The geometry of the equilibrium of forces and moments in shells

We demonstrate that the internal forces and moments in a shell structure can be described by 4 vectors in 3 dimensional space which could be used to express the equilibrium of a shell graphically. The internal forces and moments described by 2 of the 4 vectors actively carry imposed loads and imposed couples, while the other 2 describe ‘redundant’ internal forces and moments that are purely in equilibrium with themselves. Nevertheless redundant stresses and moments are important in controlling internal stresses and satisfying boundary conditions. We show that forces and moments ‘flow’ across a shell in exactly the same way that fluid flows, with a combination of irrotational flow and incompressible flow. The aim is that these ideas can help in the initial design of those shell structures which have to rely on bending moments in addition to membrane action in supporting loads, and also in the interpretation of results from an analysis using the finite element method. We demonstrate the use of the technique for the design of an umbrella structure like those built by Amancio Williams.