## 5 Illustrative Examples
### Regarding the Cantilevered Structure  
To check the validity of the proposed model, numerical examples were carried out on the cantilevered structure in Figure 6. Table 1 showed the information about the first ten modes, including the values of the natural frequencies \(f_{i}\), obtained with the present model and ANSYS shell theory, the relative differences \(\delta_{i}\) and deformation mode participations \(\mu_{i}\). The results of the present model were obtained with forty elements, equally distributed along the beam length. For the ANSYS analysis, the structure was discretized into 1120 Shell 181 4-node elements, distributed as forty elements, along the length, and twenty-eight over the cross-section. The mode participation of each deformation mode was to be calculated by the following. \[\mu_{i}=\left\{ \begin{array}{l}\frac{\chi_{i}\varphi_{\text{max}}}{\sum\limits _{j=1}^{N_{1}}\chi_{j}\varphi_{\text{max}}}\times 100\%,i=1,\ 2,\cdots,N_{1}\\ \frac{\chi_{i}\omega_{i}}{\sum\limits_{j=N_{1}+1}^{N_{1}}\chi_{j}\omega_{\text{ max}}}\times 100\%,i=N_{1}+1,N_{1}+2,\cdots,r\end{array} \right. \tag{20}\]  
where the amplitudes were derived with the values of the free end cross-section, while the relative differences were calculated based on the assumption that the results of ANSYS shell theory were accurate enough. The results in Table 1 show that the natural frequencies obtained with the present model were very close to those from the ANSYS shell theory. They also proved that warping and distortion modes played important roles in dynamics and large errors might emerge if higher-order modes are ignored. For example, the 3 rd-10 th modal would not appear when applying conventional beam theories of [PERSON] and [PERSON]. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multirow{2}{*}{\begin{tabular}{c} **Present** \\ **Model** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **ANSYS** \\ **Shell** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Relative** \\ **Error** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Warping and Distortion Mode** **Paritications (\%)** \\ \end{tabular} } \\ \cline{2-11}  & & & & & & & & & & & & & \\ \hline
1 st & 85.201 & 81.477 & 4.57 & 0 & 0.67 & 0 & 0 & 14.8 & 0 & 0 & 0 \\
2 rd & 105.45 & 101.77 & 3.63 & 0 & 0 & 0.32 & 0 & 0 & 24.0 & 0 & 0 \\
3 rd & 123.40 & 126.34 & \(-\)2.33 & 100 & 0 & 0 & 0 & 0 & 0 & 93.9 & 0 \\
4 th & 193.97 & 190.51 & 1.82 & 0 & 0 & 0 & 59.7 & 0 & 0 & 100 \\
5 th & 201.95 & 202.28 & \(-\)0.16 & 0 & 0 & 0 & 85.1 & 0 & 0 & 0 & 100 \\
6 th & 217.06 & 225.57 & 3.77 & 0 & 0 & 73.7 & 0 & 0 & 100 \\
7 th & 237.98 & 240.80 & \(-\)1.17 & 0 & 3.52 & 0 & 0 & 83.5 & 0 & 0 & 0 \\
8 th & 263.42 & 260.68 & 1.05 & 0 & 0 & 0 & 65.2 & 0 & 0 & 100 \\
9 th & 283.98 & 279.35 & 1.66 & 0 & 8.71 & 0 & 0 & 95.0 & 0 & 0 \\
10 th & 302.90 & 307.93 & \(-\)1.63 & 0 & 0 & 0 & 67.9 & 0 & 0 & 0 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the first ten natural frequencies of the cantilevered thin-walled structure. Figure 10: Convergence of the first ten natural frequencies of the cantilevered thin-walled structure, varying with the number of employed finite elements. The idea could be verified by the results demonstrated in Figures 11 and 12, where non-null amplitudes of final deformation modes have been illustrated to vary along with the longitudinal axis of the structure. Simply put, conventional modes were dominant only in the first two among the first ten modal. It should also be noted that the wave forms were always different between the conventional and the higher-order deformation modes. The phenomenon indicates that in the process of combining like terms, it was unacceptable to superpose primary deformation modes, if they belonged to the rigid displacement and the elastic deformation of the cross-section, respectively. It also showed that warping and distortion were quite independent of conventional modes.