On Energy Method for Solving the Problem of Stability and Postbuckling Bending of Beams Taking Into Account a Shear and Nonlinearly Elastic Supports

Buckling and postbuckling behavior are investigated under compression of an elastic rod with nonlinearly elastic supports. In this respect, the in-plane shear by Timoshenko’s theory is taken into account. It is proposed to use a new formulation of the energy relation to solve the problem of finding the critical load and the dependence of displacements on the compressive force. It allows in some cases to obtain a solution which gives a lower value of the critical load than the known approaches, using energy methods. The new approach is demonstrated on the example of the problem concerning buckling of a hinged beam of stepped thickness. It is found that it gives a lower value of the critical load compared to that, to which the solution leads, using Timoshenko’s approach in calculating a work of the compressive force. This means that the proposed approach gives a solution closer to the true solution. The sought functions are expressed through the deflection angle of the rod section, which is represented as a series according to some functions, satisfying the fixing conditions. The system of equations with respect to the coefficients of this series is derived from the energy relation when using different virtual deflection angles of the sections. The proposed procedure is demonstrated on the problems of postbuckling bending of beams with different fixing conditions. It is shown that for the particular cases the known solutions of linear problems, both taking the in-plane shear into account and not, are derived from the obtained results. Moreover, for verification of the method the ANSYS application software package is used. The results of a number of numerical experiments are presented. The peculiarity of the solution of the problem is revealed in determining the shape of the beam bending, caused by taking the in-plane shear into account. Further, the energy approach to solving the problem of postbuckling strain of the compressed elastic rods, having nonlinearly elastic supports which resist rotation of the sections, is considered. In this respect, the in-plane shear by Timoshenko’s theory is also taken into account. The proposed procedure is demonstrated on the example of the problem of postbuckling bending of the hinged beam.