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Authorized training centre of Singapore Institute of Engineering Technologists

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PE (Structure)

The resources contain the relevant references for a Professional Engineer working in Structural Design Industry

Resources

Topics in Steel Design

Preface vi

Chapter 1 Introduction 1-1

Chapter 2 Concepts in Structural Steel Design 2-1

Chapter 3 Tension Members 3-1

Chapter 4 Compression Members 4-1

Chapter 5 Beams 5-1

Chapter 6 Beam-Columns 6-1

Chapter 7 Simple Connections 7-1

Chapter 8 Eccentric Connections 8-1

Chapter 9 Composite Construction 9-1

Chapter 10 Plate Girders 10-1

Topics in Reinforced Concrete & Design

http://www.highlightcomputer.com/PE-Structure.pdf

Topics in Theory of Structure

CONTENTS

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

Contents VII

Theory of

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

Topics in Timber Engineering

Contents

List of Contributors vii

Preface ix

1 Timber Engineering - General Introduction 1

Sven Thelandersson

Part One Basic Properties of Wood-Based Structural Elements 13

2 Introduction: Wood as a Construction Material 15

Sven Thelandersson

3 Grading of Timber with Respect to Mechanical Properties 23

Carl-Johan Johansson

4 Structural Timber - Variability and Statistical Modelling 45

Tord Isaksson

5 Mechanical Performance and Modelling of Glulam 67

Erik Serrano

6 Engineered Wood Products for Structural Purposes 81

Frank Lam, Helmut Prion

7 Fracture Perpendicular to Grain - Structural Applications 103

Per Johan Gustafsson

8 Strength Under Long-term Loading 131

Preben Hoffmeyer

9 Effects of Climate and Climate Variations on Strength 153

Alpo Ranta-Maunus

Part Two Design Aspects of Timber Structures 169

10 Introduction: Safety and Serviceability in Timber Engineering 171

Sven Thelandersson

11 Reliability of Structures with Timber and Wood-Based Products 177

Ricardo O. Foschi

12 Design of Structures based on Glulam, LVL and Other Solid Timber Products 201

Hans J. Larsen

13 Short- and Long-term Deformations of Timber Structures 221

Annika Martensson

14 Vibrations of Timber Floors: Serviceability Aspects 241

Ian Smith

15 Design for Earthquake Resistance 267

Erol Karacabeyli and Marjan Popovski

Part Three Joints and Structural Assemblies 301

16 Introduction: Fasteners, Joints and Composite Structures 303

Hans J. Larsen

17 Joints with Dowel-type Fasteners 315

Hans J. Blass

18 Structural Adhesive Joints Including Glued-in Bolts 333

Simon Aicher

19 Trusses and Joints with Punched Metal Plate Fasteners 365

Jacob Nielsen

20 Shear Walls and Diaphragms 383

Helmut G.L. Prion, Frank Lam

21 Composite Structures 409

Ario Ceccotti

Index 429