# Post-Buckling Behaviour of Steel Structures with Different Types of Imperfections
###### Abstract  
In this paper, the stability of steel members with a complex initial geometrical imperfection pattern are analysed. This issue is extremely important in the case of slender structures, characterised by multiple close critical loads and modal interactions, which can lead to unstable post-critical paths and imperfection sensitivity. Despite the fact that the loss of stability, as a result of complex geometrical imperfections, is a very common mechanism for the destruction of slender steel structures, there is still no unambiguous and adequate research in the literature and in scientific research taking into account multimodal buckling. Therefore, in this study, special attention was focused on the analysis of the equilibrium path of the structure in the pre- and post-buckling range. This was studied by introducing a model of a structure composed of four rigid bars connected by elastic nodes. For this model, as well as for the structure with and without initial geometrical imperfections, a set of nonlinear algebraic equations of equilibrium was developed. A complex pattern of imperfections was taken into account using a linear superposition of buckling modes obtained from a linear eigenvalue problem. In order to investigate the nature of bifurcation points, the concept of minimum of potential energy was adopted. By means of numerous examples, the influence of imperfections on the structural behaviour was discussed. It was found that, for special imperfection patterns, an increase in the amplitude of initial geometrical imperfection can result in an increase in the value of the critical load defining the bifurcation point. In these cases, initial geometrical imperfections can play a positive role, resulting in stable post-buckling behaviour. This phenomenon corresponds with the so-called \"modal nudging\" which aims to improve the buckling response of slender elastic structures by introducing a small disturbance in the primary geometry of the structure, which results in equilibrium paths of greater load-carrying capacity. Among other observations, a snap-through phenomenon caused by transition from the local to the global minimum of potential energy was also noted. The observed snap-through was caused by the specific configuration of initial geometrical imperfections, which in this case played quite a dangerous role. It should be emphasised that the proposed model structure allows for a full description of the post-critical behaviour and a trace of the influence of complex imperfection configurations in a simple and clear manner. s 2022 2022  
of minimum potential energy is the disappearance of its first variation. The nature of the equilibrium path is determined by higher order variations of potential energy. The type of bifurcation point can be defined based on the analysis of the second variation of potential energy. This approach was proposed by [PERSON] [1], who was considered to be the father of modern engineering mechanics. In the classical theory of stability, the type of equilibrium path can be illustrated by a ball moving on a differently shaped surface as stable, unstable and neutral. In problems with many degrees of freedom, there is also another case, called the metastable point of equilibrium. This state is characterised by temporary equilibrium and local minimum of potential energy. Note that the nature of a bifurcation point can be defined by the higher-order variation of potential energy and can be symmetric or asymmetric. In engineering practice, i.e., in the current codes [2], the most common concept is the one of critical stress, which was proposed originally by the Swiss mathematician [PERSON], who in 1757 derived the formula of critical load. This formula describes the compressive load at which, due to initial geometrical imperfection, a slender column suddenly bends. This phenomenon is called buckling and can take different forms with respect to the bar cross-sectional characteristic. In case of slender, thin-walled bar members, a spatial loss of stability is observed.