The academic contribution of the work is the development of a theory of the stress state of circular arched structures. Providing the necessary parameters for the strength and rigidity characteristics of parts of mechanical engineering structures leads to large ratios of axial moments of inertia and large overall dimensions of cross-sections. A mechanical engineering object must meet regulatory requirements for strength and rigidity, but there is still a danger of losing the stability of a flat bending shape. Having lost stability, the rod experiences two bends and torsion. Large lateral movements often lead to various accidents. Therefore, the problem of preventing such phenomena remains relevant. The paper presents a sequence of actions to solve the boundary value problem of the stability of a plane-bending form of mechanical engineering structures. A system of two differential stability equations for the indicated structural elements – circular arches – has been integrated. Fundamental orthonormal functions for differential stability equations for a circular rod are presented in two versions. It is proposed to solve stability problems using the boundary element method.