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M. Reza Eslami Buckling and Postbuckling of Beams, Plates, and Shells 123
M. Reza Eslami Mechanical Engineering Department Amirkabir University of Technology Tehran Iran ISSN 2522-560X ISSN 2522-5618 (electronic) Structural Integrity ISBN 978-3-319-62367-2 ISBN 978-3-319-62368-9 (ebook) https://doi.org/10.1007/978-3-319-62368-9 Library of Congress Control Number: 2017952504 Springer International Publishing AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
This book is dedicated to my dear daughter and son, Golnaz and Sam
Preface The author is pleased to present Buckling and Post-buckling of Beams, Plates, and Shells. This book serves a wide range of readers, in particular, graduate students, PhD candidates, professors, scientists, researchers in various industrial and government institutes, and engineers. Thus, the book should be considered not only as a graduate textbook, but also as a reference book to those working or interested in areas of structural stability under mechanical and/or thermal loads. The book is self-contained, so that the reader should not need to consult other sources while studying the topic. The necessary mathematical concepts and numerical methods are presented in the book and the reader may easily follow the subjects based on these basic tools. It is expected, however, that the reader should have some basic knowledge in the classical mechanics and theory of elasticity. In the context of continuum mechanics, a structural element is either modelled by the elasticity theory or the flexural theory, where the latter theory is employed when the structure is thin and consequently lumped in the thickness direction. When such element is under loads producing compressive stress, the problem of stability becomes important. The most general type of elements falling into this category are beams, plates, and shells. The beam elements are either straight or curved, plates are either rectangular or circular, and the shells are cylindrical, spherical, and conical. This book covers the stability of all these structures. The applied loads may be either mechanical or thermal, where this book covers the stability of all the above structures under both mechanical and thermal loads. The structural element may be assumed to be made of homogeneous/isotropic material, or the functionally graded materials. Both types of material are discussed in this book. The structure may experience bifurcation phenomenon, snap through, or it may follow the post-buckling path, where these types of behavior are discussed in the book. The collection of materials in this book is therefore the most comprehensive, as of today, of the subject of structural stability. It covers all areas of structural type, loading type, and the material type. vii
viii Preface The book contains 8 chapters, where the chapters cover the stability of all major areas of the flexural theory. Chapter 1 gives a brief discussion on the concept of stability. A structure under loads that produce compressive stresses may loose its stability, provided that the compressive stresses bring the structure into a certain condition. The structural instability may be in form of bifurcation (buckling), snap through, and finite disturbance buckling which occurs when a structure reaches the bifurcation point and then a sharp drop of the applied load occurs before reaching to a stable post-buckling path. Proper examples are given in this chapter to familiar the readers with the concept of stability. Chapter 2 deals with the stability of straight beams. The basic governing equations, such as the kinematical relations, the constitutive law, the equations of motion, and the stability of straight beams are first discussed and derived. Then, the stability of beams under the thermal and thermo-electrical loads are presented. Buckling and post-buckling of the piezo-fgm beams and FGMP beams are discussed in detail and the thermally post-buckling of beams on elastic foundation is presented in the following. The chapter concludes with the dynamic buckling of the FGM beams under thermal loads. By setting the proper value for the power law index, the results are reduced to those of the isotropic/homogeneous beams. The third chapter presents buckling and post-buckling of the curved beams and rings. The basic governing equations are given and derived at the beginning of the chapter, and then the stability of arcs under external uniform pressure and external concentrated force are presented. Arcs under thermal force and axial force are discussed and closed form solutions are given for these types of loads. The buckling and post-buckling of rings under external pressure and thermal loads, employing the numerical method based on the general differential quadrature, are presented at the end of chapter. The material of the arcs and ring is assumed to be functionally graded, where by setting the proper value for the power law index, the results are reduced to those of the isotropic/homogeneous arcs. Chapter 4 contains the stability of rectangular plates, which are frequently used in the engineering design problems. The basic governing equations, such as the kinematical relations, the constitutive laws, the equilibrium, and stability equations are presented and derived in the chapter and then the existence of bifurcation of rectangular plates is discussed. It is proved that the bifurcation path of the FGM rectangular plates under thermal loads depends upon the type of boundary conditions. Buckling of plates under thermal and in-plane compressive forces are discussed, and the thermal stability of the piezo-fgm beams and the beams on elastic foundation are given in the following. Closed form solutions are derive for each section. The effect of geometric imperfection on the stability of rectangular plates is then presented. The material of the plate is assumed to be functionally graded, where by setting the proper value for the power law index, the results are reduced to those of the isotropic/homogeneous rectangular plates.
Preface ix The stability of circular plates is subject of the fifth chapter. The basic general governing equations are initially derived and presented. Thermal buckling of circular and annular plates under different plate theories are presented and the stability of circular plates on elastic foundation is discussed in detail, where closed form solutions are derived for the buckling loads. Thermal buckling and post-buckling of rotating circular plates and thermal buckling and post-buckling of geometrically imperfect circular plates are then presented at the end of chapter. Closed form solutions are derived whenever possible and the material of the plate is assumed to be functionally graded, where by setting the proper value for the power law index, the results are reduced to those of the isotropic/homogeneous plates. Cylindrical shells, as a widely used element in many structural systems, is treated in the six chapter. The basic governing equations, including the kinematic and constitutive laws, the equilibrium, and the stability equations are discussed and derived. Then, the mechanical and thermal buckling loads of cylindrical shells are calculated and presented in closed form solutions. Thermal buckling loads for the piezo-fgm cylindrical shells for different types of temperature distributions which are mostly encountered in the engineering design problems are derived in closed form solutions. Dynamic thermal buckling and post-buckling of the piezo-fgm cylindrical shells is treated in the next section and the chapter concludes with the discussion of stability of cylindrical shells on elastic foundation. Chapter 7 brings the stability problems of spherical shells as one of the naturally and inheritably stable structural element. Similar to the other chapters, it starts with the presentation of the basic governing equations. For this special type of shells, the behavior and nature of deep and shallow spherical shells are quite different. Thus, both theories of the deep and shallow spherical shells are derived and presented at the beginning of the chapter. Stability of isotropic/homogeneous spherical shells under the mechanical and thermal loads are discussed and closed form solutions are derived and the results are extended to those of the shallow and deep FGM shells. The effect of geometrically imperfection is discussed and the stability of piezo-fgm shells is derived and the effects of piezo-control on thermal buckling of the shallow and deep shells are shown in the next section. Buckling and post-buckling of the shallow piezo-fgm spherical shells concludes the chapter. The stability of conical shells under the mechanical and thermal loading conditions are the subject of last chapter. The basic governing equations are derived and given at the beginning of the chapter. Buckling loads associated with the mechanical and thermal loads are discussed and the buckling of piezo-fgm conical shells under thermal loads is discussed at the end of chapter. At the end of all chapters there are a number of problems for the students to solve. Also, at the end of each chapter, there is a list of relevant references. The book is prepared over some 44 years of teaching the graduate courses and research of the graduate students. During this long period of time, the results of class work assignments and student research are carefully gathered and put into this volume of work. The author takes this opportunity to thank all his students who made possible to provide this piece of work.
x Preface The author s special thank is for his previous PhD student, Dr. Y. Kiani, now an assistant professor at Shahrekord University. His contribution to develop this work is outstanding. Many chapters of this book is prepared with detail comments and help of Dr. Kiani. Tehran, Iran February 2017 M. Reza Eslami
Contents 1 Concept of Stability... 1 1.1 Introduction... 1 1.2 Type of Instability... 3 1.3 General References... 6 2 Buckling and Post-buckling of Beams... 7 2.1 Introduction... 7 2.2 Kinematic Relations... 7 2.3 Equilibrium Equations... 11 2.4 Stability Equations... 12 2.5 Thermal Buckling of FGM Beams... 13 2.5.1 Introduction... 13 2.5.2 Functionally Graded Timoshenko Beams... 14 2.5.3 Existence of Bifurcation Type Buckling... 15 2.5.4 Thermal Buckling... 15 2.5.5 Types of Thermal Loads... 17 2.5.6 Results and Discussion... 20 2.6 Thermo-Electrical Buckling of Beams... 23 2.6.1 Introduction... 23 2.6.2 Piezoelectric FGM Beam... 24 2.6.3 Governing Equations... 24 2.6.4 Existence of Bifurcation Type Buckling... 27 2.6.5 Stability Equations... 28 2.6.6 Types of Thermal Loads... 29 2.6.7 Results and Discussion... 31 2.7 Postbuckling of Piezo-FGM Timoshenko Beams... 33 2.7.1 Introduction... 33 2.7.2 Governing Equations... 34 2.7.3 Clamped Clamped Boundary Conditions... 39 xi
xii Contents 2.7.4 Simply Supported-Simply Supported Boundary Conditions... 41 2.7.5 Results and Discussion... 43 2.8 Vibration of Thermo-Electrically Post-buckled FGPM Beams... 48 2.8.1 Introduction... 48 2.8.2 Governing Equations... 51 2.8.3 Finite Elements Model... 55 2.8.4 Result and Discussions... 58 2.9 Vibration of Thermally Post-buckled Beams on Elastic Foundation... 64 2.9.1 Introduction... 64 2.9.2 Governing Equations... 65 2.9.3 Types of Thermal Loading... 70 2.9.4 Results and Discussion... 71 2.10 FGM Beams, Thermal Dynamic Buckling... 91 2.10.1 Fundamental Equations of the FGM Beam.... 92 2.10.2 Governing Equations... 93 2.10.3 Numerical Investigation... 97 2.11 Problems... 104 References... 105 3 Buckling and Post-buckling of Curved Beams and Rings... 111 3.1 Introduction... 111 3.2 Strain-Displacement Relations and Constitutive Law... 112 3.3 Equilibrium Equations... 114 3.4 Stability Equation... 115 3.5 Stability of Arches, Uniform Pressure... 116 3.5.1 Introduction... 116 3.5.2 General Formulation... 118 3.5.3 Existence of Bifurcation Type Instability... 120 3.5.4 Critical Bifurcation Loads... 120 3.5.5 Limit Load Instability Analysis... 123 3.5.6 Result and Discussion... 124 3.5.7 Studying the Bifurcation Behavior... 130 3.6 Stability of Arches; Concentrated Force... 133 3.6.1 Introduction... 133 3.6.2 Governing Equations... 135 3.6.3 Bifurcation Analysis... 140 3.6.4 Limit Load Type of Instability... 144 3.6.5 Results and Discussion... 145 3.6.6 Studying the Bifurcation Phenomenon... 156 3.7 Thermal Buckling of Arches... 156 3.7.1 Governing Equations... 158 3.7.2 Displacements and Internal Forces... 161 3.7.3 Stress and Strain... 163
Contents xiii 3.7.4 Nonlinear Buckling Analysis... 164 3.7.5 Results and Discussions... 166 3.8 Postbuckling of Rings... 169 3.8.1 Governing Equations... 172 3.8.2 Prebuckling Analysis... 175 3.8.3 Stability Equations... 176 3.8.4 Postbuckling Analysis... 178 3.8.5 Solution Procedure... 181 3.8.6 Numerical Results and Discussion... 184 3.9 Problems... 186 References... 187 4 Buckling of Rectangular Plates... 189 4.1 Introduction... 189 4.2 Kinematic Relations and the Constitutive Law... 190 4.3 Equilibrium Equations... 192 4.4 Stability Equations... 195 4.5 Existence of Bifurcation Type Buckling... 197 4.6 Thermal Buckling of Rectangular Plates... 197 4.6.1 Introduction... 197 4.6.2 Governing Equations and Boundary Conditions... 199 4.6.3 Pre-buckling Loads... 200 4.6.4 Result and Discussions... 203 4.7 Rectangular Plates; In-Plane Compressive Load... 216 4.7.1 Introduction... 216 4.7.2 Governing Equations... 217 4.7.3 Buckling Analysis... 218 4.7.4 Results and Discussion... 220 4.8 Thermoelastic Buckling of Piezo-Controlled Plates... 224 4.8.1 Introduction... 224 4.8.2 Fundamental Equations... 224 4.8.3 Thermal Buckling... 230 4.8.4 Result and Discussion... 231 4.9 FGM Plates on Pasternak Elastic Foundation... 232 4.9.1 Introduction... 232 4.9.2 Governing Equations... 232 4.9.3 Existence of Bifurcation Type Buckling... 234 4.9.4 Stability Equations... 235 4.9.5 Solution of the Stability Equation... 236 4.9.6 Babnov Galerkin Solution (BGS)... 237 4.9.7 Power Series Solution (PSS)... 238 4.9.8 Semi-Levy Solution (SLS)... 240 4.9.9 Types of Thermal Loading... 240 4.9.10 Result and Discussions... 242
xiv Contents 4.10 Sandwich Plates on the Pasternak Elastic Foundation... 247 4.10.1 Introduction... 247 4.10.2 Governing Equations... 247 4.10.3 Mechanical Buckling... 252 4.10.4 Thermal Buckling... 253 4.10.5 Results and Discussion... 254 4.11 Imperfect Plates on Elastic Foundation... 260 4.11.1 Introduction... 260 4.11.2 Sandwich FGM Plates... 261 4.11.3 Governing Equations... 262 4.11.4 Solving Equations... 264 4.11.5 Results and Discussion... 268 4.11.6 Comparative Studies... 268 4.11.7 Parametric Studies... 270 4.12 Problems... 274 References... 274 5 Buckling and Post-buckling of Circular/Annular Plates... 279 5.1 Introduction... 279 5.2 Kinematic Relations and Constitutive Law... 280 5.3 Equilibrium Equations... 282 5.4 Stability Equations... 283 5.5 Thermal Buckling of Circular and Annular Plates... 284 5.5.1 Introduction... 284 5.5.2 Governing Equations... 286 5.5.3 Existence of Bifurcation Type Buckling... 287 5.5.4 Solving the Stability Equation for Annular Plates... 288 5.5.5 Solving the Stability Equation for Circular Plates... 290 5.5.6 Types of Thermal Loading... 291 5.5.7 Results and Discussions... 293 5.6 Thermal Buckling of Shear Deformable Annular Plates.... 298 5.6.1 Introduction... 298 5.6.2 Fundamental Equations of FG Annular Plate... 298 5.6.3 Equilibrium Equations... 301 5.6.4 Stability Equations... 302 5.6.5 Decoupling the Stability Equations... 304 5.6.6 Non-dimensionalizing and Solving the Stability Equations... 306 5.6.7 Numerical Investigation... 308 5.6.8 Parametric Studies... 309 5.7 Circular Plate on Partial/Complete Foundation... 317 5.7.1 Introduction... 317 5.7.2 Governing Equations... 318 5.7.3 Stability Equations... 320 5.7.4 Solving the Stability Equation... 321
Contents xv 5.7.5 Exterior Region, Contact-Less Domain... 323 5.7.6 Continuity and Boundary Conditions... 323 5.7.7 Results and Discussions... 325 5.8 Thermal Buckling of Annular Plates on Pasternak Medium... 329 5.8.1 Introduction... 329 5.8.2 Governing Equations... 330 5.8.3 Results and Discussions... 335 5.9 Thermo-Inertial Stability of Circular Plates... 343 5.9.1 Introduction... 343 5.9.2 Governing Equations... 344 5.9.3 Bifurcation-Type Buckling and Pre-buckling State... 345 5.9.4 Stability Equations... 346 5.9.5 Solving the Stability Equation... 348 5.9.6 Analytical Solution... 349 5.9.7 Power Series Solution... 350 5.9.8 Nonlinear Analysis... 351 5.9.9 Results and Discussions... 352 5.10 Thermal Postbuckling of Imperfect Circular FGM Plates... 358 5.10.1 Fundamental Equations of the FG Circular Plates... 358 5.10.2 Temperature Profile... 362 5.10.3 Equilibrium Equations... 363 5.10.4 Results and Discussion... 365 5.10.5 Comparison Study... 366 5.10.6 Parametric Studies... 366 5.11 Problems... 376 References... 377 6 Buckling of Circular Cylindrical Shells... 381 6.1 Introduction... 381 6.2 Kinematical Relations and the Constitutive Laws... 382 6.3 Equilibrium Equations... 384 6.4 Stability Equations... 385 6.5 Mechanical Buckling, Timoshenko Technique... 387 6.5.1 Derivations... 388 6.5.2 Numerical Results... 392 6.6 Thermal Buckling of FGM Cylindrical Shell... 393 6.6.1 Introduction... 393 6.6.2 Derivations... 393 6.6.3 Prebuckling Analysis... 394 6.6.4 Types of Thermal Loading... 397 6.6.5 Results and Discussion... 399 6.7 Thermal Buckling; Imperfect Wan-Donnell Model... 402 6.7.1 Introduction... 402 6.7.2 Fundamental Equations... 403 6.7.3 Axisymmetric Imperfections... 404
xvi Contents 6.7.4 Thermal Buckling... 409 6.7.5 Result and Discussion... 411 6.8 Thermal Buckling; Piezoelectric FGM Shells... 417 6.8.1 Introduction... 417 6.8.2 Fundamental Equations... 418 6.8.3 Thermal Buckling... 426 6.8.4 Result and Discussion... 428 6.9 Dynamic Thermal Postbuckling; Piezoelectric Shells... 432 6.9.1 Introduction... 432 6.9.2 Kinematical and Constitutive Equations... 432 6.9.3 Equations of Motion... 433 6.9.4 Postbuckling Analysis... 435 6.9.5 Numerical Solution... 437 6.9.6 Result and Discussion... 439 6.10 Mechanical Buckling, Shell on Elastic Foundation... 444 6.10.1 Introduction... 444 6.10.2 Governing Equations... 445 6.10.3 Mechanical Buckling Analysis... 448 6.10.4 Results and Discussion... 451 6.11 Problems... 459 References... 460 7 Buckling of Spherical Shells... 465 7.1 Introduction... 465 7.2 Kinematic Relations and the Constitutive Law... 466 7.3 Equilibrium Equations... 471 7.4 Stability Equations... 473 7.5 Isotropic Shallow Shells, Mechanical Load... 478 7.5.1 Introduction... 478 7.5.2 Derivations... 479 7.5.3 Mechanical Buckling Load... 480 7.6 Thermal Buckling, Isotropic Spherical Shells... 481 7.6.1 Introduction... 481 7.6.2 Derivations... 481 7.6.3 Results and Discussion... 484 7.7 Perfect Shallow FGM Spherical Shells... 486 7.7.1 Introduction... 486 7.7.2 Derivations... 487 7.7.3 Results and Discussion... 492 7.8 Perfect Deep FGM Spherical Shells... 494 7.8.1 Introduction... 494 7.8.2 Derivations... 495 7.8.3 Results and Discussion... 497
Contents xvii 7.9 Imperfect FGM Spherical Shells... 499 7.9.1 Introduction... 499 7.9.2 Derivation... 500 7.9.3 Results and Discussion... 502 7.10 Piezoelectric Shallow and Deep FGM Shells... 509 7.10.1 Introduction... 509 7.10.2 Derivations... 509 7.10.3 Results and Discussion... 513 7.11 Nonlinear Analysis of Piezo-FGM Shallow Shells... 522 7.11.1 Derivations... 522 7.11.2 Thermomechanical Analysis... 525 7.11.3 Results and Discussion... 529 7.12 Problems... 534 References... 535 8 Buckling of Conical Shells... 539 8.1 Introduction... 539 8.2 Kinematic Relations and the Constitutive Law... 540 8.3 Equilibrium Equations... 543 8.4 Stability Equations... 544 8.5 Mechanical Instability of Truncated Conical Shells... 545 8.5.1 Introduction... 545 8.5.2 Derivations... 545 8.5.3 Results and Discussion... 548 8.6 Thermal Instability of Conical Shells... 555 8.6.1 Introduction... 555 8.6.2 Governing Equations... 557 8.6.3 Prebuckling Deformations and Bifurcation Concept... 560 8.6.4 Nonlinear Bending Approach... 560 8.6.5 Linear Bending Approach... 561 8.6.6 Linear Membrane Approach... 562 8.6.7 Stability Equations... 563 8.6.8 Numerical Result and Discussion... 565 8.6.9 Parametric Studies... 567 8.7 Thermal Buckling of Piezo-FGM Conical Shells... 572 8.7.1 Introduction... 572 8.7.2 Governing Equations... 573 8.7.3 Prebuckling Analysis... 577 8.7.4 Stability Equation... 578 8.7.5 Solution Procedure... 579 8.7.6 Results and Discussion... 580 8.8 Problems... 585 References... 586