Free vibration analysis of pre-stressed FGM Timoshenko beams under large transverse deflection by a variational method

A theoretical study on free vibration behavior of pre-stressed functionally graded material (FGM) beam is carried out. Power law variation of volume fraction along the thickness direction is considered. Geometric non-linearity is incorporated through von Karman non-linear strain-displacement relationship. The governing equation for the static problem is obtained using minimum potential energy principle. The dynamic problem for the pre-stressed beam is formulated as an eigenvalue problem using Hamilton's principle. Three classical boundary conditions with immovable ends are considered for the present work, namely clamped-clamped, simply supported-simply supported and clamped-simply supported. Four different FGM beams, namely Stainless Steel-Silicon Nitride, Stainless Steel-Zirconia, Stainless Steel-Alumina and Titanium alloy-Zirconia, are considered for generation of results. Numerical results for non-dimensional frequency parameters of undeformed beam are presented. The results are presented in non-dimensional pressure-displacement plane for the static problem and in non-dimensional frequency-displacement plane for the dynamic problem. Comparative frequency-displacement plots are presented for different FGMs and also for different volume fraction indices. (C) 2016, Karabuk University. Publishing services by Elsevier B.V.