# Post-Buckling Behaviour of Steel Structures with Different Types of Imperfections
## 3 Nonlinear Model of the Structure with a Complex Geometrical Imperfection Pattern
### Model with Initial Geometrical Imperfections  
In the proposed model, initial geometrical imperfections are introduced in the form of initial nodal displacements \(u_{n}^{i}\) which induce initial rotations of bars \(\varphi_{n}^{i}\) and respective node rotations \(\phi_{n}^{i}\) (superscript \"\(i\)\" denote imperfections). These deformations are kinematically admissible. Therefore, they do not induce any stress. As was mentioned before, elastic deformations appear when compression force \(P\) is active. It induces moments in elastic nodes \(M_{n}=k_{n}\cdot\phi_{n}^{e}\).  
In order to determine geometrical relationships in the model with initial geometrical imperfections, a representative segment was analysed too (Figure 4).  
Next, geometrical relations are considered in two stages. First, the initial state when compression force \(P\) is not active (\(P=0\)), the initial bar and hinge rotations, being initial geometrical imperfections, can be described as follows:  
\[\varphi_{n}^{i}=asin\frac{u_{n}^{i}-u_{n-1}^{i}}{l_{n}}\text{ and }\varphi_{n+1}=asin \frac{u_{n+1}^{i}-u_{n}^{i}}{l_{n+1}} \tag{9}\]  
\[\phi_{n}^{i}=asin\frac{u_{n}^{i}-u_{n-1}^{i}}{l_{n}}-asin\frac{u_{n+1}^{i}-u_{ n}^{i}}{l_{n+1}} \tag{10}\]  
In the second state, when compression force \(P\) is active (\(P>0\)), displacements are the sum of the initial and elastic part and take the form:  
\[u_{n}=u_{n}^{i}+u_{n}^{e}\text{ }and\text{ }\varphi_{n}=\varphi_{n}^{i}+ \varphi_{n}^{e} \tag{11}\]  
Thus, the total hinge rotation can be expressed as:  
\[\begin{array}{ll}\phi_{n}=\phi_{n}^{i}+\phi_{n}^{e}&=asin\frac{u_{n}^{i}-u_{ n-1}^{i}}{l_{n}}-asin\frac{u_{n+1}^{i}-u_{n}^{i}}{l_{n+1}}\\ &+asin\frac{u_{n}^{e}-u_{n-1}^{e}}{l_{n}}-asin\frac{u_{n+1}^{e}-u_{n}^{e}}{l_{ n+1}}\\ &=asin\frac{u_{n}-u_{n-1}}{l_{n}}-asin\frac{u_{n+1}-u_{n}}{l_{n+1}}\end{array} \tag{12}\]  
Now, the set of non-linear equilibrium equations take the form:  
\[k_{n}\cdot\phi_{n}^{e}=P\cdot\left(u_{n}^{i}+u_{n}^{e}\right)=P\cdot u_{n} \tag{13}\]  
For \(n=1\), \(2\), \(3\) the form Formula (13) takes the form of three nonlinear algebraic equations of equilibrium:  
Figure 4: Model structure with initial imperfections.  
\[\begin{cases}k_{1}\big{[}asin\frac{u_{1}^{e}}{L}-as\frac{u_{2}^{e}-u_{1}^{e}}{L} \big{]}=P\big{(}u_{1}^{i}+u_{1}^{e}\big{)}=Pu_{1}\\ k_{2}\big{[}asin\frac{u_{2}^{e}-u_{1}^{e}}{L}-as\frac{u_{2}^{e}-u_{2}^{e}}{L} \big{]}=P\big{(}u_{2}^{i}+u_{2}^{e}\big{)}=Pu_{2}\\ k_{3}\big{[}asin\frac{u_{3}^{e}-u_{3}^{e}}{L}-as\frac{-u_{3}^{e}}{L}\big{]}=P \big{(}u_{3}^{i}+u_{3}^{e}\big{)}=Pu_{3}\end{cases} \tag{14}\]  
where \(u_{n}^{e}\) is unknown, and \(u_{n}^{i}\) is the linear combination of eigenvectors obtained as a result of the solution of a linear eigenvalue problem (7).  
In order to investigate various configurations of imperfections, mode participation factors \(\alpha_{m}\) are introduced. Thus, initial node displacements can be computed as follows:  
\[u_{n}^{i}=\alpha_{m}\cdot u_{m}^{e} \tag{15}\]  
Changing the values of mode participation factor \(\alpha_{m}\), it is possible to obtain any combination of imperfections.