Optimum structure to carry a uniform load between pinned supports: exact analytical solution

Recent numerical evidence indicates that a parabolic funicular is not necessarily the optimal structural form to carry a uniform load between pinned supports. When the constituent material is capable of resisting equal limiting tensile and compressive stresses, a more efficient structure can be identified, comprising a central parabolic section and networks of truss bars emerging from the supports. In the current article, a precise geometry for this latter structure is identified, avoiding the inconsistencies that render the parabolic form non-optimal. Explicit analytical expressions for the geometry, stress and virtual-displacement fields within and above the structure are presented. Furthermore, a suitable displacement field below the structure is computed numerically and shown to satisfy the Michell-Hemp optimality criteria, hence formally establishing the global optimality of this new structural form.