Spline Semidiscretization Analysis for Orthotropic Plates and Shells

This paper presents a numerical analysis procedure, called spline semidiscretization procedure, for the unified analysis of orthotropic and/or isotropic thin plates and shallow shells of rectangular projection with the two opposite edges in the y direction simply supported. The sine and cosine functions may thus be employed as the displacement trial functions in the y direction. By semidiscretization through dividing plate and shell into N equal subintervals, the $B_3$ spline function, consisting of the $(N+3)$ local $B_3$ spline functions (the first and last three local $B_3$ spline functions have been modified for accommodating to any type of boundary conditions) with respect to the $(N+1)$ points and two extended additional points in the x direction, can then be used as the displacement trial function in the x direction. Governing equations of an orthothopic shallow shell subjected to the distributed, linearly distributed, concentrated loads or their combinations are derived based on its potential energy functional. Unified formulas for the determination of displacements and internal forces of the orthotropic and/or isotropic thin plates and shallow shells are obtained. In comparison to the conventional finite element method, with the displacement trial functions having the good properties with piecewise polynomial as well as orthogonality and decoupling, the present procedure has remarkably fewer unknowns to be solved (more precisely, a term by term analysis involving only much smaller matrices can be conducted), and thus it is computatively more efficient. Likewise, the computational program, with minimal preparation of input data, can be very easily developed through the present formulation. Numerical results indicate that the present method can render a very high accuracy. The fast convergence shown in numerical examples demonstrates the reliability of the results.