Dynamic Green Function Solution of Beams Under Moving Loads With Elastically Supports

The dynamic analysis of bridges simulated as Euler-Bernoulli beam models with elastic supports subjected to mobile loads are analyzed by conventional methods to obtain a new solution for displacement. Generally, these beam supports can be characterized by springs with a given stiffness, which considerably influence the structure's dynamic behavior and even attenuate the dynamic amplification. The solutions proposed until now are defined only on span but not supports. In this paper, we used Green's function, considering boundary and continuity conditions and shear force to study the global behavior of the beam. A new displacement formula is proposed for the beam to support a span according to the velocity of the mobile load, the beam rigidity, and the stiffness of supports. A further study leads to the present two new formulas, which directly give displacements at the level of supports according only to the beam rigidity and supports stiffness and to the load value at any time. The result of this analysis shows that several combined factors influence the vibratory behavior of the beam when it is supported on elastically supports, namely the stiffness of the supports, the rigidity of the beam, its length, the value of the mobile load, and its velocity. The evolution of support stiffness leads to classical boundary conditions. A study of coupling between the beam and supports is presented, with the study of the comportment in function to the ratio between the beam rigidity and spring stiffness.