## 3 Matrix Structural Analysis for Exact Solutions of Shear Deformable Beams  
### Relationship between Element-End Displacements and Forces  
A shear deformable Timoshenko beam-column as shown in Figure 2 is analyzed. The beam-column has a length \(L\), flexural rigidity \(EI\), shear rigidity \(GA\), and is subjected to an axial load \(P\). There are many classical researches on the equilibrium analysis of beam-columns considering the second-order effect. This paper focuses on the element-end displacements and forces based on the matrix approach. The derivations for the element-end displacements and forces of the beam-column in Figure 2 follow from the classical research, and they are presented in Appendix A. The small displacement assumption and the Timoshenko beam theory (TBT) are used in the analysis. The element deformation is solved exactly from the governing differential Equation (A8). Therefore, the use of one element per member for the exact solution could be allowed. Figure 2: Equilibrium analysis of axial-loaded Timoshenko beam–column: (a) analyzed element; (b) equilibrium of element short segment with shear force \(Q\)(x) normal to the deformed element axis; (c) element short segment with shear force \(V\)(x) normal to the undeformed element axis. Based on the derivations, the element-end translational displacement/rotation angle and the corresponding shear force/bending moment of the beam-column can be expressed in matrix forms as (see Appendix A)  
\[\left( \begin{array}{c}\delta_{1}\\ \theta_{1}\\ \delta_{2}\\ \theta_{2}\\ \end{array} \right)=\left[ \begin{array}{cccc}1&0&0&1\\ 0&\chi\lambda/L&1&0\\ \cos\lambda&\sin\lambda&L&1\\ -\chi\lambda\sin\lambda/L&\chi\lambda\cos\lambda/L&1&0\\ \end{array} \right]\left( \begin{array}{c}\xi_{1}\\ \xi_{2}\\ \xi_{3}\\ \xi_{4}\\ \end{array} \right) \tag{7}\]  
\[\left( \begin{array}{c}V_{1}\\ M_{1}\\ V_{2}\\ M_{2}\\ \end{array} \right)=EI\chi\left[ \begin{array}{cccc}0&0&\left(\lambda/L \right)^{2}&0\\ \left(\lambda/L\right)^{2}&0&0&0\\ 0&0&-\left(\lambda/L\right)^{2}&0\\ -\left(\lambda/L\right)^{2}\cos\lambda&-\left(\lambda/L\right)^{2}\sin\lambda &0&0\\ \end{array} \right]\left( \begin{array}{c}\xi_{1}\\ \xi_{2}\\ \xi_{3}\\ \xi_{4}\\ \end{array} \right) \tag{8}\]  
where \(\xi_{1}\), \(\xi_{2}\), \(\xi_{3}\), and \(\xi_{4}\) are combination factors for the element deformation; \(\lambda\) denotes an axial force factor for the Timoshenko beam, defined as Equation (9), in which a coefficient \(\chi\) is defined as Equation (10) for the shear deformation effect. The axial compressive force can be expressed as Equation (11) based on Equation (9). \[\lambda=\sqrt{\frac{PL^{2}}{EI[1-P/\left(k_{\rm s}GA\right)]}}=\sqrt{\frac{PL^{ 2}}{\chi EI}} \tag{9}\]  
\[\chi=1-P/\left(k_{\rm s}GA\right) \tag{10}\]  
\[P=\chi\lambda^{2}EI/L^{2}. \tag{11}\]  
Combining Equations (10) and (11), the coefficient \(\chi\) can be expressed as a function of the axial force factor \(\lambda\) and the dimensionless factor \(\varphi\) (Equation (12)). The axial compressive force \(P\) can then be expressed as Equation (13) based on Equation (11). \[\chi=\frac{1}{1+\lambda^{2}EI/\left(k_{\rm s}GAL^{2}\right)}=\frac{1}{1+ \lambda^{2}\varphi/12} \tag{12}\]  
\[P=\frac{\lambda^{2}EI/L^{2}}{1+\lambda^{2}\varphi/12}. \tag{13}\]  
Combining Equations (7) and (8), the relationship between the element-end displacement vector (translations/rotation angles) and force vector (shear forces/bending moments) is formulated as  
\[\left( \begin{array}{c}V_{1}\\ M_{1}\\ V_{2}\\ M_{2}\\ \end{array} \right)=\left[ \begin{array}{c}\mathbf{K_{\rm TB,2 nd}}\end{array}  \right]\left( \begin{array}{c}\delta_{1}\\ \theta_{1}\\ \delta_{2}\\ \theta_{2}\\ \end{array} \right)= \tag{14}\] \[EI\chi\left[ \begin{array}{cccc}0&0&\left(\lambda/L\right)^{2}&0 \\ \left(\lambda/L\right)^{2}&0&0&0\\ 0&0&-\left(\lambda/L\right)^{2}&0\\ -\left(\lambda/L\right)^{2}\cos\lambda&-\left(\lambda/L\right)^{2}\sin\lambda &0&0\\ \end{array} \right]\left[