Stability of Shallow Arches Under Multiple Loads

For elastic structures that exhibit nonlinear pre-buckling deformations, various types of interaction curves and surfaces (i.e., stability boundaries) are possible. Shallow elastic arches subjected to three independent concentrated loads are analyzed. Several sets of boundary conditions (pinned, clamped, and clamped-pinned) and initial shapes (sinusoidal and circular) are considered, and critical loads (for proportional loading) are determined by a standard technique. The stability boundaries in the three-dimensional loading space are then constructed. These boundaries are comprised of limit points and bifurcation points. For structures in which no pre-buckling deformations occur, the stability boundaries are always concave toward the origin of the loading space. However, for the arches treated here, it is demonstrated that no such general convexity property exists.