The Finite Element Method in Thin Shell Theory: Application to Arch Dam Simulations

206 Pages·1982·4.37 MB·English

by  Michel Bernadou

Progress in Scientific Computing
M. Bernardou · J. M. Boisserie
The Finite Element
Method in Thin
Shell Theory:
Application to Arch
Dam Simulation

Progress in Scientific Computing
Vol. 1
Edited by
S. Abarbanel
R. Glowinski
G. Golub
H.-O. Kreiss
Springer Science+Business Media, LLC
M. Bernadou
J. M. Boisserie
The Finite Element
Method in Thin Shell
Theory:
Application to Arch Dam
Simulations
1982 SpringerScience+BusinessMedia, LLC
Authors:
Hichel Bernadou
INRIA
Domaine de Voluceau-Rocquencourt
B.P. 105
F-78153 Le Chesnay Cedex
FRANCE
Jean-Harie Boisserie
E.D.F.-D.E.R.
6, Quai Watier
F-78400 Chatou
FRANCE
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Bernadou, Hichel:
The finite element method in thin shell theory
application to arch dam stimulations /
H. Bernadou ; J. H. Boisserie .
11
Boston; Basel; Stuttgart : Birkhauser, 1982.
(Progress in scientific computing ; Vol.1)
ISBN 978-0-8176-3070-6 ISBN 978-1-4684-9143-2 (eBook)
DOI 10.1007/978-1-4684-9143-2
NE: Boisserie, Jean-Harie.; GT
Library of Congress Cataloging in Publication Data
Bernadou, H. (Michel), 1943-
The Finite element method in thin shell theory.
(Progress in scientific computing ; v. )
Bibliography: p.
Includes index.
1. Finite element method. 2. Shells (Engineering)
3. Arch dams--~mthematical models. I. Boisserie, J.-M.
(Jean-Harie), 1932- 11. Title. 111. Series.
TA347.F5B47 627' .82 82-4293
AACR2
All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any
means, electronic, mechanical, photocopying, recording or otherwise,
without prior permission of the copyright owner.
©Springer Science+Business Media New York, 1982
Originally published by Birkhäuser Boston in 1982.
TABLE OF CONTENTS
Preface ix
PART I : NUMERICAL ANALYSIS OF A LINEARTHIN SHELL MODEL
Introduction
1 - The Continuous Problem 5
1.1 - Definition of the middle surface 5
e
1.2 - Geometrical definition of the undeformed shell 9
1.3 - The linear model of W.T. KOlTER 10
1.4 - Two equivalent formulations of the shell problem 16
1.5 - Other expressions for the bilinear form and the
linear form 18
1.6 - Existence and uniqueness of a solution 21
2 - The Discrete Problem 27
......,.
2.1 - The finite element space V 29
h
2.2 - The discrete problem 33
2.3 - Examples of error estimates 37
2.4 - Uathematical studies of the convergence and of
the error estimates 39
3 - Implementation 65
3.I - Interpolation modules 65
3.2 - Energy functional and second member modules when
the spaces X and X are constructed using
hl h2
ARGYRIS triangles 80
3.3 - Energy functional and second member modules when
the spaces X and X are constructed using
hl h2
the complete HSIEH-CLOUGH-TOCHER triangle 82
vi
3.4 - Energy functional and second member modules when the
spaces ~I and ~2 are constructed using triangles of
type (2) and complete HSIEH-CLOUGH-TOCHER triangles,
respectively 84
3.5 - Energy functional and second member modules when the
spaces ~I and X are constructed using reduced
h2
HSIEH-CLOUGH-TOCHER triangles 85
3.6 - Energy functional and second member modules when the
spaces X and ~2 are constructed using triangles of
hl
type (1) and reduced HSIEH-CLOUGH-TOCHER triangles,
respectively 87
PART II : APPLICATION TO ARCH DAM SIMULATIONS
Introduction 89
4 - Geometrical definition of the dam 91
4.1 - A pre-project of a dam 91
4.2 - Definition of the middle surface 97
4.3 - Calculation of the geometrical parameters of the
middle surface 101
4.4 - Definition of the arch dam thickness 106
5 - Variational formulation of the arch dam problem 109
5.1 - Gravitational loads (due to the weight of the dam) 109
5.2 - Hydrostatic loads (due to the water pressure) III
5.3 - Thermal loads 114
5.4 - Variational formulation of the arch dam problem 119
5.5 - Another expression for the linear form f(.) 121
6 - Implementation - Presentation of results
6.1 - Values of physical constants 123
6.2 - Triangulation 124
6.3 - How to take into account boundary conditions 125
6.4 - How to take into account symmetry conditions 132
6.5 - Solution method 139
6.6 - Calculation of the displacements 144
6.7 - Calculation of the stresses at any point of the dam
physical components 144
vii
6.8 - Calculation of the stresses on the upstream and
downstream walls of the dam ; physical components 149
7 - Numerical experiments 150
7.1 - The effect of different kinds of loads 150
7.2 - The effect of changes of triangulation 157
7.3 - The effect of changes of numerical integration scheme 158
Bibliography 167
Glossary of symbols 173
Index 189
ix
PREFACE
~his Monograph has two objectives : to analyze a finite element
method useful for solving a large class of thin shell problems, and to
show in practice how to use this method to simulate an arch dam problem.
The first objective is developed in Part I. We record the defini-
tion of a general thin shell model corresponding to the W.T. KOlTER
linear equations and we show the existence and the uniqueness for a
solution. By using a conforming finite element method, we associate
a family of discrete problems to the continuous problem ; prove the
convergence of the method ; and obtain error estimates between exact
and approximate solutions. We then describe the implementation of some
specific conforming methods.
The second objective is developed in Part 2. It consists of
applying these finite element methods in the case of a representative
practical situation that is an arch dam problem. This kind of problem
is still of great interest, since hydroelectric plants permit the rapid
increase of electricity production during the day hours of heavy
consumption. This regulation requires construction of new hydroelectric
plants on suitable sites, as well as permanent control of existing dams
that may be enlightened by numerical stress analysis.
ACKNOWLEDGEMENTS
The authors take this opportunity to express their gratitude to
Professors J.L. LIONS, P.G. CIARLET and R. GLOWINSKI for providing all
facilities for the development of their researches in an excellent
scientific atmosphere at the "Institut National de Recherche en
Informatique et en Automatique" (INRIA). They are also indebted to Pro
J.T. ODEN, who has been kind enough to read the manuscript in its
entirety and to suggest various improvements ; and to M. LEROY for
x
supplying specifications of GRAND'MAISON arch dam project and for his
constant interest.
Many thanks are due to "Electricite de France, Direction des
Etudes et Recherches", for constant support and computing facilities.
The authors gratefully appreciate the excellent typing of
Mrs. DESNOUS, as well as the kind assistance of the staff of BIRKP~USER
Boston, in particular that of Ms. K. STEINBERG.