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Summary The paper deals with two main advantages in the analysis of slender elastic structures both achieved through the mixed (stress and displacement) format with respect to the more commonly used displacement one: (i) the smaller error in the extrapolations usually employed in the solution strategies of nonlinear problems and (ii) the lower polynomial dependence of the problem equations on the finite element degrees of freedom when solid finite elements are used. The smaller extrapolation error produces a lower number of iterations and larger step length in path‐following analysis and a greater accuracy in Koiter asymptotic method. To focus on the origin of the phenomenon, the two formats are derived for the same finite element interpolation. The reduced polynomial dependence improves the Koiter asymptotic strategy in terms of both computational efficiency, accuracy and simplicity. Copyright © 2016 John Wiley & Sons, Ltd.

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...