General Equations for Buckling of Thin, Shallow Arches of Any Shape

General differential equilibrium and stability equations are derived for thin, prismatic quasi‐shallow arches with an arbitrary centroidal‐axis shape under dead loading. Sanders‐type strain‐displacement relationships are used in the derivation, subject to Donnell‐type quasi‐shallow approximations and Kirchhoff assumptions. No simplifications about arch shallowness are made other than those connected with the Donnell approximations. A general criterion is obtained that determines whether the arch equilibrium becomes critical with respect to snap‐through or bifurcation buckling. Bifurcation occurs when a particular coefficient vanishes, which takes place when the arch membrane‐stress resultant reaches the value necessary to buckle an equivalent straight bar. The equations derived here reduce to those for circular arches as a special case. Discretized versions of the differential equations for pin‐ended arches are obtained by using series expansions for the various variables. The veracity of the resulting nonlinear algebraic equilibrium equations is confirmed by the fact that when the appropriate initial‐curvature approximations are introduced, they agree with corresponding equations obtained by previous authors using different, but less general, methods.