Spline-Discretization-Based Free Vibration Analysis for Orthotropic Plates

Extensive numerical analysis indicates that the conventional spline functions semidiscretization analysis for orthotropic plates and shells reported by Li et al. (Journal of Engineering Mechanics, 2003) cannot be directly employed in the vibration analysis of orthotropic rectangular plates with the free $\left(F\right)$ and/or guided $\left(G\right)$ edges. The modified spline function based method then is presented in this paper to cope with the vibration analysis of isotropic and orthotropic rectangular plates with possible combinations of the free $\left(F\right)$ , simply supported $\left(S\right)$ , clamped $\left(C\right)$ , and guided $\left(G\right)$ edges at the four bounds. Through dividing the rectangular plate into $N$ and $M$ equal subintervals along the $x$ and $y$ directions, the modified spline functions consisting of $N+3$ and $M+3$ local $B_3$ spline functions with respect to $N+1$ points and two extended additional points in the $x$ direction and $M+1$ points and two extended additional points in the $y$ direction, in which the first and last three local $B_3$ spline functions have been modified for accommodating to any one of the $F$ , $S$ , $C$ , and $G$ boundary conditions, are used as the displacement trial functions. Employing the modified spline functions, only one unknown coefficient is required to be determined for the deflection and its first and second derivatives, respectively, at a certain bound. The modified spline function matrix based characteristic equation of the orthothopic rectangular plate is derived using its potential energy functional. In comparison to the finite-element method and the spline finite strip method (SFSM), the present method has remarkably fewer numbers of degrees of freedom, and thus it is computatively more efficient. Likewise, the computational program with minimal preparation of input data can be easily developed through the present formulation. The results indicate that the present method can render both very high accuracy and fast convergence.