# Post-Buckling Behaviour of Steel Structures with Different Types of Imperfections
## 4 Numerical Examples  
From among many cases studied by the author, the examples shown in Figure 1 are considered. This model consists of four and three elastic nodes. In this example, the length of the bar was assumed \(L=1\), and compression load \(P=1\). Numerical examples are solved for various amplitudes of initial imperfections \(u_{n}^{i}\) (Equation (15)) introduced as a linear combination of eigenvectors \(u_{m}^{e}\) obtained from the linear eigenvalue problem (Equation (7)) scaled by buckling mode proportionality factor \(\alpha_{m}\). Keeping in mind that the eigenvector is normalised, thus the displacements are not real, the buckling mode proportionality factor should be assumed so as to reflect real imperfections. This can be achieved by the measurement of imperfection in situ, based on tolerances given in codes or assumed as arbitrary amplitudes. In this example, equilibrium paths in the model considered are analysed, taking into account different configurations of geometrical imperfections. The increasing values of imperfections associated with three successive forms of the eigenvector are considered. The amplitudes of the mode proportionality factor \(\alpha_{m}\), used to create different configurations of the initial vector of geometrical imperfections are presented in Table 1.  
Moreover, based on the analysis of potential energy, the nature of equilibrium states is described. It should be noted that the structure is in equilibrium when potential energy  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No. **} & \multicolumn{2}{c}{**Symbol of the**} & \multicolumn{2}{c}{**Amplitudes of the Buckling Mode Proportionality**} \\  & **Imperfection Pattern** & \(\mathbf{\alpha_{1}}\) & \(\mathbf{\alpha_{2}}\) & \(\mathbf{\alpha_{3}}\) \\ \hline
**1. ** & \(m1\) & 0.001 & 0 & 0 \\
**2. ** & \(m2\) & 0 & 0.001 & 0 \\
**3. ** & \(m3\) & 0 & 0 & 0.001 \\
**4. ** & \(s1\) & 0.045 & - & - \\  
reaches a minimum. The necessary condition for the occurrence of minimum potential energy is the disappearance of its first variation. This condition can be written as:  
\[\delta\Pi=0 \tag{16}\]  
In the case of the example considered, potential energy can be described according to the formula:  
\[ \begin{array}{l}\Pi=\frac{1}{2}\big{(}k_{1}\cdot\big{(}\varphi_{1}^{e}\big{)}^ {2}+k_{2}\cdot\big{(}\varphi_{2}^{e}\big{)}^{2}+k_{3}\cdot\big{(}\varphi_{3}^{ e}\big{)}^{2}\big{)}\\ -\lambda PL(cos\big{(}\varphi_{1}^{i}\big{)}+cos\big{(}\varphi_{2}^{i}\big{)}+ cos\big{(}\varphi_{3}^{i}\big{)}-cos\big{(}\varphi_{1}^{e}\big{)}+cos( \varphi_{2}^{e})+cos\big{(}\varphi_{3}^{e}\big{)})\end{array}  \tag{17}\]  
where \(\lambda\) is a load multiplier. The nature of the equilibrium path is determined by higher order variations of potential energy. The type of bifurcation point can be determined based on the analysis of the second variation of potential energy as stable, neutral or unstable, according to:  
\[\delta^{2}\Pi=\left\{ \begin{array}{l}>0\;\;stable\\ =0\;neutral\\ <0\;unstable\end{array} \right. \tag{18}\]  
Numerical examples are described by symbols composed of small letters of the Latin alphabet and numbers. The letter means the amplitude of the adopted buckling mode proportionality factor (m--small, s--medium, d--large), and the number--the form of a buckling mode, according to which the imperfection vector is determined, shown in Figure 3. Thus, symbol m1 denotes an example in which imperfections are introduced according to the shape of the first buckling mode with a small amplitude \(\alpha_{1}=0.001\). Figure 5 presents the equilibrium paths obtained for the structure with small amplitudes of initial geometrical imperfections developed in the shape of the first, second and third buckling modes, respectively (\(m1\), \(m2\) and \(m3\)). It is worth noting that in all three cases, a stable equilibrium path is observed, where there are points of bifurcation and the accompanying unstable equilibrium path.