Analysis of the Tunneling Blast Safety Criterion Based on Longitudinal Shocks

. In modern tunneling construction, the blasting seismic efect seriously threatens the safety and stability of the existing engineering construction. Meanwhile, due to the complexity of blasting loading, there is a very large diference from real blasting by using the sample triangle dynamic loading model or trapezoid dynamic loading model to analyze the blasting vibration problem. In this paper, based on the analysis of the pressure change, volume expansion, fracture development, and blasting gas motion, an accurate blasting loading model was proposed. Ten, adopting plane longitudinal shock theory, the coupled loading of multiple holes can be obtained. Subsequently, the 3D simulation analysis of the tunnel blasting shows an error of 2% compared with on-site monitoring data, meeting engineering requirements. Finally, based on the simulation results, regression prediction function of the velocity curve and efective tensile stress curve under diferent safety criteria are established, achieving accurate prediction of the damage degree of the host rock and liner line under diferent engineering requirements.


Introduction
With the swift advancement of blasting technology, as well as its excellent adaptability and afordability in various geological conditions, tunnel engineering construction by blasting accounts for over 95% of mountain tunnel construction [1]. Te drilling and blasting method has become the most commonly used construction method in tunnel engineering construction. However, although the blasting method has brought signifcant changes to tunnel engineering construction, vibration disasters are also a major concern, especially due to the seismic efect of blasting, which is the foremost disaster [2][3][4][5][6]. Terefore, predicting the propagation law of blasting vibration waves has become a crucial aspect of blasting problems research [7]. At present, although scholars have made several eforts in studying the blasting propagation law, most of the studies are based on on-site monitoring data and simplifed empirical formulas, which are too idealistic and lack accuracy in refecting the actual engineering blasting vibration situation [8][9][10]. At the same time, some scholars also use new research methods to study the blasting vibration law, such as using neural network technology [11][12][13], signal noise reduction [14,15], transient fnite element analysis [16], coupled Lagrange and Euler [17], Jones-Wilkins-Lee (JWL) equation of state [18], and wavelet packet decomposition [19] to summarize the blasting vibration law. However, these methods ignore the infuence of vibration frequency and duration, resulting in certain limitations. Terefore, owing to its high accuracy and afordability, numerical simulation technology has increasingly developed into an important means for engineering analysis and prediction. Many scholars worldwide have employed diferent numerical simulation software to simulate and analyze blasting [20][21][22][23][24][25][26][27]. Nonetheless, most of these simulations are too simplistic to refect the actual situation of piecewise diferential blasting and do not well analyze the interaction of loads between blasting holes.
Tis paper presents the establishment of a mathematical calculation and analysis model by analyzing the change in blasting gas pressure, the expansion of blasting hole volume, the development of surrounding rock fssures, and the movement of blasting gas in a single blasting hole, as well as the accurate form of blasting dynamic load change over time. Using the exact blast load form of a single hole, the plane longitudinal shock wave interaction model is employed to determine the blasting load relationship under the interaction of porous holes. Finally, the fnite element numerical simulation software FLAC3D is utilized to construct a 3D tunnel model, and the blasting safety criterion is obtained through the efective stress curve formula.

Expression of the Blasting Load Velocity
Rock is considered an incompressible medium, and it is assumed that no additional energy loss occurs during blasting. Terefore, the blasting vibration wave of a single hole in the medium propagates outward in the form of longitudinal wave concentric circles, as depicted in Figure 1.
According to the dynamic gradient theory, the propagation velocity in surrounding rock is shown in equation (28).
where V s � ln(L + r � (r/2r 0 ), V is the circular propagation velocity of blasting, ρ 0 is the density of the explosive, Q is the energy of blasting, ρ r is the density of the rock, L 0 is the charge length of the blasting hole, r 0 is the radius of the blasting hole, and r is the distance from the blasting center.

Accurate Load Model for Blasting Analysis
Te load exerted on the wall of the blasting hole after blasting is an intricate process. Following the principle of explosive initiation and action, the entire blasting process can be roughly classifed into four stages: rapid increase of blasting pressure, expansion of blasting hole volume, rapid ejection of gas along the hole mouth, and completion of excavation of fracture.

Te Rise Stage of the Blasting Load.
After blasting, the blasting gas pressure continues to increase, resulting in a continuous increase in the dynamic load acting on the wall of the blasting hole. Tis study shows that when the blasting wave reaches the bottom cross section of the blasting hole, the blasting load attains the maximum, and the maximum load can be analyzed by the Chapman-Jouerger model [29].
where P D is the bursting pressure, V D is the velocity of the explosion gas, c is the specifc heat capacity ratio of the explosion gas, and c � 3.0 in this paper. For the uncoupled charge, the initial burst pressure can be expressed as follows: where a is the diameter of the explosive cartridge and b is the diameter of the blasting hole.
Te rise time of the load is as follows:

Volume Expansion of the Blast Hole.
Within the blasting hole, prior to the ejection of flling material, the blasting gas causes the surrounding rock to expand, leading to the outward movement of the flling material, and an increase in the cavity volume. Te volume increase can be determined by the following formula: where r(t) is the radius of the blasting hole as a function of time, u(t) is the expansion rate of the hole wall with time, ω(η) is the width of the crack, L a is the length of the crack, and y(t) is the displacement of the flling. According to the gas fxing rate, the relationship between the gas pressure in the blasting chamber and the volume change is as follows [30]: Substituting equation (5), the rule of volume change with time, into equation (6) yields where V 0 is the initial volume of the blasting hole: A, B, R 1 , R 2 , and ω are the parameters of the blasting materials, and E 0 is the initial energy of the explosive.

Blasting Gas Rapid Overfow.
Once the flling material in the blasting hole is ejected or in nonflling blasting, highpressure blasting gas will overfow rapidly along the hole mouth. According to gas dynamics theory, the blasting hole at this stage can be simplifed into the bottle-like structure as shown in Figure 2.
In the fgure, V 0 , P 0 , ρ 0 , T 0 , ] 0 are the initial volume, initial pressure, initial density, initial temperature, and initial velocity of gas in the late blasting stage, respectively, and P e , ρ e , T e , ] e are the pressure, density, temperature, and velocity at the outlet, respectively. Based on gas dynamics theory, the airfow formula is as follows: Assuming that the entire gas overfow process is an adiabatic process, the frst law of thermodynamics is written as follows: Put the fow equation (8) into equation (9), and the following is given by the following expression:

Whole Process Analysis of the Blasting Load.
At the end of blasting, the blasting energy, exerted by the blasting load, causes the surrounding rock fssure of the blasting hole to open, and the excavation of the surrounding rock mass is completed, resulting in the instantaneous scattering of the blasting gas and a rapid reduction of gas pressure to zero. Trough the decomposition of the abovementioned whole blasting process and step-by-step analysis, the specifc form of the blasting load action can be obtained, as shown in Figure 3.

Longitudinal Shock Wave Interaction Analysis
Trough the abovementioned analysis, it can be seen that the blasting load in a single hole is in a nonlinear form, so the dynamic load can be approximately decomposed into the relation between amplitude P ′ and its second derivative P ″ , which is expressed in the following form [31]: Te plane wave in the Lagrangian coordinate system can be expressed as follows: where c 0 � −0.1 × 10 5 s − 1 is the coefcient of wave propagation and G � (1/ρ 0 c 2 0 )[1 + (1/2)ρ 0 ]. After calculation, where the subscripts 1 and 2 represent the two interacting longitudinal shock waves. A large number of experimental studies have shown that, generally, the blasting crushing zone in a rock mass is 3-5 times the radius of the blasting hole, while the blasting fracture zone is 10-15 times the radius of the blasting hole [32]. In the piecewise diferential blasting of a tunnel, the direct blasting hole is usually 40 mm, and the distance between two blasting holes is approximately 60 cm, so the shock wave of the blasting hole will only be afected by the shock wave of adjacent holes. Te load inside the blasting under the action of adjacent blasting holes established according to the longitudinal shock wave model is as follows:  with a coil diameter of 32 mm, a coil length of 200 mm, and a coil weight of 150 g. Te conventional blasting method was employed, with blasting holes, including cut holes, auxiliary holes, caving holes, peripheral holes, and bottom holes arranged in the tunneling end face. Among them, the cut hole is located in the middle and lower positions of the end face, and the vertical wedge cut is adopted. Te spacing range of the holes was mostly 0.5-0.7 m, and the hole depth of the surrounding rock was 3.0 m. Te vault of the peripheral hole utilized the interval charging method, while continuous uncoupled charging was employed at other locations, with a charging length of 2.5 m. Microdiferential section blasting was used to carry out the blasting, with the specifc segmented form shown in Figure 4; the segmented time interval is shown in Table 1, and the specifc blasting parameters are shown in Table 2.

Physical and Mechanical Parameters.
Te lithology of the surrounding rock is conglomerate, which belongs to the Houcheng Formation of the Jurassic middle system. Te color is purplish red, and the structure is gravel. Te gravel composition in the rock is andesite, fused tuf, rhyolite, siliceous rock, and clay rock. Te gravel diameter is generally 2-60 mm, subribbed-subrounded, with good roundness and general sorting. Te intergravel backfll is composed of rock debris, fne sand, and stable accessory mineral sands, such as quartz and feldspar, which are homogeneous with the gravel composition. Iron and carbonate cement between the gravel and backfll. Te physical and mechanical parameters of the conglomerate are shown in Table 3.

Numerical Simulation Model and Data Analysis
Based on the actual engineering geological structure in the feld, 3D numerical values are applied. Te simulation software FLAC3D establishes the numerical model of the tunnel, as shown in Figure 5. Te whole model has 372,200 units and 386,527 nodes. Te model is 228 m long in the Xdirection, 203 m wide in the Y-direction, and 159 m high in the Z-direction.

Te Dynamic Parameters Applied in the Simulation.
In the numerical simulation, based on the site monitoring data and repeated trial calculation, the measured results are compared with the trial calculation results, and the damping value that is suitable for the engineering site is obtained. Te local damping coefcient of the selected calculation model in the rock and soil mass is 0.015.

Simulation Accuracy Verifcation.
In the simulation, based on the on-site monitoring results, the simulation duration is 2 s, and a total of 1324133 steps are calculated. After calculation, the velocity curve with the same location as the on-site monitoring site is extracted and compared with the actual on-site monitoring speed curve, as shown in Figures 6 and 7.
Trough the analysis of Figure 6, it can be concluded that the maximum peak intensity is 2.505 cm/s. In Figure 7, the peak intensity of feld monitoring data at the same location is 2.51 cm/s, and the error rate is 2%. Te arrival time of each peak of the two groups of curves was basically the same, which was between 0.3 s and 0.7 s. It can be seen from the comparative analysis that the numerical simulation based on this method is more accurate for analyzing the vibration of the surrounding rock around the blasting location and meets the analysis requirements.

Analysis of the Safety Criteria.
Te velocity curve and efective tensile stress curve of the rock surrounding the tunnel and along the tunnel design line are shown in Figures 8-11, respectively, and the distances from the blasting position along the tunnel design line are 10 m, 20 m, 30 m, 50 m, and 80 m, as shown in Table 4.
By summarizing the data in Table 4, the relation curves between the peak velocity and efective tensile stress in the surrounding rock and along the tunnel design line were drawn (typically at a distance of 20 m highlighted in bold in Table 4). Te details are shown in Figures 12 and 13. Figure 12 shows that the regression function of the peak velocity and efective tensile stress in the surrounding rock are as follows: According to the strength test, the dynamic tensile strength of sandstone is 4 MPa, so the surrounding rock will be damaged when the peak velocity is greater than 12.7 cm/s. Figure 13 shows that the regression function of the peak velocity and efective tensile stress along the tunnel design line are as follows:

Shock and Vibration 7
Te dynamic tensile strength is 4 MPa, so the rock mass will be damaged when the peak velocity is greater than 5.85 cm/s along the tunnel design line.

Conclusion
Tis paper conducted theoretical research on the longitudinal wave shocks in tunnel blasting hole and conducts a comparison analysis between the numerical modelling result and in situ monitoring data. Te following conclusion can be drawn: (1) By establishing computational and analytical models for the changes in blasting pressure inside the blasting hole, volume expansion of the blasting hole, development of the surrounding rock fractures, and movement of blasting gas, the exact form of the timevarying blasting dynamic load was obtained. Trough analysis and on-site monitoring verifcation, it can be seen that the variation of the dynamic load conforms more to the actual load variation than traditional triangular and trapezoidal load analysis, with signifcantly improved accuracy. (2) Te horizontal longitudinal shock wave interaction model was employed to analyze the porous load, and the simulation results were compared with actual monitoring data. Te results show that the simulation error is within 2%, indicating that the model meets the requirements of practical analysis. (3) By extracting the vibration velocity curve and effective tensile stress curve along the surrounding rock mass and tunnel design line, a blasting vibration safety criterion feld was established, which can comprehensively refect the damage situation of any point in the vibration feld.
However, the proposed model in this paper did not take into account the complex geological information in a wide range of geological conditions and instead viewed the rock stratum as a homogeneous entity, which may not be representative of all tunneling construction scenarios. Terefore, it is necessary to consider the rock stratum conditions of the tunnel in future research and provide more practical applications to validate the efectiveness of the model.

Data Availability
Te data that support the fndings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.  Shock and Vibration