Isogeometric form finding of membrane shells by optimised Airy stress function

A two-stage form-finding procedure, based on Isogeometric Analysis (IgA), is proposed to determine the configuration of shells having a prescribed planar footprint so as to carry applied loads in a state of purely membrane stresses. The boundary-value problem of a membrane shell is described by Pucher's equation in terms of Airy stress function, external loads and shell mid-plane elevation. Within the IgA framework, the weak form of the problem is derived and the resulting integral equation is discretised by approximating the relevant fields as a linear combination of control point values and B-Spline basis functions. Taking external loads as input, control point values of the Airy potential are computed via a nonlinear programming (NLP) routine aimed at minimising the thrusts at edge supports while ensuring no-traction membrane stresses fulfilling the relevant boundary conditions. The corresponding shell form is then obtained in terms of control point heights based on the stress field computed at the previous stage. Compared to the traditional practice of manually prescribing the Airy stress function, incorporation of the proposed NLP routine into the present IgA-based form-finding procedure proves to be an effective and versatile tool to assist the designer in computing a suitable Airy stress field able to account for usual static as well as user-defined functional constraints, regardless of the domain geometry complexity. Two numerical examples substantiate the presented approach.