An equi-stress hole for a stringer plate with cracks

Based on the equivalent strength criterion, an effective solution to the inverse elastic problem of the determining of an optimal shape of the hole contour is proposed for an elastic infinite plate reinforced by stringers. The plate is weakened by two rectilinear cracks. According to the Irwin-Orowan theory of quasi-brittle fracture, the stress intensity factor in the vicinity of the cracks' tips is adopted as a parameter characterizing the stress state in this same region. The criterion determining the optimal shape of the hole is represented as a condition of the absence of the stress concentration on the hole surface and a requirement that the stress intensity factors are zero in the vicinity of the crack tips. An apparatus of the theory of analytic functions and theory of singular integral equations is used. The formulated problem is reduced to a conditional extremum problem. A closed system of algebraic equations is obtained, which allows minimization of the stress state and stress intensity factors depending on the geometric and mechanical characteristics of the stringer plate. The action of the stringers is replaced by unknown equivalent concentrated forces at the points where the stringers join the plate.