NURBS-Based Grid Shell Form Finding on Domains with Topologically Arbitrary Boundaries

In architecture, special attention is paid to the shapes of thin curved surface structures known as shells. Ideally, shells should have shapes that can support their self-weight without bending. These shapes rely solely on in-plane stresses flowing along the surface, resulting in highly efficient thin structures. The process of finding the shape of a shell is called form finding. In the context of form finding of shells, the computation of another surface, called the Airy stress function, often plays a key role. An Airy stress function is a smooth and continuous surface whose horizontal projection matches the shell, with stress distribution information encoded in its curvatures. However, some form-finding problems, particularly those involving topologically complex boundary curves, cannot be easily solved due to a limitation of the Airy stress function. By construction, it cannot represent stress distributions that transmit net forces between disjoint domain boundaries. Formally, the Airy stress function is incomplete: It cannot represent all valid stress fields. It requires extension to capture the case of interacting boundaries. In this paper, we address the limitation of the Airy stress function by reintroducing a previously overlooked additional stress function originally presented by Schaefer [1953] and Gurtin[1963]. In combination with the Airy stress function, this formulation was shown by Gurtin[1972] to represent all possible stress states, regardless of the topological complexity of the domain boundary. Using several examples, we demonstrate that topologically complex boundaries with interacting forces can be solved using this stress function inserted in combination with the Airy stress function.