This book presents a method which is capable of evaluating the deformation characteristics of thin shell structures A free vibration analysis is chosen as a convenient means of studying the displacement behaviour of the shell, enabling it to deform naturally without imposing any particular loading conditions. The strain-displacement equations for thin shells of arbitrary geometry are developed. These relationships are expressed in general curvilinear coordinates and are formulated entirely in the framework of tensor calculus. The resulting theory is not restricted to shell structures characterized by any particular geometric form, loading or boundary conditions. The complete displacement and strain equations developed by Flugge are approximated by the curvilinear finite difference method and are applied to computing the natural frequencies and mode shapes of general thin shells. This approach enables both the displacement components and geometric properties of the shell to be approximated numerically and accurately. The selection of an appropriate displacement field to approximate the deformation of the shell within each finite difference mesh is discussed in detail. In addition, comparisons are made between the use of second and third-order finite difference interpolation meshes.
Inhalt
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.