Plate and Shell Structures

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.

• Preface xviiNotation xixPart 1 Fundamentals: Theory and Modelling 11 General Information 31.1 Introduction 31.2 Review of Theories Describing Elastic Plates and Shells 61.3 Description of Geometry for 2D Formulation 91.4 Definitions and Assumptions for 2D Formulation 161.5 Classification of Shell Structures 21References 242 Equations for Theory of Elasticity for 3D Problems 26Reference 303 Equations of Thin Shells According to the Three-Parameter Kirchhoff–Love Theory 313.1 General Equations for Thin Shells 313.2 Specification of Lame Parameters and Principal Curvature Radii for Typical Surfaces 383.3 Transition from General Shell Equations to Particular Cases of Plates and Shells 423.4 Displacement Equations for Multi-Parameter Plate and Shell Theories 453.5 Remarks 47References 474 General Information about Models and Computational Aspects 484.1 Analytical Approach to Statics, Buckling and Free Vibrations 494.2 Approximate Approach According to the Finite Difference Method 514.3 Computational Analysis by Finite Element Method 544.4 Computational Models – Summary 55Reference 555 Description of Finite Elements for Analysis of Plates and Shells 565.1 General Information on Finite Elements 565.2 Description of Selected FEs 585.3 Remarks on Displacement-based FE Formulation 69References 70Part 2 Plates 736 Flat Rectangular Membranes 756.1 Introduction 756.2 Governing Equations 766.3 Square Membrane under Unidirectional Tension 816.4 Square Membrane under Uniform Shear 836.5 Pure In-Plane Bending of a Square Membrane 856.6 Cantilever Beam with a Load on the Free Side 886.7 Rectangular Deep Beams 946.8 Membrane with Variable Thicknesses or Material Parameters 97References 1017 Circular and Annular Membranes 1027.1 Equations of Membranes – Local and Global Formulation 1027.2 Equations for the Axisymmetric Membrane State 1047.3 Annular Membrane 105References 1098 Rectangular Plates under Bending 1108.1 Introduction 1108.2 Equations for the Classical Kirchhoff–Love Thin Plate Theory 1108.3 Derivation of Displacement Equation for a Thin Plate from the Principle of Minimum Potential Energy 1178.4 Equation for a Plate under Bending Resting on a Winkler Elastic Foundation 1188.5 Equations of Mindlin–Reissner Moderately Thick Plate Theory 1198.6 Analytical Solution of a Sinusoidally Loaded Rectangular Plate 1228.7 Analysis of Plates under Bending Using Expansions in Double or Single Trigonometric Series 1278.8 Simply Supported or Clamped Square Plate with Uniform Load 1318.9 Rectangular Plate with a Uniform Load and Various Boundary Conditions – Comparison of STSM and FEM Results 1358.10 Uniformly Loaded Rectangular Plate with Clamped and Free Boundary Lines – Comparison of STSM and FEM Results 1398.11 Approximate Solution to a Plate Bending Problem using FDM 1438.12 Approximate Solution to a Bending Plate Problem using the Ritz Method 1518.13 Plate with Variable Thickness 1538.14 Analysis of Thin and Moderately Thick Plates in Bending 155References 1599 Circular and Annular Plates under Bending 1609.1 General State 1609.2 Axisymmetric State 1629.3 Analytical Solution using a Trigonometric Series Expansion 1649.4 Clamped Circular Plate with a Uniformly Distributed Load 1669.5 Simply Supported Circular Plate with a Concentrated Central Force 1699.6 Simply Supported Circular Plate with an Asymmetric Distributed Load 1719.7 Uniformly Loaded Annular Plate with Static and Kinematic Boundary Conditions 174References 177Part 3 Shells 17910 Shells in the Membrane State 18110.1 Introduction 18110.2 General Membrane State in Shells of Revolution 18210.3 Axisymmetric Membrane State 18310.4 Hemispherical Shell 18610.5 Open Conical Shell under Self Weight 19310.6 Cylindrical Shell 19510.7 Hemispherical Shell with an Asymmetric Wind Action 199References 20411 Shells in the Membrane-Bending State 20511.1 Cylindrical Shells 20511.2 Spherical Shells 22111.3 Cylindrical and Spherical Shells Loaded by a Uniformly Distributed Boundary Moment and Horizontal Force 22911.4 Cylindrical Shell with a Spherical Cap – Analytical and Numerical Solution 23211.5 General Case of Deformation of Cylindrical Shells 23711.6 Cylindrical Shell with a Semicircular Cross Section under Self Weight – Analytical Solution of Membrane State 23811.7 Cylindrical Scordelis-Lo Roof in the Membrane-Bending State – Analytical and Numerical Solution 24211.8 Single-Span Clamped Horizontal Cylindrical Shell under Self Weight 246References 25412 Shallow Shells 25612.1 Equations for Shallow Shells 25612.2 Pucher’s Equations for Shallow Shells in the Membrane State 26012.3 Hyperbolic Paraboloid with Rectangular Projection 26212.4 Remarks on Engineering Applications 266References 26713 Thermal Loading of Selected Membranes, Plates and Shells 26813.1 Introduction 26813.2 Uniform Temperature Change along the Thickness 27013.3 Linear Temperature Change along the Thickness – Analytical Solutions 275References 286Part 4 Stability and Free Vibrations 28714 Stability of Plates and Shells 28914.1 Overview of Plate and Shell Stability Problems 28914.2 Basis of Linear Buckling Theory, Assumptions and Computational Models 29114.3 Description of Physical Phenomena and Nonlinear Simulations in Stability Analysis 29814.4 Analytical and Numerical Buckling Analysis for Selected Plates and Shells 30114.5 Snap-Through and Snap-Back Phenomena Observed for Elastic Shallow Cylindrical Shells in Geometrically Nonlinear Analysis 319References 32115 Free Vibrations of Plates and Shells 32315.1 Introduction 32315.2 Natural Transverse Vibrations of a Thin Rectangular Plate 32515.3 Parametric Analysis of Free Vibrations of Rectangular Plates 32815.4 Natural Vibrations of Cylindrical Shells 33315.5 Remarks 337References 338Part 5 Aspects of FE Analysis 33916 Modelling Process 34116.1 Advantages of Numerical Simulations 34116.2 Complexity of Shell Structures Affecting FEM 34216.3 Particular Requirements for FEs in Plate and Shell Discretization 343References 34617 Quality of FEs and Accuracy of Solutions in Linear Analysis 34717.1 Order of Approximation Function versus Order of Numerical Integration Quadrature 34717.2 Assessment of Element Quality via Spectral Analysis 34717.3 Numerical Effects of Shear Locking and Membrane Locking 35017.4 Examination of Element Quality – One-Element and Patch Tests 35417.5 Benchmarks for Membranes and Plates 35717.6 Benchmarks for Shells 35917.7 Comparison of Analytical and Numerical Solutions, Application of Various FE Formulations 361References 36218 Advanced FE Formulations 36518.1 Introduction 36518.2 Link between Variational Formulations and FE Models 36618.3 Advanced FEs 373References 383A List of Boxes with Equations 387B List of Boxes with Data and Results for Examples 389Index 391