Finite element analysis for axisymmetric elastic stress problems.

This thesis describes the development and the applicability of a finite element sequence of computer programs for the elastic stress analysis of axisymmetric structures. The following topics are discussed:- (i) the need for an asymmetric element, (ii) the mathematical theory outlined in physical terms, (iii) the philosophy adopted in writing computer programs, and (iv) an ideal problem used to test the sequence of programs, Particular attention has heen paid to accuracy, minimization of errors and ease of-running programs. Thus, in determining the fifteen integrals required to obtain the element stiffnesses, algebraic equations v/ere used in preference to numerical integration and thus avoid both truncation error and round-off error. Care was taken in the formulation of these equations to allow for the possibility of one or more vertices to lie on the axis of symmetry and to allow for two vertices to have the same radius. The latter facility is not available in treatments by other authors. To enable the unsymmetric matrices used in the formation of element stiffness to be inverted; to improve the accuracy of the application of the overall stiffness matrix to the load vector; and in fitting surfaces by the method of least squares, special subroutines were written in a form suitable for partial pivoting. To avoid large ratios of matrix element size, which would prevent inversion on even the largest word length computer, a technique of changing origin and scaling v/as adopted in fitting surfaces by the method of least squares. The technique of changing origin was also used in the formation of certain of the integrals to obtain a greater degree of accuracy on the digital computers with their inherent round-off error. Axisymmetric analysis was eased by separating the computation into a sequence of discrete programs. Adopting this procedure (a) reduced the requirements for computer time . and space thus minimizing cost and restrictions imposed by normal operating procedures; (b) allowed decisions on restraint and external loading to be deferred until the relevant values were required and available; (c) allowed modifications to be made to any program in the sequence without having a feedback effect on any of the other programs, and, finally, (d) facilitated parametric studies to be made of the effect of varying geometry, material properties, restraints, and loading. The novel fitting of fourth order surfaces to displacements to obtain strains and stresses, a feature unique to this sequence of programs, produces a| high order of accuracy. The high order of accuracy was illustrated by the problem used to test the sequence of computer programs, where with three quarters of a million matrix elements, the resulting accumulation of round-off error only amounted to 0,001%. There was very good agreement between finite element results and Lame, the general error (except v/here the loads were applied) being about 0.03%, so that the accumulated round-off error was insignificant even with one thousand finite elements. In addition to solving small-deflect ion static-elastic problems, the described sequence of computer programs may be utilised to solve large displacement problems by adopting an iterative 'procedure and, by taking advantage of the mid-side nodes, small modifications would permit analysis of pseudo dynamic problems