Shape optimization of shells: an R-Funicularity based approach

Shells with a funicular shape with respect to a given load should be able to bear it without introducing bending. Usually, this is not possible due to boundary conditions and bending stiffness; then, it is useful to measure how far the actual structural behaviour is from the funicular one by means of the generalized eccentricity. This measure was used in 2018 to define the Relaxed Funicularity (R-Funicularity), a concept according to which a shell is still funicular when the generalized eccentricity belongs to an admissibility domain. Later, a shape optimization procedure aimed at finding R-Funicular shells was proposed and implemented in a MATLAB based tool named R-Fun Optimization, integrated with the Finite Element analysis solver SAP2000. In this work, R-Fun Optimization is extensively discussed and used to optimize the shape of a bi-parabolic shell roof with respect to six objective functions somehow depending on the generalized eccentricity. The results obtained with three different boundary conditions are showed and compared to those obtained in literature with different shape optimization approaches. Also, a graphical representation of the objective functions over the variables domain is given and used to discuss their performances. Finally, a few numerical examples where the geometry is described by means of B-spline are proposed and, in one case, the isogeometric refinement is applied in order to improve the local control of the shape in the optimization process.