Structural Analysis with Finite Elements

The finite element method has become an indispensible tool in structural analysis, and tells an unparalleled success story. With success, however, came criticism, because it was noticeable that knowledge of the method among practitioners did not keep up with success. Reviewing engineers complain that the method is increasingly applied without an understanding of structural behavior.

Often a critical evaluation of computed results is missing, and frequently a basic understanding of the limitations and possibilities of the method are nonexistent. But a working knowledge of the fundamentals of the finite element method and classical structural mechanics is a prerequisite for any sound finite element analysis. Only a well trained engineer will have the skills to critically examine the computed results.

Finite element modeling is more than preparing a mesh connecting the elements at the nodes and replacing the load by nodal forces. This is a popular model but this model downgrades the complex structural reality in such a way that—instead of being helpful—it misleads an engineer who is not well acquainted with finite element techniques.

The object of this book is therefore to provide a foundation for the finite element method from the standpoint of structural analysis, and to discuss questions that arise in modeling structures with finite elements. What encouraged us in writing this book was that—thanks to the intensive
research that is still going on in the finite element community—we can explain the principles of finite element methods in a new way and from a new perspective by making ample use of influence functions. This approach should appeal in particular to structural engineers, because influence functions are a genuine engineering concept and are thus deeply rooted in classical structural mechanics, so that the structural engineer can use his engineering knowledge and insight to assess the accuracy of finite element results or to discuss the modeling of structures with finite elements.

Just as a change in the elastic properties of a structure changes the Green’s functions or influence functions of the structure so a finite element mesh effects a shift of the Green’s functions. We have tried to concentrate on ideas, because we considered these and not necessarily the technical details to be important. The emphasis should be on structural mechanics and not on programming the finite elements, and therefore we have also provided many illustrative examples.

Finite element technology was not developed by mathematicians, but by engineers (Argyris, Clough, Zienkiewicz). They relied on heuristics, their intuition and their engineering expertise, when in the tradition of medieval craftsmen they designed and tested elements without fully understanding the exact background. The results were empirically useful and engineers were grateful because they could suddenly tackle questions which were previously unanswerable. After these early achievements self-confidence grew, and a second epoch followed that could be called baroque: the elements became more and more complex (some finite element programs offered 50 or more elements) and enthusiasm prevailed. In the third phase, the epoch of “enlightment” mathematicians became interested in the method and tried to analyze the method with mathematical rigor. To some extent their efforts were futile or extremely difficult, because engineers employed “techniques” (reduced integration, nonconforming elements, discrete Kirchhoff elements) which had no analogy in the calculus of variations. But little by little knowledge increased, the gap closed, and mathematicians felt secure enough with the method that they could provide reliable estimates about the behavior of some elements.

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