Optimal Shape of an Arch under a Central Point Load for Maximum Buckling Load

This technical note presents the optimal shape of an arch carrying a central point load to two pinned supports for maximum in-plane bifurcation buckling load. The arch shapes considered are triangular, parabolic, circular, catenary, and ogival. It is assumed that the arches are elastic and inextensible and their flexural rigidity is uniform throughout the arch length. The Hencky bar-chain model was used for elastic buckling analysis. It is found that the optimal arch shape depends on the arch apex height to length ($h/L$) ratio. The optimal arch shape changes from circular to catenary to parabolic and finally to triangular from $h/L=0.1$ to 1. This shows that the funicular triangular arch shape for the central point load is not the optimal arch shape against in-plane buckling when $h/L$ is less than 0.528. For maximum buckling load, one may use either the circular, catenary, or parabolic arch with $h/L$ approximately equal to 0.3.