Buckling of Rib-Stiffened Plates: An Asymptotic Approach

A perturbation scheme is developed to study the buckling of rib-stiffened plates. Only "weak" stiffeners are considered. The method of solution is suggested by a slight variation in the standard model and developed, initially, for a simpler problem, the vibration of a string with a bead attached. The traditional model of the bead as a point mass is altered to include the width of the bead, which serves as our small parameter ε. A set of coupled integral equations for the eigenfrequencies and eigenmodes results. The effects of the bead are shown to be O(ε) and, hence, small. Rescaling, we find the first corrections to the eigenfrequencies and eigenmodes. In the rib-stiffened plate problems, the stiffening beams are usually modeled as line segments, i.e., delta functions, along which the plate density and rigidity are varied. By altering the model as in the bead-string problem, analytical expressions, in the form of generalized asymptotic series, are found for the effects of the stiffeners on the buckling loads and modes of the plate. Here, ε is the ratio of the area covered by the stiffening material to the area of the plate. Solutions for the primary buckling loads and modes are found in the form of perturbation expansions near the bifurcation points for small displacements. Optimization of the lowest buckling load with respect to the placement of the stiffeners is elementary for the unattached stiffeners and tractable for the attached ones.