## 4 Sectional Deformation Modes
For example, among the out-of-plane family with non-null amplitudes in Figure 6, modes 1-4 were like terms (with equal or opposite amplitude values), and so were modes 5 and 7, which were to be assembled together  
Figure 4: A cantilevered thin-walled structure with geometry and material parameters as shown for modal analyses. with their amplitudes, as corresponding weights. The assembled modes are designated as primary deformation modes. As shown in Figure 7, two primary deformation modes were obtained through the process. Obviously, the former represented a rigid rotation of the whole cross-section about \(y\)-axis. It is a conventional deformation mode in the beam theory of [PERSON], and here it was numbered as sectional deformation mode I. The other mode was left alone and was numbered as sectional deformation mode V, which represents the first warping deformation (out-of-plane distortion caused by uneven longitudinal extensions, see [PERSON] et al. [31]) of the cross-section. The case confirmed that a modal shape of a thin-walled structure might be a mixture of rigid displacement and elastic deformation of the cross-section. Figure 5: Deformed cross-sections of the free end for the first four modal shapes of the thin-walled structure with a rectangular hollow cross-section: (**a**) The 1 st modal, (**b**) the 2 rd modal, (**c**) the 3 rd modal, and (**d**) the 4 th modal. Figure 6: Amplitudes of basis deformation modes participating in the first modal of the free end cross-section of a cantilevered thin-walled structure. It should be pointed out that not all primary sectional deformation modes could be identified as final deformation mode. For example, among in-plane mode family demonstrated in Figure 8, modes 1-4 were assembled as a conventional deformation mode representing a rigid translation of the cross-section, being numbered as sectional deformation mode VIII, but not for the rest of the primary deformation modes i, ii and iii (named as secondary deformation modes describing elastic deformation profile of the cross-section). These secondary modes were of no clear physical significance and might not be efficient for a reduced model. For this reason, the three secondary modes were assembled again to form a final distortion mode. This new mode, like the first distortion mode, was associated with rigid translation of the cross-section, along the \(x\)-axis and could also be observed in actual experiments. In this sense, it was endued with structural meaning, and numbered as sectional deformation mode XI. Figure 8: Assemblage of in-plane mode family for the first modal of a thin-walled structure with a rectangular hollow cross-section. Figure 7: Assemblage of out-of-plane mode family for the first modal of a thin-walled structure with a rectangular hollow cross-section. In this way, a total of 14 final sectional deformation modes were identified through the \"combining like terms\" operation carried out on the first four modal shapes (as shown in Figure 9). Among these modes, modes I, II, III, VIII, IX, and X corresponded to the conventional rigid displacements of the whole cross-section. The other eight modes were newly defined to describe elastic deformation of the cross-section with a minimum total. In general, these new deformation modes were structurally meaningful and linearly independent. Additionally, their arrangements were in accordance with the sequence they appeared in, which showed the capability of a hierarchy. The proposed simplified approach, although involved with cross-section analysis, was different from both the theory of [PERSON] et al. [28] for obtaining GBT deformation modes or that of [PERSON] et al. [29] for formulating the higher order model. Both of these theories have to solve the generalized eigenvalue problem to uncouple the governing differential equations or the basis functions, which is quite demanding. In fact, the authors focused on avoiding this in the process of identifying sectional deformation modes.