EXACT SOLUTIONS FOR THERMAL POST-BUCKLING OF SHEAR DEFORMABLE BEAMS

An exact and closed form solution is obtained for the nonlinear static responses of beams subjected to uniform in-plane thermal loading. The equations governing the axial and transverse deformations of FGM beams are derived based on the nonlinear first-order shear deformation beam theory (FBT). The three equations are reduced to a single nonlinear fourth-order integral-differential equation governing the transverse deformations. The equation and the corresponding boundary conditions lead to differential eigenvalue problem. The nonlinear equation is directly solved without any approximation and a closed-form solution for thermal post-buckling deformation is obtained as a function of the applied thermal load. By using the solution, one can obtain the analytical relation of critical buckling load between the shear deformable beams and the classical ones. To demonstrate the influence of in-plane loading, as well as transverse shear deformation and boundary conditions, numerical examples based on the exact solutions are presented, and the post-buckling responses of the beams are discussed. The exact solutions obtained herein can be served as benchmarks to verify and improve various approximate theories and numerical methods.