## 4 Illustrative Examples
### Cross-Section Analysis
#### 4.1.1 An I-Shaped Cross-Section  
An I-shaped cross-section with material parameters of \(E=200\) GPa, \(\
u=0.3\) and \(\rho=7850\) kg/m\({}^{3}\) was considered to exhibit the identification of cross-section deformation modes. The material was linear, elastic and isotropic. This example also shed light on the versatility of the proposed approach in  
Figure 7: The numbering of deformation modes identified for the illustrative thin-walled structure.  
coping with open cross-sections. The cross-section had a web with dimensions 80 mm \(\times\) 2 mm and two flanges with dimensions 75 mm \(\times\) 2 mm. A cantilevered beam, with cross-section and a length \(L\) = 450 mm, was modelled using the proposed theory. The cross-section discretization was performed with six natural nodes and one artificial node on the middle point of the web. Figure 8 displays the output of the identification by handling the first four eigenvectors of the thin-walled structure using the proposed approach. These deformation modes were classified into three families: Modes i, ii, iii, I, II and III belonged to the Classic family, while modes iv and IV belonged to the Primary family and the others belonged to the Secondary family.  
The results were compared with the first 20 GBT deformation modes [38]. As shown in Figure 9, the proposed approach captured most of the GBT modes by dealing with just the first four eigenvectors of the thin-walled structure. Actually, these cross-section deformation modes are qualified to reproduce the 3D deformation of the first 10 vibration modes of the thin-walled structure, which covered most engineering cases. When the other deformation modes with lower priority are needed, all one needs to do is handle more eigenvectors. The quality is conductive to obtain a selective set of deformation modes according to the actual needs of the structural analysis, instead of calculating all of them from the very beginning. Thus, the proposed approach is convenient and efficient in adapting different demands.  
Figure 8: The cross-section deformation modes identified from the first four eigenvectors of the illustrative thin-walled structure applying the proposed approach.  
Figure 9: The generalized beam theory (GBT) deformation modes of the illustrative thin-walled structure derived with GBTUL 2.0.  
There were also some differences between the two sets of cross-section deformation modes. For example, the in-plane and out-of-plane modes were tied together to describe the structural behaviours related to flexure and rotation in GBT, while they were identified separately in the proposed approach. However, the difference will not influence the accuracy of the thin-walled model, since the modes 2, 3, 4, 10, 11 and 12 tied together in Figure 9 were equal to the linear combination of modes i, ii, iv, I, II and III, respectively. Besides, the pattern to exhibit the structural interpretation of deformation modes was not quite the same. For example, modes 7 and 8 were separately given in Figure 9, while the two were assembled with weights 1 and -1 in Figure 8. It is predictable that another mode assembled with weights 1 and 1 can be identified by handling higher order eigenvectors. This difference was independent of the model accuracy, but the modes shown in Figure 8 were closer to the actual structural behaviours.  
Overall, the proposed approach can capture the cross-section deformation modes of an open cross-section with a convenient and efficient way. The derived deformation modes were generally consistent with GBT, though there were some differences that independent of the model accuracy.