Geometrically nonlinear analysis of discretized structures by the group theoretic approach

This thesis is concerned with the static and dynamic response analysis of large, geometrically nonlinear structures with symmetry by using a reduction technique, the Group Theoretic Approach (GTA).; In engineering practice, there are many structures that have symmetric properties. The GTA introduced provides a way in which the abstract symmetric group ideas from group theory can be applied to engineering problems, and can dramatically reduce the problem of the analysis of a large scale structure to a smaller problem which captures identical solutions to those for the original structure.; The emphasis has been on developing and using the GTA for the static and dynamic analysis of large scale geometrically nonlinear structures with symmetry.; In construction of a reduced problem for the original structure, group representation theory and projection operator theory were adopted to find symmetry modes for the configuration space of the structure. The symmetry modes were actually a set of basis vectors in the sub-space for a reduced problem, and they were used to form the transformation matrix by which the original structure was reduced to a smaller problem.; A relatively large collection of linear and geometrically nonlinear symmetric structures were treated by using the GTA. The significant numerical savings, as well as the identical solution offered by the GTA, have been demonstrated by consideration of several typical numerical problems. Various kinds of engineering structures, such as trusses, frames and membrane structures, were evaluated. It is observed that the solutions computed by the GTA in the reduced sub-space were identical to those obtained in the original full space when true symmetry exists in the structure. When the symmetry is not perfect, the GTA yields approximations, the accuracy depending upon the amount of asymmetry.; This investigation has demonstrated that the GTA is mathematically very elegant and rigorous. The GTA has a great potential in analysis of any symmetric problem which can be solved by the finite element method. This investigation also has opened a door to much wider application of the GTA.