The Koiter method recovers the equilibrium path of an elastic structure using a reduced model, obtained by means of a quadratic asymptotic expansion of the finite element model. Its main feature is the possibility of efficiently performing sensitivity analysis by including a posteriori the effects of the imperfections in the reduced nonlinear equations. The state-of-art treatment of geometrical imperfections is accurate only for small imperfection amplitudes and linear pre-critical behaviour. This work enlarges the validity of the method to a wider range of practical problems through a new approach, which accurately takes into account the imperfection without losing the benefits of the a posteriori treatment. A mixed solid-shell finite element is used to build the discrete model. A large number of numerical tests, regarding nonlinear buckling problems, modal interaction, unstable post-critical and imperfection sensitive structures, validates the proposal.
1155Standard path-following approaches, aimed at recovering the equilibrium path for a single loading case and assigned imperfections, are not suitable for this purpose because of the high computational burden of the single run [13] and are unusable if no information about the worst imperfection shapes is available. For these reasons, the FE implementation of asymptotic methods [14-24] based on Koiter's theory of elastic stability [25] has recently become [26][27][28][29][30][31][32][33][34][35][36] more and more attractive. The Koiter method consists of the construction of a reduced model, in which the FE model is replaced by its second-order asymptotic expansion using the initial path tangent, m buckling modes and the corresponding second-order modes, named quadratic correctives. In this way, once the reduced model is built, the equilibrium path of the structure can be obtained by solving the nonlinear reduced system of m equations in m + 1 unknowns, which represent the modal amplitudes and the load factor. The coefficients of the reduced system are evaluated using strain energy variations up to the fourth order. Shell structures can require a very large number of FE DOFs to avoid significant discretization errors, while m is usually at most a few tens. Clearly, the convenience of the method with respect to the standard path-following strategy is evident.Since the first proposals [6,15,37,38], the method has been continuously enhanced in terms of both accuracy and computational efficiency. In particular, a mixed (stress-displacement) formulation is required to avoid an interpolation locking phenomenon in the evaluation of the coefficients of the reduced system [6,[38][39][40][41] and to make the asymptotic expansion accurate for a wider range, avoiding the extrapolation locking [14,27] common in the displacement-based approach and providing accurate results also for nonlinear pre-critical behaviours. Geometrically exact shells and beams [42,43] or corotational approaches [32,44] have been proposed to achieve structural model objectivity. Both the strategies ma...
