Free Vibration of Statically Deformed Bi-directionally Graded Timoshenko Beam on Elastic Foundation

In the present work, geometric nonlinearity is taken into consideration to study the free vibration response of nonlinearly deflected Timoshenko beam with bi-directional gradation, resting on an elastic foundation of Winkler type. Exponential variation of material properties is considered along the thickness and length direction. In order to carry out the present numerical study, the entire physical domain of analysis is converted into a normalized domain and suitable number of computational points are created on it to conduct a whole domain analysis. The adopted mathematical formulation is displacement based and semi-analytical. A unique methodology is used for the formulation, in which the static analysis is taken first under large amplitude static force, then a subsequent eigenvalue analysis is performed on the known statically deformed beam profile. Appropriate energy principles are utilized for both analyses to derive the equation of motion in matrix form, which are solved with the help of the algorithm of direct substitution method. Generated results are validated with the results of previously published paper. New sets of result are also presented as benchmark results to represent the impact of bi-directional gradation and elastic foundation on nonlinear vibration response