Papadrakakis, Manolis (1978). Gradient and relaxation nonlinear techniques for the analysis of cable supported structures. (Unpublished Doctoral thesis, City, University of London)
Abstract
The purpose of this work is to investigate the efficiency of numerical nonlinear solution procedures when applied to the static analysis of cable supported structures. Gradient and relaxation methods are developed and compared with existing nonlinear solution techniques. In order to obtain a more general picture of the performances of the above methods, stiffness methods with Newton Raphson iterative schemes have also been included in the comparative study.
Chapter 1 examines the behaviour and characteristics of cable supported structures and investigates the analytical requirements for static analysis. A state of the art of numerical solution techniques used to analyse these structures is presented. An extensive review of published work in relation to the analysis of single unstiffened cables, dual cables and cable networks is also presented.
Chapter 2 approaches the solution of the structural problem through total energy formulations. Three basic energy formulations are discussed with particular emphasis given to the total potential energy formulation. The principles of the unconstrained minimization method are considered and different search techniques for approximating the minimum are discussed. Expressions for the gradient vector of the total potential energy are obtained and the tangent stiffness matrix is evaluated as the matrix of the second partial derivatives of the total potential energy formulation. Different scaling techniques are reviewed and the effects of the termination criterion used, for different methods of analysis, on the final accuracy of the methods is also discussed.
In Chapter 3 there is an extensive theoretical treatment of gradient methods for the nonlinear solution of structural problems. Particular emphasis is given to the conjugate gradient algorithm and the modifications proposed by various investigators since it first appeared in 1952. A number of one dimensional linear searches are studied which approximate the minimum along the p direction and determine the scalar parameter a for the next iteration. And extensions of the conjugate gradient algorithm for the evaluation of the scalar parameter e, as proposed by Sorenson and Polak and Ribiere are discussed, Finally, the memory gradient method which employs a two dimensional linear search for a simultaneously evaluation of a and e is also presented.
Chapter 4 examines the efficiency of the methods discussed in Chapter 3 when applied to the nonlinear solution of a number of test problems. The problems are selected to have varying numbers of degrees of freedom and the respective stiffness matrices to have differing condition numbers in order to study the response of the methods for different structural characteristics. The Fletcher and Reeves method with Davidon's linear search with a cubic equation to approximate the minimum, Stanton's algorithm for bracketing the solution and the regular falsibisection algorithm to approximate the minimum, a combined algorithm of Davidon and Stanton's techniques, Buchholdt's method, Polak and Ribiere's algorithm, Sorenson's version, the memory gradient method and a number of linearized conjugate gradient algorithms are developed and their convergence characteristics are compared. The effects of scaling and reinitialization are also studied.
In Chapter 5 there is a theoretical investigation of relaxation methods and in particular the dynamic relaxation and the successive overelaxation methods. A rigorous examination of the characteristic properties of dynamic relaxation is carried out. The method is treated as a standard eigenvalue problem for error vectors and expressions for the iteration parameters are developed with respect to the minimum and maximum eigenvalue of the current stiffness matrix. A theoretical comparison of a number of pure iterative methods is performed and relationships between the iteration or scalar parameters of the conjugate gradient method, the dynamic relaxation method, the jacobi semiiterative method, and the Tchebycheff methods, are established. This suggests that all these methods in fact belong to the same family of methods called "three term recursion formulae". A combined conjugate gradient and Tchebycheff type method is also studied. A method for the automatic evaluation of the dynamic relaxation parameters is developed by the author which can guarantee convergence for almost any arbitrary initial estimate of the minimum and maximum eigenvalues of the current stiffness matrix. The concept of using kinetic energy damping instead of viscous damping in the dynamic relaxation iterative process is also examined. Finally, the successive overelaxation method is modified to be applicable to the nonlinear analysis of structural problems, and two ongoing processes for automatic evaluation of the optimum overelaxation parameter w , proposed by Carre and Hageman, are also examined.
Chapter 6 is devoted to a theoretical and numerical investigation of the problem of finding the minimum and maximum eigenvalues of a symmetric matrix. The power method, the steepest descent method, the conjugate gradient method, and the coordinate relaxation method, are among the techniques examined and compared in this Chapter. Several other modifications to the initial conjugate gradient algorithm are also studied, including the modification proposed by Fried for the evaluation of the scalar parameters and the one proposed by Geradin. An orthogonalization process is also applied to alleviate the dependency of the convergence of the method on the initial approximation for the final eigenvector.
In Chapter 7 numerical studies of the relaxation methods discussed in Chapter 5 are performed. Alternative forms of the dynamic relaxation methods with an "a priori" evaluation of the iteration parameters (using one of the methods discussed in Chapter 6), with automatic adjustment of the relaxation parameters based on the method developed in Chapter 5, and with the incorporation of kinetic damping, are applied for different test problems. Techniques to avoid the occurrence of instability of the method, when the current maximum eigenvalue of the iteration matrix becomes greater than the estimated maximum eigenvalue, are also developed and compared. Finally, the efficiency of the successive overelaxation method, with both constant and adjustable relaxation parameters is examined and compared with the efficiency of the dynamic relaxation method.
In Chapter 8 a review of methods operating through the formulation of the overall stiffness matrix is carried out. The efficiency of these methods is dependent on both the method employed to perform the linear solution when this is necessary and the nonlinear technique used to approximate the nonlinear equilibrium position in each iteration. A compact store elimination scheme, proposed by Jennings, is studied in conjunction with the Gaussian elimination procedure. Three different classes of nonlinear techniques are discussed together with the area in which each one has proved to be more suitable.
Chapter 9 performs a general comparative study of the convergence characteristics of the best methods from each classification {gradient, relaxation and stiffness methods}, and examines the advantages and disadvantages involved in the application of the methods to the nonlinear elastic analysis of cable supported structures with members being allowed to slacken. The computer time required to obtain a certain degree of accuracy, the storage requirements and the cost involved are all examined and compared in an effort to select the most suitable method for each particular class of problem.
In Chapter 10 the ultimate load carrying capacity of cable structures is studied, with members being allowed to slacken and with the inclusion of nonlinear stressstrain relationships. Two different solution procedures are employed: the stiffness method with or without the compact store elimination scheme in conjunction with Newton Raphson iteration, and Stanton's conjugate gradient algorithm. The convergence of the methods are tested for different values of the termination parameters and load increments. A continuous stressstrain curve as proposed by Jonatowski is used and provision for the cable members to reload following a different path is also included. Finally, Chapter 11 reviews the general conclusions resulting from the experience gained from the theoretical and numerical treatment of the methods discussed in this work, together with suggestions for further research.

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