Finite Element Analysis of Hyperbolic Cooling Towers

The analysis of thin shells of revolution in general has always occupied an important place in the theory of structures, and recently the problem of hyperbolic cooling towers has attracted many investigators due to the wide use of such shells in industry. Until the early 1960's these towers were of moderate size, probably not exceeding 76m (250ft) height. In this range, the structural safety and stability were not of primary concern because, the minimum wall thickness and reinforcement were sufficient to provide the required safety. It was not necessary to use very rigorous methods to analyse the problem. The analysis involved the following assumptions, i) flexural stresses were ignored (membrane tneory), ii) The geometry of the shell was assumed to be perfect and to be idealised as a set of straight sided conical frusta, and iii) The boundary conditions at the base were taken to be fixed or continuously hinged with full tangential restraint.

Contenu

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.