Membrane Solution for a Paraboloid under Self-Weight

Stress flows in a predictable pattern, and structural optimization is achieved by matching the natural flow of stress with the structural shape. The geometry of the parabolic shape simulates the natural flow of stress, and is therefore highly efficient in the conveyance of stress. However, despite its importance, the membrane solution of a parabolic dome has never been solved. Designers have been reliant on numerical methods, such as finite elements, or older techniques such as graphical solutions. For this reason, a closed-form membrane solution for a parabolic dome is derived. The solution solves for the meridian and hoop stresses, in the vertical and horizontal directions of the dome for the case of uniformly distributed loads, such as the self-weight of a uniformly thick shell. Finite element analysis (FEA) was also used to undertake a full shell analysis (i.e., membrane and bending behavior) to examine the edge effects that are not captured in the membrane solution. From this study, the benefits of the parabolic dome were found to be similar to the catenary dome; i.e., the stresses in the meridian and hoop directions are compressive, boundary effects are largely minimal, andstresses flow primarily in-plane (membrane action).