Theory of Structures: Fundamentals, Framed Structures, Plates and Shells

This book provides the reader with a consistent approach to theory of structures on the basis of applied mechanics. It covers framed structures as well as plates and shells using elastic and plastic theory, and emphasizes the historical background and the relationship to practical engineering activities. This is the first comprehensive treatment of the school of structures that has evolved at the Swiss Federal Institute of Technology in Zurich over the last 50 years.

The many worked examples and exercises make this a textbook ideal for in-depth studies. Each chapter concludes with a summary that highlights the most important aspects in concise form. Specialist terms are defined in the appendix.

There is an extensive index befitting such a work of reference. The structure of the content and highlighting in the text make the book easy to use. The notation, properties of materials and geometrical properties of sections plus brief outlines of matrix algebra, tensor calculus and calculus of variations can be found in the appendices.

This publication should be regarded as a key work of reference for students, teaching staff and practising engineers. Its purpose is to show readers how to model and handle structures appropriately, to support them in designing and checking the structures within their sphere of responsibility.

Preface  V

I INTRODUCTION

1 THE PURPOSE AND SCOPE OF THEORY OF STRUCTURES  1

1.1 General  1

1.2 The basis of theory of structures  1

1.3 Methods of theory of structures  2

1.4 Statics and structural dynamics  3

1.5 Theory of structures and structural engineering  3

2 BRIEF HISTORICAL BACKGROUND  5

II FUNDAMENTALS

3 DESIGN OF STRUCTURES  11

3.1 General  11

3.2 Conceptual design  11

3.3 Service criteria agreement and basis of design  14

3.4 Summary  26

3.5 Exercises  27

4 STRUCTURAL ANALYSIS AND DIMENSIONING  29

4.1 General  29

4.2 Actions  29

4.2.1 Actions and action effects  29

4.2.2 Models of actions and representative values  30

4.3 Structural models  31

4.4 Limit states  31

4.5 Design situations and load cases  32

4.6 Verifications  33

4.6.1 Verification concept  33

4.6.2 Design values  33

4.6.3 Verification of structural safety  34

4.6.4 Verification of serviceability  35

4.7 Commentary  35

4.8 Recommendations for the structural calculations  36

4.9 Recommendations for the technical report  38

4.10 Summary  40

4.11 Exercises  41

5 STATIC RELATIONSHIPS  43

5.1 Force systems and equilibrium  43

5.1.1 Terminology  43

5.1.2 Force systems  44

5.1.3 Equilibrium  45

5.1.4 Overall stability  45

5.1.5 Supports  47

5.1.6 Hinges  50

5.1.7 Stress resultants  51

5.2 Stresses  53

5.2.1 Terminology  53

5.2.2 Uniaxial stress state  53

5.2.3 Coplanar stress states  54

5.2.4 Three-dimensional stress states  57

5.3 Differential structural elements  61

5.3.1 Straight bars  61

5.3.2 Bars in single curvature  62

5.4 Summary  68

5.5 Exercises  69

6 KINEMATIC RELATIONSHIPS  71

6.1 Terminology  71

6.2 Coplanar deformation  72

6.3 Three-dimensional deformation state  74

6.4 Summary  76

6.5 Exercises  77

7 CONSTITUTIVE RELATIONSHIPS  79

7.1 Terminology  79

7.2 Linear elastic behaviour  81

7.3 Perfectly plastic behaviour  83

7.3.1 Uniaxial stress state  83

7.3.2 Three-dimensional stress states  84

7.3.3 Yield conditions  85

7.4 Time-dependent behaviour  90

7.4.1 Shrinkage  90

7.4.2 Creep and relaxation  91

7.5 Thermal deformations  94

7.6 Fatigue  94

7.6.1 General  94

7.6.2 S-N curves  95

7.6.3 Damage accumulation under fatigue loads  96

7.7 Summary  98

7.8 Exercises  99

8 ENERGY METHODS  101

8.1 Introductory example  101

8.1.1 Statically determinate system  101

8.1.2 Statically indeterminate system  103

8.1.3 Work equation  104

8.1.4 Commentary  105

8.2 Variables and operators  105

8.2.1 Introduction  105

8.2.2 Plane framed structures  107

8.2.3 Spatial framed structures  109

8.2.4 Coplanar stress states  110

8.2.5 Coplanar strain state  111

8.2.6 Slabs  111

8.2.7 Three-dimensional continua  113

8.2.8 Commentary  114

8.3 The principle of virtual work  115

8.3.1 Virtual force and deformation variables  115

8.3.2 The principle of virtual deformations  115

8.3.3 The principle of virtual forces  115

8.3.4 Commentary  116

8.4 Elastic systems  118

8.4.1 Hyperelastic materials  118

8.4.2 Conservative systems  119

8.4.3 Linear elastic systems  125

8.5 Approximation methods  128

8.5.1 Introduction  128

8.5.2 The RITZ method  129

8.5.3 The GALERKIN method  132

8.6 Summary  134

8.7 Exercises  135

III LINEAR ANALYSIS OF FRAMED STRUCTURES 9 STRUCTURAL ELEMENTS AND TOPOLOGY  137

9.1 General  137

9.2 Modelling of structures  137

9.3 Discretised structural models  140

9.3.1 Description of the static system  140

9.3.2 Joint equilibrium  141

9.3.3 Static determinacy  142

9.3.4 Kinematic derivation of the equilibrium matrix  144

9.4 Summary  147

9.5 Exercises  147

10 DETERMINING THE FORCES  149

10.1 General  149

10.2 Investigating selected free bodies  150

10.3 Joint equilibrium  154

10.4 The kinematic method  156

10.5 Summary  158

10.6 Exercises  158

11 STRESS RESULTANTS AND STATE DIAGRAMS  159

11.1 General  159

11.2 Hinged frameworks  160

11.2.1 Hinged girders  161

11.2.2 Hinged arches and frames  163

11.2.3 Stiffened beams with intermediate hinges  165

11.3 Trusses  166

11.3.1 Prerequisites and structural topology  166

11.3.2 Methods of calculation  169

11.3.3 Joint equilibrium  169

11.3.4 CREMONA diagram  171

11.3.5 RITTER method of sections  172

11.3.6 The kinematic method  173

11.4 Summary  174

11.5 Exercises  175

12 INFLUENCE LINES  177

12.1 General  177

12.2 Determining influence lines by means of equilibrium conditions  178

12.3 Kinematic determination of influence lines  179

12.4 Summary  183

12.5 Exercises  183

13 ELEMENTARY DEFORMATIONS  185

13.1 General  185

13.2 Bending and normal force  185

13.2.1 Stresses and strains  185

13.2.2 Principal axes  187

13.2.3 Stress calculation  189

13.2.4 Composite cross-sections  190

13.2.5 Thermal deformations  192

13.2.6 Planar bending of curved bars  193

13.2.7 Practical advice  194

13.3 Shear forces  194

13.3.1 Approximation for prismatic bars subjected to pure bending  194

13.3.2 Approximate coplanar stress state  196

13.3.3 Thin-wall cross-sections  197

13.3.4 Shear centre  199

13.4 Torsion  200

13.4.1 Circular cross-sections  200

13.4.2 General cross-sections  201

13.4.3 Thin-wall hollow cross-sections  204

13.4.4 Warping torsion  207

13.5 Summary  216

13.6 Exercises  218

14 SINGLE DEFORMATIONS  221

14.1 General  221

14.2 The work theorem  222

14.2.1 Introductory example  222

14.2.2 General formulation  223

14.2.3 Calculating the passive work integrals  223

14.2.4 Systematic procedure  226

14.3 Applications  226

14.4 MAXWELL’s theorem  230

14.5 Summary  231

14.6 Exercises  231

15 DEFORMATION DIAGRAMS  233

15.1 General  233

15.2 Differential equations for straight bar elements  233

15.2.1 In-plane loading  233

15.2.2 General loading  235

15.2.3 The effect of shear forces  235

15.2.4 Creep, shrinkage and thermal deformations  235

15.2.5 Curved bar axes  235

15.3 Integration methods  236

15.3.1 Analytical integration  236

15.3.2 MOHR’s analogy  238

15.5 Exercises  243

16 THE FORCE METHOD  245

16.1 General  245

16.2 Structural behaviour of statically indeterminate systems  245

16.2.1 Overview  245

16.2.2 Statically determinate system  246

16.2.3 System with one degree of static indeterminacy  247

16.2.4 System with two degrees of static indeterminacy  249

16.2.5 In-depth analysis of system with one degree of static indeterminacy  250

16.2.6 In-depth analysis of system with two degrees of static indeterminacy  253

16.3 Classic presentation of the force method  254

16.3.1 General procedure  254

16.3.2 Commentary  255

16.3.3 Deformations  257

16.3.4 Influence lines  259

16.4 Applications  262

16.5 Summary  272

16.6 Exercises  274

17 THE DISPLACEMENT METHOD  277

17.1 Independent bar end variables  277

17.1.1 General  277

17.1.2 Member stiffness relationship  277

17.1.3 Actions on bars  278

17.1.4 Algorithm for the displacement method  280

17.2 Complete bar end variables  281

17.2.1 General  281

17.2.2 Member stiffness relationship  282

17.2.3 Actions on bars  283

17.2.4 Support force variables  283

17.3 The direct stiffness method  284

17.3.1 Incidence transformation  284

17.3.2 Rotational transformation  285

17.3.3 Algorithm for the direct stiffness method  286

17.4 The slope-deflection method  290

17.4.1 General  290

17.4.2 Basic states and member end moments  292

17.4.3 Equilibrium conditions  293

17.4.4 Applications  294

17.4.5 Restraints  298

17.4.6 Influence lines  303

17.4.7 CROSS method of moment distribution  305

17.5 Summary  309

17.6 Exercises  310

18 CONTINUOUS MODELS  311

18.1 General  311

18.2 Bar extension  311

18.2.1 Practical examples  311

18.2.2 Analytical model  312

18.2.3 Residual stresses  314

18.2.4 Restraints  315

18.2.5 Bond  316

18.2.6 Summary  320

18.3 Beams in shear  321

18.3.1 Practical examples  321

18.3.2 Analytical model  321

18.3.3 Multi-storey frame  321

18.3.4 VIERENDEEL girder  323

18.3.5 Sandwich panels  324

18.3.6 Summary  326

18.4 Beams in bending  326

18.4.1 General  326

18.4.2 Analytical model  327

18.4.3 Restraints  327

18.4.4 Elastic foundation  329

18.4.5 Summary  332

18.5 Combined shear and bending response  333

18.5.1 General  333

18.5.2 Shear wall - frame systems  334

18.5.3 Shear wall connection  338

18.5.4 Dowelled beams  342

18.5.5 Summary  344

18.6 Arches  345

18.6.1 General  345

18.6.2 Analytical model  345

18.6.3 Applications  346

18.6.4 Summary  350

18.7 Annular structures  350

18.7.1 General  350

18.7.2 Analytical model  351

18.7.3 Applications  352

18.7.4 Edge disturbances in cylindrical shells  353

18.7.5 Summary  354

18.8 Cables  354

18.8.1 General  354

18.8.2 Analytical model  355

18.8.3 Inextensible cables  357

18.8.4 Extensible cables  358

18.8.5 Axial stiffness of laterally loaded cables  360

18.8.6 Summary  360

18.9 Combined cable-type and bending response  361

18.9.1 Analytical model  361

18.9.2 Bending-resistant ties  362

18.9.3 Suspended roofs and stress ribbons  363

18.9.4 Suspension bridges  368

18.9.5 Summary  368

18.10 Exercises  369

19 DISCRETISED MODELS  371

19.1 General  371

19.2 The force method  372

19.2.1 Complete and global bar end forces  372

19.2.2 Member flexibility relation  372

19.2.3 Actions on bars  374

19.2.4 Algorithm for the force method  374

19.2.5 Comparison with the classic force method  376

19.2.6 Practical application  376

19.2.7 Reduced degrees of freedom  376

19.2.8 Supplementary remarks  379

19.3 Introduction to the finite element method  381

19.3.1 Basic concepts  381

19.3.2 Element matrices  381

19.3.3 Bar element rigid in shear  381

19.3.4 Shape functions  385

19.3.5 Commentary  386

19.4 Summary  386

19.5 Exercises  387

IV NON-LINEAR ANALYSIS OF FRAMED STRUCTURES

20 ELASTIC-PLASTIC SYSTEMS  389

20.1 General  389

20.2 Truss with one degree of static indeterminacy  389

20.2.1 Single-parameter loading  389

20.2.2 Dual-parameter loading and generalisation  395

20.3 Beams in bending  398

20.3.1 Moment-curvature diagrams  398

20.3.2 Simply supported beams  399

20.3.3 Continuous beams  403

20.3.4 Frames  404

20.3.5 Commentary  405

20.4 Summary  406

20.5 Exercises  407

21 LIMIT ANALYSIS  409

21.1 General  409

21.2 Upper- and lower-bound theorems  410

21.2.1 Basic concepts  410

21.2.2 Lower-bound theorem  410

21.2.3 Upper-bound theorem  411

21.2.4 Compatibility theorem  411

21.2.5 Consequences of the upper- and lower-bound theorems  411

21.3 Static and kinematic methods  412

21.3.1 General  412

21.3.2 Simply supported beams  413

21.3.3 Continuous beams  415

21.3.4 Plane frames  416

21.3.5 Plane frames subjected to transverse loads  421

21.4 Plastic strength of materials  426

21.4.1 General  426

21.4.2 Skew bending  426

21.4.3 Bending and normal force  428

21.4.4 Bending and torsion  432

21.4.5 Bending and shear force  434

21.5 Shakedown and limit loads  435

21.6 Dimensioning for minimum weight  437

21.6.1 General  437

21.6.2 Linear objective function  438

21.6.3 FOULKES mechanisms  438

21.6.4 Commentary  440

21.7 Numerical methods  441

21.7.1 The force method  441

21.7.2 Limit load program  442

21.7.3 Optimum design  444

21.8 Summary  446

21.9 Exercises  447

22 STABILITY  449

22.1 General  449

22.2 Elastic buckling  449

22.2.1 Column deflection curve  449

22.2.2 Bifurcation problems  453

22.2.3 Approximation methods  454

22.2.4 Further considerations  460

22.2.5 Slope-deflection method  465

22.2.6 Stiffness matrices  469

22.3 Elastic-plastic buckling  471

22.3.1 Concentrically loaded columns  471

22.3.2 Eccentrically loaded columns  474

22.3.3 Limit loads of frames according to second-order theory  477

22.4 Flexural-torsional buckling and lateral buckling  480

22.4.1 Basic concepts  480

22.4.2 Concentric loading  482

22.4.3 Eccentric loading in the strong plane  483

22.4.4 General loading  485

22.5 Summary  488

22.6 Exercises  489

V PLATES AND SHELLS

23 PLATES  491

23.1 General  491

23.2 Elastic plates  491

23.2.1 Stress function  491

23.2.2 Polar coordinates  493

23.2.3 Approximating functions for displacement components  496

23.3 Reinforced concrete plate elements  496

23.3.1 Orthogonal reinforcement  496

23.3.2 General reinforcement  500

23.4 Static method  501

23.4.1 General  501

23.4.2 Truss models  501

23.4.3 Discontinuous stress fields  505

23.4.4 Stringer-panel model  511

23.5 Kinematic method  512

23.5.1 Applications in reinforced concrete  512

23.5.2 Applications in geotechnical engineering  517

23.6 Summary  520

23.7 Exercises  522

24 SLABS  525

24.1 Basic concepts  525

24.1.1 General  525

24.1.2 Static relationships  525

24.1.3 Kinematic relationships  531

24.2 Linear elastic slabs rigid in shear with small deflections  533

24.2.1 Fundamental relationships  533

24.2.2 Methods of solution  535

24.2.3 Rotationally symmetric problems  536

24.2.4 Rectangular slabs  539

24.2.5 Flat slabs  543

24.2.6 Energy methods  546

24.3 Yield conditions  547

24.3.1 VON MISES and TRESCA yield conditions  547

24.3.2 Reinforced concrete slabs  550

24.4 Static method  557

24.4.1 Rotationally symmetric problems  557

24.4.2 Moment fields for rectangular slabs  560

24.4.3 Strip method  563

24.5 Kinematic method  567

24.5.1 Introductory example  567

24.5.2 Calculating the dissipation work  568

24.5.3 Applications  569

24.6 The influence of shear forces  572

24.6.1 Elastic slabs  572

24.6.2 Rotationally symmetric VON MISES slabs  574

24.6.3 Reinforced concrete slabs  575

24.7 Membrane action  575

24.7.1 Elastic slabs  575

24.7.2 Perfectly plastic slab strip  577

24.7.3 Reinforced concrete slabs  578

24.8 Summary  581

24.9 Exercises  583

25 FOLDED PLATES  587

25.1 General  587

25.2 Prismatic folded plates  588

25.2.1 Sawtooth roofs  588

25.2.2 Barrel vaults  589

25.2.3 Commentary  593

25.3 Non-prismatic folded plates  594

25.4 Summary  594

25.5 Exercises  594

26 SHELLS  595

26.1 General  595

26.2 Membrane theory for surfaces of revolution  596

26.2.1 Symmetrical loading  596

26.2.2 Asymmetric loading  600

26.3 Membrane theory for cylindrical shells  601

26.3.1 General relationships  601

26.3.2 Pipes and barrel vaults  602

26.3.3 Polygonal domes  604

26.4 Membrane forces in shells of any form  606

26.4.1 Equilibrium conditions  606

26.4.2 Elliptical problems  607

26.4.3 Hyperbolic problems  608

26.5 Bending theory for rotationally symmetric cylindrical shells  613

26.6 Bending theory for shallow shells  615

26.6.1 Basic concepts  615

26.6.2 Differential equation for deflection  616

26.6.3 Circular cylindrical shells subjected to asymmetric loading  617

26.7 Bending theory for symmetrically loaded surfaces of revolution  620

26.7.1 Basic concepts  620

26.7.2 Differential equation for deflection  620

26.7.3 Spherical shells  621

26.7.4 Approximation for shells of any form  623

26.8 Stability  623

26.8.1 General  623

26.8.2 Bifurcation loads  624

26.8.3 Commentary  626

26.9 Summary  627

26.10 Exercises  628

APPENDIX

A1 DEFINITIONS  631

A2 NOTATION  637

A3 PROPERTIES OF MATERIALS  643

A4 GEOMETRICAL PROPERTIES OF SECTIONS  645

A5 MATRIX ALGEBRA  649

A5.1 Terminology  649

A5.2 Algorithms  650

A5.3 Linear equations  652

A5.4 Quadratic forms  652

A5.5 Eigenvalue problems  653

A5.6 Matrix norms and condition numbers  654

A6 TENSOR CALCULUS  655

A6.1 Introduction  655

A6.2 Terminology  655

A6.3 Vectors and tensors  656

A6.4 Principal axes of symmetric second-order tensors  658

A6.5 Tensor fields and integral theorems  658

A7 CALCULUS OF VARIATIONS  661

A7.1 Extreme values of continuous functions  661

A7.2 Terminology  661

A7.3 The simplest problem of calculus of variations  662

A7.4 Second variation  663

A7.5 Several functions required  664

A7.6 Higher-order derivatives  664

A7.7 Several independent variables  665

A7.8 Variational problems with side conditions  665

A7.9 The RITZ method  666

A7.10 Natural boundary conditions  667

REFERENCES  669

NAME INDEX  671

SUBJECT INDEX  673