## 3 Identification of Deformation Modes
### Cross-Section Deformation Modes
#### Concrete Implementation
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**PDM \(i\)**} & \multirow{2}{*}{**Participation \(\xi_{i}\)**} & **Correlation** & \multirow{2}{*}{**Weight \(\lambda_{i,j}\)**} & **Residual** & **Correlation** & \multirow{2}{*}{**Weight \(\lambda_{i,j}\)**} & **Residual** \\  & & & & & & **Participation \(\xi_{i}\)** & **Coefficient \(\tau_{N,j}\)** & **Participation \(\xi_{i}\)** \\ \hline
1 & 0.1085 & 1 & 1 & 0 & / & / & 0 \\
2 & 0.1085 & \(-1.000\) & \(-1.000\) & 0 & / & / & 0 \\
3 & 0.1085 & \(-1.000\) & \(-1.000\) & 0 & / & / & 0 \\
4 & 0.1085 & 1.000 & 1.000 & 0 & / & / & 0 \\
5 & 0.0022 & / & / & 0.0022 & 1 & 1 & 0 \\
6 & 0.0022 & / & / & 0.0022 & 1.000 & 1.000 & 0 \\
7 & 0.0022 & / & / & 0.0022 & \(-1.000\) & \(-1.000\) & 0 \\
8 & 0.0022 & / & / & 0.0022 & \(-1.000\) & \(-1.000\) & 0 \\
11 & 0.0017 & / & / & 0.0017 & 0.5874 & 0.7181 & 0.0004 \\
12 & 0.0017 & / & / & 0.0017 & \(-0.5874\) & \(-0.7181\) & 0.0004 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The participations, correlation coefficients and weights of the out-of-plane preliminary deformation modes for the first eigenvector of the illustrative thin-walled structure. Figure 4: The assembling of the out-of-plane preliminary deformation modes by handling the first eigenvector of the illustrative thin-walled structure: (**a**) a classic mode and (**b**) a warping mode. number of amplitude function vectors to be processed. Table 2 shows the results, where the retained EAFV are separated into two teams. Within the first team, \(\chi_{29}\)(1) was selected as the standard reference, and the others were decomposed into components using DAFC 1. As shown in the fifth column, the residual participations of \(\chi_{37}\)(1) and \(\chi_{39}\)(1) still exceeded 0.001. In this case, they were absorbed into the second team for further analysis. Within the second team, \(\chi_{13}\)(1) was selected as the standard reference. Subsequently, the other amplitude function vectors were decomposed and assigned weights using DAFC 1 and AWDM, respectively. The assembling result is shown in Figure 5.  
It can be seen that the residual participations of \(\chi_{37}\)(1) and \(\chi_{39}\)(1) still exceeded 0.001 but was not used for further analysis. This was because the residual component was too weak to provide the information of a structural behaviour. In practice, up to three new deformation modes can be identified from an eigen matrix. Actually, deformation modes related to the residual components are deemed to play more important roles in other eigenvectors with higher orders. Notice that the amplitude function vectors of Dominant family may not lead to classic mode, but instead may lead to Primary mode in some eigenvectors. For example, the correlation coefficients were \(-\gamma_{1,2}=\gamma_{1,3}=-\gamma_{1,4}=1\) in the third eigen matrix, which indicated a primary warping mode. In some cases, the participations of the Dominant family may be lower than the threshold 0.001. For example, there were no EAFV among the Dominant family within the fourth eigen matrix. Then, the in-plane PDM were divided into teams to assemble distortion modes, separately. Moreover, it can be seen that mutual correlation coefficients of amplitude functions belonging to the Dominant family were not 1 or \(-1\), so information from the two classic modes (or one classic and one primary mode)