## 4 Applications and Discussions
### Comparison with the Simplified Expression by [PERSON] et al. The element stability functions \(T_{\text{TB}}\), \(Q_{\text{TB}}\), \(S_{\text{TB}}\), and \(C_{\text{TB}}\) are the basic properties for the element load-deformation responses. These stability functions correspond to the element-end shear force or bending moment for a unit element-end translation or rotation angle. The stability functions are transcendental functions of the axial force factor \(\lambda\) for given cross-sectional property factor \(\varphi\). The different expressions of the element stability stiffness matrix in Equations (5) and (15) are essentially defined by the different expressions of the element stability functions. In the simplified expression (Equation (5)), the element stability functions are modified from the element stability functions \(T_{\text{EB}}\), \(Q_{\text{EB}}\), \(S_{\text{EB}}\), and \(C_{\text{EB}}\) of axial-loaded Euler-Bernoulli beam-columns, which are functions of the axial force factor \(\lambda_{\text{EB}}\) for Euler-Bernoulli beams. In addition, the relationship between the axial force factor \(\lambda\) and \(\lambda_{\text{EB}}\) (for Timoshenko and Euler-Bernoulli beams, respectively) can be derived based on Equations (9) and (12). \[\lambda_{EB}=\sqrt{\frac{PL^{2}}{EI}}=\sqrt{\chi}\lambda=\frac{\lambda}{\sqrt{ 1+\lambda^{2}\varphi/12}}. \tag{21}\]  
The element stability functions \(T_{\text{Ek}}\), \(Q_{\text{Ek}}\), \(S_{\text{Ek}}\), and \(C_{\text{Ek}}\) by [PERSON] et al. [18] is a simplified expression that is not derived exactly, so it could result in errors in some situations comparing with the exact stiffness matrix (Equation (15)). Figure 3 shows the comparison of the exact (Equation (15)) and simplified (Equation (5)) expressions of the element stability functions as the axial force factor \(\lambda\) varies. In Figure 3a-c, the cross-sectional property factor \(\varphi\) is fixed to typical given values of 0, 0.5, and 1, respectively, which corresponds to an increase in the shear deformation effect. Figure 3 shows that the element stability functions \(T_{\text{TB}}\), \(Q_{\text{TB}}\), and \(S_{\text{TB}}\) decreases (while \(C_{\text{TB}}\) increases) with the increasing axial force factor \(\lambda\). As shown in Figure 3a, for a cross-sectional property factor of \(\varphi\) = 0 (which corresponds to zero shear deformation effect, i.e., the situation of Euler-Bernoulli beam-columns), the element stability functions by [PERSON] et al. [18] (Equation (5)) give the same result as the derived exact stiffness matrix (Equation (15)). In addition, for an axial force factor of \(\lambda\) = 0, Figure 3b,c also shows that the two formulations give the same result. This is because the simplified form (Equation (5)) for the element stiffness matrix by [PERSON] et al. [18] is defined based on these conditions. However, when the shear deformation effect is considered, the simplified form  
by [PERSON] et al. [18] varies considerably with the derived exact stiffness matrix (Equation (15)), and the variation may be significant for a relatively large axial force factor. Therefore, for the condition of a relatively large axial force factor, the exact element stability stiffness matrix of Timoshenko beam-columns derived in this paper is recommended to be used because the simplified form by [PERSON] et al. [18] could lead to large errors. As an example, consider the elastic buckling problem of a fixed-clamped Timoshenko column. The exact buckling load is solved by Case 3 in Section 4.1. Next, the stiffness matrix in Equation (5) by [PERSON] et al. [18] is used to obtain the buckling load.