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4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Agalovian, L. A., author. [Asimptoticheskaia teoriia anizotropnykh plastin i obolochek. English] Asymptotic theory of Anisotropic plates and shells / by Lenser Aghalovyan (National Academy of Sciences, Armenia) ; translated by D. Prikazchikov. pages cm Translation of: Asimptoticheskaia teoriia anizotropnykh plastin i obolochek. Includes bibliographical references and index. ISBN (hardcover : alk. paper) 1. Elastic plates and shells. 2. Anisotropy. 3. Asymptotic expansions. 4. Plates (Engineering)--Mathematical models. 5. Shells (Engineering)--Mathematical models. I. Title. QA935.A '12--dc British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore

5 Preface Asymptotic methods seem to be one of the natural approaches to analyze the stressstrain field arising in beams, plates and shells, justified by the geometric features. However, despite the long history of asymptotic methods, until recent decades the majority of results in the area has been obtained through ad hoc assumptions and expansions along the transverse coordinate. One of the reasons, in our opinion, is the fact that introduction of a small asymptotic parameter led to singularly perturbed governing equations having the small parameter in the coefficients of senior derivatives, with the mathematical theory of such equations arising relatively recently. For the same reason a number of problems for validation of technical theories of beams, plates and shells, especially, for anisotropic structures, remained open, though it has been clear that these problems may be resolved through the full analysis within the three-dimensional elasticity theory. This monograph contains results on the theory of anisotropic beams, plates and shells relying on a unified approach, namely, on the asymptotic method of integration of the governing equations of 3D elasticity. The underlying links of the proposed methodology with existing applied theories and with the well-known Saint-Venant principle are revealed. Special attention is paid to the boundary layer solutions, along with associated velocities of attenuation. A class of boundary value problems is then pointed out, for which the Saint-Venant principle is satisfied exactly, i.e. it is shown that the principle is a generic property of the solution obtained through rigorous mathematics. The proposed asymptotic approach is later applied to non-classical boundary value problems for plates and shells, when the face boundary conditions are different from that adopted in the classical theories for plates and shells. This leads to a progress in analysis of beams, plates and shells resting on elastic foundations and enables novel formulations for the modulus of a foundation for layered and inhomogeneous foundations. The asymptotic method of this monograph is also applied to a number of dynamic boundary value problems for thin walled structures, including these interacting with various physical fields. The equations and figures in this book are numbered within each chapter. In case v

6 vi Asymptotic Theory of Anisotropic Plates and Shells of a reference to a formula from another chapter a triple numeration is employed, with the first number denoting the number of the chapter. The numeration of tables is continuous. The author has had close collaborative links with Prof. A.L. Goldenveizer for a number of years, which are very gratefully recognized. Fruitful discussions of some of the results obtained by the author with Prof. S.A. Ambartsumyan, are acknowledged. The author is also indebted to Prof. P.E. Tovstik for useful remarks and comments. L. A. Aghalovyan

7 Preface for the English Edition Every author is pleased when his research is honored and becomes available to a wider range of readers. The Russian edition of the monograph has become known for demonstration of the capacity of asymptotic methods for solving new classes of problems for beams, plates and shells composed of both traditional elastic and composite materials. The author is therefore hopeful that the English edition to be of interest, stimulating further developments in the area of thin walled structures, especially, those interacting with various physical-mechanical fields. A new Chapter 11 has been added to this English edition, dealing with 3D dynamic problems for anisotropic plates. Some novel classes of problems which may be successfully treated by asymptotic methods are pointed out. The literature has also been updated. It is a pleasure for the author to express his sincere gratitude to Prof. J. Kaplunov, without whom this book would hardly become possible. His idea was supported by Prof. Yang Chen, and Dr. D.A. Prikazchikov, who has acted as a translator. Kind attitude of the editors of World Scientific Publishing Mr. Roh-Suan Tung and Ms. E.H. Chionh is also gratefully acknowledged. Finally, the author feels indebted to Dr. Lusine Ghulghazaryan for help with technical preparation of the manuscript. L. A. Aghalovyan September 2014 vii

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9 Foreword Beams, plates, and shells are components of nearly all modern structures. Their geometry is typically characterized by having one of the dimensions significantly deviating from the others. In case of the beams one of the dimensions usually exceeding the other two, whereas in case of plates and shells the situation is opposite. This geometric feature should have general effect on the methods of analysis of these solid structures. A comprehensive analysis of the stress-strain field should rely on the formulation of 3D elasticity in the narrow region. Since this analysis is often hardly possible, and at the same time motivated by the above-mentioned geometric feature of beams, plates, and shells, a number of approximate applied theories has arisen, relying on certain assumptions. Adoption of a hypothesis leads at the end to some form of variation of the sought-for quantities over the transverse coordinate (thickness). This may be interpreted as an approximation of the associated stress and displacement components. Therefore, the current state of art (within analytical framework) is two-fold. It includes on the one hand a number of well-established technical theories of beams, plates and shells, with the range of applicability of these typically subject of refinement. On the other hand, there is always a correctly formulated problem of 3D elasticity, which often does not have a straightforward solution. Among the methods applied to this problem, and also to justification of the applied theories, we mention the power series, expansion along the Legendre polynomials, variational approach and combinations of methods. Surprisingly, despite their relatively long history, the asymptotic methods have not been applied until relatively recently, even though the presence of a small parameter is trivially observed. This could possibly be explained by the fact that the perturbation by a small parameter occurred to be singular, with the mathematical theory of such equations developed intensively only from late 1940s. Similar types of singularly perturbed equations were arising earlier in hydromechanics, celestial mechanics, electrical engineering, and other areas of natural sciences. Though all of the early treatments of such problems were intuitive, it should be mentioned that most of the adopted assumptions have been verified later through rigorous mathematics. It therefore seems logical that most of the terminology involving boundary layers, matching and others, was transferred to mathematical literature. Let us discuss ix

10 x Asymptotic Theory of Anisotropic Plates and Shells briefly some of the specific distinctions between regularly and singularly perturbed equations. The singularity of the perturbation leads to certain peculiarities of the properties of the solution as a function of a small parameter. In case of regular perturbation the resulting solution is a continuous function of a small parameter, whereas in case of singular perturbation the solution is discontinuous. For example, in case of a regularly perturbed equation, e.g. u εu = 0 (0.1) the solution u = C 1 +C 2 e εx is continuous in ε, whereas in case of a singularly perturbed equation (containing small parameter as a coefficient within senior derivative) εu u = 0 (0.2) the solution u = C 1 +C 2 e x/ε exhibits a discontinuity at ε = 0, x 0. The solution of initial or boundary value problems for a regularly perturbed problem may be found as series in small parameter, i.e. u = ε s u s. It is also known from the very beginning that in view of certain conditions, the solution for small ε iscloseto the solution atε = 0, orthe so-calleddegeneratingorunperturbed solution. The problem is therefore reduced to further refinement of this solution using higher order approximations. In order to determine the solution for a regularly perturbed equation, the original operator has to be split only once. This is achieved by substitution of the sought-for quantities as series in small parameter into the governing equations along with the boundary and initial conditions. For example, in case of the first equation (0.1) one obtains u s = u s 1. It should be mentioned that direct correspondence holds between the order of the equation and the number of conditions. In case of singular perturbation the operator is usually split several times, with the corresponding solutions being both continuous and discontinuous (of boundary layer type). Since the solution of a singularly perturbed problem contains regular component and the boundary layer part, the procedure usually involves several steps: 1) analysis of the degenerated problem including higher order approximations, corresponding to the first splitting of the operator (this part is usually referred to as the outer problem or outer solution within the framework of beams, plates and shells); 2) determination of the boundary layer solutions using the second splitting of the original perturbed operator; 3) matching of the obtained qualitatively different solutions through boundary and initial conditions. At the moment the theory of singularly perturbed equations with the perturbation caused by the presence of a small parameter at senior derivatives is relatively well-established, see e.g. the monographs of Friedrichs (1955); Vishik and Lusternik (1957); Vishik and Lyusternik (1960); Goldenveizer (1959, 1960); Vazov (1968); Trenogin (1970); Nayfeh (1973); Vasilieva and Butuzov (1973); Lomov (1981). It should be noted that singularly perturbed boundary value problems for plates and s=0

11 Foreword xi shells contain the small parameter as a coefficient of only a part of senior derivative (problems for singularly degenerating areas). In the classical case the degenerated system is of lower order than the perturbed one. As an example, we mention the classical shell theory. However, in case of plates it is no longer the case, when the degenerated system is of lower spatial dimension. This effect implies infinite number of boundary value functions, forming a countable set, which leads to novel mathematical problems, e.g. the problem of basic properties of the boundary layer solutions. The development of asymptotic techniques for theories of plates and shells was initially carried out by several eminent scientists including K.O. Friedrichs, A. Green, A.L. Goldenveizer, E. Reiss, I.I. Vorovich, within the framework of isotropic plates and shells. As will be shown below, the approach may be extended to anisotropic plates and shells. The asymptotic method allows verification of existing applied theories, and also provides further insights on refinements along with tackling novel classes of boundary value problems. The above-mentioned novel problems include the case of general anisotropy(containing 21 material constants), the classes of boundary value problems involving the face boundary conditions with prescribed values of displacements or mixed boundary conditions of elasticity, etc. This monograph is concerned with development of theories for anisotropic beams, plates and shells relying on the method of asymptotic integration of equations of 3D elasticity. The underlying links of asymptotic theories with technical theories and the Saint-Venant principle are revealed. The limiting transition from the 3D formulation of elasticity to 2D approximate theories is studied for several classes of boundary conditions. The distribution of stress field in the near-edge vicinity is investigated. The corresponding dependence of the velocity of attenuation on elastic parameters is discussed. Interaction of the outer solution with the boundary layer solution is analyzed along with the cases of degeneration of edge effects. Solutions for a new class of mixed boundary value problems for anisotropic beams, plates and shells are presented. Using the same asymptotic method, some more novel results for elastic foundations have been established. In view of the vast literature on justification of asymptotic methods, this issue is only partly discussed in the current monograph with the main attention drawn to the results following from the method.

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13 Contents Preface Preface for the English Edition Foreword 1. Plane Problem for a Rectangular Elastic Strip Outer Solution for a Rectangular Isotropic Strip Boundary Layer Solution. Decay of Edge Effects Interaction between the Boundary Layer and Outer Solution Relation to the Saint-Venant s Principle Relation to the Mesnager-Timoshenko Solution Comparison with the Bernoulli-Coulomb-Euler Theory Matching of the Outer Solution with the Boundary Layer for Mixed Edge Boundary Conditions Displacements Prescribed at the Edges of a Rectangle Asymptotic Analysis of an Orthotropic Rectangle The Winkler-Fuss Model The Second and the Mixed Boundary Value Problems Special Cases Boundary Layer Matching of the Outer Solution and the Boundary Layer Asymptotic Solution of the Outer Problem for a Two-Layer Anisotropic Rectangular Beam Special Cases Boundary Layer for a Two-Layer Beam Models of Elastic Foundation The Winkler-Fuss Model The Modulus of a Foundation with Variable Elastic Parameters.. 79 v vii ix xiii

14 xiv Asymptotic Theory of Anisotropic Plates and Shells 3. Direct Asymptotic Integration of 3D Elasticity Equations for Orthotropic Plates Governing Equations Outer Solution Relation to the Kirchhoff Plate Theory Relation to the Iterative Ambartsumyan Theory Two Types of Boundary Layer and Related Governing Equations Matching of the Outer Solution and the Boundary Layer for an Orthotropic Plate Boundary Conditions at the Edges First Boundary Value Problem. Iterative Matching Variational Matching Two Types of Mixed Boundary Conditions (a Simply Supported Plate) Boundary Conditions at Clamped Edge Refinement of the Classical Theory for Bending and Extension of Anisotropic Plates Elastic Plates of General Anisotropy Governing Equations Iterative Procedure for the Outer Solution Applicability of the Kirchhoff Hypothesis Special Features of the Boundary Layer Matching of the Outer Solution with the Boundary Layer Non-Classical Boundary Value Problems for Anisotropic Plates Formulations of the Non-Classical Boundary Value Problems The Outer Solution First Boundary Value Problem Second Boundary Value Problem Mixed Boundary Value Problems Boundary Layer for Non-Classical Boundary Value Problems Boundary Layer for the First Boundary Value Problem Boundary Layer for the Second Boundary Value Problem Boundary Layer for the Mixed Boundary Value Problem Matching of the Outer Solution with the Boundary Layer Two-Layer Anisotropic Plates. The Modulus of a Layered Foundation Formulation of Boundary Value Problems. Solution of the Outer Problem

15 Contents xv 7.2 Analysis of the First Boundary Value Problem. Modulus of an Orthotropic Foundation Second Boundary Value Problem Mixed Boundary Value Problems Modules of Non-Homogeneous Layered Foundations Asymptotic Analysis of the Outer Problem for an Orthotropic Shell Governing Equations and Boundary Conditions The Long Wave Outer Solution Comparison of the Asymptotic Outer Solution with Membrane and Full Shell Theories Optimal Constitutive Relations The Effect of Anisotropy on the Accuracy of Kirchhoff-Love Theory Special Cases of Asymptotic Behavior of Elastic Shells Boundary Layer in Orthotropic Shells Boundary Layer Matching of the Outer Solution with Boundary Layer Refinement of 2D Boundary Conditions Accuracy of Technical Shell Theories Non-Classical Boundary Value Problems for Anisotropic Shells Governing Equations and the Outer Solution of the Boundary Value Problems for an Elastic Shell Mixed Boundary Value Problems Non-Classical Boundary Value Problems for a Two-Layer Shell The Outer Problem for a Two-Layer Shell Spatial Dynamic Problems for Anisotropic Plates Formulation of the Forced Vibration Problems for Anisotropic Plates Forced Vibrations of Orthotropic Plates Mathematically Exact Solutions Boundary Layer of an Orthotropic Plate Matching of the Outer Solution and the Boundary Layer Forced Vibrations of Layered Orthotropic Plates Forced Vibrations of Plates of General Anisotropy Bibliography 347