Computational Tensor Analysis of Shell Structures

1. Introduction.- 2. General Theory.- 2.1 A Summary of the Tensorial Quantities Required in the Formulation of a Shell Theory.- 2.1.1 Base Vectors.- 2.1.2 Metric Tensors.- 2.1.3 Coordinate Transformations.- 2.1.4 Christoffel Symbols.- 2.1.5 Covariant Derivatives.- 2.2 Surface Geometry.- 2.2.1 Curvilinear Coordinates on a Surface.- 2.2.2 Geometry of a Curved Surface.- 2.3 The Strain Tensor.- 2.4 The Stress Tensor.- 2.5 The Constitutive Equations.- 2.6 The Theory of Shells.- 2.6.1 Shell Geometry.- 2.6.2 Deformation Characteristics.- 2.6.3 The Change in Curvature Tensor.- 2.6.4 The Strain-Displacement Equations.- 2.6.5 Interpretation and Discussion of the Strain-Displacement Equations.- 3. Numerical Fundamentals.- 3.1 The Curvilinear Finite Difference Method.- 3.2 The Numerical Computation of the Surface Geometric Quantities.- 3.2.1 Base Vectors and Metric Tensors.- 3.2.2 The Christoffel Symbol.- 3.2.3 The Curvature Tensor.- 3.2.4 Covariant Derivative of the Curvature Tensor.- 3.3 The Principle of Virtual Displacements.- 3.4 Discretization and Displacement Fields.- 3.5 The Numerical Implementation of the General Surface Stress, Strain and Displacement Components.- 3.5.1 The General Surface Displacement Components.- 3.5.2 The General Strain Tensor.- 3.5.3 The General Stress Tensor.- 3.6 Boundary Conditions.- 3.7 The Numerical Solution of the Eigenvalue Problem.- 4. Numerical Implementation.- 4.1 A Second Order Implementation.- 4.1.1 Second Order CFD Approximation.- 4.1.2 Numerical Integration Scheme.- 4.1.3 Numerical Examples.- 4.2 A Third Order Implementation.- 4.2.1 Third Order CFD Approximation.- 4.2.2 Boundary Conditions.- 5. Numerical Applications.- 5.1 Simply Supported Plate.- 5.2 Cantilever Plate.- 5.3 Spherical Cap on a Square Base.- 5.4 Cylindrical Panel.- 5.5 Curved Fan Blade.- 5.6 Conical Shell Panel.- 5.7 Cylindrical Tank.- 6. Summary.- References.- Appendix A: Displacement Transformations.- Appendix B: Finite Difference Expressions.- Appendix C: Numerical Integration of the Stiffness Matrix.- Appendix D: Application of the CFD method to the Analysis of Beam Bending Problems with Fixed Edges.- Appendix E: Transformation of the Generalized Eigenvalue Problem to Standard Form.- Appendix F: Numerical Results.- Simply supported plate.- Cantilever plate.- Curved fan blade.- Spherical shell.- Cylindrical panel.- Cylindrical tank.- Conoidal shell.