# Development of Phenomenological Model for RC Circular Columns
## 4 Empirical Equations for Model Parameters
### Empirical Equations  
(1) The empirical prediction equations  
Through a regression analysis, the empirical prediction equations were obtained as follows:  
\[M_{y}=\exp(-14.709)\cdot d^{3.02}\cdot(v+1)^{1.677}\cdot p_{l}^{0.569}\cdot f _{c}^{0.274}\cdot f_{ly}^{0.513} \tag{3}\]  
\[\theta_{y}=\exp(-11.569)\cdot p_{l}^{0.456}\cdot(s/d)^{(-0.218)}\cdot f_{ls}^{ 1.292}\cdot c^{0.218}\cdot\alpha_{sp}^{0.277}\cdot(v+1)^{(-0.788)}\cdot\left(V_ {p}/V_{n}\right)^{0.13} \tag{4}\]  
\[M_{c}/M_{y}=\exp(-0.303)\cdot d_{s}^{0.156}\cdot(s/d)^{(-0.034)}\cdot d_{l}^{ (-0.102)}\cdot f_{ly}^{(0.093)}\cdot\lambda_{sp}^{(-0.057)}\cdot\left(V_{p}/V_ {n}\right)^{(-0.035)} \tag{5}\]  
\[\theta_{p}=\exp(-1.786)\cdot s_{n}^{(-0.77)}\cdot\lambda_{sp}^{0.431}\cdot d_ {s}^{0.916}\cdot(v+1)^{(-1.782)}\cdot d_{l}^{(-0.688)} \tag{6}\]  
\[\Lambda=\exp(2.028)\cdot\lambda_{sp}^{0.495}\cdot s_{n}^{(-0.745)}\cdot d_{s} ^{0.898}\cdot d^{(-0.653)}\cdot p_{l}^{(-0.354)} \tag{7}\]Most of the circular columns in the database were not loaded to softening stage, and \(\theta_{\rm pc}\) could not be measured, so the regression method was not used to obtain the empirical equation for \(\theta_{\rm pc}\). According to reference [8], it has been shown that there is an empirical relationship between the initial stiffness and the softening stiffness, i.e., \(K_{c}=-0.08K_{0}\), and after determining \(M_{y},\theta_{y},M_{c}\), the estimation Equation (8) can be obtained. In addition, according to the test results of 15 adequately loaded components, 0.1 is set as the upper limit value of \(\theta_{\rm pc}\)[8]. The larger value of \(\theta_{\rm pc}\) is generally due to the fact that the loading is not sufficiently adequate, and the deterioration in the components is not yet apparent. \[\theta_{pc}=\left(\theta_{y}\cdot M_{c}\left/M_{y}\right.\right)/0.08\leq 0.1 \tag{8}\]  
Evaluation of the prediction equations  
The coefficients of determination (R\({}^{2}\)) of the regression equations for model parameters \(M_{y},\theta_{y},\theta_{\rm p}\), and \(\Lambda\) are 0.98, 0.65, 0.54, and 0.50, respectively, which indicates that the regression equations are well fitted; the coefficient of determination of \(M_{\rm c}/M_{y}\) is 0.36, which indicates that this parameter has a poorer correlation with the design parameters. For each model parameter of each component, the ratio between the calibrated and predicted values was calculated. The statistics of this ratio are presented in Table 4; \(\mu\), \(\widetilde{\mu}\), and \(\sigma\) are the mean, median, and standard deviation, respectively. Table 4 shows that the ratios' mean values corresponding to the model parameters \(M_{y},\theta_{y},M_{\rm c}/M_{y},\theta_{\rm p},\Lambda\) are 1.01, 1.02, 1.02, 1.09, and 1.13, and the median values are 0.99, 1.01, 1.02, 0.99, and 1.0, respectively, which indicates that the prediction equations can predict the hysteretic model parameters relatively well. The standard deviations of the ratios corresponding to the model parameters \(M_{y},\theta_{y},M_{\rm c}/M_{y},\theta_{\rm p},\Lambda\) are 0.18, 0.22, 0.11, 0.44, and 0.57, respectively, so it can be seen that the prediction accuracies of \(M_{y},\theta_{y},M_{\rm c}/M_{y}\) are high, and the dispersion is small. In contrast, the dispersion of \(\theta_{\rm p},\Lambda\) is relatively large. It should be noted that the statistical data of \(\theta_{\rm pc}\) is from reference [8], and its corresponding mean and median values are 1.2 and 1.0, while the logarithmic standard deviation is 0.72. Since, after the peak point of the backbone curve, the cracking of the components, the slipping of the reinforcement, and the shear deformation are more serious, the standard deviation and dispersion are relatively large, and the accuracy of the prediction result of \(\theta_{\rm pc}\) will be smaller than that of the other parameters. The model parameters of the 80 columns in the database were predicted using the aforementioned empirical prediction equations.