# Study on Vibration Effects and Optimal Delay Time for Tunnel Cut-Blasting Beneath Existing Railways
## 2 Numerical Simulation of Blasting Vibration of Tunnel Crossing Under the Existing Railway
### Validation of Numerical Simulation Model  
At the tunnel blasting construction site, the TC-4850 Blasting vibration meter (Chengdu, China) was employed to monitor the peak particle velocity (PPV) at surface measurement points directly above the tunnel face. During the tunnel blasting process, the charge per segment in the cutting holes section significantly exceeds that of other blasthole segments, resulting in relatively more concentrated blast energy in this area. Consequently, monitoring data from the cutting holes section were selected for linear regression analysis to verify the accuracy of the numerical simulation results.  
The peak particle velocity (PPV) of blasting is mostly based on semi-empirical formulas. According to Safety Regulations for Blasting (GB6722-2014) [30], [PERSON]'s formula is used for the prediction of peak vibration velocity. Based on regression analysis of vibration monitoring data from the Zhongliangshan Tunnel site, the K and \(\alpha\) parameters of the tunnel's surrounding rock were derived. Subsequently, a blasting vibration velocity prediction model was established to enable accurate prediction of peak vibration velocities at various locations. The curve fitting results are illustrated in Figure 3, and the model was validated through comparative analysis with [PERSON]'s empirical formula. The equation is  
\[\mathrm{V}=\mathrm{K}(\mathrm{\rho})^{\alpha}=\mathrm{K}(\mathrm{Q}^{1/3}/ \mathrm{R})^{\alpha} \tag{2}\]  
where V is the vibration velocity, K is the coefficient related to the blasting conditions, Q is the single segment detonation quantity, R is the distance of monitoring points, and \(\alpha\) is the vibration attenuation coefficient.  
Taking logarithms on both sides of [PERSON]'s formula, it can be transformed into the following,  
\[\mathrm{lgV}=\mathrm{lgK}+\mathrm{\alpha lg}(\mathrm{Q}^{1/3}/\mathrm{R}) \tag{3}\]  
The following linear regression equation is obtained:  
\[\mathrm{Y}=\mathrm{a}\mathrm{X}+\mathrm{b} \tag{4}\]  
Linear regression analysis of the site monitoring data (Figure 3a) yielded K = 219.28, \(\alpha\) = 1.598, and correlation coefficient R\({}^{2}\) = 0.9997. Consequently, the [PERSON]'s equation for the site is  
\[\mathrm{V}=219.28(\mathrm{Q}^{1/3}/\mathrm{R})^{1.598} \tag{5}\]  
The numerical simulation results of 25-45 m surface nodes on the unexcavated side behind the tunnel face are recorded and compared with the calculation results of [PERSON]'s formula derived from regression analysis. Table 4 shows the monitoring information of  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Density** & **A/GPa** & **R1** & **R2** & \(\omega\) & **E\({}_{0}\)/GPa** & **V** \\ \hline
1.63 & 3.71 & 4.15 & 0.95 & 0.3 & 0.07 & 1.0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of model and state equation of the dynamite materials.  
each measuring point. The comparison results are shown in Figure 3b. As illustrated in the figure, the theoretical values calculated using [PERSON]'s formula exhibit excellent agreement with the numerical simulation results, thereby validating the high credibility of the numerical simulation methodology in predicting blast-induced ground vibration characteristics. However, due to the complexity of the geological conditions on site, there are differences between the measurement results and the numerical simulation results.