## 3 Numerical Examples
### Numerical Example 3  
Shown in Figure 6, the cross-section of Example 3 adopts a T-shaped geometry with additional flanges, aimed at assessing the structural integrity and vibration properties under various loading conditions. This example focuses on practical applications such as structural joints and overhangs, demonstrating the utility of the model in predicting the dynamic responses of beams in complex framing and connection scenarios. Applying an axial load of P = 0 kN, we computed the six natural frequencies for the simply supported thin-walled beam in modes \(n=1\), 2, 3, as shown in Table 5. Figure 7 demonstrates that this example exhibits behavior analogous to Examples 1 and 2. Natural \begin{table}
\begin{tabular}{l c c c} \hline \hline
**L (m)** & \(n=1\) & \(n=2\) & \(n=3\) \\ \hline
3 & 3087.0 kN & 12,269.4 kN & 27,522.9 kN \\
10 & 299.2 kN & 1126.6 kN & 2505.1 kN \\
50 & 34.2 kN & 67.3 kN & 122.5 kN \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Axial load at a natural frequency of zero for the different beam lengths and mode numbers of numerical example 2. Figure 6: Cross-section for numerical example 3. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**External Mode** & \(L=3\) **m** & \(L=10\) **m** & \(L=50\) **m** \\ \hline \(n=1\) & 121.81 & 11.38 & 0.77 \\  & 798.36 & 92.60 & 3.84 \\  & 1180.57 & 141.80 & 5.90 \\  & 3206.05 & 1609.39 & 321.88 \\  & 5364.62 & 2487.69 & 2397.82 \\  & 5867.06 & 4392.26 & 4212.44 \\ \hline \(n=2\) & 485.29 & 44.16 & 2.16 \\  & 2200.81 & 336.88 & 15.28 \\  & 3085.72 & 509.32 & 23.51 \\  & 4651.50 & 2735.56 & 643.75 \\  & 8983.38 & 3218.77 & 2409.41 \\  & 10,729.24 & 4893.86 & 4235.57 \\ \hline \(n=3\) & 1089.47 & 98.76 & 4.37 \\  & 3701.60 & 673.49 & 34.14 \\  & 4996.54 & 1001.67 & 52.45 \\  & 6222.42 & 3078.47 & 965.63 \\  & 12,488.01 & 4828.16 & 2428.49 \\  & 16,093.86 & 5600.63 & 4273.68 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Natural frequencies (rad/s) of Timoshenko–Vlasov beam studied as numerical example 2.  
frequencies progressively decrease with an increasing axial load, eventually reaching zero at the elastic buckling threshold. The critical axial load values corresponding to the onset of buckling are provided in Table 6. Notably, the critical axial load decreases with longer beam lengths while increasing with higher mode numbers, once again reflecting the same phenomena observed in the previous two examples. The calculation results in Appendix E indicate that, compared with the traditional theory, the error decreases with an increasing beam length in the Timoshenko-Vlasov model. However, a trend of increasing error is observed at \(L=3\) m. Furthermore, the results presented in Table 6 show that, for \(L=50\) m, the axial load causing buckling in the Timoshenko-Vlasov beam remains nearly constant across different modes. First, the Timoshenko-Vlasov theory accounts for shear deformation and warping effects, which become particularly significant in thin-walled or non-standard beams. These effects, often neglected in the traditional classical stability theory of columns, reduce the sensitivity of the critical buckling load to higher modes by increasing the stiffness of the beam in certain directions. Consequently, the load required for buckling initiation does not increase significantly with the mode number, resulting in a nearly constant axial load across modes. Additionally, unlike traditional beam theories that assume decoupled behavior between bending, torsion, and axial loads, the Timoshenko-Vlasov model captures the interactions between these forces. In long beams or beams with complex geometries (such as the T-shaped cross-section in this example), the coupling of torsional and bending deformations  
Figure 7: The impact of axial load variations on the lowest natural vibration frequency (rad/s) of numerical example 3: (**a**) \(L=3\), (**b**) \(L=10\), and (**c**) \(L=50\). with axial loads can alter the relationship between the load and mode number.