Derivation of particular solutions for linear loading in the thick plate theory

Abstract

In this paper, the particular solutions in the theory of thick plates are derived. These new particular solutions are the analytical solutions of the governing differential equations of equilibrium due to linear domain loading. Uniform domain loading can be considered as a special case of the present formulation. The expressions for the displacement and the traction kernels are derived and given in explicit form. Also the necessary kernels for the internal stress resultants are also derived and given. The derived particular solutions are veri®ed analytically. A new set of boundary integral equation formulation for thick plates using the new particular solutions are outlined. IntroductionMany practical applications in engineering structures, such as side walls of water tanks and retaining walls (see for example, [1]), can be modeled using plates under linear domain loading. In the ®eld of structural design, the most popular method to analyze such structures is the ®nite element method [2] which requires the discretization of the problem domain leading to high computational requirements. In the past two decades, the boundary element method (BEM) has emerged as a powerful tool to solve engineering problems [3,4,5]. The main advantage of the BEM is the boundary only discretization of the considered problem. In the presence of body force terms, such as domain loading, a domain discretization is required to deal with such terms, which losses the main advantage of the BEM. Many researchers have studied methods of transformation of such domain integrals to the boundary. The ®rst technique is the direct application of Green's identities. Cruse [6], with the help of representing the fundamental solution in terms of the Galerkin tensor, used this technique to deal with body forces in elasticity problems. Stippes and Rizzo [7] extended the formulation in Ref. [6] to thermal stress problems. Hartmann and Zotemantel [8] used similar technique to transform domain integrals due to uniform loading in thin plates to the boundary. Vander Weee Èn [9] and El-Zafrany [10, 11] used Green's second identity to treat uniform loading in thick plates. Costa and Brebbia [12] applied the same technique for thin plates on elastic foundations and Rashed et al. [13, 14] for thick plates resting on elastic foundations. Antes and Steinfeld [15] used the Green ®rst identity to transform domain integrals due to body forces to the boundary for static and dynamic analysis of elasticity problems. Recently Rashed [16] extended the formulation in Ref. [9] to treat thick plates under linear loading.A logical extension to the above mentioned technique is the multiple reciprocity method (MRM) [17]. The MRM employs Green's second identity in a recursive way to transform domain integrals to a series of boundary integrals. In case of uniform body forces, only one application of Green's second identity is enough to transform such domain integrals to the boundary. Neves and Brebbia [18] used the MRM to treat body forces for elasticity problems. Recently, Rashed ...