The Romans used arched forms to support downward loads. Everybody knows that. The differential equations for a curved shell are more complicated than for a flat plate. They are fundamentally of a different sort. For example, curved shells introduce a new variable, stretching of the wood in first order. (Flat plates will have stretching for large displacements, but not to first order.) The new equations involve two more orders of differentiation because of the new variable of stretching.
Because you say you are an engineer, I expect that you know about wave equations on differ