Structural Analysis with the Finite Element Method Linear Statics Volume 2. Beams, Plates and Shells

Lecture Notes on Numerical Methods in Engineering and Sciences

Aims and Scope of the Series This series publishes text books on topics of general interest in the field of computational engineering sciences. The books will focus on subjects in which numerical methods play a fundamental role for solving problems in engineering and applied sciences. Advances in finite element, finite volume, finite differences, discrete and particle methods and their applications are examples of the topics covered by the series. The main intended audience is the first year graduate student. Some books define the current state of a field to a highly specialised readership; others are accessible to final year undergraduates, but essentially the emphasis is on accessibility and clarity. The books will be also useful for practising engineers and scientists interested in state of the art information on the theory and application of numerical methods. Series Editor Eugenio Oñate International Center for Numerical Methods in Engineering (CIMNE) School of Civil Engineering, Technical University of Catalonia (UPC), Barcelona, Spain Editorial Board Francisco Chinesta, Ecole Nationale Supérieure d'Arts et Métiers, Paris, France Charbel Farhat, Stanford University, Stanford, USA Carlos Felippa, University of Colorado at Boulder, Colorado, USA Antonio Huerta, Technical University of Catalonia (UPC), Barcelona, Spain Thomas J.R. Hughes, The University of Texas at Austin, Austin, USA Sergio R. Idelsohn, CIMNE-ICREA, Barcelona, Spain Pierre Ladeveze, ENS de Cachan-LMT-Cachan, France Wing Kam Liu, Northwestern University, Evanston, USA Xavier Oliver, Technical University of Catalonia (UPC), Barcelona, Spain Manolis Papadrakakis, National Technical University of Athens, Greece Jacques Périaux, CIMNE-UPC Barcelona, Spain & Univ. of Jyväskylä, Finland Bernhard Schrefler, Università degli Studi di Padova, Padova, Italy Genki Yagawa, Tokyo University, Tokyo, Japan Mingwu Yuan, Peking University, China Titles: 1. E. Oñate, Structural Analysis with the Finite Element Method.

Linear Statics. Volume 1. Basis and Solids, 2009 2. K. Wiśniewski, Finite Rotation Shells. Basic Equations and

Finite Elements for Reissner Kinematics, 2010 3. E. Oñate, Structural Analysis with the Finite Element Method. Linear Statics. Volume 2. Beams, Plates and Shells, 2013 4. E.W.V. Chaves. Notes on Continuum Mechanics. 2013

Structural Analysis with the Finite Element Method Linear Statics Volume 2. Beams, Plates and Shells

Eugenio Oñate International Center for Numerical Methods in Engineering (CIMNE)

School of Civil Engineering Universitat Politècnica de Catalunya (UPC) Barcelona, Spain

ISBN: 978-1-4020-8742-4 (HB) ISBN: 978-1-4020-8743-1 (e-book) Depósito legal: B-18335-2012 A C.I.P. Catalogue record for this book is available from the Library of Congress Lecture Notes Series Manager: Mª Jesús Samper, CIMNE, Barcelona, Spain Cover page: Pallí Disseny i Comunicació, www.pallidisseny.com Printed by: Artes Gráficas Torres S.A., Morales 17, 08029 Barcelona, España www.agraficastorres.es Printed on elemental chlorine-free paper

Structural Analysis with the Finite Element Method. Linear Statics. Volume 2. Beams, Plates and Shells Eugenio Oñate First edition, 2013

International Center for Numerical Methods in Engineering (CIMNE), 2013 Gran Capitán s/n, 08034 Barcelona, Spain www.cimne.com No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

To my family

Foreword

The present volume continues the objective of Volume 1: to present theFinite Element Method (FEM) for solid and structural mechanics from abalanced perspective that interweaves theory, formulation, physical mod-eling and computer implementation. The challenging balance is made pos-sible by the three-decade practical experience of the author in teachingFEM courses while building and leading a large research center (CIMNE)that conducts advanced research in computational mechanics for a widevariety of engineering applications. That experience is put to good use tosupport of the educational goals addressed by this book series. A majordifference between the first and second volume is in the level and detail ofcoverage.

The goal of the first volume was to offer an introductory coverage of FEM.The material concentrated on solid and structural mechanics to maintain aclear application focus. The problem class is restricted to linear static anal-ysis. This volume addressed the fundamentals steps of the Direct StiffnessMethod (DSM) version of FEM, while introducing related mathematicalconsiderations such as consistency, accuracy and convergence. These are inturn translated into practical modeling rules. Emphasis is on the physicalinterpretation of the method as a “divide and conquer” technique. Accord-ingly, the exposition level is that appropriate to a first course in FEM atthe master level. The assumed preparation level of a student taking sucha course is expected to include multivariate calculus, linear algebra, and abasic knowledge of structural analysis at the Mechanics of Materials level,as well as some familiarity with computer programming concepts. Becauseof the inevitable space and time limitations of a first-course treatment,the contents of Volume 1 were restricted to simple structural elements,notably axially loaded members and two-dimensional solids. Those mod-els are sufficient, however, to illustrate the primary steps of DSM, as wellas to provide specific examples that teach the most important modelingrules.

A key virtue of DSM is that its steps are applicable to any finite elementmodel formulated within the framework of the stiffness equations. Conse-quently, those steps need not be repeated in a more advanced treatmentsuch as that presented in this volume. The author is free to directly pro-ceed to more difficult problems by focusing on more advanced elementformulation techniques and associated modeling rules. The problems ad-dressed in the volume involve the classical structural components thatcarry bending actions: beams, plates and shell, as well as combinationssuch as stiffened plates and shells. The coverage still targets on linearstatic problems.

Consideration of bending effects brings about a new set of formulationand modeling difficulties. Foremost is locking: a pathological overstiffnessendemic to certain “thin” configurations, and which must be overcome toavoid unsafe designs. Remedies to those difficulties, however, can in turnproduce undesirable side effects. For example reduced integration to curelocking may give rise to numerical instabilities; e.g., mesh “hourglassing”.The delicate interplay between diagnosing and curing requires more ad-vanced mathematical tools, which go beyond those deemed sufficient forVolume 1. Accordingly, the present treatment is intended for follow-upcourses that cover more advanced models in structural mechanics, as wellas a more detail coverage of the computer implementation. The followingChapter summary give a more specific idea of the contents of the presentvolume.

Chapters 1-4 cover structural beam members. The Euler-Bernoulli andTimoshenko models of plane beams are presented in Chapters 1 and 2, re-spectively. Chapter 3 deals with composite laminate farbrications of planebeans. Chapter 4 addresses 3D beams, focusing on composite fabrica-tion and the problem of cross section warping under torsion and shear.Rotation-free beam element models are introduced as advanced topic.

Chapters 5-7 deal with flat plate structural components. The Kirchhoffand Reissner-Mindlin Model are covered in Chapters 5 and 6 respectively,while Chapter 7 addresses composite laminates fabrication of plate walls.Rotation free plate elements are covered again; at the plate level thisrepresents a still ongoing research topic.

Chapters 8 through 10 deal with shell structures. The coverage includesfacet element models, axisymmetric configurations, doubly curved shellmodels, culminating with the treatment of stiffened shells.

Chapter 11 covers prismatic structures. This is a specialized form of shells(e.g., folded thin roofs) which deserves special treatment on account of its

VIII Foreword

industrial importance as well as ubiquity in various engineering branches(Aerospace, Civi, Marine and Mechanical).

Chapter 12 present a computer implementation dubbed MATfem, sup-ported by the MATLAB high level programming language. This may beviewed as a “unification chapter” that brings together models formulatedin the previous chapters, as well as in Volume 1, into a MATLAB frame-work. This material was organized in collaboration with Professor Zarate.

Each of the formulation oriented chapters (1 through 11) covers its titlematerial in three stages. First, classical theories for the pertinent struc-tural configuration is summarized in a form suitable for FEM formulation.Second, the construction of finite element models based on those theories.Third, advanced material pertaining to the topic is included; for example,how to overcome difficulties such as locking. Some of the advanced materialis still the matter of ongoing research by the author’s team at CIMNE. Animportant example are rotation-free elements for beam, plates and shells,as well as the treatment of prismatic structures.

The volume concludes with seven Appendices that summarize relevantreference material for the benefit of the reader.

Several features that clearly distinguish this volume from other texts ata similar level should be noted. The rich treatment of composite fabrica-tion in Chapters 3, 4 and 7 does not have a counterpart in other FEMtextbooks. The treatment of rotation-free beam, plate and shell elementsin Chapters 1, 2, 5 and 9 reflects the long-term involvement of the authorin that research thrust, which offers significant promise in its extension toextremely large deformations as occur in important fabrication processes.Finally, the MATfem implementation in Chapter 12 stands out for thecareful attention to modularity and completeness.

While as noted Volume 1 was specifically oriented to an introductorycourse, the present volume can be used in two contexts:

1. A textbook that support advanced FEM courses. Since the overallcontent is too extensive to be covered in a one-semester course, theinstructor will likely need to select specific presentation topics, andperhaps designate others as launching pads for course team projects.The instructor will have to provide exercitation problems that enhancethe student comprehension of the material, and well as exam problemsthat test that understanding.

Foreword IX

2. As a reference monograph for advanced topics that are not adequatelytreated aside from the specialized literature. An instance would berotation free elements for complex plates and shell assemblies.

In summary, the present volume ably complements and supplements thefirst one with a wealth of material that provide both instructors and re-searchers with a wealth of possibilities.

Carlos FelippaProfessor in the University of Colorado at BoulderNovember 2012

X Foreword

Preface

This book complements the content of the first volume: Structural Analysiswit the FEM. Basis and solids (Springer/CIMNE, 2009). The scope of thesecond volume covers the finite element analysis of “structural elements”such as beams, plates and shells. Similarly, as in Volume 1, the study isrestricted to linear static analysis only.The book is addressed to undergraduate students and readers that areexposed to the FEM analysis of beams, plates and shells for the first time.No previous experience on the FEM is strictly necessary. However, someknowledge of basic FEM concepts, such as mesh discretization, displace-ment interpolation, shape functions, numerical integration, element ma-trices and vectors, assembly of the stiffness equations, etc., as described inVolume 1, will facilitate the understanding of the underlying ideas behindthe FEM.Throughout the text emphasis has been put in the study of beam, plateand shell structures with composite laminated material, this being a possi-bility of increasing interest for practical applications. For didactic reasons,however, the homogeneous and isotropic material case is considered first,and in most cases in a separate chapter, in order to facilitate the study toan inexperienced reader.As in Volume 1, the study of each structural model is introduced with thedetailed description of the underlying theory. This includes the generalstructural mechanics assumptions, the kinematic description, the stressand stress fields, the constitutive relationship and the expression of thevirtual work principle in terms of stress resultants and generalized strainsdefined over the “reference geometry” of the element, i.e. a line for a beam,a plane for a plate or a flat shell and a curved surface for a general shellstructure.Chapter 1 introduces the FEM analysis of two-dimensional (2D) planeslender beams following the classical Euler-Bernoulli beam theory. Thepopular two-noded Euler-Bernoulli beam element is derived in some de-tail. The chapter concludes with the formulation of two rotation-free beamelements. These elements have the vertical deflection as the only nodal

variable and are an interesting alternative to standard slender beam ele-ments.Chapter 2 describes the formulation of beam elements based on the moreadvanced Timoshenko beam theory. This theory accounts for the effect ofshear deformation and is therefore applicable to slender and thick beams.Timoshenko beam elements suffer from the so-called shear locking defectthat leads to overstiff situations for slender beams. A number of proceduresto avoid shear locking are described.Chapter 3 is devoted to the FEM analysis of beams with composite lami-nated material. Emphasis is put in studying the particularities introducedby the composite material in the mechanical behaviour of the beam, themore important one being the introduction of the axial deformation modein addition to the standard bending mode typical of homogeneous planebeams. Higher order beam theories that are able to reproduce the complex-ity of the in-plane displacement field across the thickness are described. Aneffective 2-noded composite laminated Timoshenko beam element basedon the so-called refined zigzag theory is detailed.Chapter 4 studies three-dimensional (3D) Timoshenko and Euler-Bernoullibeams under arbitrary loading. Both the classical Saint-Venant theory al-lowing for a uniform torsion field and the more complex theory for non-uniform torsion of beams with thin-walled open section are described.Both theories are generalized for homogeneous and composite material. Anumber of two-noded beam elements adequate for each theory consideredare presented.Chapter 5 focusses on the FEM analysis of homogeneous thin plates fol-lowing the classical Kirchhoff plate theory. The difficulties for satisfyingthe continuity condition for the deflection slopes across the element sidesare detailed. A number of conforming and non-conforming quadrilateraland triangular thin plate elements are presented. Part of the chapter isdevoted to the study of two families of rotation-free thin plate triangles.Chapter 6 extends the FEM analysis to thick and thin homogeneous platesthat follow the more advanced Reissner-Mindlin plate theory. This can beviewed as an extension of Timoshenko beam theory and, thus, it accountsfor transverse shear deformation effects. Different techniques to avoid theshear locking defect are detailed. A collection of Reissner-Mindlin quadri-lateral and triangular plate elements is presented.Chapter 7 is devoted to composite laminated plates. The effect of the axialdisplacement induced by the changes in the material properties across thethickness in the governing equations of the plate problem are detailed. An

XII Preface

accurate 4-noded composite laminated plate quadrilateral based on therefined zigzag theory is presented.Chapter 8 introduces the FEM study of shell structures via flat shell ele-ments. The basic equations of Reissner-Mindlin flat shell theory expressedin the local axes of the element are identical to those of a composite lam-inated plate as studied in the previous chapter. The equations for a flatshell element are formulated for homogeneous and composite laminatedmaterial. The particularity of the assembly of the element equations inglobal axes are detailed. The formulation of thin shell elements followingKirchhoff theory is also explained. The chapter concludes with the de-scription of some higher order theories for composite laminated flat shellelements, including the refined zigzag theory.Chapter 9 deals with axisymmetric shell structures. Reissner-Mindlin as-sumptions for the shell kinematics are studied first. Both homogeneousand composite laminated materials are considered. A simple locking-free2-noded troncoconical axisymmetric shell element is presented in some de-tail. The formulation of curved axisymmetric shell elements is also studiedand the combined effect of shear and membrane locking is discussed. Thederivation of thin axisymmetric shell elements following Kirchhoff assump-tions is explained and two interesting rotation-free thin axisymmetric shellelements are presented. The axisymmetric formulation is particularized forcircular plates, arches and shallow shells. The derivation of axisymmet-ric shell elements via the degeneration of 3D axisymmetric solid elementsis also explained. The chapter concludes with the study of higher ordertheories for composite laminated axisymmetric shells.Chapter 10 studies the analysis of 3D shell structures of arbitrary shapeusing degenerated solid elements. The degeneration process is explainedand the element matrices and vectors are obtained. Different techniquesfor deriving degenerated shell elements free of shear and membrane lockingare presented. Several procedures for the explicit integration of the elementstiffness matrix across the thickness are explained. The basic concepts ofthe isogeometry approach and the derivation of isogeometric shell elementsare studied. The chapter concludes with the formulation of stiffened shellelements by coupling beam and shell elements.Chapter 11 presents the derivation of finite strip and finite prism methodsfor analysis of prismatic plate/shell and 3D solid structures, respectively.These elements combine a finite element approximation across the trans-verse section of the structure with Fourier series expansions for represent-ing the longitudinal response. Both procedures can therefore be considered

XIIIPreface

as a particular class of reduced order models. The finite strip formulationis detailed for plates, folded plates with straight and circular plant andaxisymmetric shells with non arbitrary loading. The simple 2-noded stripallows one to analyze complex prismatic shell structures, such as box-girder bridges, in an extremely simple form. The finite prism formulationis derived for prismatic solids with straight and circular plant and axysym-metric solids with arbitrary loading.Chapter 12 explains the programming of some of the elements studied inthe book for beam, plate and shell analysis using MATLAB. The elementsare implemented in the MAT-Fem code that can be freely downloadedfrom the web. The general structure of the code is explained as well as thebasic concepts for programming the stiffness matrix and the equivalentnodal force for the element. The essential parts of the MAT-Fem code arelisted for each element considered.The book concludes with a number of appendices with details of topicsof general interest such as the properties of selected materials, the equi-librium equations for a solid, some numerical integration quadratures fortriangular, quadrilateral and hexahedral elements, the derivation of theshear correction parameters, the shear center and the warping functionsin beams, the stability conditions for beam and plate elements based onthe assumed strain technique, the analytical solution for circular platesand the expression of the shape functions for some C0 triangular andquadrilateral elements.I want to express my gratitude to Dr. Francisco Zarate who was responsiblefor writing the computer program Mat-fem presented in Chapter 12 andalso undertook the task of writing this chapter.The content of the book is an expanded version of the course on FiniteElement Analysis of Structures which I have taught at the School of CivilEngineering in the Technical University of Catalonia (UPC) since 1979. Iwant to express my thanks to my colleagues in the Department of Con-tinuum Mechanics and Structural Analysis at UPC for their support andcooperation over many years. Special thanks to Profs. Benjamın Suarez,Miguel Cervera and Juan Miquel, Drs. Francisco Zarate and Daniel diCapua and Mr. Miguel Angel Celigueta with whom I have shared theteaching of the mentioned course during many years.Many examples included in the book are the result of problems solvedby academics and research students at UPC and CIMNE in cooperationwith companies which are acknowledged in the text. I thank all of them

XIV Preface

for their contributions. Special thanks to the GiD team at CIMNE forproviding many pictures shown in the book.Many thanks also to my colleagues and staff at CIMNE for their coo-peration and support during so many years that has made possible thepublication of this book.Finally, my special thanks to Mrs. Marıa Jesus Samper from CIMNE forher excellent work in the typing and editing of the manuscript.

Eugenio OnateBarcelona, November 2012

XVPreface

Contents

1 SLENDER PLANE BEAMS. EULER-BERNOULLI THEORY 11.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 CLASSICAL BEAM THEORY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Displacement field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.3 Strain and stress fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.4 Bending moment-curvature relationship . . . . . . . . . . . . . . . . 31.2.5 Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 THE 2-NODED EULER-BERNOULLI BEAM ELEMENT . . . . . 61.3.1 Approximation of the deflection, curvature and bending

moment fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 Discretized equilibrium equations for the element.

Element stiffness matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.3 Equivalent nodal force vector for the element . . . . . . . . . . . 101.3.4 Global equilibrium equations . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 ROTATION-FREE EULER-BERNOULLI BEAM ELEMENTS . 201.4.1 Cell-centred beam (CCB) element . . . . . . . . . . . . . . . . . . . . . 211.4.2 Cell-vertex beam (CVB) element . . . . . . . . . . . . . . . . . . . . . . 281.4.3 Examples of application of CCB and CVB rotation-free

beam elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.5 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2 THICK/SLENDER PLANE BEAMS. TIMOSHENKOTHEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 TIMOSHENKO PLANE BEAM THEORY . . . . . . . . . . . . . . . . . . . 38

2.2.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2.2 Strain and stress fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2.3 Resultant stresses and generalized strains . . . . . . . . . . . . . . . 39

2.2.3.1 Computation of the shear correction parameter . . 412.2.4 Principle of virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3 TWO-NODED TIMOSHENKO BEAM ELEMENT. . . . . . . . . . . . 432.3.1 Approximation of the displacement field . . . . . . . . . . . . . . . . 43

XVIII Contents

2.3.2 Approximation of the generalized strains and theresultant stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3.3 Discretized equations for the element . . . . . . . . . . . . . . . . . . 452.4 LOCKING OF THE NUMERICAL SOLUTION . . . . . . . . . . . . . . 47

2.4.1 Substitute transverse shear strain matrix . . . . . . . . . . . . . . . 552.5 MORE ON SHEAR LOCKING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.6 SUBSTITUTE SHEAR MODULUS FOR THE TWO-NODED

TIMOSHENKO BEAM ELEMENT . . . . . . . . . . . . . . . . . . . . . . . . . 582.7 QUADRATIC TIMOSHENKO BEAM ELEMENT . . . . . . . . . . . . 582.8 ALTERNATIVES FOR DERIVING LOCKING-FREE

TIMOSHENKO BEAM ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . 622.8.1 Reinterpretation of shear locking . . . . . . . . . . . . . . . . . . . . . . 622.8.2 Use of different interpolations for deflection and rotation . 632.8.3 Linked interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.8.4 Assumed transverse shear strain approach . . . . . . . . . . . . . . 68

2.9 EXACT TWO-NODED TIMOSHENKO BEAM ELEMENT . . . . 812.10 ROTATION-FREE BEAM ELEMENT ACCOUNTING FOR

TRANSVERSE SHEAR DEFORMATION EFFECTS . . . . . . . . . 872.10.1 Iterative computation of the nodal deflections and the

element shear angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922.10.2 Performance of the CCB+1 element . . . . . . . . . . . . . . . . . . . 93

2.11 BEAMS ON ELASTIC FOUNDATION . . . . . . . . . . . . . . . . . . . . . . 942.12 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3 COMPOSITE LAMINATED PLANE BEAMS . . . . . . . . . . . . . . 983.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.2 KINEMATICS OF A PLANE LAMINATED BEAM . . . . . . . . . . . 993.3 BASIC CHARACTERISTICS OF COMPOSITE MATERIALS . 100

3.3.1 About the properties of fibers . . . . . . . . . . . . . . . . . . . . . . . . . 1013.3.2 About the properties of the matrix . . . . . . . . . . . . . . . . . . . . 1023.3.3 Approximation of the properties of the composite . . . . . . . 102

3.4 STRESSES AND RESULTANT STRESSES . . . . . . . . . . . . . . . . . . 1033.5 GENERALIZED CONSTITUTIVE MATRIX . . . . . . . . . . . . . . . . . 1033.6 AXIAL-BENDING COUPLING AND NEUTRAL AXIS . . . . . . . 1053.7 THERMAL STRAINS AND INITIAL STRESSES . . . . . . . . . . . . . 1063.8 COMPUTATION OF THE SHEAR CORRECTION

PARAMETER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.9 PRINCIPLE OF VIRTUAL WORK . . . . . . . . . . . . . . . . . . . . . . . . . 1123.10 TWO-NODED COMPOSITE LAMINATED TIMOSHENKO

BEAM ELEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.11 SHEAR LOCKING IN COMPOSITE LAMINATED BEAMS . . . 1143.12 EXACT TWO-NODED TIMOSHENKO BEAM ELEMENT

WITH COMPOSITE LAMINATED SECTION . . . . . . . . . . . . . . . 1163.13 COMPOSITE LAMINATED EULER-BERNOULLI BEAM

ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173.14 HIGHER ORDER COMPOSITE LAMINATED BEAM

THEORIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173.15 LAYER-WISE THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.16 ZIGZAG THEORIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Contents XIX

3.17 REFINED ZIGZAG THEORY (RZT) . . . . . . . . . . . . . . . . . . . . . . . . 1223.17.1 Zigzag displacement field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223.17.2 Strain and stress fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1233.17.3 Computation of the zigzag function . . . . . . . . . . . . . . . . . . . . 1243.17.4 Generalized constitutive relationship . . . . . . . . . . . . . . . . . . . 1253.17.5 Virtual work expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.18 TWO-NODED LRZ COMPOSITE LAMINATED BEAMELEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.19 STUDY OF SHEAR LOCKING AND CONVERGENCE FORTHE LRZ COMPOSITE LAMINATED BEAM ELEMENT . . . . 1283.19.1 Shear locking in the LRZ beam element . . . . . . . . . . . . . . . . 1283.19.2 Convergence of the LRZ beam element . . . . . . . . . . . . . . . . . 130

3.20 EXAMPLES OF APPLICATION OF THE LRZ COMPOSITELAMINATED BEAM ELEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.20.1 Three-layered laminated thick cantilever beam under end

point load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.20.2 Non-symmetric three-layered simple supported thick

beam under uniform load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1363.20.3 Non-symmetric ten-layered clamped slender beam under

uniformly distributed loading . . . . . . . . . . . . . . . . . . . . . . . . . 1403.20.4 Modeling of delamination with the LRZ element . . . . . . . . . 143

3.21 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4 3D COMPOSITE BEAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1504.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1504.2 BASIC DEFINITIONS FOR A 3D COMPOSITE BEAM. . . . . . . 152

4.2.1 Local and global axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1524.2.2 Constitutive behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534.2.3 Resultant constitutive parameters and neutral axis . . . . . . 1554.2.4 Principal inertia axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1564.2.5 Summary of steps for defining the local coordinate system 1574.2.6 Computation of the shear center . . . . . . . . . . . . . . . . . . . . . . 1584.2.7 Properties of the shear center . . . . . . . . . . . . . . . . . . . . . . . . . 159

4.3 3D SAINT-VENANT COMPOSITE BEAMS . . . . . . . . . . . . . . . . . 1624.3.1 Displacement and strain fields. Timoshenko theory . . . . . . 1624.3.2 Stresses, resultant stresses and generalized constitutive

matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1654.3.3 Generalized constitutive matrix . . . . . . . . . . . . . . . . . . . . . . . 1664.3.4 Computation of shear stresses due to torsion and the

warping function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714.3.5 Thin-walled closed sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

4.3.5.1 Torsion of multi-cellular sections . . . . . . . . . . . . . . . 1754.3.6 Virtual work expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

4.4 FINITE ELEMENT DISCRETIZATION. 2-NODEDTIMOSHENKO 3D BEAM ELEMENT . . . . . . . . . . . . . . . . . . . . . . 1804.4.1 Definition of neutral axis and element matrices . . . . . . . . . . 1804.4.2 Stiffness and force transformations . . . . . . . . . . . . . . . . . . . . . 183

4.5 QUASI-EXACT TWO-NODED 3D TIMOSHENKO BEAMELEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

XX Contents

4.6 CURVED TIMOSHENKO BEAM ELEMENTS . . . . . . . . . . . . . . . 1884.7 3D EULER-BERNOUILLI BEAMS. SAINT-VENANT THEORY 1894.8 PLANE GRILLAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1914.9 EXAMPLES OF THE PERFORMANCE OF 3D

TIMOSHENKO BEAM ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . 1954.10 THIN-WALLED BEAMS WITH OPEN SECTION . . . . . . . . . . . . 196

4.10.1 Geometric description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1974.10.2 Kinematic assumptions. Timoshenko theory . . . . . . . . . . . . 2004.10.3 Warping function and strains and stresses due to torsion . 2014.10.4 Resultant stresses and generalized constitutive equation . . 2044.10.5 Virtual work expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2074.10.6 Two-noded Timoshenko beam element with thin-walled

open section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2084.10.7 Computation of stresses due to torsion in thin-walled

open sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2104.11 THIN-WALLED OPEN TIMOSHENKO BEAM ELEMENTS

ACCOUNTING FOR SHEAR STRESSES DUE TO TORSION . 2154.11.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2154.11.2 Finite element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 2184.11.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

4.12 DEGENERATED 3D BEAM ELEMENTS . . . . . . . . . . . . . . . . . . . 2244.12.1 Description of geometry and displacement field . . . . . . . . . . 2264.12.2 Strain field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2274.12.3 Stiffness matrix and equivalent nodal force vector for the

element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2304.13 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

5 THIN PLATES. KIRCHHOFF THEORY . . . . . . . . . . . . . . . . . . . 2335.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2335.2 KIRCHHOFF PLATE THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

5.2.1 Main assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2345.2.2 Displacement field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2355.2.3 Strain and stress fields and constitutive equation . . . . . . . . 2375.2.4 Bending moments and generalized constitutive matrix . . . . 2395.2.5 Principle of virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2405.2.6 Equilibrium equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2415.2.7 The boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

5.3 FORMULATION OF THIN PLATE ELEMENTS . . . . . . . . . . . . . 2465.4 RECTANGULAR THIN PLATE ELEMENTS . . . . . . . . . . . . . . . . 247

5.4.1 Non-conforming 4-noded MZC rectangle . . . . . . . . . . . . . . . . 2475.4.2 12 DOFs plate rectangle proposed by Melosh . . . . . . . . . . . 2615.4.3 Conforming BFS plate rectangle . . . . . . . . . . . . . . . . . . . . . . . 262

5.5 TRIANGULAR THIN PLATE ELEMENTS . . . . . . . . . . . . . . . . . . 2645.5.1 Non-conforming thin plate triangles . . . . . . . . . . . . . . . . . . . . 2645.5.2 Conforming thin plate triangles . . . . . . . . . . . . . . . . . . . . . . . 268

5.6 CONFORMING THIN PLATE QUADRILATERALSOBTAINED FROM TRIANGLES . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

5.7 CONFORMING THIN PLATE ELEMENTS DERIVED FROMREISSNER-MINDLIN FORMULATION . . . . . . . . . . . . . . . . . . . . . 272

Contents XXI

5.8 ROTATION-FREE THIN PLATE TRIANGLES . . . . . . . . . . . . . . 2735.8.1 Formulation of rotation-free triangles by a combined

finite element and finite volume method . . . . . . . . . . . . . . . . 2745.8.2 Cell-centred patch. BPT rotation-free plate triangle . . . . . . 278

5.8.2.1 Boundary conditions for the BPT element . . . . . . 2805.8.3 Cell-vertex patch. BPN rotation-free plate triangle . . . . . . . 282

5.9 PATCH TESTS FOR KIRCHHOFF PLATE ELEMENTS . . . . . . 2855.10 COMPARISON OF KIRCHHOFF PLATE ELEMENTS . . . . . . . 2865.11 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

6 THICK/THIN PLATES. REISSNER-MINDLIN THEORY . . 2916.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2916.2 REISSNER-MINDLIN PLATE THEORY . . . . . . . . . . . . . . . . . . . . 292

6.2.1 Displacement field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2936.2.2 Strain and stress fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2936.2.3 Stress–strain relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2966.2.4 Resultant stresses and generalized constitutive matrix . . . . 2986.2.5 Virtual work principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

6.3 FINITE ELEMENT FORMULATION . . . . . . . . . . . . . . . . . . . . . . . 3016.3.1 Discretization of the displacement field . . . . . . . . . . . . . . . . . 3016.3.2 Discretization of the generalized strains and resultant

stress fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3026.3.3 Derivation of the equilibrium equations for the element . . . 3036.3.4 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3046.3.5 The boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

6.4 PERFORMANCE OF REISSNER–MINDLIN PLATEELEMENTS FOR THIN PLATE ANALYSIS . . . . . . . . . . . . . . . . . 3086.4.1 Locking. Reduced integration. Constraint index . . . . . . . . . 3086.4.2 Mechanisms induced by reduced integration . . . . . . . . . . . . 3096.4.3 Requirements for the ideal plate element . . . . . . . . . . . . . . . 311

6.5 REISSNER-MINDLIN PLATE QUADRILATERALS BASEDON SELECTIVE/REDUCED INTEGRATION . . . . . . . . . . . . . . . 3126.5.1 Four-noded plate quadrilateral (Q4) . . . . . . . . . . . . . . . . . . . 3126.5.2 Eight-noded Serendipity plate quadrilateral (QS8) . . . . . . . 3146.5.3 Nine-noded Lagrange plate quadrilateral (QL9) . . . . . . . . . 3156.5.4 Nine-noded hierarchical plate quadrilateral (QH9) . . . . . . . 3166.5.5 A generalization of the 9-noded hierarchical plate

quadrilateral (QHG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3176.5.6 Nine-noded heterosis plate quadrilateral (QHET) . . . . . . . . 3186.5.7 Higher order Reissner–Mindlin plate quadrilaterals with

12 and 16 nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3196.6 REISSNER-MINDLIN PLATE ELEMENTS BASED ON

ASSUMED TRANSVERSE SHEAR STRAIN FIELDS . . . . . . . . . 3196.6.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3196.6.2 Selection of the transverse shear strain field . . . . . . . . . . . . . 3226.6.3 Derivation of the substitute transverse shear strain matrix 323

6.7 REISSNER–MINDLIN PLATE QUADRILATERALS BASEDON ASSUMED TRANSVERSE SHEAR STRAIN FIELDS . . . . . 3276.7.1 4-noded plate quadrilateral with linear shear field (QLLL) 327

XXII Contents

6.7.2 8-noded Serendipity plate quadrilateral with assumedquadratic transverse shear strain field (QQQQ-S) . . . . . . . . 330

6.7.3 9-noded Lagrange plate quadrilateral with assumedquadratic transverse shear strain field (QQQQ-L) . . . . . . . . 332

6.7.4 Sixteen DOFs plate quadrilateral (QLQL) . . . . . . . . . . . . . . 3346.7.5 4-noded plate quadrilateral of Tessler-Hughes . . . . . . . . . . 3356.7.6 8-noded plate quadrilateral proposed by Crisfield . . . . . . . . 3386.7.7 Higher order 12 and 16-noded plate quadrilaterals with

assumed transverse shear strain fields . . . . . . . . . . . . . . . . . . 3396.8 REISSNER-MINDLIN PLATE TRIANGLES . . . . . . . . . . . . . . . . . 339

6.8.1 6-noded quadratic triangle with assumed linear shearstrains (TQQL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

6.8.2 Quadratic/Linear Reissner–Mindlin plate triangle (TLQL) 3416.8.3 Linear plate triangle with nine DOFs (TLLL) . . . . . . . . . . . 344

6.9 MORE PLATE TRIANGLES BASED ON ASSUMED SHEARSTRAIN FIELDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

6.10 REISSNER-MINDLIN PLATE ELEMENTS BASED ONLINKED INTERPOLATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

6.11 DISCRETE–KIRCHHOFF PLATE ELEMENTS . . . . . . . . . . . . . . 3496.11.1 3-noded DK plate triangle (DKT) . . . . . . . . . . . . . . . . . . . . . 3496.11.2 DK plate quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

6.12 DK ELEMENTS ACCOUNTING FOR SHEARDEFORMATION EFFECTS: DST ELEMENT . . . . . . . . . . . . . . . 355

6.13 PATCH TESTS FOR REISSNER-MINDLIN PLATE ELEMENTS3576.14 EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

6.14.1 Performance of some plate elements based on assumedtransverse shear strain fields . . . . . . . . . . . . . . . . . . . . . . . . . . 358

6.14.2 Simple supported plate under uniform load. Adaptivesolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

6.14.3 Effect of shear deformation in a plate simply supportedat three edges under a line load acting on the free edge . . . 366

6.15 EXTENDED ROTATION-FREE PLATE TRIANGLE WITHSHEAR DEFORMATION EFFECTS . . . . . . . . . . . . . . . . . . . . . . . . 3686.15.1 Iterative solution scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3726.15.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3736.15.3 Examples of performance of the BPT+1 element . . . . . . . . 374

6.16 LIMITATIONS OF THIN PLATE THEORY . . . . . . . . . . . . . . . . . 3796.17 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

7 COMPOSITE LAMINATED PLATES . . . . . . . . . . . . . . . . . . . . . . . 3827.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3827.2 BASIC THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

7.2.1 Displacement field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3837.2.2 Strain and generalized strain vectors . . . . . . . . . . . . . . . . . . . 3847.2.3 Stress-strain relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3857.2.4 Resultant stresses and generalized constitutive matrix . . . . 392

7.3 COMPUTATION OF TRANSVERSE SHEAR CORRECTIONPARAMETERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

7.4 PRINCIPLE OF VIRTUAL WORK . . . . . . . . . . . . . . . . . . . . . . . . . 399

Contents XXIII

7.5 COMPOSITE LAMINATED PLATE ELEMENTS . . . . . . . . . . . . 4007.5.1 Displacement interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4007.5.2 Generalized strain matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 4017.5.3 Stiffness matrix and equivalent nodal force vector . . . . . . . 4027.5.4 Simply supported sandwich square plate under uniform

loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4037.6 HIGHER ORDER COMPOSITE LAMINATED PLATE

ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4057.7 LAYER-WISE TLQL TRIANGULAR PLATE ELEMENT . . . . . 406

7.7.1 Displacement field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4067.7.2 Generalized strain matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 4077.7.3 Element stiffness matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4087.7.4 Simple supported multilayered square plates under

sinusoidal load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4097.8 COMPOSITE LAMINATED PLATE ELEMENTS BASED ON

THE REFINED ZIGZAG THEORY . . . . . . . . . . . . . . . . . . . . . . . . . 4107.8.1 Refined zigzag theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

7.8.1.1 Definition of the zigzag functions . . . . . . . . . . . . . . 4127.8.1.2 Strain and stress fields . . . . . . . . . . . . . . . . . . . . . . . 4147.8.1.3 Computation of the zigzag function . . . . . . . . . . . . 4157.8.1.4 Resultant constitutive equations . . . . . . . . . . . . . . . 4177.8.1.5 Principle of virtual work . . . . . . . . . . . . . . . . . . . . . . 4187.8.1.6 Element matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

7.8.2 QLRZ plate element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4207.8.2.1 Convergence study for the QLRZ plate element . . 4207.8.2.2 Simply supported of square and circular

multilayered plates . . . . . . . . . . . . . . . . . . . . . . . . . . . 4227.9 FAILURE THEORIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

7.9.1 Ultimate stress of a lamina under simple loading conditions4277.9.2 Failure stress of a lamina under combined loading conditions4287.9.3 The Tsai-Wu failure criterion . . . . . . . . . . . . . . . . . . . . . . . . . 4297.9.4 The reserve factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4317.9.5 Some conclusions on failure criteria of laminates . . . . . . . . . 432

7.10 MODELING OF DELAMINATION VIA ZIGZAG THEORY . . . 4337.11 EDGE DELAMINATION IN COMPOSITE LAMINATES . . . . . . 4347.12 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

8 ANALYSIS OF SHELLS WITH FLAT ELEMENTS . . . . . . . . . 4388.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4388.2 FLAT SHELL THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4398.3 REISSNER-MINDLIN FLAT SHELL THEORY . . . . . . . . . . . . . . . 441

8.3.1 Displacement field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4418.3.2 Strain field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4438.3.3 Stress field. Constitutive relationship . . . . . . . . . . . . . . . . . . . 4458.3.4 Resultant stresses and generalized constitutive matrix . . . . 4488.3.5 Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

8.4 REISSNER-MINDLIN FLAT SHELL ELEMENTS . . . . . . . . . . . . 4518.4.1 Discretization of the displacement field . . . . . . . . . . . . . . . . . 4518.4.2 Discretization of the generalized strain field . . . . . . . . . . . . . 452

XXIV Contents

8.4.3 Derivation of the element stiffness equations . . . . . . . . . . . . 4548.5 ASSEMBLY OF THE STIFFNESS EQUATIONS . . . . . . . . . . . . . 4568.6 NUMERICAL INTEGRATION OF THE STIFFNESS MATRIX

AND THE EQUIVALENT NODAL FORCE VECTOR . . . . . . . . 4598.7 BOUNDARY CONDITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4618.8 DEFINITION OF THE LOCAL AXES . . . . . . . . . . . . . . . . . . . . . . 463

8.8.1 Definition of local axes from an element side . . . . . . . . . . . . 4638.8.2 Definition of local axes by intersection with a coordinate

plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4658.8.3 Definition of a local axis parallel to a global one . . . . . . . . . 466

8.9 COPLANAR NODES. TECHNIQUES FOR AVOIDINGSINGULARITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4678.9.1 Selective assembly in local axes . . . . . . . . . . . . . . . . . . . . . . . 4688.9.2 Global assembly with six DOFs using an artificial

rotational stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4698.9.3 Drilling degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . 4708.9.4 Flat shell elements with mid-side normal rotations . . . . . . . 4738.9.5 Quasi-coplanar nodes in smooth shells . . . . . . . . . . . . . . . . . 473

8.10 CHOICE OF REISSNER-MINDLIN FLAT SHELL ELEMENTS 4758.11 SHEAR AND MEMBRANE LOCKING . . . . . . . . . . . . . . . . . . . . . . 4768.12 THIN FLAT SHELL ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

8.12.1 Kinematic, constitutive and equilibrium equations . . . . . . . 4798.12.2 Derivation of thin flat shell element matrices . . . . . . . . . . . . 4808.12.3 Selection of thin flat shell elements . . . . . . . . . . . . . . . . . . . . 4828.12.4 Incompatibility between membrane and bending fields . . . 483

8.13 BST AND BSN ROTATION-FREE THIN FLAT SHELLTRIANGLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4838.13.1 BST rotation-free shell triangle . . . . . . . . . . . . . . . . . . . . . . . . 4838.13.2 Enhanced BST rotation-free shell triangle . . . . . . . . . . . . . . 4918.13.3 Extension of the BST and EBST elements for kinked and

branching shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4938.13.4 Extended EBST elements with transverse shear deformation4948.13.5 Basic shell nodal patch element (BSN) . . . . . . . . . . . . . . . . 494

8.14 FLAT SHALLOW SHELL ELEMENTS . . . . . . . . . . . . . . . . . . . . . . 4988.15 FLAT MEMBRANE ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 5018.16 HIGHER ORDER COMPOSITE LAMINATED FLAT SHELL

ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5028.16.1 Layer-wise TLQL element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5038.16.2 Composite laminated flat shell elements based on the

refined zigzag theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5068.17 EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

8.17.1 Comparison of different flat shell elements . . . . . . . . . . . . . . 5118.17.2 Adaptive mesh refinement of analysis cylindrical shells . . . 5118.17.3 Examples of application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

8.18 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

Contents XXV

9 AXISYMMETRIC SHELLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5259.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5259.2 GEOMETRICAL DESCRIPTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 5279.3 AXISYMMETRIC SHELL THEORY BASED ON REISSNER–

MINDLIN ASSUMPTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5309.3.1 Displacement field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5309.3.2 Strain vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5329.3.3 Stresses and resultant stresses . . . . . . . . . . . . . . . . . . . . . . . . . 535

9.3.3.1 Constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . 5369.3.3.2 Shear correction factor . . . . . . . . . . . . . . . . . . . . . . . 5399.3.3.3 Layered composite material . . . . . . . . . . . . . . . . . . . 5399.3.3.4 Initial stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540

9.3.4 Principle of virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5419.4 TRONCOCONICAL REISSNER-MINDLIN ELEMENTS . . . . . . 543

9.4.1 Displacement and strain interpolation . . . . . . . . . . . . . . . . . . 5449.4.2 Local stiffness matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5469.4.3 Transformation to global axes . . . . . . . . . . . . . . . . . . . . . . . . . 547

9.5 SHEAR AND MEMBRANE LOCKING . . . . . . . . . . . . . . . . . . . . . . 5499.5.1 Transverse shear locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5499.5.2 Membrane locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5509.5.3 Other techniques to avoid locking in Reissner-Mindlin

troncoconical shell elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 5519.6 INTEGRATION RULES FOR THE LINEAR AND

QUADRATIC REISSNER-MINDLIN TRONCOCONICALELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5539.6.1 Quadrature for the 2-noded Reissner-Mindlin

troncoconical element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5539.6.2 Quadrature for the 3-noded Reissner-Mindlin

troncoconical element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5559.7 APPLICATIONS OF THE TWO-NODED REISSNER-

MINDLIN TRONCOCONICAL ELEMENT . . . . . . . . . . . . . . . . . . 5569.7.1 Clamped spherical dome under uniform pressure . . . . . . . . 5569.7.2 Toroidal shell under internal pressure . . . . . . . . . . . . . . . . . . 5569.7.3 Cylindrical tank with spherical dome under internal

pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5569.7.4 Elevated water tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558

9.8 CURVED AXISYMMETRIC SHELL ELEMENTS OF THEREISSNER-MINDLIN FAMILY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5619.8.1 Displacement and load generalized strain fields . . . . . . . . . . 5619.8.2 Computation of curvilinear derivatives and curvature radius562

9.9 AXISYMMETRIC THIN SHELL ELEMENTS BASED ONKIRCHHOFF ASSUMPTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5649.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5649.9.2 Basic formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5659.9.3 Troncoconical shell elements based on Kirchhoff theory . . . 566

9.9.3.1 Two-noded Kirchhoff troncoconical element . . . . . 5669.9.3.2 Curved Kirchhoff axisymmetric shell elements . . . 568

9.10 AXISYMMETRIC MEMBRANE ELEMENTS . . . . . . . . . . . . . . . . 5699.11 ROTATION-FREE AXISYMMETRIC SHELL ELEMENTS . . . . 571

XXVI Contents

9.11.1 Cell-centred rotation-free troncoconical element (ACC) . . . 5729.11.1.1 ACC element matrices . . . . . . . . . . . . . . . . . . . . . . . . 5729.11.1.2 Boundary conditions for the ACC element . . . . . . 574

9.11.2 Cell-vertex rotation-free troncoconical element . . . . . . . . . . 5779.11.2.1 ACV element matrices . . . . . . . . . . . . . . . . . . . . . . . . 5779.11.2.2 Boundary conditions for the ACV element . . . . . . 579

9.11.3 Example. Analysis of a dome under internal pressure . . . . . 5809.12 AXISYMMETRIC PLATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5809.13 PLANE ARCHES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5859.14 SHALLOW AXISYMMETRIC SHELLS AND ARCHES . . . . . . . 5899.15 MORE ABOUT MEMBRANE LOCKING . . . . . . . . . . . . . . . . . . . 5919.16 AXISYMMETRIC SHELL ELEMENTS OBTAINED FROM

DEGENERATED AXISYMMETRIC SOLID ELEMENTS . . . . . 5939.17 HIGHER ORDER COMPOSITE LAMINATED

AXISYMMETRIC SHELL ELEMENTS. . . . . . . . . . . . . . . . . . . . . . 5989.17.1 Layer-wise theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5989.17.2 Zigzag theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

9.17.2.1 Displacement, strain and stress fields . . . . . . . . . . . 5999.17.2.2 Computation of the zigzag function . . . . . . . . . . . . 6019.17.2.3 Two-noded zigzag axisymmetric shell element . . . 602

9.18 FINAL REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

10 CURVED 3D SHELL ELEMENTS AND SHELL STIFFNERS60510.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60510.2 DEGENERATED SHELL ELEMENTS. BASIC CONCEPTS . . . 60710.3 COORDINATE SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

10.3.1 Global cartesian coordinate system x,y, z . . . . . . . . . . . . . . 60910.3.2 Nodal coordinate system v1i , v2i , v3i . . . . . . . . . . . . . . . . . . 60910.3.3 Curvilinear parametric coordinate system ξ, η, ζ . . . . . . . . . 61110.3.4 Lamina (local) coordinate system x′,y′, z′ . . . . . . . . . . . . . . 61410.3.5 Covariant and contravariant coordinate systems . . . . . . . . . 615

10.4 GEOMETRIC DESCRIPTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61610.5 DISPLACEMENT FIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61710.6 LOCAL STRAIN MATRIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61910.7 STRESS VECTOR AND CONSTITUTIVE EQUATION . . . . . . . 62310.8 STIFFNESS MATRIX AND EQUIVALENT NODAL FORCE

VECTOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62610.9 COMPUTATION OF STRESS RESULTANTS . . . . . . . . . . . . . . . . 62910.10 FOLDED CURVED SHELLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63010.11 TECHNIQUES FOR ENHANCING THE PERFORMANCE

OF DEGENERATED SHELL ELEMENTS . . . . . . . . . . . . . . . . . . . 63110.11.1 Selective/reduced integration techniques . . . . . . . . . . . . . . . 63210.11.2 Assumed fields for the transverse shear and membrane

strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63410.11.3 4-noded degenerated shell quadrilateral (Q4) with

assumed linear transverse shear strain field . . . . . . . . . . . . . 63510.12 EXPLICIT THICKNESS INTEGRATION OF THE

STIFFNESS MATRIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63610.13 CONTINUUM-BASED RESULTANT (CBR) SHELL THEORY 637

Contents XXVII

10.13.1 Geometric and kinematic description . . . . . . . . . . . . . . . . . . 63910.13.2 Computation of the CBR shell Jacobian . . . . . . . . . . . . . . . 63910.13.3 CBR Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640

10.13.3.1 Displacement derivatives . . . . . . . . . . . . . . . . . . . . . 64010.13.3.2 Generalized strains . . . . . . . . . . . . . . . . . . . . . . . . . . 641

10.13.4 PVW, stress resultants and generalized constitutive matrix64210.13.5 Enhanced transverse shear deformation matrix . . . . . . . . . 64410.13.6 CBR shell elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644

10.13.6.1 CBR element stiffness matrix and equivalentnodal force vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 646

10.14 CBR-S SHELL ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64710.15 QL9 CBR-S SHELL ELEMENT WITH ASSUMED

MEMBRANE AND TRANSVERSE SHEAR STRAINS . . . . . . . . 64910.15.1 Assumed quadratic membrane strain field . . . . . . . . . . . . . . 65010.15.2 Assumed quadratic transverse shear strain field . . . . . . . . . 652

10.16 DK CURVED SHELL ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . 65310.16.1 Semi-Loof curved shell element . . . . . . . . . . . . . . . . . . . . . . . 653

10.17 PERFORMANCE OF DEGENERATED SHELL ELEMENTSWITH ASSUMED STRAIN FIELDS . . . . . . . . . . . . . . . . . . . . . . . . 654

10.18 DEGENERATED FLAT SHELL AND PLATE ELEMENTS . . . 65710.19 SHELL ELEMENTS BASED ON SIX AND SEVEN

PARAMETER MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65710.20 ISOGEOMETRIC SHELL ANALYSIS . . . . . . . . . . . . . . . . . . . . . . 658

10.20.1 Isogeometric analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65810.20.2 NURBS as a basis for isogeometric FE shell analysis . . . . 66210.20.3 Example. Pinched cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . 666

10.21 SHELL STIFFENERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66710.22 SLAB-BEAM BRIDGES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67010.23 FINAL REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674

11 PRISMATIC STRUCTURES. FINITE STRIP ANDFINITE PRISM METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67511.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67511.2 ANALYSIS OF A SIMPLY SUPPORTED BEAM BY

FOURIER SERIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67611.3 BASIC CONCEPTS OF FINITE STRIP AND FINITE PRISM

METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68011.4 FINITE STRIP METHOD FOR RECTANGULAR

REISSNER-MINDLIN PLATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68411.4.1 Reduced integration for Reissner-Mindlin plate strips . . . . 688

11.5 FINITE STRIP METHOD FOR STRAIGHT PRISMATICFOLDED PLATE STRUCTURES . . . . . . . . . . . . . . . . . . . . . . . . . . . 69011.5.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69011.5.2 Assembly of strip equations. Transformation to global axes 69411.5.3 Equivalent nodal force vector . . . . . . . . . . . . . . . . . . . . . . . . . 69511.5.4 Flat shell strips. Two-noded strip with reduced integration 69611.5.5 Simplification for composite laminated plate strip element 697

11.6 ANALYSIS OF CURVED PRISMATIC SHELLS BY THEFINITE STRIP METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697

XXVIII Contents

11.6.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69711.6.2 Circular plate strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702

11.7 AXISYMMETRIC SHELLS UNDER ARBITRARY LOADING . 70311.8 TRONCOCONICAL STRIP ELEMENTS BASED ON

KIRCHHOFF SHELL THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70611.9 THE FINITE PRISM METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70611.10 AXISYMMETRIC SOLIDS UNDER ARBITRARY LOADING . 71311.11 INTERMEDIATE SUPPORTS WITH RIGID DIAPHRAGMS . 71411.12 EXTENSION OF FINITE STRIP AND FINITE PRISM

METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71611.13 EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

11.13.1 Simply supported square plate under uniformlydistributed loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

11.13.2 Curved simply supported plate . . . . . . . . . . . . . . . . . . . . . . . 71811.13.3 Circular plate under eccentric point load . . . . . . . . . . . . . . . 72011.13.4 Pinched cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72111.13.5 Simply supported curved box girder-bridge . . . . . . . . . . . . . 72111.13.6 Two-span cellular bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72311.13.7 Simple supported slab-beam bridge over a highway . . . . . . 72311.13.8 Circular bridge analyzed with finite strip and finite

prism methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72311.13.9 Simply supported thick box girder-bridge analyzed with

the finite prism method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72611.14 FINAL REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727

12 PROGRAMMING THE FEM FOR BEAM, PLATE ANDSHELL ANALYSIS IN MAT-fem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72912.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72912.2 MAT-fem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73012.3 DATA FILES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732

12.3.1 Material data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73212.3.2 Mesh topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73312.3.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73412.3.4 Point and surface loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735

12.4 START . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73612.5 STIFFNESS MATRIX AND EQUIVALENT NODAL FORCE

VECTOR FOR SELF-WEIGHT AND DISTRIBUTED LOAD . . 73712.5.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73712.5.2 Point loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740

12.6 PRESCRIBED DISPLACEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . 74012.7 SOLUTION OF THE EQUATIONS SYSTEM . . . . . . . . . . . . . . . . 74112.8 NODAL REACTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74112.9 RESULTANT STRESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741

12.9.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74112.9.2 Computation of the stresses at the nodes . . . . . . . . . . . . . . . 742

12.10 POSTPROCESSING STEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74412.11 GRAPHICAL USER INTERFACE . . . . . . . . . . . . . . . . . . . . . . . . . 745

12.11.1 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74512.11.2 Program execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748

Contents XXIX

12.11.3 Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74912.12 2-NODED EULER-BERNOUILLI BEAM ELEMENT . . . . . . . . . 750

12.12.1 Stiffness matrix and equivalent nodal force vector . . . . . . . 75012.12.2 Computation of bending moment . . . . . . . . . . . . . . . . . . . . . 75012.12.3 Example. Clamped slender cantilever beam under end

point load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75112.13 2-NODED TIMOSHENKO BEAM ELEMENT . . . . . . . . . . . . . . . 753

12.13.1 Stiffness matrix and equivalent nodal force vector . . . . . . . 75412.13.2 Computation of bending moment and shear force . . . . . . . 75512.13.3 Example. Thick cantilever beam under end-point load . . . 755

12.14 4-NODED MZC THIN PLATE RECTANGLE . . . . . . . . . . . . . . . 75712.14.1 Element stiffness matrix and equivalent nodal force vector 75712.14.2 Computation of bending moments . . . . . . . . . . . . . . . . . . . . 75812.14.3 Example. Clamped thin square plate under uniform

loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75812.15 Q4 REISSNER-MINDLIN PLATE RECTANGLE . . . . . . . . . . . . 762

12.15.1 Stiffness matrix and equivalent nodal force vector . . . . . . . 76312.15.2 Computation of resultant stresses . . . . . . . . . . . . . . . . . . . . . 76412.15.3 Example. Thick clamped square plate under uniformly load765

12.16 QLLL REISSNER-MINDLIN PLATE QUADRILATERAL. . . . . 76612.16.1 Stiffness matrix and equivalent nodal force vector . . . . . . . 76712.16.2 Computation of resultant stresses . . . . . . . . . . . . . . . . . . . . . 76712.16.3 Example. Thick clamped plate under uniform distributed

load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76812.17 TQQL REISSNER-MINDLIN PLATE TRIANGLE . . . . . . . . . . . 769

12.17.1 Stiffness matrix and equivalent nodal force vector . . . . . . . 76912.17.2 Computation of stress resultants . . . . . . . . . . . . . . . . . . . . . . 77112.17.3 Example. Thick clamped square plate under uniform load 772

12.18 4-NODED QLLL FLAT SHELL ELEMENT . . . . . . . . . . . . . . . . . 77212.18.1 Generalized constitutive matrix . . . . . . . . . . . . . . . . . . . . . . . 77512.18.2 Stiffness matrix and equivalent nodal force vector . . . . . . . 77512.18.3 Computation of local resultant stresses . . . . . . . . . . . . . . . . 77612.18.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777

12.18.4.1 Clamped hyperbolic shell under uniform loading 77712.18.5 Scordelis roof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781

12.19 2-NODED REISSNER-MINDLIN TRONCOCONICAL SHELLELEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78312.19.1 Generalized constitutive matrix . . . . . . . . . . . . . . . . . . . . . . . 78412.19.2 Stiffness matrix and equivalent nodal force vector . . . . . . . 78512.19.3 Resultant stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78612.19.4 Example. Thin spherical dome under uniform external

pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78712.20 FINAL REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790

A BASIC PROPERTIES OF MATERIALS . . . . . . . . . . . . . . . . . . . . 791

B EQUILIBRIUM EQUATIONS FOR A SOLID . . . . . . . . . . . . . . . 793B.1 EQUILIBRIUM AT A BOUNDARY SEGMENT . . . . . . . . . . . . . . 794

XXX Contents

C NUMERICAL INTEGRATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795C.1 1D NUMERICAL INTEGRATION . . . . . . . . . . . . . . . . . . . . . . . . . . 795C.2 NUMERICAL INTEGRATION IN 2D . . . . . . . . . . . . . . . . . . . . . . . 796

C.2.1 Numerical integration in quadrilateral domains . . . . . . . . . . 796C.2.2 Numerical integration over triangles . . . . . . . . . . . . . . . . . . . 797

C.3 NUMERICAL INTEGRATION OVER HEXAEDRA . . . . . . . . . . 798

D COMPUTATION OF THE SHEAR CORRECTIONPARAMETER FOR BEAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800

E PROOF OF THE SINGULARITY RULE FOR THESTIFFNESS MATRIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803

F COMPUTATION OF THE SHEAR CENTER AND THEWARPING FUNCTION IN THIN-WALLED OPENCOMPOSITE BEAM SECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 805

G STABILITY CONDITIONS FOR REISSNER-MINDLIN PLATE ELEMENTS BASED ON ASSUMEDTRANSVERSE SHEAR STRAINS . . . . . . . . . . . . . . . . . . . . . . . . . . 808

H ANALYTICAL SOLUTIONS FOR ISOTROPICTHICK/THIN CIRCULAR PLATES . . . . . . . . . . . . . . . . . . . . . . . . 813H.1 GOVERNING EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813H.2 CLAMPED PLATE UNDER UNIFORM LOADING q . . . . . . . . . 813H.3 SIMPLY SUPPORTED PLATE UNDER UNIFORM LOAD q . . 814H.4 CLAMPED PLATE UNDER POINT LOAD AT THE CENTER 815H.5 SIMPLY SUPPORTED PLATE UNDER CENTRAL POINT

LOAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815H.6 INFLUENCE OF THICKNESS IN THE SOLUTION FOR

UNIFORM LOADED PLATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816

I SHAPE FUNCTIONS FOR SOME C0 CONTINUOUSTRIANGULAR AND QUADRILATERAL ELEMENTS . . . . 817I.1 TRIANGULAR ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817

I.1.1 3-noded triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817I.1.2 6-noded triangle (straight sides) . . . . . . . . . . . . . . . . . . . . . . . 817

I.2 QUADRILATERAL ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 818I.2.1 4-noded rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818I.2.2 8-noded Serendipity rectangle . . . . . . . . . . . . . . . . . . . . . . . . . 818I.2.3 9-noded Lagrangian rectangle . . . . . . . . . . . . . . . . . . . . . . . . . 818

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819

Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851

Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859