## 3 Identification of Deformation Modes
### Cross-Section Deformation Modes
#### Concrete Implementation
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**PDM \(i\)** & **Participation \(\xi_{i}\)** & **Correlation** & **Weight \(\lambda_{29i}\)** & **Residual** & **Correlation** & **Weight \(\lambda_{29i}\) & **Weight \(\lambda_{29i}\)** & **Residual** \\  & & **Coefficient \(r_{29i}\)** & **Participation \(\xi_{i}\)** & **Coefficient \(r_{29i}\)** & **Weight \(\lambda_{29i}\)** & **Participation \(\xi_{i}\)** \\ \hline
29 & 0.0453 & 1 & 1 & 1 & 0 & / & / & / \\
34 & 0.0147 & 0.9997 & 0.3270 & 0.0004 & / & / & / \\
37 & 0.0291 & 0.9959 & 0.6377 & 0.0059 & \(-0.9935\) & \(-2.274\) & 0.0016 \\
39 & 0.0291 & \(-0.9959\) & \(-0.6377\) & 0.0059 & 0.9935 & 2.274 & 0.0016 \\
41 & 0.1547 & 0.9995 & 3.393 & 0.0002 & / & / & / \\
13 & 0.0024 & / & / & 0.0024 & 1 & 1 & 0 \\
14 & 0.0024 & / & / & 0.0024 & \(-1.000\) & \(-1.000\) & 0 \\
17 & 0.0026 & / & / & 0.0026 & \(-1.000\) & \(-1.000\) & 0 \\
18 & 0.0026 & / & / & 0.0026 & 1.000 & 1.000 & 0 \\
33 & 0.0010 & / & / & 0.0010 & \(-0.9062\) & \(-0.4038\) & 0.0002 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The participations, correlation coefficients and weights of the in-plane preliminary deformation modes for the first eigenvector of the illustrative thin-walled structure. Figure 5: The assembling of in-plane preliminary deformation modes for the first eigenvector of the illustrative thin-walled structure: **(a)** A classic mode and **(b)** a distortion mode. participated in the eigen matrix, that is, the scenarios of DAFC 2. For example, in handling the third eigen matrix, the correlation coefficients were calculated as \(\gamma_{25,13}=-\gamma_{25,14}=-\gamma_{25,15}=\gamma_{25,16}=0.9908\) and \(\gamma_{25,26}=-\gamma_{25,27}=-\gamma_{25,28}=1\). The results support \(\chi_{13}(^{3})=-\chi_{14}(^{3})=-\chi_{15}(^{3})=\chi_{16}(^{3})\) and \(\chi_{25}(^{3})=\chi_{26}(^{3})=-\chi_{27}(^{3})=-\chi_{28}(^{3})\). Therefore, only \(\chi_{13}(^{3})\) and \(\chi_{26}(^{3})\) were handled, and the results were applicable to the other PDM. According to the signs of the benchmark points, the classic mode was involved in the rotation of about \(z\). Thus, in Equation (19), the coefficient can be determined as \(\mu_{1}=-4/3\). Applying DAFC 2, both \(\chi_{13}(^{3})\) and \(\chi_{26}(^{3})\) were decomposed into two components, respectively. Figure 6 shows the results, where the components \(\chi_{13,1}\) and \(\chi_{25,1}\) represent the longitudinal variation of the rotation mode, while \(\chi_{13,1}\) and \(\chi_{25,1}\) correspond to a primary modes. Thus, all the other EAFV were decomposed into two components using DAFC 2, and their weights were calculated through AWDM. The above procedure was carried out on the first several eigen matrices one by one until the number of identified deformation modes satisfied the requirements. Obviously, the number of the identified deformation modes is unlikely to exceed the number of PDM. Besides, the procedure can be implemented with the help of a computer in a more efficient way. Because the deformation modes are assembled in the light of eigenvectors, their amplitudes should be normalized after the identification. The normalized criteria were simply expressed as rendering the maximum value of each deformation mode to be 1.