Exact Static Solution of Grillwork with Periodic Supports

The rectangular grillwork with simply supported boundary condition and periodically distributed internal supports is investigated by using the double U-transformation technique. The equivalent grillwork with cyclic periodicity in two (x and y) directions can be produced by using the image method. The equilibrium equations of nodes of the equivalent grillwork possess cyclic periodicity in the two directions. In the process of analysis for the equivalent grillwork, the double U-transformation is used twice. First, by applying the double U-transformation once, the simultaneous equations for the support reactions are derived. Second, by applying the double U-transformation again, the simultaneous equations can be uncoupled into a set of single-degree-of-freedom equations, thus leading to the exact solution. As an example, a square grillwork with 6 × 6 mesh and 2 × 2 internal supports is worked out by means of the formulas derived in this study. It can be proven that the explicit closed-form results satisfy the nodal equilibrium equations, the support conditions, and the boundary conditions exactly.