## 4 Applications and Discussions  
### Elastic Buckling Analysis of Shear Deformable Columns  
In this section, classical application examples of a Timoshenko column with typical element-end boundary conditions are presented, and the results are listed in Table 1.  
For a pinned-pinned Timoshenko column (Case 1), the unrestrained element-end deformations are the element-end rotations. Therefore, the buckling state is corresponding to the condition that the determinant of the submatrix for the element-end rotations (second and fourth row/column of the element stiffness matrix [\(\mathbf{K_{\rm TB}}\)], which is the structural stiffness matrix in this case) equals zero, and the buckling condition in this case can be solved as  
\[\text{det}\left[ \begin{array}{cc}S_{\rm TB}&C_{\rm TB}\\ C_{\rm TB}&S_{\rm TB}\end{array} \right]=0\Leftrightarrow S_{\rm TB}-C_{\rm TB }=0\Rightarrow\lambda_{\rm cr}=\pi\Rightarrow P_{\rm cr}=\frac{\pi^{2}EI/L^{ 2}}{1+\pi^{2}\varphi/12}. \tag{16}\]  
For a fixed-free Timoshenko column (Case 2), the unrestrained element-end deformations are the translation and rotation at the free end. Therefore, the buckling state is corresponding to the condition that the determinant of the stiffness matrix at the free end (first two or last two row/column of the element stiffness matrix [\(\mathbf{K_{\rm TB}}\)], which is the structural stiffness matrix in this case) equals zero, and the buckling condition in this case can be solved as  
\[\text{det}\left[ \begin{array}{cc}T_{\rm TB}&-Q_{\rm TB}\\ -Q_{\rm TB}&S_{\rm TB}\end{array} \right]=0\Leftrightarrow T_{\rm TB}S_{\rm TB}-Q _{\rm TB}{}^{2}=0\Rightarrow\lambda_{\rm cr}=\pi/2\Rightarrow P_{\rm cr}= \frac{\pi^{2}EI/4L^{2}}{1+\pi^{2}\varphi/48}. \tag{17}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case**} & **Boundary** & **Conditions for** & **Axial Force Factor at** & **Buckling Axial** \\  & **Conditions** & **Buckling** & **Buckling State (\(\mathbf{\lambda_{\rm cr}}\))** & **Force (\(\mathbf{P_{\rm cr}}\))** \\ \hline
1 & Pinned–pinned & \(S_{\rm TB}-C_{\rm TB}=0\) & \(\lambda_{\rm cr}=\pi\) & \(P_{\rm cr}=\frac{\pi^{2}EI/L^{2}}{1+\pi^{2}\varphi/12}\) \\
2 & Fixed–free & \(T_{\rm TB}S_{\rm TB}-Q_{\rm TB}{}^{2}=0\) & \(\lambda_{\rm cr}=\pi/2\) & \(P_{\rm cr}=\frac{\pi^{2}EI/L^{2}}{1+\pi^{2}\varphi/48}\) \\
3 & Fixed–clamped & \(T_{\rm TB}=0\) & \(\lambda_{\rm cr}=\pi\) & \(P_{\rm cr}=\frac{\pi^{2}EI/L^{2}}{1+\pi^{2}\varphi/12}\) \\
4 & Fixed–pinned & \(S_{\rm TB}=0\) & \(\tan\lambda_{\rm cr}=\chi\lambda_{\rm cr}=\frac{\lambda_{\rm cr}}{1+\lambda_{ \rm cr}\varphi/12}\) & \(P_{\rm cr}=\frac{\lambda_{\rm cr}^{2}EI/L^{2}}{1+\lambda_{\rm cr}\varphi/12}\) \\
5 & Fixed–fixed & \(\phi_{\rm TB}=0\) & \(\lambda_{\rm cr}=2\pi\) & \(P_{\rm cr}=\frac{4\pi^{2}EI/L^{2}}{1+\pi^{2}\varphi/3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Elastic buckling analysis of a Timoshenko column with typical boundary conditions. For a fixed-clamped Timoshenko column (Case 3), the only unrestrained element-end deformation is the translational displacement at the sliding clamped end. The relationship between this unrestrained translational displacement and the corresponding element-end shear force is defined by \(T_{\text{TB}}\). Therefore, the buckling state is corresponding to the condition that \(T_{\text{TB}}=0\). Based on Figure 3 in Section 4.3, \(T_{\text{TB}}\) decreases to 0 as \(\lambda\) increases to \(\pi\). Therefore, the buckling condition in this case can be solved as  
\[T_{\text{TB}}=0\Rightarrow\lambda_{\text{cr}}=\pi\Rightarrow P_{\text{cr}}= \frac{\pi^{2}EI/L^{2}}{1+\pi^{2}\varphi/12}. \tag{18}\]  
For a fixed-pinned Timoshenko column (Case 4), the only unrestrained element-end deformation is the rotation at the pinned end. The relationship between this unrestrained rotation and the corresponding element-end bending moment can be defined by \(S_{\text{TB}}\). Therefore, the buckling state is corresponding to the condition that \(S_{\text{TB}}=0\).