# Analysis and Application of Vibration Damping Hole in Tunnel Working Face Based on Grey Correlation Analysis

Authors: Baofu Duan; Zhang Zhengxin; Zhang Chunwu; Zongjun Sun
Journal: Shock and Vibration
 (2022)

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## Abstract

In the process of tunnel blasting construction, certain vibration hazards will be caused to adjacent buildings. Therefore, it is necessary to explore feasible vibration reduction measures in the construction process. Combined with field measurement and numerical calculation, the law of vibration reduction effect of vibration reduction holes in the blasting construction of subway tunnels is analyzed. The gray correlation theory is used to determine the influence degree of vibration reduction hole parameters on the vibration reduction effect; each parameter is clarified, and the primary and secondary relationship of each parameter in the vibration reduction effect is clarified. Among them, the effect of damping hole spacing is particularly obvious. Combined with numerical calculation and comprehensive analysis of on-site construction effects, it is found that during the construction process, the distance between the vibration damping holes is 120 mm, the distance between the vibration damping holes and the explosion source is 2000 mm, the diameter of the vibration damping holes is more than 120 mm, the depth of the vibration damping holes is 1200–1500 mm, and double-row vibration-damping hole spacing is 50–100 mm, which can achieve a better vibration-damping effect. The research results will provide a reference vibration reduction technology for the blasting construction of urban shallow tunnels and have certain reference value and guiding significance for the blasting scheme design and vibration control of similar projects.


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## Body

## 1. Introduction

At present, in the construction of urban rail transit, mining excavation has become a common construction method for subway tunnel crossing hard rock because of its advantages of economy and efficiency. However, in the blasting engineering close to important buildings, the mining method will cause very serious harm.In this regard, some scholars have studied a variety of measures to reduce the impact of blasting vibration. On the one hand, the blasting parameters are studied. For example, Liu et al. [1] used LS-DYNA dynamic finite element software to calculate the impact of tunnel excavation on adjacent tunnels under different explosive consumption and put forward some measures to control vibration. Xie et al. [2] used MIDAS/GTS dynamic finite element software to simulate and analyze the variation law of the existing tunnel under different planned footage and different blasting load surfaces. Yao and He et al. [3] obtained the influence of the change of support system, rock wall reinforcement measures, and construction measures on the advance tunnel through two-dimensional and three-dimensional numerical simulation. Li et al. [4] considered the deep mines in Linyi Mining Area as the research object and found that the relationship between principal stresses is σH > σv > σh. In this stress state, the lateral earth pressure coefficient is greater than 1, and the magnitude of the three principal stresses increases with the increase of depth.On the other hand, reasonable vibration reduction measures and related vibration blasting laws are studied. For example, Qiang [5] studied the propagation and attenuation law of seismic waves in fault stratum according to the propagation characteristics of stress waves. Wang [6] analyzed the attenuation law of transverse and longitudinal vibration velocity of the tunnel according to the propagation law of vibration wave by collecting the measured data of the impact of drilling and blasting method on surrounding buildings. Song [7] established a nonlinear finite element model of soil-tunnel interaction based on FLAC finite difference software, and the effects of tunnel depth, lining thickness, and tunnel diameter on the dynamic response of the tunnel were discussed. Comprehensive analysis found that under the action of strong earthquakes, the bearing capacity of the tunnel dropped sharply, the lining is damaged, and there is a large residual internal force. Tang and Huang et al. [8] studied the impact of blasting construction of shallow buried large section tunnel under complex geological conditions. They creatively applied the guide hole near the cutting hole on the basis of spaced charging. Finally, through analysis, it was found that this method has a good vibration reduction effect on blasting construction.The research of scholars has proved that, in the propagation of seismic waves, adding vibration reducing holes on the propagation path of waves can change the propagation path of waves and make waves reflect, refract, and diffract, which will greatly reduce the energy carried by waves and reduce the harm caused by vibration. Moreover, the construction technology of vibration reducing holes is simple and easy to popularize, but it is rare to set vibration reducing holes at the tunnel face. The author will use the actual parameters in tunnel construction to conduct qualitative research on the damping hole, and the results obtained can be directly applied in practical engineering. For example, the damping hole parameters with the best damping effect can be selected according to the vibration speed limit of the protected building, which avoids the increase of cost caused by blind excavation according to experience, and has certain significance for the layout of damping holes in practical engineering.


## 2. Numerical Simulation

### 2.1. Engineering Background

The name of the project is work area 1 of lot 6111 of phase II project of Shenzhen Urban Rail Transit Line 6. In this paper, the Shen-Mei section is selected as the research object. The single tunnel and single line in this section start from the new shaft, with a total length of 200 m and a buried depth of 23.5 m∼38.5 m. The clear distance between the concealed excavation sideline of the right line of the tunnel and the Shuxiang building is 5.8 m. The foundation of the secret base is a concrete ring beam, the building is a steel structure slab house, the clear distance between the tunnel top and the secret base is at least 30 m, and the clamping stratum is slightly weathered granite. The main rock stratum of the tunnel body in the Shen-Mei section is slightly weathered granite and moderately weathered granite. Figure1 is the schematic diagram.Figure 1
Location of on-site monitoring points.


### 2.2. Basic Data Selection of Numerical Model

In this paper, a three-dimensional numerical calculation model is established with the help of MIDAS/GTS NX finite element simulation software. The boundary range of numerical simulation of tunnel engineering is 3∼5 times the tunnel opening [9], the buried depth of the tunnel is 30 m, and the depth of the cut hole is 1.3 m. In order to reduce the error caused by the model size on the calculation results, and considering the calculation capacity of the computer, the size of the model is 60 m × 60 m × 70 m. After comprehensive consideration, the total analysis time of linear time history analysis is 0.1 s, the time interval is 0.01 s, and the total steps are 100 steps. There are 277545 units in the model. The Mohr–Coulomb criterion is selected for the constitutive equation, and an independent ground surface spring is added to each node on the boundary of the numerical model to simulate the elastic boundary of the actual rock and soil mass. The stratum selected in the numerical calculation model is shown in Figure 2, and the physical and mechanical parameters of the rock stratum [10] are shown in Table 1.Figure 2
Numerical model.Table 1
Physical and mechanical parameters of rock and soil layer.
Serial numberFormation nameElastic modulus (MPa)Bulk density (kN/m3)Poisson’s ratioμCohesion (kPa)Internal friction angle (°)Test weight (kN/m3)1Miscellaneous fill4.516.580.35281618.62Silty clay5.815.10.31222117.83Completely weathered granite4.5600.263530194Intensely weathered granite5.11500.254032.6205Moderately weathered granite1.624500.22100040246Slightly weathered granite40.5322000.1815004626.5


### 2.3. Eigenvalue Analysis

During the analysis of the MIDAS/GTS NX numerical model, the eigenvalue analysis should be carried out before the dynamic time history analysis, and the inherent dynamic characteristics of the structure should be initially analyzed in combination with the natural vibration characteristics of the structure. Through the eigenvalue analysis of the numerical calculation model, the vibration mode, natural frequency, mass, and stiffness of the structure can be obtained [11]. Considering all factors, the model in this paper is finally determined, and the direct integration method is used to analyze the structural dynamic response. Through the eigenvalue analysis of the numerical model, it is concluded that the two main vibration periods of the building are 1.0944 s and 1.0893 s, respectively; the corresponding mass participation coefficients are 28.21% and 32.12%, respectively; and the natural vibration frequency also conforms to the frequency range specified in GB6722-2014, which is not discussed in this paper. In this paper, only the amplitude and vibration velocity are discussed.


### 2.4. Application of Blasting Load

In this paper, the dynamite data used in the project are imported into the dynamic load data generator of MIDAS/GTS NX to obtain a schematic diagram of the final load curve, as shown in Figure3, and the dynamite data are provided in Table 2.Figure 3
Numerical model.Table 2
Dynamite parameter.
Explosion speed (cm/s)Explosive density (g/cm3)Maximum number of charges (kg)Load factorDrilling diameter (mm)Charge diameter (mm)42001.21.61004238


### 2.5. Size Selection and Feasibility Verification of Numerical Calculation Model

#### 2.5.1. Size Selection of Numerical Calculation Model

The model mainly changed five sets of data. By changing one set of data, the other set of data remained unchanged for control experiments. The main model dimensions are shown in Table3. The schematic diagram of the arrangement of the vibration damping holes is shown in Figures 4 and 5 (the dimensions in the figure are the basic dimensions of each model, and the detailed parameters are shown in Table 3).Table 3
Model size of damping hole.
ParameterDistance between the damping hole and the explosion source (mm)Damping hole diameter (mm)Vibration damping hole spacing (mm)Depth of damping hole (mm)Double-row damping holes (mm)Change the distance between the damping hole and the explosion source1300809015000150080901500017008090150001900809015000Change the diameter of the damping hole190060901500019008090150001900100901500019001209015000Change the damping hole spacing190080601500019008090150001900801201500019008015015000Change the depth of the damping hole1900809012000190080901500019008090180001900809021000Switch to double-row damping holes19008090150050190080901500100190080901500150190080901500200Figure 4
General arrangement of vibration damping holes.Figure 5
Layout of local damping holes.


#### 2.5.2. Feasibility Verification of Numerical Calculation Model

According to the research of several articles [12, 13], the vertical vibration velocity can be used to replace the three vector vibration velocities in practical engineering to simplify the control standard. Therefore, only the Z direction is selected as the research, as shown in Table 4. It can be seen from the table that, except for the peak vibration velocity in the Z direction of the measuring point directly above the tunnel, the numerical simulation results of all other measuring points are greater than the field monitoring data. The reason for this phenomenon is that the numerical model has made considerable assumptions and optimization under complex geological conditions. For example, the fractures of underground rocks and strata have made isotropic assumptions, while ignoring the influence of groundwater. The error of the field monitoring and numerical simulation results in the table is within 10%, which ensures that the numerical calculation can well restore the actual situation of the field, and makes full preparations for the later research on the parameters of the damping hole, so as to ensure the correctness of the later numerical calculation model.Table 4
Comparison of monitoring data and numerical simulation results.
Measuring pointMaximum single-stage dose (kg)Burst distance (m)Maximum vibration velocity inZ direction (cm/s)Combined velocity maximum vibration velocity (cm/s)On-site monitoring resultsNumerical simulation resultsError (%)On-site monitoring resultsNumerical simulation resultsError (%)Measuring point directly above the tunnel1.6300.3260.3153.370.3130.3284.79Shuxiang building measuring point310.2490.2687.630.2640.2785.30Security room measuring point460.007850.008629.800.008460.009239.10


### 2.6. Analysis of Numerical Simulation Results

Based on the comparison between Section 2.5 and the actual data and the research of scholars, the peak vibration velocity in theZ direction of five factors is obtained, as shown in Table 5. Five groups of curves are formed according to 21 groups of data, and the measuring point directly above the tunnel with large vibration velocity and small error is selected to analyze the change of vibration velocity, as shown in Figures 6–10.(1)
It can be clearly seen from Table5 that the peak vibration velocity of the tunnel face without damping holes is higher than that of the tunnel face with damping holes. Therefore, the setting of damping holes has an obvious effect on reducing the vibration wave velocity generated by blasting.It can be found from Figure6 that the peak vibration velocity decreases with the increase of the hole diameter, indicating that the vibration reduction effect exists with the increase of the hole diameter. From the reduction range, it can be found that the vibration reduction effect is still decreasing when the hole diameter is greater than 120 mm. Therefore, it can be seen that the vibration reduction hole with a larger hole diameter can be arranged in the project.(2)
It can be seen from Figure7 that the peak vibration velocity generally decreases with the increase of the distance between the vibration damping hole and the explosion source, while the lowest peak vibration velocity appears at the measuring point directly above the tunnel at about 1700 mm. It can be seen that there is also a certain threshold for this factor. It is not that the closer the vibration damping hole is to the explosion source, the better the vibration damping effect is. It is recommended to set the vibration damping hole at about 1700 mm.(3)
It can be seen from Figure8 that the curve of the measuring point changes greatly, and the peak vibration velocity fluctuates with the increase of hole spacing. There is a minimum vibration velocity at 120 mm, a peak value at 60–90 mm, and an upward trend after 150 mm. It can be seen that the change of hole spacing will have a more complex vibration velocity change, so this factor will have a great impact on the peak vibration velocity.(4)
It can be found from Figure9 that when the hole depth reaches about 1300 mm, the vibration speed reaches the lowest. Then, with the increase of hole depth, the peak vibration speed continues to rise, and the rising momentum does not decrease. Here, it is mainly because the deeper the vibration damping hole is, there will be a certain cavity effect, which will increase the vibration speed.(5)
It can be seen from Figure10 that the peak vibration velocity of double row damping holes decreases first and then increases with the increase of row spacing; the lowest vibration velocity appears at about 50 mm, and then the curve shows a gentle trend at about 200 m. The row spacing of 50 mm is rarely used in practical engineering. After 150 mm, the peak vibration velocity changes little, and the vibration velocity does not change when the row spacing continues to increase. Therefore, the change of this factor will not have a great impact on the vibration velocity.Table 5
Summary table of the peak vibration speed in theZ direction of each influencing factor.
Influencing factorsParameter (mm)Shuxiang building measuring pointSecurity bureau measuring pointMeasuring point directly above the tunnelPeak vibration velocity inZ direction (cm/s)Peak vibration velocity inZ direction (cm/s)Peak vibration velocity inZ direction (cm/s)No damping holes00.26870.00860.3149Vibration hole diameter600.25670.00810.2903800.24550.00720.27901000.24030.00580.26601200.23200.00540.2475The distance between the vibration damping hole and the explosion source13000.26530.00840.303715000.26040.00820.294717000.25350.00800.281719000.25000.00790.2859Vibration damping hole spacing600.25330.00760.2697900.24550.00720.27901200.23480.00630.26721500.24290.00690.2737Vibration damping hole depth12000.24550.00720.279015000.25800.00780.289918000.26070.00820.280321000.24140.00710.2745Double row vibration damping hole row spacing500.24550.00720.27901000.25420.00770.28451500.26010.00800.29672000.21400.00610.2437Figure 6
Peak vibration velocity of damping hole diameter.Figure 7
The distance between the damping hole and the explosion source vs. peak vibration velocity.Figure 8
Peak vibration velocity of the damping hole spacing.Figure 9
Peak vibration velocity of the damping hole depth.Figure 10
Peak vibration velocity of double-row damping hole row spacing.


## 2.1. Engineering Background

The name of the project is work area 1 of lot 6111 of phase II project of Shenzhen Urban Rail Transit Line 6. In this paper, the Shen-Mei section is selected as the research object. The single tunnel and single line in this section start from the new shaft, with a total length of 200 m and a buried depth of 23.5 m∼38.5 m. The clear distance between the concealed excavation sideline of the right line of the tunnel and the Shuxiang building is 5.8 m. The foundation of the secret base is a concrete ring beam, the building is a steel structure slab house, the clear distance between the tunnel top and the secret base is at least 30 m, and the clamping stratum is slightly weathered granite. The main rock stratum of the tunnel body in the Shen-Mei section is slightly weathered granite and moderately weathered granite. Figure1 is the schematic diagram.Figure 1
Location of on-site monitoring points.


## 2.2. Basic Data Selection of Numerical Model

In this paper, a three-dimensional numerical calculation model is established with the help of MIDAS/GTS NX finite element simulation software. The boundary range of numerical simulation of tunnel engineering is 3∼5 times the tunnel opening [9], the buried depth of the tunnel is 30 m, and the depth of the cut hole is 1.3 m. In order to reduce the error caused by the model size on the calculation results, and considering the calculation capacity of the computer, the size of the model is 60 m × 60 m × 70 m. After comprehensive consideration, the total analysis time of linear time history analysis is 0.1 s, the time interval is 0.01 s, and the total steps are 100 steps. There are 277545 units in the model. The Mohr–Coulomb criterion is selected for the constitutive equation, and an independent ground surface spring is added to each node on the boundary of the numerical model to simulate the elastic boundary of the actual rock and soil mass. The stratum selected in the numerical calculation model is shown in Figure 2, and the physical and mechanical parameters of the rock stratum [10] are shown in Table 1.Figure 2
Numerical model.Table 1
Physical and mechanical parameters of rock and soil layer.
Serial numberFormation nameElastic modulus (MPa)Bulk density (kN/m3)Poisson’s ratioμCohesion (kPa)Internal friction angle (°)Test weight (kN/m3)1Miscellaneous fill4.516.580.35281618.62Silty clay5.815.10.31222117.83Completely weathered granite4.5600.263530194Intensely weathered granite5.11500.254032.6205Moderately weathered granite1.624500.22100040246Slightly weathered granite40.5322000.1815004626.5


## 2.3. Eigenvalue Analysis

During the analysis of the MIDAS/GTS NX numerical model, the eigenvalue analysis should be carried out before the dynamic time history analysis, and the inherent dynamic characteristics of the structure should be initially analyzed in combination with the natural vibration characteristics of the structure. Through the eigenvalue analysis of the numerical calculation model, the vibration mode, natural frequency, mass, and stiffness of the structure can be obtained [11]. Considering all factors, the model in this paper is finally determined, and the direct integration method is used to analyze the structural dynamic response. Through the eigenvalue analysis of the numerical model, it is concluded that the two main vibration periods of the building are 1.0944 s and 1.0893 s, respectively; the corresponding mass participation coefficients are 28.21% and 32.12%, respectively; and the natural vibration frequency also conforms to the frequency range specified in GB6722-2014, which is not discussed in this paper. In this paper, only the amplitude and vibration velocity are discussed.


## 2.4. Application of Blasting Load

In this paper, the dynamite data used in the project are imported into the dynamic load data generator of MIDAS/GTS NX to obtain a schematic diagram of the final load curve, as shown in Figure3, and the dynamite data are provided in Table 2.Figure 3
Numerical model.Table 2
Dynamite parameter.
Explosion speed (cm/s)Explosive density (g/cm3)Maximum number of charges (kg)Load factorDrilling diameter (mm)Charge diameter (mm)42001.21.61004238


## 2.5. Size Selection and Feasibility Verification of Numerical Calculation Model

### 2.5.1. Size Selection of Numerical Calculation Model

The model mainly changed five sets of data. By changing one set of data, the other set of data remained unchanged for control experiments. The main model dimensions are shown in Table3. The schematic diagram of the arrangement of the vibration damping holes is shown in Figures 4 and 5 (the dimensions in the figure are the basic dimensions of each model, and the detailed parameters are shown in Table 3).Table 3
Model size of damping hole.
ParameterDistance between the damping hole and the explosion source (mm)Damping hole diameter (mm)Vibration damping hole spacing (mm)Depth of damping hole (mm)Double-row damping holes (mm)Change the distance between the damping hole and the explosion source1300809015000150080901500017008090150001900809015000Change the diameter of the damping hole190060901500019008090150001900100901500019001209015000Change the damping hole spacing190080601500019008090150001900801201500019008015015000Change the depth of the damping hole1900809012000190080901500019008090180001900809021000Switch to double-row damping holes19008090150050190080901500100190080901500150190080901500200Figure 4
General arrangement of vibration damping holes.Figure 5
Layout of local damping holes.


### 2.5.2. Feasibility Verification of Numerical Calculation Model

According to the research of several articles [12, 13], the vertical vibration velocity can be used to replace the three vector vibration velocities in practical engineering to simplify the control standard. Therefore, only the Z direction is selected as the research, as shown in Table 4. It can be seen from the table that, except for the peak vibration velocity in the Z direction of the measuring point directly above the tunnel, the numerical simulation results of all other measuring points are greater than the field monitoring data. The reason for this phenomenon is that the numerical model has made considerable assumptions and optimization under complex geological conditions. For example, the fractures of underground rocks and strata have made isotropic assumptions, while ignoring the influence of groundwater. The error of the field monitoring and numerical simulation results in the table is within 10%, which ensures that the numerical calculation can well restore the actual situation of the field, and makes full preparations for the later research on the parameters of the damping hole, so as to ensure the correctness of the later numerical calculation model.Table 4
Comparison of monitoring data and numerical simulation results.
Measuring pointMaximum single-stage dose (kg)Burst distance (m)Maximum vibration velocity inZ direction (cm/s)Combined velocity maximum vibration velocity (cm/s)On-site monitoring resultsNumerical simulation resultsError (%)On-site monitoring resultsNumerical simulation resultsError (%)Measuring point directly above the tunnel1.6300.3260.3153.370.3130.3284.79Shuxiang building measuring point310.2490.2687.630.2640.2785.30Security room measuring point460.007850.008629.800.008460.009239.10


## 2.5.1. Size Selection of Numerical Calculation Model

The model mainly changed five sets of data. By changing one set of data, the other set of data remained unchanged for control experiments. The main model dimensions are shown in Table3. The schematic diagram of the arrangement of the vibration damping holes is shown in Figures 4 and 5 (the dimensions in the figure are the basic dimensions of each model, and the detailed parameters are shown in Table 3).Table 3
Model size of damping hole.
ParameterDistance between the damping hole and the explosion source (mm)Damping hole diameter (mm)Vibration damping hole spacing (mm)Depth of damping hole (mm)Double-row damping holes (mm)Change the distance between the damping hole and the explosion source1300809015000150080901500017008090150001900809015000Change the diameter of the damping hole190060901500019008090150001900100901500019001209015000Change the damping hole spacing190080601500019008090150001900801201500019008015015000Change the depth of the damping hole1900809012000190080901500019008090180001900809021000Switch to double-row damping holes19008090150050190080901500100190080901500150190080901500200Figure 4
General arrangement of vibration damping holes.Figure 5
Layout of local damping holes.


## 2.5.2. Feasibility Verification of Numerical Calculation Model

According to the research of several articles [12, 13], the vertical vibration velocity can be used to replace the three vector vibration velocities in practical engineering to simplify the control standard. Therefore, only the Z direction is selected as the research, as shown in Table 4. It can be seen from the table that, except for the peak vibration velocity in the Z direction of the measuring point directly above the tunnel, the numerical simulation results of all other measuring points are greater than the field monitoring data. The reason for this phenomenon is that the numerical model has made considerable assumptions and optimization under complex geological conditions. For example, the fractures of underground rocks and strata have made isotropic assumptions, while ignoring the influence of groundwater. The error of the field monitoring and numerical simulation results in the table is within 10%, which ensures that the numerical calculation can well restore the actual situation of the field, and makes full preparations for the later research on the parameters of the damping hole, so as to ensure the correctness of the later numerical calculation model.Table 4
Comparison of monitoring data and numerical simulation results.
Measuring pointMaximum single-stage dose (kg)Burst distance (m)Maximum vibration velocity inZ direction (cm/s)Combined velocity maximum vibration velocity (cm/s)On-site monitoring resultsNumerical simulation resultsError (%)On-site monitoring resultsNumerical simulation resultsError (%)Measuring point directly above the tunnel1.6300.3260.3153.370.3130.3284.79Shuxiang building measuring point310.2490.2687.630.2640.2785.30Security room measuring point460.007850.008629.800.008460.009239.10


## 2.6. Analysis of Numerical Simulation Results

Based on the comparison between Section 2.5 and the actual data and the research of scholars, the peak vibration velocity in theZ direction of five factors is obtained, as shown in Table 5. Five groups of curves are formed according to 21 groups of data, and the measuring point directly above the tunnel with large vibration velocity and small error is selected to analyze the change of vibration velocity, as shown in Figures 6–10.(1)
It can be clearly seen from Table5 that the peak vibration velocity of the tunnel face without damping holes is higher than that of the tunnel face with damping holes. Therefore, the setting of damping holes has an obvious effect on reducing the vibration wave velocity generated by blasting.It can be found from Figure6 that the peak vibration velocity decreases with the increase of the hole diameter, indicating that the vibration reduction effect exists with the increase of the hole diameter. From the reduction range, it can be found that the vibration reduction effect is still decreasing when the hole diameter is greater than 120 mm. Therefore, it can be seen that the vibration reduction hole with a larger hole diameter can be arranged in the project.(2)
It can be seen from Figure7 that the peak vibration velocity generally decreases with the increase of the distance between the vibration damping hole and the explosion source, while the lowest peak vibration velocity appears at the measuring point directly above the tunnel at about 1700 mm. It can be seen that there is also a certain threshold for this factor. It is not that the closer the vibration damping hole is to the explosion source, the better the vibration damping effect is. It is recommended to set the vibration damping hole at about 1700 mm.(3)
It can be seen from Figure8 that the curve of the measuring point changes greatly, and the peak vibration velocity fluctuates with the increase of hole spacing. There is a minimum vibration velocity at 120 mm, a peak value at 60–90 mm, and an upward trend after 150 mm. It can be seen that the change of hole spacing will have a more complex vibration velocity change, so this factor will have a great impact on the peak vibration velocity.(4)
It can be found from Figure9 that when the hole depth reaches about 1300 mm, the vibration speed reaches the lowest. Then, with the increase of hole depth, the peak vibration speed continues to rise, and the rising momentum does not decrease. Here, it is mainly because the deeper the vibration damping hole is, there will be a certain cavity effect, which will increase the vibration speed.(5)
It can be seen from Figure10 that the peak vibration velocity of double row damping holes decreases first and then increases with the increase of row spacing; the lowest vibration velocity appears at about 50 mm, and then the curve shows a gentle trend at about 200 m. The row spacing of 50 mm is rarely used in practical engineering. After 150 mm, the peak vibration velocity changes little, and the vibration velocity does not change when the row spacing continues to increase. Therefore, the change of this factor will not have a great impact on the vibration velocity.Table 5
Summary table of the peak vibration speed in theZ direction of each influencing factor.
Influencing factorsParameter (mm)Shuxiang building measuring pointSecurity bureau measuring pointMeasuring point directly above the tunnelPeak vibration velocity inZ direction (cm/s)Peak vibration velocity inZ direction (cm/s)Peak vibration velocity inZ direction (cm/s)No damping holes00.26870.00860.3149Vibration hole diameter600.25670.00810.2903800.24550.00720.27901000.24030.00580.26601200.23200.00540.2475The distance between the vibration damping hole and the explosion source13000.26530.00840.303715000.26040.00820.294717000.25350.00800.281719000.25000.00790.2859Vibration damping hole spacing600.25330.00760.2697900.24550.00720.27901200.23480.00630.26721500.24290.00690.2737Vibration damping hole depth12000.24550.00720.279015000.25800.00780.289918000.26070.00820.280321000.24140.00710.2745Double row vibration damping hole row spacing500.24550.00720.27901000.25420.00770.28451500.26010.00800.29672000.21400.00610.2437Figure 6
Peak vibration velocity of damping hole diameter.Figure 7
The distance between the damping hole and the explosion source vs. peak vibration velocity.Figure 8
Peak vibration velocity of the damping hole spacing.Figure 9
Peak vibration velocity of the damping hole depth.Figure 10
Peak vibration velocity of double-row damping hole row spacing.


## 3. Grey Correlation Analysis of Damping Hole Parameters on Vibration Reduction Effect

### 3.1. Grey Relational Analysis Principle

In this section, the grey correlation analysis method is used to study the sensitivity of five damping hole factors affecting the vibration speed and analyze the influence degree of different factors on the vibration speed, so as to determine which influencing factor can affect the vibration speed most. Grey correlation analysis can determine the correlation degree of relevant factors. Under the condition of less data, less information, and unclear relationship, it can judge the correlation degree of relevant factors and comparative factors with the help of the analysis results. Grey correlation analysis theory is an important part of grey system theory, and correlation degree is the correlation between reference factors and influencing factors. The greater the correlation, the greater the value of the correlation degree, and the more sensitive the influencing factor is to comparative factors [14–16]. The basic principles and specific steps of grey correlation analysis are as follows:(1)
Determining the number sequence matrix of relevant factor variables and the data sequence matrix of system characteristic variables: set the number sequence of relevant factors asX (the influencing factors of the vibration reduction effect will be used as X below) and the reference number sequence as y (the peak vibration velocity in the Z direction will be used as Y):(1)X=X1,X2,…,Xm∂;=x11x12…x1nx21x22…x2n⋮⋮⋮⋮xm1xm2…xmn,Y=Y1,Y2,…,Ym∂;=y11y12…y1ny21y22…y2n⋮⋮⋮⋮ym1ym2…ymn.(2)
Dimensionless matrix: because the dimension levels of each factor in each sequence are different, and the associated information is scattered, it is necessary to dimensionless the sequence of relevant factor variablesX and the sequence of system characteristic variables Y [17, 18], so that the overall comparative analysis can be carried out. Then,(2)X′j=Xi′1,Xi′2,…,Xi′n.Thus,(3)Xi′j=Xij−minXijmaxXij−minXij.(3)
Obtaining the difference information of the matrix:(4)Δij=Yi′j−Xi′j.The minimum and maximum values are obtained in a matrix of dissimilarities:(5)Δmax=Δijmax,Δmin=Δijmin.(4)
Solving the correlation coefficient matrix and grey correlation degree: the correlation coefficient is calculated as follows:(6)γij=Δmin+ζΔmaxΔij+ζΔmax.In the formula,ζ is the resolution coefficient, which is generally taken as ζ = 0.5.If the data of any column and each row is greater than all the columns at the same time, the factor corresponding to the column change is the optimal factor; if the data of any row and each column is greater than all the columns at the same time, the characteristic corresponding to the change of row is the optimal characteristic; if there is no optimal factor and optimal characteristic, the average value of the correlation coefficient can be taken as the correlation degree.The degree of correlation can be calculated by the following formula:(7)Ai=1n∑j=1nrij.The correlation degree is kept in the range of (0, 1), and the result of the correlation degree mainly reflects the ability of the influencing factors to affect the whole among all factors. By comparing the data inAi, they are arranged from large to small. The larger the correlation, the better the correlation. The higher the value, the stronger the ability to affect the whole.


### 3.2. Grey Correlation Analysis of Influencing Factors of Damping Effect of Damping Hole

According to the calculation results of numerical experiments, the related factor variables and system characteristic variables of each influencing factor are arranged as shown in Table6. Here, X1 is the vibration damping hole diameter (mm), X2 is the distance between the vibration damping hole and the explosion source (mm), X3 is the vibration damping hole spacing (mm), X4 is the vibration damping hole depth (mm), X5 is the double row vibration-damping hole row spacing (mm), Y1 is the peak vibration velocity in the Z direction of the measuring point of Shuxiang Building (mm), Y2 is the peak vibration velocity in the Z direction of the measuring point in the Security Bureau (mm), and Y3 is the peak vibration velocity in the Z direction at the measuring point just above the tunnel (mm).Table 6
Correlation factor variables and system characteristic variables.
Serial numberRelated factor variablesSystem characteristic variablesX1X2X3X4X5Y1Y2Y30000000.26870.00860.31491601900120120000.25670.00810.29032801900120120000.24550.00720.27931001900120120000.24030.00580.26641201900120120000.2320.00540.24755801300120120000.26530.00840.30376801500120120000.26040.00820.29477801700120120000.25350.0080.28178801900120120000.250.00790.2759980190060120000.25330.00760.26971080190090120000.24550.00720.27911801900120120000.23480.00630.267212801900150120000.24290.00690.273713801900120120000.24550.00720.27914801900120150000.2580.00780.289915801900120180000.26070.00820.280316801900120210000.24140.00710.2745178019001201200500.24550.00720.2791880190012012001000.25420.00770.28451980190012012001500.26010.0080.29672080190012012002000.2140.00610.2437According to the grey correlation principle and the above specific steps, the data in Table6 are converted into interval relative values, the matrix is dimensionless, the difference matrix is obtained [19], and finally, the grey absolute correlation calculation results are obtained as shown in Table 7.Table 7
Results of grey absolute association analysis.
CorrelationVibration hole diameterThe distance between the vibration damping hole and the explosion sourceVibration damping hole spacingVibration damping hole depthDouble row vibration-damping hole row spacingShuxiang building measuring point0.78470.87180.97860.59090.5191Security bureau measuring point0.80420.84800.98890.59710.5204Measuring point directly above the tunnel0.83020.82060.95050.60540.5221Σ2.41912.54042.91801.79341.5616The grey absolute correlation matrix corresponding to Table7 is as follows:(8)A=0.78470.87180.80420.84800.83020.82060.97860.59090.98890.59710.95050.60540.51910.52040.5221.


### 3.3. Result Analysis

In matrixA, it is found that the data in the third column are larger than each column at the same time, so the vibration damping hole corresponding to the third column is the optimal factor. Hence, the vibration damping hole spacing corresponding to the third column has the most significant influence on the vibration damping effect.It can also be seen from matrixA that the sensitivity of the damping hole diameter and distance from the explosion source corresponding to the first and second columns is significantly higher than that of the fourth and fifth columns. Therefore, the damping effect is mainly related to the damping hole spacing, diameter, and distance from the explosion source. Among them, the hole spacing of the damping holes has the greatest influence on the damping effect, which is also in line with the conclusions of most scholars.


### 3.4. Engineering Examples

Through grey correlation analysis, it is found that the hole spacing, distance from the explosion source, and hole diameter have a great influence on the vibration reduction effect. However, it is necessary to further verify the accuracy of the results through examples. According to the engineering background of the first section, the test will reflect the magnitude of the vibration reduction effect and verify the accuracy of the analysis by changing the five influencing factors.From the curve in Section2.6, it can be found that the upper and lower amplitudes of the peak vibration velocity can reflect the vibration reduction effect well. Therefore, the concept of vibration reduction rate η is introduced here to reflect the ability of vibration reduction amplitude:(9)η=v1−v2v2.In the formula,η is the vibration reduction rate, %; v1 is the vibration velocity of the measuring point without vibration damping holes, cm/s; and v2 is the vibration velocity of the measuring point when there is a damping hole, cm/s.To prevent the influence of the farther away from the explosion source, the smaller the vibration speed and the greater the change of the vibration reduction rate, so the measurement point of the secret bureau is removed. Because the distance from the explosion source is closer, the larger the vibration speed, the smaller the vibration reduction rate, and the smaller the change. Therefore, the measurement point just above the tunnel is removed and finally the measurement point of the Shuxiang building is selected, see Table8 and Figure 11.Table 8
Field test data and corresponding damping rate.
Influencing factorsParameter (mm)Shuxiang building measuring pointSpeed (cm/s)Vibration reduction rate (%)No damping holes00.328Vibration hole diameter600.27316.7800.27516.31000.27915.01200.28214.2The distance between the vibration damping hole and the explosion source13000.2911.515000.28812.417000.2814.719000.27317.0Vibration damping hole spacing600.25522.4900.26319.91200.27117.61500.2814.8Vibration-damping hole depth12000.3212.115000.3125.018000.30110.621000.29310.6Double row vibration-damping hole row spacing500.27914.91000.28612.91500.3154.12000.3241.2Figure 11
Each factor and the corresponding damping rate.From Figure11, it is found that the damping rate of each parameter can change greatly. The maximum variation of the damping rate of damping hole diameter is 16.7%. The maximum variation of the damping rate of the distance between the damping hole and the explosion source is 15.1%. The maximum variation of the damping rate of the distance between the damping holes is 21.1%, and the maximum variation of the damping rate of the depth of the damping holes is 10.1%. The maximum variation of damping rate of double row damping hole row spacing is 13.6%. According to the analysis of 2.6, although the hole depth and double row spacing vibration reduction rate also change greatly, it is not suitable for the project. Through a comprehensive comparison of the vibration reduction rate, the vibration reduction rate of the spacing between the vibration reduction holes is the largest and has the greatest impact on the vibration reduction effect, which is also in line with the conclusion in Section 3.3.It can be seen from Figure11 that the increased range of damping rate of damping hole diameter is small, and the growth trend of damping rate does not decrease, which is in line with the analysis of Figure 6. The damping rate of the distance between the damping hole and the explosion source is generally smooth, and the change of the damping rate is small, which is in line with the change law of Figure 7. The damping rate of the spacing between the damping holes is the largest, and the overall change is obvious, which is in line with the above conclusions. The overall vibration reduction rate of the depth of the vibration reduction hole has little change, which is in line with the analysis results in Figure 9. The damping rate of double row damping holes changes greatly, but the overall damping rate is low, and the subsequent reduction of damping rate is not obvious, which is in line with the analysis in Figure 10.


## 3.1. Grey Relational Analysis Principle

In this section, the grey correlation analysis method is used to study the sensitivity of five damping hole factors affecting the vibration speed and analyze the influence degree of different factors on the vibration speed, so as to determine which influencing factor can affect the vibration speed most. Grey correlation analysis can determine the correlation degree of relevant factors. Under the condition of less data, less information, and unclear relationship, it can judge the correlation degree of relevant factors and comparative factors with the help of the analysis results. Grey correlation analysis theory is an important part of grey system theory, and correlation degree is the correlation between reference factors and influencing factors. The greater the correlation, the greater the value of the correlation degree, and the more sensitive the influencing factor is to comparative factors [14–16]. The basic principles and specific steps of grey correlation analysis are as follows:(1)
Determining the number sequence matrix of relevant factor variables and the data sequence matrix of system characteristic variables: set the number sequence of relevant factors asX (the influencing factors of the vibration reduction effect will be used as X below) and the reference number sequence as y (the peak vibration velocity in the Z direction will be used as Y):(1)X=X1,X2,…,Xm∂;=x11x12…x1nx21x22…x2n⋮⋮⋮⋮xm1xm2…xmn,Y=Y1,Y2,…,Ym∂;=y11y12…y1ny21y22…y2n⋮⋮⋮⋮ym1ym2…ymn.(2)
Dimensionless matrix: because the dimension levels of each factor in each sequence are different, and the associated information is scattered, it is necessary to dimensionless the sequence of relevant factor variablesX and the sequence of system characteristic variables Y [17, 18], so that the overall comparative analysis can be carried out. Then,(2)X′j=Xi′1,Xi′2,…,Xi′n.Thus,(3)Xi′j=Xij−minXijmaxXij−minXij.(3)
Obtaining the difference information of the matrix:(4)Δij=Yi′j−Xi′j.The minimum and maximum values are obtained in a matrix of dissimilarities:(5)Δmax=Δijmax,Δmin=Δijmin.(4)
Solving the correlation coefficient matrix and grey correlation degree: the correlation coefficient is calculated as follows:(6)γij=Δmin+ζΔmaxΔij+ζΔmax.In the formula,ζ is the resolution coefficient, which is generally taken as ζ = 0.5.If the data of any column and each row is greater than all the columns at the same time, the factor corresponding to the column change is the optimal factor; if the data of any row and each column is greater than all the columns at the same time, the characteristic corresponding to the change of row is the optimal characteristic; if there is no optimal factor and optimal characteristic, the average value of the correlation coefficient can be taken as the correlation degree.The degree of correlation can be calculated by the following formula:(7)Ai=1n∑j=1nrij.The correlation degree is kept in the range of (0, 1), and the result of the correlation degree mainly reflects the ability of the influencing factors to affect the whole among all factors. By comparing the data inAi, they are arranged from large to small. The larger the correlation, the better the correlation. The higher the value, the stronger the ability to affect the whole.


## 3.2. Grey Correlation Analysis of Influencing Factors of Damping Effect of Damping Hole

According to the calculation results of numerical experiments, the related factor variables and system characteristic variables of each influencing factor are arranged as shown in Table6. Here, X1 is the vibration damping hole diameter (mm), X2 is the distance between the vibration damping hole and the explosion source (mm), X3 is the vibration damping hole spacing (mm), X4 is the vibration damping hole depth (mm), X5 is the double row vibration-damping hole row spacing (mm), Y1 is the peak vibration velocity in the Z direction of the measuring point of Shuxiang Building (mm), Y2 is the peak vibration velocity in the Z direction of the measuring point in the Security Bureau (mm), and Y3 is the peak vibration velocity in the Z direction at the measuring point just above the tunnel (mm).Table 6
Correlation factor variables and system characteristic variables.
Serial numberRelated factor variablesSystem characteristic variablesX1X2X3X4X5Y1Y2Y30000000.26870.00860.31491601900120120000.25670.00810.29032801900120120000.24550.00720.27931001900120120000.24030.00580.26641201900120120000.2320.00540.24755801300120120000.26530.00840.30376801500120120000.26040.00820.29477801700120120000.25350.0080.28178801900120120000.250.00790.2759980190060120000.25330.00760.26971080190090120000.24550.00720.27911801900120120000.23480.00630.267212801900150120000.24290.00690.273713801900120120000.24550.00720.27914801900120150000.2580.00780.289915801900120180000.26070.00820.280316801900120210000.24140.00710.2745178019001201200500.24550.00720.2791880190012012001000.25420.00770.28451980190012012001500.26010.0080.29672080190012012002000.2140.00610.2437According to the grey correlation principle and the above specific steps, the data in Table6 are converted into interval relative values, the matrix is dimensionless, the difference matrix is obtained [19], and finally, the grey absolute correlation calculation results are obtained as shown in Table 7.Table 7
Results of grey absolute association analysis.
CorrelationVibration hole diameterThe distance between the vibration damping hole and the explosion sourceVibration damping hole spacingVibration damping hole depthDouble row vibration-damping hole row spacingShuxiang building measuring point0.78470.87180.97860.59090.5191Security bureau measuring point0.80420.84800.98890.59710.5204Measuring point directly above the tunnel0.83020.82060.95050.60540.5221Σ2.41912.54042.91801.79341.5616The grey absolute correlation matrix corresponding to Table7 is as follows:(8)A=0.78470.87180.80420.84800.83020.82060.97860.59090.98890.59710.95050.60540.51910.52040.5221.


## 3.3. Result Analysis

In matrixA, it is found that the data in the third column are larger than each column at the same time, so the vibration damping hole corresponding to the third column is the optimal factor. Hence, the vibration damping hole spacing corresponding to the third column has the most significant influence on the vibration damping effect.It can also be seen from matrixA that the sensitivity of the damping hole diameter and distance from the explosion source corresponding to the first and second columns is significantly higher than that of the fourth and fifth columns. Therefore, the damping effect is mainly related to the damping hole spacing, diameter, and distance from the explosion source. Among them, the hole spacing of the damping holes has the greatest influence on the damping effect, which is also in line with the conclusions of most scholars.


## 3.4. Engineering Examples

Through grey correlation analysis, it is found that the hole spacing, distance from the explosion source, and hole diameter have a great influence on the vibration reduction effect. However, it is necessary to further verify the accuracy of the results through examples. According to the engineering background of the first section, the test will reflect the magnitude of the vibration reduction effect and verify the accuracy of the analysis by changing the five influencing factors.From the curve in Section2.6, it can be found that the upper and lower amplitudes of the peak vibration velocity can reflect the vibration reduction effect well. Therefore, the concept of vibration reduction rate η is introduced here to reflect the ability of vibration reduction amplitude:(9)η=v1−v2v2.In the formula,η is the vibration reduction rate, %; v1 is the vibration velocity of the measuring point without vibration damping holes, cm/s; and v2 is the vibration velocity of the measuring point when there is a damping hole, cm/s.To prevent the influence of the farther away from the explosion source, the smaller the vibration speed and the greater the change of the vibration reduction rate, so the measurement point of the secret bureau is removed. Because the distance from the explosion source is closer, the larger the vibration speed, the smaller the vibration reduction rate, and the smaller the change. Therefore, the measurement point just above the tunnel is removed and finally the measurement point of the Shuxiang building is selected, see Table8 and Figure 11.Table 8
Field test data and corresponding damping rate.
Influencing factorsParameter (mm)Shuxiang building measuring pointSpeed (cm/s)Vibration reduction rate (%)No damping holes00.328Vibration hole diameter600.27316.7800.27516.31000.27915.01200.28214.2The distance between the vibration damping hole and the explosion source13000.2911.515000.28812.417000.2814.719000.27317.0Vibration damping hole spacing600.25522.4900.26319.91200.27117.61500.2814.8Vibration-damping hole depth12000.3212.115000.3125.018000.30110.621000.29310.6Double row vibration-damping hole row spacing500.27914.91000.28612.91500.3154.12000.3241.2Figure 11
Each factor and the corresponding damping rate.From Figure11, it is found that the damping rate of each parameter can change greatly. The maximum variation of the damping rate of damping hole diameter is 16.7%. The maximum variation of the damping rate of the distance between the damping hole and the explosion source is 15.1%. The maximum variation of the damping rate of the distance between the damping holes is 21.1%, and the maximum variation of the damping rate of the depth of the damping holes is 10.1%. The maximum variation of damping rate of double row damping hole row spacing is 13.6%. According to the analysis of 2.6, although the hole depth and double row spacing vibration reduction rate also change greatly, it is not suitable for the project. Through a comprehensive comparison of the vibration reduction rate, the vibration reduction rate of the spacing between the vibration reduction holes is the largest and has the greatest impact on the vibration reduction effect, which is also in line with the conclusion in Section 3.3.It can be seen from Figure11 that the increased range of damping rate of damping hole diameter is small, and the growth trend of damping rate does not decrease, which is in line with the analysis of Figure 6. The damping rate of the distance between the damping hole and the explosion source is generally smooth, and the change of the damping rate is small, which is in line with the change law of Figure 7. The damping rate of the spacing between the damping holes is the largest, and the overall change is obvious, which is in line with the above conclusions. The overall vibration reduction rate of the depth of the vibration reduction hole has little change, which is in line with the analysis results in Figure 9. The damping rate of double row damping holes changes greatly, but the overall damping rate is low, and the subsequent reduction of damping rate is not obvious, which is in line with the analysis in Figure 10.


## 4. Conclusion

(1)
The analysis shows that the vibration damping effect can be greatly improved by setting the vibration damping holes on the face of the tunnel, and a method of damping vibration from the blast source is proposed. The research results will provide a reference vibration reduction technology for the blasting construction of urban shallow tunnels and have certain reference value and guiding significance for the blasting scheme design and vibration control of similar projects.(2)
The comprehensive analysis shows that the vibration damping hole spacing, diameter, and the distance from the vibration damping hole to the explosion source have obvious effects on the vibration damping effect, and the remaining factors have little influence, among which the vibration damping hole spacing plays the main role.(3)
In the actual project, it is recommended to select 120 mm for the hole spacing of the vibration damping hole; the recommended distance from the vibration damping hole to the explosion source is 2000 mm; the recommended diameter of the vibration damping hole is more than 120 mm; the recommended hole depth is 1200–1500 mm; and the recommended row spacing of double row damping holes is 50–100 mm.In this paper, only a single vibration reduction measure is studied, and only the research method of changing a single variable is used. However, in the actual project, the field situation is more complicated, and it is difficult to ensure that only a single factor is changed. The next step will be to study the simultaneous changes of various damping hole parameters and various damping methods.



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*Source: 8442130-2022-06-27.xml*