## Appendix A Equilibrium Analysis of Shear Deformable Timoshenko Beam-Column  
To establish the equilibrium conditions, a short segment of the element with length \(dx\) is analyzed, as depicted in Figure 2b,c, where the shear force is shown using \(Q(x)\) (which is defined to be normal to deformed element axis) and \(V(x)\) (which is defined to be normal to the undeformed element axis), respectively. The different compositions of shear force in Figure 2b,c give the relationship between \(Q(x)\) and \(V(x)\):  
\[V(x)=Q(x)-P\frac{dy}{dx} \tag{1}\]  
where \(x\) denotes the longitudinal coordinate of the element; \(y\) denotes the translational displacement. The equilibrium of bending moment (using Figure 2b) and shear force (using Figure 2c) give Equations (1) and (2), respectively. \[\frac{dM}{dx}=Q \tag{2}\]  
\[\frac{dV}{dx}=-\omega(x)\Rightarrow\frac{dQ}{dx}=P\frac{d^{2}y}{dx^{2}}- \omega(x) \tag{3}\]  
where \(M\) denotes the bending moment; \(P\) denotes the axial compressive force; \(\dot{\omega}(x)\) denotes distributed lateral load. According to the Timoshenko beam theory, the bending moment \(M\) and the shear force \(Q\) are given by  
\[M=EI\frac{d\phi}{dx} \tag{4}\]  
\[Q=k_{\mathrm{s}}GA\left(\phi+\frac{dy}{dx}\right) \tag{5}\]  
where \(\phi\) denotes the rotation angle in the Timoshenko beam (defined as positive in the anticlockwise direction); \(GA\) denotes the cross-sectional shear rigidity; \(k_{\mathrm{s}}\) denotes the shear correction coefficient to account for the difference in the constant state of shear stress in the Timoshenko beam theory and the parabolic variation of the actual shear stress through the depth of the cross-section [6; 19]. By substituting Equations (1) and (4) into Equation (3), a relationship between \(\phi\) and \(y\) is derived as  
\[EI\frac{d^{3}\phi}{dx^{3}}=P\frac{d^{2}y}{dx^{2}}-\omega(x). \tag{6}\]  
By substituting Equation (5) into Equation (3), a governing equation is derived as  
\[\frac{d\phi}{dx}=-[1-P/(k_{\mathrm{s}}GA)]\frac{d^{2}y}{dx^{2}}-\omega(x)/(k_ {\mathrm{s}}GA). \tag{7}\]  
Combining Equations (6) and (7), a fourth-order governing differential equation is derived as Equation (8) for null distributed translational load (\(\dot{\omega}=0\)). \[\frac{d^{4}y}{dx^{4}}+\frac{P}{EI[1-P/(k_{\mathrm{s}}GA)]}\frac{d^{2}y}{dx^{2 }}=0\Rightarrow\frac{d^{4}y}{dx^{4}}+\left(\frac{\lambda}{L}\right)^{2}\frac {d^{2}y}{dx^{2}}=0 \tag{8}\]  
where \(\lambda\) 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. \[\lambda=\sqrt{\frac{PL^{2}}{EI[1-P/(k_{\mathrm{s}}GA)]}}=\sqrt{\frac{PL^{2}}{ \chi EI}} \tag{9}\]  
\[\chi=1-P/(k_{\mathrm{s}}GA) \tag{10}\]  
The solution of \(y(x)\) in the governing differential Equation (8) is given by a linear combination of trigonometric functions and the linear polynomial function (Equation (11)), using the deformation combination factors \(\xi_{1}\), \(\xi_{2}\), \(\xi_{3}\), and \(\xi_{4}\). The rotation angle \(\phi\) can then be solved as Equation (12) based on Equation (7). \[\frac{dM}{dx}=Q \tag{2}\]0  
\[\frac{dM}{dx}=Q \tag{2}\]1  
The element bending moment \([M(x)]\) and shear force \([Q(x)\) and \(V(x)]\) can then be formulated as Equations (13), (14), and (15) based on Equations (44), (45), and (11), respectively. \[\frac{dM}{dx}=Q \tag{2}\]2  
\[\frac{dM}{dx}=Q \tag{2}\]3  
\[\frac{dM}{dx}=Q \tag{2}\]4  
For a matrix structural analysis, the element-end displacements and corresponding forces are discussed in matrix forms. Based on Equations (11) and (12), the element-end displacement vector (translations/rotation angles) is formulated in a matrix-form relationship with the four deformation combination factors, as shown in Equation (16). \[\left( \begin{array}{c}\delta_{1}\\ \delta_{1}\\ \delta_{2}\\ \delta_{2}\\ \end{array} \right)=\left( \begin{array}{c}y(0)\\ -\phi(0)\\ y(L)\\ -\phi(L)\\ \end{array} \right)=\left[