mechanics of materials volume 2 e j hearn

MECHANICS OF MATERIALS 2 An introduction to the Mechanics of Elastic and Plastic Deformation of Solids and Structural Materials THIRD EDITION E. J. HEARN PhD; BSc(Eng) Hons; CEng; FIMechE; FIProdE; FIDiagE University of Warwick United Kingdom la== EINEMANN Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 225 Wildwood Avenue, Woburn, MA 01801-2041 A division of Reed Educational and Professional Publishing Ltd -& A member of the Reed Elsevier plc group OXFORD AUCKLAND BOSTON MELBOURNE NEW DELHI First published 1977 Reprinted with corrections 1980, 1981, 1982 Second edition 1985 Reprinted with corrections 1989 Reprinted 1992, 1995, 1996 Third edition 1997 Reprinted 1 ?99 0 E.J. Hem 1977, 1985, 1997 All rights reserved. No part of this publication may be reproduced in any materiai form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England WlP 9HE. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to the publishers British Library Cataloguing in Publication Data Hem, E. J. (Edwin John) Mechanics of materials. - 3rd ed. 1. materials 1. Strength of materials 2. Strains and stress 3. Deformations (Mechanics) 4. Elasticity I. Title 620.1 12 An introduction to the mechanics of elastic and plastic deformation of solids and structural ISBN 0 7506 3266 6 Library of Congress Cataloguing in Publication Data Heam, E. J. (Edwin John) Mechanics of materials 1: an introduction to the mechanics of elastic and plastic deformation of solids and structural materialsE. J. Heam. - 3rd ed. p. cm. Includes bibliographical references and index. ISBN 0 7506 3266 6 1. Strength of materials. I. Title TA405.H3 620.1’123-dc21 96-49967 CIP Typeset by Laser Words, Madras, lndia Rinted and bound in Great Britain Also of interest ASHBY Materials Selection in Mechanical Design ASHBY & JONES Engineering Materials 1 Engineering Materials 2 BRANDES & BROOK Smithells Metals Reference Book, 7th Edition BRYDSON Plastics Materials, 6th Edition CAMPBELL Castings CHARLES, CRANE & FURNESS Selection and Use of Engineering Materials, 2nd Edition CRAWFORD Plastics Engineering, 2nd Edition HEARN Mechanics of Materials 1 HULL & BACON Introduction to Dislocatioils, 3rd Edition JONES Engineering Materials 3 LLEWELLYN Steels: Metallurgy & Applications SMALLMAN & BISHOP Metals and Materials CONTENTS Introduction xv Notation xvii 1 Unsymmetrical Bending 1 Summary Introduction I .1 1.2 1.3 1.4 1.5 I .6 1.7 Momenta1 ellipse 1.8 Stress determination 1.9 Alternative procedure for stress determination 1.10 Alternative procedure using the momenta1 ellipse 1.1 1 Dejections Examples Problems Product second moment of area Principal second moments of area Mohr’s circle of second moments of area Land’s circle of second moments of area Rotation of axes: determination of moments of area in terms of the principal values The ellipse of second moments of area 2 Struts Summary Introduction 2.1 Euler’s theory 2.2 Equivalent strut length 2.3 2.4 Euler “validity limit” 2.5 Rankine or Rankine-Gordon formula 2.6 Perry- Robertson formula 2.7 2.8 Struts with initial curvature 2.9 Struts with eccentric load 2.10 Laterally loaded struts 2.1 1 Comparison of Euler theory with experimental results British Standard procedure (BS 449) Alternative procedure for any strut-loading condition 8 9 11 11 11 13 15 16 24 28 28 30 31 35 36 37 38 39 41 41 42 46 48 V vi Contents 2.1 2 Struts with unsymmetrical cross-section Examples Problems 3 Strains Beyond the Elastic Limit Summary Introduction 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.1 1 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 Plastic bending of rectangular-sectioned beams Shape factor - symmetrical sections Application to I-section beams Partially plastic bending of unsymmetrical sections Shape factor - unsymmetrical sections Dejections of partially plastic beams Length of yielded area in beams Collapse loads - plastic limit design Residual stresses after yielding: elastic-perfectly plastic material Torsion of shafts beyond the elastic limit - plastic torsion Angles of twist of shafts strained beyond the elastic limit Plastic torsion of hollow tubes Plastic torsion of case-hardened shafts Residual stresses after yield in torsion Plastic bending and torsion of strain-hardening materials (a) Inelastic bending (b) Inelastic torsion Residual stresses - strain-hardening materials Influence of residual stresses on bending and torsional strengths Plastic yielding in the eccentric loadirzg of rectangular sections Plastic yielding and residual stresses under axial loading with stress concentrations Plastic yielding of axially symmetric components (a) Thick cylinders - collapse pressure (b) Thick cylinders - “auto-frettage ’’ (c) Rotating discs Examples Problems 4 Rings, Discs and Cylinders Subjected to Rotation and Thermal Gradients Summary 4.1 4.2 Rotating solid disc 4.3 4.4 4.5 Thin rotating ring or cylinder Rotating disc with a central hole Rotating thick cylinders or solid shafs Rotating disc of uniform strength 49 50 56 61 61 62 64 65 67 67 69 69 69 71 73 75 77 77 79 79 80 80 83 84 84 85 86 87 87 89 94 96 109 117 117 118 119 122 124 125 Contents 4.6 Combined rotational and thermal stresses in uniform discs and thick cylinders Examples Problems 5 Torsion of Non-Circular and Thin- Walled Sections Summary 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.1 1 Rectangular sections Narrow rectangular sections Thin-walled open sections Thin-walled split tube Other solid (non-tubular) shafts Thin-walled closed tubes of non-circular section (Bredt-Batho theory) Use of “equivalent J” for torsion of non-circular sections Thin-walled cellular sections Torsion of thin-walled stifSened sections Membrane analogy EfSect of warping of open sections Examples Problems 6 Experimental Stress Analysis Introduction 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.1 1 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 Brittle lacquers Strain gauges Unbalanced bridge circuit Null balance or balanced bridge circuit Gauge construction Gauge selection Temperature compensation Installation procedure Basic measurement systems D.C. and A.C. systems Other types of strain gauge Photoelasticity Plane-polarised light - basic polariscope arrangements Temporary birefringence Production of fringe patterns Interpretation of fringe patterns Calibration Fractional fringe order determination - compensation techniques Isoclinics-circular polarisation Stress separation procedures Three-dimensional photoelasticity vii 126 129 136 141 141 142 143 143 145 145 147 149 150 151 152 153 154 160 166 166 167 171 173 173 173 175 175 176 177 179 180 181 182 183 184 185 186 187 188 190 1 90 VI11 Contents 6.22 Rejective coating technique 6.23 Other methods of strain measurement Bibliography 7 Circular Plates and Diaphragms Summary A. CIRCULAR PLATES 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.1 1 7.12 7.13 7.14 Stresses Bending moments General equation for slope and dejection General case of a circular plate or diaphragm subjected to combined uniformly distributed load q (pressure) and central concentrated load F Uniformly loaded circular plate with edges clamped Uniformly loaded circular plate with edges freely supported Circular plate with central concentrated load F and edges clamped Circular plate with central concentrated load F and edges freely supported Circular plate subjected to a load F distributed round a circle Application to the loading of annular rings Summary of end conditions Stress distributions in circular plates and diaphragms subjected to lateral pressures Discussion of results - limitations of theory Other loading cases of practical importance B. BENDING OF RECTANGULAR PLATES 7.15 7.16 Rectangular plates with simply supported edges carrying uniformly distributed loads Rectangular plates with clamped edges carrying uniformly distributed loads Examples Problems 8 Introduction to Advanced Elasticity Theory 8.1 Types of stress 8.2 8.3 8.4 The Cartesian stress components: notation and sign convention 8.2.1 Sign conventions The state of stress at a point Direct, shear and resultant stresses on an oblique plane 8.4.1 8.4.2 Line of action of resultant stress Line of action of normal stress 190 192 192 193 193 195 195 197 198 199 200 202 203 205 206 208 208 209 21 1 212 213 213 214 215 218 220 220 220 22 1 22 1 224 226 227 Contents ix 8.5 8.6 8.7 8.8 8.9 8.10 8.1 1 8.12 8.13 8.14 8.15 8.16 8.17 8.18 8.19 8.20 8.21 8.22 8.23 8.24 8.25 8.26 8.27 8.4.3 8.4.4 Principal stresses and strains in three dimensions - Mohr 's circle representation Graphical determination of the direction of the shear stress r,, on an inclined plane in a three-dimensional principal stress system The combined Mohr diagram for three-dimensional stress and strain systems Application of the combined circle to two-dimensional stress systems Graphical construction for the state of stress at a point Construction for the state of strain on a general strain plane State of stress-tensor notation The stress equations of equilibrium Principal stresses in a three-dimensional Cartesian stress system 8.13.1 Solution of cubic equations Stress invariants - Eigen values and Eigen vectors Stress invariants Reduced stresses Strain invariants Alternative procedure for determination of principal stresses 8.1 8.1 Evaluation of direction cosines for principal stresses Octahedral planes and stresses Deviatoric stresses Deviatoric strains Plane stress and plane strain 8.22.1 Plane stress 8.22.2 Plane strain The stress-strain relations The strain-displacement relationships The strain equations of transformation Compatibility The stress function concept 8.27.1 Forms of Airy stress function in Cartesian coordinates 8.27.2 Case 1 - Bending of a simply supported beam by a uniformly 8.27.3 The use of polar coordinates in two dimensions 8.27.4 Forms of stress function in polar coordinates 8.27.5 Case 2 - hi-symmetric case: solid shaft and thick cylinder radially loaded with uniform pressure 8.27.6 Case 3 - The pure bending of a rectangular section curved beam 8.27.7 Case 4 - Asymmetric case n = 1. Shear loading of a circular arc cantilever beam 8.27.8 Case 5 - The asymmetric cases n >, 2 -stress concentration at a circular hole in a tension$eld Line of action of shear stress Shear stress in any other direction on the plane distributed loading 227 227 228 229 230 232 234 235 235 236 242 242 243 244 246 247 247 248 249 25 1 25 3 254 255 255 256 257 259 26 1 263 265 267 27 1 272 273 273 274 276 X Contents 8.27.9 Other useful solutions of the biharmonic equation Examples Problems 9 Introduction to the Finite Element Method 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.1 1 Introduction Basis of the finite element method Applicability of the finite element method Formulation of the Jinite element method General procedure of the Jinite element method 9.4.1 Identification of the appropriateness of analysis by the jinite element method 9.4.2 Identification of the type of analysis 9.4.3 Idealisation 9.4.4 Discretisation of the solution region 9.4.5 Creation of the material model 9.4.6 Node and element ordering 9.4.7 Application of boundary conditions 9.4.8 Creation of a data file 9.4.9 Computer, processing, steps 9.4.10 Interpretation and validation of results 9.4.1 1 Modification and re-run Fundamental arguments 9.5.1 Equilibrium 9.5.2 Compatibility 9.5.3 Stress-strain law 9.5.4 Forceldisplacement relation The principle of virtual work A rod element 9.7.1 Formulation of a rod element using fundamental equations 9.7.2 Formulation of a rod element using the principle of virtual work equation A simple beam element 9.8.1 Formulation of a simple beam element using fundamental equations 93.2 Formulation of a simple beam element using the principle of virtual work equation A simple triangular plane membrane element 9.9.1 Formulation of a simple triangular plane membrane element using the principle of virtual work equation Formation of assembled stcfiess matrix by use of a dof. correspondence table Amlieation of boundarv conditions and uartitioninn r, " 279 283 290 300 300 300 302 303 303 303 305 305 305 312 312 316 317 318 318 319 319 319 321 322 322 323 324 324 32% 334 3 34 339 343 344 347 349 Contents xi 9.12 Solution for displacements and reactims Bibliography Examples Problems 10 Contact Stress, Residual Stress and Stress Concentrations Summary 10.1 Contact stresses Introduction 10.1.1 General case of contact between two curved surfaces 10.1.2 Special case I - Contact of parallel cylinders 10.1.3 Combined normal and tangential loading 10.1.4 Special case 2 - Contacting spheres 10.1.5 Design considerations 10.1.6 Contact loading of gear teeth 10.1.7 Contact stresses in spur and helical gearing 10.1.8 Bearing failures Introduction 10.2.1 Reasom for residual stresses (a) Mechanical processes (b) Chemical treatment (c) Heat treatment (d) Welds (e) Castings 10.2 Residual stresses 10.2.2 The injuence of residual stress on failure 10.2.3 Measurement of residual stresses The hole-drilling technique X-ray difiaction 10.2.4 Summary of the principal effects of residual stress Introduction 10.3.1 Evaluation of stress concentration factors 10.3.2 St. Venant 's principle 10.3.3 Theoretical considerations of stress concentrations due to 10.3 Stress concentrations concentrated loads (a) Concentrated load on the edge of an infinite plate (b) Concentrated load on the edge of a beam in bending 10.3.4 Fatigue stress concentration factor 10.3.5 Notch sensitivity 10.3.6 Strain concentration - Neuber 's rule 10.3.7 Designing to reduce stress concentrations (a) Fillet radius (b) Keyways or splines 349 350 350 375 381 38 1 382 382 385 386 388 389 390 39 1 392 393 394 394 395 395 397 398 400 401 402 402 404 407 408 408 408 413 420 422 422 423 423 424 425 426 427 427 [...]... been obtained from eqn (1.14) Alternatively, however, the problem may be solved by realising that the N.A and the plane of the external bending moment are conjugate diameters of an ellipse? - the momenta1 Conjugate diameters of an ellipse: two diameters of an ellipse are conjugate when each bisects all chords parallel to the other diameter Two diameters y = rnlx and y = m2x are conjugate diameters of. .. tan0 l.4 k ,2 = - tan0 (1 .24 ) k ,2 where k, and k, are the radii of gyration about the principal axes and hence the semi-axes of the momental ellipse The N.A can now be added to the diagram to scale The second moment of area of the section about the N.A is then given by Ah2, where h is the perpendicular distance between the N.A and a tangent AA to the ellipse drawn parallel to the N.A (see Fig 1 1 1... simple loading cases, theoretical expressions are derived to cover the mechanical behaviour of these components Because of the reliance of such expressions or certain basic assumptions, the text also includes a chapter devoted to the important experimental stress and strain measurement techniques in use today with recommendations for further reading Building upon the fundamentals established in Mechanics. .. is a revised and extended third edition of the highly successful text initially published in 1977 intended to cover the material normally contained in degree and honours degree courses in mechanics of materials and in courses leading to exemption from the academic requirements of the Engineering Council It should also serve as a valuable reference medium for industry and for post-graduate courses Published... gyration of the surface is required about any other axis, e. g the N.A., then it is given by the distance between the N.A and the tangent AA which is parallel to the N.A (see Fig 1.1I) Thus ~ N A=h The ellipse is then termed the momenta1 ellipse and is extremely useful in the solution of unsymmetrical bending problems as described in $ 1.lo 1.8 Stress determination Having determined both the values of the... thin-walled shells subjected to pressure or selj-weight 12. 4 Bending stresses at discontinuities in thin shells 1 2. 5 Viscoelasticity References Examples Problems 517 518 521 527 527 527 Appendix 1 npical mechanical and physical properties for engineering metals Appendix 2 Typical mechanical properties of non-metals Appendix 3 Other properties of non-metals 534 535 Index 537 536 INTRODUCTION This text is... Now let there be a tensile stress a on the element of area d A Then force F on the element = a d A the direction of the force being parallel to the 2 axis The moment of this force about the X axis is then a d A y total moment = M , = adAy Now, by definition, the latter being termed the product second moment of area (see $1 l): Mx = P Z x y +QZxx (1 .20 ) Similarly, considering moments about the Y... fracture are also covered Each chapter of both books contains a summary of essential formulae which are developed within the chapter and a large number of worked examples The examples have been selected to provide progression in terms of complexity of problem and to illustrate the logical way in which the solution to a difficult problem can be developed Graphical solutions have been introduced where appropriate... With at least one of the principal axes being an axis of symmetry the second moments of area about the principal axes I , and I , can easily be determined With unsymmetrical sections (e. g angle-sections, Z-sections, etc.) the principal axes are not easily recognized and the second moments of area about the principal axes are not easily found except by the use of special techniques to be introduced in... Number of coils or leaves of spring Equivalent J or effective polar moment of area Autofrettage pressure Radius of elastic-plastic interface Thick cylinder radius ratio R2/R1 Ratio elastic-plastic interface radius to internal radius of thick cylinder R,/RI Resultant stress on oblique plane Normal stress on oblique plane Shear stress on oblique plane Direction cosines of plane Direction cosines of line . direction on the plane distributed loading 22 7 22 7 22 8 22 9 23 0 23 2 23 4 23 5 23 5 23 6 24 2 24 2 24 3 24 4 24 6 24 7 24 7 24 8 24 9 25 1 25 3 25 4 25 5 25 5 25 6 25 7 25 9 26 1 26 3. Basis of the finite element method Applicability of the finite element method Formulation of the Jinite element method General procedure of the Jinite element method 9.4.1 Identification of. 6th Edition CAMPBELL Castings CHARLES, CRANE & FURNESS Selection and Use of Engineering Materials, 2nd Edition CRAWFORD Plastics Engineering, 2nd Edition HEARN Mechanics of Materials