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.