{VCH}Vorlagen/Titelei_Kindmann_Kraus.3d – Seiten 1–421. 02. 2011, Seite 1

{VCH}Vorlagen/Titelei_Kindmann_Kraus.3d – Seiten 1–421. 02. 2011, Seite 1

Rolf Kindmann, Matthias KrausSteel StructuresDesign using FEM

{VCH}Vorlagen/Titelei_Kindmann_Kraus.3d – Seiten 1–421. 02. 2011, Seite 1

{VCH}Vorlagen/Titelei_Kindmann_Kraus.3d – Seiten 1–421. 02. 2011, Seite 1

Rolf Kindmann/Matthias Kraus

Steel StructuresDesign using FEM

{VCH}Vorlagen/Titelei_Kindmann_Kraus.3d – Seiten 1–421. 02. 2011, Seite 1

Prof. Dr.-Ing. Rolf KindmannRuhr-Universit�t BochumLehrstuhl f�r Stahl-, Holz- und LeichtbauUniversit�tsstraße 15044801 Bochum

Dr.-Ing. Matthias KrausIngenieursoziet�t SKPPrinz-Friedrich-Karl-Str. 3644135 Dortmund

Language Polishing by Paul Beverley, London

Cover: SIGNAL IDUNA PARK, Dortmund, � Professor Rolf Kindmann

Library of Congress Card No.:applied for

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available onthe Internet at <http://dnb.d-nb.de>.

� 2011 Wilhelm Ernst & Sohn, Verlag f�r Architektur und technische Wissenschaften GmbH & Co. KG,Rotherstraße 21, 10245 Berlin, Germany

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – byphotoprint, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from thepublishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be consideredunprotected by law.

Coverdesign: Sophie Bleifuß, BerlinHerstellung: pp030 – Produktionsb�ro Heike Praetor, BerlinPrinting and Binding: Betz-Druck GmbH, Darmstadt

Printed in the Federal Republic of Germany.Printed on acid-free paper.

ISBN 978-3-433-02978-7Electronic version available. O-book ISBN 978-3-433-60077-1

4

Preface

Steel structures are usually beam or plate structures consisting of thin-walled cross sections. For their design, deformations, internal forces and moments as well as stresses are needed, and the stability of the structures is of great importance. Generally, the finite element method (FEM) is used for structural analysis and as a basis for the verification of sufficient load-bearing capacity.

This book presents the relevant procedures and methods needed for calculations, computations and verifications according to the current state of the art in Germany and the rest of Europe. In doing so the following topics are treated in detail:

determination of cross-section properties, stresses and plastic cross section bearing capacity finite element method for linear and nonlinear calculations of beam structures solution of eigenvalue problems (stability) for flexural, lateral torsional, torsional and plate buckling verification of sufficient load-bearing capacity finite element method for open and hollow cross sections

The basis of the calculations and verifications are the German standard DIN 18800 and the German version of Eurocode 3. They are widely comparable, however, the final version of Eurocode 3 has just been published and the corresponding national annexes have to be considered.

This book has evolved from the extensive experience of the authors in designing and teaching steel structures. It is used as lecture notes for the lecture “Computer-oriented Design of Steel Structures” on the Masters’ programme “Computational Engineering”

German books – see [25], [31] and [42] – and therefore, the references at the end of the book contain many publications in the German language. Further information can be found at www.kindmann.de, www.rub.de/stahlbau and www.skp-ing.de.

The authors would like to thank Mr Florian Gerhard for the translations, Mr Paul

Matthias Kraus Rolf Kindmann

at the Ruhr-University Bochum, Germany. Large parts of the contents were taken from

Beverley for language polishing and Mr Peter Steinbach for the drawing of figures.

© 2011 Ernst & Sohn GmbH & Co. KG. Published by Ernst & Sohn GmbH & Co. KG.Steel Structures. Design using FEM. First Edition. Rolf Kindmann, Matthias Kraus.

Content

1 Introduction 11.1 Verification Methods 11.2 Methods to Determine the Internal Forces and Moments 21.3 Element Types and Fields of Application 41.4 Linear and Nonlinear Calculations 61.5 Designations and Assumptions 71.6 Fundamental Relationships 131.7 Limit States and Load Combinations 161.8 Introductory Example 191.9 Content and Outline 231.10 Computer Programs 24

2 Cross Section Properties 252.1 Overview 252.2 Utilisation of Symmetry Properties 292.3 Standardisation Part I: Centre of Gravity, Principal Axes and

Moments of Inertia 31

2.4 Calculation of Standardised Cross Section Properties Part I 402.4.1 Separation of the Cross Section into Partial Areas 402.4.2 Partial Areas of Thin-Walled Rectangles 432.4.3 Basic Cross Sections and Elementary Compound Cross Section

Shapes 46

2.4.4 Tabular Calculation of Cross Section Properties 512.4.5 Numeric Integration / Fibre and Stripe Model 532.5 Standardisation Part II: Shear Centre, Warping Ordinate and

Warping Constant 58

2.6 Warping Ordinate 632.7 Shear Centre M 67

3 Principles of FEM 723.1 General Information 723.2 Basic Concepts and Methodology 723.3 Progress of the Calculations 783.4 Equilibrium 803.4.1 Preliminary Remarks 803.4.2 813.4.3 83

Virtual Work Principle

© 2011 Ernst & Sohn GmbH & Co. KG. Published by Ernst & Sohn GmbH & Co. KG.

Principle of Minimum of Potential Energy

Steel Structures. Design using FEM. First Edition. Rolf Kindmann, Matthias Kraus.

ContentVIII

3.4.4 Differential Equations 843.5 Basis Functions for the Deformations 873.5.1 General 873.5.2 Polynomial Functions for Beam Elements 873.5.3 Trigonometric and Hyperbolic Functions for Beam Elements 913.5.4 Basis Functions for Plate Buckling 953.5.5 One-Dimensional Functions for Cross Sections 993.5.6 Two-Dimensional Functions for Cross Sections 103

4 FEM for Linear Calculations of Beam Structures 1084.1 Introduction 1084.2 Beam Elements for Linear Calculations 1084.2.1 Linking Deformations to Internal Forces and Moments 1084.2.2 Axial Force 1104.2.3 Bending 1134.2.4 Torsion 116

1204.3 Nodal Equilibrium in the Global Coordinate System 1234.4 Reference Systems and Transformations 1264.4.1 Problem 1264.4.2 Beam Elements in the X-Z Plane 1314.4.3 Beam Elements in a Three-Dimensional X-Y-Z COS 1344.4.4 Loads 1384.4.5 Warping Moment and Derivative of the Angle of Twist 1394.4.6 Finite Elements for Arbitrary Reference Points 1464.5 Systems of Equations 1474.5.1 Aim 1474.5.2 Total Stiffness Matrix 1474.5.3 Total Load Vector 1494.5.4 Geometric Boundary Conditions 1514.6 Calculation of the Deformations 1534.7 Determination of the Internal Forces and Moments 1544.8 Determination of Support Reactions 1564.9 Loadings 1574.9.1 Concentrated Loads 157

1574.9.3 Settlements 1584.9.4 Influences of Temperature 1594.10 Springs and Shear Diaphragms 1594.11 Hinges 164

4.2.5 Arbitrary Stresses

4.9.2 Distributed Loads

Content IX

5 FEM for Nonlinear Calculations of Beam Structures 1685.1 General 1685.2 Equilibrium at the Deformed System 1685.3 Extension of the Virtual Work 1715.4 Nodal Equilibrium with Consideration of the Deformations 1785.5 Geometric Stiffness Matrix 1805.6 Special Case: Bending with Compression or Tension Force 1855.7 Initial Deformations and Equivalent Geometric Imperfections 1895.8 Second Order Theory Calculations and Verification Internal Forces 1935.9 Stability Analysis / Critical Loads 2015.10 Eigenmodes / Buckling Shapes 2035.11 Plastic Hinge Theory 2065.12 Plastic Zone Theory 2105.12.1 Application Areas 2105.12.2 Realistic Calculation Assumptions 2105.12.3 Influence of Imperfections 2135.12.4 Calculation Example 214

6 Solution of Equation Systems and Eigenvalue Problems 2176.1 Equation Systems 2176.1.1 Problem 2176.1.2 Solution Methods 2186.1.3 Gaussian Algorithm 2196.1.4 Cholesky Method 2206.1.5 Gaucho Method 2206.1.6 Calculation Example 2226.1.7 Additional Notes 2246.2 Eigenvalue Problems 2246.2.1 Problem 2246.2.2 Explanations for Understanding 2256.2.3 Matrix Decomposition Method 2306.2.4 Inverse Vector Iteration 2366.2.5 Combination of the Solution Methods 241

7 Stresses According to the Theory of Elasticity 2457.1 Preliminary Remarks 2457.2 Axial Stresses due to Biaxial Bending and Axial Force 2477.3 Shear Stresses due to Shear Forces 2507.3.1 Basics 2507.3.2 Calculation Formula for 254

ContentX

7.3.3 Open Cross Sections 2557.3.4 Closed Cross Sections 2607.4 Stresses due to Torsion 2617.4.1 General 2617.4.2 Arbitrary Open Cross Sections 2647.4.3 Closed Sections 2707.5 Interaction of All Internal Forces and Verifications 2707.6 Limit Internal Forces and Moments on the Basis of the Theory of

Elasticity272

8 Plastic Cross Section Bearing Capacity 2738.1 Effect of Single Internal Forces 2738.2 Limit Load-Bearing Capacity of Cross Sections 2758.2.1 Preliminary Remarks 2758.2.2 Plastic Cross Section Reserves 2778.2.3 Calculation Methods and Overview 2818.3 Limit Load-Bearing Capacity of Doubly-Symmetric I-Cross

Sections 288

8.3.1 Description of the Cross Section 2888.3.2 Perfectly Plastic Internal Forces Spl 2898.3.3 Equilibrium between Internal Forces and Partial Internal Forces 2918.3.4 Combined Internal Forces N, My, Mz, Vy and Vz 2938.3.5 Interaction Conditions of DIN 18800 and Comparison with the

PIF-Method296

8.4 Computer-Oriented Methods 3038.4.1 Problem Definition 3038.4.2 Strain Iteration for a Simple Example 3048.4.3 Strain Iteration for Internal Forces 3078.4.4 Consideration of the Internal Forces 3148.4.5 Examples / Benchmarks 317

9 Verifications for Stability and according to Second Order Theory

319

9.1 Introduction 3199.2 Definition of Stability Cases 3219.3 Verification according to Second Order Theory 3239.4 Verifications for Flexural Buckling with Reduction Factors 3299.4.1 Preliminary Remarks 3299.4.2 Axial Compression 3309.4.3 Uniaxial Bending with Compression Force 3389.4.4 Modified Reduction Factors 340

Content XI

9.5 Calculation of Critical Forces 3429.5.1 Details for the Determination 3429.5.2 Replacement of Structural Parts by Springs 3489.5.3 Compression Members with Springs 3529.6 Verifications for Lateral Torsional Buckling with Reduction

Factors360

9.6.1 Preliminary Remarks 3609.6.2 Beams Not Susceptible to Lateral Torsional Buckling 3609.6.3 Scheduled Centric Compression 3629.6.4 Uniaxial Bending without Compression Force 3649.6.5 Uniaxial Bending with Axial Compression Force 3689.6.6 Reduction Factors according to Eurocode 3 3699.6.7 Accuracy of Reduction Factors 3739.7 3759.8 Verifications with Equivalent Imperfections 3819.8.1 Verification Guidance 3819.8.2 Equivalent Geometric Imperfections 3819.9 Calculation Examples 3939.9.1 Single-Span Beam with Cantilever 3939.9.2 Beam with Scheduled Torsion 3969.9.3 Two Hinged Frame – Calculation in the Frame Plane 3999.9.4 Two Hinged Frame – Stability Perpendicular to the Frame Plane 4049.9.5 Frame Considering Joint Stiffness 413

10 FEM for Plate Buckling 42010.1 Plates with Lateral and In-Plane Loading 42010.2 Stresses and Internal Forces 42010.3 Displacements 42210.4 Constitutive Relationships 42310.5 Principle of Virtual Work 42510.6 Plates in Steel Structures 42810.7 Stiffness Matrix for a Plate Element 42910.8 Geometric Stiffness Matrix for Plate Buckling 43210.9 Plates with Longitudinal and Transverse Stiffeners 43410.10 Verifications for Plate Buckling 43810.11 Determination of Buckling Values and Eigenmodes with FEM 44810.12 45110.12.1 Single Panel with Constant x and 1.5 45110.12.2 Beam Web with Longitudinal Stiffeners 45410.12.3 Web Plate of a Composite Bridge with Shear Stresses 45710.12.4 Web Plate with High Bending Stresses 459

Calculation of Critical Moments

Calculation Examples

ContentXII

11 FEM for Cross Sections 46111.1 Tasks 46111.2 Principle of Virtual Work 46411.3 One-Dimensional Elements for Thin-Walled Cross Sections 46911.3.1 Virtual work 46911.3.2 Element Stiffness Relationships 47211.3.3 Equation Systems 47411.3.4 Calculation of Cross Section Properties and Stresses 47611.3.5 Compilation 47911.4 Two-Dimensional Elements for Thick-Walled Cross Sections 48011.4.1 Preliminary Remarks 48011.4.2 Virtual Work for Thick-Walled Cross Section Elements 48211.4.3 Element Geometry 48411.4.4 Transformation Relationships 48611.4.5 Stiffness Relationships 48811.4.6 Numerical Integration 49011.4.7 Cross Section Properties and Stresses 49311.4.8 Performance of the Approximate Solutions 49511.4.9 Special Case: Rectangular Elements 49711.5 Calculation Procedure 501

50311.6.1 Preliminary Remarks 50311.6.2 50311.6.3 50811.6.4 Rectangular Solid Cross Section 51111.6.5 Doubly Symmetric I-Profile 51811.6.6 Crane Rail 525

References 528

Index 534

Single-Celled Box Girder Cross Section Bridge Cross Section with Trapezium Stiffeners

11.6 Calculation Exa mples

1 Introduction

1.1 Verification Methods

For civil engineering structures the ultimate limit state (structural safety) and ser-viceability limit state have to be verified, see for example DIN 18800 Part 1. Since components for steel constructions are usually rather slender and thin-walled, struc-tural safety verifications for constructions susceptible to losing stability regarding flexural, lateral torsional and plate buckling are of major significance and therefore constitute a main focal subject in static calculations. In this context, the determination of internal forces and moments, deformations and critical loads is a central task. Its solution is treated in this book using the finite element method (FEM).

The calculations and verifications have to meet the legal requirements as well as the state of the art. For steel structures the basic standard DIN 18800 and corresponding engineering standards, or Eurocode 3, have to be taken into consideration. Table 1.1 contains a compilation of the verification methods according to DIN 18800 and the verifications as they are generally applied. Eurocode 3 contains similar regulations.

Table 1.1 Verification procedures according to DIN 18800 and common verifications

Verification procedure

Calculation of stresses Sd

Calculation of resistances Rd

Verifications

Elastic-Elastic

Elastic theory stresses and

Elastic theory design value of yield

stress fy,d

Verification of stresses: R,d = fy,d

R,d = fy,d / 3v R,d = fy,d

Elastic-Plastic

Elastic theory internal forces and moments N, My, etc.

Plastic theory utilisation of the plastic bearing capacity of the

cross sections

e.g. My Mpl,y,d or using interaction conditions or the

partial internal forces method

Plastic-Plastic

Plastic theory internal forces and moments according

to the plastic hinge or plastic zone theory

Plastic theory utilisation of the plastic bearing capacity of the cross sections and the

static system

According to the plastichinge theory or according to the plastic zone theory (with

computer programs)

The use of a verification method implies that the individual cross section parts (webs and flanges) can carry the compression stresses, so that no buckling occurs and a suf-ficient rotation capacity is provided. Assistance for the checking of the b/t relations can be found in profile tables; see for example [29]. If only longitudinal axial stresses and shear stresses occur, it is 2 2

v 3 . The verification of the equivalent/ R,d and / R,d >

© 2011 Ernst & Sohn GmbH & Co. KG. Published by Ernst & Sohn GmbH & Co. KG.

stress (verification method Elastic-Elastic) is only required for

Steel Structures. Design using FEM. First Edition. Rolf Kindmann, Matthias Kraus.