Funicularity in elastic domes: Coupled effects of shape and thickness

An historical overview is presented concerning the theory of shell structures and thin domes. Early conjectures proposed, among others, by French, German, and Russian Authors are discussed. Static and kinematic matrix operator equations are formulated explicitly in the case of shells of revolution and thin domes. It is realized how the static and kinematic matrix operators are one the adjoint of the other, and, on the other hand, it can be rigorously demonstrated through the definition of stiffness matrix and the application of virtual work principle. In this context, any possible omission present in the previous approaches becomes evident. As regards thin shells of revolution (thin domes), the elastic problem results to be internally statically-determinate, in analogy to the case of curved beams, being characterized by a system of two equilibrium equations in two unknowns. Thus, the elastic solution can be obtained just based on the equilibrium equations and independently of the shape of the membrane itself. The same cannot be affirmed for the unidimensional elements without flexural stiffness (ropes). Generally speaking, the static problem of elastic domes is governed by two parameters, the constraint reactions being assumed to be tangential to meridians at the dome edges: the shallowness ratio and the thickness of the dome. On the other hand, when the dome thickness tends to zero, the funicularity emerges and prevails, independently of the shallowness ratio or the shape of the dome. When the thickness is finite, an optimal shape is demonstrated to exist, which minimizes the flexural regime if compared to the membrane one.