Structures Theory and Analysis

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Structures Theory and Analysis

~macmillan ~ international

HIGHER EDUCATION

~RED GLOBE \<ffi7 PRESS

M.S. WILLIAMS J.D. TODD

© M. S. Williams 2000, under exclusive licence to Springer Nature Limited 2019

All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission.

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Published by RED GLOBE PRESS

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Contents

Preface xi

Acknowledgements xii

Notation xiii

1 Introducing structures 1.1 What is a structure? 2 1.2 Elements of the structural system 3

1.2.1 Loads on structures 3 1.2.2 Supports 5 1.2.3 Members, joints and structural actions 7 1.2.4 Structural materials 13

1.3 Structural systems 14 1.3.1 Pin-jointed frames 15 1.3.2 Moment frames 17 1.3.3 High-rise frames 21 1.3.4 Cable and arch structures 22 1.3.5 Continuum structures 26

1.4 Structural analysis and design 27 1.5 The role of computers in structural analysis 29 1.6 Further reading 29

2 Plane statics 31 2.1 Basic structural principles 32

2.1.1 Forces and displacements 32 2.1.2 Sign convention 32 2.1.3 Equilibrium of forces 33 2.1.4 Free body diagrams 34 2.1.5 The force polygon 35 2.1.6 Statical determinacy 35 2.1.7 The principle of superposition 36 2.1.8 Compatibility of displacements 36

2.2 Determination of reactions 37 2.2.1 Stability and determinacy of reactions 38 2.2.2 Calculation of reactions 40

2.3 Internal forces in structures 45 2.3.1 Sign convention for internal forces 46 2.3.2 Relations between load, shear force and bending moment 49

2.4 Shear force and bending moment diagrams for beams 50 2.4.1 Use of superposition 54

v

vi ~ Contents

2.4.2 Relating bending moment diagrams to deflected shapes 56 2.4.3 Minimisation of bending moments 58

2.5 Problems 59

3 Statically determinate structures 62 3.1 Plane pin-jointed trusses 63

3.1.1 Stability and determinacy 63 3.1.2 Resolution at joints 66 3.1.3 Method of sections 69 3.1.4 Introduction to matrix methods 71

3.2 Three-dimensional pin-jointed trusses 74 3.2.1 Tension coefficients 75

3.3 Statically determinate moment frames 78 3.3.1 Stability and determinacy 78 3.3.2 Calculation of internal forces 79 3.3.3 Qualitative analysis 85

3.4 Cable and arch structures 87 3.4.1 Suspension cables 87 3.4.2 Parabolic arches 90 3.4.3 Three-pinned arches 91

3.5 Problems 92

4 Stress and strain 97 4.1 Stress 98

4.1.1 Direct stress 98 4.1.2 Shear stress 99

4.2 Strain 101 4.3 One-dimensional elasticity 104

4.3.1 Stress-strain relations 104 4.3.2 Statically determinate systems of parallel bars 106 4.3.3 Redundant systems of parallel bars 108 4.3.4 Thermal stresses and strains 109 4.3.5 Strain energy 111

4.4 Two-dimensional elasticity 112 4.4.1 Resolution of plane stresses 112 4.4.2 Mohr's circle for stress 115 4.4.3 Mohr's circle for strain 119 4.4.4 Qualitative analysis of plane stress systems 121 4.4.5 Relations between elastic constants 123 4.4.6 Pressure vessels 125 4.4.7 Plane strain problems 126

4.5 Elementary plasticity 128 4.5.1 Yielding 128 4.5.2 One-dimensional plasticity problems 129

4.6 Further reading 132 4.7 Problems 132

5 Bending of beams 135 5.1 The basics of bending behaviour 136

Contents~~ 5.2 Elastic bending stresses 137

5.2.1 Elementary bending theory 137 5.2.2 Bending properties of plane areas 139 5.2.3 Composite beams 145 5.2.4 Strain energy 147

5.3 Elastic deflections 147 5.3.1 Direct integration method for simple beams 147 5.3.2 Analysis of statically indeterminate beams 149 5.3.3 Macaulay's notation for continuous beams 150 5.3.4 Moment-area methods 153 5.3.5 Use of standard solutions 154 5.3.6 Qualitative analysis 157

5.4 Elasto-plastic bending 159 5.4.1 Plastic moment of a section 160 5.4.2 Residual deformations 162 5.4.3 Influence of axial load 164

5.5 Problems 165

6 Torsion and shear 168 6.1 Torsion of circular sections 169

6.1.1 Elastic torque--stress relationships 169 6.1.2 Torsion combined with other loads 173 6.1.3 Yielding in torsion 175

6.2 Torsion of thin-walled non-circular sections 176 6.2.1 Closed sections 176 6.2.2 Thin plates and open sections 179

6.3 Shear stresses in bending 181 6.3.1 Elastic shear stress distributions 181 6.3.2 Application to typical structural sections 184 6.3.3 Unsymmetrical sections and shear centre 187

6.4 Shear deflections 188 6.5 Problems 190

7 Virtual work and influence lines 193 7.1 The principle of virtual work 194 7.2 Application to statically determinate problems 195

7.2.1 Statically determinate reactions 195 7.2.2 Deflections of statically determinate trusses 197

7.3 Forces in redundant trusses 199 7.3.1 Single redundancy 199 7.3.2 Multiple redundancies 201 7.3.3 Temperature effects and lack of fit 203

7.4 Flexural problems 204 7.4.1 The virtual work equation for flexure 204 7.4.2 Analysis of indeterminate beams and frames 206 7.4.3 Curved elements 208

7.5 Betti's reciprocal theorem 209 7.6 Influence lines 210

7.6.1 Determination of influence lines for statically determinate beams 210

viii I Contents ~'mt

7.6.2 Influence lines for redundant beams 213

7.6.3 Applications of influence lines 215

7.7 Further reading 216

7.8 Problems 217

8 Moment distribution 221

8.1 Outline of method 221

8.1.1 Sign convention 223

8.1.2 Fixed-end moments 224

8.1.3 Stiffness, carry-over and distribution factors 225

8.2 Application to statically indeterminate structures 227

8.2.1 Continuous beams 227

8.2.2 Beams with support movements 231

8.2.3 Rigid-jointed frames 233

8.3 Problems 236

9 The stiffness matrix method 239

9.1 The slope-deflection method 240

9.1.1 Deflected shapes of rigid-jointed frames 240

9.1.2 Sign convention and notation 243

9.1.3 The slope-deflection equations 244 9.1.4 Application to indeterminate structures 245

9.2 Stiffness matrices for bars and trusses 249 9.2.1 Sign convention and notation for computer applications 250

9.2.2 Local stiffness matrix for an axially loaded element 251 9.2.3 Global stiffness matrix for a simple two-bar structure 252 9.2.4 Boundary conditions 254 9.2.5 Global stiffness matrices for plane trusses 255

9.3 Stiffness matrices for beams and rigid-jointed frames 258 9.3.1 Local stiffness matrices for 2D and 3D beam elements 259 9.3.2 Global stiffness matrices for plane rigid-jointed frames 261

9.4 Applications and special cases 266 9.4.1 Hand calculations using stiffness matrices 266 9.4.2 Members loaded between the joints 267 9.4.3 Frames with internal pins 269 9.4.4 Grillages 271 9.4.5 Stiffness matrix computer programs 273 9.4.6 Checking stiffness matrix analyses 274

9.5 Further reading 281 9.6 Problems 281

10 The finite element method 286 10.1 Outline of method 287 10.2 A simple plane stress finite element: the constant-strain triangle 288

10.2.1 Shape and displacement functions 288 10.2.2 Element strain and stress matrices 292 10.2.3 Element stiffness matrix 294 1 0.2.4 Assembly of structure stiffness matrix 296 10.2.5 Elements loaded between the nodes 298

Contents I ix KKC

10.3 Other element types 300 10.3.1 Other plane stress triangles 300 10.3.2 Plane stress quadrilaterals 303 10.3.3 Plate bending elements 305 1 0.3.4 Three-dimensional solid elements 305

10.4 Using finite element computer programs 306 1 0.4.1 Choice of element type and mesh density 308 1 0.4.2 Checking finite element analyses 310 1 0.4.3 A cautionary tale 312

10.5 Further reading 312 10.6 Problems 312

11 Buckling and instability 315 11.1 Stable and unstable equilibria 316 11.2 Buckling of struts 318

11.2.1 Buckling behaviour 318 11.2.2 Euler buckling load for a perfect pin-ended strut 318 11.2.3 Struts with other end conditions 321 11.2.4 Effect of initial imperfections 326 11.2.5 Interaction of buckling and yielding 329 11.2.6 Application to design 332

11.3 Energy methods for struts 334 11.3.1 Potential energy of a buckled strut 334 11.3.2 Rayleigh's method 335

11.4 Lateral-torsional buckling of beams 338 11.5 Other forms of instability 340 11.6 Further reading 340 11.7 Problems 341

12 Plastic analysis of structures 344 12.1 Plastic collapse of structures 345 12.2 The development of a plastic hinge 346 12.3 Plastic analysis of beams 350

12.3.1 Collapse mechanisms 350 12.3.2 Statical method of analysis 353 12.3.3 Virtual work approach 356 12.3.4 Load factor 357 12.3.5 Theorems used in plastic analysis 358 12.3.6 Minimisation problems 360

12.4 Plastic analysis of frames 361 12.4.1 Rectangular portal frames 361 12.4.2 Analysis of more complex structures 364 12.4.3 Combination of mechanisms 367

12.5 Comparison of elastic and plastic design approaches 370 12.6 Further reading 370 12.7 Problems 371

13 Structural dynamics 374 13.1 Single-degree-of-freedom systems 376

13.1.1 Undamped free vibrations 377

x I Contents

13.1.2 Damped free vibrations 13.1 .3 Forced vibration 13.1.4 General dynamic loading

13.2 Multi-degree-of-freedom systems 13.2.1 Equations of motion 13.2.2 Free vibrations of MDOF systems 13.2.3 Forced vibration response

13.3 Earthquake loading 13.4 Further reading 1 3.5 Problems

Appendix A: Axis and Sign Convention Appendix 8: Glossary Answers to Problems Index

379 383 389 392 392 396 399 403 406 406

410 416 422 427

Preface

This book is the successor to Joseph Todd's Structural Theory and Analysis, first published in 1974 and

reprinted frequently throughout the 1970s and 1980s. It is not a new edition of that text, but rather

a completely new book, radically different in content, organisation and style. We have, however,

tried to keep as close as possible to the principles that underpinned Todd's original text: that the

material should be presented in as clear and concise a way as possible; that both qualitative

understanding of structural behaviour and a firm grasp of the mathematical theory are important;

and that a sound knowledge of basic structural theory is an essential prerequisite to the introduction

of computer methods.

The book is deliberately broad, aiming to cover in reasonable detail most of the structures topics

that an undergraduate is likely to meet in a civil or general engineering degree course. Given the

broad scope it is not, of course, possible to go into enormous depth on all topics. Reading lists are

therefore included for topics where more in-depth study may be desirable.

The opening chapter sets the scene by introducing the topic of structures in an entirely non­

mathematical way. Using well-known structures as examples, it explains how the most common

structural forms work and provides the motivation for the remainder of the book. The following five

chapters lay the foundations of structural theory, covering statically determinate structures and the

basics of the theory of elasticity. The book then moves on to cover analysis methods appropriate for

indeterminate structures, where both statics and elasticity are required in order to obtain solutions. It is in this form of analysis that the use of computer methods is now near-universal. We have

therefore kept the account of traditional methods such as virtual work and moment distribution as

concise as possible and gone into rather more detail on the computer-oriented stiffness matrix and

finite element approaches. The book concludes with three chapters on topics of fundamental

importance to structural engineers- stability, plastic analysis and structural dynamics.

With the exception of Chapter l, which is entirely qualitative, an attempt has made to integrate

mathematical analysis techniques with explanations of structural behaviour throughout. It is our

belief that these two must be taught in parallel - a qualitative appreciation of complex structural

behaviour cannot be developed without knowledge of the relevant theory, and conversely the

choice of an appropriate analytical approach must be guided by a physical understanding of what is

going on. The issue of sign conventions in structural engineering is a particularly thorny one - there is no

convention that pleases everyone, or that entirely eliminates the occasional occurrence of

unexpected and irritating minus signs. We have used a consistent sign convention as far as possible

in this text and, to minimise confusion, have explained it in detail in Appendix A. Where numerical

examples are given, the units are either kiloNewtons and metres, or Newtons and millimetres; we

have tried to avoid mixing the two systems within a single example. A glossary of terms frequently

used in structural engineering has been provided in Appendix B, though every effort has been made

to keep jargon to a minimum.

M.S. Williams

J.D. Todd

xi

xii

Acknowledgements

We wish to thank Malcolm Stewart and Christopher Glennie of Macmillan Press, for their help and encouragement during the preparation of the text. We are grateful to colleagues in the Department of Engineering Science and at New College, Oxford, for their advice and support, particularly Guy Houlsby, David Clarke and Michael Burden. The finite element mesh in Figure l 0.17 is reproduced by kind permission of Charles Augarde. We are grateful to Manti Mendi for his assiduous checking of the worked examples and problems, though the authors alone are responsible for any remaining errors. Lastly, we are indebted to E.J. Milner-Gulland, whose unfailing support was essential to the completion of this project.

Notation

The notation used accords with accepted conventions as far as possible. This has the unfortunate result that a few symbols have more than one meaning. However, the correct meaning should be obvious from the context. SI units are given in square brackets.

A

Ae a B

b

F F f

fn G H h I

J K K k k kA ks I le M M

Cross-sectional area [m2]

Area enclosed by mean perimeter of closed, thin-walled section [m2]

Acceleration [m/s2]. or Robertson constant (empirical factor used in strut design) Element strain matrix Section breadth [m] Damping coefficient [Ns/m] Carry-over factor for bar ij - ratio of moment at end j to that at end i Element stress matrix Section depth [m] Distribution factor for bar ij Young's modulus (constant of proportionality between direct stress and strain) [Pa] Equilibrium matrix Extension [ m] Concentrated load (or W, or P) [N] Load vector (global coordinates) Load vector (local coordinates) Natural frequency [Hz] Shear modulus (constant of proportionality between shear stress and strain) [Pa] Horizontal reaction [N] or warping constant [m6]

Height [m] Second moment of area [m4]

Polar second moment of area [m4 ]

Bulk modulus (constant relating mean stress to volumetric strain) [Pa] Stiffness matrix (global coordinates) Stiffness Stiffness matrix (local coordinates) Axial stiffness [N /m] Bending stiffness [Nm/rad] Length [m] Effective length [ m] Bending moment [Nm] Mass matrix Critical moment at which lateral-torsional buckling occurs [Nm] Displacement-related moment at end i of bar ij [Nm] Fixed-end moment at end i of bar ij [Nm] Plastic moment [Nm]

xiii

xiv I Notation

r

ry s s S;

Sij

T

T

t tij

u u u U, V, W

w X X

X, y, Z

y, z Ze Zp Z;

a y 8 s s

v

Yield moment [Nm] Mass [kg], or modular ratio of two materials (= EI/E2) Concentrated load (or W, or F) [N] Critical buckling load for a strut. or collapse load for a structure [N] Euler buckling load for a pin -ended strut [N] Rayleigh buckling load for a strut [N] Pressure [Pa] Shear flow (shear stress x thickness) [N/m] Radius of curvature [m] Radius [m] Radius of gyration about the minor axis [m] Shear force [N] Shape function matrix ith shape function for a finite element Rotational stiffness of bar ij when loaded by a moment at end i [Nm/rad] Axial force [N], or natural period of vibration [s] Transformation matrix between local and global coordinate systems Thickness [m] Bar force vector Tension coefficient for bar ij (axial force divided by length) [N/m] Strain energy [ J] Displacement vector (global coordinates) Displacement vector (local coordinates) Displacement components in x, y and z directions (or Ux, Uy, u,) [m] Velocity [m/s] Acceleration [m/s2]

Horizontal ground displacement [m] Displacement components in x, y and z directions (or u, v, w) [m] Vertical reaction [N], or volume [ m3], or potential energy [ J] Concentrated load (or P, or F) [N] Uniformly distributed load [N/m], or displacement component in z direction [m] Nodal coordinate vector for a finite element Coordinate vector for an arbitrary point within a finite element Axis directions Coordinates of centroid of cross-section [m] Elastic modulus [m3]

Plastic modulus [m3]

Generalised coordinate for the ith mode [ m] Coefficient of thermal expansion [;oc], or shape factor(= Zp/Ze) Shear strain, or safety factor (= yield load 7 working load) Deflection of a point [m], or logarithmic decrement (a measure of damping) Direct strain Element strain vector Principal strains Damping ratio (proportion of critical damping) Perry factor for a strut (also known as the initial curvature parameter) Angle of rotation [rads], or temperature [°C] Slenderness ratio for a strut(= le/ry). or load factor (=collapse load 7 working load) Poisson's ratio

Notation I xv h~~-. -

a Direct stress [Pa] 11 Element stress vector a1, a2 Principal stresses [Pa] ac Critical axial stress at buckling [Pa] ah Hoop, or circumferential stress [Pa] a1 Longitudinal stress [Pa] a, Radial stress [Pa] ay Yield stress [Pa] r Shear stress [Pa] ry Yield stress in shear [Pa] ¢ Angle of twist, or angle between local and global axes [ rads] 'II Modal matrix (comprising a set of mode shapes written as column vectors) t!J Mode shape vector Q Frequency ratio (=loading frequency-;- natural frequency) w Circular frequency [rad/s] Wct Damped circular natural frequency [ rad/s] Wn Undamped circular natural frequency [rad/s]