Free Vibration of Initially Deflected Axially Functionally Graded Non-Uniform Timoshenko Beams on Elastic Foundation

The present paper studies the effect of elastic foundation on free vibration of initially deflected non-uniform axially functionally graded (AFG) thick beam on elastic foundation on the basis of Timoshenko beam theory. The elastic foundation is idealized as a set of parallel linear springs. Formulation is carried out through displacement based energy principle. First, the static problem is solved to find out the unknown displacement field by using minimum total potential energy principle. Solution methodology involves an iterative technique known as direct substitution with relaxation scheme. Secondly, subsequent dynamic problem is set up as an eigenvalue problem on the basis of the known displacement field. The governing set of equations in dynamic problem is obtained by using Hamilton's principle and solved with the help of Matlab's intrinsic solver. The results of the present method are validated with previously published articles. Frequency vs. displacement curve corresponding to different combination of system parameters are presented in non-dimensional plane and are capable of serving as benchmark results.