# Post-Buckling Behaviour of Steel Structures with Different Types of Imperfections
###### Abstract
In the profile, warping occurs, combined with elastic couplings involving bending and torsion. [PERSON] developed a torsion theory in which restrained warping is included [3]. This theory is also called \"warping torsion\" or \"non-uniform torsion\". Note that the loss of stability is limited by the occurrence of initial geometrical imperfections. In typical structural elements, geometrical imperfections can take global and local forms. Global imperfections refer mainly to the bending or twisting of rod axes. On the other hand, local imperfections may occur in the form of deviations of the dimensions of individual cross-sectional walls, or the shape of the contour of the cross-section associated with the local deformation of walls, or the symmetric and asymmetric \"opening\" and \"closing\" of the cross-section. These imperfections are particularly dangerous when their shape coincides with particular forms of the local or distortion buckling mode [4]. Taking into account the actual geometrical imperfections, both global and local, is possible using the finite element method [5; 6]. Issues connected with the accuracy of the computational models which take into account the initial geometrical imperfections and residual stresses were dealt with in [7]. The influence of local geometrical imperfections of thin-walled sections on the stability of columns and short beams was analysed in [8]. There are many methods of introducing geometrical imperfections into the FEM numerical model. One of the simplest methods is the equivalent load method in which the force induces in the numerical model deformation corresponding to initial geometrical imperfections [9; 10]. This method is especially suitable for generating global geometrical imperfections, but its disadvantage is the creation of additional stresses, which in fact are not accompanied by the formation of geometrical imperfections. An alternative method of imperfection modelling is a concept based on the equivalent bar stiffness proposed by [11]. In [12] the author shows that geometrically exact beam finite elements can be successfully employed to determine, accurately, the behaviour of steel I-section beams undergoing large displacements and finite rotations. For this purpose, a two-node, geometrically exact, beam element was presented and validated, which can handle arbitrary initial configurations (e.g., curved configurations), plasticity, geometric imperfections and residual stresses. A very realistic model of construction imperfections, both global and local, can be obtained by introducing a distorted geometry of the FE mesh by measuring the actual imperfections \"in situ\". This approach requires the use of highly advanced measuring devices (such as 3D scanners) to create 3D imperfection maps and convert them to FEM programs. Unfortunately, in the case of civil engineering structures, this method is still rarely used because of a relatively high cost. It has one more drawback, namely it allows the creation of an initial imperfections pattern only for a given structural element. Therefore, it only covers one particular case. Due to the high randomness of imperfections, the results obtained on the basis of deterministicallydefined imperfections are not always reliable and do not allow potentially dangerous configurations to be identified. An alternative method of taking into account imperfections may be a method of shaping deformed geometry as a linear superposition of buckling modes obtained from a linear solution of the eigenvalue problem [13]. In this approach, the amplitude of buckling modes can be scaled by proportional coefficient factors determined arbitrarily based on standard tolerances or on a limited number of measurements of real imperfections. Nonlinear FE formulations based on von Karman kinematics with initial geometrical imperfections are taken into account in [14].