A Finite Element Formulation for the Large Deflection Random Response of Thermally Buckled Structures

The effects of temperature and acoustic loading are included in a theoretical finite element large deflection formulation for thin, isotropic plate and beam type structures. Thermal loads are applied as steady-state temperature distributions, and acoustic loads are taken to be stationary and Gaussian with zero mean and uniform magnitude and phase over the surface of the structure. Material properties are considered to be independent of temperature. Also, inplane and rotary inertia terms are assumed to be neglegible, and all inplane edge conditions are taken to be immovable. For the random vibration analysis, cross correlation terms are included. The nature of the loads leads to the solution of two separate problems. First, the problem of thermal postbuckling is solved to determine the deflections and stresses due to the thermal load only. These deflections and stresses are then used as initial deflections and stresses for the random vibration analysis. Since both analyses are nonlinear, iterative techniques are used to solve each. The solution technique used for the thermal postbuckling analysis is that of Newton-Raphson iteration. This method is found to always converge; whereas, direct iteration fails to converge. For the large deflection random vibration analysis, the linear mode shapes of the thermally buckled structure are used to reduce the equations of motion to a system of nonlinear modal equations. An equivalent linearization technique is then used to iteratively solve for the mean square deflections. Instead of using direct iteration, an underrelaxation technique is employed to reduce the number of iterations required for a converged solution. In addition to obtaining mean square deflections, the boundary for stable random vibrations for the thermally buckled structure (snap-through boundary) is predicted by considering the incremental equations of motion. Solutions obtained using these analysis methods are compared with previous solutions to assess the accuracy of the finite element formulation. The thermal postbuckling solution is compared with a 25-mode classical solution for a square plate clamped on all edges, and the random vibration solution is compared with 100-mode classical beam solutions. The present study shows that the infinite element method can be used to analyze structures subjected to combined thermal-acoustic loads