# Post-Buckling Behaviour of Steel Structures with Different Types of Imperfections
###### Abstract
The authors proposed a predictor-corrector reduced-order modelling method for the nonlinear buckling analysis of thin-walled structures with initial geometrical imperfections based on Koiter theory. The main innovation of this method is the implementation of initial geometrical imperfection, instead of equivalent loads. This makes the Koiter-Newton reduced-order modelling method applicable for pre- and post-buckling, even for relatively large imperfection amplitudes. The idea of increasing buckling capacity with modal geometrical imperfections was discussed by [15]. They note that not all imperfections are detrimental to buckling problems. A recent design trend called \"modal nudging\" takes advantage of geometrical imperfections to change the primary structure to equilibrium paths with greater bearing capacity. In this article, the authors proposed an automated procedure based on a Koiter-inspired reduced-order model that allows the equilibrium paths of structures with geometrical imperfections to be estimated rapidly according to the first linearized buckling modes. Fast random search is used to select a geometrical imperfection that pushes the structure onto a favourable equilibrium path. In this case, a geometrical imperfection may play a beneficial role. \"Modal nudging\" aims to improve the buckling response of slender elastic structures by introducing a small change to the base geometry, capable of pushing the structure onto equilibrium paths with higher bearing capacity, both for unstable and stable equilibrium pathways. However, the authors note that it is not enough to guarantee a response that is insensitive to imperfections. In fact, designers should be aware that small random imperfections could push the structure back onto unstable paths. The \"modal nudging\" was also used by [16], as a design strategy that intends to improve the load-carrying capacity of a structure, by nudging the structure to follow a specific equilibrium path. They proved that \"modal nudging\" by small deterministically chosen geometric changes lead the steel frames onto their higher load-carrying paths. This is why it is so important in engineering practice to study different imperfection configurations. The nonlinear finite element model used to simulate equilibrium paths is presented in [17]. Special interest is focused on the identification of critical points which contribute to the understanding and quantification of stability processes in a nonlinear system. The paper discusses the methods of stochastic and sensibility analyses which are frequently applied to the assessment of the safety and reliability of supporting structural systems. It is pointed out that the failures of the slender structural system caused by stability loss very often causes its decrease in safety and reliability. Therefore, reliable methods of stability analysis are important instruments to ensure the general effectiveness of modern structural systems. The snap-through phenomenon is very important in the stability analyses of steel members with initial imperfection. It is also observed in real engineering structures as well as during experiments. This issue is studied by [18] on the example of geometrically nonlinear truss structures. By tracing their equilibrium paths, using an arc length approach within a finite element analysis, snap-through behaviour is demonstrated. The snap-through phenomenon of the Mises truss taking into account spatial asymmetric and symmetric buckling with an out-of-plane lateral linear spring is also investigated in [19]. In the paper, the tangent stiffness matrix, for the primary equilibrium trajectory, is obtained in a closed-form and eigenvalues of the stiffness matrix are also derived analytically. They are used for the detection, classification and sequencing of critical points, in the primary equilibrium path of the Mises truss.