## 2 Shear Spring Model
### Pinching4 Model  
The structural analysis program employed for simulating the shear spring in this study is OpenSees [25], and the shear spring element is modeled using a zero-length element with the Pinching4 material [23]. This study aims to develop predictive equations for the shear spring parameters in the Pinching4 model. Thus, this section provides a brief overview of the parameters required for the Pinching4 model, as detailed in [PERSON] et al., 2003 [23]. As depicted in Figure 2, the Pinching4 model requires parameters for constructing a four-point backbone curve, typically determined by idealizing pushover test results. The four points are defined as the cracking point \((D_{cr},\ F_{cr})\), yield point \((D_{yi},\ F_{yi})\), maximum force point \((D_{mx},F_{mx})\), and ultimate point \((D_{ul},\ F_{ul})\), where \(D\) and \(F\) denote displacement and force, respectively, at each state. Additionally, pinching behavior is modeled using three parameters, \(\kappa_{rd},\kappa_{rf}\), and \(\kappa_{uf}\), characterizing reloading displacement, reloading force, and unloading force, respectively. These parameters are defined as the ratio of the force or displacement at the pinching point to the maximum force or displacement of the current cycle, as shown in Figure 2a. Stiffness and strength degradation, indicative of reduced member performance under cyclic loading, are captured by three parameters \((k_{i},\ (d_{max})_{i},\ \text{and}\ (f_{max})_{i})\), as illustrated in Figure 2b-d. Here, \(i\) denotes the current displacement increment, and thus \(k_{i}\), \((d_{max})_{i}\), and \((f_{max})_{i}\) represent the current unloading stiffness, deformation, and envelope maximum strength, respectively. The parameter \((d_{max})_{i}\) defines the end of the reload cycle for increasing deformation demand. These parameters are calculated using Equations (1)-(3). Figure 1: Idealized modeling approaches for beam–column elements (adapted from [PERSON] et al., 2010) [27]: (**a**) rigid-plastic hinge model; (**b**) inelastic spring model with hysteretic properties; (**c**) finite-length hinge zone at member ends; (**d**) fiber-section model along the member length; and (**e**) finite element model. \[k_{i} =k_{0}\times(1-\delta k_{i}) \tag{1}\] \[\left(d_{max}\right)_{i} =\left(d_{max}\right)_{0}\times(1+\delta d_{i})\] (2) \[\left(f_{max}\right)_{i} =\left(f_{max}\right)_{0}\times(1-\delta f_{i}) \tag{3}\]  
Here, \(k_{0}\) is the initial unloading stiffness and \(\left(d_{max}\right)_{0}\) is the maximum historic deformation demand. \(\left(f_{max}\right)_{0}\) is the initial envelope maximum strength for the case of no damage. \(\delta k_{i}\), \(\delta d_{i}\) and \(\delta f_{i}\) are the stiffness damage value, the reloading stiffness damage value, and the strength damage value at \(t_{i}\), respectively. The parameters \(\delta k_{i}\), \(\delta d_{i}\), and \(\delta f_{i}\) are defined using Equations (4)-(6). \[\delta k_{i}=\left(gK1\cdot\left(\widetilde{d}_{max}\right)^{gK2}+gK2\cdot \left(\frac{E_{i}}{E_{mono}}\right)^{gK4}\right)\leq gKLim \tag{4}\]  
\[\delta d_{i}=\left(gD1\cdot\left(\widetilde{d}_{max}\right)^{gD2}+gD2\cdot \left(\frac{E_{i}}{E_{mono}}\right)^{gD4}\right)\leq gDLim \tag{5}\]  
\[\delta f_{i}=\left(gF1\cdot\left(\widetilde{d}_{max}\right)^{gF2}+gF2\cdot \left(\frac{E_{i}}{E_{mono}}\right)^{gF4}\right)\leq gFLim \tag{6}\]  
Here, \(\widetilde{d}_{max}\) is the ratio of maximum to minimum historic displacement demands defining failure. \(E_{i}\) represents the accumulated hysteretic energy, and \(E_{mono}\) is the energy required to achieve the deformation that defines failure. The parameter used with \(\left(\widetilde{d}_{max}\right)\)  
Figure 2: Pinching4 material model parameters (adapted from [PERSON] et al., 2003) [(23)]: (**a**) backbone curve and pinching; (**b**) unloading stiffness degradation; (**c**) strength degradation; (**d**) and reloading stiffness degradation.