Arch structures are architecturally pleasing and an efficient means of load transfer. Their utilisation may range from the main structural system of a bridge to smaller components in roof structures.  

Similar to the behaviour of a straight beam, an arch loaded in its plane is subject to out-of-plane buckling if the load factor exceeds a critical value.

To establish the corresponding load-deflection behaviour, equilibrium equations must be formulated for the deformed structure. 

For general loading and boundary conditions, closed form solution of the resulting non-linear differential equations leads to great mathematical difficulties.  Most existing solutions are therefore based on simplifying assumptions of small deformations which enable linearisation of the equations and hence their solution.

For free-standing slender arches little guidance for design for stability is found in literature. While a considerable amount of theoretical research has been carried out on the buckling of curved members, most of this research has concentrated on the establishment of the bifurcation load corresponding to elastic flexural-torsional instability. Literature on the post-buckling behaviour of arch structures is scarce.

In a Final Year student project by Ibel, a parametric study on the elastic-plastic collapse loads of free standing metal arches was conducted using program SIGMA.  The results of the study, which was carried out for a range of arch slenderness values,  were presented in a format typical of column curves that are used for the design of straight members.  Curves for different arch cross sections were plotted and compared with those of steel design standards.