This book presents the design of steel structures using finite element methods (FEM)according to the current state of the art in Germany and the rest of Europe. After a short introduction on the basics of the design, this book illustrates the FEM with a focus on internal forces, displacements, critical loads and modal shapes. Next to finite element procedures for linear calculations considering the stress states of normal force, biaxial bending and warping torsion, non-linear calculations and the stability cases of flexural buckling, lateral torsional buckling and plate buckling are concentrated on significantly. In this context, design procedures for stability according to the standard Eurocode 3 is introduced and discussed. In addition, important fundamental issues are covered, such as the determination of cross-section properties as well as the elastic and plastic cross-section resistance. Complementary, finite element procedures for cross sections are dealt with, which will have an increasing importance in the future. This book has evolved within the teaching activities of the authors in the lecture ?Computer-oriented Design of Steel Structures? on the Masters? Programme ?Computational Engineering? at the University of Bochum. It covers the total variety of demands needed to be discussed for the safe, economic and modern design of steel structures.

Dr.-Ing. Matthias Kraus studied civil engineering at the Technical University of Darmstadt. In 2001 he changed to the University of Bochum as researcher, where he received his doctorial degree in 2005 and then worked as chief engineer at the Institute of Steel and Composite Structures. Since 2010 Dr. Kraus has worked at Schrmann-Kindmann and Partners, Dortmund, as chief engineer, he also has adjunct lectuerships at the university of Bochum.

Introduction	p. 1
Verification Methods	p. 1
Methods to Determine the Internal Forces and Moments	p. 2
Element Types and Fields of Application	p. 4
Linear and Nonlinear Calculations	p. 6
Designations and Assumptions	p. 7
Fundamental Relationships	p. 13
Limit States and Load Combinations	p. 16
Introductory Example	p. 19
Content and Outline	p. 23
Computer Programs	p. 24
Cross Section Properties	p. 25
Overview	p. 25
Utilisation of Symmetry Properties	p. 29
Standardisation Part I: Centre of Gravity, Principal Axes and Moments of Inertia	p. 31
Calculation of Standardised Cross Section Properties Part I	p. 40
Separation of the Cross Section into Partial Areas	p. 40
Partial Areas of Thin-Walled Rectangles	p. 43
Basic Cross Sections and Elementary Compound Cross Section Shapes	p. 46
Tabular Calculation of Cross Section Properties	p. 51
Numeric Integration / Fibre and Stripe Model	p. 53
Standardisation Part II: Shear Centre, Warping Ordinate and Warping Constant	p. 58
Warping Ordinate	p. 63
Shear Centre M	p. 67
Principles of FEM	p. 72
General Information	p. 72
Basic Concepts and Methodology	p. 72
Progress of the Calculations	p. 78
Equilibrium	p. 80
Preliminary Remarks	p. 80
Virtual Work Principle	p. 81
Principle of Minimum of Potential Energy	p. 83
Differential Equations	p. 84
Basis Functions for the Deformations	p. 87
General	p. 87
Polynomial Functions for Beam Elements	p. 87
Trigonometric and Hyperbolic Functions for Beam Elements	p. 91
Basis Functions for Plate Buckling	p. 95
One-Dimensional Functions for Cross Sections	p. 99
Two-Dimensional Functions for Cross Sections	p. 103
FEM for Linear Calculations of Beam Structures	p. 108
Introduction	p. 108
Beam Elements for Linear Calculations	p. 108
Linking Deformations to Internal Forces and Moments	p. 108
Axial Force	p. 110
Bending	p. 113
Torsion	p. 116
Arbitrary Stresses	p. 120
Nodal Equilibrium in the Global Coordinate System	p. 123
Reference Systems and Transformations	p. 126
Problem	p. 126
Beam Elements in the X-Z Plane	p. 131
Beam Elements in a Three-Dimensional X-Y-Z COS	p. 134
Loads	p. 138
Warping Moment and Derivative of the Angle of Twist	p. 139
Finite Elements for Arbitrary Reference Points	p. 146
Systems of Equations	p. 147
Aim	p. 147
Total Stiffness Matrix	p. 147
Total Load Vector	p. 149
Geometric Boundary Conditions	p. 151
Calculation of the Deformations	p. 153
Determination of the Internal Forces and Moments	p. 154
Determination of Support Reactions	p. 156
Loadings	p. 157
Concentrated Loads	p. 157
Distributed Loads	p. 157
Settlements	p. 158
Influences of Temperature	p. 159
Springs and Shear Diaphragms	p. 159
Hinges	p. 164
FEM for Nonlinear Calculations of Beam Structures	p. 168
General	p. 168
Equilibrium at the Deformed System	p. 168
Extension of the Virtual Work	p. 171
Nodal Equilibrium with Consideration of the Deformations	p. 178
Geometric Stiffness Matrix	p. 180
Special Case: Bending with Compression or Tension Force	p. 185
Initial Deformations and Equivalent Geometric Imperfections	p. 189
Second Order Theory Calculations and Verification Internal Forces	p. 193
Stability Analysis / Critical Loads	p. 201
Eigenmodes / Buckling Shapes	p. 203
Plastic Hinge Theory	p. 206
Plastic Zone Theory	p. 210
Application Areas	p. 210
Realistic Calculation Assumptions	p. 210
Influence of Imperfections	p. 213
Calculation Example	p. 214
Solution of Equation Systems and Eigenvalue Problems	p. 217
Equation Systems	p. 217
Problem	p. 217
Solution Methods	p. 218
Gaussian Algorithm	p. 219
Cholesky Method	p. 220
Gaucho Method	p. 220
Calculation Example	p. 222
Additional Notes	p. 224
Eigenvalue Problems	p. 224
Problem	p. 224
Explanations for Understanding	p. 225
Matrix Decomposition Method	p. 230
Inverse Vector Iteration	p. 236
Combination of the Solution Methods	p. 241
Stresses According to the Theory of Elasticity	p. 245
Preliminary Remarks	p. 245
Axial Stresses due to Biaxial Bending and Axial Force	p. 247
Shear Stresses due to Shear Forces	p. 250
Basics	p. 250
Calculation Formula for ¿	p. 254
Open Cross Sections	p. 255
Closed Cross Sections	p. 260
Stresses due to Torsion	p. 261
General	p. 261
Arbitrary Open Cross Sections	p. 264
Closed Sections	p. 270
Interaction of All Internal Forces and Verifications	p. 270
Limit Internal Forces and Moments on the Basis of the Theory of Elasticity	p. 272
Plastic Cross Section Bearing Capacity	p. 273
Effect of Single Internal Forces	p. 273
Limit Load-Bearing Capacity of Cross Sections	p. 275
Preliminary Remarks	p. 275
Plastic Cross Section Reserves	p. 277
Calculation Methods and Overview	p. 281
Limit Load-Bearing Capacity of Doubly-Symmetric I-Cross Sections	p. 288
Description of the Cross Section	p. 288
Perfectly Plastic Internal Forces S_pl	p. 289
Equilibrium between Internal Forces and Partial Internal Forces	p. 291
Combined Internal Forces N, M_y, M_z, V_y and V_z	p. 293
Interaction Conditions of DIN 18800 and Comparison with the PIF-Method	p. 296
Computer-Oriented Methods	p. 303
Problem Definition	p. 303
Strain Iteration for a Simple Example	p. 304
Strain Iteration for ¿ Internal Forces	p. 307
Consideration of the ¿ Internal Forces	p. 314
Examples / Benchmarks	p. 317
Verifications for Stability and according to Second Order Theory	p. 319
Introduction	p. 319
Definition of Stability Cases	p. 321
Verification according to Second Order Theory	p. 323
Verifications for Flexural Buckling with Reduction Factors	p. 329
Preliminary Remarks	p. 329
Axial Compression	p. 330
Uniaxial Bending with Compression Force	p. 338
Modified Reduction Factors ¿	p. 340
Calculation of Critical Forces	p. 342
Details for the Determination	p. 342
Replacement of Structural Parts by Springs	p. 348
Compression Members with Springs	p. 352
Verifications for Lateral Torsional Buckling with Reduction Factors	p. 360
Preliminary Remarks	p. 360
Beams Not Susceptible to Lateral Torsional Buckling	p. 360
Scheduled Centric Compression	p. 362
Uniaxial Bending without Compression Force	p. 364
Uniaxial Bending with Axial Compression Force	p. 368
Reduction Factors according to Eurocode 3	p. 369
Accuracy of Reduction Factors	p. 373
Calculation of Critical Moments	p. 375
Verifications with Equivalent Imperfections	p. 381
Verification Guidance	p. 381
Equivalent Geometric Imperfections	p. 381
Calculation Examples	p. 393
Single-Span Beam with Cantilever	p. 393
Beam with Scheduled Torsion	p. 396
Two Hinged Frame - Calculation in the Frame Plane	p. 399
	p. 404
Frame Considering Joint Stiffness	p. 413
FEM for Plate Buckling	p. 420
Plates with Lateral and In-Plane Loading	p. 420
Stresses and Internal Forces	p. 420
Displacements	p. 422
Constitutive Relationships	p. 423
Principle of Virtual Work	p. 425
Plates in Steel Structures	p. 428
Stiffness Matrix for a Plate Element	p. 429
Geometric Stiffness Matrix for Plate Buckling	p. 432
Plates with Longitudinal and Transverse Stiffeners	p. 434
Verifications for Plate Buckling	p. 438
Determination of Buckling Values and Eigenmodes with FEM	p. 448
Calculation Examples	p. 451

Beam Web with Longitudinal Stiffeners	p. 454
Web Plate of a Composite Bridge with Shear Stresses	p. 457
Web Plate with High Bending Stresses	p. 459
FEM for Cross Sections	p. 461
Tasks	p. 461
Principle of Virtual Work	p. 464
One-Dimensional Elements for Thin-Walled Cross Sections	p. 469
Virtual work	p. 469
Element Stiffness Relationships	p. 472
Equation Systems	p. 474
Calculation of Cross Section Properties and Stresses	p. 476
Compilation	p. 479
Two-Dimensional Elements for Thick-Walled Cross Sections	p. 480
Preliminary Remarks	p. 480
Virtual Work for Thick-Walled Cross Section Elements	p. 482
Element Geometry	p. 484
Transformation Relationships	p. 486
Stiffness Relationships	p. 488
Numerical Integration	p. 490
Cross Section Properties and Stresses	p. 493
Performance of the Approximate Solutions	p. 495
Special Case: Rectangular Elements	p. 497
Calculation Procedure	p. 501
Calculation Examples	p. 503
Preliminary Remarks	p. 503
Single-Celled Box Girder Cross Section	p. 503
Bridge Cross Section with Trapezium Stiffeners	p. 508
Rectangular Solid Cross Section	p. 511
Doubly Symmetric I-Profile	p. 518
Crane Rail	p. 525
References	p. 528
Index	p. 534
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