Two-Dimensional Model
A general two-dimensional theory will be developed in the same lines as the three-dimensional one considered before. The two-dimensional domain will be denoted by S and its boundary by Γ
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AND
LECTURES· No.
121
E. R. de ARANTES e OLIVEIRA UNIVERSITY OF LISBON
FOUNDATIONS OF THE MATHEMATICAL THEORY OF STRUCTURES
SPRINGER-VERLAG WIEN GMBH
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©
1975 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien-New York in 1975
ISBN 978-3-211-81312-6 DOI 10.1007/978-3-7091-4328-5
ISBN 978-3-7091-4328-5 (eBook)
1. INTRODUCTION
The differential equations of Mathematical Physics are very often as sociated to variational principles which state that the solution of the corresponding equation, under given boundary conditions, makes a certain functional stationary on a certain space of functions. Those methods which replace the problem of solving the equation by the equivalent problem of seeking the function which makes the functional stationary are called variational methods. A classical variational technique is the Ritz method, which reduces the problem of the minimization of a given functional F on a given space C to the minimization of the same functional
F on a finite-dimensional subspace C I of
C. The finite element method is also a variational method in which the elements of C I are piecewise defined on a given domain. It does not always coincide with the Ritz method, however, because the finite-dimensional set CI , on which
F is made stationary, is generally not contained in C • It became thus necessary to generalize the
old theory to cover the new situations. New convergence the-
I. Introduction
6
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orems were namely stated and demonstrated. The presentation of such theory, using the concepts of functional analysis is the aim of Chapter 3. The theory supposes that an extremum principle exists. Other variational principles (even Galerkin's method) can be used for establishing sequences of approximations, but the convergence of such sequences cannot be proved, within the frame of the present theory, without the help of an extremum principle. Although the new convergence theorems were established with the aim of being applied to the finite element method, the theory has a much more general scope. Structural and non- structural applications can indeed be considered and even the linear assumption is not necessary. Although non- structural applications can be covered, one of the most interesting applications of the convergence theorems appears in the theory of structures as it is shown in Chapter 4. Chapters 5 and 6 respectively introduce the three- and two-dimensional models of the theory of structures. The discrete model is finally considered in Chapter 7. A short account of the evolution of the finite element theory and of the papers which the author has been publishing on the subject will be presented now.
The Finite Element Method
7
The finite elem
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