A Variational Approach to Structural Analysis

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A Wiley-Interscience Publication JOHN WILEY & SONS, INC.



A Wiley-Interscience Publication JOHN WILEY & SONS, INC.

Copyright 2002 by John Wiley & Sons, New York. 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, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ @ WILEY.COM. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold with the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional person should be sought. This title is also available in print as 0-471-39593-5. Some content that appears in the print version of this book may not be available in this electronic edition. For more information about Wiley products, visit our web site at www.Wiley.com

To Christina and Edward



xi1 7


2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14

Variational Notation / 7 The Gradient / 10 Integration by Parts / 11 Stokess Theorem / 13 Greens Theorem in the Plane / 15 Adjoint Equations / 16 Meaning of 2 / 19 Total Differentials / 20 Legendre Transformation / 21 Lagrange Multipliers / 24 Differential Equations of Equilibrium / 27 Strain-Displacement Relations / 29 Compatibility Conditions of Strain / 33 Thermodynamic Considerations / 35 Problems / 38vii






3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20

Virtual Work Denition / 40 Generalized Coordinates / 41 Virtual Work of a Deformable Body / 42 Thermal Stress, Initial Strain, and Initial Stress / 47 Some Constitutive Relationships / 48 Accounting for All Work / 51 Axially Loaded Members / 53 The Unit-Displacement Method / 60 Finite Elements for Axial Members / 65 Coordinate Transformations / 71 Review of the Simple Beam Theory / 74 Shear Stress in Simple Beams / 92 Shear Deection in Straight Beams / 95 Beams with Initial Curvature / 99 Thermal Strain Correction in Curved Beams / 112 Shear and Radial Stress in Curved Beams / 114 Thin Walled Beams of Open Section / 121 Shear in Open Section Beams / 147 Slope-Deection Equations / 155 Approximate Methods / 165 Problems / 173196



4.1 4.2 4.3 4.4 4.5 4.6

Complementary Virtual Work Denition / 196 Complementary Virtual Work of a Deformable Body / 197 Symmetry / 210 The Unit Load Method / 217 Force Elements / 235 Generalized Force-Displacement Transformations Problems / 242






5.1 5.2 5.3 5.4

Conservative Forces and Potential Functions Stationary Potential Energy / 271 Castiglianos First Theorem / 274 Complementary Energy / 277





5.5 5.6 5.7 5.8 5.9

Stationary Complementary Potential Energy Engesser-Crotti Theorem / 282 Variational Statements / 287 The Galerkin Method / 290 Derived Variational Principles / 300 Problems / 305






6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12

Linear-Stability Analysis / 318 Geometric Measure of Strain / 322 A Beam with Initial Curvature Revisited / 330 Thin Walled Open Beams Revisited / 337 Some Stability Concepts / 349 Energy Criterion of Stability / 350 Stiffness / 353 Stiffening and Unstiffening Models / 360 Bifurcation Analysis / 369 Imperfection Analysis / 372 Circulatory Dynamic Stability / 377 Instationary Dynamic Stability / 384 Problems / 388396 401



My objective in writing this book has been to provide a discourse on the treatment of variational formulations in deformable structures for upper-level undergraduate through graduate-level students of aeronautical, civil, and mechanical engineering, as well as engineering mechanics. Its self-containment is also designed to be useful to practicing professional engineers who need to review related topics. Emphasis is placed on showing both the power and the pitfalls of virtual methods. Today, analysis in structures, heat transfer, acoustics, and electromagnetics currently depends on the nite element method or the boundary element method. These methods, in turn, depend on virtual methods or their generalization represented by techniques such as the Galerkin method. The notes for this book come from a graduate aerospace course in structures that I have taught at the University of Southern California for the last eighteen years. The students come from such diverse backgrounds as mechanical engineering, structural engineering, physics, uid dynamics, control theory, and, occasionally, astronomy. At least half of an average class is made up of graduates working in the industry full-time. The majority view the course as a terminal course in structures, but only about half have a rm understanding of structures or plan to make a career in the subject. All, however, need an understanding of virtual methods and structures to survive in the industrial world. The control theory expert must be able to convey his or her needs to the structural-modeling group. It is highly embarrassing to nd out three years into a project that there has been a lack of communication between design groups. Without the necessity of memorizing numerous formulas, virtual methods provide a logical, unied approach to obtaining solutions to problemsxi



in mechanics. Thus students have a readily understood method of analysis valid across all areas of mechanics, from structures to heat transfer to electricalmechanical, that will remain with them throughout their careers. Students often want to be taught the nite element method without having any basic understanding of mechanics. Although my notes touch in an extremely basic way on nite element methods (FEM), I try to impress on students that FEM codes are nothing more than modern slide rules that are only effective when the underlying mechanics is fully understood. All too often, I have observed in my twenty-odd years as a principal engineer for the worlds largest FEM developer that even in industry, there is a blind use and acceptance of FEM results without consideration of whether the model is a good mathematical representation of the physics. That said, however, I should also state that I have encountered clients with a far better understanding of the use of the code and its application to physics than the code developers. The text is divided into six chapters, with discussion basically limited to springs, rods, straight beams, curved beams, and thin walled open beams (which represent a specialized form of shells). Springs and rods were selected because the reader can easily understand them, and they provide a means in nonlinear discussions to get closed solutions with very general physical interpretations. The three classes of beams were chosen for several reasons. First, my years in aerospace and my current interaction with both auto-industry and aircraftindustry clients indicate that beams are still a very important structural component, as they are used extensively in both automobile and aircraft structures and to model such complex nonlinear problems as blade-out conditions on aircraft engines. Second, beams can be used to clearly demonstrate the ease with which virtual methods can be applied to determine governing equations. Third, beams can be used to clearly demonstrate when and why virtual methods will yield inconsistent results. Fourth, beams yield equations that can generally be solved by the student. And fth, the physics of the solution can be discussed in relationship to what an FEM code can perform. The rst two chapters are mainly introductory material, introducing the variational notation used and reviewing the equilibrium and compatibility equations of mechanics. Since variational methods rely heavily on integration by parts and on the variational operator functioning in a manner similar to a total differential, these techniques are discussed in great detail. Though the text itself does not extensively use the concept of adjoint operators, a section is included here in an attempt to bridge the gap between basic mechanics courses, which make no mention of the subject, and advanced texts, which assume extensive knowledge of the subject. Legendre transformations are introduced for later use in establishing the duality between variational methods and complementary virtual methods. Lagrange multipliers are discussed, and a possible physical interpretation is given. The third chapter covers virtual work. It uses kinematical formulations for the determination of the required strain relationships for straight, curved, and thin walled beams. The importance of accounting for all work is emphasized,