Timoshenko beam-column with generalized end conditions on elastic foundation: Dynamic-stiffness matrix and load vector

The dynamic-stiffness matrix and load vector of a Timoshenko beam-column resting on a two-parameter elastic foundation with generalized end conditions are presented. The proposed model includes the frequency effects on the stiffness matrix and load vector as well as the coupling effects of: (1) bending and shear deformations along the member; (2) translational and rotational lumped masses at both ends; (3) translational and rotational masses uniformly distributed along its span; (3) axial load (tension or compression) applied at both ends; and (4) shear forces along the span induced by the applied axial load as the beam deforms according to the “modified shear equation” proposed by Timoshenko. The dynamic analyses of framed structures can be performed by including the effects of the imposed frequency (ω>0) on the dynamic-stiffness matrix and load vector while the static and stability analyses can be carried out by making the frequency ω=0. The proposed model and corresponding dynamic-stiffness matrix and load vector represent a general solution capable to solve, just by using a single segment per element, the static, dynamic and stability analyses of any elastic framed structure made of prismatic beam-columns with semi-rigid connections resting on two-parameter elastic foundations. Analytical results indicate that the elastic behavior of framed structures made of beam-columns is frequency dependent and highly sensitive to the coupling effects just mentioned. Three comprehensive examples are presented to show the capacities and validity of the proposed method and the obtained results are compared with the finite element method and other analytical approaches.