# Effects of Vibration on Adjacent Pipelines under Blasting Excavation
## 4 Numerical Simulation and Dynamic Response Analysis
### Blasting Load Application  
The test used equivalent blast loads to reflect the blasting process. The equivalent load applied to the elastic boundary reflected the blast effect, and this loading method can effectively save calculation time. The equivalent load was mainly made up of progressive parabolic-type, exponential decay-type, and linear decay-type (triangular loads) loads [26]. The test adopted triangular equivalent blasting.  
For uncoupled charges, the initial mean pressure, \(P_{0}\), of blast holes is obtained by Equation (12) [27; 28; 29; 30]:  
\[P_{0}=\frac{\rho_{e}D^{2}}{2(\gamma+1)}\bigg{(}\frac{d_{c}}{d_{b}}\bigg{)}^{2 \gamma} \tag{12}\]  
where \(\rho_{e}\) is the density of the explosive, \(D\) is the explosive velocity, \(\gamma\) is the isentropic index [31], \(d_{c}\) is the explosive diameter, and \(d_{b}\) is the blast hole's diameter.  
The load applied equivalently to the elastic boundary in the cluster hole is given by Equation (13):  
\[P_{1}=kP_{0}\bigg{(}\frac{r_{0}}{r_{1}}\bigg{)}^{2+\frac{\mu}{1+\mu}}\bigg{(} \frac{r_{1}}{r_{2}}\bigg{)}^{2-\frac{\mu}{1-\mu}} \tag{13}\]  
where \(k\) is the impact factor when the cluster cutting holes are detonated, which is related to the distribution and number of blasting holes, \(r_{0}\) is the hole's radius, \(r_{1}\) is the crushing zone's radius under the column charge, and \(r_{2}\) is the fractured zone's radius [26]. \(\mu\) is the Poisson's ratio.  
In this paper, the tunnel-blasting explosives used were No. 2 rock emulsion explosives. \(\rho_{e}=1.016\) g/cm\({}^{3}\) and \(D=4000\) m/s were input into the above formula to calculate the equivalent to the elastic boundary of the load as 156 MPa. The load rise time was 0.4 ms,  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multirow{2}{*}{**Units**} & \multicolumn{3}{c}{**Type**} \\ \cline{3-5}  & & **Pipeline** & **Surface Soil** & **Sintered Tuff** \\ \hline Density & g/cm\({}^{3}\) & 7.62 & 1.64 & 2.712 \\ Elastic modulus & GPa & 200 & 0.07 & 53.56 \\ Shear modulus & GPa & 60 & \(6.37\times 10^{-3}\) & 18.34 \\ Cohesion & MPa & & 0.08 & 6.4 \\ Tensile stress & MPa & 275 & 0.02 & 2.7 \\ Poisson’s ratio & & 0.28 & 0.30 & 0.28 \\ Frictional angle & \({}^{\circ}\) & & 37 & 47 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Material parameters.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Size of the Grid** **(cm)**} & **T3** & **T4** & **T5** & **T6** & **-** \\  & **Velocity** & **Velocity** & **Velocity** & **Velocity** & **Time** \\  & **(cm/s)** & **(cm/s)** & **(cm/s)** & **(cm/s)** & **(cm/s)** & **(h)** \\ \hline
10–25 & 1.16 & 1.27 & 1.28 & 1.12 & 58 \\
25–50 & 1.21 & 1.25 & 1.32 & 1.21 & 29 \\
50–75 & 1.26 & 1.29 & 1.35 & 1.17 & 14 \\
50–100 & 1.24 & 1.31 & 1.34 & 1.20 & 5 \\
75–100 & 1.29 & 1.28 & 1.41 & 1.19 & 1.2 \\ Results of field monitoring & 1.27 & 1.33 & 1.36 & 1.24 & - \\ \hline \hline \end{tabular}
\end{table}
Table 5: Calculation accuracy for different grid sizes.  
and the total load time was 3.4 ms [32]. The load curve form is shown in Figure 7. Although empirical equations and some simplifications for calculating the equivalent load method were proposed during the conception of this model, it was found through trial and error that using an equivalent load can improve calculation efficiency and achieve the desired surface vibration calculation accuracy [33,34].