General shell equations

Hulls are essential elements of lightweight structures, whether aerospace or land-based. This article summarizes the set of equations that govern them. These formulations, developed in both statics and dynamics, provide a general equation for thin elements. These systems can be applied to a wide range of elements within the linear elastic framework used in aerospace and civil engineering, and form the basis for analytical and numerical developments of thin structures in space. The equations presented thus complement the classical strength-of-materials forms dedicated to straight beams, curved beams and plates.

The calculation of left-handed hulls, and in particular non-developable hulls, provides a coherent framework for dealing with all structural surfaces. The local description of the mean surface is therefore the fundamental geometric framework of hull theory. The Beltrami system, expressed in terms of internal flows (visseur), completes this description, generating the equations of hull statics and the general Lagrangian formulation of displacements, taken on the mean surface (Reissner system). The explicit introduction of inertial forces then enables us to explain the general equations of linear hull dynamics, formulated in the modal vibration and shock analyses. Particular attention is paid to invariant special cases, especially shells of revolution, providing reference cases and common examples for applications.