Membrane Theory of a Paraboloid with an Oculus

The membrane solution for the parabolic dome, or paraboloid, has been recently developed. This solution encompasses equations for both meridian and hoop stresses, for the case of gravity load and uniform shell wall thickness. Remaining to be addressed are the membrane stress equations for domes featuring an oculus, defined as a concentric circular hole at the dome’s apex. The membrane equations for this case have been derived and compared to a finite-element solution, revealing a close alignment between the proposed closed-form solutions and the finite-element approximations. The only notable deviation occurs at the base, a result of boundary effects. However, these boundary effects do not impact the stresses around the opening.