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.

In architecture, shapes of surfaces that can withstand gravity with no bending action are considered ideal for shell structures. Those shells have special geometries through which they can stream gravitational force toward the ground via stresses strictly tangent to the surface, making them highly efficient. The process of finding these special forms is called form-finding. Recently, [Miki and Mitchell 2022] presented a method to reliably produce mixed tension-compression continuum shells, a type of shells known to be especially difficult to form-find. The key to this method was to use the concept of the Airy stress function to derive a valid bending-free shell shape by iterating on both the shell shape and the Airy stress function; this turns a problem that is over-constrained in general into a problem with many solutions. In [Miki and Mitchell 2022], it was proposed that the method could also be used to design grid shells by tracing curves on a continuum shell such that the resulting grid has bars that are both bending-free and form flat panels, a property useful for construction of real grid shells made of glass and steel. However, this special type of grid is not guaranteed to exist in general on a mixed-tension compression shell, even when the shell is in bending-free equilibrium [Miki and Mitchell 2023]. Additional conditions must be imposed on the shell shape to guarantee the existence of simultaneously bending-free and conjugate grid directions. The current study resolves the existence issue by adding alignment conditions. We consider several practical curve alignment conditions: alignment with the lines of curvature of the shell, approximate alignment with a bidirectional set of user-prescribed guide curves, and exact alignment with a single direction of user-prescribed guide curves. We report that the variable projection method originally used to solve the form-finding problem in the work of [Miki and Mitchell 2022] can be successfully extended to solve the newly introduced alignment conditions, and conclude with results for several practical design examples. To our knowledge, this is the first method that can take a user-input grid and find a "nearby" grid that is both flat-panelled and in bending-free equilibrium for the general case of mixed tension-compression grid shells.

Achieving a pure-compression stress state is considered central to the form-finding of shell structures. However, the pure-compression assumption restricts the geometry of the structure's plan in that any free boundary edges cannot bulge outward. Allowing both tension and compression is essential so that overhanging leaves can stretch out toward the sky. When performing tension-compression mixed form-finding, a problem with boundary condition (BC) compatibility arises. Since the form-finding equation is hyperbolic, boundary information propagates along the asymptotic lines of the stress function. If conflicting BC data is prescribed at either end of an asymptotic line, the problem becomes ill-posed. This requires a user of a form-finding method to know the solution in advance. By contrast, pure-tension or pure-compression problems are elliptic and always give solutions under any BCs sufficient to restrain rigid motion. To solve the form-finding problem for tension-compression mixed shells, we focus on the Airy's stress function, which describes the stress field in a shell. Rather than taking the stress function as given, we instead treat both the stress function and the shell as unknowns. This doubles the solution variables, turning the problem to one that has an infinity of different solutions. By enforcing equilibrium in the shell interior and prescribing the correct matching pairs of BCs to both the stress function and the shell, a stress function and shell can be simultaneously found such that equilibrium is satisfied everywhere in the shell interior and thus automatically has compatible BCs by construction. The problem of a potentially over-constrained form-finding is thus avoided by expanding the solution space and creating an under-determined problem. By varying inputs and repeatedly searching for stress function-shell pairs that fall within the solution space, a user is allowed to interactively explore the possible forms of tension-compression mixed shells under the given plan of the shell.