## 4 Illustrative Examples
### Beam Analysis
#### 4.2.1 Concerning the Accuracy and Efficiency  
In order to verify the accuracy and efficiency of the proposed models, comparisons regarding the natural frequencies were carried out with ANSYS shell theory. Selectively, several sets of cross-section deformation modes were constructed considering the hierarchy and used to derive the corresponding higher order models. One set includes all the 48 PDM and yielded the proposed beam 1; another set has six Classic modes and results in the classic beam; the third set has six Classic modes and three Primary modes, corresponding to the proposed beam 2; the fourth set has all the modes of set 3 and two Secondary modes VI and VII included, reaching the proposed beam 3; the last set added three Secondary modes v, vi and vii on the basis of set 4, and corresponds to the proposed beam 4. Table 3 shows the results, where the relative errors are calculated based on the assumption that the values derived with ANSYS shell model are exact enough. Table 3 shows a clear trend of gradual improvement on the accuracy regarding the natural frequencies as the number of employed cross-section deformation modes increases. It is also noticeable that the proposed beam model applying 14 cross-section deformation modes had a good agreement with the numerical results obtained from ANSYS shell model regarding the first 10 natural frequencies, while the classic beam model was not competent to perform the dynamic analysis, even for the first  
Figure 11: The first four warping modes of the single-cell rectangular cross-section derived by [PERSON] et al. [37]: (**a**) Shear-lag mode 1, (**b**) shear-lag mode 2, (**c**) shear-lag mode 3 and (**d**) a higher order mode. two natural frequencies. Meanwhile, the results support that it is essential to consider cross-section deformation modes in the analysis of thin-walled structures. It can also be seen that the set of deformation modes, including only Classic and Primary modes, were able to form a higher order model to catch the first eight natural frequencies (the sixth not included) of a thin-walled structure with relative errors lower than 4.90%. The ability can be improved to calculate the first 10 natural frequencies with relative errors lower than 4.15% when eight Secondary modes are added to the proposed beam model. These results specify the accuracy of the proposed model, as well as the rational hierarchy of the proposed numbering system. One can also have an insight of the hierarchy of the identified cross-section deformation modes through their participation in the eigenvectors of the thin-walled structure. Figure 12 shows their amplitude functions varying along the beam axis concerning the first four eigenvectors. A clear distinction can be observed among the participations of Classic, Primary and Secondary modes. Generally, the participation of Classic mode is dozens of times of that of Primary mode, and so is that of Primary mode and Secondary mode. Just because of this character, a rational hierarchy becomes possible in organizing identified cross-section deformation modes. And because of the hierarchy, one is able to select several cross-section deformation modes with high priority to develop a reduced beam model based on actual demands. This effectively contributes to the improvement of efficiency in computation, since fewer degrees of freedom should be dealt with in the thin-walled model without a significant loss of accuracy.