Solutions to Problems of Isotropic Rectangular Plates

Tables of characteristic functions representing normal modes of vibration of a beam

1. Equations of Linear Elasticity in Cartesian Coordinates.- 1.1 Stresses.- 1.2 Displacements.- 1.3 Strains.- 1.4 Isotropy and Its Elastic Constants.- 1.5 Equilibrium Equations.- 1.6 Stress-Strain Relations.- 1.7 Linear Strain-Displacement Relations.- 1.8 Compatibility Equations.- 1.9 Summary.- 1.10 References.- 1.11 Problems.- 2. Derivation of the Governing Equations for Beams and Rectangular Plates.- 2.1 Assumptions of Plate Theory.- 2.2 Derivation of the Equilibrium Equations for a Plate.- 2.3 Derivation of Plate Moment-Curvature Relations and Integrated Stress Resultant- Displacement Relations.- 2.4 Derivation of the Governing Equations for a Plate.- 2.5 Boundary Conditions.- 2.6 Stress Distribution within a Plate.- 2.7 References.- 2.8 Problems.- 3. Beams and Rods.- 3.1 General Remarks.- 3.2 Development of the Governing Equations.- 3.3 Solutions for the Beam Equation.- 3.4 Stresses in Beams - Rods - Columns.- 3.5 Example: Clamped-Clamped Beam with a Constant Lateral Load, q(x) = -q0.- 3.6 Example: Cantilevered Beam with a Uniform Lateral Load, q(x) = -q0.- 3.7 Example: Simply Supported Beam with a Uniform Load over Part of Its Length.- 3.8 Beam with an Abrupt Change in Stiffness.- 3.9 Beam Subjected to Concentrated Loads.- 3.10 Solutions by Green's Functions.- 3.11 Tapered Beam Solution Using Galerkin's Method.- 3.12 Problems.- 4. Solutions to Problems of Rectangular Plates.- 4.1 Some General Solutions to the Biharmonic Equation.- 4.2 Double Series Solution (Navier Solution).- 4.3 Single Series Solution (Method of M. Levy).- 4.4 Example of Plate with Edges Supported by Beams.- 4.5 Summary.- 4.6 References.- 4.7 Problems.- 5. Thermal Stresses in Plates.- 5.1 General Considerations.- 5.2 Derivation of the Governing Equations for a Thermoelastic Plate.- 5.3 Boundary Conditions.- 5.4 General Treatment of Plate Nonhomogeneous Boundary Conditions.- 5.5 Thermoelastic Effects on Beams.- 5.6 Self-Equilibration of Thermal Stresses.- 5.7 References.- 5.8 Problems.- 6. Circular Plates.- 6.1 Introduction.- 6.2 Derivation of the Governing Equations.- 6.3 Axially Symmetric Circular Plates.- 6.4 Solutions for Axially Symmetric Circular Plates.- 6.5 Circular Plate, Simply Supported at the Outer Edge, Subjected to a Uniform Lateral Loading, p0.- 6.6 Circular Plate, Clamped at the Outer Edge, Subjected to a Uniform Lateral Loading, p0.- 6.7 Annular Plate, Simply Supported at the Outer Edge, Subjected to a Stress Couple, M, at the Inner Boundary.- 6.8 Annular Plate, Simply Supported at the Outer Edge, Subjected to a Shear Resultant, Q0, at the Inner Boundary.- 6.9 General Remarks.- 6.10 Problems.- 7. Buckling of Columns and Plates.- 7.1 Derivation of the Plate Governing Equations for Buckling.- 7.2 Buckling of Columns Simply Supported at Each End.- 7.3 Column Buckling with Other Boundary Conditions.- 7.4 Buckling of Plates Simply Supported on All Four Edges.- 7.5 Buckling of Plates with Other Loads and Boundary Conditions.- 7.6 References.- 7.7 Problems.- 8. The Vibrations of Beams and Plates.- 8.1 Introduction.- 8.2 Natural Vibrations of Beams.- 8.3 Natural Vibrations of Plates.- 8.4 Forced Vibrations of Beams and Plates.- 8.5 References.- 8.6 Problems.- 9. Energy Methods in Beams, Columns and Plates.- 9.1 Introduction.- 9.2 Theorem of Minimum Potential Energy.- 9.3 Analysis of Beams Subjected to a Lateral Load.- 9.4 The Buckling of Columns.- 9.5 Vibration of Beams.- 9.6 Minimum Potential Energy for Rectangular Plates.- 9.7 The Buckling of a Plate under Uniaxial Load, Simply Supported on Three Sides, and Free on an Unloaded Edge.- 9.8 Functions to Assume in the Use of Minimum Potential Energy for Solving Beam, Column, and Plate Problems.- 9.9 Problems.- 10. Cylindrical Shells.- 10.1 Cylindrical Shells under General Loads.- 10.2 Circular Cylindrical Shells under Axially Symmetric Loads.- 10.3 Edge Load Solutions.- 10.4 A General Solution for Cylindrical Shells under Axially Symmetric Loads.- 10.5 Sample Solutions.- 10.6 Circular Cylindrical Shells under Asymmetric Loads.- 10.7 Shallow Shell Theory (Donnell'1. Equations of Linear Elasticity in Cartesian Coordinates.- 1.1 Stresses.- 1.2 Displacements.- 1.3 Strains.- 1.4 Isotropy and Its Elastic Constants.- 1.5 Equilibrium Equations.- 1.6 Stress-Strain Relations.- 1.7 Linear Strain-Displacement Relations.- 1.8 Compatibility Equations.- 1.9 Summary.- 1.10 References.- 1.11 Problems.- 2. Derivation of the Governing Equations for Beams and Rectangular Plates.- 2.1 Assumptions of Plate Theory.- 2.2 Derivation of the Equilibrium Equations for a Plate.- 2.3 Derivation of Plate Moment-Curvature Relations and Integrated Stress Resultant- Displacement Relations.- 2.4 Derivation of the Governing Equations for a Plate.- 2.5 Boundary Conditions.- 2.6 Stress Distribution within a Plate.- 2.7 References.- 2.8 Problems.- 3. Beams and Rods.- 3.1 General Remarks.- 3.2 Development of the Governing Equations.- 3.3 Solutions for the Beam Equation.- 3.4 Stresses in Beams - Rods - Columns.- 3.5 Example: Clamped-Clamped Beam with a Constant Lateral Load, q(x) = -q0.- 3.6 Example: Cantilevered Beam with a Uniform Lateral Load, q(x) = -q0.- 3.7 Example: Simply Supported Beam with a Uniform Load over Part of Its Length.- 3.8 Beam with an Abrupt Change in Stiffness.- 3.9 Beam Subjected to Concentrated Loads.- 3.10 Solutions by Green's Functions.- 3.11 Tapered Beam Solution Using Galerkin's Method.- 3.12 Problems.- 4. Solutions to Problems of Rectangular Plates.- 4.1 Some General Solutions to the Biharmonic Equation.- 4.2 Double Series Solution (Navier Solution).- 4.3 Single Series Solution (Method of M. Levy).- 4.4 Example of Plate with Edges Supported by Beams.- 4.5 Summary.- 4.6 References.- 4.7 Problems.- 5. Thermal Stresses in Plates.- 5.1 General Considerations.- 5.2 Derivation of the Governing Equations for a Thermoelastic Plate.- 5.3 Boundary Conditions.- 5.4 General Treatment of Plate Nonhomogeneous Boundary Conditions.- 5.5 Thermoelastic Effects on Beams.- 5.6 Self-Equilibration of Thermal Stresses.- 5.7 References.- 5.8 Problems.- 6. Circular Plates.- 6.1 Introduction.- 6.2 Derivation of the Governing Equations.- 6.3 Axially Symmetric Circular Plates.- 6.4 Solutions for Axially Symmetric Circular Plates.- 6.5 Circular Plate, Simply Supported at the Outer Edge, Subjected to a Uniform Lateral Loading, p0.- 6.6 Circular Plate, Clamped at the Outer Edge, Subjected to a Uniform Lateral Loading, p0.- 6.7 Annular Plate, Simply Supported at the Outer Edge, Subjected to a Stress Couple, M, at the Inner Boundary.- 6.8 Annular Plate, Simply Supported at the Outer Edge, Subjected to a Shear Resultant, Q0, at the Inner Boundary.- 6.9 General Remarks.- 6.10 Problems.- 7. Buckling of Columns and Plates.- 7.1 Derivation of the Plate Governing Equations for Buckling.- 7.2 Buckling of Columns Simply Supported at Each End.- 7.3 Column Buckling with Other Boundary Conditions.- 7.4 Buckling of Plates Simply Supported on All Four Edges.- 7.5 Buckling of Plates with Other Loads and Boundary Conditions.- 7.6 References.- 7.7 Problems.- 8. The Vibrations of Beams and Plates.- 8.1 Introduction.- 8.2 Natural Vibrations of Beams.- 8.3 Natural Vibrations of Plates.- 8.4 Forced Vibrations of Beams and Plates.- 8.5 References.- 8.6 Problems.- 9. Energy Methods in Beams, Columns and Plates.- 9.1 Introduction.- 9.2 Theorem of Minimum Potential Energy.- 9.3 Analysis of Beams Subjected to a Lateral Load.- 9.4 The Buckling of Columns.- 9.5 Vibration of Beams.- 9.6 Minimum Potential Energy for Rectangular Plates.- 9.7 The Buckling of a Plate under Uniaxial Load, Simply Supported on Three Sides, and Free on an Unloaded Edge.- 9.8 Functions to Assume in the Use of Minimum Potential Energy for Solving Beam, Column, and Plate Problems.- 9.9 Problems.- 10. Cylindrical Shells.- 10.1 Cylindrical Shells under General Loads.- 10.2 Circular Cylindrical Shells under Axially Symmetric Loads.- 10.3 Edge Load Solutions.- 10.4 A General Solution for Cylindrical Shells under Axially Symmetric Loads.- 10.5 Sample Solutions.- 10.6 Circular Cylindrical Shells under Asymmetric Loads.- 10.7 Shallow Shell Theory (Donnell's Equations).- 10.8 Inextensional Shell Theory.- 10.9 Membrane Shell Theory.- 10.10 Examples of Membrane Theory.- 10.11 References.- 10.12 Problems.- 11. Elastic Stability of Shells.- 11.1 Buckling of Isotropic Circular Cylindrical Shells under Axially Symmetric Axial Loads.- 11.2 Buckling of Isotropic Circular Cylindrical Shells under Axially Symmetric Axial Loads and an Internal Pressure.- 11.3 Buckling of Isotropic Circular Cylindrical Shells under Bending.- 11.4 Buckling of Isotropic Circular Cylindrical Shells under Lateral Pressures.- 11.5 Buckling of Isotropic Circular Cylindrical Shells in Torsion.- 11.6 Buckling of Isotropic Circular Cylindrical Shells under Combined Axial Loads and Bending Loads.- 11.7 Buckling of Isotropic Circular Cylindrical Shells under Combined Axial Load and Torsion.- 11.8 Buckling of Isotropic Circular Cylindrical Shells under Combined Bending and Torsion.- 11.9 Buckling of Isotropic Circular Cylindrical Shells under Combined Bending and Transverse Shear.- 11.10 Buckling of Isotropic Circular Cylindrical Shells under Combined Axial Compression, Bending and Torsion.- 11.11 Buckling of Isotropic Spherical Shells under External Pressure.- 11.12 Buckling of Anisotropic and Sandwich Cylindrical Shells.- 11.13 References.- 11.14 Problems.- 12. The Vibration of Cylindrical Shells.- 12.1 Governing Differential Equations for Natural Vibrations.- 12.2 Hamilton's Principle for Determining the Natural Vibrations of Cylindrical Shells.- 12.3 Reference.- Appendix 1. Properties of Useful Engineering Materials.- Appendix 2. Answers to Selected Problems.

Solutions to Problems of Isotropic Rectangular Plates

References

Tables of characteristic functions representing normal modes of vibration of a beam

The Behavior of Thin Walled Structures: Beams, Plates, and Shells

Related Papers (5)

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