Expansion of the Dynamic Stiffness Matrix of a Timoshenko Beam-Column on a Two Parameter Elastic Foundation

The theory of beams on elastic foundation occupies a prominent place in contemporary structural mechanics. The governing equations of motion of dynamic systems are generally partial differential equations that are difficult to solve in mathematical terms. This work is facilitated through the use of matrix methods and modern computational techniques. The matrix methods are based on the concept of replacing the actual continuous structure by an equivalent model made up of discrete structural elements having known elastic and inertial properties expressible in matrix form. The purpose of the present paper is to expand the dynamic stiffness matrix of a Timoshenko beam-column on two-parameter elastic foundation. So, first, a brief review of the literature related to soil-structure interaction is presented. Then, the method of dynamic analysis of the structures with distributed mass and elasticity is discussed. In the discussion, the procedure of derivation of dynamic matrices using exact differential equation of vibration is studied. Afterward, the Timoshenko beam theory which takes into account the effect the rotary inertia of mass and shear distortion can be modeled is presented. Furthermore, foundation model including two-parameter models of elastic foundation are studied, and the governing differential equation of vibration of each presented model is derived. Then, the differential equation of the model is solved in order to derive the dynamic stiffness matric of a Timoshenko beam-column on two-parameter elastic foundation. Then, expansion of the obtained dynamic matrix is carried out using power series. We use program Mathematica for this purpose. The result of this expansion is a two-variable power series of some useful matrices in terms of vibration frequency and axial force which its coefficients are the relevant first, second and higher order stiffness, geometrical and mass matrices.