# Prediction of Tunnel Blasting Vibration Velocity Considering the Influence of Free Surface
## 5 Engineering Applications
### Formula Fitting  
Considering the structural health and service life of the existing tunnel, the construction unit set 5 cm/s as the blasting vibration velocity threshold. To obtain a suitable vibration velocity prediction formula to guide the design of subsequent blasting parameters, in the existing tunnel near the blast source side of the arch waist, a test was conducted every 4 m to test the vibration velocity (Figure 9), The vibration velocity of the measurement point is the maximum in the vertical tunnel direction, and the vertical vibration velocity curve is shown in Figure 10.  
At the same time, as the vertical test results are summarized in Table 3, the first test explosion excavation in the middle guide tunnel, the largest section of MS3, produced a delayed charge of 42.6 kg, and the MS1 and MS3 charges were close to 33.6 kg and 42.6 kg, respectively. Obviously, by Figure 10. it can be seen that the cutting blasthole was subject to the strongest clamping restraint, inducing the largest peak vibration velocity.  
Figure 8: Schematic diagram of on-site drilling.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Type** & **Blasthole Type** & **Detonator Series** & \begin{tabular}{c} **Blasthole** \\ **Number/Pieces** \\ \end{tabular} & \begin{tabular}{c} **Charge Weight** \\ **per Hole/kg** \\ \end{tabular} & \
\begin{tabular}{c} **Total Charge** \\ **/kg** \\ \end{tabular} \\ \hline \multirow{4}{*}{
\begin{tabular}{c} The middle guide \\ tunnel \\ \end{tabular} } & Cutting blasthole & 1 & 10 & 2.4 & 33.6 \\ \cline{2-6}  & & 3 & 14 & 3.0–3.3 & 42.6 \\ \cline{2-6}  & & 5 & 12 & 2.4–3.3 & 34.2 \\ \cline{2-6}  & Breaking blasthole & 7 & 12 & 2.1–3.3 & 37.0 \\ \cline{2-6}  & & 9 & 14 & 0.9–2.7 & 28.5 \\ \cline{2-6}  & & 11 & 17 & 0.9–1.8 & 18.6 \\ \cline{2-6}  & Contour blasthole & 11 & 6 & 0.9–1.5 & 8.1 \\ \cline{2-6}  & & 13 & 32 & 0.6–1.2 & 24.6 \\ \hline Total & & & 121 & & 216.1 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Statistical table of blasting parameters for the middle guide tunnel.  
In engineering applications, we paid more attention to radial and vertical vibration, while the regularity of the tangential vibration was poor [24]. Therefore, the prediction formula was used to fit the test data in radial and vertical directions, as in Equation (8). The fitted regression coefficients \(R^{2}\) = 0.91 in the vertical direction and \(R^{2}\) = 0.70 in the radial direction are satisfactory for engineering applications. In addition, it is clear that the larger the vibration velocity, the better the fitting effect of the modified formula.  
\[\begin{split}\text{Vertical direction}:&\ \ln v=9.27+3.25\ln\left(\frac{\sqrt{Q}}{R}\right)+0.43\ln\left(\frac{W_{\text{ min}}}{mR}\right)\\ \text{Radial direction}:&\ \ln v=13.64+2.95\ln\left( \frac{\sqrt{Q}}{R}\right)+0.58\ln\left(\frac{W_{\text{min}}}{mR}\right)\end{split} \tag{8}\]  
To evaluate the accuracy of the modified formula, the modified formula and the Sadovsky formula were used for prediction analysis. To save space in the article, only the prediction results in the vertical direction are presented, as shown in Table 4. It can be seen that the average relative tolerance calculated by the modified formula is 13.32% and the relative tolerance of the cutting blasthole is 18.15%, which is much smaller than the average relative tolerance calculated by the Sadovsky formula of 24.79% and the relative tolerance of the cutting blasthole of 56.48%, indicating that the correct formula significantly improves the prediction effect of the peak blast vibration velocity and more truly reflects the engineering reality.