Unconventional finite element method for nonlinear analysis of beams and plates
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In this thesis, mixed finite element models of beams and plates bending are developed to include other variables (i.e., the membrane forces and shear forces) in addition to the bending moments and vertical deflection, and to see the effect of it on the nonlinear analysis. Models were developed based on the weighted residual method. The effect of inclusion of additional variables is compared with other mixed models to show the advantage of the one type of model over other models. For beam problems the Euler-Bernoulli beam theory and the Timoshenko beam theory are used. And for the plate problems the classical plate theory and the first-order shear deformation plate theory are used. Each newly developed model is examined and compared with other models to verify its performance under various boundary conditions. In the linear convergence study, solutions are compared with analytical solutions available and solutions of existing models. For non-linear equation solving direct method and Newton-Raphson method are used to find non-liner solutions. Then, converged solutions are compared with available solutions of the displacement models. Noticeable improvement in accuracy of force-like variables (i.e., shear resultant, membrane resultant and bending moments) at the boundary of elements can be achieved by using present mixed models in both linear and nonlinear analysis. Post processed data of newly developed mixed models show better accuracy than existing displacement based and mixed models in both of vertical displacement and force-like variables. Also present beam and plate finite element models allow use of relatively lower level of interpolation function without causing severe locking problems.