A simple approach to investigate vibratory behaviour of thermally stressed laminated structures

The aim of the present paper is to propose a simple method to investigate the vibratory behaviour of laminated composite structures subjected to large thermal loads (may be beyond critical temperature Tcr). von Karman type non-linear strain-displacement relationships are employed to derive non-linear finite element equations of motion. These finite element equations are based on secant stiffness rather than tangential stiffness. The secant stiffness matrix is separated into three parts, i.e., (i) linear stiffness matrix independent of field variables, (ii) non-linear stiffness matrix depending linearly on field variables and (iii) non-linear matrix depending quadratically on field variables. Linear thermal buckling and free vibration analyses are performed as a first step to compute the critical temperatures, natural frequency and corresponding mode shapes. Assuming the mode shape corresponding to fundamental frequency as the spatial distribution, large-order non-linear finite element equations are reduced to a single second-order ordinary non-linear differential equation. A direct numerical integration method is employed to compute the non-linear frequencies of thermally stressed structures. To demonstrate the method, vibratory behaviour of thermally stressed laminated beams is investigated. The proposed method is validated by comparing the non-linear frequencies of beams (not subjected to initial stress) obtained using the present method with those available in the literature. The influence of difference in buckling mode shape and vibration mode shape for certain boundary conditions on the non-linear behaviour is also studied. Some interesting observations regarding the finiteness of amplitude in the post-buckling regime are also made