# Post-Buckling Behaviour of Steel Structures with Different Types of Imperfections
## 4 Numerical Examples
Figure 5: Equilibrium path obtained for small amplitudes of the geometrical imperfections in the shape of the first, second and third buckling mode, respectively (\(m1\), \(m2\), and \(m3\)); displacement of the node: (**a**) No. 1, (**b**) No. 2, (**c**) No. 3. When imperfection related to the first buckling, the bifurcation point corresponds almost exactly to the first eigenvalue obtained for the ideal structure. However, when imperfections are developed in accordance with the second buckling mode, the bifurcation point corresponds to the second eigenvalue, etc. It can be seen that the shape of the initial geometrical imperfection strongly determines the nature of the equilibrium path. Figure 6 shows the equilibrium paths obtained for examples in which geometrical imperfections are determined according to the small, medium and large amplitude of the first buckling mode (\(m1\), \(s1\) and \(d1\)). In all three cases, the bifurcation point associated only with the first critical load is observed. However, it can be observed that an increase in the amplitude of the initial geometrical imperfection results in an increase in the value of the critical load. Because the structure under consideration is primarily characterised by a stable equilibrium path, an increase in the imperfection amplitude somehow defies the snap-through of the structure to the configuration described by the unstable equilibrium path, and in this situation plays a positive role. In the case considered, the displacement both before and after the snap-through has the shape of the first buckling mode (Figure 7). Moreover, based on the analysis of the second-order variation of potential energy, it is found that before snap-through the equilibrium path is stable, and afterwards it turns into an unstable one. This is illustrated in the figures by a ball moving on a differently shaped surface. To determine the contribution of particular buckling mode in the displacement vector for each load increment, factor \(\rho_{m}\) is calculated according to the following relationship:  
\[\mathbf{\rho}=[\mathbf{u}^{\alpha\tau}]^{-1}\mathbf{u} \tag{19}\]  
Figure 8 shows the contribution of an individual buckling mode in the displacement vector. Coefficients \(\rho_{1}\), \(\rho_{2}\) and \(\rho_{3}\) illustrate the influence of the first, second and third buckling mode on the displacement vector, respectively. It is worth noting that in the example considered, in the whole range of the equilibrium path, the first buckling mode has a significant share in the displacement vector. Thus, the initial imperfection, determined by the first buckling mode, strongly influences the shape of the total displacement vector. Figure 6: Equilibrium path obtained for small, medium and large amplitudes of the geometrical imperfections in the shape of the first buckling mode, respectively (\(m1\), \(s1\) and \(d1\)); displacement of the node: (**a**) No. 1, (**b**) No. 2, (**c**) No. 3. In the subsequent examples, initial geometric imperfections with three different amplitudes, developed according to the second or third buckling mode, are taken into account. For all imperfection configurations, only one bifurcation point is observed. However, the imperfection in the shape of the second buckling mode is associated with a second eigenvalue (Figure 9). In turn, for imperfections developed in accordance with the third buckling mode, the bifurcation point is associated with the third eigenvalue (Figure 10). As in the first case, an increase in the amplitude of the initial geometrical imperfection results in an increase in the value of the critical load. The displacements before and after a snap-through for all the initial amplitudes of the geometrical imperfections developed according to the second and third buckling forms have the shape of the second (Figure 11) and the third buckling form, respectively (Figure 12). Figure 8: Coefficient \(\rho\) for geometrical imperfections developed according to the first buckling mode: (**a**) small (\(m1\)), (**b**) large (\(d1\)).