Elasto-Plastic and Damage Analysis of Plates and Shells

Transcription

1 Elasto-Plastic and Damage Analysis of Plates and Shells

2 George Z. Voyiadjis Pawel Woelke Elasto-Plastic and Damage Analysis of Plates and Shells With 82 Figures and 14 Tables 123

3 Dr. George Z. Voyiadjis Dr. Pawel Woelke Department of Civil and Environmental Applied Science Dept. Engineering Weidlinger Associates, Inc. CEBA Building, Taylor Hall 375 Hudson Street, 12 FL Room 3508-B New York, NY Louisiana State University USA Baton Rouge, LA USA ISBN: e-isbn: Library of Congress Control Number: c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: Kirchner, Erich Printed on acid-free paper springer.com

4 In Memory of Professor Maciej Bieniek

5 Professor Maciej Bieniek The subject of shell mechanics was extensively studied by the senior author s advisor and friend, Professor Maciej Bieniek, to whom this book is dedicated. Maciej P. Bieniek was born in Wilno, Poland, in His academic and professional career began in June 1945, at Cracow Polytechnic, the first technical university to open in Poland after the war. He later transferred to Gdansk University of Technology, from which he graduated in 1948 with the equivalent of an M.Sc. degree in civil engineering, majoring in Structures and Transportation. He remained in Gdansk, where and upon completion of his research on viscoelastic wave propagation, he received his Ph.D. degree in Applied Mechanics in In 1958, Bieniek was appointed a Visiting Scholar in the Department of Civil Engineering and Engineering Mechanics at Columbia University and the following year, Visiting Associate Professor. In 1960, he was named an Associate Professor of Civil Engineering. Three years later, Bieniek became a Professor of Civil Engineering at the University of Southern California. His research interests at that time were in the areas of structural dynamics, mechanics of shells, random vibrations, and elasto-plastic wave propagation. Bieniek returned to Columbia in 1969 as a Professor of Civil Engineering. In 1979, he received the Great Teacher Award of the Society of Columbia Graduates. He chaired the department from 1979 to In 1989, he was named Renwick Professor and, upon his retirement in 1993, Renwick Professor Emeritus. Bieniek supervised 33 doctoral students, including the senior author of this book. He also authored and co-authored numerous scientific publications in refereed technical journals. His achievements cover highly advanced areas in continuum mechanics, vibrations, structural dynamics, shell mechanics, and reliability of structures. He was known for his innovative approach to research with an emphasis on applicability to engineering practice. Throughout his career, Bieniek was active as a consultant. At Weidlinger Associates, he was responsible for development of the two dynamic finite element codes: TRANAL, for buried structures, and EPSA, for submerged shells. Both are widely used by Weidlinger Associates as well as various government agencies. He also served as a consultant to the Port Authority of New York and New Jersey and to the Triborough Bridge and Tunnel Authority on New York City s major suspension bridges. His significant contributions were recognized in 1991, by the vii

6 viii Professor Maciej Bieniek Roebling Award of the Metropolitan Section of the American Society of Civil Engineers (ASCE). In 1995, Bieniek was named an Honorary Member of the ASCE, one of the highest honors an engineer can receive. Maciej P. Bieniek died on January 23, 2006, in Atlanta, GA. Baton Rouge, Louisiana New York George Z. Voyiadjis Pawel Woelke

7 Preface Shells and plates are critical structures in numerous engineering applications. Analysis and design of these structures is of continuing interest to the scientific and engineering communities. Accurate and conservative assessments of the maximum load carried by a structure, as well as the equilibrium path in both the elastic and inelastic range, are of paramount importance to the engineer. The elastic behavior of shells has been closely investigated, mostly by means of the finite element method. Inelastic analysis however, especially accounting for damage effects, has received much less attention from researchers. In this book, we present a computational model for finite element, elasto-plastic, and damage analysis of thin and thick shells. Formulation of the model proceeds in several stages. First, we develop a theory for thick spherical shells, providing a set of shell constitutive equations. These equations incorporate the effects of transverse shear deformation, initial curvature, and radial stresses. The proposed shell equations are conveniently used in finite element analysis. AsimpleC 0 quadrilateral, doubly curved shell element is developed. By means of a quasi-conforming technique, shear and membrane locking are prevented. The element stiffness matrix is given explicitly, making the formulation computationally efficient. We represent the elasto-plastic behavior of thick shells and plates by means of the non-layered model, using an Updated Lagrangian method to describe a small-strain geometric non-linearity. For the treatment of material non-linearities, we adopt an Iliushin s yield function expressed in terms of stress resultants, with isotropic and kinematic hardening rules. Finally, we incorporate the damage effects modeled through the evolution of porosity into the yield function, giving a generalized and convenient yield surface expressed in terms of the stress resultants. As the elastic stiffness matrix is derived explicitly, and a non-layered model in which integration through the thickness is unnecessary is used, the current stiffness matrix is also given explicitly and numerical integration is not performed at any stage of the analysis. This makes the model mathematically consistent, accurate for a variety of applications, and economical in terms of computational power. We would like to acknowledge the editing assistance of Ann Rae Jonas. ix

8 Contents 1 Introduction Shell Structures Motivation and Scope BasicAssumptions... 5 References Shell Constitutive Equations Introduction Thickness of the Shell Initial Curvature and Radial (Transverse Normal) Stresses Plate Constitutive Equations StressesandStressResultantsinaThinPlate Equilibrium Equations and Governing Differential EquationofPlate Transverse Shear and Transverse Normal Stresses in a Plate Coordinate Transformation Strains in Spherical Coordinates Theoretical Formulation of the Shell Equations Assumed Out-of-Plane Stress Components Displacement Field Stress Components Stress Couples and Stress Resultants on the Middle Surface Average Displacements ū, v, w and Rotations φ θ,φ φ Equilibrium Equations and Boundary Conditions The Non-Linear Nature of the Stress Distribution TheEquivalentFormulationforThickPlates Examples Thick Sphere Subjected to Uniform Pressures Thick Cylinder Subjected to Uniform Pressures Summary References xi

9 xii Contents 3 Shell Element Based on the Refined Theory of Thick Spherical Shells Introduction Shear Locking Membrane Locking Mesh Instabilities FiniteElementFormulation Shell Constitutive Equations Displacements and Boundary Conditions Element Displacement and Strain Fields Quasi- ConformingMethod StrainEnergyandStiffnessMatrix NumericalExamples ThePatchTest Cantilevered Beam Morley s Hemispherical Shell (Morley and Moris, 1978) Pinched Cylinder with Diaphragms Scordellis-Lo Roof Pinched Cylinder Summary References Geometrically Non-linear Finite Element Analysis of Thick Plates and Shells Introduction Updated Lagrangian Description Shell Kinematics Local Coordinates SurfaceCoordinates BaseCoordinates Explicit Tangent Stiffness Matrix NumericalExample Summary References Elasto-Plastic Geometrically Non-linear Finite Element Analysis of Thick Plates and Shells Introduction Yield Criterion and Hardening Rule Iliushin s Yield Function (Iliushin, 1956) Influence of the Shear Forces DevelopmentofthePlasticHinge Bauschinger Effect and Kinematic Hardening Rule Explicit Elasto-Plastic Tangent Stiffness Matrix with Large Displacements NumericalExamples...106

10 Contents xiii Simply Supported Elasto-Plastic Beam Simply Supported Plate Cylindrical Shell Subjected to Ring of Pressure Spherical Dome Subjected to Ring of Pressure Summary References Elasto-Plastic Geometrically Non-linear Finite Element Analysis of Thick Plates and Shells With Damage Due to Microvoids Introduction YieldandDamageCriterion Explicit Tangent Stiffness Matrix NumericalExamples Clamped Square Plate Subjected to a Central Point Load Spherical Dome Subjected to Ring of Pressure Summary References Non-linear Post Buckling Finite Element Analysis of Plates and Shells Introduction Element Tangent Stiffness Matrix Element Stiffness in Local Coordinates Initial Surface Coordinates for Large Deformation Analysis TransformationofElementStiffnessMatrix SolutionAlgorithm NumericalExamples The Williams Toggle Frame Simply Supported Circular Plate Subjected to Edge Pressure Rectangular Plate Subjected to In-Plane Load Cylindrical Shell Under a Central Load Spherical Shell Subjected to Central Load Summary References Determination of Transverse Shear Stresses and Delamination in Composite Laminates Using Finite Elements Introduction Kinematics of the Shell Lamina Constitutive Equations FailureCriteriaforCompositeLaminates ImplementationandNumericalExamples Laminated Composite Strip under Three-Point Bending Composite Cylinder under Internal Pressure Cylindrical Shell Subjected to Ring of Pressure Summary References...183

11 xiv Contents 9 Numerical Methods and Computational Algorithms Introduction Linear Elastic Analysis System of Linear Algebraic Equations Non-linear Analysis System of Non-linear Algebraic Equations Modified Newton-Raphson Method Combined Incremental/IterativeSolutions The Arc-Length Technique Integrating the Rate Equations Return to the Yield Surface. 194 References Appendix Index...201