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.
Highlights
A membrane shell supports loads by in-plane stresses and its curvatures
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
This paper has presented the first form-finding algorithm which can take an initial shape as the input and adjust it under loads in various directions
Summary
A membrane shell supports loads by in-plane stresses and its curvatures. The absence of bending moments allows this type of structure to span a considerable distance with minimal material. The method of designing self-supporting surfaces in [6] goes one step further; it can take a 3D planar-facet mesh as the input and derive the best-fit polyhedral Airy stress function, which is equivalent to the reciprocal diagrams (see [7]) in the thrust network analysis Those two interactive methods are incompatible with horizontal loads, which are crucial for shell structures in seismic hazard zones [8]. To develop a more fundamental form-finding method, this paper revisits the governing Pucher’s equation, which describes the relation among the external loads (in vertical and horizontal directions), the shape of the shell, and the Airy stress function.
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