Mechanics of Materials 1 Part 1 potx

MECHANICS OF MATERIALS I An Introduction to the Mechanics of Elastic and Plastic Deformation of Solids and Structural Materials THIRD EDITION E. J. HEARN Ph.D., B.Sc. (Eng.) Hons., C.Eng., F.I.Mech.E., F.I.Prod.E., F.1.Diag.E. University of Warwick United Kingdom EINEMANN OXFORD AUCKLAND BOSTON JOHANNESBURG MELBOURNE NEW DELHI 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 First published 1977 Reprinted with corrections 1980, 1981, 1982 Second edition 1985 Reprinted with corrections 1988 Reprinted 1989, 1991, 1993, 1995, 1996 Third edition 1997 Reprinted 1998, 1999,2000 0 E. J. Hearn 1977, 1985, 1997 All rights reserved. No part of this publication may be reproduced in any material 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 WIP 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 A catalogue record for this book is available from the British Library ISBN 0 7506 3265 8 Library of Congress Cataloguing in Publication Data Hearn, E. J. (Edwin John) Mechanics of materials 1: an introduction to the mecahnics of elastic and plastic deformation of solids and structural components/E. J. Hearn. - 3rd ed. p. cm. Includes bibliographical references and index. ISBN 0 7506 3265 8 1. Strength of materials. I. Title TA405.H3 96-49967 620.1'23-dc21 CIP Printed and bound in Great Britain by Scotprint, Musselburgh FOR EVERY TITLGTHAT WE PUBLISH. BU176RWORTHdlEINEMANN WU PAY FUR BTCV TO PLANT AND CARE FOR A TREE. Also of interest ASHBY Materials Selection in Mechanical Design ASHBY & JONES Engineering Materials 1 Engineering Materials 2 CAMPBELL Castings CHARLES, CRANE & FURNESS Selection and Use of Engineering Materials CRAWFORD Plastics Engineering HEARN Mechanics of Materials 2 HULL & BACON Introduction to Dislocations, 3rd Edition JONES Engineering Materials 3 LLEWELLYN Steels: Metallurgy & Applications SMALLMAN & BISHOP Metals and Materials INTRODUCTION This text is the suitably revised and extended third edition of the highly successful text initially published in 1977 and 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 in two volumes, the text should also prove valuable for students studying mechanical science, stress analysis, solid mechanics or similar modules on Higher Certificate and Higher Diploma courses in the UK or overseas and for appropriate NVQ* programmes. The study of mechanics of materials is the study of the behaviour of solid bodies under load. The way in which they react to applied forces, the deflections resulting and the stresses and strains set up within the bodies, are all considered in an attempt to provide sufficient knowledge to enable any component to be designed such that it will not fail within its service life. Typical components considered in detail in this volume include beams, shafts, cylinders, struts, diaphragms and springs and, in most simple loading cases, theoretical expressions are derived to cover the mechanical behaviour of these components. Because of the reliance of such expressions on certain basic assumptions, the text also includes a chapter devoted to the important experimental stress and strain measurement techniques in use today with recom- mendations for further reading. Each chapter of the text 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. In order to provide clarity of working in the worked examples there is inevitably more detailed explanation of individual steps than would be expected in the model answer to an examination problem. All chapters (with the exception of Chapter 16) conclude with an extensive list of problems for solution of students together with answers. These have been collected from various sources and include questions from past examination papers in imperial units which have been converted to the equivalent SI values. Each problem is graded according to its degree of difficulty as follows: A Relatively easy problem of an introductory nature. A/B Generally suitable for first-year studies. B Generally suitable for second or third-year studies. C More difficult problems generally suitable for third year studies. *National Vocational Qualifications xv xvi Introduction Gratitude is expressed to the following examination boards, universities and colleges who have kindly given permission for questions to be reproduced: City University East Midland Educational Union Engineering Institutions Examination Institution of Mechanical Engineers Institution of Structural Engineers Union of Educational Institutions Union of Lancashire and Cheshire Institues University of Birmingham University of London C.U. E.M.E.U. E.I.E. and C.E.I. 1.Mech.E. 1.Struct.E. U.E.I. U.L.C.I. U.Birm. U.L. Both volumes of the text together contain 150 worked examples and more than 500 problems for solution, and whilst it is hoped that no errors are present it is perhaps inevitable that some errors will be detected. In this event any comment, criticism or correction will be gratefully acknowledged. The symbols and abbreviations throughout the text are in accordance with the latest recommendations of BS 1991 and PD 5686t. As mentioned above, graphical methods of solution have been introduced where appro- priate since it is the author’s experience that these are more readily accepted and understood by students than some of the more involved analytical procedures; substantial time saving can also result. Extensive use has also been made of diagrams throughout the text since in the words of the old adage “a single diagram is worth 1000 words”. Finally, the author is indebted to all those who have assisted in the production of this volume; to Professor H. G. Hopkins, Mr R. Brettell, Mr R. J. Phelps for their work asso- ciated with the first edition and to Dr A. S. Tooth’, Dr N. Walke?, Mr R. Winters2 for their contributions to the second edition and to Dr M. Daniels for the extended treatment of the Finite Element Method which is the major change in this third edition. Thanks also go to the publishers for their advice and assistance, especially in the preparation of the diagrams and editing, to Dr. C. C. Perry (USA) for his most valuable critique of the first edition, and to Mrs J. Beard and Miss S. Benzing for typing the manuscript. E. J. HEARN t Relevant Standards for use in Great Britain: BS 1991; PD 5686 Other useful SI Guides: The Infernational System of Units, N.P.L. Ministry of Technology, H.M.S.O. (Britain). Mechty, The International System of Units (Physical Constants and Conversion Factors), NASA, No SP-7012, 3rd edn. 1973 (U.S.A.) Metric Practice Guide, A.S.T.M. Standard E380-72 (U.S.A.). 1. $23.27. 2. $26. 3. $24.4 Dr. A. S. Tooth, University of Strathclyde, Glasgow. D. N. Walker and Mr. R. Winters, City of Birmingham Polytechnic. Dr M. M. Daniels, University of Central England. NOTATION Quantity Angle Length Area Volume Time Angular velocity Velocity Weight Mass Density Force Moment Pressure Stress Strain Shear stress Shear strain Young's modulus Shear modulus Bulk modulus Poisson's ratio Modular ratio Power Coefficient of linear expansion Coefficient of friction Second moment of area Polar moment of area Product moment of area Temperature Direction cosines Principal stresses Principal strains Maximum shear stress Octahedral stress A V t 0 V W m P F or P or W M P 0 E z Y E G K m V Si Unit rad (radian) m (metre) mm (millimetre) mz m3 s (second) rad/s m/s N (newton) kg (kilogram) kg/m3 N Nm Pa (Pascal) N/m2 bar ( = lo5 N/m2) N/m2 N/m2 N/m2 N/m2 N/m2 - - - - W (watt) m/m "C m4 m4 m4 "C N/m2 N/mz N/mZ - - - xvii xviii Notation Quantity Symbol Deviatoric stress Deviatoric strain Hydrostatic or mean stress Volumetric strain Stress concentration factor Strain energy Displacement Deflection Radius of curvature Photoelastic material fringe value Number of fringes Body force stress Radius of gyration Slenderness ratio Gravitational acceleration Cartesian coordinates Cylindrical coordinates Eccentricity Number of coils or leaves of spring Equivalent J or effective polar moment of area Autofrettage pressure PA Radius of elastic-plastic interface RP Thick cylinder radius ratio R2/R1 K m Ratio elastic-plastic interface radius to internal radius of thick cylinder R,/R1 Resultant stress on oblique plane Normal stress on oblique plane Shear stress on oblique plane Direction cosines of plane Direction cosines of line of action of resultant stress Direction cosines of line of action of shear stress Components of resultant stress on oblique plane Shear stress in any direction 4 on oblique plane Invariants of stress Invariants of reduced stresses Airy stress function SI Unit N/m2 N/mz - - - J m m m N/m2/fringe/m N/m3 - m4 N/m2 or bar m - N/m2 N/m2 N/m2 - N/m2 Notation xix Quantity ‘Operator’ for Airy stress function biharmonic equation Strain rate Coefficient of viscosity Retardation time (creep strain recovery) Relaxation time (creep stress relaxation) Creep contraction or lateral strain ratio Maximum contact pressure (Hertz) Contact formulae constant Contact area semi-axes Maximum contact stress Spur gear contact formula constant Helical gear profile contact ratio Elastic stress concentration factor Fatigue stress concentration factor Plastic flow stress concentration factor Shear stress concentration factor Endurance limit for n cycles of load Notch sensitivity factor Fatigue notch factor Strain concentration factor Griffith‘s critical strain energy release Surface energy of crack face Plate thickness Strain energy Compliance Fracture stress Stress Intensity Factor Compliance function Plastic zone dimension Critical stress intensity factor “J” Integral Fatigue crack dimension Coefficients of Paris Erdogan law Fatigue stress range Fatigue mean stress Fatigue stress amplitude Fatigue stress ratio Cycles to failure Fatigue strength for N cycles Tensile strength Factor of safety SI Unit S S - N/mz (N/m2)- m N/mZ N/mZ Nm m Nm mN-’ N/m2 N/m3I2 m N/m3I2 m N/m2 N/m2 N/m2 - - N/m2 N/m2 - xx No tu t ion Quantity Elastic strain range Plastic strain range Total strain range Ductility Secondary creep rate Activation energy Universal Gas Constant Absolute temperature Arrhenius equation constant Larson-Miller creep parameter Sherby-Dorn creep parameter Manson-Haford creep parameter Initial stress Time to rupture Constants of power law equation Symbol SI Unit - N/m2 S - CONTENTS Introduction Notation xv XVii 1 Simple Stress and Strain 1.1 Load 1.2 Direct or normal stress (a) 1.3 Direct strain (E ) 1.4 Sign convention for direct stress and strain 1.5 Elastic materials - Hooke’s law 1.6 Modulus of elasticity - Young’s modulus 1.7 Tensile test 1.8 Ductile materials 1.9 Brittle materials 1.10 Poisson’s ratio 1.1 1 Application of Poisson’s ratio to a two-dimensional stress system 1.12 Shear stress 1.1 3 Shear strain 1.14 Modulus of rigidity 1,15 Double shear 1.16 Allowable working stress -factor of safety 1.17 Load factor 1.18 Temperature stresses 1.19 Stress concentrations -stress concentration factor 1.20 Toughness 1.21 Creep and fatigue Examples Problems Bibliography 2 Compound Bars Summary 2.1 Compound bars subjected to external load V 1 1 2 2 2 3 3 4 8 8 9 10 11 11 12 12 12 13 13 14 14 15 17 25 26 27 27 28 [...]... torque 8 .17 Combined bending, torsion and direct thrust 8 .18 Combined bending, torque and internal pressure Examples Problems 15 4 15 4 15 5 15 6 15 7 15 8 15 9 15 9 16 0 16 2 16 4 16 5 16 6 17 3 17 6 17 6 17 7 17 9 18 0 18 1 18 2 18 2 18 2 18 2 18 3 18 4 18 4 18 6 18 6 18 7 18 7 18 8 18 9 18 9 19 0 19 5 Contents 9 Thin Cylinders and Shells Summary 9 .1 Thin cylinders under internal pressure 9 .1. 1 Hoop or circumferential stress 9 .1. 2 Longitudinal... concentrated load offset from the centre 6.4 Built-in beam carrying a non-uniform distributed load 6.5 Advantages and disadvantages of built-in beams 6.6 Effect of movement of supports Examples Problems 92 94 94 97 10 2 10 5 10 6 10 6 10 8 11 2 11 2 11 2 11 5 11 8 11 9 12 3 13 8 14 0 14 0 14 1 14 1 14 2 14 3 14 5 14 6 14 6 14 7 15 2 Contents Vlll 7 Shear Stress Distribution Summary Introduction 7 .1 Distribution of shear stress... fits 10 .14 Compound cylinder -different materials 10 .15 Uniform heating of compound cylinders of different materials 10 .16 Failure theories -yield criteria 10 .17 Plastic yielding - “auto-frettage” 10 .18 Wire-wound thick cylinders Examples Problems ix 19 8 19 8 19 8 19 9 19 9 200 20 1 202 203 203 204 205 206 208 213 215 215 216 217 219 220 22 1 22 1 222 223 224 226 226 229 229 230 23 1 233 233 234 236 25 1. .. io0 x 10 -3 = 10 .6 x 10 -6m 210 x 10 9 extension of section (3) = 11 3.2 x lo6 x 250 x 10 -3 = 75.8 x 10 -6m 210 x 10 9 400 x 10 -3 210 x 10 9 = 215 .6 x m + total extension = (75.8 10 .6 + 215 .6 )10 -6 = 302 x m = 0.302mm Example 1. 2 (a) A 25 mm diameter bar is subjected to an axial tensile load of 10 0kN Under the action of this load a 200mm gauge length is found to extend 0 .19 x 10 -3mm Determine the modulus of. .. arrangements 16 .14 Temporary birefringence 16 .15 Production of fringe patterns 16 .16 Interpretation of fringe patterns 16 .17 Calibration 402 403 403 403 403 404 406 410 41 1 41 1 412 413 414 414 416 417 427 430 430 43 1 43 5 437 437 437 438 439 440 4 41 443 444 445 446 446 448 449 450 Contents 16 .18 16 .19 16 .20 16 . 21 16.22 16 .23 Fractional fringe order determination - compensation techniques Isoclinics - circular... 11 Strain Energy Summary Introduction 1 1 .1 Strain energy - tension or compression 1 1.2 Strain energy -shear 1 1.3 Strain energy -bending 1 1.4 Strain energy - torsion 1 1.5 Strain energy of a three-dimensionalprincipal stress system 1 1.6 Volumetric or dilatational strain energy 1 1.7 Shear or distortional strain energy 1 1.8 Suddenly applied loads 1 1.9 Impact loads -axial load application 1 1 .10 ... 397 4 01 4 01 40 1 Contents xii 15 .1 15.2 15 .3 15 .4 15 .5 15 .6 15 .7 15 .8 15 .9 15 .10 15 .1 1 15 .12 15 .13 Maximum principal stress theory Maximum shear stress theory Maximum principal strain theory Maximum total strain energy per unit volume theory Maximum shear strain energy per unit volume (or distortion energy) theory Mohr 's modijied shear stress theory for brittle materials Graphical representation of. .. circuit 16 .4 Null balance or balanced bridge circuit 16 .5 Gauge construction 16 .6 Gauge selection 16 .7 Temperature compensation 16 .8 Installation procedure 16 .9 Basic measurement systems 16 .10 D.C and A.C systems 16 .11 Other types of strain gauge 16 .12 Photoelasticity 16 .13 Plane-polarised light - basic polariscope arrangements 16 .14 Temporary birefringence 16 .15 Production of fringe patterns 16 .16 Interpretation... applications 1 1 .11 Castigliano’sfirst theorem for deflection 1 1 .12 “Unit-load method 1 1 .13 Application of Castigliano’s theorem to angular movements 1 1 .14 Shear deflection Examples Problems ” 12 Springs Summary Introduction 12 .1 Close-coiled helical spring subjected to axial load W 12 .2 Close-coiled helical spring subjected to axial torque T 12 .3 Open-coiled helical spring subjected to axial load W 12 .4... analysis 14 .15 Analytical determination of principal strains from rosette readings 14 .16 Alternative representations of strain distributions at a point 14 .1I Strain energy of three-dimensional stress system Examples Problems 15 Theories of Elastic Failure Summary Introduction xi 326 326 326 327 329 329 3 31 332 334 338 342 358 3 61 36 1 36 1 362 363 363 363 364 365 366 3 61 310 312 374 375 318 38 1 383 385 . 16 6 17 3 17 6 17 6 17 7 17 9 18 0 18 1 18 2 18 2 18 2 18 2 18 3 18 4 18 4 18 6 18 6 18 7 18 7 18 8 18 9 18 9 19 0 19 5 Contents 9 Thin Cylinders and Shells ix 19 8 Summary 9 .1 Thin cylinders. disadvantages of built-in beams 6.6 Effect of movement of supports Examples Problems 76 77 78 78 79 88 92 92 94 94 97 10 2 10 5 10 6 10 6 10 8 11 2 11 2 11 2 11 5 11 8 11 9 12 3 13 8 14 0. 16 .15 Production of fringe patterns 16 .16 Interpretation of fringe patterns 16 .17 Calibration 402 403 403 403 403 404 406 410 41 1 41 1 412 41 3 414 414 41 6 417 427