Handbook of Thin Plate Buckling and Postbuckling Handbook of Thin Plate Buckling and Postbuckling Frederick Bloom Department of Mathematics Northern Illinois University DeKalb, Illinois Douglas Coffin Institute of Paper Science and Technology Atlanta, Georgia CHAPMAN & HALL/CRC Boca Raton London New York Washington, D.C. Library of Congress Cataloging-in-Publication Data Bloom, Frederick, 1944- Handbook of thin plate buckling and postbuckling / Frederick Bloom, Douglas Coffin. p. cm. Includes bibliographical references and index. ISBN 1-58488-222-0 1. Plates (Engineering). 2. Buckling (Mechanics). I. Coffin, Douglas. II. Title. TA660.P6 B55 2000 624.1¢7765—dc21 00-050441 CIP This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. © 2001 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-222-0 Library of Congress Card Number 00-050441 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper Contents Preface 1Introduction:PlateBucklingandthevonKarman Equations 1.1BucklingPhenomena 1.2ThevonKarmanEquationsforLinearElasticIsotropic andOrthotropicThinPlates 1.2.1RectilinearCoordinates 1.2.2PolarCoordinates 1.3BoundaryConditions 1.3.1BoundaryConditionsontheDeflection 1.4TheLinearEquationsforInitialBuckling 1.5Figures:PlateBucklingandthevonKarmanEquations 2InitialandPostbucklingBehaviorof(Perfect)Thin RectangularPlates 2.1PlateswithLinearElasticBehavior 2.1.1IsotropicSymmetry:InitialBuckling 2.1.2RectilinearOrthotropicSymmetry: InitialBuckling 2.1.3PostbucklingBehaviorofIsotropicandRectilinear OrthotropicLinearElasticRectangularPlates 2.2PlateswithNonlinearElasticBehavior:The Johnson-UrbanikGeneralizationofthe vonKarmanEquations 2.3PlateswhichExhibitElastic-Plasticor ViscoelasticBehavior 2.4ComparisonsofInitialandPostbucklingBehavior ofRectangularPlates 2.4.1VariationswithRespecttoBoundaryConditions 2.4.2VariationswithRespecttoSymmetry 2.4.3VariationswithRespecttoConstitutiveResponse 2.5InitialBuckling/PostbucklingFigures,Graphs,and Tables:RectangularPlates 3InitialandPostbucklingBehaviorofThin CircularPlates 3.1PlateswithLinearElasticBehavior 3.1.1IsotropicSymmetry:InitialBucklingand PostbucklingBehavior 3.1.2CylindricallyOrthotropicSymmetry:Initial BucklingandPostbucklingBehavior 3.1.3RectilinearOrthotropicSymmetry:Initial BucklingandPostbucklingBehavior 3.2PlateswhichExhibitNonlinearElastic, Viscoelastic,orElastic-PlasticBehavior 3.3ComparisonsofInitialandPostbuckling BehaviorofCircularPlates 3.4InitialBuckling/PostbucklingFigures,Graphs,and Tables:CircularPlates 4InitialandPostbucklingBehaviorof(Perfect)Thin AnnularPlates 4.1PlateswithLinearElasticBehavior 4.1.1IsotropicSymmetry:InitialBuckling 4.1.2CylindricallyOrthotropicSymmetry: InitialBuckling 4.1.3IsotropicandCylindricallyOrthotropic Symmetry:PostbucklingBehavior 4.1.4RectilinearOrthotropicSymmetry:Initial BucklingandPostbucklingBehavior 4.2PlateswhichExhibitNonlinearElastic, Viscoelastic,orElastic-PlasticBehavior 4.3ComparisonsofInitialandPostbuckling BehaviorofAnnularPlates 4.4InitialBuckling/PostbucklingFigures,Graphs,and Tables:AnnularPlates 5PostbucklingBehaviorofImperfectPlates andSecondaryPlateBuckling 5.1ImperfectionSensitivity 5.1.1AnExampleofImperfectionBifurcation 5.1.2RectangularPlates 5.1.3CircularandAnnularPlates 5.2SecondaryBucklingforThinPlates 5.2.1SecondaryBucklingforCircularPlates 5.2.2SecondaryBucklingforRectangularPlates 5.3ImperfectionBucklingandSecondaryBucklingFigures, Graphs,andTables 6GeneralizedvonKarmanEquationsforElastic PlatesSubjecttoHygroexpansiveorThermal StressDistributions 6.1RectilinearCoordinates 6.2PolarCoordinates 6.3BoundaryConditions 6.4ThermalBendingandBucklingEquationsandBoundary Conditions 7ThermalBending,Buckling,andPostbuckling ofRectangularandCircularPlates 7.1SmallDeflectionTheory 7.2LargeDeflectionTheory 7.3ApplicationsoftheBerger’sApproximation 7.4ThermalBending,Buckling,andPostbucklingFigures, Graphs,andTablesI 8OtherAspectsofHygrothermalandThermalBuckling 8.1Hygroexpansive/ThermalBucklinginthe PresenceofImperfections 8.2BucklingofPlatesofVariableThickness 8.3ViscoelasticandPlasticBuckling 8.4ThermalBending,Buckling,andPostbucklingFigures, Graphs,andTablesII References Preface IntheSpringof1996,theauthorsinitiatedaresearchprogramattheIn- stitute of Paper Science and Technology (Atlanta, Ga.) whose goal was to model cockle, i.e., the hygroscopic buckling of paper. In the course of researching the aforementioned problem, an extensive examination was made of the expansive literature on the buckling and postbuckling behavior of thin plates in an effort to understand how such phenomena were controlled by the wide variety of factors listed below. A compara- tivestudyofplatebucklingwasthenproducedandpublishedasapairof IPST reports; this book presents the essential content of these reports. The essential factors which influence critical buckling loads, initial mode shapes, and postbuckling behavior for thin plates are studied, in detail, with specific examples, throughout this treatise; among the factors discussed which affect the occurrence of initial buckling and the initial mode shapes into which plates buckle are the following: (i) Aspect Ratios: Included in our discussions will be examples of buckling for rectangular plates, circular plates, and annular plates; we will look at how, e.g., critical buckling loads for the various types of plates are affected by changes in the aspect ratio, associated with the plate, given that all other factors (support conditions, load conditions, etc.) are held fixed. For a rectangular plate, the aspect ratio is just the ratio of its sides; for a circular plate it is just the plate radius while, for an annular region, it is the ratio of the inner to the outer radius. (ii) Support Conditions: Theedge(s)ofaplatemaybesupported in many different ways; mathematically, an edge support condition is reflectedinaconstraintontheout-of-planedeflectionofthatplatealong its edge(s). The basic types of support that will be considered are those which correspond to having the edges(s) of the plate clamped, free, or simply supported. For a rectangular plate or an annular plate, one edge (or pair of edges) may be supported in one manner while the other edge (or pair) of edges can be supported in a different manner. (iii) LoadConditions: Theedge(s)ofathinplatemaybesubjected 1 to compressive, tensile, or shear loadings or to some combinations of such loadings along different edges or pairs of edges. The effect of using different load conditions on critical buckling loads and mode shapes for rectangular, circular, and annular plates are illustrated throughout the book. (iv) Material Symmetry: For thin plates exhibiting a particular constitutiveresponse,thematerialsymmetryexhibitedbytheplatemay haveamarkedeffecton,e.g.,themagnitudeofthelowestbucklingload, when all other factors, such as aspect ratio or edge support conditions are held constant. The basic types of material symmetry that will be examined in this work are isotropic symmetry, rectilinearly orthotropic symmetry, and cylindrically orthotropic symmetry. (v) Constitutive Behavior: Our examination of the influence of constitutiveresponseoninitialbucklingofthinplateswasguidedbythe behavior observed with respect to the deformation of paper, i.e., that under various combinations of loading conditions, paper may exhibit linearelastic, nonlinearelastic, viscoelastic, orelastic-plastic. Examples of all four kinds of such constitutive response are presented throughout this treatise. (vi) HygroscopicorThermalStressDistributions: Thinplates are often exposed to variations in temperature or moisture or hygroex- pansivestrainswhichresultinlocalcompressivestressdistributionsthat can cause buckling even in the absence of applied external loading. Beyond the factors delineated above, which influence both critical buckling loads and mode shapes for thin plates and also influence the subsequent postbuckling behavior, two other important factors come into play with respect to postbuckling behavior. (vii) Initial Imperfections: Thin plates often possess some small initial transverse deflection (e.g., a paper sheet with an initial cockle which is wetted, dried, and experiences further local buckling) or may be subjected to a small normal loading (transverse) to the initially flat unbuckledstateoftheplate; thepresenceofeitherorbothofthesecon- ditionsamountstoan“initialimperfection”whichmayhaveaprofound effect on the subsequent postbuckling behavior of the plate. (viii) Secondary Buckling: After a thin plate has buckled into an initial mode shape, it is possible that, with increasing load, instead of obtainingabuckledshapewhichdeformscontinuouslyduringpostbuck- ling,asthemagnitudeoftheloadingincreases,acriticalloadlargerthan the initial buckling load is reached at which the buckle pattern changes suddenly and a new mode shape appears; such phenomena have been 2 exhibited with respect to plate buckling and will be examined with par- ticular emphasis on the unsymmetric wrinkling that has been observed in circular plates. Throughoutthistreatise,severalresultsappearwhichcannotbefound in earlier books on plate buckling, e.g., the discussion of the Johnson- Urbanik generalization of the von Karman Equations for plates exhibit- ing nonlinear elastic behavior; other results appear which have not pre- viously been published elsewhere, most notably the analysis of initial bucklingforannularrectilinearlyorthotropicthinplates,whichwaspre- sented in the Ph.D. Thesis of the second author. The authors wish to express their gratitude to the Institute of Paper ScienceandTechnologyanditsmembercompaniesforthesupportwhich made possible the production of the IPST reports that form the basis for the present work. Frederick Bloom Douglas Coffin DeKalb, Illinois Atlanta, Georgia April 2000 April 2000 3 THIS BOOK IS DEDICATED TO the memory of Sidney Bloom (1912-1999) and to the memory of his grandson Scott Bloom (1969-1997) 4