## 4 Sectional Deformation Modes  
The basis deformation modes above were established on the basis of purely kinematic concepts. Though they could serve as basis functions in Equation (2), the problem lied in that their large amount of deformation modes made for a non-reduced model and that the physical interpretation was not intuitive enough to couple with the beam structural behaviors. A set of final deformation modes, which exhibited well-defined structural significance and ranked hierarchically, according to their participations, was very important for a one-dimensional higher-order model. In this section,  
Figure 3: Basis deformation modes of a thin-walled, rectangular, hollow cross-section with eight discretization nodes. a procedure to assemble basis deformation modes was developed, similar to the concept of combining like terms in algebra. This procedure was based on the assumption of considering the cross-section deformation in dynamic systems as the supposition of rigid displacement and elastic deformation of the cross-section, whose amplitudes vary along the longitudinal axis, with a fixed proportion relationship. In this sense, the participation of each basis deformation mode in a final sectional deformation mode can be determined with the data extracted from the modal shapes of certain cross-sections which were derived from the presented model. For convenience, the object cross-section always chose the free end of a cantilevered thin-walled structure, in free vibration. Its slenderness ratio needed to be kept between three and six, since thin-walled structures (Figure 4), in this range, are prone to perform more obvious cross-sectional deformation (see [PERSON] et al. [30]). Take the structure in Figure 4 as an example to demonstrate the process. Substituting the set of basis functions shown in Figure 3 into the presented model and employing the boundary conditions in Figure 4, the dynamic model of the thin-walled structure was established for modal analysis. The model was solved in MATLAB with the first few modal shapes extracted and participations of each basis deformation mode obtained for subsequent use. It should be noted that the number of modal shapes to be studied was related to the required number of final deformation modes, which largely influenced the accuracy of the proposed model. In most cases, four to eight modal shapes were essential and might have been enough for thin-walled structures with simple cross-sections. Figure 5 shows the deformed cross-section of the free end for the first 4 modals, where the blue curves indicate an undeformed cross-section and the red ones describe a deformed cross-section. The \(x\) and \(y\) axes spanned the cross-section plane, while \(z\) axis dedicated the cross-section deformation along the longitudinal direction. For the ease of implementation, basis deformation modes were separated into in-plane and out-of-plane ones, which were to be extracted from the four modal shapes illustrated in Figure 5. In fact, the procedure could be completed to solve the free vibration system, by applying the proposed finite element, since its feature of variable separation, and the amplitudes of each basis deformation mode participating in the modal shapes were obtained simultaneously. Figure 6 demonstrated the amplitudes of basis deformation modes for the first modal of the cantilevered structure. In Figure 6, the eight amplitudes on the left, corresponded to the out-of-plane mode family, while the twenty-four on the right represented the participations of the in-plane mode family (in accordance with the sequence shown in Figure 3). Here the sign of amplitude value stood for whether the contributing deformation was identical with the predicted positive direction shown in Figure 3. Among each family, a process similar to combining like terms was carried out to linearly superpose basis deformation modes, with equal or opposite amplitudes.