The dome and the ring: Verification of an old mathematical model for the design of a stiffened shell roof

We study an old mathematical model, developed before the computer era, for analyzing the strength of a stiffened shell roof. The specific problem considered is a textbook example presented in K. Girkmann: Flächentragwerke, 3rd edition, 1954. Here the roof consists of a spherical dome and a stiffening ring of rectangular cross section attached to the edge of the dome. The problem is to compute the resultant force and moment acting at the junction of the dome and the ring. We approach the old model for solving the problem in two different ways. First we carry out a historical study, where we look for possible improvements of the old model while limiting ourselves to manual computations only. We find a variant of the model which, despite being about as simple as the original one, is considerably more accurate in comparison with recent numerical solutions based on FEM and axisymmetric 3D elastic formulation of the problem. The second approach in our study is to carry out an a posteriori error analysis of our refined old model. The analysis is based on variational methods and on the Hypercircle theorem of the linear theory of elasticity. The error analysis confirms, and largely also explains, the observed — rather high — accuracy of the refined old mathematical model.