MECHANICS OF OPTIMAL STRUCTURAL DESIGN Minimum Weight Structures

David W. A. Rees School of Engineering and Design, Brunei University, Uxbridge, UK

©WILEY A John Wiley and Sons, Ltd., Publication

Contents

Preface xi

Glossary of Terms xv

Key Symbols xix

Chapter 1 Compression of Slender Struts 1 1.1 Introduction 1 1.2 Failure Criteria 1 1.3 Solid Cross-Sections 3 1.4 Thin-Walled, Tubular Sections 6 1.5 Thin-Walled, Open Sections 13 1.6 Summary of Results 24 References 25 Exercises 25

Chapter 2 Compression of Wide Struts 29 2.1 Introduction 29

29 31 37 57 60 61 61

65 65 66 69 76 88

References 89 Exercises 89

2.2 2.3 2.4 2.5 2.6

Failure Criteria Cellular Sections Open Sections Corrugated Sandwich Panel Summary of Results

References Exercise

Chapter 3 Bending of Slender Beams 3.1 3.2 3.3 3.4 3.5

Introduction Solid Cross-Sections Thin-Walled, Tubular Sections Open Sections Summary of Results

vi Contents

Chapter 4 Torsion of Bars and Tubes 91 4.1 Introduction 91 4.2 Solid Cross-Sections 92 4.3 Thin-Walled, Open Sections 99 4.4 Thin-Walled, Closed Tubes 109 4.5 Multi-Cell Tubes 121 References 130 Exercises 130

Chapter 5 Shear of Solid Bars, Tubes and Thin Sections 135 5.1 Introduction 135 5.2 Bars of Solid Section 136 5.3 Thin-Walled Open Sections 143 5.4 Thin-Walled, Closed Tubes 159 5.5 Concluding Remarks 170 References 171 Exercise 171

Chapter 6 Combined Shear and Torsion in Thin-Walled Sections 173 6.1 Introduction 173 6.2 Thin-Walled, Open Sections 173 6.3 Thin-Walled, Closed Tubes 177 6.4 Concluding Remarks 189 References 190 Exercises 190

Chapter 7 Combined Shear and Bending in Idealised Sections 193 7.1 Introduction 193 7.2 Idealised Beam Sections 193 7.3 Idealised Open Sections 201 7.4 Idealised Closed Tubes 210 References 221 Exercises 221

Chapter 8 Shear in Stiffened Webs 223 8.1 Introduction 223 8.2 Castellations in Shear 223 8.3 Corrugated Web 226 8.4 Flat Web with Stiffeners 231 References 237 Exercises 237

Chapter 9 Frame Assemblies 239 9.1 Introduction 239 9.2 Double-Strut Assembly 239 9.3 Multiple-Strut Assembly 244

Contents vii

9.4 Cantilevered Framework 247 9.5 Tetrahedron Framework 253 9.6 Cantilever Frame with Two Struts 256 9.7 Cantilever Frame with One Strut 259 References 264 Exercises 264

Chapter 10 Simply Supported Beams and Cantilevers 265 10.1 Introduction 265 10.2 Variable Bending Moments 265 10.3 Cantilever with End-Load 271 10.4 Cantilever with Distributed Loading 281 10.5 Simply Supported Beam with Central Load 292 10.6 Simply Supported Beam with Uniformly Distributed Load 303 10.7 Additional Failure Criteria 316 References 322 Exercises 323

Chapter 11 Optimum Cross-Sections for Beams 325 11.1 Introduction 325 11.2 Approaching Optimum Sections 326 11.3 Generalised Optimum Sections 328 11.4 Optimum Section, Combined Bending and Shear 330 11.5 Solid, Axisymmetric Sections 331 11.6 Fully Optimised Section 341 11.7 Fully Optimised Weight 345 11.8 Summary 355 References 356 Exercises 356

Chapter 12 Structures under Combined Loading 357 12.1 Introduction 357 12.2 Combined Bending and Torsion 357 12.3 Cranked Cantilever 359 12.4 Cranked Strut with End-Load 362 12.5 Cranked Bracket with End-Load 365 12.6 Portal Frame with Central Load 368 12.7 Cantilever with End and Distributed Loading 371 12.8 Centrally Propped Cantilever with End-Load 377 12.9 End-Propped Cantilever with Distributed Load 385 12.10 Simply Supported Beam with Central-Concentrated and Distributed

Loadings 390 12.11 Centrally Propped, Simply Supported Beam with Distributed Load 395 References 400 Exercises 400

viii Contents

Chapter 13 Encastre Beams 403 13.1 Introduction 403 13.2 Central-Concentrated Load 403 13.3 Uniformly Distributed Load 418 13.4 Combined Loads 437 References 463 Exercises 463

Chapter 14 Plastic Collapse of Beams and Frames 465 14.1 Introduction 465 14.2 Plane Frames 466 14.3 Beam Plasticity 468 14.4 Collapse of Simple Beams 474 14.5 Encastre Beams 478 14.6 Continuous Beams 481 14.7 Portal Frames 486 14.8 Effect of Axial Loading upon Collapse 497 14.9 Effect of Shear Force upon Collapse 500 14.10 Effect of Hardening upon Collapse 505 References 507 Exercises 507

Chapter 15 Dynamic Programming 511 15.1 Introduction 511 15.2 Single-Span Beam 511 15.3 Two-Span Beam 513 15.4 Three-Span Beam 515 15.5 Design Space 517 Reference 520 Exercises 520

Appendix A Mechanical Properties 521 A.l Non-Metals 521 A.2 Metals and Alloys 522 References 524

Appendix В Plate Buckling Under Uniaxial Compression 525 B.l Wide and Slender Struts 525 B.2 Plates with Supported Sides 527 B.3 Inelastic Buckling 530 B.4 Post-Buckling 533 References 534

Appendix С Plate Buckling Under Biaxial Compression and Shear 537 C.l Biaxial Compression 537 C.2 Pure Shear 539

Contents ix

C.3 Inelastic Shear Buckling References

Appendix D Secondary Buckling D.l Buckling Modes D.2 Local Compressive Buckling D.3 Global Buckling D.4 Local Shear Buckling References

Bibliography

Index

541 541

543 543 544 545 547 547

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