Structural Analysis with Finite Elements

Friedel Hartmann Casimir Katz

Structural Analysis with

Finite Elements

With 408 Figures and 26 Tables

123

Friedel Hartmann

University of Kassel Structural Mechanics

Kurt-Wolters-Str. 3

Germany

friedelhartmann@uni-kassel.de

Casimir Katz

SOFiSTiK AG

85764 Oberschleissheim

Germany

casimir.katz@sofistik.de

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ISBN-10 3-540-49698- x Springer Berlin Heidelberg New York

ISBN-13 978-3-540-49698-4 Springer Berlin Heidelberg New York

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Preface

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 prac-

titioners did not keep up with success. Reviewing engineers complain that the

method is increasingly applied without an understanding of structural behav-

ior. 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 inten-

sive 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

Preface

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 in-

tuition 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 sec-

ond epoch followed that could be called baroque: the elements became more

and more complex (some ﬁnite element programs oﬀered 50 or more ele-

ments) and enthusiasm prevailed. In the third phase, the epoch of “enlight-

ment” mathematicians became interested in the method and tried to analyze

the method with mathematical rigor. To some extent their eﬀorts were futile

or extremely dicult, because engineers employed “techniques” (reduced inte-

gration, 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.

We thus recognize that mathematics is an essential ingredient of ﬁnite ele-

ment technology.

One of the aims of this book is to teach structural engineers the theoretical

foundations of the ﬁnite element method, because this knowledge is invaluable

in the design of safe structures.

This book is an extended and revised version of the original German ver-

sion. We have dedicated the web page http://www.winfem.de to the book.

From this page the programs WINFEM (ﬁnite element program with focus on

inﬂuence functions and adaptive techniques), BE-SLABS (boundary element

analysis of slabs) and BE-PLATES (boundary element analysis of plates) can

be downloaded by readers who want to experiment with the methods. Addi-

tional information can also be found on http://www.sofistik.com .

FriedelHartmann@uni-kassel.de

Casimir.Katz@soﬁstik.de

Kassel

Friedel Hartmann

Munich August 2003

Casimir Katz

Acknowledgement. We thank Thomas Graetsch, who wrote the program WINFEM

and provided many illustrative examples for the approximation of inﬂuence functions

with ﬁnite elements, and Marc Damashek and William J. Gordon for their help in

preparing the manuscript. The permission of Oxford University Press to reprint the

picture on page 145 is greatly acknowledged.

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