# Macro-Modelling Approach for the In-Plane Cyclic Response of Cold-Formed Steel Partition Walls
## 3 Development of the Proposed Analytical Model
### Hysteresis Modelling of CFS Partition Walls  
Figure 6 summarizes an entire procedure that can determine the lateral force-displacement relation of a CFS partition wall erected using typical Korean construction details. The flow chart consists of several sub-routines related to the structural behavior of all components in the partition wall: screw connections, GBs, and CFS studs. The procedure starts to compute relative displacements at the corner of GBs and screw connections with input dimensions of the partition wall. With horizontal displacements imposed to the partition wall selected with a sufficiently small value, the sub-routine 1 determines the behaviors of all elements before a gap closing at the corner of GBs. In this step, load-bearing capacities of the partition wall depend on only the behavior of the screw connections. The sub-routine 2 is carried out when the imposed horizontal displacement is larger than \(\delta_{h,gap,GB}\). The contact properties at the critical surface of GBs are defined in this step. If an imposed horizontal displacement is less than the \(\delta_{h,gap,ST}\), the sub-routine 3 is carried out where the bearing force of the stud is still not developed. Otherwise, the sub-routine 4 is performed to determine the bearing behavior of the stud. The lateral forces corresponding to the selected horizontal displacement are finally calculated with the sub-routine 5 based on the force equilibrium condition at all the components. Finally, an imposed horizontal displacement is increased and the sub-routines 1 to 5 are iterated until it becomes sufficiently larger than the targeted displacement for the analysis. This study simplifies the nonlinear hysteretic behavior of a CFS partition wall using discretized lumped springs of influential sub-components, as shown in Figure 7. Three spring assemblies are introduced to represent the hysteresis of the sub-components including studs, screw connections, and GBs which are, respectively, denoted as Element 1, Element 2 and Element 3 in the figure. The monotonic nonlinear behavior of all Elements is described in the previous Sections. Element 1 and Element 3 are constructed with a primary spring and a gap element while Element 2 is made of only a primary spring element. The primary spring in each sub-component is assigned to a one-degree-of-freedom element with zero-length and its backbone curve is adopted from the monotonic force-displacement relations suggested in Figures 3-5. The primary springs are placed at the mid-height of a partition wall and connected to top and bottom RC slabs with rigid links. The springs are implemented to activate in the in-plane direction of a partition wall while its out-of-plane behavior is not considered in this study. Gap elements are including in Element 1 and Element 3 to capture the gap closing at GB-to-RC slab interfaces and stud-to-track connections. Pinching effects in unloading and reloading are reflected in the hysteresis model of all the screw connections. This study utilizes a well-known Wayne-Stewart model to simulate the cyclic response of all the screw connections including complex pinched hysteretic responses accounting for degradations under cyclic loading as shown in Figure 8. Nine parameters are required to define the Wayne-Stewart  
Figure 6: Flow chart for the monotonic behavior of a CFS partition wall. Figure 7: Schematization of the modelling approach for the CFS partition wall. model: initial stiffness, \(k_{0}\), post-yield stiffness factor, \(r_{1}\), post capping stiffness factor, \(r_{2}\), yield strength, \(V_{y}\), capping strength, \(V^{\prime}\), unloading stiffness factor, \(r_{un}\), intercept strength, \(V_{os}\), pinching power factor, \(\alpha\), and softening factor, \(\beta\). From the tri-linear monotonic curve depicted in Figure 3c, \(V^{\prime}\) and \(V_{y}\) are defined as the maximum load and 75% of \(V^{\prime}\). The initial and post-yield stiffness are, respectively, used for parameters, \(k_{0}\) and \(r_{1}\).