Nonlinear analysis of nonprismatic Timoshenko beam for different geometric nonlinearity models

Abstract

This paper deals with the statics and stability of a nonprismatic beam described according to Timoshenko theory, with various geometric nonlinearity models taken into account. The investigated models differed in the complexity of the nonlinear part of the strain tensor. In the simplest model only one nonlinear deformation component, i.e. (W′)2/2, was taken into account. Most of the works on geometrically nonlinear beam models, known to the present authors, analyze the simplest model. As demonstrated here this model yields correct results only in for beams with nonslidable supports. An analysis of slidable systems carried out in this paper indicates big differences between the solutions obtained using the different nonlinearity models and shows that in the case of the simplest model the solutions differ considerably from the ones obtained by, e.g., FEM. It also shown that when certain additional strain tensor elements are taken into account, this, although correct from the mathematical point of view, leads to incorrect solutions. One original contribution of this paper is the application of the approximation method to solve the nonlinear problem. The method uses the Chebyshev series whose expansion coefficients are determined from a certain system of recurrence equations. The method enables one to solve equations with variable coefficients. As shown in the previous papers by the author, in the case of solutions to linear problems this method leads to very accurate (also in comparison with analytical solutions) results. The other original contribution is the demonstration of the influence of the particular nonlinear strain tensor components on the solutions to the analyzed problems.