The Behavior of Thin Walled Structures: Beams, Plates, and Shells

Transcription

1 The Behavior of Thin Walled Structures: Beams, Plates, and Shells

2 MECHANICS OF SURFACE STRUCTURES Editors: W. A. Nash and G. /E. Oravas I. P. Seide, Small Elastic Deformations of Thin Shells ISBN V. Pane, Theories of Elastic Plates ISBN J04-X 3. J. L. Nowinski, Theory of Thermoelasticity with Applications ISBN X 4. S. Lukasiewicz, Local Loads in Plates and Shells ISBN V. Flirt, Statistics, Formfinding and Dynamics of Air-Supported Membrane Structures ISBN Yeh Kai-yuan, ed., Progress in Applied Mechanics. The Chien Wei-zang Anniversary Volume ISBN R. Negrutiu, Elastic Analysis of Slab Structures ISBN J. R. Vinson, The Behavior of Thin Walled Structures: Beams, Plates, and Shells ISBN

3 The Behavior of Thin Walled Structures: Beams, Plates, and Shells By Jack R. Vinson Department of Mechanical Engineering University of Delaware Newark, Del., USA Kluwer Academic Publishers Dordrecht / Boston / London

4 Library of Congress Cataloging-in-Publication Data Vi'nson, Jack R., The behavior of thin walled structures. (Mechanics of surface structures j 8) Includes bibliographies and i n d ~ x. 1. Thin-walled structures. 1. Title. II. Series: Mechanics of surface structures ; v. 8. TA660.TSVS ' ISBN-13: : / e-isbn-13: Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. All Rights Reserved <D 1989 by Kluwer Academic Publishers Softcover reprint of the hardcover 1 st edition 1989 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owners.

5 To my beautiful wife, Midge, for her love, encouragement, patience, and direct assistance that made this textbook possible.

6 Contents Preface Xl 1. Equations of Linear Elasticity in Cartesian Coordinates Stresses Displacements Strains Isotropy and Its Elastic Constants Equilibrium Equations Stress-Strain Relations Linear Strain-Displacement Relations Compatibility Equations Summary References Problems 8 2. Derivation of the Governing Equations for Beams and Rectangular Plates Assumptions of Plate Theory Derivation of the Equilibrium Equations for a Plate Derivation of Plate Moment-Curvature Relations and Integrated Stress Resultant-Displacement Relatiohs Derivation of the Governing Equations for a Plate Boundary Conditions Stress Distribution within a Plate References Problems Beams and Rods General Remarks Development of the Governing Equations Solutions for the Beam Equation Stresses in Beams - Rods - Columns 28 vii

7 Vlll Contents 3.5 Example: Clamped-Clamped Beam with a Constant Lateral Load, q(x) = - qo Example: Cantilevered Beam with a Uniform Lateral Load, q(x) = -qo Example: Simply Supported Beam with a Uniform Load over Part of Its Length Beam with an Abrupt Change in Stiffness Beam Subjected to Concentrated Loads Solutions by Green's Functions Tapered Beam Solution Using Galerkin's Method Problems Solutions to Problems of Rectangular Plates Some General Solutions to the Biharmonic Equation Double Series Solution (Navier Solution) Single Series Solution (Method of M. Levy) Example of Plate with Edges Supported by Beams Summary References Problems Thermal Stresses in Plates General Considerations Derivation of the Governing Equations for a Thermoelastic Plate Boundary Conditions General Treatment of Plate Nonhomogeneous Boundary Conditions Thermoelastic Effects on Beams Self-Equilibration of Thermal Stresses References Problems Circular Plates Introduction Derivation of the Governing Equations Axially Symmetric Circular Plates Solutions for Axially Symmetric Circular Plates Circular Plate, Simply Supported at the Outer Edge, SUbjected to a Uniform Lateral Loading, Po Circular Plate, Clamped at the Outer Edge, Subjected to a Uniform Lateral Loading, Po Annular Plate, Simply Supported at the Outer Edge, SUbjected to a Stress Couple, M, at the Inner Boundary Annular Plate, Simply Supported at the Outer Edge, Subjected to a Shear Resultant, Qo, at the Inner Boundary 79

8 Contents ix 6.9 General Remarks Problems Buckling of Columns and Plates Derivation of the Plate Governing Equations for Buckling Buckling of Columns Simply Supported at Each End Column Buckling with Other Boundary Conditions Buckling of Plates Simply Supported on All Four Edges Buckling of Plates with Other Loads and Boundary Conditions References Problems The Vibrations of Beams and Plates Introduction Natural Vibrations of Beams Natural Vibrations of Plates Forced Vibrations of Beams and Plates References Problems Energy Methods in Beams, Columns and Plates Introduction Theorem of Minimum Potential Energy Analysis of Beams Subjected to a Lateral Load The Buckling of Columns Vibration of Beams Minimum Potential Energy for Rectangular Plates The Buckling of a Plate under Uniaxial Load, Simply Supported on Three Sides, and Free on an Unloaded Edge Functions to Assume in the Use of Minimum Potential Energy for Solving Beam, Column, and Plate Problems Problems Cylindrical Shells Cylindrical Shells under General Loads Circular Cylindrical Shells under Axially Symmetric Loads Edge Load Solutions A General Solution for Cylindrical Shells under Axially Symmetric Loads Sample Solutions Circular Cylindrical Shells under Asymmetric Loads Shallow Shell Theory (Donnell's Equations) Inextensional Shell Theory Membrane Shell Theory Examples of Membrane Theory References Problems 153

9 x Contents 11. Elastic Stability of Shells Buckling of Isotropic Circular Cylindrical Shells under Axially Symmetric Axial Loads Buckling of Isotropic Circular Cylindrical Shells under Axially Symmetric Axial Loads and an Internal Pressure Buckling of Isotropic Circular Cylindrical Shells under Bending Buckling of Isotropic Circular Cylindrical Shells under Lateral Pressures Buckling of Isotropic Circular Cylindrical Shells in Torsion Buckling of Isotropic Circular Cylindrical Shells under Combined Axial Loads and Bending Loads Buckling of Isotropic Circular Cylindrical Shells under Combined Axial Load and Torsion Buckling of Isotropic Circular Cylindrical Shells under Combined Bending and Torsion Buckling of Isotropic Circular Cylindrical Shells under Combined Bending and Transverse Shear Buckling of Isotropic Circular Cylindrical Shells under Combined Axial Compression, Bending and Torsion Buckling ofisotropic Spherical Shells under External Pressure Buckling of Anisotropic and Sandwich Cylindrical Shells References Problems The Vibration of Cylindrical Shells Governing Differential Equations for Natural Vibrations Hamilton's Principle for Determining the Natural Vibrations of Cylindrical Shells Reference 170 Appendix 1. Properties of Useful Engineering Materials 171 Appendix 2. Answers to Selected Problems 173 Index 179

10 Preface This book is intended primarily as a teaching text, as well as a reference for individual study in the behavior of thin walled structural components. Such structures are widely used in the engineering profession for spacecraft, missiles, aircraft, land-based vehicles, ground structures, ocean craft, underwater vessels and structures, pressure vessels, piping, chemical processing equipment, modern housing, etc. It presupposes that the reader has already completed one basic course in the mechanics or strength of materials. It can be used for both undergraduate and graduate courses. Since beams (columns, rods), plates and shells comprise components of so many of these modern structures, it is necessary for engineers to have a working knowledge of their behavior when these structures are subjected to static, dynamic (vibration and shock) and environmental loads. Since this text is intended for both teaching and self-study, it stresses fundamental behavior and techniques of solution. It is not an encyclopedia of all research or design data, but provides the reader the wherewithal to read and study the voluminous literature. Chapter 1 introduces the three-dimensional equations oflinear elasticity, deriving them to the extent necessary to treat the following material. Chapter 2 presents, in a concise way, the basic assumptions and derives the governing equations for classical Bernoulli-Euler beams and plates in a manner that is clearly understood. In Chapter 3, the solutions for beam problems are treated for a variety of commonly occurring static loads. In this chapter, Green's functions are developed and Galerkin's method are employed to illustrate these two powerful methods of solution. In Chapter 4, both the N avier and Levy methods of solution for flat plates are shown, along with numerous solutions and tabulations of results for problems frequently encountered. Because thermal loadings are so often involved in practical structures, a comprehensive discussion and treatment of thermal stresses and deformations is given in Chapter 5 for beams and plates, including the complexities of nonhomogeneous boundary conditions that result. The important fact that in many cases stresses due to thermal gradients are self equilibrating is illustrated. The analogous moisture effects associated with structures composed of polymeric materials is also discussed. Chapter 6 introduces the theory and treats numerous problems associated with circular plates, because they are encountered in so many structural applications. Xl

11 Xll Preface In Chapter 7 and 8, the eigenvalue problems of buckling and vibration are treated in a systematic way to enable the reader to analyze and determine critical buckling loads and natural frequencies to insure the structural integrity of thin walled columns, beams and plate structures. Chapter 9 deals with energy methods for beams, columns, and plates in great detail. Energy solutions not only provide alternative approaches to solutions previously discussed, but provide the useful means to obtain approximate solutions to problems involving complicated loads or geometries which would be difficult or impossible to obtain otherwise. Chapters 10, 11 and 12 introduce and treat many practical problems of cylindrical shells, which are one of the most commonly utilized shell configurations. Through mastering these chapters, the reader will understand the important effects of curvature and the 'bending boundary layer', and hence, can study the literature on other shell geometries more easily. Also, membrane theory, inextensional theory and Donnell's equations are treated as systematic approximations of the full set of equations. In Appendix I, useful material property data are provided systematically for easy reference and use in solving problems for many metallic and polymeric isotropic materials at various temperatures. Appendix 2 provides answers to many of the problems at the end of each chapter. Throughout the text, the structures considered are of homogeneous isotropic materials, such as metals and polymers, in order that the concentration be on the structural mechanics, without the complications made necessary in dealing with structures composed of anisotropic composite materials. Those considerations are treated in other texts, such as The Behavior of Structures Composed of Composite Materials by Vinson and Sierakowski, published by Martinus Nijhoff, Newark, Delaware JACK R. VINSON