Optimal archgrids: a variational setting

Radosław Tomasz Czubacki , Tomasz Denis Lewiński

Abstract

The paper deals with the variational setting of the optimal archgrid construction. The archgrids, discovered by William Prager and George Rozvany in 1970s, are viewed here as tension-free and bending-free, uniformly stressed grid-shells forming vaults unevenly supported along the closed contour of the basis domain. The optimal archgrids are characterized by the least volume. The optimization problem of volume minimization is reduced to the pair of two auxiliary mutually dual problems, having mathematical structure similar to that known from the theory of optimal layout: the integrand of the auxiliary minimization problem is of linear growth, while the auxiliary maximization problem involves test functions subjected to mean-square slope conditions. The noted features of the variational setting governs the main properties of the archgrid shapes: they are vaults over a subregion of the basis domain being the effective domain of the minimizer of the auxiliary problem. Thus, the method is capable of cutting out the material domain from the design domain; this process is built in within the theory. Moreover, the present paper puts forward new methods of numerical construction of optimal archgrids and discusses their applicability ranges.