The stability problems of bars with plane curved axes are rather manifold, and can be systematized as follows:
- the axis can be circular or can have any other shape; - the load causes central compression before buckling, but during buckling it can exhibit
three kinds of behaviour: it can maintain its original direction, it can always remain perpendicular to the (deformed) bar axis (hydrostatic pressure), or it can always pass through a given point ('central load');
- the boundary conditions of the bar can be manifold: the arch can be independent (its ends can be clamped or hinged; there can be also a hinge at then-middle cross section); or the arch can be connected to other structural elements (arches with hangers or struts);
- their cross section can be full (closed) or thin-walled (open); in the latter case the cross section may also deform in its own plane;
- the loss of stability may occur either in the plane of the arch (by bifurcation: steep arches; by limit point or snapping through: flat arches), or perpendicularly to the plane of the arch (by bifurcation); the post-critical behaviour can also be manifold;
— depending on what we neglect in the computation: with the buckling phenomena in the plane of the arch we can consider from the bending, compression or shear deformations either only the first, or also the second, or all three; we can consider or neglect the horizontal displacements of the points of the axis of the arch.