Form-Finding of Membrane Shells via Isogeometric Analysis

A numerical form-finding procedure, based on Isogeometric Analysis (IgA), is proposed to determine the elevation of a shell having a prescribed covered area so as to carry applied loads in a pure membrane state of stress. The boundary-value problem of a membrane shell is described by Pucher’s equation in terms of external load, Airy potential and shell mid-surface height. Membrane stress components at a point are computed via second derivatives of the Airy potential. Within the IgA framework, the equivalent weak form of the problem is derived and the resulting integral equation is discretized by approximating the relevant fields as a linear combination of control point values and B-Spline basis functions. Once the external load is prescribed, control point values of the Airy potential are computed via a nonlinear optimization routine aimed at minimizing thrusts at edge supports while ensuring a no-traction principal stress state fulfilling boundary conditions. The resulting shell form is then computed in terms of control point heights. Two numerical examples show the effectiveness of the proposed approach.