Instabilities can introduce highly non-linear effects into structural problems. The instabilities, not clearly associated with a change in a parameter, result in a stochastic variation of the responses. This process variation can be distinguished from the effects of the parameter variation by mapping the response variation onto a predictable space and a residual space, where the predictable space contains the possible effects of the parameter variation, and the residual space contains the process effects. This study discusses the sources (mechanics) of the response variation in this class of problems, the use of response surfaces to distinguish between effects driven by design variable changes and bifurcations, and the visualization of unstable zones in the structure. Analytical problems, a headform impact problem, and an occupant safety study clarify the use of the proposed methodology. Copyright © 2005 John Wiley & Sons, Ltd.