Authors: O. Zienkiewicz and Y. K. Cheung

Publisher: McGraw Hill, NY, 1970

Condition: Very Good

The powerful method of 'finite elements' permits almost all problems of structural stress analysis, or the analysis of such field problems as heat transfer and fluid flow, to be presented in a mathematical form suitable for solution on a digital computer. This is indispensable if complex structures are to be economically designed, not only to serve the advanced needs of aeronautics, space flight, turbine design, and nuclear technology, but also for use in such general engineering fields as dam and bridge building.

The Finite Element Method is the first comprehensive textbook on a subject which until now has been presented

mainly in specialist papers. Although it begins with first principles and is a relatively simple treatment of a wide subject, the book takes the reader up to the frontiers of present-day research. It also includes many examples of solutions to practical problems. such as those relating to the design of dams, nuclear reactors, and turbines as well as those concerned with rock mechanics and seepage in civil engineering projects. A final chapter gives details of typical computer programmes, written in FORTRAN language, with comments on data preparation and digital solutions.

The book is intended for final year honours and graduate level courses in structural and aeronautical engineering in which modern methods of analysis are superseding the more classical approaches Come familiarity with matrix algebra. which is the subject of a special appendix. is necessary for the proper understanding and application of the method. The book will also appeal to a broad range of practising engineers working in the fields of aeronautical, civil, and mechanical engineering and also to industrial firms concerned with aircraft structures, turbines. space, and nuclear technology.

**About the author :**

* Professor O. C. Zienkiewicz was educated in Poland and read for his B.Sc. degree a Imperial College, London. In 1943 he joined the team of Sir Richard Southwell -pioneer of the relaxation methods and was awarded his Ph.D. in 1945. After working for some two years as a consulting engineer, he lectured at the University of Edinburgh before becoming in 1957 Professor of Civil Engineering at Northwestern University, Evanston, Illinois, U.S.A. He returned to Britain to take up the position of Head of the Civil Engineering Department at the University of Wales, Swansea. In 1965 the degree of D.Sc. was conferred upon him by the University of London. He is the author of some 60 scientific papers dealing with various problems of applied mechanics and engineering.

* Y. K. Cheung, who worked with Professor Zienkiewicz on parts of this book, graduated at the South China Institute of Technology and from 1958 practised as a structural engineer. He joined the University of Wales, Swansea, as research assistant in 1961, was awarded his Ph.D. in 1964, and is now a lecturer in civil engineering there. His research work has been mainly concerned with the field of finite element analysis and he is the author of several papers on this subject.

__PREFACE__

The engineer of the present day is faced with solving structural and other problems of growing complexity. Classical mathematics, despite its ever-increasing sophistication, is capable of solving only severely idealized situations while placing at the same time a heavy burden on his skilled time, which could more usefully be employed in the process of design. Fortunately, the rapid development of digital computers, with a progressively greater capacity and a decreasing cost of performing arithmetical operations, has come to his rescue, allowing the use of relatively simple numerical formulations and revolutionizing his approach to the process of analysis.

It is now often no more expensive to perform a more accurate analysis instead of producing approximate calculations of doubtful validity. With the aid of such methods as the one described in this book, solution of previously intractable problems has become possible. Expensive experimental models now often used in the design of important structures are rapidly becoming displaced by more economic computation.

With this progress in the field of analysis, automatic optimization of component design is rapidly becoming a reality. On the other hand, such new devices as a computer 'sketch pad', by which the designer can interact with the machine, are being developed. Both will allow the future engineer to make the best use of his creative and scientific talents.

Before numerical, computer-based solutions of real problems dealing with complex continua can be solved, it is necessary to limit their infinite degrees of freedom to a finite, if large, number of unknowns. Such a process of discretization was first successfully performed by the now well-known method of finite differences.

Now an alternative approach, that of the finite elements, appears to offer considerable advantages and its relatively simple logic makes it ideally suited for the computer. Many papers illustrating the application

this process have been published, but it is felt that a fairly comprehensive, simple presentation is called for to make the procedures more widely understood. This is being attempted in the present volume.

The finite element method was developed originally as a structural analysis, and the major part of the applications which illustrated belong to this field. However, the wider basis of the me will be stressed with applicability to such diverse problems as that of heat conduction, fluid flow, etc.

Although the book is primarily intended for the engineering profession it is hoped that it may be of interest to mathematicians, who may someday develop a calculus of finite elements in parallel with that of tinni difference calculus. Because of this emphasis, mathematical demands made of the reader will not be exacting. An elementary knowledge of differential calculus coupled with some rudiments of matrix algebra are the basic requirements. For uninitiated readers a brief summary of the principles of matrix algebra is included in the Appendix.

The first chapter of the book has relatively little to do with 'finite elements'. It summarizes the basic principles of stiffness analysis of structures in a simple way so that reference to other structural textbooks is superfluous. It is a characteristic of the finite element process, whether used in a structural context or to describe other phenomena, that the standard procedures of structural assembly can always be followed.

Chapter 2 describes the essentials of the finite element formulation of elastic problems based on assumed displacement patterns. A careful study of this chapter lays the foundations of the method which, in Chapters 3 to 9, is applied to a variety of elasticity problems. It is important to note here two things. First, that the method is a general one based on an approximate solution of an extremum problem. Second, that, contrary to the well-known Ritz process, quantities with obvious physical meaning are chosen as the variable parameters.

The first fact permits an immediate extension to non-structural problems—some of which are dealt with in Chapter 10. The second allows the engineer to maintain at all times a direct physical 'contact' with the real problem being examined.

Obviously, the finite element method, because of its tremendous utility, is in rapid process of evolution. No book of this type can, therefore, hope to be complete. Although Chapter 14 is intended to throw light on some possible future developments, it is nevertheless hoped that as a text this work will remain of some permanent value, outlining the basic principles as well as some immediate applications.

Since simplicity of presentation has been the guiding motif in writing this text, it should also appeal to the beginner as well as the more experienced practitioner of the art, whose interest may be in the discussion such topics as the use of numerical versus closed form integration and reference to other technical details. For the beginner, some indication o the preparation of a typical computer program is given in Chapter 13.

some knowledge of Fortran computer language will be useful, but clearly since this is a rapidly developing field, the reader will need to keep abreast of new programming techniques.

As the engineer will be a primary user of the text, practical examples have been included whenever possible. The great majority of these refer to civil engineering problems with which the authors have been associated. Clearly, applications in all other branches of engineering can equally be envisaged-the major use of the methodology being in the field of 'aero-space' engineering.