ANALYTICAL SOLUTIONS FOR VIBRATION OF THERMAL BUCKLED BEAMS

Analytical solution is obtained for the dynamic responses in the vicinity of the buckled configuration of beams subjected to a uniform in-plane thermal loading. The equations governing the axial and transverse deformations of FGM beams are derived based on the classical beam theory. Then, the two equations are reduced to a single nonlinear fourth-order integral-differential equation governing the transverse deformations. By assuming that the amplitude of beam’s vibration and the additional strains induced in it are infinitesimal, and its response harmonic, the non-linear partial differential equations are reduced to two sets of coupled ordinary differential equations; one for the nonlinear static response, and the other for linear vibrations of the beam superimposed upon the buckled configuration. The nonlinear equation is directly solved and analytical solutions for static response and natural frequency are obtained as a function of the applied thermal load. The exact solutions obtained herein can be served as benchmarks to verify and improve various approximate theories and numerical methods.