Thin, elastic, shallow shells having uniform thickness and rectangular boundaries are investigated. The boundary conditions are either simply supported or claped, and the shell is subjected to a uniformly distributed load applied over either the full shell area or a central region. The thickness, material properties, edge lengths, and surface area of the shell are specified, and the objective is the determination of the shell shape which will maximize the buckling load.

Marguerre's two, coupled, non-linear equations of equilibrium are used to describe prebuckling deformations and stresses. Considering small vibrations about the equilibrium state, two, coupled, linear equations of motion are derived. Subsequently, by recognizing that at buckling the lowest frequency of vibration goes to zero, the buckling equations are obtained. Finally, the Lagrange multiplier technique is employed to formulate an augmented objective function, and the calculus of variations is applied in order to derive the governing set of equations. The resulting system of equations is solved numerically by the finite difference method.

Results for shells with various surface areas are presented. For each surface area the investigation is performed on shells having either clamped or simply supported boundary conditions and either a square or a rectangular boundary. The applied uniform load covers either the full shell area or a partial central region. The shell form, buckling load, and buckling modes of the optimal forms are compared with those of the reference form (double sine) having the same surface area, and changes are noted. Also, comparisons with respect to forms, buckling load, and type of buckling are made between the optimal form of a shell subjected to a full uniform load and the optimal form of the same shell subjected to a partial uniform load.

In some cases, the buckling load of the optimal form is sensitive to imperfections in the form or in the loading distribution as well as to changes in the design. In these cases, some of the apparent advantages of the optimum form may be diminished. Thus, the frequencies of vibration at buckling, the corresponding buckling modes, and the presence of adjacent equilibrium states are monitored in order to evaluate the sensitivity of the optimal form to imperfections and to design changes.