Optimization in a Class of Hole Shapes in Plate Structures

An analytical/numerical procedure has been developed which yields the optimum geometry for a constrained hole shape in a large plate under prescribed boundary stresses at infinity. The optimality criterion is based on the minimization of a certain stress integral taken around the hole boundary. Muskhelishvili’s method is used to first obtain the symbolic stress expressions for a given mapping function with unknown coefficients. These stress expressions are then squared and integrated around the hole boundary to obtain the stress integral. A sample problem is worked out in detail to demonstrate this procedure. The optimum shape of a square-like (double-barrel shape) hole with rounded corners was determined by restricting the mapping function to only one specific unknown coefficient. Modification of this shape by introduction of an additional term is also discussed. Numerical stress-concentration values for this case are compared with those from other sources. Most of the algebraic and numerical calculations presented in this paper were performed using MACSYMA, a symbolic manipulation package developed at M.I.T. and in regular use at the David W. Taylor Naval Ship R&D Center.