Thin Plates and Shells

Theory, Analysis, and Applications

Eduard Ventsel Theodor Krauthammer The Pennsylvania State University University Park, Pennsylvania

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Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved.

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To: Liliya, Irina, and Masha and Nina, Yoaav, Adi, and Alon

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Preface

Thin-walled structures in the form of plates and shells are encountered in many branches of technology, such as civil, mechanical, aeronautical, marine, and chemical engineering. Such a widespread use of plate and shell structures arises from their intrinsic properties. When suitably designed, even very thin plates, and especially shells, can support large loads. Thus, they are utilized in structures such as aerospace vehicles in which light weight is essential. In preparing this book, we had three main objectives: first, to offer a comprehensive and methodical presentation of the fundamentals of thin plate and shell theories, based on a strong foundation of mathematics and mechanics with emphasis on engineering aspects. Second, we wanted to acquaint readers with the most useful and contemporary analytical and numerical methods for solving linear and nonlinear plate and shell problems. Our third goal was to apply the theories and methods developed in the book to the analysis and design of thin plate-shell structures in engineering. This book is intended as a text for graduate and postgraduate students in civil, architectural, mechanical, chemical, aeronautical, aerospace, and ocean engineering, and engineering mechanics. It can also serve as a reference book for practicing engineers, designers, and stress analysts who are involved in the analysis and design of thin-walled structures. As a textbook, it contains enough materal for a two-semester senior or graduate course on the theory and applications of thin plates and shells. Also, a special effort has been made to have the chapters as independent from one another as possible, so that a course can be taught in one semester by selecting appropriate chapters, or through equivalent self-study. The textbook is divided into two parts. Part I (Chapters 1–9) presents plate bending theory and its application and Part II (Chapters 10–20) covers the theory, analysis, and principles of shell structures.

Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved.

The book is organized in the following manner. First, the general linear theories of thin elastic plates and shells of an arbitrary geometry are developed by using the basic classical assumptions. Deriving the general relationships and equations of the linear shell theory requires some familiarity with topics of advanced mathematics, including vector calculus, theory of differential equations, and theory of surfaces. We tried to keep a necessary rigorous treatment of shell theory and its principles and, at the same time, to make the book more readable for graduate students and engineers. Therefore, we presented the fundamental kinematic and static relationships, and elements of the theory of surfaces, which are necessary for constructing the shell theory, without proof and verification. The detailed derivation and proof of the above relationships and equations are given in Appendices A–E so that the interested reader can refer to them. Later on, governing differential equations of the linear general theory are applied to plates and shells of particular geometrical forms. In doing so, various approximate engineering shell theories are presented by introducing some supplementary assumptions to the general shell theory. The mathematical formulation of the above shell theories leads, as a rule, to a system of partial differential equations. A solution of these equations is the focus of attention of the book. Emphasis is given to computer-oriented methods, such as the finite difference and finite element methods, boundary element and boundary collocation methods, and to their application to plate and shell problems. Nevertheless, the emphasis placed on numerical methods is not intended to deny the merit of classical analytical methods that are also presented in the book, for example, the Galerkin and Ritz methods. A great attempt has been made to emphasize the physical meanings of engineering shell theories, mathematical relationships, and adapted basic and supplementary assumptions. The accuracy of numerical results obtained with the use of the above theories, and possible areas of their application, are discussed. The main goal is to help the reader to understand how plate and shell structures resist the applied loads and to express this understanding in the language of physical rather than purely mathematical aspects. To this end, the basic ideas of the considered plate and shell models are demonstrated by comparisons with more simple models such as beams and arches, for which the main ideas are understandable for readers familiar with strength of materials. We believe that understanding the behavior of plate and shell structures enables designers or stress analysts to verify the accuracy of numerical structural analysis results for such structures obtained by available computer code, and to interpret these results correctly. Postgraduate students, stress analysts, and engineers will be interested in the advanced topics on plate and shell structures, including the refined theory of thin plates, orthotropic and multilayered plates and shells, sandwich plate and shell structures, geometrically nonlinear plate, and shell theories. Much attention is also given to orthotropic and stiffened plates and shells, as well as to multishell structures that are commonly encountered in engineering applications. The peculiarities of the behavior and states of stress of the above thin-walled structures are analyzed in detail. Since the failure of thin-walled structures is more often caused by buckling, the issue of the linear and nonlinear buckling analysis of plates and shells is given much attention in the book. Particular emphasis is placed on the formulation of elastic stability criteria and on the analysis of peculiarities of the buckling process for thin

Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved.

shells. Buckling analysis of orthtropic, stiffened, and sandwich plates and shells is presented. The important issues of postbuckling behavior of plates and shells—in paticular, the load-carrying capacity of stiffened plates and shells—are discussed in detail. Some considerations of design stability analysis for thin shell structures is also provided in the book. An introduction to the vibration of plates and shells is given in condensed form and the fundamental concepts of dynamic analysis for free and forced vibrations of unstiffened and stiffened plate and shell structures are discussed. The book emphasizes the understanding of basic phenomena in shell and plate vibrations. We hope that this materal will be useful for engineers in preventing failures and for acousticians in controlling noise. Each chapter contains fully worked out examples and homework problems that are primarily drawn from engineering practice. The sample problems serve a double purpose: to help readers understand the basic principles and methods used in plate and shell theories and to show application of the above theories and methods to engineering design. The selection, arrangement, and presentation of the material have been made with the greatest care, based on lecture notes for a course taught by the first author at The Pennsylvania State University for many years and also earlier at the Kharkov Technical University of Civil Engineering, Ukraine. The research, practical design, and consulting experiences of both authors have also contributed to the presented material. The first author wishes to express his gratitude to Dr. R. McNitt for his encouragement, unwavering support, and valuable advice in bringing this book to its final form. Thanks are also due to the many graduate students who offered constructive suggestions when drafts of this book were used as a text. A special thanks is extended to Dr. I. Ginsburg for spending long hours reviewing and critiquing the manuscript. We thank Ms. J. Fennema for her excellence in sketching the numerous figures. Finally, we thank Marcel Dekker, Inc., and especially, Mr. B. J. Clark, for extraordinary dedication and assistance in the preparation of this book. Eduard Ventsel Theodor Krauthammer

Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved.

Contents

Preface PART I.

THIN PLATES

1 Introduction 1.1 General 1.2 History of Plate Theory Development 1.3 General Behavior of Plates 1.4 Survey of Elasticity Theory References 2 The Fundamentals of the Small-Deflection Plate Bending Theory 2.1 Introduction 2.2 Strain–Curvature Relations (Kinematic Equations) 2.3 Stresses, Stress Resultants, and Stress Couples 2.4 The Governing Equation for Deflections of Plates in Cartesian Coordinates 2.5 Boundary Conditions 2.6 Variational Formulation of Plate Bending Problems Problems References 3 Rectangular Plates 3.1 Introduction 3.2 The Elementary Cases of Plate Bending 3.3 Navier’s Method (Double Series Solution)

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3.4 Rectangular Plates Subjected to a Concentrated Lateral Force P 3.5 Levy’s Solution (Single Series Solution) 3.6 Continuous Plates 3.7 Plates on an Elastic Foundation 3.8 Plates with Variable Stiffness 3.9 Rectangular Plates Under Combined Lateral and Direct Loads 3.10 Bending of Plates with Small Initial Curvature Problems References 4 Circular Plates 4.1 Introduction 4.2 Basic Relations in Polar Coordinates 4.3 Axisymmetric Bending of Circular Plates 4.4 The Use of Superposition for the Axisymmetric Analysis of Circular Plates 4.5 Circular Plates on Elastic Foundation 4.6 Asymmetric Bending of Circular Plates 4.7 Circular Plates Loaded by an Eccentric Lateral Concentrated Forc 4.8 Circular Plates of Variable Thickness Problems References 5 Bending of Plates of Various Shapes 5.1 Introduction 5.2 Elliptical Plates 5.3 Sector-Shaped Plates 5.4 Triangular Plates 5.5 Skew Plates Problems References 6 Plate Bending by Approximate and Numerical Methods 6.1 Introduction 6.2 The Finite Difference Method (FDM) 6.3 The Boundary Collocation Method (BCM) 6.4 The Boundary Element Method (BEM) 6.5 The Galerkin Method 6.6 The Ritz Method 6.7 The Finite Element Method (FEM) Problems References 7 Advanced Topics 7.1 Thermal Stresses in Plates 7.2 Orthotropic and Stiffened Plates 7.3 The Effect of Transverse Shear Deformation on the Bending of Elastic Plates

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7.4 Large-Deflection Theory of Thin Plates 7.5 Multilayered Plates 7.6 Sandwich Plates Problems References 8 Buckling of Plates 8.1 Introduction 8.2 General Postulations of the Theory of Stability of Plates 8.3 The Equilibrium Method 8.4 The Energy Method 8.5 Buckling Analysis of Orthotropic and Stiffened Plates 8.6 Postbuckling Behavior of Plates 8.7 Buckling of Sandwich Plates Problems References 9 Vibration of Plates 9.1 Introduction 9.2 Free Flexural Vibrations of Rectangular Plates 9.3 Approximate Methods in Vibration Analysis 9.4 Free Flexural Vibrations of Circular Plates 9.5 Forced Flexural Vibrations of Plates Problems References PART II.

THIN SHELLS

10 Introduction to the General Linear Shell Theory 10.1 Shells in Engineering Structures 10.2 General Definitions and Fundamentals of Shells 10.3 Brief Outline of the Linear Shell Theories 10.4 Loading-Carrying Mechanism of Shells References 11 Geometry of the Middle Surface 11.1 Coordinate System of the Surface 11.2 Principal Directions and Lines of Curvature 11.3 The First and Second Quadratic Forms of Surfaces 11.4 Principal Curvatures 11.5 Unit Vectors 11.6 Equations of Codazzi and Gauss. Gaussian Curvature. 11.7 Classification of Shell Surfaces 11.8 Specialization of Shell Geometry Problems References

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12 The General Linear Theory of Shells 12.1 Basic Assumptions 12.2 Kinematics of Shells 12.3 Statics of Shells 12.4 Strain Energy of Shells 12.5 Boundary Conditions 12.6 Discussion of the Governing Equations of the General Linear Shell Theory 12.7 Types of State of Stress for Thin Shells Problems References 13 The Membrane Theory of Shells 13.1 Preliminary Remarks 13.2 The Fundamental Equations of the Membrane Theory of Thin Shells 13.3 Applicability of the Membrane Theory 13.4 The Membrane Theory of Shells of Revolution 13.5 Symmetrically Loaded Shells of Revolution 13.6 Membrane Analysis of Cylindrical and Conical Shells 13.7 The Membrane Theory of Shells of an Arbitrary Shape in Cartesian Coordinates Problems References 14 Application of the Membrane Theory to the Analysis of Shell Structures 14.1 Membrane Analysis of Roof Shell Structures 14.2 Membrane Analysis of Liquid Storage Facilities 14.3 Axisymmetric Pressure Vessels Problems References 15 Moment Theory of Circular Cylindrical Shells 15.1 Introduction 15.2 Circular Cylindrical Shells Under General Loads 15.3 Axisymmetrically Loaded Circular Cylindrical Shells 15.4 Circular Cylindrical Shell of Variable Thickness Under Axisymmetric Loading Problems References 16 The Moment Theory of Shells of Revolution 16.1 Introduction 16.2 Governing Equations 16.3 Shells of Revolution Under Axisymmetrical Loads 16.4 Approximate Method for Solution of the Governing Equations (16.30)

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16.5

Axisymmetric Spherical Shells, Analysis of the State of Stress at the Spherical-to-Cylindrical Junction 16.6 Axisymmetrically Loaded Conical Shells 16.7 Axisymmetric Deformation of Toroidal Shells Problems References 17 Approximate Theories of Shell Analysis and Their Applications 17.1 Introduction 17.2 The Semi-Membrane Theory of Cylindrical Shells 17.3 The Donnel–Mushtari–Vlasov Theory of Thin Shells 17.4 Theory of Shallow Shells 17.5 The Theory of Edge Effect Problems References 18 Advanced Topics 18.1 Thermal Stresses in Thin Shells 18.2 The Geometrically Nonlinear Shell Theory 18.3 Orthotropic and Stiffened Shells 18.4 Multilayered Shells 18.5 Sandwich Shells 18.6 The Finite Element Representations of Shells 18.7 Approximate and Numerical Methods for Solution of Nonlinear Equations Problems References 19 Buckling of Shells 19.1 Introduction 19.2 Basic Concepts of Thin Shells Stability 19.3 Linear Buckling Analysis of Circular Cylindrical Shells 19.4 Postbuckling Analysis of Circular Cylindrical Shells 19.5 Buckling of Orthotropic and Stiffened Cylindrical Shells 19.6 Stability of Cylindrical Sandwich Shells 19.7 Stability of Shallow Shells Under External Normal Pressure 19.8 Buckling of Conical Shells 19.9 Buckling of Spherical Shells 19.10 Design Stability Analysis Problems References 20 Vibrations of Shells 20.1 Introduction 20.2 Free Vibrations 20.3 Free Vibrations 20.4 Free Vibrations 20.5 Free Vibrations

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of of of of

Cylindrical Shells Conical Shells Shallow Shells Stiffened Shells

20.6 Forced Vibrations of Shells Problems References Appendix A. Some Reference Data A.1 Typical Properties of Selected Engineering Materials at Room Temperatures (U.S. Customary Units) A.2 Typical Properties of Selected Engineering Materials at Room Temperatures (International System (SI) Units) A.3 Units and Conversion Factors A.4 Some Useful Data A.5 Typical Values of Allowable Loads A.6 Failure Criteria Appendix B. Fourier Series Expansion B.1 Dirichlet’s Conditions B.2 The Series Sum B.3 Coefficients of the Fourier Series B.4 Modification of Relations for the Coefficients of Fourier’s Series B.5 The Order of the Fourier Series Coefficients B.6 Double Fourier Series B.7 Sharpening of Convergence of the Fourier Series References Appendix C.1 C.2 C.3 C.4

C. Verification of Relations of the Theory of Surfaces Geometry of Space Curves Geometry of a Surface Derivatives of Unit Coordinate Vectors Verification of Codazzi and Gauss Equations

Appendix D. Derivation of the Strain–Displacement Relations D.1 Variation of the Displacements Across the Shell Thickness D.2 Strain Components of the Shell Appendix E.

Verification of Equilibrium Equations

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