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Full Description
Over the past decade or so much has been written on the various attempts to produce efficient, accurate and reliable Mindlin plate finite elements. In the late sixties, a degenerated, Mindlin-type, curved shell element was developed and subsequently many improvements in such elements have been made. Reliability and efficiency in use has always been a major objective. Degenerated shell elements have enjoyed widespread popularity despite certain potential defects, including shear and membrane lock- ing behaviour and spurious mechanisms. After introducing the basic foundations of Mindlin-type elements, this book describes these defects and also gives the reasons for their occurrence. Furthermore, the author proposes an approach to overcome these defects. A series of linear benchmark tests are proposed to illustrate the performance of the assumed strain element formulations. The formula- tions and applications for material non-linearity are also presented. Both isotropic and anisotropic material models are included together with the results for both static and transient dynamic analyses. Two associated programs are fully documented and provided on floppy discs with test examples.
Source codes for the two associated programs are provided: one is for static analysis and the other for dynamic analysis, and the programs can be compiled and run on either a mini or mainframe coniputer via a terminal. The author hopes that this book may provide further impetus in the important research area of plate and shell element technology.
Contents
1 Introduction.- 1.1 Changing Trends in Shell Analysis.- 1.2 Reliability of Finite Element Analysis.- 1.3 Shell Element Formulations.- 1.4 Objective and Layout.- 2 Degenerations of Three-Dimensional Theory.- 2.1 Introduction.- 2.2 Three-Dimensional Theory.- 2.3 Plate Theory.- 2.3.1 Thin Plate Theory (Kirchhoff hypothesis).- 2.3.2 Thick Plate Theory (Mindlin-Reissner Hypothesis).- 2.4 Three-Dimensional Degenerated Curved Shell.- 2.4.1 Definition of Strains.- 2.4.2 Definition of Stresses.- 2.4.3 The Total Potential Energy.- 3 Defects of Mindlin Plate and Degenerated Shell Elements.- 3.1 Introduction.- 3.2 Formulation of Mindlin Plate Bending Elements.- 3.2.1 Formulation of Mindlin Isoparametric Bending Elements.- 3.2.2 Finite Element Representations of Boundary Conditions.- 3.3 Formulation of Degenerated Shell Elements.- 3.3.1 Coordinate Systems.- 3.3.2 Element Geometry.- 3.3.3 Displacement Field.- 3.4 Defects of Mindlin Plate and Degenerated Shell Elements.- 3.4.1 Locking Phenomena.- 3.4.2 Reduced and Selective Integration.- 3.4.3 Alternative Methods Avoiding the Locking Problems.- 4 Assumed Strain Finite Element Plate Formulations.- 4.1 Introduction.- 4.2 Essence of Shear Locking.- 4.3 Reinterpretation of Selectively Integrated Elements.- 4.4 Elimination of Shear Locking.- 4.4.1 Assumed Transverse Shear Strain Fields.- 4.4.2 Location of the Sampling Points for the Shear Strains.- 4.4.3 The Evaluation of the Stiffness Matrix.- 5 Linear Benchmark Tests for Plate Elements.- 5.1 Introduction.- 5.2 Eigen-Analyses of the Stiffness Matrices.- 5.3 Patch Tests.- 5.3.1 Patch Tests for Bending.- 5.3.2 Patch Tests for Twisting.- 5.3.3 Patch Tests for Shear.- 5.4 Locking Tests.- 5.5 Long Cantilever.- 5.6 Convergence Tests.- 5.6.1 Simply Supported Square Plates under Uniform Load.- 5.6.2 Clamped Square Plate under Uniform Load.- 5.6.3 Clamped Circular Plates.- 5.7 Skew Plates.- 5.7.1 Razzaque's 60 Skew Plate.- 5.7.2 Morley's 30 Skew Plate.- 5.8 Stress Resultant Study.- 6 Assumed Strain Finite Element Shell Formulations.- 6.1 Introduction.- 6.2 Shear Locking, Membrane Locking and Selective Integration.- 6.2.1 Shear Locking in Shells.- 6.2.2 Essence of Membrane Locking.- 6.3 Elimination of Shear Locking.- 6.4 Elimination of Membrane Locking.- 6.4.1 Orthogonal Curvilinear Coordinate System (r, s, t or ri).- 6.4.2 Assumed Membrane Strains.- 6.5 Assumed Strain Degenerated Shell Elements.- 6.5.1 Location of Sampling Points for the Membrane Strains.- 6.5.2 The Evaluation of the Stiffness Matrix.- 6.6 Discussion.- 7 Linear Benchmark Tests for Shell Elements.- 7.1 Introduction.- 7.2 Eigen-Analysis of the Stiffness Matrices.- 7.3 Patch Tests.- 7.3.1 Bending, Twisting and Shear Patch Tests.- 7.3.2 Plane Stress Patch Tests.- 7.4 Locking Tests.- 7.4.1 Shear Locking Tests.- 7.4.2 Membrane Locking Tests.- 7.5 Conclusions.- 8 Formulations and Applications for Elasto-Plastic Shell Analyses.- 8.1 Introduction.- 8.2 Laminated Plate and Shell Model.- 8.3 Shear Correction Factors.- 8.4 The Anisotropic Yield Criterion.- 8.4.1 Generalised Huber-Mises Yield Criterion.- 8.4.2 Determination of Anisotropic Parameters.- 8.4.3 Relation between Elasto-Plastic Stresses and Strains.- 8.4.4 Tangent Stiffness Matrix.- 8.5 Numerical Examples.- 8.5.1 Clamped Square Plate.- 8.5.2 Clamped Quadratic Shell.- 9 Formulations and Applications for Elasto-Plastic Dynamic Shell Analyses.- 9.1 Introduction.- 9.2 Dynamic Equilibrium Equations.- 9.3 Modelling of Mass Matrix.- 9.3.1 Consistent Mass Matrix.- 9.3.2 Lumped Mass Matrix.- 9.4 Newmark's Time Stepping Scheme.- 9.5 Numerical Examples.- 9.5.1 Rectangular Plate.- 9.5.2 Simply Supported Square Plate.- 9.5.3 Clamped Circular Plate.- 9.5.4 Spherical Shell Caps.- Appendix 1 Software Description for Elasto-Plastic Static Analysis.- A1.1 Introduction.- A1.2 Glossary of Variable Names.- A1.2.1 Main Arrays.- A1.2.2 Main Variables.- A1.3 Program Overview.- A1.4 File Handling.- A1.4.1 Subroutine INPUT.- A1.4.2 User Instructions.- A1.4.3 Files.- A1.5 Documented Example - Clamped Quadratic Shell.- A1.5.1 Input Data File.- A1.5.2 Output Data File.- A1.6 Stiffness Evaluation Module.- A1.6.1 Subroutine for Evaluation of Stiffness Matrix.- A1.6.2 Evaluation of Strain-Displacement Matrix.- A1.6.3 Subroutine BSAMP.- A1.6.4 Subroutine FUNC.- A1.6.5 Subroutine FRAME.- A1.6.6 Subroutine SFR1.- A1.6.7 Subroutine SFR3.- A1.6.8 Subroutine TBMAT.- A1.6.9 Subroutine XDIC.- Appendix 2 Software Descriptions for Elasto-Plastic Transient Analysis.- A2.1 Introduction.- A2.2 Glossary of Variable Names.- A2.2.1 Main Arrays.- A2.2.2 Main Variables.- A2.3 Program Overview.- A2.4 File Handling.- A2.4.1 Subroutine INPUTD.- A2.4.2 Subroutine INTIME.- A2.4.3 User Instructions.- A2.4.4 Files.- A2.5 Documented Example - Thin Spherical Cap.- A2.5.1 Input Data File.- A2.5.2 Output Data File.- Author Index.
