Strongly non-linear stochastic response of a system with random initial imperfections

Strongly non-linear responses of structures with random imperfections of Gaussian type in their geometrical form under slowly increasing loading are investigated. Large displacements as a source of non-linearities are taken into account. Imperfections are considered as stochastic functions of space coordinates. The finite element method is supposed as a basis. A stochastic version of the arc length method of stochastic non-linear algebraic systems in linearised formulation is proposed. A closed interaction between deterministic and stochastic parts of response is demonstrated. Several numerical tests of theoretical results on simple Mieses frames modelling imperfect shallow shells have been carried out. Analytical and numerical results make it possible to demonstrate non-conventional properties typical for a randomly imperfect structure with a tendency to various types of snap-through effects.