# Macro-Modelling Approach for the In-Plane Cyclic Response of Cold-Formed Steel Partition Walls
## 3 Development of the Proposed Analytical Model
### Structural Characteristics of CFS Partition Walls  
#### 3.2.1 Screw Connections  
Screw connections play an important role determining the initial lateral behavior of a CFS partition wall since the racking behavior results in relative displacements with CFS stud  
Figure 3: Monotonic behavior of a screw connection: (**a**) deformed shapes of the gypsum board (GB); (**b**) relative deformation; (**c**) load-bearing capacity. at the screw connection can be considered as two different phases: (1) before and (2) after the contact between a GB and an RC slab occurs. During a displacement, \(\delta_{v}\) is less than the initial gap, \(g_{gb}\) between the slab and the edge of the GB; relative vertical deformations at the screw connection occur only in the axial direction of a CFS stud. From the triangle shown at 1 in Figure 3a, a relative vertical deformation, \(\Delta_{v,t}\) at the top corner of a GB can be computed by:  
\[\Delta_{v,t}=\delta_{v}(1/\cos\theta) \tag{4}\]  
A relative vertical deformation, \(\Delta_{v,b}\) at the bottom corner of a GB can be presented in triangles 5 and 6, and calculated from:  
\[\Delta_{v,b}=(W-w_{b})\tan\theta=(W-\frac{\Delta_{v,t}}{\tan\theta})\tan\theta \tag{5}\]  
Then, a relative vertical deformation, \(\Delta_{v,i}\) at the \(i\)th screw connection is calculated under the assumption that the \(\Delta_{v,i}\) is distributed linearly between \(\Delta_{v,t}\) and \(\Delta_{v,b}\):  
\[\Delta_{v,i}=\frac{(\Delta_{v,t}-\Delta_{v,b})}{H}I_{i}+\Delta_{v,b} \tag{6}\]  
where \(I_{i}\) is the distance of the \(i\)th screw connection from the bottom slab. If a displacement \(\delta_{v}\) becomes equal to the initial gap, \(g_{gb}\), \(\Delta_{v,i}\) cannot be increased whereas relative horizontal deformations, \(\Delta_{h,t}\) and \(\Delta_{h,b}\) at the top and bottom corners of a GB are initiated as shown in Figure 3b. With the assumption that relative horizontal deformations, \(\Delta_{h,i}\) s are distributed like the figure, they are determined with the additional horizontal displacement, \(\delta^{\prime}_{h}\) occurring after the gap closes:  
\[\Delta_{h,i}=\frac{\delta^{\prime}_{h}}{H}\Big{(}I_{i}-\frac{H}{2}\Big{)}\cos\theta \tag{7}\]  
\[\delta^{\prime}_{h}=\delta_{h}-\delta_{h,gap,GB} \tag{8}\]  
where \(\delta_{h,gap,GB}\) is the horizontal displacement at the initiation of the gap closing. Plugging \(\delta_{v}=g_{gb}\) into Equation (3) gives:  
\[\Big{[}\Big{(}2g_{gb}+H\Big{)}^{2}-W^{2}\Big{]}\delta_{h}^{2}-\Big{(}2H^{2}W \Big{)}\delta_{h}+\Big{[}\Big{(}2g_{gb}+H\Big{)}^{2}H^{2}-H^{4}\Big{]}=0 \tag{9}\]  
The value of \(\delta_{h}\) satisfying this quadratic equation becomes \(\delta_{h,gap,GB}\) like the following equation:  
\[\delta_{h,gap,GB}=\frac{H^{2}W+\sqrt{H^{4}W^{2}-4g_{gb}H^{2}(H+g_{gb})[\left(2 g_{gb}+H\right)^{2}-W^{2}]}}{\left[\left(2g_{gb}+H\right)^{2}-W^{2}\right]} \tag{10}\]  
Based on \(\Delta_{v,i}\) and \(\Delta_{h,i}\) calculated from two different deformation phases of a GB, the total relative deformation, \(\Delta_{t,i}\) of the screw connection is conservatively calculated by the square-root-of-sum-of-squares (SRSS) combination. As shown in the free body diagram of Figure 3c, the lateral force, \(F_{SC}\) of all the screw connections of a CFS partition wall is derived from the force equilibrium condition indicating that the vertical section locating the screws along with the axial direction of studs might be critical for the racking behavior of GBs. Therefore, \(F_{SC}\) can be obtained from the summation of internal shear forces corresponding to screw tilting and bearing forces, \(f_{i}\) corresponding to \(\Delta_{t,i}\):  
\[F_{SC}=\left(\frac{W}{H}\right)\sum_{i}f_{i} \tag{11}\]where the value of \(f_{i}\) utilizes the force-deformation relation which can be fundamentally defined from experimental results of the screw connection. A partition wall has several screw connections.