Plate and Shell Structures: Selected Analytical and Finite Element Solutions

Plate and Shell Structures: Selected Analytical and Finite Element Solutions Maria Radwañska, Anna Stankiewicz, Adam Wosatko, Jerzy Pamin Cracow University of Technology, Poland   Comprehensively covers the fundamental theory and analytical and numerical solutions for different types of plate and shell structures   Plate and Shell Structures: Selected Analytical and Finite Element Solutions not only provides the theoretical formulation of fundamental problems of mechanics of plates and shells, but also several examples of analytical and numerical solutions for different types of shell structures. The book contains advanced aspects related to stability analysis and a brief description of modern finite element formulations for plates and shells, including the discussion of mixed/hybrid models and locking phenomena.   Key features: 52 example problems solved and illustrated by more than 200 figures, including 30 plots of finite element simulation results. Contents based on many years of research and teaching the mechanics of plates and shells to students of civil engineering and professional engineers. Provides the basis of an intermediate-level course on computational mechanics of shell structures.   The book is essential reading for engineering students, university teachers, practitioners and researchers interested in the mechanics of plates and shells, as well as developers testing new simulation software.

Table of Contents:
Preface xvii Notation xix Part 1 Fundamentals: Theory and Modelling 1 1 General Information 3 1.1 Introduction 3 1.2 Review of Theories Describing Elastic Plates and Shells 6 1.3 Description of Geometry for 2D Formulation 9 1.4 Definitions and Assumptions for 2D Formulation 16 1.5 Classification of Shell Structures 21 References 24 2 Equations for Theory of Elasticity for 3D Problems 26 Reference 30 3 Equations of Thin Shells According to the Three-Parameter Kirchhoff–Love Theory 31 3.1 General Equations for Thin Shells 31 3.2 Specification of Lame Parameters and Principal Curvature Radii for Typical Surfaces 38 3.3 Transition from General Shell Equations to Particular Cases of Plates and Shells 42 3.4 Displacement Equations for Multi-Parameter Plate and Shell Theories 45 3.5 Remarks 47 References 47 4 General Information about Models and Computational Aspects 48 4.1 Analytical Approach to Statics, Buckling and Free Vibrations 49 4.2 Approximate Approach According to the Finite Difference Method 51 4.3 Computational Analysis by Finite Element Method 54 4.4 Computational Models – Summary 55 Reference 55 5 Description of Finite Elements for Analysis of Plates and Shells 56 5.1 General Information on Finite Elements 56 5.2 Description of Selected FEs 58 5.3 Remarks on Displacement-based FE Formulation 69 References 70 Part 2 Plates 73 6 Flat Rectangular Membranes 75 6.1 Introduction 75 6.2 Governing Equations 76 6.3 Square Membrane under Unidirectional Tension 81 6.4 Square Membrane under Uniform Shear 83 6.5 Pure In-Plane Bending of a Square Membrane 85 6.6 Cantilever Beam with a Load on the Free Side 88 6.7 Rectangular Deep Beams 94 6.8 Membrane with Variable Thicknesses or Material Parameters 97 References 101 7 Circular and Annular Membranes 102 7.1 Equations of Membranes – Local and Global Formulation 102 7.2 Equations for the Axisymmetric Membrane State 104 7.3 Annular Membrane 105 References 109 8 Rectangular Plates under Bending 110 8.1 Introduction 110 8.2 Equations for the Classical Kirchhoff–Love Thin Plate Theory 110 8.3 Derivation of Displacement Equation for a Thin Plate from the Principle of Minimum Potential Energy 117 8.4 Equation for a Plate under Bending Resting on a Winkler Elastic Foundation 118 8.5 Equations of Mindlin–Reissner Moderately Thick Plate Theory 119 8.6 Analytical Solution of a Sinusoidally Loaded Rectangular Plate 122 8.7 Analysis of Plates under Bending Using Expansions in Double or Single Trigonometric Series 127 8.8 Simply Supported or Clamped Square Plate with Uniform Load 131 8.9 Rectangular Plate with a Uniform Load and Various Boundary Conditions – Comparison of STSM and FEM Results 135 8.10 Uniformly Loaded Rectangular Plate with Clamped and Free Boundary Lines – Comparison of STSM and FEM Results 139 8.11 Approximate Solution to a Plate Bending Problem using FDM 143 8.12 Approximate Solution to a Bending Plate Problem using the Ritz Method 151 8.13 Plate with Variable Thickness 153 8.14 Analysis of Thin and Moderately Thick Plates in Bending 155 References 159 9 Circular and Annular Plates under Bending 160 9.1 General State 160 9.2 Axisymmetric State 162 9.3 Analytical Solution using a Trigonometric Series Expansion 164 9.4 Clamped Circular Plate with a Uniformly Distributed Load 166 9.5 Simply Supported Circular Plate with a Concentrated Central Force 169 9.6 Simply Supported Circular Plate with an Asymmetric Distributed Load 171 9.7 Uniformly Loaded Annular Plate with Static and Kinematic Boundary Conditions 174 References 177 Part 3 Shells 179 10 Shells in the Membrane State 181 10.1 Introduction 181 10.2 General Membrane State in Shells of Revolution 182 10.3 Axisymmetric Membrane State 183 10.4 Hemispherical Shell 186 10.5 Open Conical Shell under Self Weight 193 10.6 Cylindrical Shell 195 10.7 Hemispherical Shell with an Asymmetric Wind Action 199 References 204 11 Shells in the Membrane-Bending State 205 11.1 Cylindrical Shells 205 11.2 Spherical Shells 221 11.3 Cylindrical and Spherical Shells Loaded by a Uniformly Distributed Boundary Moment and Horizontal Force 229 11.4 Cylindrical Shell with a Spherical Cap – Analytical and Numerical Solution 232 11.5 General Case of Deformation of Cylindrical Shells 237 11.6 Cylindrical Shell with a Semicircular Cross Section under Self Weight – Analytical Solution of Membrane State 238 11.7 Cylindrical Scordelis-Lo Roof in the Membrane-Bending State – Analytical and Numerical Solution 242 11.8 Single-Span Clamped Horizontal Cylindrical Shell under Self Weight 246 References 254 12 Shallow Shells 256 12.1 Equations for Shallow Shells 256 12.2 Pucher’s Equations for Shallow Shells in the Membrane State 260 12.3 Hyperbolic Paraboloid with Rectangular Projection 262 12.4 Remarks on Engineering Applications 266 References 267 13 Thermal Loading of Selected Membranes, Plates and Shells 268 13.1 Introduction 268 13.2 Uniform Temperature Change along the Thickness 270 13.3 Linear Temperature Change along the Thickness – Analytical Solutions 275 References 286 Part 4 Stability and Free Vibrations 287 14 Stability of Plates and Shells 289 14.1 Overview of Plate and Shell Stability Problems 289 14.2 Basis of Linear Buckling Theory, Assumptions and Computational Models 291 14.3 Description of Physical Phenomena and Nonlinear Simulations in Stability Analysis 298 14.4 Analytical and Numerical Buckling Analysis for Selected Plates and Shells 301 14.5 Snap-Through and Snap-Back Phenomena Observed for Elastic Shallow Cylindrical Shells in Geometrically Nonlinear Analysis 319 References 321 15 Free Vibrations of Plates and Shells 323 15.1 Introduction 323 15.2 Natural Transverse Vibrations of a Thin Rectangular Plate 325 15.3 Parametric Analysis of Free Vibrations of Rectangular Plates 328 15.4 Natural Vibrations of Cylindrical Shells 333 15.5 Remarks 337 References 338 Part 5 Aspects of FE Analysis 339 16 Modelling Process 341 16.1 Advantages of Numerical Simulations 341 16.2 Complexity of Shell Structures Affecting FEM 342 16.3 Particular Requirements for FEs in Plate and Shell Discretization 343 References 346 17 Quality of FEs and Accuracy of Solutions in Linear Analysis 347 17.1 Order of Approximation Function versus Order of Numerical Integration Quadrature 347 17.2 Assessment of Element Quality via Spectral Analysis 347 17.3 Numerical Effects of Shear Locking and Membrane Locking 350 17.4 Examination of Element Quality – One-Element and Patch Tests 354 17.5 Benchmarks for Membranes and Plates 357 17.6 Benchmarks for Shells 359 17.7 Comparison of Analytical and Numerical Solutions, Application of Various FE Formulations 361 References 362 18 Advanced FE Formulations 365 18.1 Introduction 365 18.2 Link between Variational Formulations and FE Models 366 18.3 Advanced FEs 373 References 383 A List of Boxes with Equations 387 B List of Boxes with Data and Results for Examples 389 Index 391

About the Author :
Maria Radwañska, recently professor emeritus, worked for over 40 years at the Faculty of Civil Engineering of Cracow University of Technology, first at the Institute of Structural Mechanics, then at the Institute for Computational Civil Engineering. She has done research on statics and stability of elastic and elastic-plastic bars, plates and shells. She was a teacher of structural mechanics, theory of plates and shells, theory of stability and computational methods (including finite element method) at the level of doctoral, graduate and undergraduate studies, as well as at training courses for professional engineers. She is the author or co-author of seven monographs, books and textbooks, and of numerous journal papers. She was a member of the research team of Prof. Zenon Waszczyszyn, who implemented the FE Code ANKA for buckling and nonlinear analysis of structures and wrote in 1994 the Elsevier book on FEM for stability of structures. She received numerous awards from the Minister of Science and Higher Education of Poland. Anna Stankiewicz, lecturer at the Institute for Computational Civil Engineering, Cracow University of Technology. She has taught computational methods, engineering graphics, computer science, mechanics of materials and structures, and since 2008 succeeded M. Radwañska as lecturer of the theory of plates and shells (both in the courses in Polish and English). She was the coordinator and lecturer in training courses on Shell Structures – modelling and FEM analysis and on Computational Methods in Civil Engineering, organized for professional engineers in years 2010-15 within EU Human Capital Operational Programme. Currently she is also involved in the research on thermoplasticity. Adam Wosatko, assistant professor at the Institute for Computational Civil Engineering, Cracow University of Technology. He has taught applied mathematics and numerical methods, computational methods, engineering graphics, computer science, mechanics of materials and structures, and the theory of plates and shells (both in the courses in Polish and English). He was the lecturer in training courses on Shell Structures – modelling and FEM analysis and on Computational Methods in Civil Engineering, organized for professional engineers in years 2010-14 within EU Human Capital Operational Programme. In 2014-15 post-doc at the Department of Civil and Environmental Engineering of University of Waterloo, Canada. Currently also involved in the research on the mechanics of structures in fire. Jerzy Pamin, professor of Cracow University of Technology (CUT), since 2006 head of the Institute for Computational Civil Engineering. He was Ph.D. student at Faculty of Civil Engineering, Delft University of Technology, the Netherlands, where he obtained a Ph.D. in 1994 on the basis of broadly cited dissertation on Gradient-Dependent Plasticity in Numerical Simulation of Localization Phenomena, written under the supervision of R. de Borst. Since then he has been employed at CUT, in 1998-99 post-doc at Koiter Institute Delft, in 2002 Humboldt Fellow at the University of Kaiserslautern, Germany. In 2004 he obtained a D.Sc. degree at CUT for the monograph on Gradient-enhanced continuum models: formulation, discretization and applications, for which he won the M.T. Huber award of Division for Technical Sciences of the Polish Academy of Sciences in 2007. His field of expertise is mechanics of generalized continua and FE simulations in mechanics of materials and structures. He has taught mechanics of materials and structures, computational methods and advanced FEM (both in the courses in Polish and English), and coordinated of an undergraduate civil engineering course in English.