ABSTRACT
The structural stability of a simple benchmark problem – the snap through of a shallow cylindrical roof – is revisited. The problem is analysed in a robust manner using a technique known as generalised path-following, which combines the mathematical domains of finite element analysis and numerical continuation. Using this technique, the well-known arc-length method for tracing equilibrium paths in load-displacement space is extended to explore other interesting paths on a two-dimensional solution manifold in three-dimensional load-displacement-parameter space. These paths include unconventional equilibrium paths traced by varying a model parameter, e.g. shell thickness, as well as critical paths that describe points where the tangential stiffness matrix is singular – that is where buckling, snap-through or other instabilities occur. For the chosen benchmark problem, a localised region of stable equilibrium exists on one of the unstable equilibrium branches. The evolution of this stable region with variations in shell thickness is explored by tracing the locus of critical limit points that separate the stable and unstable portions of the equilibrium curve. Hence, without resorting to computationally expensive parametric studies, we establish a two-dimensional manifold of stable equilibria in displacement-load-thickness space, which can be graphically interpreted as an island of stability.
