A form-finding method for membrane shells with radial basis functions. (15th January 2022) - Chiang, Yu-Chou; Borgart, Andrew
Abstract: The equilibrium of a membrane shell is governed by Pucher's equation that is described in terms of the relations among the external load, the shape of the shell, and the Airy stress function. Most of the existing funicular form-finding algorithms take a discretized stress network as the input and find the shape. When the resulting shape does not meet the user's expectation, there is no direct clue on how to revise the input. The paper utilizes the method of radial basis functions, which is typically used to smoothly approximate arbitrary scalar functions, to represent C ∞ smooth shapes and stress functions of shells. Thus, the boundary value problem of solving Pucher's equation can be converted into a least-squares regression problem, without the need of discretizing the governing equation. When the provided shape or stress function admits no solution, the algorithm recommends users how to tweak the input in order to find an approximate solution. The external load in this method can easily incorporate vertical and horizontal components. The latter part might not always be negligible, especially for the seismic hazard zones. This paper identifies that the peripheral walls are preferable to allow the membrane shells to carry horizontal loads in various directions without deviating from their original shapes. When there are no sufficient supports, the algorithm can also suggest the potential stress eccentricities, which could inform the design of reinforcing beams. Graphical abstract: Highlights: The form-finding process starts from either a provisional shape or stress function. The shape and stress functions are smoothly represented by radial basis functions. Horizontal loads are compatible with this form-finding method. Boundary conditions can be switched on by increasing the weight in the least squares. Boundary restrains are important for a shell subjected to multiple load cases.