# Post-Buckling Behaviour of Steel Structures with Different Types of Imperfections
## 4 Numerical Examples
Figure 7: Configuration of displacement before and after snap-through for initial amplitudes of geometrical imperfections developed according to the first buckling mode: (**a**) small (\(m1\)), (**b**) large (\(d1\)). This phenomenon is confirmed by the analysis of coefficient \(\rho\), which obtains non-zero values before and after snap-through only for the second buckling mode (Figure 13) for both small and large initial amplitudes of the geometrical imperfections developed in accordance with the second buckling mode. Similarly, when imperfections in the form of the third buckling mode are introduced into the structure model, non-zero values of coefficient \(\rho\) are obtained only for the third buckling mode (Figure 14). Figure 10: Equilibrium path obtained for small, medium and large amplitudes of the geometric imperfections in the shape of third buckling mode, respectively (\(m3\), \(s3\) and \(d3\)); displacement of the node: (**a**) No. 1, (**b**) No. 2, (**c**) No. 3. Figure 9: Equilibrium path obtained for small, medium and large amplitudes of the geometric imperfections in the shape of second buckling mode, respectively (\(m2\), \(s2\) and \(d2\)); displacement of the node: (**a**) No. 1, (**b**) No. 2, (**c**) No. 3. Figure 11: Configuration of displacement before and after snap-through for initial amplitudes of geometrical imperfections developed according to the second buckling mode: (**a**) small (\(m2\)), (**b**) large (\(d2\)). Figure 12: Configuration of displacement before and after snap-through for initial amplitudes of geometrical imperfections developed according to the third buckling mode: (**a**) small (\(m3\)), (**b**) large (\(d3\)). Figure 5:0 Coefficient \(\rho\) for geometrical imperfections developed according to the second buckling mode: (**a**) small (\(m2\)), (**b**) large (\(d2\)). Next, analyses were carried out taking into account examples in which initial geometrical imperfections developed in the form of two different buckling modes. Figure 15 shows equilibrium paths obtained for geometrical imperfections developed according to the first low-amplitude buckling mode and second or third medium amplitude buckling mode, respectively (\(m1s2\) or \(m1s3\)). In turn, Figure 16 shows the equilibrium paths for geometrical imperfections corresponding with a large amplitude of the first buckling mode and medium amplitude of the second or third buckling mode (\(d1s2\) or \(d1s3\)). In all cases, two bifurcation points are observed: the first is associated with the first and the second with the second or third eigenvalue. It is observed that the introduction of geometrical imperfections developed according to two different buckling mode results in two or three snap-throughs associated with transition from the local to the global minimum of potential energy. The nature of a bifurcation point is defined by the second-order variation of potential energy and is observed to be stable before the first snap-through, and next, in case of a higher snap-through, unstable or metastable. A metastable bifurcation point is characterised by local minimum of potential energy and temporary equilibrium state. This phenomenon is illustrated by a ball moving on a differently shaped surface. The analysis of displacements before and after a snap-through shows that initial imperfection, which is a superposition of two different buckling modes, strongly influences the shape of the displacement vector. Figure 5:1 Coefficient \(\rho\) for geometrical imperfections developed according to the third buckling mode: (a) small (\(m3\)), (b) large (\(d3\)). However, it is still visible that at the first bifurcation point, the most influential is the first buckling mode, at the second bifurcation point--the second buckling mode, and at third bifurcation point--the third buckling mode (Figures 17 and 18).