# Construction of a Dimensional Damage Model of Reinforced Concrete Columns under Explosion Loading
## 3 Numerical Simulations to Obtain the Undetermined Coefficient
### Numerical Simulation Results and Analysis
Figure 5: Damage diagram of the reinforced concrete column. (**a**) is the damage diagram for the Z4-1 condition; (**b**) is the damage diagram for the Z5-1 condition. The proportional detonation distance \(Z\) is introduced, and the proportional detonation distance is defined as the ratio of the detonation distance to the third power of the explosion equivalent, that is,  
\[Z=L/m^{1/3} \tag{8}\]  
In Figures 6a and 7a, the curve abscissa can be expressed as \(Z\), and in Figures 6b and 7b, the curve abscissa can be expressed as \(1/Z\). By observing the five curves in Figure 6a, it can be found that under the same explosion equivalent, the disturbance in the column decreases with the increase in the proportional explosion distance \(Z\) in a similar and nonlinear manner. In Figure 6b, after modifying the abscissa of the curve to \(1/Z\), it can be clearly observed that under the same explosive equivalent, the linear relationship between the disturbance \(\xi\) in the reinforced concrete column and \(1/Z\) is generally linear, and the greater the explosive equivalent, the greater the slope of the linear line fitted by the disturbance \(\xi\) and \(1/Z\) in the column. Without considering the simulation conditions of \(Z4\)-\(1\) and \(Z5\)-\(1\), the data are less when \(L\) = 0.1 m. Therefore, the relationship between the disturbance \(\xi\) in the column and the proportional detonation distance \(Z\) is not analyzed when \(L\) = 0.1 m. By observing the four curves in Figure 7a, it can be found that under different explosion distances, the disturbance in the column decreases similarly and nonlinearly with the increase in the proportional explosion distance \(Z\); in Figure 7b, after adjusting the abscissa of the curve to \(1/Z\), it can be observed that the nonlinear relationship between the disturbance \(\xi\) and \(1/Z\) in the reinforced concrete column still shows a similar change trend under the same explosion distance. Figure 6: The relationship between the disturbance and the explosion distance and the explosion equivalent in the column under different explosion equivalents. (**a**) is the change curve of \(\xi\), and \(\mathrm{L}/\mathrm{m}^{1/3}\); (**b**) is the change curve of \(\xi\), and \(\mathrm{m}^{1/3}/\mathrm{L}\). Figure 7: The relationship between the disturbance in the column and the explosion distance and the explosion equivalent under different explosion distances. (**a**) is the change curve of \(\xi\), and \(\mathrm{L}/\mathrm{m}^{1/3}\); (**b**) is the change curve of \(\xi\) and \(\mathrm{m}^{1/3}/\mathrm{L}\). Based on the five curves obtained by linear fitting, the mean square error (MSE) can be used to evaluate the change in the data. The mean square error refers to the average value of the square of the distance between the predicted value \(f(x)\) of the model and the real value y of the sample [32]. The formula is shown in Formula (9). \[\text{MSE}=\frac{1}{n}\sum_{i=1}^{n}\left(y_{i}-f(x_{i})\right)^{2} \tag{9}\]  
where \(y_{i}\) and \(f(x_{i})\) represent the true value and predicted value of the ith sample, respectively, and n is the number of samples. The closer the mean square error MSE is to 0, the smaller the fitting error is. The mean square error values of the five linear fitting functions are 6.48 \(\times\) 10\({}^{-5}\), 6 \(\times\) 10\({}^{-3}\), 4.3 \(\times\) 10\({}^{-3}\), 1.96 \(\times\) 10\({}^{-4}\) and 1.8 \(\times\) 10\({}^{-3}\), respectively. The maximum mean square error value is 6 \(\times\) 10\({}^{-3}\), which is close to 0, which proves that the fitting error of the five linear fitting curves is low and the fitting effect is good. That is to say, it is proven that there is a certain linear relationship between the perturbation \(\xi\) in the column and the reciprocal of the proportional explosion distance \(Z\) under the same explosion equivalent. Thus, the \(\alpha\) value in Formula (7) is 1/3 and the \(\beta\) value is 1.