Dynamic response and stability of an inclined Euler beam under a moving vertical concentrated load

Abstract

This paper is concerned with the dynamic response and stability of an inclined beam under a moving vertical concentrated load. The governing equation for transverse motion of the beam is presented and it contains the effect of the compressive axial component of the vertical load on the bending stiffness capacity of the beam. This moving load problem on inclined beam is no longer simply a dynamic problem, but it involves a structural stability problem when the load is large and the speed of the moving load is low. Furthermore, with the presence of the axial load component, this problem is no longer symmetrical. In other words, the ascending moving load and descending moving load produce different beam responses. For solving this time-dependent problem, we decouple the formulation and adopt the semi-discrete finite element method in the space domain. The equation of motion is then solved by using the Newmark method. The present numerical approach and the assumptions adopted were verified by considering a zero angle of inclination (i.e. the horizontal beam case). Based on this developed formulation, numerical solutions on the moving load problem were obtained for various angles of beam inclination. The natural frequencies of the beam become lower as the moving load moves up the inclined beam due to the compressive axial force component that degrades the elastic bending stiffness capacity. When the compressive axial force component is large enough and the speed of the moving load is low, a structural stability problem may surface.