Non-linear dynamic stability of damped Beck's column with variable cross-section

In this paper the non-linear dynamic stability of Beck's column with variable mass and stiffness properties in the presence of damping (both internal and external) is investigated using a complete non-linear dynamic analysis. This approach permits the examination of the global stability of the system in contrast to the static non-linear one, which, though more economical in computational cost, is associated only with the loss of local stability via flutter or divergence. The governing equations describing the dynamic response are derived in terms of the displacements taking also into account the axial deformation, which has a striking influence on the critical load. Since the cross-sectional properties of the beam vary along its axis, the resulting coupled non-linear differential equations have variable coefficients. Their solution is achieved using the analog equation method (AEM) of Katsikadelis. Besides its accuracy and effectiveness, this method overcomes the shortcoming of a FEM solution, which may experience lack of convergence. Interesting conclusions are drawn. The important, however, finding is that the inclusion of the axial deformation affects highly the critical load of Beck's column with varying cross-sectional properties, while it leaves it unaltered for Beck's column with uniform cross-section.