## 3 Identification of Deformation Modes
### Cross-Section Deformation Modes
#### Concrete Implementation  
For a better implementation, the amplitude function vectors of each eigen matrix were divided into two groups (representing the out-of-plane and in-plane PDM, respectively) and separately handled using the basic algorithms. Besides, the order of the eigen matrix and the type of PDM were considered in planning the sequence: The eigen matrix of lower order has priority than those of higher order, and within each eigen matrix, amplitude functions are dealt with in the sequence of Dominant family, Integral family and Selective family. For a better grasp of the procedure, a cantilevered thin-walled structure with an illustrative cross-section and a beam length of \(L=6.4\) m was modelled to illustrate the concrete implementation. The procedure first focused on the out-of-plane PDM of the first eigen matrix. By checking the participation, it was discovered that only 10 of the 12 out-of-plane PDM had participations exceeding 0.001 (as show in Table 1). The corresponding amplitude function vectors were recognized as EAFV for further analysis. Next, the correlation coefficients were calculated, and the results support that can be selected as the standard reference since it has the biggest participation among the amplitude functions of the Dominant family. As shown in the third column of Table 1, the correlation coefficients \(|\gamma_{1,i}|=1.0000\) (\(i=1,2,3,4\)) indicate that amplitude functions of these PDM were collinear. According to the signs of the benchmark points, it can be determined that the related classic mode was the bending about \(x\) axis, since \(\gamma_{1,2}=\) -1, \(\gamma_{1,3}=\) -1 and \(\gamma_{1,4}=1\). Thus, they were assembled to derive the first classic mode by employing the weights (the fourth column of Table 1) calculated with AWDM. The result is shown in Figure 4a. Then, the implementation goes on with the other six EAFV. Among them, \(\chi_{5}{}^{(1)}\) was selected as the standard reference since it had the biggest participation (see the fifth column of Table 1). The calculation of correlation coefficients indicated that not all of them were collinear, which gave top priority to the Integral family. Thus, DAFC 1 was applicable for the further decomposition. Then, \(\chi_{10}{}^{(1)}\) and \(\chi_{12}{}^{(1)}\) were decomposed into two components, respectively. The first component was collinear with \(\chi_{5}{}^{(1)}\), and the other component should be ignored since the residual participation was below 0.001. The weights of the first components were derived and shown in the seventh column. Consequently, there was no EAFV left. The assembling of the new deformation mode is shown in Figure 4b. The assembling of the in-plane PDM needed to handle more data since more amplitude functions were involved. It was essential to determine the EAFV by checking the participations. Then, EAFV were classified into teams where the absolute correlation coefficients were bigger than \(\gamma_{1}=0.9\). Among each team, the standard reference was selected in the light of the biggest participation. Then, the correlation coefficients were calculated to carry out DCAF, which states \(\gamma_{25,26}=\gamma_{25,27}=\gamma_{25,28}=1\), which indicates the Classic mode of translation along the \(y\) axis, according to the signs of benchmark points. The weights were obtained through AWDM. After screening out these dominant modes, the left 24 EAFV were divided into two teams: \(\chi_{29}{}^{(1)}\sim\chi_{32}{}^{(1)}\), \(\chi_{34}{}^{(1)}\), and \(\chi_{36}{}^{(1)}\sim\chi_{44}{}^{(1)}\) as the first team, and \(\chi_{13}{}^{(1)}\sim\chi_{20}{}^{(1)}\), \(\chi_{33}{}^{(1)}\) and \(\chi_{35}{}^{(1)}\) as the other team. Besides, one should note that some amplitude function vectors are exactly the same in mathematics. This property contributes to further reducing the