The Particular Solutions for Thin Plates Resting on Pasternak Foundations Under Arbitrary Loadings

摘要

Analytical particular Solutions of splines and monomials are obtained for problems of thin plate resting on Pasternak foundation Under arbitrary loadings, which are governed by a fourth-order partial differential equation (PDEs). These analytical particular Solutions arc valuable when the arbitrary loadings are approximated by augmented polyharmonic splines (APS) constructed by splines and monomials. In Our derivations, the real coefficient operator in the governing equation is decomposed into two complex coefficient operators whose particular Solutions are known in literature. Then. we use the difference trick to recover the analytical particular solutions of the original operator In addition, we show that the derived particular solution of spline with its first few directional derivatives are bounded as r -> 0. This solution procedure may have the potential in obtaining analytical particular Solution.; of higher order PDEs constructed by products of Helmholz-type operators. Furthermore, we demonstrate the usages of these analytical particular Solutions by few numerical cases in which the homogeneous Solutions are complementarily solved by the method of fundamentals solutions (MFS).