Optimal arch shape solution under static vertical loads

In this paper, a new analytical solution for the optimal shape of a plane-statically determined arch subjected to uniform vertical loads is presented. The classical problem of a catenary subjected to the self-weight is extended to an inverted catenary subjected to the self-weight and to a constant vertical load distribution. In this condition, the authors demonstrate that a class of analytical solutions exists and that unlike previously proposed solutions it corresponds to the minimum ratio of the self-weight of the arch to the total applied load. Finally, existence conditions for such a solution are derived.