## 3 Predictive Equations for Shear Spring Parameters
### Development of Predictive Equations
\[D_{cr}=e^{-1.82}\cdot b^{-1.69}\cdot h^{-1.89}\cdot s^{0.93}\cdot L^{1.83}\cdot f _{c}^{\prime 0.02}\cdot f_{yl}^{0,69}\cdot f_{yt}^{1.48}\cdot\rho_{l}^{0.33} \cdot\rho_{t}^{1.06}\cdot\rho_{Pn}^{-0.33}\cdot\kappa_{c}^{-3.58} \tag{12}\]  
\[D_{yi}=e^{-7.25}\cdot b^{-0.98}\cdot h^{-0.89}\cdot L^{1.90}\cdot f_{c}^{0.28} \cdot f_{yt}^{1.77}\cdot\rho_{l}^{0.52}\cdot\rho_{t}^{0.47}\cdot\rho_{Pn}^{-0. 12}\cdot\kappa_{c}^{-1.99} \tag{13}\]  
\[D_{mx}=e^{-8.25}\cdot b^{-0.55}\cdot h^{-0.90}\cdot L^{1.88}\cdot f_{c}^{ \prime 1.22}\cdot f_{yt}^{0.36}\cdot\rho_{l}^{0.15}\cdot\rho_{t}^{0.07}\cdot \rho_{Pn}^{-0.53}\cdot\kappa_{c}^{3.10} \tag{14}\]  
\[F_{cr}=e^{2.82}\cdot b^{-0.16}\cdot h^{1.43}\cdot f_{c}^{\prime-0.20}\cdot \rho_{l}^{0.47}\cdot\rho_{t}^{0.43}\cdot\rho_{Pn}^{0.19}\cdot\kappa_{c}^{-2.54} \tag{15}\]  
\[F_{yi}=e^{1.06}\cdot b^{0.42}\cdot h^{0.99}\cdot f_{c}^{\prime-0.05}\cdot \rho_{l}^{0.59}\cdot\rho_{l}^{0.22}\cdot\rho_{Pn}^{0.12}\cdot\kappa_{c}^{-0.87} \tag{16}\]  
\[F_{mx}=e^{-0.47}\cdot b^{0.35}\cdot h^{1.03}\cdot f_{c}^{\prime 0.28}\cdot\rho_{l}^{ 0.46}\cdot\rho_{l}^{0.17}\cdot\rho_{Pn}^{0.14}\cdot\kappa_{c}^{-0.68} \tag{17}\]  
\[\kappa_{nl}=e^{1.98}\cdot A_{g}^{-0.12}\cdot b^{0.15}\cdot cc^{0.16}\cdot f_{ c}^{\prime 0.19}\cdot f_{yl}^{-0.35}\cdot f_{yt}^{-0.03}\cdot\rho_{l}^{-0.15}\cdot\rho_{t}^{ 0.13}\cdot\rho_{Pn}^{-0.03}-1 \tag{18}\]  
\[\kappa_{rf}=e^{-1.66}\cdot A_{g}^{0.13}\cdot f_{c}^{\prime 0.48}\cdot f_{yl}^{-0. 35}\cdot f_{yt}^{0.01}\cdot\rho_{l}^{-0.19}\cdot\rho_{t}^{-0.02}\cdot\rho_{Pn}^ {0.04}\cdot\kappa_{c}^{0.86}-1 \tag{19}\]  
\[\kappa_{uf}=e^{-1.95}\cdot A_{g}^{-0.05}\cdot f_{c}^{\prime-0.50}\cdot f_{yl}^ {-0.01}\cdot f_{yt}^{1.03}\cdot\rho_{l}^{0.10}\cdot\rho_{t}^{0.38}\cdot\rho_{Pn }^{-0.36}\cdot\kappa_{c}^{-2.42}-1 \tag{20}\]  
\[\alpha_{rk}=e^{-1.73}\cdot A_{g}^{0.58}\cdot L^{-0.71}\cdot{f_{c}^{\prime}}^{ -1.04}\cdot f_{yt}^{0.48}\cdot\rho_{l}^{0.17}\cdot\rho_{t}^{-0.17}\cdot\rho_{Pn }^{-0.50}-1 \tag{21}\]  
\[D_{yi}=e^{-7.25}\cdot b^{-0.98}\cdot h^{-0.89}\cdot L^{1.90}\cdot f_{c}^{0.28} \cdot f_{yt}^{1.77}\cdot\rho_{l}^{0.52}\cdot\rho_{t}^{0.47}\cdot\rho_{Pn}^{-0. 12}\cdot\kappa_{c}^{-1.99} \tag{13}\]0  
\[D_{yi}=e^{-7.25}\cdot b^{-0.98}\cdot h^{-0.89}\cdot L^{1.90}\cdot f_{c}^{0.28} \cdot f_{yt}^{1.77}\cdot\rho_{l}^{0.52}\cdot\rho_{t}^{0.47}\cdot\rho_{Pn}^{-0. 12}\cdot\kappa_{c}^{-1.99} \tag{13}\]1The \(R^{2}\) values for the pinching parameters (0.6259 to 0.8989) and the degradation parameters (0.4264 to 0.6415) are relatively lower, suggesting a moderate predictive accuracy. This can be attributed to the inherently nonlinear nature of pinching and degradation behaviors, which are influenced by a multitude of factors, including concrete damage mechanisms, such as the degradation of the crack crack, the crack crack, and the crack crack. \[D_{yi}=e^{-7.25}\cdot b^{-0.98}\cdot h^{-0.89}\cdot L^{1.90}\cdot f_{c}^{0.28} \cdot f_{yt}^{1.77}\cdot\rho_{l}^{0.52}\cdot\rho_{t}^{0.47}\cdot\rho_{Pn}^{-0. 12}\cdot\kappa_{c}^{-1.99} \tag{13}\]2 Results of regression analysis. state, crack width, transverse reinforcement stress, and the opening and closing of concrete cracks [31]. Figure 7 represents a statistical comparison of the errors between predicted and calibrated parameters. The near-zero median error suggests minimal bias in the predictions. Consistent with the high \(R^{2}\) values observed in the regression analysis, the backbone curve parameters demonstrate low standard deviations, indicative of high prediction precision. However, for the unloading force pinching parameter (\(\kappa_{uf}\)) and degradation parameters (\(\alpha_{uk},\alpha_{rk},\alpha_{st}\)), the standard deviation values are either close to or exceed 1. This implies greater variability in the prediction errors for these parameters compared to the backbone curve parameters.