Classic Reprints Series

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2001; 52: (DOI: /nme.339) Classic Reprints Series Displacement and equilibrium models in the nite element method by B. Fraeijs de Veubeke, Chapter 9, Pages of Stress Analysis, Edited by O. C. Zienkiewicz and G. S. Holister, Published by John Wiley & Sons, 1965 INTRODUCTORY REMARKS BY O. C. ZIENKIEWICZ A few years ago we took the editorial decision to introduce a series called Classic Reprints in which from time-to-time we shall record papers which have contributed signicantly to the development of the subject and which are not easy to locate in standard archival journals. We opened the series with the paper by Courant of 1943 to celebrate 50 years of his work and to allow access for our readers to this well-known contribution in which the linear triangular element was used for the rst time. The present reprint of the paper by Professor Fraeijs de Veubeke which was presented in 1963 and published in 1965 is the second of this series. The reason for the choice of this paper is not only that it contains very many seminal ideas but also because the location of this paper is not widely accessible. Baudouin Fraeijs de Veubeke Correspondence to: O. C. Zienkiewicz, Department of Civil Engineering, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, U.K. olek@ Copyright? 2001 John Wiley & Sons, Ltd.

2 288 O. C. ZIENKIEWICZ The paper was published in 1965 as part of a book in which both numerical and experimental methods of stress analysis were presented by various authorities dealing with the subject. Ten of these papers were concerned with numerical analysis and in the appendix to this introduction we give the list of contents listing chapters one to ten and dealing with nite dierence, nite element methods and the boundary integral procedures. One of the reasons for that particular publication was that at the time the subject of numerical analysis was still at the cross roads and the three alternatives of nite dierences, nite elements and boundary integral methods were making inroads into the computer-based activity. Which procedure was the best? This was obviously the question which at that time we felt needed to be addressed. This question of course did not possess a single answer. Professor Fraeijs de Veubeke, at that time a colleague of Professor Massonnet, was suggested to us by himas a very suitable presenter of the subject. His paper was certainly not disappointing; indeed, the archival value of his chapter which we reproduce here was so great that it has been widely cited. Unfortunately, the book itself after limited printing of 3000 copies was exhausted and is now no longer in print. Thus I ampresenting the paper here to allow those who cannot nd its origin to read it. The paper (or rather chapter) was well ahead of its time and in addition to presenting a comprehensive look at the formulation of nite elements, it included many ideas expressed and recorded for the rst time. These are: 1. The introduction of the so-called equilibrating elements based on complementary energy principle. 2. The realization that the standard potential energy formulation and the complementary energy formulation provide bounds to the energy of the system, the rst from the lower and the second from the upper limit. The bounding thus is an extremely useful practical measure for assessing the accuracy of any particular solution. 3. The chapter introduces for the rst time a quadratic element (here the two-dimensional six-node triangle) which later was to become extremely popular. 4. It introduces mixed formulations which are for the rst time discussed in detail. 5. Of particular importance is the introduction of the limitation principle for mixed formulation which failed to draw the attention of many later investigators and was only incorporated into texts in the late 1980s. Others may nd dierent aspects of this paper to comment on. It seems to me, however, that the ve items mentioned indicate that the paper deserves to be put in the classic category. In some detail perhaps it is worthwhile to mention that the ideas introduced for achieving the complementary energy formulations were not widely developed in the form suggested here. Later de Veubeke and Zienkiewicz [1] introduced an alternative way of approaching the same subject via stress function formulations. However, it is of importance to realize that the idea of bounds is unique and for the rst time introduced here seriously. Whilst introduction of a quadratic element is novel it proceeded by several years the use of such elements and isoparametric forms with polynomial powers going as high as four. However, it is the quadratic triangle which provides one of the most versatile and useful elements and perhaps deserves the name of the Veubeke triangle just as the linear triangle is often associated with the name of Turner. Finally, the limitation principle which, as I mentioned earlier, waited for some years before being recognized provides to me the reason why so many mixed formulations have proved

3 DISPLACEMENT AND EQUILIBRIUM MODELS 289 to be less than miraculous. To many it seems that the addition of free stress variable would provide a great freedomof representing that of the stresses in complex structure and thus greatly improved the accuracy. This only was to be the case if the limitation principle is not violated. What the principle says is simply that if the mixed formulation is capable of producing the same approximation of that produced by direct displacement form then it will in fact reproduce that formexactly and give identical and therefore not improved results. Any student considering for instance the addition of many higher order polynomials representing stress distribution in a linear triangle will soon nd out that although the formulation is correct nothing is gained by its use as all the higher-order terms disappear. Fraeijs de Veubeke contributed many other papers during his career and indeed established a very successful Department of Aeronautics at the University of Liege which at present continues as one of the European centres of nite element analysis. Unfortunately, Fraeijs died at a relatively early age in 1976 and his contributions were cut o. This note has been written by his friend in sincere appreciation of his various contributions and of his personality. APPENDIX Contents Part I. Numerical methods Chapter 1 The nite-dierence approach 3 D. N. de G. Allen and D. W. Windle Chapter 2 Two-dimensional stress analysis and plate exure by nite dierences 20 O. C. Zienkiewicz Chapter 3 Stress analysis of shells I Introduction: a lead fromplate theory 41 D. N. de G. Allen Chapter 4Stress analysis of shells II Basic (plane strain) theory 52 D. N. de G. Allen and G. M. Birtwistle Chapter 5 Stress analysis of shells III Basic (plane stress) theory 72 D. N. de G. Allen and G. M. Birtwistle Chapter 6 Stress analysis of shells IV Boundary conditions 79 D. N. de G. Allen and G. M. Birtwistle Chapter 7 The nite element method in structural mechanics 85 R. W. Clough Chapter 8 Finite element procedures in the solution of plate and shell problems 120 O. C. Zienkiewicz Chapter 9 Displacement and equilibrium models in the nite element method 145 B. Fraeijs de Veubeke Chapter 10 Numerical use of integral procedures 198 C. E. Massonnet REFERENCES 1. de Veubeke B Fraeijs, Zienkiewicz OC. Strain energy bounds in nite element analysis. Journal of Strain Analysis 1967; 2: O. C. ZIENKIEWICZ Department of Civil Engineering, University of Wales Swansea, Swansea SA2 8PP, U.K.

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