Finite deformation analysis using the finite element method

An analysis of the finite deformation of an elastic body using the finite element method is investigated. The governing nonlinear equations of equilibrium are derived through the principle of virtual work using a Lagrangian description. A general incremental virtual work equation is obtained, and then linearized to permit the use of direct solution techniques. A residual loading term is defined which represents the nonsatisfaction of equilibrium of the solution obtained at the end of an increment using the linear incremental virtual work equation. The residual loading term is used to control the divergence of the linearized incremental solution from the exact equilibrium solution, through the self-correcting solution technique. The finite element method is introduced in general for three dimensional analysis, and is then specialized for two dimensional, plane elasticity analysis. Two eight degree of freedom rectangular finite elements are developed using a bilinear assumed displacement field. The first element is numerically integrated using Gaussian quadrature, while the second employs a nonuniform integration scheme in order to improve this element's performance. Four finite deformation problems are analysed using the procedure presented in this thesis, and the results are compared with available closed form solutions. The problems analysed are those of a uniformly loaded infinite plate strip having either simply supported longitudinal edges or fixed longitudinal edges, a cantilever beam under a uniformly distributed load, and lastly a cantilever beam with a para-bolically distributed end load. Excellent agreement was obtained between the finite element analysis results and the closed form solutions.