Static Timoshenko Beam‐Columns on Elastic Media

Two approaches are used to derive differential equations, stiffness coefficients, and fixed‐end forces for the analysis of structural systems composed of Timoshenko beam‐columns that may be supported by an elastic foundation. The analysis is for the evaluation of the critical static axial load and buckling mode‐shapes of a structure with consideration of elastic media, bending, and shear deformations. 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, which consists of the bending and shear slope. In the second approach, the shear component of the axial load, however, is calculated only from the bending slope. Analytical expressions for a typical simple beam are derived to show the influence of a foundation parameter on the buckling modes. It is observed that the critical axial loads are significantly reduced when shear deformations are considered for smaller slenderness ratios, but they are remarkably increased when the elastic media are included for all slenderness ratios. The first approach yields less buckling load than the second, and the difference between them becomes more pronounced for beams with low slenderness ratios and higher modes. Numerical examples are provided that illustrate the application.