• Author / Uploaded
• js kalyana rama

Citation preview

J.H. Argyris

Energy Theorems and Structural Analysis

EN ER GY THEOREMS AN D STR UC TU RA L ANALYSIS A Generalised Discourse with Applications on Energy Principles of Structural Analysis Including the Effects of Temperature and Non-Linear Stress-Strain Relations.

by J. H. ARGYR IS, D.Sc. (Eng) Professor of Aeronautical Structures, University of London, Imperial College of Science and TechnoiOKY

Co-auth or of Part II

S. KELSEY, B.Sc. (Eng) Lecturer in Aeronautical Structures, Imperial College of Science and Technology

Springer Science+ Business Media, LLC

First published by Butterworth & Co. (Publishers) Ltd.

Originally published in a series of articles in AIRCRAFT ENGINEERING Oct., Nov., 1954; Feb., March, April, May, 1955.

ISBN 978-1-4899-5852-5 ISBN 978-1-4899-5850-1 (eBook) DOI 10.1007/978-1-4899-5850-1

J. H. Argyris 1960. Originally published by Plenum Press in 1960. Softcover reprint of the hardcover 1st edition 1960

PREFACE

T

HE present work was originally published as a series of articles in Aircraft Engineering between October 1954 and May 1955. The purpose of these papers was two-fold. Firstly to generalize and extend but at the same time abo to unify the fundamental energy principles of analysis of elastic structures. Although much of the corresponding theory has been available for a number of years, to the best of the author's knowledge it has not been given before in such generality. As an example, whilst keeping within the small deflection theory the argumeats have been developed ab initio to include non-linear elasticity and arbitrary initial strains e.g. thermal strains. The first assumption introduces naturally the twin concepts of work and complementary work first put forward by Engesser. The author has attempted in this connexion to refer to all relevant and historically important papers. Since the appearance of the present articles, a few papers have been published which touch upon the same subject but suffer, unfortunately, from a rather incomplete list of references. Secondly, the writer developed in considerable detail practical methods of analysis of complex structures-in particular for aeronautical engineering applications. The most important contributions are the matrix methods of analysis. Since they are only cursorily referred to in the Introduction, it may be appropriate here to describe their use and origin in greater detail. The matrix formulation besides providing an elegant and concise expression of the theory of such structures, is ideally suited for modern automatic computation because of the systematic ordering of numerical operation which the matrix calculus affords. The necessary programming for the digital computer is simplified since it can be preprograp1med to carry out matrix operations with only simple orders as to location and size of the matrix concerned and the operation to be performed. The specific programming for a particular problem may therefore be written comparatively quickly and easily and, moreover, follows closely the algebraic analysis. As developed here, the matrix methods of analysis follow from particular forms of the two fundamental energy principles applicable to structures made up as an assembly of discrete elements. The one principle leads to an analysis in terms if displacements as unknowns (displacement method), while the second leads to an analysis in terms of forces (force method). Besides revealing more clearly the duality of the two methods, this derivation shows also the close connexion between the aproximate methods (like the Rayleigh-Ritz method) for continuous systems and the matrix methods for finite assemblies. This is particularly valuable in providing suitable techniques for establishing the basic propertiesstiffness and flexibility-of the individual elements of a complex structure where these elements have to be assigned simplified stress or strain patterns. But in stressing the advantages of a unified approach to these diverse problems, a word of caution is necessary against carrying over into the modern methods too many ideas associated with practical calculations

by the established or classical methods. The ability to tackle successfully problems in which the number of unknowns is measured in hundreds carries with it the necessity of rethinking one's practical approach if maximum advantage is to be gained from modern computational techniques. In the force method of analysis the choice of basic system and of the redundant forces must be governed primarily by the requirements of simplicity and standardization, in orJer to reduce the m:mual preparation of data to a minimum, and reduce the proba!Jility of errors. At the time of publication of the original articles it was intended to reprint them as a single volume and to follow up the Parts I and II, contained here, with further parts dealing specifically with the practical application of the matrix methods. Unfortunately it was not possible, for a number of reasons, to complete this plan and the articles have for some time teen unavailable. Sir.ce there appears to be a persistent interest in them the present reprint has been produced to meet the deficiency. Grateful thanks and acknowledgment are due to the Editor of Aircraft Engineering for permission to reprint the articles in this form. The method of reproduction has not permitted complete rearrangemc:1t of the text into book form, so that the divisions into monthly instalments are still marked by blank spaces. However, errors in the text have been corrected as far as possible, and the pages have been renumbered consecutively to make for easier reference. Grateful tha:1.ks are due to Miss J. A. Bergg for her care and skill in effecting these changes. The author would also like to thank here those correspondents who have written to point out textual errors and misprints. A list of references to further work is also appended. These are all concerned with the matrix methods of analysis whose basic theory is developed here. In particular, Ref. 6 is an expanded and developed form of part of the work which was initially planned for the original series.

1.

2. 3. 4. 5. 6.

FURTHER REFERENCES TO RECE1'11T WORK J. H. Argyris and S. Kelsey. "Structural Analysis by the Matrix Force Method, with applications to Aircraft Wings". Wissenschaft/iche Gese/lschaft fur Luftfahrt, Jahrbuch 1956, p. 78. J. H. Argyris and S. Kelsey. "The Matrix Force Method of Structural Analysis and some new applications". Brit. Aeron. Research Council, R. & M. 3034, February, 1956. J. H. Argyris. "Die Matrizen-Theorie der Statik". lngenieur Archiv, Vol. 25, No.3, p. 174, 1957. J. H. Argyris. "On the Analysis of Complex Elastic Structures". Applied Mechanics Reviews, Vol. I I, No.7, 1958. J. H. Argyris and S. Kelsey. "Note on the Theory of Aircraft Structures". Zeitschrift fur Flugwissenschaften, Vol. 7, No. 3, 1959. J. H. Argyris and S. Kelsey. "The Analysis of Fuselages of Arbitrary Cross-Section and Taper". Aircraft Engineering, Vol. XXXI, No. 361, p. 62; No. 362, p. 101; No. 363, p. 133; No. 364, p. 169; No. 365, p. 192; No. 366, p. 244; No. 367, p. 272; 1959. (To be published in book form by Butterworths Scientific Publications.)

Part I. General Theory By J. H. Argyris

T

1. INTRODUCTION

HE increasing complexity of aircraft structures and the many exact or approximate methods available for their analysis demand an integrated view of the whole subject, not only in order to simplify their applications but also to discover some more general truths and methods. There are also other reasons demanding a more comprehensive discussion of the basic theory. We mention only the increasing attention paid to temperature stresses and the realization of the importance of nonlinear effects. When viewed from all these aspects the idea of presenting a unified analysis appears more than necessary. With this present paper we set out to develop a comprehensive system for the determination of stresses and deformations in elastic structures based on two fundamental energy principles. Although much of the theory given has naturally been known for many years we believe that some of the theorems and the generality of the results are new. The loading systems considered are of an arbitrary nature and include ab initio the effect of temperature or other initial strains. Neither do we restrict ourselves to elastic bodies obeying Hooke's law but take account of purely elastic nonlinear stress-strain laws. This is possibly not of very great importance at present but may have wider applications in the future. No problems of stability will be touched upon in the present series of articles and any other considerations of large-deflexion theory are, in general, omitted. Thus the purpose is to investigate, within the small-deflexion theory, the stresses and deformations in elastic bodies not necessarily obeying a linear stressstrain law and under any load and temperature distribution. Dynamic effects are initially not considered and hence it is assumed for the present that the loads and temperature are of the quasi-static type. When investigating thermal strain effects we ought strictly to base the analysis on thermodynamic considerations. These are, however, only slightly touched upon here. As in all theoretical work, we start by discussing the exact implications and equations derived from the initial assumptions, but we do not restrict ourselves here to this aspect. On the contrary, we pay close attention to approximate methods of analysis based on the physical concepts of work and strain energy. In particular we attempt to give upper and lower bounds to overall properties of the structure such as its stiffness. No attempt is mad!! to estimate the error of stress and deformations at any particular point. This series of papers originally arose12 • 13 from lectures given by the author since 1949-50 at the Imperial College, University of London. Naturally, the scope of the present work has grown beyond the narrower concept of undergraduate teaching, but the basis of the analysis dates back to that time. It is appropriate here to point out that certain of the basic ideas originate with Engesser2 who unfortunately does not seem to have followed them up. We refer, of course, to the two complementary concepts of work and complementary work. If we consider an ordinary load displacement diagram, then, even if we restrict ourselves to small displacements, this may be curvilinear, if the material follows a non-linear stress-strain law. Work is the area between the displacement axis and the curve, while complementary work is that included between the force axis and the curve. Thus, the two areas complement each other in the rectangular area (force) x (displacement) which would be the work if the ultimate force were acting with its full intensity from the beginning of the displacement. Naturally, in the case of a body following Hooke's law, the two complementary areas are equal, but it is still useful for the purpose of analysis to keep them apart. Since writing a previous paper12 on the subject the author has had the opportunity of consulting the most interesting latest book 9 of Stephen Timoshenko. There a reference is made to the work of Westergaard, 11 who indeed has developed further the basic ideas of Engesser, but not on quite such a general basis as here. Since approximate methods figure prominently in this paper reference ought to be made to the work of Prager and Synge. They too set out to develop systematically the determination of upper and lower limits to strain energy, restricting themselves, however, to Hooke's law and excluding temperature effects. Moreover, it appears that although many of their

2

GENERAL REFERENCES (I) Biezeno, C. B., and Gramme!, R. Technische Dynamik, 1st ed., Springer, Berlin, 1939. (2) Engesser, F. Z. Architek u. lng. Verein Hannover, Vol. 35, pp. 733-774, 1899. (3) Lord Rayleigh. Theory of Sound, 2nd ed., Vols. I and II, Macmillan, London, 1892 and 1896. (4) Maxwell, J. C. Phil. Mag., Vol. 27, p. 294, 1864. (5) Mohr, 0. Z. Arch. u. lng. Verein Hannover, 1874, p. 509, and 1875, p. 17. (6) Mueller-Breslau, H. Die neueren Methoden der Festigkeitslehre und der Statik der Baukonstruktilmen, 1st ed., Korner, Leipzig, 1886. 19 ~7. Southwell, R. V. Introduction to the Theory of Elasticity, 2nd ed., Clarendon Press, Oxford,

19~8/

Timoshenko, S., and Goodier, J. N. Theory of Elasticity, 2nd ed., MacGraw-Hill, New York,

(9) Timoshenko, S. History