THEORY OF ELASTIC STABILITY Second Edition Stephen P. Timoshenko James M. Gere DOVER PUBLICATIONS, INC. Mineola, New York Copyright Copyright O 1961 by Stephen P. Timoshenko and James M. Gere. Copyright O renewed 1989 by James M. Gere. All rights reserved. Bibliographical Note This Dover edition, first published in 2009, is an unabridged republication of the second edition of the work, originally published by the McGraw-Hill Book Company, Inc., New York and London, in 1961. Library of Congress Cataloging-in-PublicationD ata Timoshenko, Stephen, 1878-1972. Theory of elastic stability / Stephen P. Timoshenko, James M. Gere. - Dover ed. p. cm. Originally published: New York : McGraw-Hill Book, 1961. Includes bibliographical references and index. ISBN-13: 978-0-486-47207-2 ISBN-10: 0-486-47207-8 1. Elasticity. 2. Strain and stresses. I. Gere, James M. 11. Title. Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501 PREFACE TO THE SECOND EDITION Since the first edition of this book, the subject of stability of structures has steadily increased in importance, especially in the design of metal structures. As a result, many engineering schools now offer courses in this subject, usually as part of a curriculum in applied mechanics. This book is intended primarily to serve the needs of the beginning student of the subject, and the emphasis is on fundamental theory rather than specific applications. In this second edition the authors have attempted to bring up to date the subject matter of the first edition and at the same time to maintain the presentation which was characteristic of the earlier work. The book begins with the analysis of beam-columns (Chap. 1) and then proceeds to elastic buckling of bars (Chap. 2). The latter chapter has been enlarged to include buckling under the action of nonconservative forces, periodically varying forces, and impact. In addition, the discussion of the determination of critical loads of columns by successive approxima- tions has been expanded. The material on inelastic buckling of bars has been augmented by the introduction of the tangent modulus and placed in a new chapter (Chap. 3). Chapter 4 describes experiments on buckling of bars and is about the same as in the earlier edition, since it was felt that the original material still retains its inherent value. A new chapter on torsional buckling (Chap. 5) has been added to the book, and the chapter on lateral buckling of beams (Chap. 6) has been extensively revised. Chapter 7 deals with the bucklihg of rings, curved bars, and arches and contains several additions. The chapters dealing with bending of plates and shells (Chaps. 8 and 10) are substantially unchanged and are included in the book as prerequisites to Chaps. 9 and 11 on buckling of plates and shells. In Chap. 9 (buckling of plates) several new cases of buckling are considered, and some tables for calcu- lating critical stresses have been added. All the important material from the first edition has been retained in this chapter as well as in Chap. 11 (buckling of shells). The material added in Chap. 11 consists of discussion of postbuckling behavior of compressed cylindrical shells and some new material on buckling of curved sheet panels, stiffened cylindrical shells, and spherical shells. v Vi PREFACE TO THE SECOND EDITION Numerous footnote references are given throughout the text as an aid to the student who wishes to pursue some aspect of the subject further. The authors take this opportunity to thank Mrs. Thor H. Sjostrand and Mrs. Richard E. Platt for assistance in preparing the manuscript and reading the proofs for this second edition. Stephen P. Timoshenko James M. Gere CONTENTS . . . . . . . . . . . . . . . Preface to the Second Edition v . . . . . . . . . . . . . . . Preface to the First Edition Vii . . . . . . . . . . . . . . . . . . . . . Notations xv . . . . . . . . . . . . . . . . Chapter 1 Beam-cohnns 1 . . . . . . . . . . . . . . . . . . 1.1 Introduction 1 . . . . . . . . . 1.2 Differential Equations for Beam-columns 1 . . . . . . . . 1.3 Beam-column with Concentrated Lateral Load 3 1.4 Several Concentrated Loads . . . . . . . . . . . J . 7 1.5 Continuous Lateral Load . . . . . . . . . . . . . . 9 1.6 Bending of a Beam-column by Couples . . . . . . . . . . 12 1.7 Approximate Formula for Deflections. . . . . . . . . . . 14 1.8 Beam-columns with Built-in Ends . . . . . . . . . . . . 15 1.9 Beam-columns with Elastic Restraints . . . . . . . . . . 17 1.10 Continuous Beams with Axial Loads . . . . . . . . . . . 19 1.1 1 Application of Trigonometric Series . . . . . . . . . . . 24 . . . . . . . . 1.12 The Effect of Initial Curvature on Deflections 31 1.13 Determination of Allowable Stresses . . . . . . . . . . . 37 . . . . . . . . . . Chapter 2 Elastic Buckling of Bars and Frames 46 . . . . . . . . . . . . . . 2.1 Euler's Column Formula 46 2.2 Alternate Form of the Differential Equation for Determining Critical Loads . . . . . . . . . . . . . . . . . . . . 51 . . . 2.3 The Use of Beam-column Theory in Calculating Critical Loads 59 2.4 Buckling of Frames . . . . . . . . . . . . . . . . 62 . . . . . . . . . . . . 2.5 Buckling of Continuous Beams 66 . . . . . . 2.6 Buckling of Continuous Beams on Elastic Supports 70 2.7 Large Deflectionsof Buckled Bars (the Elastica) . . . . . . . 76 2.8 The Energy Method . . . . . . . . . . . . . . . . 82 2.9 Approximate Calculation of Critical Loads by thp Energy Method . . 88 2.10 Buckling of a Bar on an Elastic Foundation . . . . . . . . . 94 2.1 1 Buckling of a Bar with Intermediate Compressive Forbes . . . . . 98 2.12 Buckling of a Bar under Distributed Axial Loads . . . . . . . 1 00 2.13 Buckling of a Bar on an Elastic Foundation under Distributed Axial Loads . . . . . . . . . . . . . . . . . . . . 107 2.14 Buckling of Bars with Changes in Cross Section . . . . . . . 1 13 2.15 The Determination of Critical Loads by Successive Approximations . . 116 2.16 Bars with Continuously Varying Cross Section . . . . . . . . 1 25 . . . . . . . 2.17 The Effect of Shearing Force on the Critical Load 132 xi xii CONTENTS 2.18 Buckling of Built-up Columns . . . . . . . . . . . . . 135 2.19 Buckling of Helical Springs . . . . . . . . . . . . . 1 42 2.20 Stability of a System of Bars . . . . . . . . . . . . 1 44 2.21 The Case of Nonconservative Forces . . . . . . . . . . . 1 52 2.22 Stability of Prismatic Bars under Varying Axial Forces . . . . . 1 58 . Chapter 3 Inelastic Buckling of Bars . . . . . . . . . . . . 1 63 3.1 Inelastic Bending . . . . . . . . . . . . . . . . 163 3.2 Inelastic Bending Combined with Axial Load . . . . . . . . 1 67 3.3 Inelastic Buckling of Bars (Fundamental Case) . . . . . . . . 1 75 3.4 Inelastic Buckling of Bars with Other End Conditions. . . . . . 182 . Chapter 4 Experiments and Design Formulas . . . . . . . . . 1 85 4.1 Column Tests . . . . . . . . . . . . . . . . . 185 4.2 Ideal-column Formulas as a Basis of Column Design . . . . . . 1 92 4.3 Empirical Formulas for Column Design . . . . . . . . . 1 95 4.4 Assumed Inaccuracies as a Basis of Column Design . . . . . . 1 97 4.5 Various End Conditions . . . . . . . . . . . . . . 2 02 4.6 The Deaigh of Built-up Columns . . . . . . . . . . . . 2 06 . Chapter 6 Torsional Buckling . . . . . . . . . . . . . . 21 2 5.1 Introduction . . . . . . . . . . . . . . . . . . 2 12 5.2 Pure Torsion of Thin-walled Bars of Open Cross Section . . . . . 2 12 5.3 Nonuniform Torsion of Thin-walled Bars of Open Cross Section . . . 2 18 5.4 Torsional Buckling . . . . . . . . . . . . . . . . 2 25 5.5 Buckling by Torsion and Flexure . . . . . . . . . . . . 229 5.6 Combined Torsional and Flexural Buckling of a Bar with Continuous Elastic Supports . . . . . . . . . . . . . . . . . 2 37 5.7 Torsional Buckling under Thrust and End htoments . . . . . . 2 44 . Chapter 6 Lateral Buckling of Beams . . . . . . . . . . . . 2 51 6.1 Differential Equations for Lateral Buckling . . . . . . . . . 2 51 6.2 Lateral Buckling of Beams in Pure Bending . . . . . . . . . 2 53 6.3 Lateral Buckling of a Cantilever Beam . . . . . . . . . . 2 57 6.4 Lateral Buckling of Simply Supported I Beams . . . . . . . . 2 62 6.5 Lateral Buckling of Simply Supported Beam of Narrow Rectangular Cross Section . . . . . . . . . . . . . . . . . . 2 68 6.6 Other Cases of Lateral Buckling . . . . . . . . . . . . 2 70 6.7 Inelastic Lateral Buckling of I Beams . . . . . . . . . . 2 72 . Chapter 7 Buckling of Rings, Curved Bars, and Arches . . . . . . 2 78 7.1 Bending of a Thin Curved Bar with a Circular Axis . . . . . . 2 78 7.2 Application of Trigonometric Series in the Analysis of a Thin Circular Ring . . . . . . . . . . . . . . . . . . . . 282 7.3 Effect of Uniform Pressure on Bending of a Circular Ring . . . . . 2 87 7.4 Buckling of Circular Rings and Tubes under Uniform External Pressure 289 7.5 The Design of Tubes under Uniform External Pressure on the Basis of Assumed Inaccuracies . . . . . . . . . . . . . . . 2 94 7.6 Buckling of a Uniformly Compressed Circular Arch . . . . . . 2 97 7.7 Arches of Other Forms . . . . . . . . . . . . . . . 3 02 . . . . . . . . . . . 7.8 Buckling of Very Flat Curved B m 305 CONTENTS xiii . . . . . . . . . . . . . 7.9 Buckling of a Bimetallic Strip 310 . . . . . . 7.10 Lateral Buckling of a Curved Bar with Cicular Axis 313 . . . . . . . . . . . . . . Chapter 8 Bending of Thin Plates 319 8.1 Pure Bending of Plates . . . . . . . . . . . . . . . 3 19 8.2 Bending of Plates by Distributed Lateral Load . . . . . . . . 3 25 8.3 Combined Bending and Tension or Compreasion of Plates . . . . . 3 32 8.4 Strain Energy in Bending of Plates . . . . . . . . . . . 3 35 . . . 8.5 Deflections of Rectangular Plates with Simply Supported Edges 340 8.6 Bending of Plates with a Small Initial Curvature . . . . . . . 3 44 8.7 Large Deflections of Plates . . . . . . . . . . . . . . 3 46 Chapter 9. Buckling of Thin Plates . . . . . . . . . . . . . 34 8 9.1 Methods of Calculation of Critical Loads . . . . . . . . . 3 48 9.2 Buckling of Simply Supported Rectangular Plates Uniformly Compressed in One Direction . . . . . . . . . . . . . . . . . 3 51 9.3 Buckling of Simply Supported Rectangular Plates Compressed in Two Perpendicular Directions . . . . . . . . . . . . . . 3 56 9.4 Buckling of Uniformly Compressed Rectangular Plates Simply Supported along Two Opposite Sides Perpendicular to the Direction of Compression . . and Having Various Edge Conditions along the Other Two Sides 360 9.5 Buckling of a Rectangular Plate Simply Supported along Two Opposite Sides and Uniformly Compressed in the Direction Parallel to Those Sidea . 370 9.6 Buckling of a Simply Supported Rectangular Plate under Combined Bending and Compression . . . . . . . . . . . . . . 3 73 9.7 Buckling of Rectangular Plates under the Action of Shearing Stresses . 379 9.8 Other Cases of Buckling of Rectangular Plates . . . . . . . . 385 9.9 Buckling of Circular Plates . . . . . . . . . . . . . 3 89 . . . . . . . . . . . 9.10 Buckling of Plates of Other Shapes 392 . . . . . . . . . . 9.11 Stability of Plates Reinforced by Ribs 394 . . . . . . . . 9.12 Buckling of Plates beyond Proportional Limit 408 9.13 Large Deflections of Buckled Plates . . . . . . . . . . . 4 11 9.14 Ultimate Strength of Buckled Plates . . . . . . . . . . . 4 18 9.15 Experiments on Buckling of Plates . . . . . . . . . . . 4 23 . . . . 9.16 Practical Applications of the Theory of Buckling of Plates 429 . Chapter 10 Bending of Thin Shells . . . . . . . . . . . . . 4 40 10.1 Deformation of an Element of a Shell . . . . . . . . . 4 40 10.2 Symmetrical Deformation of a Circular Cylindrical Shell . . . . 4 43 . . . . . 10.3 Inextensional Deformation of a Circular Cylindrical Shell 445 10.4 General Case of Deformation of a Cylindrical Shell . . . . . . . 4 48 10.5 Symmetrical Deformation of a Spherical Shell . . . . . . . . 4 53 . Chapter 11 Buckling of Shells . . . . . . . . . . . . . . 457 11.1 Symmetrical Buckling of a Cylindrical Shell under the Action of IJniform Axial Compression . . . . . . . . . . . . . . . . 4 57 11.2 Inextensional Forms of Bending of Cylindrical Shells Due to Instability . 461 11.3 Buckling of a Cylindrical Shell under the Action of Uniform Axial Pressure 462 11.4 Experiments with Cylindrical Shells in Axial Compression . . . . . 4 68 11.5 Buckling of a Cylindrical Shell under the Action of Uniform External . . . . . . . . . . . . . . . . . Lateral Pressure 474 XiV CONTENTS . . . . . . 11.6 Bent or Eccentrically Compressed Cylindrical Shells 482 11.7 Axial Compression of Curved Sheet Panels . . . . . . . . . 4 85 11.8 Curved Sheet Panels under Shear or Combined Shear and Axial Stress . 488 11.9 Buckling of a Stiffened Cylindrical Shell under Axial Compression . . 490 11.10 Buckling of a Cylindrical Shell under Combined Axial and Uniform Lateral Pressure . . . . . . . . . . . . . . . . 4 95 11.11 Buckling of a Cylindrical Shell Subjected to Torsion . . . . . . 5 00 11.12 Buckling of Conical Shells . . . . . . . . . . . . . . 5 09 . . . . . . 11.13 Buckling of Uniformly Cornpressed Spherical Shells 512 . . . . . . . . . . . . . . . . . . . . . Appendix 521 TTaabbllee AA--21 .. TTaabbllee ooff tthhee FFuunnccttiioonnss t+~(pb),a )$n db h)(,px )b ) .. .. .. .. .. .. .. 55 2219 Table A.3 . Properties of Sections . . . . . . . . . . . . 5 30 Name Index . . . . . . . . . . . . . . . . . . . . 531 . . . . . . . . . . . . . . . . . . . Subject Index 636 CHAPTER 1 BEAM-COLUMNS 1.1. Introduction. In the elementary theory of bending, it is found that stresses and deflections in beams are directly proportional to the applied loads. This condition requires that the change in shape of the beam due to bending must not affect the action of the applied loads. For example, if the beam in Fig. 1-la is subjected to only lateral loads, such as and Q2, the presence of the small deflections and and slight Q1 61 62 changes in the vertical lines of action of the loads will have only an insig- nificant effect on the moments and shear forces. Thus it is ~ossibleto make calculations for deflections, stresses, moments, etc., on the basis of the initial configuration of the beam. Under these conditions, and also if Hooke's law holds for the material, the deflections are proportional to the acting forces and the principle of superposition is valid; i.e., the final deformation is obtained by summation of the deformations produced by the individual forces. Conditions are entirely different when both axial and lateral loads act simultaneously on the beam (Fig. 1-lb). The bending moments, shear forces, stresses, and deflections in the beam will not be proportional to the magnitude of the axial load. Furthermore, their values will be dependent upon the magnitude of the deflections produced and will be sensitive to even slight eccentricities in the application of the axial load. Beams subjected to axial compression and simultaneously supporting lateral loads are known as beam-columns. In this first chapter, beam- columns of symmetrical cross section and with various conditions of support and loading will be analyzed.' 1.2. Differential Equations for Beam-columns. The basic equations for the analysis of beam-columns can be derived by considering the beam in Fig. 1-2a. The beam is subjected to an axial compressive force P and to a distributed lateral load of intensity q which varies with the dis- tance x along the beam. An element of length dx between two cross sections taken normal to the original (undeflected) axis of the beam is For an analysis of beams subjected to axial tension see Timoshenko, "Strength of Materials," 3d ed., part 11, p. 41, D. Van Nostrand Company, Inc., Princeton, N.J., 1956. 1 2 THEORY OF ELASTIC STABILITY shown in Fig. 1-2b. The lateral load may be considered as having con- stant intensity q over the distance dz and will be assumed positive when in the direction of the positive y axis, which is downward in this case. The shearing force V and bending moment M acting on the sides of the element are assumed positive in the directions shown. The relations among load, shearing force V, and bending moment are obtained from the equilibrium of the element in Fig. 1-2b. Summing forces in the y direction gives -V+qdz+(V+dV) - 0 Taking moments about point n and assuming that the angle between the axis of the beam and the horizontal is small, we obtain If terms of second order are neglected, this equation becomes If the effects of shearing deformations and shortening of the beam axis are neglected, the expression for the curvature of the axis of the beam is The quantity EZ represents the flexural rigidity of the beam in the plane of bending, that is, in the xy plane, which is assumed to be a plane of sym- metry. Combining Eq. (1-3) with Eqs. (1-1) and (I-?), we can express the differential equation of the axis of the beam in the following alternate forms : EZ-d 3y + P -d y = -V dxa dx and Equations (1-1) to (1-5) are the basic differential equations for bending of