Strength of Materials Strength of Materials G. H. RYDER M.A.(Cantab.), A.M.I.Mech.E. Principal Lecturer Royal Military College of Science, Shrivenham THIRD EDITION IN SI UNITS M THIRD EDITION IN SI UNITS © G. H. RYDER 1969 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission First published 1953 Reprin ted with amendmen ts 1955 Second Edition 1957 Reprinted with amendments 1958 Third Edition 1961 Reprinted 1963 Reprinted with additions 1965 Third Editions in Sl units 1969 Reprinted 1970 Reprinted 1971 Reprinted with corrections 1973 Reprinted 1974,1975, 1977,1979, 1980 (twice), 1982,1983 Demy 8vo, xii + 340 pages 287 line illustrations Published by THE MACMILLAN PRESS LTD London and Basingstoke Companies and representatives throughout the world ISBN 978-0-333-10928-1 ISBN 978-1-349-15340-4 (eBook) DOI 10.1007/978-1-349-15340-4 The paperback edition of this book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser Preface The principal feature of this edition is the introduction of the Systeme International d'Unites (SI), under which the United Kingdom is adopting the metric system. Also the opportunity has been taken to bring the notation up to date, by the use of sigma and tau for stresses, epsilon for strains, for example. It sets out to cover in one volume the whole of the work required up to final degree standard in Strength of Materials. The only prior know ledge assumed is of elementary applied mechanics and calculus. Conse quently, it should prove of value to students preparing for the Higher National Certificate and professional institution examinations, as well as those following a degree, or diploma course. The main aim has been to give a clear understanding of the principles underlying engineering design, and a special effort has been made to indicate the shortest analysis of a wide variety of problems. Each chapter, starting with assumptions and theory, is complete in itself and is built up logically to cover all aspects of the particular theory. In this way the student is made aware of the limitations from the start, and, although he may leave sections of a chapter to be digested later, it should enable him to avoid making errors in principle. Separate paragraph numbers are used for each chapter to enable quick reference to be made, and equation numbers quoted in worked examples are from the current paragraph except where stated. A summary of formulae, methods, and underlying principles is given at the end of each chapter; specialized works of reference have been quoted for the use of readers wishing to extend their knowledge of a particular branch of the work. Examples worked out in the text, and problems given at the end of each chapter, are typical of National Certificate and Degree standard. The aim has been to present a diversity of problems without undue overlapping. Acknowledgement is made to the Senate of the University of London for permission to use questions from their examination papers, which have been marked U.L. Numerical answers are given to all the problems. 1969 G. H. RYDER v Contents Chaptn Page INTRODUCTION Strength of Materials. Conditions of Equilibrium. Stress- Strain Relations. Compatibility. SI Units xi I DIRECT STRESS Load. Stress. Principle of St. Venant. Strain. Hooke'. Law. Modulus of Elasticity (Young's Modulus). Tensile Test. Factor of Safety. Strain Energy, Resilience. Im pact Loads. Varying Cross-section and Load. Compound Bars. Temperature Stresses. Elastic Packings. Stress Concentrations II SHEAR STRESS Shear Stress. Complementary Shear Stres.. Shear Strain. Modulus of Rigidity. Strain Energy. Cottered Joints. Riveted Joints. Eccentric Loading 24 III COMPOUND STRESS AND STRAIN Oblique Stress. Simple Tension. Note on Diagrams. Pure Shear. Pure Normal Stresses on Given Planes. General Two-dimensional Stress System. Principal Planes. Principal Stresses. Shorter Method for Principal Stresses. Maximum Shear Stress. Mohr's Stress Circle. Poisson's Ratio. Two-dimensional Stress System. Principal Strains in Three Dimensions. Principal Stresses Determined from Principal Strains. Analysis of Strain. Mohr's Strain Circle. Volumetric Strain. Strain Energy. Shear Strain Energy. Theories of Failure. Graphical Representation. Conclusions 34 IV ELASTIC CONSTANTS Elastic Conltants. Bulk Modulus. Relation between E and G 65 V SHEARING FORCE AND BENDING MOMENT Shearing Force. Bending Moment. Types of Load. Types of Support. Relations between w, F and M. Concentrated Loads. Uniformly Distributed Loads. Combined Loads. Varying Distributed Loads. Graphics1 Method 71 VI BENDING STRESS Pure Bending. Moments of Inertia. Graphical Determina tion of Moment of Inertia. Bending Stresses. Stress Concentrations in Bending. Combined Bending and Direct Stress. Middle Third Rule for Rectangular Sections. Middle Quarter Rule for Circular Sections. Composite Beams. Reinforced Concrete Beams. Principal Moments of Inertia. Unsymmetrics1 Bending 86 VII SHEAR STI\J!SS IN BEAMS Variation of Shear Stress. Rectangular Section. I-Section. Principal Stresses in I-Beams. Pitch of Rivets in Built-up Girders. Solid Circular Section. Thin Circular Tube. MilceUaneoul Sections. Shear Centre 117 vii viii CONTBNTS Chapt. .. Page VIII TORSION Circular Shafts. Strain Energy in Torsion. Shafts of Varying Diameter. Stresa Concentrations in Torsion. Shafts under Action of Varying Torque. Compound Shafts. Torsion Beyond the Yield Point. Combined Bending and Twisting. Rectangular Shafts. Torsion of Thin Tubular Sections. Torsion of Thin-Walled Cellular Sections. Torsion of Thin Rectangular Members. Torsion of Thin Open Sections .. 130 IX DEFLECTION OP BEAMS Strain Energy due to Bending. Application to Impact. De· flection by Calculus. Macaulay's Method. Moment-Area Method. Method of Deflection Coefficients. Deflection due to Shear. Deflection by Graphical Method .. 1 S2 X BUILT-IN AND CONTINUOUS BEAMS Moment-Area Method for Built-in Beams. Macaulay Method. Continuous Beams. Beams on Elastic Foundations. Portsl Frames 178 XI BENDING OP CURVBD BARS AND RIGID FRAMES Stresaes in Bars of Small Initial Curvature. Stresses in Bars of Large Initial Curvature. Deflection of Curved Bars (Direct Method). Deflection from Strain Energy (Castigliano's Theorem). Portal Frame by Strain Energy .. 19S XII PLASTIC THEORY OF BENDING Bending Beyond the Yield Stress. Assumptions in the Plastic Theory. Moment of Resistance at a Plastic Hinge. Collapse Loads. Combined Bending and Direct Stress. Portal Frames-Collapse Loads 209 XIII SPRINGS Close-coiled Helical Springs. Open-coiled Helical Springs. Leaf Springs. Flat Spiral Springs • • • • . . • • 225 XIV STRUTS Definition. Pin-ended (Hinged) Strut Axially Loaded. Direction-fixed at Both Ends. Partial Fixing of the Ends. Direction-fixed at One End and Free at the Other. Direc tion-fixed at One End and Position-fixed at the Other. Strut with Eccentric Load. Strut with Initial Curvature. Limi tations of Euler Theory. Rankine-Gordon Formula. Johnson's Parabolic Formula. Perry-Robertson Formula. Straight-Line Formulae. Strut with Lateral Loading. Tie with Lateral Loading. Struts of Varying Cross-Section- Energy Method 238 CONTBNTS IX Chapter Page XV CYLINDERS AND SPHERES Thin Cylinder under Internal Pressure. Thin Spherical Shell under Internal Pressure. Cylindrical Shell with Hemi spherical Ends. Volumetric Strain on Capacity. Tube under Combined Loading. Wire Winding of Thin Cylinders. Rotational Stresses in Thin Cylinders. Thick Cylinders. Internal Pressure only. Plastic Yielding of Thick Tubes. Compound Tubes. Hub Shrunk on Solid Shaft. Thick Spherical Shells 259 XVI ROTATING DISCS AND CYLINDERS Disc of Uniform Thickness. Solid Disc. Disc with Central Hole. Long Cylinder. Disc of Uniform Strength. Tem perature Stresses in Uniform Disc. Plastic Collapse of Rotating Discs .. 287 XVII CIRCULAR PLATES Circular Plates Symmetrically Loaded. Solid Circular Plate. Annular Ring, Loaded Round Inner Edge 295 XVIII VIBRATIONS AND CRITICAL SPEEDS Linear Vibrations. Torsional Oscillations-Single Inertia. Torsional Oscillations-Two Inertias. Torsional Oscilla tions of Spring. Transverse Vibrations-Single Mass. Transverse Vibrations of Uniform Beam. Transverse Vibrations-Combined Loading. Energy Method for Frequency. Whirling of Shafts. Whirling of Eccentrically Mounted Mass 302 XIX MATERIAL TESTING AND EXPERIMENTAL METHODS Tensile Tests. Compression Tests. Hardness Tests. Impact Tests. Effect of Carbon Content. Effect of Tempering. Creep. Fatigue. Extensometers. Electrical Resistance Strain Gauges. Photo-Elastic Stress Analysis. Brittle Lacquers 320 Appendix-TABLE OF ELASTIC CONSTANTS 337 Illdex 338 Notation A,a Area, constants B, b Width. D,d Diameter, depth. E Young's Modulus. e Eccentricity, extension. F Shearing force. f Frequency of vibration. G Modulus of rigidity. g Acceleration due to gravity. h Distance, height. I Moment of inertia. J Polar moment of inertia. K Bulk modulus, radius of gyration. k Stress concentration factor, stiffness of shaft, spring, or beam L, 1 Length. Load factor. M Bending moment, mass. m Modular ratio, mass. P Load. p Pressure or compressive stress. R, r Radius, reaction. S Shape factor. T Torque. t Thickness, temperature, time. U Strain energy-resilience. u Radial shift. V,v Volume. W Concentrated load w Distributed load, weight per unit length. X,x Co-ordinate; extension. Y,y Co-ordinate; deflection. Z Section modulus. z Co-ordinate; intercept. a Coefficient of thermal expansion, angle. S Deflection. e Direct Strain. (J Slope of beam, twist of shaft cp Shear strain, chord angle. p Density. fJ' Direct stress. Stresses in Directions OX, OY, OZ fJ'x,fJ'y,fJ'z fJ'1,fJ'2,fJ'3 Principal stresses. T Shear stress. v Poisson's ratio. CJ) Angular velocity. A A. Sign for maximum (e.g., M). x Introduction Strength of Materials is the study of the behaviour of structural and machine members under the action of external loads, taking into account the internal forces created and the resulting deformations. Analysis is directed towards determining the limiting loads which the member can stand before failure of the material or excessive deformation occurs. To this end three basic sets of relations can be obtained, as set out in the following paragraphs. Throughout the text it will be shown how these conditions are brought into play. It will not always be necessary to apply all the conditions, as simplified analysis may be suggested by symmetry or approximations. In other cases relations will be obtained by indirect methods, e.g. by strain energy or virtual work, which themselves incorporate certain of the basic conditions. Conditions of Equilibrium. The external forces and reactions on a member (including inertia forces if necessary) must form a system in equilibrium, and are therefore related by a certain number of equations, known as the conditions of equilibrium, depending on the configura tion.- In a general three-dimensional system six such equations are obtained, in a coplanar system three, reducing to two if the forces are parallel or concurrent. These equations can be obtained by resolving or taking moments, and the number of unknown forces or reactions which can thereby be determined is equal to the number of such equations. Stress-Strain Relations. It will be shown subsequently that for a given material there are relations betWeen the strains (i.e. deformation) in a member and the stresses (i.e. internal forces) producing them. These stresses and strains can be analysed by methods to be developed, and equations connecting them can be obtained. The number of such relations depends on the complication of the system in a similar manner to that of the preceding paragraph. Compatibility. Sometimes a number of relations can be obtained between the strains or deformations to ensure that the system derived from any assumptions made is compatible, i.e. the deformations can exist concurrently. Such conditions clearly arise where a number of parts have to fit together, as in the analysis of compound Uhrs, beams, and cylinders. • See author's Mec/umic. Applied to Engine"';",. xi