Finite Element Analysis of Hyperbolic Cooling Towers

1. Introduction.- 2. General Theory of Thin Elastic Shells.- 3. Linear Analysis of Thin Shells of Revolution Using Finite Element Method.- 3.1: Introduction.- 3.2: Geometric Parameters and Relations of Shells of Revolution.- 3.2.1: Geometric Characteristics of Hyperboloid of Revolution.- 3.3: Basic Equations of Elasticity for Shells of Revolution.- 3.3.1: Kinematic Equations.- 3.3.2: Equilibrium Equations.- 3.3.3: Constitutive Equations.- 3.4: Fourier Analysis.- 3.4.1: Representation of Distribution of Wind Pressure on Cooling Tower Shells.- 3.5: Finite Element Formulation.- 3.5.1: Idealisation of Geometry of a Shell of Revolution.- 3.5.2: Geometric Derivation of a Doubly Curved Elemext.- 3.5.3: Element Displacement Function.- 3.5.4: Determination of the Equations of Equilibrium.- 3.5.5: Element Local and Global Coordinates, and Displacement Transformation Matrix.- 3.5.6: Derivation of Element Stiffness Matrix and Equivalent Load Vector.- 3.5.6.1: General and Nodal Displacements.- 3.5.6.2: General Strains and Stress Resultants.- 3.5.6.3: Strain Energy.- 3.5.6.4: Potential Energy of External Loads.- 3.5.6.5: Total Potential Energy.- 3.5.6.6: Equilibrium.- 3.5.7: Assembly of Elements.- 3.5.8: Application of Boundary Condition and Solution of the System of Equations.- 3.5.9: Calculation of the Stress Resultants.- 3.6: Examples.- 4. Hyperbolic Cooling Towers on Discrete-Flexible Support System.- 4.1: Introduction.- 4.2: Continuous-Flexible Boundary Analysis.- 4.3: Discrete-Flexible Boundary Analysis.- 4.3.1: Derivation of Self-Equilibrated Edge Loading "Shell under Asymmetrical Applied Forces".- 4.3.2: Derivation of Self-Equilibrated Edge Loading "Shell under Axisymmetrical Applied Forces".- 4.3.3: Final Solution.- 4.4: Results, Discussions and Conclusions.- 5. Hyperbolic Cooling Towers with Geometric Imperfection.- 5.1: Introduction.- 5.2: Method of Geometric Simulation of Imperfection.- 5.2.1: Geometric Simulation of a Meridional Axisymmetrical Imperfection.- 5.2.2: Geometric Simulation of a Patch Imperfection.- 5.3: The Method of the Equivalent Load Simulation of Geometric Imperfections.- 5.3.1: The Equivalent Normal Load of an Axisymmetric Geometrical Imperfection.- 5.3.2: The Equivalent Normal Load of a Non-Axisymmetrical Geometrical Imperfection.- 5.3.3: The Higher Order Approximation of the Equivalent Load.- 5.3.4: Derivation of the Equivalent Load for a Shell of Revolution under Wind and Dead Loading with Axisymmetric Imperfection.- 5.3.5: Derivation of the Equivalent Load for a Shell of Revolution under Wind and Dead Loading with NonAxisymmetrical Imperfection.- 5.4: Results, Discussions and Conclusions.- 6. Non-Linear Analysis of Hyperbolic Cooling Towers.- 6.1: Introduction.- 6.2: Structural Non-Linearity.- 6.3: Equilibrium Equation of Non-Linear Systems.- 6.4: Numerical Techniques.- 6.4.1: Newton-Raphson Method.- 6.4.2: Simple Iteration.- 6.4.3: Linearised Incremental Method.- 6.5: Theory and Formulation of Geometric Non-Linear Shells of Revolution.- 6.5.1: Kinematic Equations.- 6.5.2: Idealisation of the Geometry and the Displacement Functions.- 6.5.3: Derivation of Strain Energy Expression.- 6.5.4: Derivation of the Element Non-Linear Strain Energy.- 6.6: Overall Solution Procedure.- 6.7: Results and Conclusion.- References.- Appendix 1 Elasticity Matrix.- Appendix 2.- Appendix 3 Force Density Vectors.- Appendix 4 Derivation of Stiffness Matrix of the Support Columns at Interface.- Appendix 5 Derivation of Equivalent Nodal Forces of Distributed Edge Loading.