as an approximation of the geometry of the participating cross sec
tions. For individual analyses these can also be input directly as a cross section attribute.
The cross section analysis is then influenced as follows:
S Bending with axial force
The area of the axial reinforcement A s is reduced in the analysis according to the angle
y of the reinforcement rod as follows:
A s
red + As cos y
y is calculated for each reinforcement layer from the corresponding distance to the
edge.
S Shear force Vz
The modification of the shear force in the local z-direction due to haunching may be
clearly seen in the following figure.
Fc yo

DV
z N
zu

1
yu
Ft
Fig. B12 Section forces on haunched cross section

M 1 + M * Nz u
M
DV + z 1 (tan Yu ) tan Yo) ) N tan Yo

FAGUS7 B-31
Vol. B Working with FAGUS

. 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 me
thods (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:
S 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 in
ertia of the individual walls.
S 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.
A k
u
t ef
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 corre
sponding 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 Schubwnde eingegeben wurden, wird die Torsionbeanspruchung
durch Aufteilung in Scheibenkrfte behandelt. Sind alle eingegebenen Stahlbetonschub
wnde miteinander verbunden, so werden die Scheibenkrfte infolge V,T aus einer
elastischen Schubflussberechnung entnommen (gemss Kapitel C3: Dnnwandige
Elemente). In der Querschnittslegende wird dieses Modell wie folgt angezeigt:

Fr die so modellierten Querschnitte knnen die Schubspannungen am Querschnitt

dargestellt werden (gemss Kapitel LEERER MERKER).
Bei nicht zusammenhngen 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 con
tact 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
Si + T i
2Aef
. 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

B-32 FAGUS7
 B 2 Input of Cross Section Data

wever, can be made using thin-walled elements

Fig. B13 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.
S 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 in
fluence 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).
S 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 cor
responding list in STATIK.
S 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.
S Overwriting the cross section values: Since the values listed here GI x, GAy, GA z are
only approximately calculated in FAGUS, they can be overwritten by the user. STATIK
then uses this user-defined value.
S 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).

FAGUS7 B-33
Vol. B Working with FAGUS

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 de
termined 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. B14 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. B15 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:
S Haunching (see Fig. B-12)
S Prestressing (see slope of tendons, B 2.3.4)
S Consider shear force and torsion as internal axial force (Analysis Parameters)

B-34 FAGUS7
 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:
S Input section forces (with respect to axis point, centre of shear)
S Statically-equivalent section forces with respect to centre of gravity
S Modified section forces (due to haunching, prestressing, internal axial forces)

Fig. B16 Echo print including transformed section forces

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 coordi
nates of these two points are not the same the additional torsion must be taken into ac
count.

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') > command SHEARCENTER=ACTIVE

Vz

T
Vy
S S
M

Fig. B17 Lcross section with centre of gravity S and centre of shear M

. 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
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
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.

FAGUS7 B-35
Vol. B Working with FAGUS

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 force, as well as for shear force and torsion. In the case
of combined loading the following analyses are carried out:

S Dimensioning of the axial reinforcement for bending with axial force

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

S 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 act
ive, the dimensioning is peformed successively for all loadings. The result of each dimen
sioning step is used as starting value for the next dimensioning step.

List field to select cross Management of

section variant member diameter

New input line Delete 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 Interactive

of the reinforcement reinforcement adjustment
areas

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 accord
ing 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-36 FAGUS7
 B 3 Design and Analysis

Automatic adjustment of the required reinforcement areas

The dimensioning provides the required reinforcement area, which corresponds to an effi
ciency factor = 1. The efficiency depends among other things on the selected analysis para
meters and also, of course, on the given loading. At the end of the dimensioning the rein
forcement 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:
S For distributed line reinforcement layers the result is output in the form (As = mm2).
S For point reinforcement, the required reinforcement area is converted into a (fictitious)
diameter,e.g. 4
f
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 effi
ciency factor of exactly1.

Interactive selection of diameter and adjustment of the layer

Firstly, the required reinforcemement areas are determined as described above. Then fol
lows 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 stir
rup is sought and the reinforcement is adjusted according to the input attributes (Ch.
B 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 configura
tion. 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 ad
missible 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:

S 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!
S 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 cross
sections are dimensioned automatically under "uniaxial bending".

FAGUS7 B-37
Vol. B Working with FAGUS

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 geo
mety of each layer. For beam cross sections it is recommended to input at least one line re
inforcement 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.
S Circular line reinforcement is dimensioned for tension and compression.
S 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 recom
mended 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
S Shear wall forces (for details on analysis see C 4.1)
S In the graphical presentation a positive shear wall force points from the starting point
to the end point of the shear wall
S Efficiency of concrete compression diagonals (column with the heading eff c)
S Required stirrup area (column with the heading a sw )
Stirrup dimensioning for different codes is described in Ch. C.
S Required additional axial reinforcement for each shear wall side: (heading A sl2 )
In compliance with the common reinforced concrete codes, for the dimensioning of the
stirrups the classical truss model shown below is used (a, q from EC 2):
z(cot
a ) cot q)

Fc

z
Fw V + Si
Stirruptensionforce Concretecompressionforce
a q
Fs
Fig. B18 Truss geometry

B-38 FAGUS7
 B 3 Design and Analysis

The corresponding descriptions in the codes are usually based on a single web, i.e. the crit
ical in-plane force in this case is the same as the corresponding shear force

S i + Vsd

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 res
ults in the following tensile force to be resisted per unit length F w.

V sd
Fw +
z sin a(cot
q ) cot a)

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 for
mula:

Vsd
s w
A sw +
zfywd cot q

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

A V sd 2
a sw + s sw + (unit: mm or selected by user)
sw zf ywd cot q m

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

V sd dimensioning value of the in-plane force: is given by the FAGUS shear wall model
z inner lever arm defined by the length of the corresponding shear wall.
f ywd dimensioning value of the elastic limit of the stirrup reinforcing steel: from the assi
gned stirrups or fromthe standard default value: > Variant Properties > Models
q slope of the concrete compression diagonal: Analysis Parameters > Further Values.
a 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 differ
ent 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 VRd2

The determination of V Rd2 depends on the code and is described in Part C. For practical rea
sons in FAGUS the efficiency factor is always output.

VRd,max
eff c + v 1.00
Vd

FAGUS7 B-39
Vol. B Working with FAGUS

Axial reinforcement due to force in shear wall

. Basically, the shear force analysis is completely separate from the dimensioning for ben
ding. Regarding output two settings are possible in the Analysis Parameters:
S Standard Method
A sl2 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.
S 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 gsF s
F s + 0.5Si(cot
q ) cot a) A sl2 +
fy
A sl2
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 reinforce
ment area due to M,N. The additionaly required value A sl2 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 de
termined as follows:

F si + Si(cot
q ) cot a) (sum of all tensile axial forces in the shear wall)
N * + N ) S
F si
M *y + M y
)
S
Fsi

Dz si
M *z + M z
)
S
Fsi

Dy si F si : normal force contribution of the i-th shear wall
Dy si
,
Dz si : distance from middle of shear wall to centre of gravity of cross section.

Fig. B19 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.
Tu ef
N(V y,Vz, T) + V 2)


y V z (cot
q ) cot a) ) cot q
2
2A 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 re
quired reinforcement areas is presented.

B-40 FAGUS7
 B 3 Design and Analysis

The curve A results from the analysis for bending with normal force. If there are several rein
forcement layers, the reinforcement areas for each half of the beam is summarized and writ
ten on the corresponding half. Curve B still contains the A sl2 portion of all shear walls
Dx z cot q
2
initial value determined in
point 1 and horizontally in
1 2 creased in arrow direction
Asw1 (stirrups) tensile flange forces:
to point 2.
contribution of shear force
1 2
support fans 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) ) S
A sl2

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 sl2 is used for the sum
mation (earthquake analyses, etc.).

S
A sl2 + Maximum(S
A sl2 top, S
A sl2 bottom)

As total(M, N, V) + As total(M, N) ) 2S
A sl2

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 + 0.5Vz(cot
q ) cot a).
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
S
A sl2.

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

curve determined using standard method

FAGUS7 B-41
Vol. B Working with FAGUS

B 3.3 The Tab sheet 'Analyses'

All cross section analyses are carried out in this Tab sheet.

Shear flow and

moment-stiffness diagram
curvature diagram
Stress analysis (strain) and moment-

Generate serial analysis and report

Efficiency and M-/N interaction

general interaction diagram

Stress analysis and
List field for selection of desired
cross section variant

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 pa
rameters.

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
Rd

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 v 1.00. Depending on the se
lected analysis parameters (ULS, SLS) both the ultimate load and the serviceability are veri
fied.

. Comparison with earlier versions of FAGUS:

In earlier versions of FAGUS an ultimate load factor l was determined. The relationship bet
ween eff and l is:
eff + 1
l

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 FAGUS7
 B 3 Design and Analysis

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 resist
ance 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.

Yd (lX d
, Y d)

X
Xd

Fig. B20 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 pa
rameters 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 correspon
ding 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 ob
tained 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 recom
mended 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 res
ult includes the Min/Max values, which, e.g., are required for fatigue verifications.

FAGUS7 B-43
Vol. B Working with FAGUS

. 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 accord
ing 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)

w k + s r,max
( sm * cm)

where
fct,eff
ss * kt
p,eff (1 ) a e p,eff) ss
( sm * cm) + 0.6

Es Es

ss = steel stress
k 1
+ 0.4 (for long-term loading)

f ct,eff = average value of the effective concrete tensile strength

f ct,eff + fctm or smaller, if cracking is expected before 28 days
the corresponding reduction factor can be input in the Tab sheet Analysis Paramet
ers

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

Es
ae = ratio
E cm

The maximum crack spacing is:

s r,max + k3
c ) k1
k 2
k 4
f p,eff

with the following values

k 1
+ 0.8 for reinforcement bars with good bonding
k 2
+ 0.5 (for pure bending)
k 2
+ 1.0 (for pure tension, lies between k 2
+ (1 ) 2)2 1
From the national appendix
k 3
+ 3.4
k 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 c,eff + Minimum
[
2.5(h * d)
;
(h * x)3
;
h2
]
If several different diameters are encountered, the following conversion is peformed to find
the equivalent diameter:
n 1f21 ) n 2f 22
f+
n 1f 1 ) n 2f 2

B-44 FAGUS7
 B 3 Design and Analysis

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:
As ) c 21
A p
p,eff +
A c,eff

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

ff
c1 + c
p
s
(7.5)

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

for bundles (Default): f p =1.6 A p

for single 7 wire strands f p =1.75
f wire
for single 3 wire strands: f p =1.2
f wire (f wire = wire diameter)

B 3.3.3 Stress analysis with strains

With the stress analysis, with a given strain plane the three components x, y, z 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 de
termined 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 (y,z) in the integration points of the cross section

(y, z) + x * z @ xy * y @ x z

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

s(y, z) + f ((y, z))

summation over all "fibres" of the cross section

Nx + s(y, z)
dA
M + s(y, z)
z
dA
y

M + s(y, z)
y
dA
z

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:

FAGUS7 B-45
Vol. B Working with FAGUS

My M
EA + N
x EI y + x EI z + x z
y 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 (, x). 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 deforma
tions 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 220.56 kN

12.82 N/mm2
c=1
s=1

105.58 N/mm2
0.54 o/oo
10.56 kN

Fig. B21 Stressstrain 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 quantit
ies 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 cou
ple 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:
S A: Thin-walled cross section: Cross section consists solely of thin-walled elements
(building one well connected structure)
S 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. (VTmodel 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 dis
played 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 cal
culated with the usual formula shown below (The shear flow theory is explained in chap.
C 3.):

B-46 FAGUS7
 B 3 Design and Analysis

My M
s(y, z) + N ) z ) z y
A Iy Iz
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. B22 Normal stresses due to M,N

Integrated shear wall force

= Summation of distributed
shear stresses.
Shear stresses or shear flow

Fig. B23 Shear stresses due to V,T

With the button shown on the left an equivalent stress s + f (s, t) is shown: If the cross sec
tion 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: s v + s 2 ) 3t 2



s ) s2 ) 4t 2

Rankine: s R +
2
Reference point for the section forces:
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]) or shear 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 correspond
ing 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).

FAGUS7 B-47
Vol. B Working with FAGUS

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.
N In the case of non-symmetrical cross sections, with the strain planes for y0 and z=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 Interac
tion Diagram".
In the creation of a diagram a reinforcement content can be input. The existing reinforce
M
ment area is then scaled accordingly, i.e. the reinforcement areas of different input rein
forcement 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. Ar
eas 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 an
interaction diagram is created with the existing cross section (independent of input rein
forcement).
If no reinforcement content is input (incomplete or empty input line), the diagram is cre
ated for the reinforcement already existing in the cross section. With several values (separ
ated 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 in
put 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 compon
ent is constant, whereby a value not equal to zero can also be input. It is also possible to in
put several values (separated by blanks), and then in the same diagram the corresponding
curves are displayed
M The analyses are then carried out with the reinforcement already existing in the cross sec
z
N=konst. tion, 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
M
y 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 re
quired accuracy suited to his particular needs.
The input values for the component held constant must lie within admissible ranges, other
wise 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 val
ues 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 us
ing the slider. For larger numbers the curves are more rounded, but the analysis takes
longer.

B-48 FAGUS7
 B 3 Design and Analysis

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.
My The program then determines the individual points of the curve with the aid of a stress veri
fication.
My [kNm]

N = 100.0 200.0 500.0 kN

&500.0
500

o
xy _
400

&200.0
o
_
300

o &100.0
_
200

100

& = = 20.0 o/oo

o = Zugrand: 5.0 o/oo
= f ct = 0 N/mm
_ = + y = 2.19 o/oo
= y = 2.19 o/oo

0 2 4 6 8 10 12 14 16 18
[km1]
Fig. B24 Momentcurvature diagram

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

S reaching the concrete's tensile strength (if input)
S reaching the elastic limit in the extreme steel layers on the compression side
S reaching the elastic limit in the extreme steel layers on the tension side
S reaching the limiting compression strain" at the edge on the compression side defined
in the analysis parameters
S reaching the limiting strain in the steel at the edge on the tension side defined in the
analysis parameters
S 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.

FAGUS7 B-49
 Vol. B Working with FAGUS

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
EI y between the moment My and the secant stiffness EI sec for one or more given normal/axial
forces.

EI sec + M
x
If the cross section contains tendons, in general the moment My already differs from zero
My for a curvature x=0 , i.e. the curve begins at EI sec + R. For cross sections with prestrained
tendons, therefore, no diagram is created. To deal with such cases it is recommended to in
clude the prestressing force in the analysis as an external loading and create a diagram with
a corresponding axial force (without prestrained reinforcement layers).
By means of symbols on the curve it is indicated when the extreme steel layer begins to
M
yield or when the limiting strains are reached.
To compare different cross sections, it may be useful to output directly an (ideal) secant
arc tan EI sec E-modulus. This is defined as follows:
E sec + M
x I@x

I = moment of inertia of the cross section according to Ch. C 2.1.1

The program performs the analysis for a single point of the curve following the stress ana
lysis 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:

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

For N 0 0
and
M + 0  x 0 0 (N, M and
x 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 is either performed for bending 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-50 FAGUS7
 B 3 Design and Analysis

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 l k : Buckling length
Imperfection
e o : Eccentricity due to imperfection
e0 e 1 : Eccentricity due to an end moment (calculated with e 1 = M/N)
The procedure described below is suitable for isolated members with a constant normal
lk e2 force, with a constant cross section and a well defined buckling length l k as well.
Background / Theorie based on:
SIA 262: 2013 : 4.3.7 Druckglieder
EN 1992-1-1:2004 : 5.8.8 Method based on nominal curvature
Iteration with one degree of freedom: The maximum normal force is determined with an
iteration procedure where the maximum 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 sec
tions.
l2
e + e 0 ) e 1 ) e2 e 2 + x  ck c + p2

The calculation of the curvature is carried out with the real M-N-combination and the app
y lied values can be checked and compared with a normal cross section stress analysis, provi
ded that the same analysis parameters are used.
This is more accurate than certain methods where simplified techniques are used to calcu
late 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 My-
N-diagram.

Fig. B25 MyNdiagram with loading path

Elevated temperatures (fire resistance)

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

FAGUS7 B-51
Vol. B Working with FAGUS

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

Normalkraft
Moment

Fig. B26 Temperature dependend momentcurvate diagram and

corresponding MyNDiagram 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 in
ternal members (with additional integration points) to represent the variable stiffness
more accurately. The static system has six degrees of freedom per node. In general 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 dif
ferent directions etc.) only PYRUS can deliver the correct result.

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

Column with normal force Deflection Stiffness EJ

e2 In FAGUS:
Assumption that the stiffness is
constant along the column

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

B-52 FAGUS7
 B 3 Design and Analysis

B 3.4 Batch Analyses

The functions presented here should help if frequently repeated analyses with reduced in
put are carried out. Batch Analysis" refers to the stepwise, automatic working though of
analyses and cross section dimensionings. The following tasks can be solved:
S Automatic repeat of unaltered types of analysis on different cross sections.
S Investigation of construction states with active/inactive partial sections and with the ac
companying loading history.
S Creation of internal stress states.
S 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 ana
lyses. 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 in
spected in the CubusViewer.
Print entry Active/Inactive, strains
Load cross section

Analyses
Manage
ment 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 dir
ectly 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).

FAGUS7 B-53
Vol. B Working with FAGUS

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

<engl. command> <parameter>

is more or less self-explanatory. For a specific task it is best to create a small Test Task inter
actively and the subsequent modifications can then be carried out by Duplication/Mo
difying 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:
S Creation of a new Batch Analysis. The identifier is input in a separate dialogue
S Change the identifier of a Batch Analysis
S Delete a Batch Analysis
S Save
S 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) but
tons 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 ana
lyses 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 in
put lists with the aid of a Text Editor).

B-54 FAGUS7
 B 3 Design and Analysis

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 spe
cified 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 ele
ments 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 strains i. 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, sev
eral strain instructions (inputs) may be necessary and only then should the analysis be per
formed 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

FAGUS7 B-55
Vol. B Working with FAGUS

B-56 FAGUS7
 C 1 Coordinate System and Sign Conventions

Vol. C Special Topics

FAGUS7

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 co
ordinate system. For the sake of compatibility with the program STATIK the x-axis lies paral
lel to the member axis, i.e. normal to the cross section and points in the figure below to
wards 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 b (range of values:
* 45 b v 45 , signs according to the right hand rule, i.e. in the diagram below it rep
resents 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.
zq
zh z
y q
,
z q Input coordinate system of the cross
section
(= coordinate system of the FAGUS-
Graphics Editor)
yh y
,
z Local coordinate system of the cross sec
tion
or of the member (section forces, strains,
zs S b loads)
y y h
,
z h Principal axes system of the cross sec
tion
(= internal reference system for all
ys yq STATIK- analyses)
S Centroid

Fig. 1 Cross section coordinate systems

. Remark: For an angle b 0 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).

start of Mz
member

My Vy y

S N tension positive

T
Vz x

Fig. 2 Member coordinate system and section forces

FAGUS7 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 relation
ships between the kinematic quantities and the section forces are as follows:

du + N , d 2w + M y , d 2v + * M z , dr x
+ T
dx EA x dx 2 EIy dx 2 EI z dx GIx

u, v, w : displacements in the corresponding coordinate directions x,y,z

rx : 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)
Ax : area of cross section
I y,
Iz : 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:
Vy Vz
gy + , gz +
GAy GA z
g y , gz : shear strains
V y , Vz : shear forces
A y,
A z : reduced cross section areas: A y + ayA x , A z + azA x (see also C 3.1)
FAGUS assumes for arbitrary cross sections: a y + az + 1
The two material parameters E and G 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 (EA x, EI 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 pro
gram therefore allows the overwriting of these three values directly as part of the cross sec
tion/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 dif
ferent 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:
Ei
Weighting: wi +
Eref

Area: dA
A xi + w i

Moment of inertia: I + w z
dA

yi i
2

C-2 FAGUS7
 C 2 Section properties

Moment of inertia: y
dA

I zi + w i 2

ydA
zdA
Centroid: y + , z +
dA dA
si si

For the complete cross all subsections are summed up:

Area: A x + S
A xi
Moment of inertia: I y + S
Iyi , I z + S
Izi
S
y si
A xi S
z si
A xi
Centroid: ys + , zs +
Ax Ax
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 val
ues 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 b 0 0 in
an extra column entitled ... rel. to input coordinates the values Iy *, Iz *, Iyz * are output
with respect to the local coordinate system y, z of the cross section.

Reference material
By default FAGUS selects the material with the lowest weighting factor as reference. Da
durch sind die Gewichte der Teilquerschntte immer 1 v w i . Fr einen Stahl-Beton-Ver
bundtrger wird der Beton damit standardmssig 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 + Eref @ Iy(E ref)
Dazu ist In Bezug auf die Schubsteifigkeiten folgendes zu vermerken:

A z variiert zwar linear mit E ref, der verwendete Schubmodul ndert sich aber nicht im glei
chen Mass, da die Querdehnungszahlen (insbesondere von Beton und Stahl) der beteilig
ten Baustoff unterschiedlich sind.
Schubsteifigkeit allgemein : GA z + G ref @ Az(E ref)
Schubsteifigkeit im Speziellen : G Beton @ A z(EBeton) 0 G Stahl @ A z(E Stahl)

C 2.1.2 Torsional Constant

1 The relationship between torsional moment and the derivation of the member rotation
about its axis is described by the following relationship:

dr x
2 + T
dx GIx
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 calcu
lated 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.

FAGUS7 C-3
Vol. C Special Topics

Cross section without openings:

Ix Model No. Type of cross section Torsional constant for STATIK
1 rectangle Ix=1/3bh3 (10.63h/b+0.052 ....)
2 circle Ix=0.5r4
3 thin-walled (t < 0.1 l) Ix=1/3 l t3 approximation!!
4 compact Ix=0.5r4 approximation!!
5 steel section Ix=from steelwork tables

3 Remarks on Ix Model No. 3:

For polygonal subsections with a large perimeter (in comparison to area) l and t can be es
timated 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

For arbitrary polygonal cross sections and without further information the program auto
matically assigns each subsection to one of the above cases, whereby the dimensions of the
5 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 neces
sary replaced by an improved value supplied by the user, especially if the size of the tor
sional 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 contrib
ute to Ix (i.e. no reinforcement)

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

A 2o A2
I xBredt + 4 4 uo

7
ds t
t
u = circumference, t = (assumed constant) wall thickness of the equivalent box section
Cross Section with one or more Openings:
Ix Model No. Cross section type Torsional constant for STATIK
6 Axopeni ng < 30 % Axtotal Ix = Ixtotal Ixopening approximation
7 Axopeni ng > 30 % Axtotal Ix = IxBredt 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 indi
vidual subsections (1..7).
8 Thin-walled cross sections
St. Venant's torsional constant for open, thin-walled cross sections is given by the summa
tion 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" circum
9 ference. More on this can be found in Ch. C 3. In summarising it can be said that a cross sec
tion built up of thin-walled wall elements gives correct values for Ix! If therefore the tor
sional 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 FAGUS7
 C 2 Section properties

Ix Model No. Cross section type Torsional constant for STATIK-5

8 open cross section Ix = IxSaintVenant (see Ch. C 3)
9 box section Ix = IxBredt + ( IxSaintVenant ) (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 zusammenhngenden polygonal umrandeten Teil
querschnitt (mit optionalen Aussparungen) bestehen, kann die Torsionskonstante nach St.
Venant aus einer FE-Analyse bestimmt werden. Die entsprechende Schaltflche befindet
sich im Register 'Varianten' > 'Varianteneigenschaften'.
Die numerische Berechnung basiert auf folgender Theorie:
Mit der Einfhrung einer Spannungsfunktion f wird das elastische Torsionsverhalten eines
homogenen Gebietes mit folgender Gleichung beschrieben:
2f 2f
) 2 ) 2Gq + 0
y 2 z

q = Drehwinkel pro Lngeneinheit

G = Schubmodul
wobei folgende Randbedingungen gelten:
Aussenrand: f+0
Aussparungen: f +konstant entlang einer bestimmten Aussparung
Mit Hilfe der Spannungsfunktion f knnen die Schubspannungskomponenten in einem
beliebigen Punkt angegeben werden:
f f
t xy + t xz + *
z y
Die Schiebung ist
t xy
g xy +
G
Das Torsionsmoment ergibt sich aus der Integration ber die gesamte Querschnittsfl
che A:

T+2 f
dA
A

q + T fr q = G = 1  Ix + T
GI x

FAGUS7 C-5
Vol. C Special Topics

C 3 Analyses with Thin-Walled Elements

Bei einem dnnwandigen Querschnitt ist die Dicke jedes Elementes im Vergleich zur Lnge
klein und die Variation der Normalspannungen ber die Dicke des Elementes ist deshalb
vernachlssigbar.

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 fol
lowing relation ("Bisquit" formula):
V
S(z)
t(z) + V = shear force, S= stat. moment, I= moment of inertia
b
I

s
s ) ds

z


z

x y

dx b
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 g(z) in in a par
ticular section (see above) is:

t(z)
g(z) + , G = shear modulus
G
For the whole cross section one assumes an average shear strain g m. The corresponding
term in the work equation (virtual work per unit length of member dx) is:

VV = g V
where gm + V
m
GA * GA *

g m depends on G, V and the reduced area of cross section A *. With the help of the above
formulas one can then find for A * the following definition:

g mV
= g t
dA = VV

m
GI
Sb dA
2
2
2
1 +1
A* I2
Sb dA
2
2

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

C-6 FAGUS7
 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 as

sumed 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:
M S
S Each load passing through the shear centre stresses the beam according to classical
V V bending theory with bending without rotation.
S If the load does not pass through the shear centre, it can be split up into two partial
eM
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 + Ve M (e M = dis
tance between shear centre and centroid), which stresses the section shown on the left
for bending in the flanges.
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 pro
cedure 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 and S 2 about this point
vanish. Then we introduce a torsional moment Ve 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 ex
amples.

Normal stresses Shear stresses

Geometry (1) (2) (3) (4)

S1

S1

+
S1


=
es S2

M S h S2
V T V
eM



S1


b
S1
S1

Section width: b + 0.40

Height: h + 1.00 V + 1.00 T + * 0.23 V + 1.00
S 1 + 0.14117 S 1 + 0.23 S 1 + 0.0889
Centroid: e S + 0.0889
S 2 + 1.00 S 2 + 0.00 S 2 + 1.00
Distance M-S: e M + * 0.23

Fig. 4 Numerical example for Usection, shear flow and section forces

FAGUS7 C-7
Vol. C Special Topics

By integrating the shear flow one obtains for each wall element a resulting force in the sec
tion. 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 + hS 1 * e sS2 + 1.000.0889 * 0.08891.00 + 0.00 (O.K.)
Only with open cross sections can the shear stresses be directly determined using the
Vz 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).
Vy

a t
a
d
1.0 S + Gt d
S S l

Fig. 5 Determination of shear flow for closed cross sections

C-8 FAGUS7
 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
Ts

Fig. 6 Shear stresses according to St.Venant

Normal stresses Shear stresses

s t
S2

Tw
Resultant flange
forces
s S1


t

Conditions:

N=0: s
dA + 0
My=0: sz
dA + 0

Mz=0: sy
dA + 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 + S 1
lt 3 l, t : length, thickness of thin-walled element

3

FAGUS7 C-9
 Vol. C Special Topics

Warping Constant
For all open cross sections, besides the shear centre also the warping constant (warping re
sistance) is output:
(s)
w
dA (s)
ds
s+l
Iw + w(s) +
2

s For more on this see, e.g.:
S Chwalla, E.: "Einfhrung in die Baustatik", Stahlbau-Verlag
s+0
S 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:

A 2o
I xBredt + 4
ds
t

For multiple cell sub-sections the torsional constant and shear flow are given by the com
T patibility 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 + Ix
St.Venant ) Ix
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 de
pends on whether in fact in the cross section input one or more closed hollow cells are de
fined (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 follow
ing 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
29.40

0.5 0.5 0.5 0.5

0.6

26.00 26.00 26.00

Fig. 8 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 FAGUS7
 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
A sw V sd
a q s w + fywdz(cot
q ) cot a) sin a

A sw = cross-sectional area of shear reinforcement

sw = spacing of stirrups
a = inclination of stirrups (a= 90_ = vertical stirrups)
q = inclination of concrete compression strut
f ywd = design value of strength of shear reinforcement at elastic limit
fy
f ywd + g
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 conside
ration" 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 re
sistance. 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 + acw
b wzn 1

fcd
(cot
q ) cot a)(1 ) cot 2 q)

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

S Recommended value: n 1 = n
f ck
n from National Annex or n + 0.6 * ( f ck in N/mm2 )
250
S If the design stress of the shear reinforcement is below 80% of the characteristic yield
stress f yk, n 1 may be taken as:

n 1 = 0.6 for f ck v 60Nmm2 (6.10aN)

f
n 1 + 0.9 * ck w 0.5 for f ck w 60Nmm2 (6.10bN)
200

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

For non-prestressed structures:

FAGUS7 C-11
Vol. C Special Topics

a cw = 1.0

Prestressed structures:
s cp
a cw = (1 ) ) for 0 t s cp v 0.25
fcd
f cd
(6.11aN)
a cw = 1.25 for 0.25
f cd t scp v 0.5
f cd
(6.11bN)
s cp
a cw = 2.5(1 * ) for 0.5
f cd t scp t 1.00
f cd
f cd
(6.11cN)
. The values n 1 and a 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 fck f yk
(9.5N)
Summary of input parameters:
Analysis parameters (> 'Options' > 'Analysis parameters')
: in tabsheet > 'Miscellaneous' > ...
w,min : in tabsheet > 'Reinforcement' > ...

Cross section input

: 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 betrgt:

V Rd,c + [CRd,c @ k @
100 @ l @ fck  ) k1 @ scp]
b w @ d

13

(6.2.a)

Minimum
V Rd,c + [v min ) k1 @ scp]
b w @ d

C Rd,c + 0.18 gc

k + 1 ) 200

d v 2.0
A sl
l + v 0.02
bw
d
n min + 0.035 @ k  32 @ f 12
ck

f ck : charakteristische Zylinderdruckfestigkeit

C-12 FAGUS7
 C 4 Shear and Torsion for Reinforced Concrete

C 4.1.2 SIA 262 Swisscode

Stirrup
z
The stirrup area per unit length is:
b a
A sw VRd,s
s w + fsdz sin b(cot
b ) cot a) (262.38)

Check of concrete stresses:

V Rd,c + bw @ z @ k c @ f cd @ (cos a ) cot b sin a)
sin a (262.39)

FAGUS uses k c + 0.6 (262.4.2.1.7) ).

The efficiency eff c is defined as

VSd
eff c +
VRd,c

Minimum reinforcement
a sw,min + 0.2%bw , b w v 40cm

C 4.1.3 EHE-08 Spanish Code: Instruccin de Hormign Estructural

EHE-08 is based on Eurocode with following national adaptions:

z
0.5 cot
q 2.0
a q
Check of concrete stresses: see Eurocode

Shear force resistance

V u2 + Vcu ) Vsu

V cu = Contribution of concrete

In general cases V cu is not easy to evaluate as it is dependend on the longitudinal reinforce

ment (e.g. dependent on , c 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
predesign phase. Option 1 und 3 will be available in future versions.

V su = contribution of stirrups : V su + A sw
f ywd
z
sin a(cot
q ) cot a)

Stirrup area per unit length

A sw V sd * V cu
s w + fywdz(cot
q ) cot a) sin a

FAGUS7 C-13
Vol. C Special Topics

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:

b a

Checking stresses in concrete

A sw is only output if the stresses in the concrete compression diagonals
fulfil the following condition:
t R + fc
,
red
(cos a ) cot b sin a) sin a ( f c
,
red after 3 24 211)

f c
,
red + z sin b
fcw , min z = 0.4

Check at design level

t
t Rd + gR

VSd
t Sd + t Rd
bwz

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)

The inclination of the compression diagonals following Section 17.5.4 is generally ussumed
z 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 ) may not be reduced.
b + 45
One distinguishes between:
Shear Region 1 (region with minimum reinforcement: t t 012)
The design value of the action is:
t nom + 0.4t o
This value corresponds to a minimum shear reinforcement of
0.4t fy
m+ s o (s s = permissible steel stress = , = reinforcement content)
s 1.75
Normally, the minimum reinforcement is determined by the program for the material qual
ity used. For special analyses this value can be overwritten by the user.
Shear Region 2 (reduced shear cover: t 012 t t 02)
0.4t
t+ s o
s

t2
t + t o w 0.4t 0
02

i.e. for the design the reduced shear force is

V + tb wz
Shear Region 3 (full shear cover: t 02 t t 03 )

C-14 FAGUS7
 C 4 Shear and Torsion for Reinforced Concrete

In this region the whole shear force has to be resisted by the shear reinforcement alone.
A sw V
s w + z

ss
sin b(cot
) cot b) (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.
85t 1b 0
a sw,min + ss

C 4.3 Design for torsion

Torsion models Type A
V 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, there
fore, proceeds according to the rules discussed in the previous section.
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:
Tu ef
A sl + cot a
2

A ef

f yd
t ef u ef = circumference of co-acting cross section area with torsion
t ef A ef = co-acting cross section area with torsion (= width height of shear wall)
a = slope of concrete compression diagonal (a = 45 according to DIN 1045)
A ef f yd = design value of strength at the yield limit of axial reinforcement
The shear stress due to torsion is:
t+ T
A *sw 2

A ef

t ef
t ef = co-acting (equivalent) wall thickness with torsion
do
according to SIA : t ef + , d o + min(b, h)
8
according to DIN: t ef only indirectly specified, in FAGUS limited to d o6

according to EC2: t ef + A
u=
A
2(b ) h)
The required stirrups area is

A *sw + T tan a
2

Aef

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

A sl2 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

A sl2 A sw + 2

A *sw : total area of stirrups element (both top and bottom sides of beam)

C 4.3.1 berlagerung Querkraft und Torsion

Die Betonausnutzung wird fr alle Normen mit folgender Formel nachgewiesen:

FAGUS7 C-15
Vol. C Special Topics

T ed VEd
) v 1.0 EN 1992-1-1 (6.29)
T Rd,max V Rd,max

Nach DIN 1045-1 drfen die beiden Terme fr 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 V T ed V Ed
effc = Maximum { ed 2 ) Ed 2 , , )
T Rd,max VRd,max T Rd,max V Rd,max

C 4.3.2 berlagerung Querkraft und Torsion

Die Betonausnutzung wird fr alle Normen mit folgender Formel nachgewiesen:
T ed V
) Ed v 1.0 EN 1992-1-1 (6.29)
T Rd,max V Rd,max

Nach DIN 1045-1 drfen die beiden Terme fr 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 V T ed V Ed
effc = Maximum { ed 2 ) Ed 2 , , )
T Rd,max VRd,max T Rd,max V Rd,max

C-16 FAGUS7
 C 4 Shear and Torsion for Reinforced Concrete

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 Cross section Distribution of load among the shear walls
calculation of Asw1

Asw1:=
Vy Vz T

Torsion model:
AswMin(SWZ)


OR

Asw(SWZ)+Asw(SWT)

Z
SW
Asw1

OR SWY SW
Asw(SWY)+Asw(SWT) T

SWY
SWY

AswMin(SWZ)
SWZ

OR

SWZ
Asw1
Asw(SWZ[Vz])+Asw(SWZ[T])
B

SWY2



SWY2

AswMin(SWZ)
SWZ

SWZ

OR
Asw1
Asw(SWZ[Vz])+Asw(SWZ[T])

SWY1
SWY1
B

SWT1 .. SWT8
Asw(SWY[Vy])+2*Asw(SWT1[T])



SWZ

OR Asw1
SW

Asw(SWZ[Vz])+2*Asw(SWT1[T]) Y
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 Combination of different shear walls for some simple cross sections

FAGUS7 C-17
Vol. C Special Topics

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 dif
ferent 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-T S-TV S-TV2 S-I S-IV S-O S-OH

S-L S-Z S-U S-UV S-RH S-RH2 S-C

Fig. 10 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 T-I T-I2 T-I3 T-I4 T-HO T-C T-C2 T-Z

Fig. 11 Parametricised cross section of type Thinwalled (T=thinwalled)

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 be
low. 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 de

tailed output in a FAGUS manual operation details are given on each cross section element.

Asw Asw Asw Asw
Asw

Asw

1 1 1 1 1 1
G-R G-T G-TV G-I G-IV G-RH

Asw Asw Asw Asw

G-U
1
G-UV
1
G-B1
1
G-B2
1

Fig. 12 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 conver
sion for the horizontal loading Vy is carried out.

As column and wall cross sections the following parametricised cross sections are avail
able:

C-18 FAGUS7
 C 5 Overview of Parametrised Cross Sections

C-RP C-RL C-O C-RH C-OH

Fig. 13 Parametricised cross sections of type RC columns (C=Column)

W-R W-L W-Z W-U

Fig. 14 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

. Linienfrmige Bewehrungen mit diskreten Bewehrungsstben: In FAGUS kann nun

auch bei parametrisierten Querschnitten zwischen 'linienfrmig verteilter Bewehrung' und
'Einzelstben' umgeschaltet werden. Um eine Kollision" in den Bewehrungseckpunkten
zu vermeiden (z.B. C-RH), werden die vertikalen Linien in diesem Fall ohne Endstbe gene
riert. Dies fhrt aber bei den oben dargestellten Wandelementen selten zum gewnschten
Resultat. Falls man tatschlich das Modell 'Einzelstbe' verwenden mchte, sollten solche
Querschnitte mit Vorteil in normale FAGUS-Querschnitte umgewandelt werden.

. Damit die Sttzenkreisquerschnitte ebenfalls auf (in der Regel kleine) Querkraft und Tor
sionsbeanspruchungen bemessen werden knnen, wurde hierfr 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 berprfen.

FAGUS7 C-19
Vol. C Special Topics

C-20 FAGUS7
 D 1 Introduction and Overview

Vol. D Timber Cross Sections

FAGUS7

D 1 Introduction and Overview

If a cross section is built up solely by different types of wood FAGUS uses a special proce
dure according to the requirements of this material.
Only three analysis types are available in this case
S Stress analysis
S Ultimate load / Efficiency / Capacity check
S 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
S with a parametrised cross section (tabsheet Timber")
S 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

s
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
f ..k
axial forces.
f ..d
Example C24 (Coniferous tree") acc. SIA 265: char. value f m,k + 24
Nmm2

Parallel to the grain: f c,0,d + * 12
Nmm 2 , f t,0,d + 8
Nmm2 , f m,d + 14
Nmm2

Perpendicular to the grain: f c,90,d + * 1.8
Nmm 2 , f t,90,d + 0.1
Nmm2
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 mo
dulus of elasticity.
In contrast to the (concrete) analysis methodes descriped in the previous chapters, for tim
ber 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:

h M
h t
h w
Rd + gM 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: h t
+ 1 ; moisture class h w + 1 = 1). In other cases
the term h th w can directly be given in Menu 'Options' > 'Analysis parameters'

FAGUS7 D-1
Vol. D Timber Cross Sections

Design values according to DIN 1052 und EC 5:

kmodf m,k
f m,d + gM
In the German Code DIN 1052 characteristic values are pre-defined. The partial factor g M
is given by g M + 1.3.
k mod is a factor taking into account the effect of the duration of load and moisture. The pro
gram assumption have to be checked by the user and eventually adapted!
Grain direction
Each partial cross section can have its own fibre direction a. The modulus of elasticity E
(and the cross sectional properties) is calculated with
E 0E90
Ea +
E0 sin 2 a ) E 90 cos 2 a
For E 0 and E 90 mean values are used: E 0 E0,mean , E 90 E90,mean (SIA 265, 2.3.1.1)

D 3 Analysis

D 3.1 Stress Analysis

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.
M
The results are also shown if the action is too big. The material is assumed to behave elasti
cally unconditionally but in the table Max. strains and stresses there is a message indica
ting 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 some
where 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 s t
due to an axial normal forces and stresses s m due to bending moments.

The check is done in the follwing way:

st s
) mv1
f t,d f m,d
Combined bending and axial tension
s t,0,d s m,y,d s m,z,d
SIA 265 ) ) v1
f t,0,d f m,y,d f m,z,d

D-2 FAGUS7
 D 3 Analysis

In DIN 1052 and EC5 a factor k m is introduced:

s t,0,d s m,y,d sm,z,d

) ) k m v1 EC5 (6.16)
f t,0,d f m,y,d fm,z,d

s t,0,d sm,y,d sm,z,d

) k m ) v1 EC5 (6.17)
f t,0,d fm,y,d fm,z,d

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

s c,0,d 2 sm,y,d sm,z,d
SIA 265 ( ) ) ) v1
f c,0,d fm,y,d fm,z,d
DIN 1052, EC5
s c,0,d 2 sm,y,d s m,z,d
( ) ) ) km v1 EC5 (6.18)
f c,0,d fm,y,d f m,z,d
s c,0,d 2 s m,y,d s m,z,d
( ) ) km ) v1 EC5 (6.19)
f c,0,d f m,y,d f m,z,d

Stability
The stability verfication is performed in an similar way. The allowable stresses are redu
ced 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)
kc + 1 EC5 (6.25)

k ) k 2 * l 2rel

k + 0.5[1 ) b c(l rel * 0.3) ) l2rel] EC5 (6.26)

Laterial torsional buckling:

k crit
k crit + 1 for l rel,m v 0.75 EC5 (6.32)
k crit + 1.56 * 0.75
l rel,m for 0.75 t l rel,m v 1.4
k crit + 1l2rel,m for 1.4 t l rel,m

In the case where only a moment My exists about the strong axis y, the stresses should sat
isfy the following expression:
s m,d v kcritf m,d
In the case where a combination of moment My about the strong axis y and compressive
force N c exists, the stresses should satisfy the following expression :
sm,y 2 sc
( ) ) v1 EC5 (6.33)
k critfm,d k c,ztf c,0,d

SIA 265 (37) requires:

s c,0,d sm,y,d
) v1
k c,yfc,0,d kmf m,y,d

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

FAGUS7 D-3
Vol. D Timber Cross Sections

Some intermediate results:

ah 40025
l rel,m + 0.07 + 0.07 + 0.28 km + 1
b 25
i+ Jy
A
= 7.22 l + 400 + 55.42 lrel + 0.98 kc + 0.704
7.22
* 200
kN
s c,d + + * 3.2
Nmm 2
625
cm 2
sc,0,d
Compression: + 3.2 + 0.378
kc
f c,0,d 0.70412
My M
Bending stresses: s m,y,d + + 5.76
Nmm 2, s m,z,d + z + 2.918
Nmm 2
Wy Wz
s m,y,d s m,z,d
Bending check: ) + 5.76 ) 2.918 + 0.6198
f m,y,d f m,z,d 14 14
s c,0,d s m,y,d s m,z,d
Combination: ) ) +1
f c,0,d f m,y,d f m,z,d

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

s 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
f c..k FAGUS assumes an ideal interconnection between the different types of material (ass
umption of a strain plain).
f c..d
Load factor / Efficiency / Limit state:
cu 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 materi
f t..d als 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
:+
fm,d
For pure axial tension and compression the following values are used:
f d
:+
ft,d (tension) and f d
:+
fc,d compresssion

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:
S Define geometry of timber structure and loadings as usual
Check given buckling lengths and distances between lateral buckling supports
(member attributes > last tabsheet)
S Ask for the desired analysis: Tabsheet 'Calculation' > 'Special analysis' > 'Timber ..'
S View desired results in the tabsheet 'Results'

D-4 FAGUS7
 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.

FAGUS7 D-5
Vol. D Timber Cross Sections

D-6 FAGUS7
Vol. E Introductory examples
FAGUS7

E 1 Introduction
Two short sections dealing with starting the program and opening a new cross section lib
rary 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 pro
gram operation you can consult FAGUS's Help System, whose intensive use will be well
repaid. It can be started in different ways:
S By clicking on the menu Help in the program's menu bar. A list of all available Help doc
uments is given, which you can enter immediately.
S By pressing the key <F1> 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.
S 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:
S All actions to be carried out are indented and marked as follows:
" Description of action
S Everything in bold print in the description of an action has to be typed in exactly as given
except for the following exceptions.
S Special keys are denoted by <..> (e.g. <Enter>=<o>, <Esc>, <F1>, etc.)
S The mouse buttons are abbreviated to <LMB> and <RMB> (left and right)
S Buttons on the screen are shown in square brackets (e.g. [Cancel])
S Words printed in italics Click, Select, ... are previously clearly defined user actions
S 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.

FAGUS7 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 Win
dows
(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 man
agement 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 us
ing 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 FGDataon 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 FAGUS7
 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 Cubu
sExplorer 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 activ
ated 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.

FAGUS7 E-3
Vol. E Introductory examples

E 1.4 Example 1: Cross Section with Simple Outline

E 1.4.1 Task

180
Measurements
in cm

72 Materials:
6 concrete and
reinforcing steel
as specified

6
33

27
As = 26 cm2
zq
36
57

8
27

8
yq
5 * 22
12 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 FAGUS7
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 Exaggerated Status line

input fields scale

A cross section library consists in general of several cross sections, which are created, de
leted, copied, etc. using the cross section management (<F1> 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 des
ignated 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.

FAGUS7 E-5
Vol. E Introductory examples

" 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:

Legend
Openings

Check
Handle (point)

Result combinations
Drawn partial sections

Parametrised cross sections

Steel sections
Insert existing cross sections

PT Constraint lines
Thin walled partial sections

Axis point

Result points

Cross section results

Selectable objects
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 para
metricised 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 sub
section can be defined by
S the input of the polygonal outline, which can contain one or more openings that have
to lie completely within the corresponding subsection.
S taking a section from the existing rolled steel section library
S 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 FAGUS7
 " 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 con
firmed 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 out
line 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 pos
sible 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 inter
sect.
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 (clock
wise or anticlockwise) of the polygon are not prescribed by the program.

" Type in
-60 <o> 0 <o>
60 <o> 0 <o>
<R> 30 <o> 57 <o> An <R> before an input coordinate pair
stands for relative input, i.e. the coordi
<R> 0 <o> 33 <o> nates refer to the point last input, the
<R> -180 <o> 0 <o> relative point
<R> 0 <o> -33 <o>

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 <RMB> call
its Context menu and select 'Edit Geometry' there. You are once again in the above men
tioned 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

FAGUS7 E-7
Vol. E Introductory examples

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 Rect
angle 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 <o> 27 <o>
<R> 72 <o> 36 <o>
" 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 ex
ample).
" Change to the tabsheet 'Reinforcement'
Point reinforcement
Circular reinforcement
reinforcement

Line reinforcement
New reinforcement group
Delete group
Properties of group

ment
Auxiliary points to show

Distance between auxili

ary points for reinforce

Result points
Tendons

Adjust reinforcement to stirrup

Legend for reinforcement

Check

Selectability
List field to select the
reinforcement group

First we introduce point reinforcement. The bars have a spacing to the boundary in all direc
tions 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 rein
forcement. 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 FAGUS7
 " 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 <LMB>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 <RMB> 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 Graph
ics Editor of FAGUS functions without exception according to this principle, which is gener
ally formulated in the following insertion.

FAGUS7 E-9
Vol. E Introductory examples

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 rle. 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 fnf Punktbewehrungen, indem Sie ein Fenster ber diese aufziehen.

E-10 FAGUS7
 " Holen Sie mit der <RMT> das Kontextmenu zu den selektierten Objekten und whlen Sie
daraus 'Eigenschaften' . Diese Menuzeile ist nur aktiv, wenn Sie ausschliesslich gleichartige
Objekte selektiert haben! Sie knnten jetzt beliebige Attribute verndern und diese an
schliessend den selektieren Objekten zuweisen.
. Die Schaltflche [Anwenden] ist allerdings nur aktiv, wenn Objekte selektiert sind und et
was in den Dialogeinstellungen verndert wurde. In Klammern hinter 'Anwenden' steht
immer die Anzahl der selektierten Objekte, auf welche sich das Anwenden der genderten
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

Legend

Check interpolated
Check variants

cross sections
Delete new variant

Properties

List field to choose

variant

Example of a cross sec

tion with a variant section:

A cross section can have several variants, which can differ with respect to the choice of sub

Two input subsec
sections 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.
tions =

variant 'Standard' 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:

FAGUS7 E-11
Vol. E Introductory examples

In this way the influence of reinfor

cement 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 sec
tion 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:
S Adding dimensioning to the cross section as in the current task
S Set the desired figure content with the help of the layer button
S Enter a figure of the structure in the print list
S Create a text legend with all cross section data and cross section values and enter it in
the print list
S 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 la
bels, this is done using the layer group User", in which the button for a first user layer is al
ready given. You can also distribute these additions to the figures in different layers, by cre
ating 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 con
fused with making it visible):
" Press <RMB> 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 FAGUS7
 " 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)
" Click now sequentially on the points that should be included on this dimensioning line
(2) (4)
(3),(4),(5)

" Close the dimensioning line using <Esc>

(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: <Ctrl>+<Z>
With the Redo function changes, which were made using the Undo function, can be reversed stepwise. Key com
bination: <Ctrl>+<Y>

. 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 cor
responding layer button as desired. Try out the settings...
" Whlen 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 but
tons 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)

FAGUS7 E-13
Vol. E Introductory examples

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 Cubus
Viewer 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 FAGUS7
 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

Window with list of print

For quick editing of a long entries (supports moving
list of entries, or of large lines with <LMB> (drag &
files: drop)
- deactivate filling
- deactivate preview
- present picture with less
content

Preview window

FAGUS7 E-15
Vol. E Introductory examples

E 1.5 Example 2: Composite Cross Section with Variants

E 1.5.1 Task
150 Measurements
zq in cm

Concrete
8

yq
4

slab
IPE240 S275
R.S. Formed plate in
section transverse direction
(statically ineffective
for this cross sec
-Materials: Id Class EModulus tion)
C (concrete) C25/30 32.0 kN/mm2
C2 (concrete longterm) C25/30 12.0 kN/mm2
S (structural steel) S275 210 kN/mm2

-Variants: Name 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 stiff
nesses. 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 FAGUS7
 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 previ
ously 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

FAGUS7 E-17
Vol. E Introductory examples

" 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 correspond
ing rotation angles are shown in the dialogue as buttons, and the section is presented ac
cording 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 <o> 0 <o>
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 mater
ials, 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 FAGUS7
 " 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
(<RMB>)

" 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 sub
section 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 op
tion '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 <o> 0 <o>

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.

FAGUS7 E-19
Vol. E Introductory examples

E 1.5.8 Introduction of result points

Result points are intended for use in STATIK. STATIK offers the possibility, in a homogen
eous 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:
Lines through centroid or axis
points parallel to the input axes
TL AT TR

AL AA 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
PRMO For our example we want to introduce the two result points as shown:

" 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 Flle gibt, bei denen es nicht eindeutig klar ist, zu welchem Querschnittsteil ein
Resultatpunkt gehrt (z.B. auf der Grenze zwischen zwei Teilquerschnitten), muss der zum
Resultatpunkt gehrende Querschnittsteil grundstzlich eingegeben werden.

" Klicken Sie auf [Einfhren] im ResultatpunktDialog

" Fhren Sie den Resultatpunkt durch Tippen seiner Koordinaten ein:
0 <o> 4 <o>

" Geben Sie den zweiten, zum Profil gehrenden Resultatpunkt auf die gleiche Weise ein

E-20 FAGUS7
 E 2 Beispiele Analysemodul

E 2 Beispiele Analysemodul
Die folgenden Programmbeispiele sind unter der Funktion 'Serienrechnung' in der Bei
spielsammlung mit folgenden Schritten verfgbar:
1) Neue Berechnung starten (mit CubusExplorer oder mit FAGUS)
2) Beispielsammlung laden (Menu 'Hilfe' > 'Beispiele laden' > 'Bsp Analysemodul')
3) Gewnschte Norm whlen (Menu 'Einstellungen' > 'Norm')
4) Funktion 'Serienrechnung' im Register 'Analyse' whlen
5) Aus 'Datei' die gewnschte Serie ('S1' .. 'S3') whlen und Berechnung starten
6) Dokumentation mit CubusViewer betrachten und auf Wunsch ausdrucken

E 2.1 Datei: 'S1'

In 'S1' sind eine paar kleinere, voneinander unabhngige Beispiele zusammengestellt,

wobei die unten dargestellte Angaben als bersicht dienen sollen. Die genauen geome
trischen Daten und gewhlten Baustoffklassen (normabhngig) sind direkt aus der (mit
dem CubusViewer zu druckenden) Dokumentation zu entnehmen.

E 2.1.1 Trger, Bewehrungsbemessung

Querschnitt R1" Aufgabe: Bemessung fr

Myd = 75 kNm
Vyd = 84 kN
T = 10 kNm
40.00

32.00

Gesucht:
Lngsbewehrung: As1 = ?
Bgelbewehrung: Asw (Vy) = ?
Asw (T) = ?, Asw(Vy+T) = ?
30.00

E 2.1.2 Trger R2, Grenzwerte

Querschnitt R2" Aufgabe: Bemessung fr

N My
kN kNm
-0. 193.
-0. -198.
0.170 -18.
135.00

123.00

-0.222 178.
-0.643 -205.
0.279 -17.
-0.127 214.

25.00 Gesucht: As1 = ?, As2 = ?

FAGUS7 E-21
Vol. E Introductory examples

E 2.1.3 I-Trger, Bemessung fr kombinierte Beanspruchung

Querschnitt I1" Schnittkrfte:

N My Mz Vy Vz T
-100.0 80.0 10.0 100.0 100.0 80.0
0.0 100.0 0.0 200.0 130.0 90.0
-200.0 -300.0 -40.0 90.0 85.0 50.0

Gesucht:
erforderliche Lngs- und Bgelbewehrungen

E 2.1.4 Kreissttze

Querschnitt C1" Schnittkrfte:

N My Mz Vy Vz T
-7200.0 2000.0 0.0 0.0 200.0 200.0

Gesucht:
erforderliche Lngs- und Bgelbewehrungen

E 2.1.5 Sttzenquerschnitt, Steifigkeitsdiagramme

Querschnitt S1" Momenten-Steifigkeitsdiagramm:
0.75 0.60 0.30 0.60 0.75
0.10
0.50

Randabstand der Bewehrung = 6 cm

1.00

My [kNm]
0.50

E 2.1.6 Spannungsnachweise vorgespannter Trger, Schiefe Biegung

Querschnitt T1" Schnittkrfte:
My = 120.0 kNm
Mz =-12.0 kNm

Gesucht:
Randspannungen, Neutralachse, Grafik

E-22 FAGUS7
 E 2 Beispiele Analysemodul

E 2.1.7 Interaktionsdiagramme fr einbetoniertes Walzprofil

Querschnitt W1" N-My-Diagramm, N-My-Mz-Diagramm:

N= 2000.00 3000.00 4000.00 5000.00 kN

Mz [kNm]
200
2000.00
3000.00
4000.00

100
5000.00

39.00
HEA300

0 100
200
My [kNm]
40.00 400 200 0 200 400

E 2.1.8 Spannungsanalyse fr vorgespannten Brckentrger

Querschnitt B1" Beanspruchung: My = -80 MNm
1575.00

200.00
E 2.1.9 Brckentrger mit inaktiven Querschnittsteilen
Querschnitt B2" Beanspruchung: My = -80 MNm

1575.000

statisch inaktiv statisch inaktiv

200.00

Die Konsolen sind statisch inaktiv -> gleiche Resultate wie Querschnitt B1" oben

FAGUS7 E-23
Vol. E Introductory examples

E 2.1.10 Dnnwandige Querschnitte

Querschnitt D1" Querschnitt D2" Querschnitt D3"

M M

S
S

Drei Beispiele aus: Gruttman, F. Wagner W.: Ein Weggrssenverfahren zur Berechnung von Querschnittsschub
spannungen in dnnwandigen Querschnitten", Der Bauingenieur, Okt. 2001

Querschnitt D4" Querschnitt D5"

S
M
M S

Brckenquerschnitt Aus Thrlimann, B.: Torsion" Vorlesungsunterlagen

ETH Zrich

Querschnitt D6" Querschnitt D7"

M
S

Zwei Beispiele aus Aus Sattler K..: Lehrbuch der Statik", Springer Verlag, Band I/B, 1969

E-24 FAGUS7
 E 2 Beispiele Analysemodul

E 2.2 Datei 'S2': Verbundquerschnitt mit Belastungsstufen

Der links dargestellte Verbundtrger (Walzprofil HEA300, Betonplatte 100 x 15 cm) wird in
Betonplatte CS1 folgenden drei Schritten belastet:

HEA300 Schritt Belastung auf Beanspruchung

A Stahltrger allein M y1 + 150
kNm
B Stahltrger + Betonplatte M y2 + 150
kNm (zustzlich)
C Stahltrger + Betonplatte x + * 0.2
ooo (Betonplatte CS1 allein)

Untenstehend ist die unter 'S2' verfgbare Belastungsgeschichte abgebildet. Da nach dem
Laden" standardmssig alle Teilquerschnitte einer Querschnittsvariante aktiv sind, muss
in Schritt Nr. 1 zuerst die Betonplatte inaktiv gesetzt werden.

Beanspruchung / Belastungsgeschichte Berechnung mit Laststufen, (in)aktiven TeilQS, Dehnungen

Nr. Aufgabe,Analyseparameter N My Mz
[kN] [kNm] [kNm]
0 Querschnitt laden V1
1 Inaktiv setzen CS1
2 A:Gebrauchstauglichkeit 0 150.0 0
3 Aktiv setzen CS1
4 A:Gebrauchstauglichkeit 0 150.0 0
5 Dehnungen (1)
6 A:Gebrauchstauglichkeit 0 0 0
(1): 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 ein
gefgt, sodass in den Druckeintrgen auch die Zwischenstadien erscheinen.

Fr die Norm Swisscode SIA 262" mit den Baustoffklassen C25/30" und S275" sollten
folgende Spannungsbilder ersichtlich sein:

s(My1) s(My1 ) My2) s(My1 ) My2 ) x)

-5.7 -6.1

119.1 -118.5 -144.7

119.1 193.4 201

[Nmm 2] [Nmm 2] [Nmm 2]

. Desaktivieren von Querschnittselementen zu einem spteren 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 Krfte) durchgefhrt werden, damit das interne Querschnittsgleichge
wicht wieder hergestellt wird.

FAGUS7 E-25
Vol. E Introductory examples

E 2.3 Datei 'S3': Verbundquerschnitt unter Langzeitbelastung

Im Beispiel 'S3" soll der Verbundquerschnitt V1" fr eine Langzeitbelastung (Stndige

Lasten, f + 2) mit anschliessender Kurzzeitbelastung (Verkehrslasten, f + 0) analysiert
werden.

Die vorgegebene Belastungssequenz besteht aus folgenden Schritten:

0 Querschnitt V1 laden"
1 Wahl Analyseparameter Phi=2"
2 Spannungsanalyse M y + 50
kNm
3 Wahl Analyseparameter Phi=0"
4 Spannungsanalyse M y + 50
kNm (zustzlich)

Fr die Norm SIA 262" mit den Baustoffklassen C25/30" und S275" ist am Schluss der Be
rechnung folgendes Spannungs-Dehnungsbild ersichtlich (Mit Betondiagramm Typ 4)

[ooo] s
[Nmm 2]
-0.1 -3.3

0.2 51.7

Zum Vergleich:
Dehnungen und Spannungen mit konstanter Kriechzahl fr beide Schritte:

f+0 f+2

[ooo] s
[Nmm 2]
[ooo] s
[Nmm 2]

-0.1 -4.0 -0.1 -2.7

0.2 48.7 0.3 54.9

. Bemerkungen: Wie frher entsprechen die dargestellten Dehnungen den elastischen,

spannungserzeugenden Dehnungen (Sprung bei Unterkante Platte fr f 0 0 ).

Natrlich knnten die beiden Berechnungen von 'S2' und 'S3' auch aneinandergehngt
werden. Bei einer umfangreichen Berechnung mit mehreren Schritten muss vom Benutzer
sichergestellt sein, dass die Schnittkrfte auch tatschlich aufgebracht werden knnen,
andernfalls meldet sich das Programm mit Kein Gleichgewicht gefunden".

E-26 FAGUS7
 F 1 Thermal Analysis

Vol. F Fire Analyses

FAGUS7

F 1 Thermal Analysis
In this chapter the transient thermal analysis will get explained more closely. The mechani
cal 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:

t c p c p x 2
q + l Dq + l 2q ) 2q
y 2

with
q Temperature [ C]
l Heat conductivity [WmK]
Density [kgm 3]
cp Specific heat capacity [JkgK]
t Time [s]
D Laplace-operator -
Prerequisites for the validity of this description are:
1. The material properties l, , 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

Gi

20 C

Ga

Fig. F1 Integration region W with a thermal influence at the outer bound G a and an opening
with the bound G 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 W with an outer bound G a and the inner bound G i of an opening.

FAGUS7 F-1
Vol. F Fire Analyses

The initial condition is the initial temperature state of W at the time t + 0. The boundary
conditions result from the thermal influences at the bounds G a and G i and are mathemati
.
cally depicted as the net heat flux h net normal to G. 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 ) h net,r
+ a c (q g * q m) ) F
m
f
s
[(q r ) 273)4 * (q m ) 273)4]

with
ac Heat transfer coefficient [Wm 2]
qg Gas-temperature around the element [ C]
qm Surface-temperature of the element [ C]
qr Effective radiation temperature of the fire [ C]
F Configuration factor -
m Emissivity of the element -
f Emissivity of the fire -
s Stephan-Boltzmann-Constant (+ 5.68 @ 10 *8) [Wm 2K4]
For simplification these assumptions are taken:
1. The configuration factor is F + 1 by default.
2. The effective radiation temperature q r of the fire equates the gas temperature q g.
3. The progress of the gas temperature q g is given by a temperature time curve.
4. The emissivity of the fire is f + 1.
5. The heat transfer due to inner radiation inside of openings in W is neglected, so that
at the bounds G i adiabiatic boundary conditions (No heat transfer with the envi
ronment) should be chosen.

. The configuration factor F describes the radiation exchange between two surfaces. The
mathematical definition of F can be found in appendix G of EN 1991-1-2. F 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 F. The condition applies F v 1 and with the ass
umption F + 1 one remains on the safe side.
For the numeric solution of the initial-boundary value problem the Backward-Euler me
thod as well as the Method of Finite Elements is used. The differential equation is turned
into a weak formulation with the Galerkin method and the region W is discretised with tri
angular elements. The interpolation of the temperature field inside the elements is quadra
tic.

. The chosen element formulation is achieving the adiabatic boundary condition appro
ximately, i.e. if along a bound G no boundary condition is set, no transfer of heat is taking
place.

F-2 FAGUS7
 F 1 Thermal Analysis

F 1.2 Register 'Thermal Analysis'

The complete input, calculation and output is done in the tab sheet 'Thermal Analysis':

Check
Analysis Options Input Run thermal analysis
Temperature-time curves Legend Temperature along pri
ciple axis

Material Properties

Thermal bounds
Reset thermal analysis
Temperature at result points

Output Settings

Fig. F2 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 nee
ded for the same material (e.g. for comparisons), then a new material has to be added.

Every material defined in the project is avail

able.

The values defined in this area are start va

lues 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, cal
careous). The expression 'Automatic' means
that the type is chosen by the program.

Various graphic illustrations depending on the temperature.

Fig. F3 Input of thermal material properties.

. A numeric summary of the thermal material properties can be found in the legend (F 1.2.6)

FAGUS7 F-3
Vol. F Fire Analyses

Thermal values for concrete

For the description of its thermal behaviour concrete is seen as homogenous. The tempera
ture-sensitive values l 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 l c,u and a upper limit l c,o for heat conductivity. With the input of a inter
polation factor f l a linear interpolation between these two values is possible:

l c(q) + [lc,o(q) * lc,u(q)] @ fl ) l c,u(q) 0 x fl x 1

The energy consumption for evaporation is taken into consideration. Depending on the
moisture content u the specific heat capacity c p between 100 C and 200 C is adjusted ac
cordingly.
The thermal material types for mechanical analysis under elevated temperatures are:
S Siliceous aggregates,
S Calcareous aggregates,
S High strength Class 1,
S High strength Class 2,
S 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 l a, r a and c a for construction steel, reinforcement steel and prest
ressing steel are temperature dependent according to EN 1992-1-2.
Types for rebars:
S Hot rolled reinforcing steel,
S Cold worked reinforcing steel
Prestressing steel:
S Class A (Wires and strands),
S Class B (Wires and strands)
S 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
l x, r x and c x can be assigned. Nichtlineare, temperaturabhngige Beziehungen wie oben
beschrieben knnen 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 Rndern.

F-4 FAGUS7
 F 1 Thermal Analysis

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

q 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 ex
amining elements outside of the fire compartment, for example balustrades. The mat
hematical representation of this curve is

q 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.

q 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 tempe
rature 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 dele
ted.
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 <TAB> q1

until

tn <TAB> qn

FAGUS7 F-5
Vol. F Fire Analyses

Fig. F4 Temperaturetime curves

New number pairs (t i, q i) can be added to a temperature-time curve by adding a time

t i w 0
[min] with a corresponding temperature q i
[ C] in the two fields in the upper part
of the dialog and pressing 'Add/update time-temperature value'. The new time-tempera
ture 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 dia
log. The program moves the selected values t i and q i in the input fields.
A selected row can be deleted with this button.
. The temperature value of the temperature-time curve TZK_<Nr> for t w t n is constant
q n.

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 a 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 a c + 25
WmK for bounds facing the fire and
a c + 4
WmK for bounds facing away from the fire. More information can be found in EN
1991-1-2.

F-6 FAGUS7
 F 1 Thermal Analysis

Fig. F5 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 in
formation concerning the material properties have been made)
. The emissivity of the fire f and the configuration factor f are both assumed as 1.0 (F 1.1).
Other values could be taken into account by giving the product of f
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 coa
ting can consist either of standard FAGUS material (e.g. steel casing) or a fire protection ma
terial. All the materials from Steeldoc 02/06 are available. In addition the thermal values l 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. F6 Representation of the thermal bounds

F 1.2.4 Analysis Options

In Fig. F-7 the dialog 'Analysis options' is shown

FAGUS7 F-7
Vol. F Fire Analyses

Initial temperature of the cross section at time t=0.

With the standard value of 20C no thermal expan
sions 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 'addi
tional' times allows to complement this definition.
For all times defined here a mechanical analysis
can be made

Fig. F7 Input of the analysis options

Time

The time step parameters can either be set automatically (Default) or set with constant time
steps Dt. 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 'longi
tudinal 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 over
lapping elements) may not be switched off.

F-8 FAGUS7
 F 1 Thermal Analysis

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 con
trolled here.
Should the cross section have several variants, the cross section bounds of the current vari
ant 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 sa
ved 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 con
trol 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 'ele
vated temperture'.)
Window 'Output Settings'

The temperature of each FE-node is shown

(Normally these are a lot of values that over
lap, this means that one should work in a
highly zoomed in mode )

Fig. F8 Controlling the output

FAGUS7 F-9
Vol. F Fire Analyses

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 automa
tic 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 FAGUS7
 F 1 Thermal Analysis

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 a + 25Wm 2K
Moisture content u + 1.5%
Emission factor + 0.7
Heat conductivity f l + 0 (lower bound)
Exposure 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 intro
ductory example the cross-section QS_STUETZE has the suitable input.

Due to double symmetry only is modeled. Fig. F-9 shows, that the isolines calculated by
FAGUS are almost identical with the ones provided by the reference document.

Fig. F9 Calculated Crosssectiontemperatures and EN 199212,

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).

FAGUS7 F-11
Vol. F Fire Analyses

Fig. F10 EN 199212 Appendix A, Dep. 2

Fig. F11 Temperature curve in panel crosssection with x=5 cm and x=10 cm

F-12 FAGUS7
 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(q) + f c,qfck(20 )
k

1 Normal concrete
K = calcareous
Q = siliceous
2

Q K

High strenght concrete

1 = C55/67 ..
2 = C70/85 ..
3 = C90/105

Fig. F12 Reducation factors k according to EN 199212

The automatic classification for high performance concrete is done according to section 6.1
(5) of the EN norm:

S Class 1: C 55/67, C60/75

S Class 2: C 70/85, C80/95

S Class 3: ab C90/105

For user-defined materials and 'Automatic' classification the program uses the given ma
terial 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 diffe
rent concrete classes. (A high-grade concrete can give worse results under certain con
ditions)

FAGUS7 F-13
Vol. F Fire Analyses

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 s * *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.
Tension in N/mm2

Elongation [o/oo]

Fig. F13 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 FAGUS7
 F 2 Mechanical Analysis

Tension in N/mm2
0..400

500

Temperature in Celsius

600

700

Elongation[o/oo]

Fig. F14 Temperaturedependent Strain/Stressdiagramm for B500B

F 2.1.2 Thermal strain

The thermal expansions are temperature dependent and are calculated according to sec
tions 3.3.1 (concrete) and 3.4 (steel) of the EN norm.

Fig. F15 Thermal expansion according to EN 199211 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. F16 Strain distribution of a centrally loaded cross section

FAGUS7 F-15
Vol. F Fire Analyses

The total strains tot consist of the thermal expansion Q and the elastic strains s as follows:

tot + Q ) s

The stress can then be calculated with the before mentioned s * *diagramms .

s + f (s)

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 calcu
lated temperature in the rebars is crucial for the mechanical results. Because the tempera
ture gradient close to the edge of the cross section is big, a correct placement plays a de
cisive 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:
S Utilization / Load capacity / Failure time
S Stress analysis with given forces
S Stress analysis with given strain
S Moment curvature diagram
S Analysis of second order effect under axial load

For analysis under elevated temperatures the analysis parameter are given by the pro
gramm and cannot be changed.

F-16 FAGUS7
 F 2 Mechanical Analysis

F 2.3 Utilisation / Capacity / Failure time

With this analysis the utilisation or load factor is determined for each result time and re
presented 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 dur
ing 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:

S 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.)
S Numerical results in the result points
S Cross section stiffness
S Maximum strain and stresses

F 2.5 Stress analysis with given strain

With this analysis the elongations x, y, and z 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).

FAGUS7 F-17
Vol. F Fire Analyses

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:
S Enter cross section as usual. The reinforcement should be defined as a point reinforce
ment.
S Switch to tab sheet 'Thermal analysis'
S Choose calculation type 'transient'
S Adjust calculation time and result time points if needed in the dialog 'Calculation-
options'
S Define thermal bounds
S Check cross section. Following the check the FE-mesh is shown. In general a dis
cretization with around 400 elements should be enough
S Carry out thermal analysis
S Switch to the tab sheet 'Analysis'
S Activate check box
S Choose desired analysis, enter forces and carry out calculation

F-18 FAGUS7
 F 2 Mechanical Analysis

Several examples can be found in the menu 'Help' (For example validation examples ac
cording to EN) and under 'File' > 'Documents' the related explanations can be found.

FAGUS7 F-19
Vol. F Fire Analyses

F-20 FAGUS7
 G 1 Baustoffe

Vol. G Baustoffe, Analysepa

rameter
FAGUS6

G 1 Baustoffe

G 1.1 Baustoff-Dialoge
Zu Beginn einer neuen Berechnung sollten alle Baustoffe und ihre fr die Berechnung rele
vanten Eigenschaften definiert werden (Menu > Einstellungen > Baustoffe). Der Umfang
der Baustofftabelle ist einen allen Cubus-Programmen identisch. Je nach Anwendung wer
den nicht alle Werte fr die Berechnung bentigt::

ID: Die Baustoff-Zuweisung zu einzelnen Strukturelementen (Bauteil, Querschnitt, Be

wehrung) findet ber die ID statt. Wird z.B. in der oben dargestellten ersten Zeile die Be
tonklassse gewechselt, so passen sich alle Elemente mit der Referenz C automatisch der
neuen Definition an. Die Bezeichnung ist bewusst sehr kurz gehalten, damit die nu
merischen Ausgaben nicht mit langen Texten belastet werden. Der Benutzer kann eigene
Baustoffe-ID's erzeugen, allerdings nur durch Zusatz einer einstelligen Zahl (Index). Der er
ste Buchstabe wird verwendet um den Baustofftyp festzulegen.
S Beton C (Concrete)
S Betonstahl R (Reinforcment)
S Spannstahl P (Posttensioning)
S Baustahl S (Steel)
S Holz W (Wood)
S Aluminium U
S Faserverbund F (FRP)
S Mauerwerk M (Masonry)
S Spezial X
Typ: Der Baustofftyp bestimmt u.a. das eindimensionale Materialverhalten (Kap. G 1.2) und
kann nur bei der erstmaligen Eingabe gewhlt frei gewhlt werden. (Gewisse Strukturele
mente lassen nur bestimmte Baustofftypen zu. Eine Bewehrung kann z.B. nicht vom Typ
'Beton' sein)
Bauteil: Frei whlbare Bezeichnung
E: Fr Schnittkraft- und Durchbiegungsberechnungen (CEDRUS, STATIK) wird der in dieser
Kolonne aufgefhrte Wert verwendet. Bei der Initialisierung wird E vom Vorgabewert aus
der Baustoffklasse bernommen und kann dann vom Benutzer verndert werden.

FAGUS6 G-1
Vol. G Baustoffe, Analyseparameter

. E ist ein Ausgangswert fr die Steifigkeitsbestimmung. In den verschiedenen Programmen

lassen sich diese noch durch Einfhrung eines entsprechenden Faktors zustzlich an
passen z.B. CEDRUS-6 pro Zone: E + f E
E0
G: Schubmodul: In STATIK-6 fr die die Steifigkeiten GIx, verwendet.
: Fr die Bestimmung von Massen und Eigengewicht-Lastfllen verwendet.
a: Temperaturausdehnungskoeffizient
n: Querdehnungszahl (CEDRUS)

Farbe/Farbnuance : Fr Bildschirmdarstellung und gedruckte Ausgabe verwendet

Klassen:
Die Werte der vom Programm vorgegebenen Klassen knnen nicht verndert werden, aber
die Tabelle kann mit eigenen Baustoffklassen ergnzt werden. Die Klasseninformationen
sind vor allem im Zusammenhang mit der Querschnittsanalyse und Bemessung relevant.
Umfang und Bezeichnungen sind normabhngig. Normalerweise sind charakteristische
Werte einzutragen.:
Der Bemessungswert X d einer einzelnen Baustoffeigenschaft ergibt sich aus
x x k = charakteristische Baustofffestigkeit
Xd + g k
M g M = Teilsicherheitsbeiwert

Bezogen auf die verschiedenen Baustoffe sind fr X k folgende Werte einzusetzen:

f ck,
fyk,
f pk Charakt. Werte der Beton-, Betonstahl- und Spannstahlfestigkeiten
und fr g M
g c, gs, g p, ga Teilsicherheitsbeiwerte fr Beton, Bewehrungs-, Spann- u. Baustahl.
Dieser Teilsicherheitsbeiwerte sind bei den Analyseparametern definiert. Beim Baustoff
'Beton' kommt in der Regel noch ein Abminderungsbeiwert a fr die Bercksichtigung der
Dauerstandsfestigkeit hinzu.
a@f
f cd + g c ck

Fr die SIA262 wird in dieser Tabelle direkt der Wert fr h fc @ f ck eingegeben (a + 1).

h fc @ fck
f cd + gc

G 1.2 Eindimensionale Spannungs-Dehnungsbeziehungen

In diesem Kapitel werden die fr die Querschnittsanalysen verwendeten eindimensionalen

Spannungs-Dehnungsbeziehungen beschrieben.

Die Form der s * -Diagramme oft berechnungsabhngig kann deshalb via Analysepa
rameter gesteutert werden.

G 1.2.1 Beton-Druckspannungen

Bei Biegung mit Lngskraft wird die einaxiale Spannungs-Dehnungsbeziehung in jedem

Punkt des Querschnittes durch eines der untenstehend abgebildeten Diagramme
beschrieben.
Fr die Berechnung des Querschnittswiderstandes und fr Bemessungsaufgaben wird von
den meisten Normen das Diagramm Nr. 2 vorgeschlagen, wobei fr den ersten Kurvenab
schnitt eine quadratische Parabel verwendet wird, welche durch die beiden Parameter
2.0 + * 2.00
ooo und f c definiert wird. Der tangentielle E-Modul bei Belastungsbeginn
betrgt somit E co + 1000fc .

G-2 FAGUS6
 G 1 Baustoffe

s s s s

fc fc fc fc

0.4
f c E cm
Ec E co
Ec
cu 2.0
ooo cu c1 c1u c1d c2d

1: Bilinear 2: Quadratische Parabel 3: EC2, DIN1045-1 4: SIA 262

Fig. G1 SpannungsDehnungsDiagramme fr Betondruckspannungen

Fr nichtlineare Berechnungen wird in EC2 und DIN1045-1 das Diagramm Typ 3 verlangt,
welches durch untenstehende Gleichung beschrieben wird:
k
h * h 2
s c + fc @
1 ) (k * 2)
h
wobei:
h = ec /ec1 (beide e negativ einsetzen)
ec1 = - 0.0022 (Stauchung beim Erreichen des Hchstwertes der Betondruckspan-
nung f c)
k = 1.1 . Ec,nom . ec1 /fc ( f c negativ)
Ec,nom = entweder Mittelwert Ecm (Tabelle 3.2 EC2) oder entsprechender Bemessungs
wert Ecd des Elastizittsmodules
Die Norm SIA 262 bentzt fr den aufsteigenden Ast einen hnlichen Ansatz

sc k s
z * z 2 Ecd
+ mit k s + und z + c
f cd 1 ) (k s * 2)
z 400
fcd c1d

wobei nach dem Erreichen von f cd ein konstantes Plateau bis zum Bruch verwendet wird
und die Bruchstauchung auf c2d + 3
[ooo] beschrnkt wird.
. Hinweis Vorzeichen:
Analog zu den Stablngskrften sind die Zugspannungen in FAGUS-6 auch positiv. Die
oben und im Programm gewhlte Darstellung des Spannungsdehnungsdiagrammes fr
Beton entspricht den blichen Konventionen.

G 1.2.2 Beton-Zugspannungen
Im Normalfall wird der Beton unter Zug als gerissen angenommen (Zustand II), d.h. er
bernimmt keine Zugspannungen. Fr spezielle Untersuchungen, z.B. wenn die Steifigkeit
des Querschnittes eine wesentliche Rolle spielt, kann aber auch ein Diagramm Nr. 1.. 3
gewhlt werden:

s s s s

f ct f ct f ct f ct
s ct + f (r
, f ct)

0 1 2 3

Fig. G2 SpannungsDehnungsDiagramme fr Betonzugspannungen

FAGUS6 G-3
Vol. G Baustoffe, Analyseparameter

Whrend der Beton beim Diagramm Nr. 1 nach dem Erreichen von fct keine Spannungen
mehr bernimmt, bleiben diese beim Diagramm Nr. 3 auf dem Niveau s = fct konstant. fct
kann als Parameter bei der Materialeingabe eingegeben bzw. verndert werden. Der Ver
lauf der Kurven entspricht jeweils dem fr die Druckspannungen gewhlten Typ (Punkt
spiegelung bezglich Nullpunkt). Ein etwas wirklichkeitsgetreueres Materialverhalten
wird mit Nr. 2 realisiert, indem die Grsse der Betonspannung abhngig gemacht wird von
der aktuellen Randzugdehnung.
Fr Diagramm Nr. 2 wurde folgender Ansatz gewhlt:

r 2
0 v s c + fct @ (1 * ( ))
0.2%
wobei:
sct : Betonzugspannung
fct : eingegebene Betonzugfestigkeit
er : aktuelle Randdehnung auf der Zugseite des Querschnittes
0.2 % Dehnung bei der Streckgrenze eines blichen Betonstahles (S500)
Mit diesem Modell betrgt die Betonzugfestigkeit zu Beginn der Belastung s = fct und
nimmt mit gesteigerten Krmmungen quadratisch ab. Ab dem Erreichen der Fliessspan
nung 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) be
trachtet. Gewisse Korrekturmglichkeiten wurden im vorangegangenen Abschnitt be
sprochen. Ein anderer Ansatz fr die Bercksichtigung der Mitwirkung des Betons zwis
chen den Rissen wird in der Norm SIA E 166 Klebebewehrungen" beschrieben.
Vorgehen: Fr die Ermittlung der Dehnungsebene werden mittlere Dehnungen () ver
wendet, fr die Gleichgewichtsbeziehungen wird aber der Schnitt im Riss betrachtet. Das
Verhltnis zwischen mittleren und maximalen Werten (") wird durch den Verbundbeiwert
beschrieben.
+

Falls das entsprechende Kontrollfeld ("Verbundbeiwert bercksichtigen") bei den Analy
separametern aktiviert ist, so wird die Querschnitts-Analyse mit Bercksichtigung dieses
als konstant angenommenen Faktors durchgefhrt.
Im Allgemeinen ist abhngig davon, ob das Grundmaterial gerissen ist (Beton) und ob es
sich um eine innere Stabbewehrung oder eine ussere Klebebewehrung handelt. Struk
turen 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 gegenwrtigen Programmversion werden folgende nicht vernderbare Werte ver
wendet:
Verbundbeiwert der inneren Stabbewehrung : s = 0.7
Verbundbeiwert der usseren Klebebewehrung: l = 0.9
Fr Bewehrungs- und Vorspannstahl wird s verwendet. Fr Faserverbundwerkstoffe und
Baustahl, welche als Spannglied/Lamellen" eingegeben wurden, wird l eingesetzt. Das
entsprechende Querschnittselement muss zudem unter Zugbeanspruchung sein, damit
bercksichtigt wird. Bei linienfrmigen Bewehrungen gilt dies nur fr den entsprechenden
Abschnitt.

G 1.2.4 Betonstahl, Baustahl und Spannstahl

Fr Betonstahl, Baustahl und Spannstahl werden die unten dargestellten Diagramme ver
wendet.

G-4 FAGUS6
 G 1 Baustoffe

s s
f pk
fy f p0.1k
ET

Es Es

uk

Betonstahl Baustahl Spannstahl

Fig. G3 SpannungsDehnungsDiagramme fr Stahl

s SIA162 Falls eigene Baustoffklassen definiert werden, so ist darauf zu achten, dass Kennwerte
eingegeben werden. Diese werden dann entsprechend den in den Analyseparametern
f ..k definierten Widerstandsbeiwerten/Teilsicherheitsfaktoren reduziert. Die Reduktion er
f ..d folgt normabhngig entsprechend der nebenstehenden Figur. Ob mit oder ohne Verfesti
gung gerechnet werden soll, kann ebenfalls bei den Analyseparametern eingegeben wer
den.
Bei den meisten Normen ist fr die Zug- und Druckfestigkeit der gleiche Wert anzunehmen.
s Neue Normen Fr spezielle Untersuchungen (z.B. British Standard BS5400) knnen aber auch unter
schiedliche Werte gewhlt werden.
f ..k
f ..d Falls zugleich fy < ftk gewhlt wurde, so wird ET auf Zug und Druck gleich gross gewhlt
mit:
f tk * f y
ET + *
uk y

In den Materialtabellen von FAGUS-6 werden auch fr den Spannstahl fr alle Normen
gewisse Vorschlge gemacht. Bei der Wahl eines Spannstahles ist aber unbedingt darauf
zu achten, dass die vom Programm vorgegebenen Werte mit den Herstellerangaben
bereinstimmen, d.h. in der Regel sind diese anzupassen.

Die initiale Vorspannkraft wird durch Eingabe einer Vordehnung angegeben. Weitere Erk
lrungen dazu sind z.B. im Handbuch von FAGUS zu finden.

FAGUS6 G-5
Vol. G Baustoffe, Analyseparameter

G 2 Analyseparameter

G 2.1 Der Analyseparameter-Dialog

Als Analyseparameter werden alle Parameter bezeichnet. welche das Verhalten einer Quer
schnittsanalyse beeinflussen (und nicht bereits in der Querschnittsgeometrie oder den
Baustoffparametern enthalten sind).
Es knnen mehrere Stze von Analyseparametern unterhalten werden, wobei standard
mssig je eine Definitionen fr den GZG (Gebrauchstauglichkeit) und fr den GZT
(Tragsicherheit) zur Verfgung stehen. Der Dialog ist in mehrere Registerbltter aufgeteilt:

. Beispiel: Ein M-N-Interaktionsdiagramm kann wahlweise fr den Grenzzustand der Ge

brauchstauglichkeit wie auch fr den Grenzzustand der Tragsicherheit" erzeugt werden,
indem der entsprechende Analyseparametername gewhlt wird.
Vorgabewerte: Standardmssig werden fr die verschiedenen Analysen folgende Zuord
nungen verwendet:
AP Analysetyp
GZG Spannungsanalyse, Momenten-Krmmungsdiagramme
GZT Traglastanalyse, Bemessung, Interaktionsdiagramme

Der Benutzer kann die bestehenden Programmvorgaben verndern sowie eigene Defini
tionen hinzufgen.

G 2.1.1 Registerblatt Grenzdehnungen und Grenzspannungen

Die Grenztragfhigkeit eines Querschnittes gilt als erreicht, wenn die Dehnung der ussers
ten Querschnittsfaser auf der Druckseite oder der ussersten Bewehrungslage auf der Zug
seite einen bestimmten Grenzwert erreicht hat. Fr zentrischen Druck und fr Biegung sind
die Grenzdehnungen unterschiedlich, wie dies in untenstehender Figur dargestellt wird:
. Fr die Parameter cu.c, cu.b, su fehlen einheitliche Bezeichnungen. Es wurde deshalb eine
Definition gewhlt, welche fr alle untersttzten Normen einigermassen akzeptabel sein
sollte. Dabei steht der erste Index c fr concrete" bzw. s fr Stahl", u fr ultimate (grenz-)
Bedingung und der Buchstabe nach dem Punkt fr c=zentrisch bzw b=Biegung.
Die fnf Dehnungsbereiche werden mit folgenden Begriffen charakterisiert:
Bereich 1: Mittiger Zug und Zugkraft mit kleiner Ausmitte
Bereich 2: Biegung (mit Lngskraft) unter Ausnutzung der Bewehrung
Bereich 3: Biegung (mit L.-Kraft) unter Ausnutzung der Bewehrung und des Betons
Bereich 4: Biegung (mit Lngskraft) unter Ausnutzung des Betons
Bereich 5: Lngskraft innerhalb Kernbereich des Querschnittes, zentrischer Druck

G-6 FAGUS6
 G 2 Analyseparameter

cu.c cu.b

3 d
2 h
1
4
5

su sy Zug Druck
sy = Stahlstreckgrenze

Fig. G4 Grenzdehnungsebenen fr GZT

Im Grenzzustand befinden sich die Dehnungen am Querschnittsrand also auf folgender,

geschlossener Figur:

2 + Dehnungen
am
oberen
Rand

Biegung Zug
1
1 + cu.b
1 + Dehnungen
am
unteren
Rand
Druck h
Biegung
1 +
su.c
d
5 2
3,4
1 + 2 + cu.c
h
2 + cu.b 1 +
( su.c * cu.b) ) cu.b
d

Fig. G5 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 zulssigen
Grenzwert erreichen. Fr eine exakte Bestimmung des plastischen Momentes (mit voll
stndiger Plastifizierung des Querschnittes) mssten unendlich grosse Randdehnungen
vorgegeben werden knnen.

Folgende Tabelle gibt Auskunft ber die FAGUS-6 Voreinstellungen fr einige Normen:

Norm ecu.c ecu.b esu

mittiger Druck Biegung Stahlgrenzdehnung (Zug)
SIA 262 2.0 o/oo (bzw. 3.0) 3.0 o/oo 5.0 o/oo
DIN10451 2.0 o/oo 3.5 o/oo 5.0 o/oo
EC2 2.0 o/oo 3.5 o/oo (*) 20 o/oo (falls fy=ftk, sonst 10 o/oo)
OeNorm B4200 2.0 o/oo 2.0 o/oo 4 o/oo
OeNorm B4700 2.0 o/oo 3.5 o/oo 10 o/oo
(*) abhngig von Betonqualitt

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 Stahlspan
nung) am Querschnittsrand.

FAGUS6 G-7
Vol. G Baustoffe, Analyseparameter

Nachweise fr den GZG

Fr die Gebrauchstauglichkeit sind oft Rissweiten (*) oder maximale Stahlspannungen in
Abhngigkeit vom Durchmesser und Abstand nachzuweisen. Anstelle von Grenzdehnun
gen knnen auch zulssige Stahlspannungen eingegeben werden.
Von gewissen Anwendungen (CEDRUS-6) wird dieser Wert aber von aussen entsprechen
den den vorgefundenen Eingabedaten gesetzt.
(*) Rissweiten sind als Teil von Spannungsnachweisen verfgbar.

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
verwendende s * - Diagramm und die Teilsicherheitsbeiwerte ( g c
, gs
,
.. ) knnen je
doch fr jede Analyse unterschiedlich festgelegt werden.

G 2.1.3 Registerblatt Beton-Diagramm

Hier kann die gewnschte s * -Beziehung sowohl im Druckbereich sowie fr allfllig zu
bercksichtigende Zugspannungen gewhlt werden. Eine Beschreibung dazu befindet
sich in Kap. G 1.2.

G 2.1.4 Registerblatt Vorspannung:

Bei der Vorspannung kann zwischen zwei s * -Beziehungen umgeschaltet werden. Ein
weiterer Punkt betrifft die Bercksichtigung 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 fr die Bemessung der
Lngs- und Bgelbewehrung vorgegeben:

- Minimaler Lngsbewehrungsgehalt fr Sttzen

Fr Sttzenquerschnitte existieren in allen Normen Vorschriften ber minimale Be
wehrungsgehalte. Bei der Bemessung werden die eingegebenen Bewehrungen so skaliert,
dass der Minimalwert auch fr geringe Beanspruchungen nicht unterschritten wird.

- Maximaler Lngsbewehrungsgehalt (alle Querschnittstypen)

Dient als Abbruchkriterium (und somit als Schutz gegen zu lange Rechenzeiten) im Rah
men der Bemessungsiterationen.

- Minimaler Bgelbewehrungsgehalt
Im Rahmen der Bgelbemessung wird eine statisch erforderliche Bewehrung bestimmt
(Kap. LEERER MERKER). Mit Hilfe von Resultatkombinationen kann dieser Wert an
schliessend mit dem vom Benutzer gegebenen bzw. nach Norm erforderlichen Minimalw
ert verglichen werden (vgl. Kap. C 4.4)

G-8 FAGUS6
 G 2 Analyseparameter

G 2.1.6 Weitere Werte

In diesem Register sind alle brigen Einstellungen zu finden u.a.

- Kriechzahl
Falls die hier angegebene Kriechzahl grsser als Null ist, so wird die Analyse unter Bercks
ichtigung des Betonkriechens durchgefhrt

- Neigung der Betondruckdiagonalen

Der hier eingegebene Wert entspricht dem Winkel a in B-18

- Bgel-Grenzspannung
Falls eine Bemessung auf zulssige Spannungen erfolgen soll, kann hier ein Wert (der
kleiner als die Fliessspannung sein sollte) eingegeben werden.

- Kontrollfeld fr eine Berechnung mit Bercksichtigung von Verbundspannungen

Falls dieses Kontrollfeld eingeschaltet wird, findet eine Analyse mit Bercksichtigung des
Verbundes zwischen Beton und Betonstahl, bzw. allflligen Klebebewehrungen statt.

G 2.1.7 Zustzliche Parameter

Die hier dargestellten Werte sind in Form einer Tabelle abgebildet und das Ganze ist pro
grammtechnisch so realisiert, dass die Liste jederzeit relativ einfach erweitert werden kann,
z.B. fr kurzfristige benutzerdefinierte Anpassungen.

Zur Zeit sind hier Werte zur Rissberechnung fr einige Normen zu finden.

G 2.1.8 Echoprint
Zu jeder Analyse wird eine Tabelle ausgegeben, in der die verwendeten Analyseparameter
in der unten dargestellten Kurzform ausgewiesen werden.

Die Nummer des s * -Diagramms in Abschnitt c (Beton) bezieht sich auf den Typ fr das
Betondruck und Zugverhalten ( 0=kein Zug).

Die Diagramme der drei Stahltypen (Indices: s=Betonstahl, p=Vorspannstahl, und

a=Baustahl) sind in G-3 zu finden

FAGUS6 G-9