Frequency and buckling load analysis of an axial compressive bar resting on an elastic foundation

Abstract

The nonlinear differential equations of an axial compressive bar resting on an elastic foundation are derived based on Hamilton's principal and the method of 'assumed-time-mode'. Considering simply supported boundary conditions, both free transverse vibration and buckled shapes are studied. The analytical expression of the natural frequencies of the system is derived by analyzing the eigenvalue problem. The formula of buckling load of the system can be obtained when the natural frequency becomes to zero. Considering different value of foundation stiffness, the lowest buckling modes are discussed. The first three load-frequency curves are obtained by solving the two-point boundary value problem with the shooting method. The most likely free vibration modes of the system are investigated from the load-frequency curves. The results show that the most likely free vibration mode and the lowest buckling mode are the same for a given foundation stiffness parameter.