# Piecewise Function Hysteretic Model for Cold-Formed Steel Shear Walls with Reinforced End Studs
## 3 Hysteretic Model of CFS Shear Walls with Reinforced End Studs
### Modeling of Backbone Curve
#### 3.1.2 Discrete Coordinate Method  
As shown in Figure 4, the load-displacement skeleton curve was discretized into a series of coordinates (a, b, c...). Coordinate a is defined as the elastic point, where it is assumed that the skeleton curve is linear until it has reached the elastic displacement \(\Delta_{\mathrm{e}}\) corresponding to coordinate a, with the load relative to the remaining coordinates being a function of the lateral displacement and secant stiffness of the wall. It can be observed from Equations (1) and (2) that given lateral displacement  
Figure 3: Degradation trends of secant stiffness. \(\Delta\), the secant stiffness corresponding to \(\Delta\) can be obtained according to the elastic lateral stiffness \(K_{\mathrm{e}}\) and shear capacity \(F_{\mathrm{P}}\) of the wall, the discrete coordinates can be further determined, and that the load-displacement skeleton curve of the wall will be formed by connecting each discrete coordinate. Hence, \(K_{\mathrm{e}}\) and \(F_{\mathrm{P}}\) are the basis for determining the load-displacement skeleton curve of the wall. 1.3 Determination of Elastic Lateral Stiffness \(K_{\mathrm{e}}\) and Shear Capacity \(F_{\mathrm{P}}\)  
1. Determination of elastic lateral stiffness \(K_{\mathrm{e}}\)  
A simplified method for calculating the elastic lateral stiffness of the wall sheaheted with double-layer wallboards on both sides was proposed by the [PERSON] et al. [20] based on the equivalent-bracing model. The method, which avoids the high cost and time-consuming shortcomings of full-scale shear wall tests, can predict elastic lateral stiffness accurately according to sheathing material, sheathing thickness, and the shear performance of screw connections only. Consequently, this method was followed to calculate the elastic lateral stiffness of the wall with reinforced end studs. As shown in Figure 5, the simplified model is based on the following assumptions: (1) the end studs and top track are simplified as rigid bars, and the end studs are hinged to the track on the top; additionally, the bases of the end studs are assumed to be fixed ends in consideration of hold-downs; (2) the coupling of the wallboards and interior studs is equivalent to diagonal braces considering axial force only, and the equivalent braces are assumed to be the primary components for resisting lateral load; (3) the axial stiffness of the braces are cumulative. The elastic lateral stiffness of the wall with reinforced end studs can be deduced on the basis of the above assumptions, elastic theories, structural mechanics theories, and force-balance principles, as expressed in Equation (3). The detailed derivation is described in the study by [PERSON] et al. [20]. \[K_{\mathrm{e}}=\frac{P}{\Delta}=\frac{6i_{\mathrm{c}}}{H^{2}}+\frac{2L^{2}\sum \limits_{i=1}^{2}E_{\mathrm{bi}}A_{\mathrm{bi}}}{\left(L^{2}+H^{2}\right)^{1.5}} \tag{3}\]  
\[E_{\mathrm{bi}}A_{\mathrm{bi}}=\frac{1}{2}\cdot\frac{\left(H^{2}+L^{2}\right) ^{1.5}}{\frac{HL}{G_{\mathrm{bi}}I_{\mathrm{bi}}}+\frac{2d_{\mathrm{i}}}{f_{ \mathrm{bi}}n_{\mathrm{be}}}(H^{2}+HL)} \tag{4}\]  
\[S_{\mathrm{a}}=\frac{H+L}{2n_{\mathrm{s}}+n_{\mathrm{e}}-5} \tag{5}\]  
\[n_{\mathrm{se}}=\frac{H}{S_{\mathrm{a}}}+1 \tag{6}\]  
Figure 4: Discrete coordinate method.