Dynamic Timoshenko Beam‐Columns on Elastic Media

Abstract

Differential equations, stiffness coefficients, and fixed‐end forces are formulated for a Timoshenko beam‐column on an elastic foundation subjected to lateral time‐dependent excitations and static axial loads. The theoretical formulation includes shear and bending deformations as well as rotatory inertia, with emphasis on two approaches. The two approaches differ in terms of the assumed shear component of the static axial load on the cross section. The first approach is based on the assumption that the shear component of the axial load is calculated from the total slope; in the second approach, the shear component of the axial load is calculated only from the bending slope. The dynamic stiffness coefficients and fixed‐end forces are expressed in terms of nondimensional parameters associated with the effects of transverse and rotatory inertia, axial force, elastic media, and shear and bending deformations. When the individual effect is not considered, then the associated parameter can be dropped. The significance of the individual parameters on natural frequencies and dynamic response of typical beams is then extensively examined. The two approaches are also studied by comparing the response behavior and it is found that they differ appreciably with increasing axial loads and decreasing slenderness ratios. Comparison of the natural frequencies shows that the second approach gives higher values than the first.