PREPACE T0 THE FIRST EDITION PREFACE TO THE FIRST EDITION ix vi进 difficult problems of the solution of partial differential equations with To simplify the presentation,the book begins with the discussion of given boundary conditions are replaced in this case by measurements two-dimensional problems and only later,when the reader has familiar. of slopes and deflections of a properly stretched and loaded soap film. ized himself with the various methods used in the solution of problems The experiments show that in this way not only a visual picture of of the theory of elasticity,are three-dimensional problems discussed. the stress distribution but also the necessary information regarding The portions of the book that,although of practical importance,are magnitude of stresses can be obtained with an accuracy sufficient for such that they can be omitted during the first reading are put in small practical application. type.The reader may return to the study of such problems after Again,the electrical analogy which gives a means of investigating finishing with the most essential portions of the book. torsional stresses in shafts of variable diameter at the fillets and The mathematical derivations are put in an elementary form and grooves is interesting.The analogy between the problem of bending usually do not require more mathematical knowledge than is given in of plates and the two-dimensional problem of elasticity has also been engineering schools.In the cases of more complicated problems all successfully applied in the solution of important engineering problems. necessary explanations and intermediate calculations are given so In the preparation of this book the intention was to give to engi- that the reader can follow without difficulty through all the deriva- neers,in a simple form,the necessary fundamental knowledge of the tions.Only in a few cases are final results given without complete theory of elasticity.It was also intended to bring together solutions derivations.Then the necessary references to the papers in which the of special problems which may be of practical importance and to derivations can be found are always given. describe approximate and experimental methods of the solution of In numerous footnotes references to papers and books on the theory elasticity problems. of elasticity which may be of practical importance are given.These Having in mind practieal applications of the theory of elasticity, references may be of interest to engineers who wish to study some matters of more theoretical interest and those which have not at special problems in more detail.They give also a picture of the present any direct applications in engineering have been omitted in modern development of the theory of elasticity and may be of some favor of the discussion of specific cases.Only by studying such cases use to graduate students who are planning to take their work in this with all the details and by comparing the results of exact investigations field. with the approximate solutions usually given in the elementary books In the preparation of the book the contents of a previous book on strength of materials can a designer acquire a thorough under- ("Theory of Elasticity,"vol.I,St.Petersburg,Russia,1914)on standing of stress distribution in engineering structures,and learn to the same subject,which represented a course of lectures on the theory use,to his advantage,the more rigorous methods of stress analysis. of elasticity given in several Russian engineering schools,were used In the discussion of special problems in most cases the method to a large extent. of direct determination of stresses and the use of the compatibility The author was assisted in his work by Dr.L.H.Donnell and Dr. equations in terms of stress components has been applied.This J.N.Goodier,who read over the complete manuscript and to whom method is more familiar to engineers who are usually interested in the he is indebted for many corrections and suggestions.The author magnitude of stresses.By a suitable introduction of stress functions takes this opportunity to thank also Prof.G.H.MacCullough,Dr. this method is also often simpler than that in which equations of E.E.Weibel,Prof.M.Sadowsky,and Mr.D.H.Young,who assisted equilibrium in terms of displacements are used. in the final preparation of the book by reading some portions of the In many cases the energy method of solution of elasticity problems manuscript.He is indebted also to Mr.L.S.Veenstra for the prep- has been used.In this way the integration of differential equations is aration of drawings and to Mrs.E.D.Webster for the typing of the replaced by the investigation of minimum conditions of certain inte- manuscript. grals.Using Ritz's method this problem of variational calculus is reduced to a simple problem of finding a minimum of a funetion. S.TIMOSHENKO In this manner useful approximate solutions can be obtained in many UNIVERSITY OF MICHIGAN practically important cases. December,1933