Chebyshev series solution to non-linear boundary value problems in rectangular domain

Abstract

The von Kármán equations governing the behavior of moderately large deformations of rectangular plates are expressed in displacement field. A methodology based on Chebyshev polynomials approximation to analyze the non-linear boundary value problems in rectangular domain is developed. These non-linear partial differential equations of motion are linearized using quadratic extrapolation techniques. The inertia and dissipative terms are evaluated by employing Houbolt implicit time-marching scheme. The spatial discretization of the differential equations generates incompatibility, viz. greater number of equations than the unknowns. The multiple linear regression analysis, based on the least-square error norm, is employed to overcome the incompatibility and a compatible solution is obtained. Convergence study has been carried out. The clamped and simply supported immovable rectangular plates subjected to static and dynamic loadings are analyzed. Results have been compared with the results obtained by other numerical and analytical methods.