Study on Vibration Effects and Optimal Delay Time for Tunnel Cut-Blasting Beneath Existing Railways

Abstract

With the development of underground space in urban areas, the demand for tunneling through existing railways is increasing. The adverse effects of cut-blasting during the construction of tunnels under crossing existing railways are investigated. Combined with the principle of blasting seismic wave superposition, LS-DYNA numerical simulation is used to analyze the seismic wave superposition law under different superposition methods. This study also investigates the vibration reduction effect of millisecond blasting for cut-blasting under the different classes of surrounding rocks. The results show that the vibration reduction forms of millisecond blasting can be divided into separation and interference of waveform. Based on the principle of superposition of blasting seismic waves, vibration reduction through wave interference is further divided. At the same time, a new vibration reduction mode is proposed. This vibration reduction mode can significantly improve construction efficiency while improving damping efficiency. The new vibration reduction mode can increase the vibration reduction to 80% while improving construction efficiency. Additionally, there is a significant difference in the damping effect of different classes of surrounding rock on the blasting seismic wave. Poor-quality surrounding rock enhances the attenuation of seismic wave velocity and peak stress in the surrounding rock. In the Zhongliangshan Tunnel, a tunnel cut-blasting construction at a depth of 42 m, the best vibration reduction plan of Class III is 3 ms millisecond blasting, in which the surface points achieve separation vibration reduction. The best vibration reduction plan of Class V is 1 ms millisecond blasting, in which the surface points achieve a new vibration reduction mode. During the tunnel blasting construction process, electronic detonators are used for millisecond blasting of the cut-blasting. This method can reduce the vibration effects generated by blasting. The stability of the existing railway is ultimately guaranteed. This can improve construction efficiency while ensuring construction safety. This study can provide significant guidance for the blasting construction of the tunnel through the railway.

1. Introduction

With the development of railway and highway transportation, it is common for new tunnels to be built close to existing structures, such as bridges, houses, and dams. The construction of new structures will have an impact on the neighboring buildings or sites. Urban subways are generally developed after the city planning; thus, crossing existing structural spaces becomes inevitable. The commonly used methods for subway tunnel excavation are shield construction and the drilling-blasting method [1,2]. Between them, blasting is an efficient and economical excavation method in hard rock formations. However, blasting results in vibration to neighboring structures, which may damage or even destroy neighboring structures [3]. To understand the effect of blasting on existing structures, this paper will investigate controlled blasting against the background of blasting construction in the Zhongliangshan Tunnel in Chongqing, which crosses under existing railways.

The impact of blasting on surrounding buildings is a major concern in tunnel blasting. Shan et al. [4] analyzed the vibration response characteristics of tunnel blasting by taking the blasting project of the Chongli tunnel, which crosses under the existing railway, as an example. Liang et al. [5] used monitoring data and numerical simulation to study the vibration of existing railway tunnels caused by blasting of the tunnel in the neighboring area. Zhou et al. [6] used the numerical simulation method to analyze the dynamic response of subway tunnels crossing under the existing railway tunnel by the full-face excavation method and the center diaphragm method. Jiang et al. [7] analyzed the impact of the blasting vibration effect of undersea tunnels on the structure of neighboring service tunnels and the characteristics of shock wave pressure distribution in the water excited by blasting vibration. Qin et al. [8] analyzed the influence of blasting construction on the vibration velocity and stress distribution of existing subway tunnels based on numerical simulation and on-site monitoring data. Xie et al. [9] analyzed the impact of more complicated working conditions, where a new tunnel construction crosses over the existing tunnel and under the existing railway. In addition, Li et al. [10] used the numerical simulation method to discuss the dynamic response of metro station pit blasting on adjacent foundations, and analyzed the influence of the length, width, depth of adjacent pits, and the depth of the blasting pit. Generally, a portion of the energy generated during tunnel blasting is utilized in the rock fragmentation process, while the remaining energy propagates through the rock medium as elastic waves. These waves subsequently trigger the dynamic response of the surrounding rock mass and the adjacent environment.

When the intensity of vibration exceeds the limit of the load-bearing capacity of the existing structure, blasting may significantly affect the structural stability of the structure and even cause a safety threat. Therefore, reducing the vibration effect has been the focus in the field of blasting [11,12]. Wang et al. [13] carried out the inversion of key parameters based on field data and the BP neural network, and analyzed the propagation and attenuation of tunnel blasting vibration waves under different blast center distances with the help of numerical simulation. Liu et al. [14] carried out two-hole time-delayed blasting of granite plates and analyzed the evolution of the explosive stress wave and vibration wave between the two holes. Xu et al. [15] compared the surface vibration velocity and instantaneous energy obtained from millisecond delayed initiation of cut-blasting and initiation of different charging schemes, and found that the free surface condition, the amount of the single section, and the depth of the tunnel had a significant effect on the surface vibration. Yang et al. [16] used a combination of theoretical analysis and case study to investigate the vibration frequency characteristics of blasting excavation in highly stressed rock masses, and discussed the vibration frequency characteristics and their influencing factors under the action of blast load and dynamic unloading. Yu et al. [17] established an attenuation equation for the peak particle vibration velocity by taking into account the attenuation characteristics of P-waves, S-waves, and R-waves in the blasting vibration wave. Xu et al. [18] turned to signal processing through MATLAB, explored the propagation law of blasting vibration in the stratum, and proposed a vibration reduction program to reduce the peak vibration velocity. In summary, peak vibration velocity is an important index to evaluate the hazard of blasting vibration, which is the key to ensuring construction safety.

Based on the understanding of stress wave propagation and attenuation, vibration control can be realized by controlled blasting. Wang et al. [19] analyzed the propagation and attenuation characteristics of blasting vibration in rock tunnels, and proposed a formula for predicting the peak particle velocity of adjacent tunnel sections, considering the amplification effect of the blasting vibration of adjacent tunnels by diffraction and reflection. Cheng et al. [20] analyzed the vibration response of the blasting construction sequence on the peak particle velocity (PPV) of the neighboring villages. The adjustment of the blasting sequence plays a positive role in controlling and reducing the blasting stress wave, and effectively reduces the stress propagated to the neighboring structures. Zhang et al. [21] used theoretical analysis and field test methods to study the safe control of the vibration reduction effect of high-frequency blasting vibration in existing tunnels and different blasting vibration reduction control techniques, which ensures the safe operation of the existing tunnels. In summary, limiting the peak vibration velocity of the mass at the surface can effectively reduce the vibration effect generated by the blasting construction of the tunnel crossing under the existing railway, thus guaranteeing the safety of the structure.

Millisecond blasting is one of the most effective methods to reduce the vibration effect of blasting. It mainly serves to disperse blasting energy by detonating explosives within milliseconds. Li et al. [22] proposed a method for predicting and controlling the blasting vibration spectrum of high slopes, based on a technique for predicting the time history of blast vibrations, and determined the optimal delay time. Chen et al. [23] discussed the effects of millisecond blasting time, charge length, and explosion speed on blasting vibration. Selecting the differential delay time reasonably is an effective way to control the peak vibration velocity. However, the optimal differential time interval depends on multiple factors. Liu et al. [24] considered the superposition of the wave offset effect and rock fragmentation effect, theoretically analyzing the optimal delay time selection of a reasonable range. Qiu et al. [25] analyzed the influence of delay time on the vector peak particle velocity of superimposed vibration waveforms and established a calculation formula for the minimum delay time between adjacent boreholes based on vibration reduction. This support is crucial for optimizing millisecond blasting parameters, ultimately enabling effective control of blasting vibration hazards. The superposition of blasting seismic waves is affected by many factors. Ishchenko et al. [26,27] explored the superposition of blasting waves in the case of confined geometries and variable cross-sectional charges, and divided different superposition regions according to different energies. Zhang et al. [28] analyzed the blasting vibration spectra with different delay times and core distances and studied the frequency distribution and energy characteristics of each frequency band. The results show that different delay times are needed to reduce the blasting vibration in different explosion distances. Extensive research has been conducted on blast wave superposition under various conditions. However, there are few studies on the superposition effect of blasting seismic waves at the ground surface.

This study is based on published research on the superposition effect of blasting seismic waves, and explores the engineering problems of blasting in tunnels crossing under existing railways. A solution has been proposed to reduce the blasting vibration effect. This solution involves designing a millisecond delay time to ensure that the main phases (especially peak values) of seismic waves generated by different boreholes superpose and interfere with each other when propagating to the protected object. Ultimately, the overall amplitude of the synthesized vibration is significantly reduced. Meanwhile, the propagation and superposition laws of blasting seismic waves at the ground surface were analyzed. The organizational structure of this paper is as follows. Section 2 presents the tunnel project background and details the numerical simulation and monitoring setup, with a validation of the simulation model. Section 3 proposes an interference-based vibration reduction method utilizing blasting seismic wave superposition, investigates the quantitative relationship between optimal millisecond delay time and waveform parameters, and simulates the vibration reduction effects of different superposition approaches. Section 4 analyzes the stress evolution law of surrounding rocks from Class III to Class V. At the same time, optimal delay times for different classes of surrounding rocks are explored.

2. Numerical Simulation of Blasting Vibration of Tunnel Crossing Under the Existing Railway

2.1. Engineering Background

Zhongliangshan Tunnel is a part of the interval of Chongqing Metro Line 15, with a tunnel length of 4905 m and a maximum depth of 313 m. The tunnel passes through sandstones, mudstones, and greywacke, encountering complex stratigraphic conditions. Through the coal seam mining area and faults, the surrounding rock classes III~V. In addition, the tunnel runs beneath the Chongqing-Guiyang Railway, the Caiyuanba-Geleshan Connecting Line, and other existing railway lines. The relative position of the new tunnel (Zhongliangshan Tunnel) and the Chongqing-Guiyang Railway are shown in Figure 1. The smallest depth under the Chongqing-Guiyang Railway section is 42.31 m. Zhongliangshan tunnel crosses several existing railways, and the geological conditions are complex. This is of great significance for blasting in tunnels crossing under existing railways. To assess the vibration impact of drilling and blasting operations during the construction of a tunnel beneath an existing railway and to develop an effective vibration mitigation strategy, a numerical simulation of the blasting process was conducted.

2.2. Numerical Model and Parameters

For the situation involving blasting around railways, the Chinese railway specification, Safety Technical Specification for Blasting Vibration of Railway Engineering (TB 10313-2019) [29], specifies the maximum permissible vibration velocity at different vibration frequencies. To analyze the effect of blasting on the ground vibration, a model containing the tunnel is established (Figure 2). The model size is 100 × 75 × 70 (m). The tunnel excavation was carried out using the two-step method with upper and lower benches, featuring a total diameter of 12 m and with the crown positioned 42 m below the ground surface. Six cutting holes were drilled at a spacing of 1.2 m, perpendicular to the tunnel heading face, with each hole measuring 80 mm in diameter and 2.2 m in depth. Moreover, three observation points were selected at the surface for subsequent analysis. The monitoring points consist of three key locations: (1) point A, positioned vertically above the vault; (2) point B₁, located 40 m horizontally from the unexcavated side of the tunnel face; and (3) point B₂, situated 40 m horizontally from the excavated side of the tunnel face. The initiation sequence of blastholes is indicated by numerical labels in Figure 2, with sequential detonation progressing according to the assigned serial numbers.

All models employ 8-node SOLID164 elements. In the dynamic finite element simulations, Lagrange grids are adopted for rock and lining, whereas Eulerian grids are utilized for explosives and air. The base mesh size is set at 2 m, with refinement to 20 cm around the tunnel periphery and further refinement to 2 cm near the excavation face. The model’s upper surface incorporates a free boundary condition, while all other surfaces are assigned non-reflective boundaries.

The millisecond blasting process in the tunnel excavation was simulated using LS-DYNA with a coupled Lagrange–Eulerian formulation. The Lagrange algorithm is used for rock and lining, and the Euler algorithm is used for explosives and air. The top surface of the model and the tunnel contour are set as free boundaries, and the rest are non-reflective boundaries. The computational model includes explosives, slurry, geotechnical surroundings, and lining materials. The parameters of the surrounding rock are shown in

Table 1. Physical parameters of the rock model.

Density (g/cm3)	Elastic Modulus (GPa)	Poisson’s Ratio	Yield Stress (GPa)
2.4	13	0.25	8 × 10−4

, and the material model is MAT_PLASTIC_KINEMATIC. The parameters of the lining are shown in

Table 2. Physical parameters of the lining model.

Density (g/cm3)	Elastic Modulus (GPa)	Shear Modulus of Elasticity (GPa)	Poisson’s Ratio	Tensile Strength(MPa)
2.6	0.003	0.015	0.3	1.43

, and the material model is MAT_JOHNSON_HOLMQUIST_CONCRETE.

The blast explosives are simulated using high-explosive materials, which are controlled by the ∗MAT_HIGH_EXPLOSIVE_BURN keyword. The Jones–Wilkins-Lee (JWL) equation is used to simulate the relationship between pressure and specific volume in the detonation process. The P-V relationship of the JWL equation is shown in Equation (1), and the parameters are selected in

Table 3. Parameters of model and state equation of the dynamite materials.

Density (g/cm−3)	A/GPa	R1	R2	ω	E0/GPa	V
1.63	3.71	4.15	0.95	0.3	0.07	1.0

.

$$\mathrm{P}=\mathrm{A}\left(1-{\displaystyle \frac{\mathsf{\omega }}{{\mathrm{R}}_1\mathrm{V}}}\right){\mathrm{e}}^{-{\mathrm{R}}_1\mathrm{V}}+\mathrm{B}\left(1-{\displaystyle \frac{\mathsf{\omega }}{{\mathrm{R}}_2\mathrm{V}}}\right){\mathrm{e}}^{-{\mathrm{R}}_2\mathrm{V}}+{\displaystyle \frac{\mathsf{\omega }{\mathrm{E}}_0}{\mathrm{V}}}$$

(1)

where A, B, R₁, and R₂, ω are the parameters of the explosive material; P is the pressure; V is the relative volume; and E₀ is the initial specific internal energy.

2.3. 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], Sadowski’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 α 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 Sadowski’s empirical formula. The equation is

$$\mathrm{V}=\mathrm{K}(\mathsf{\rho }{)}^{\mathsf{\alpha }}=\mathrm{K}({\mathrm{Q}}^{1/3}/\mathrm{R}{)}^{\mathsf{\alpha }}$$

(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 α is the vibration attenuation coefficient.

Taking logarithms on both sides of Sadowski’s formula, it can be transformed into the following,

$$\lg \mathrm{V}=\lg \mathrm{K}+\mathsf{\alpha }\lg ({\mathrm{Q}}^{1/3}/\mathrm{R})$$

(3)

The following linear regression equation is obtained:

$$\mathrm{Y}=\mathrm{a}\mathrm{X}+\mathrm{b}$$

(4)

Linear regression analysis of the site monitoring data (Figure 3a) yielded K = 219.28, α = 1.598, and correlation coefficient R² = 0.9997. Consequently, the Sadowski’s equation for the site is

$$\mathrm{V}=219.28({\mathrm{Q}}^{1/3}/\mathrm{R}{)}^{1.598}$$

(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 Sadowski’s formula derived from regression analysis.

Table 4. Monitoring information of measuring points.

R (m)	Q (kg)	V (cm/s)	Y (lgv)	X (lgρ)
230.9	107	0.43	−0.36051	−1.68708
233.2	107	0.44	−0.36957	−1.69137
222.1	107	0.47	−0.3279	−1.67023
203.5	70.8	0.43	−0.36351	−1.69207
205.3	70.8	0.42	−0.37366	−1.69575
213.5	94.6	0.48	−0.31695	−1.67087
198.1	73.5	0.46	−0.33724	−1.67479

shows the monitoring information of each measuring point. The comparison results are shown in Figure 3b. As illustrated in the figure, the theoretical values calculated using Sadowski’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.

3. The Mechanism and Effect of Vibration Reduction Through Waveform Interference

3.1. Blast Vibration Propagation Laws and Nodal Response Characteristics

In millisecond blasting operations, the new damage in the initial detonation hole may alter the detonation dynamics of subsequent holes. To quantify this influence on vibration waveform parameters, this study conducts a comparative analysis of waveform characteristics with large delay intervals. A numerical simulation of cut-blasting scenario with 25 ms interval was conducted, wherein six cutting holes were detonated sequentially. Ground vibration data were acquired from three strategically positioned monitoring points, with time-history analyses performed on velocity components along the X, Y, and Z axes.

As shown in Figure 4, there are six independent waveforms. Therefore, the superimposed effect of seismic waves generated by the six cut-blasting holes is weak. Then, when the charge weight in each hole is equal, the waveform and duration of single-hole blasting of one monitoring node do not change with increased blasts. The damage of previous detonation holes does not affect the surface waveform characteristics by subsequent detonation. Therefore, the superposition effect of multi-hole blasting blasts can be analyzed by single-hole blast seismic waves. The PPVs at each point are shown in

Table 5. PPV at each point.

Point	PPV (cm/s)
	X	Y	Z
A	6.42 × 10−1	2.62	1.51
B1	5.78 × 10−2	5.27 × 10−1	2.78 × 10−1
B2	6.89 × 10−2	2.22 × 10−1	9.71 × 10−2

.

The vibration velocities of the blasting seismic waves in the three directions of X, Y, and Z at the monitoring point are quite different. The vibration velocity in the Y direction is obviously larger than in the other two directions. Therefore, the Y direction is selected as the main research direction, which can better analyze the vibration effect of blasting seismic waves. Comparing Figure 4b and Figure 4c, the distances from nodes B₁ and B₂ to the tunnel face are the same, but there are differences in the waveforms and durations of the blasting seismic waves at the two nodes. This is because node B₁ is located on the excavated side of the tunnel. The excavated side of the tunnel forms a new free surface, which affects the propagation of the blasting seismic waves. This leads to the waveform difference between the excavated and unexcavated sides for the same distance.

When comparing the waveform characteristics at monitoring points A (directly above the tunnel face), B₁, and B₂, the waveform at point A exhibits higher amplitudes and a more compact shape. With increasing distance from the tunnel face, the waveforms at points B₁ and B₂ become more elongated, and their duration increases. This attenuation is primarily attributed to the energy dissipation within the rock mass during propagation. Specifically, the rock medium damps the seismic waves and absorbs their energy, while seismic scattering occurs in heterogeneous rock. Consequently, the dominant frequency of the seismic wave decreases, which is the fundamental reason for the observed waveform elongation and the increase in duration. Thus, the vibration intensity significantly decreases with distance. Therefore, distance is one of the key factors in reducing the vibration effect of tunnel blasting.

3.2. The Phase Division of Seismic Waveform by Blasting

Blasting seismic waves are primarily composed of compressional waves (P-waves), shear waves (S-waves), and surface waves [31,32]. For waves generated by the same blasting source, the differences in their propagation velocities result in distinct arrival times at a given monitoring point, eventually leading to the phenomenon of wave separation in the recorded waveform [33]. In the intermediate-to-far field of a blast, the first to arrive is the P-wave, characterized by a relatively small amplitude and high frequency, which forms the initial phase of the blasting seismic wave. This is followed by the S-wave, which exhibits lower frequency. Surface waves, typically containing the highest energy, constitute the main period of the blasting seismic wave. The attenuation of ground particle vibration then forms the coda phase.

Figure 5 illustrates the blasting vibration velocity curve over time. In this analysis, the main vibration phase is defined as the interval where the amplitude of any peak or trough is at least 50% of the Peak Particle Velocity (PPV). The duration of this phase is termed the main vibration duration (Δt_m). The PPV, recognized as a key index for blasting vibration safety control [34], occurs during a specific period designated as the main period (T). Both T and Δt_m are schematically represented in Figure 5. Taking Figure 5 as an example, select the interval exceeding 50% of PPV from the vibration velocity time-history curve (2.7 ms~6.2 ms). At this time, Δt_m is 3.4 ms. Select the range of 3.8 ms~6.2 ms where the PPV is located. At this time, T is 2.4 ms.

3.3. Principle of Vibration Reduction Through Waveform Interference

Millisecond blasting is one of the most effective techniques for mitigating blast-induced vibrations. Employing appropriate millisecond delays can significantly reduce the adverse effects of blasting seismic waves on existing railway lines. The vibration reduction mechanisms in millisecond blasting primarily involve two phenomena: waveform separation and waveform interference. Waveform interference is achieved by short millisecond delay intervals, which induce a specific phase difference between successive blast waves. This phase difference enables the superposition of the seismic waves to achieve vibration reduction through destructive interference.

Based on wave interference theory [32], the superposition mechanisms of blasting seismic waves are governed by the delay interval Δt relative to vibration characteristics (including the main period T and main vibration duration Δt_m). When Δt < T, the constant phase difference in the dominant frequency band causes full overlap of the main-period waveforms (PPV points), generating significant constructive/destructive interference (Figure 6) with pronounced amplification or attenuation effects. Conversely, for intervals where T ≤ Δt ≤ Δt_m, the main-period (PPV) oscillations separate temporally, while the main vibration phases (amplitudes ≥ 50% PPV) partially overlap (Figure 7). This partial spatiotemporal decoupling substantially reduces wave interaction efficiency. Consequently, the vibration reduction achieved through main-period superposition (Δt < T) demonstrates superior efficacy due to coherent full-phase modulation of energy-intensive components.

When superposition of the dominant-period waveform components occurs, the resulting blasting seismic wave exhibits both constructive interference maxima and destructive interference minima. Consequently, it is imperative to select an appropriate inter-hole delay interval (∆t) in milliseconds to achieve vibration reduction. Let us consider two sinusoidal wave trains propagating in the same homogeneous medium. The displacements of these wave trains are expressed as:

$$A_1=\sin \left(\omega \cdot t-{\phi }_1\right)$$

(6)

$$A_2=\sin \left(\omega \cdot t-{\phi }_2\right)$$

(7)

where A₁ and A₂ are the wave amplitudes, ω is the angular frequency, and φ₁ and φ₂ are the initial phase angles.

When the two columns of waves are superimposed:

$$A=A_1+A_2=\sin \left(\omega \cdot t-{\phi }_1\right)+\sin \left(\omega \cdot t-{\phi }_2\right)$$

(8)

Treating the trigonometric functions of Equation (3):

$$A=2\sin \left(\omega \cdot t-{\displaystyle \frac{{\phi }_1+{\phi }_2}{2}}\right)\cdot \cos \left({\displaystyle \frac{{\phi }_2-{\phi }_1}{2}}\right)$$

(9)

When t ∈ (0, ∞), it always satisfies

$$-1<sin\left(\omega t+{\displaystyle \frac{{\phi }_1+{\phi }_2}{2}}\right)<1$$

(10)

Therefore, if the amplitude of the superimposed wave is less than that of any individual wave, the amplitude superposition coefficient A satisfies the constraint: −1 ≤ A ≤ 1. It needs $-{\displaystyle \frac{1}{2}}<\cos \left({\displaystyle \frac{{\phi }_2-{\phi }_1}{2}}\right)<{\displaystyle \frac{1}{2}}$. Therefore, when ${\displaystyle \frac{2}{3}}\pi <{\phi }_2-{\phi }_1<{\displaystyle \frac{4}{3}}\pi$, the peak value of the wave after the superposition of the two columns is smaller than the peak value of the single column. Then the two columns can achieve the effect of vibration reduction through waveform interference. Especially when φ₂ − φ₁ = π, destructive interference occurs as wave peaks align with troughs, resulting in complete cancellation of the synthesized wave amplitude (A = 0). For blasting seismic waves characterized by a main period T, the delay Δt must satisfy ${\displaystyle \frac{T}{3}}<\mathsf{\Delta }t<{\displaystyle \frac{2T}{3}}$ to achieve destructive interference.

3.4. Analysis of the Vibration Reduction Effect

As demonstrated in the previous section, the duration and period of single-hole blasting seismic waves remain largely unaffected by the number of blasts for a given point. Consequently, the superposition effect between columns of blasting seismic waves in multi-hole millisecond blasting can be analyzed using the parameters derived from single-hole blasting seismic waves. From Figure 4, the complete waveforms of the first detonation were extracted. Applying the calibration method described earlier, the main vibration phase duration (Δt_m) and the main period (T) were determined for each node.

Table 6. Parameters for single-hole detonation at three points.

Point	Main Vibration Duration Δtm (ms)	Main Period T (ms)
A	2.8	0.6
B1	7.3	1.8
B2	6.7	1.3

presents the relevant blasting seismic wave parameters for single-hole detonation at points A, B₁, and B₂.

To compare the vibration reduction effects at various points under different millisecond time delays, the vibration reduction rate ε is defined as:

$$\mathsf{\epsilon }={\displaystyle \frac{\left|P_2-P_1\right|}{P_1}}\times 100\%$$

(11)

where P₁ is PPV without millisecond blasting, and P₂ is PPV with millisecond blasting.

To analyze vibration reduction under varying millisecond delay conditions, experimental cases with delays of 1, 2, 3, 5, 7, 9, 12, and 15 ms were conducted. Figure 8 illustrates the variation curves of vibration reduction rate versus time difference at each monitoring point.

Table 6 provides key waveform parameters at point B₁: a main vibration phase duration (Δt_m) of 7.3 ms and dominant period (T) of 1.8 ms. When the delay time (Δt) is 1 ms, it falls within the Δt < T range in Figure 8b and satisfies ${\displaystyle \frac{T}{3}}<1 \mathrm{ms}<{\displaystyle \frac{2T}{3}}$. Therefore, point B₁ achieves vibration reduction through main-period superposition. Simultaneously, the vibration reduction effect for the 1 ms millisecond blasting is significantly greater than that for other delay times. However, as the delay time increases (T < Δt < Δt_m), the main periods of seismic waves become staggered, while the main vibration phase remains superimposed. Consequently, the vibration reduction effect deteriorates with an increasing delay time. When the delay time exceeds 7 ms (Δt > Δt_m), the vibration reduction rate no longer undergoes significant changes. Under this condition, point B₁ achieves vibration reduction through waveform separation, with the PPV of millisecond blasting approximating that of a single wave.

At Point B₂, the main vibration phase duration (Δt_m) is 6.7 ms, and the dominant period (T) is 1.3 ms. When subjected to 1 ms time delays, it falls within the Δt < T range and satisfies ${\displaystyle \frac{2T}{3}}<1 \mathrm{ms}<T$. Phase amplification in blasting seismic waves induces significant vibration reduction deterioration. For time delays of 2–5 ms (T < Δt < Δt_m), staggered principal wave cycles coupled with superimposed vibration phases yield unstable reduction rates, as exemplified by 2 ms delays achieving minimal reduction while 5 ms demonstrates maximal efficacy relative to comparable delays. Beyond the Δt_m threshold at 7 ms (Δt > Δt_m), waveform separation governs the vibration reduction mechanism, stabilizing the reduction rate despite increasing delay times, with resultant PPV values converging toward single-wave equivalence.

For point A, the main vibration phase duration (Δt_m) is 2.8 ms and the main period (T) is 0.6 ms. When the delay time exceeds 3 ms, the superposition of blasting seismic waves progressively weakens while the vibration duration remains largely unchanged. Within the 0.6 ms < Δt < 3 ms range, principal seismic wave trains exhibit staggered main periods yet maintain superimposed vibration phases, resulting in mutual wave interference that reduces vibration energy. Specifically, at 2 ms delay, the vibration reduction effect proves moderately superior to that under other cases.

Comparing Table 6 and

Table 7. Optimal delay time and vibration reduction mode at three points.

Point	Optimal Delay Time	Vibration Reduction Mode
A	2 ms	Vibration reduction through main vibration phase superposition
B1	1 ms	Vibration reduction through main-period superposition
B2	5 ms	Vibration reduction through main vibration phase superposition

, point B₁ is more suitable for vibration reduction through main-period superposition, while point a and point B₂ are more suitable for vibration reduction through main vibration phase superposition.

In summary, the vibration reduction effect can be more reliably predicted through main-period superposition analysis. When the condition ${\displaystyle \frac{T}{3}}<\mathsf{\Delta }t<{\displaystyle \frac{2T}{3}}$ is satisfied, blasting seismic waves undergo mutual interference, enabling effective vibration reduction via main-period superposition. In contrast, vibration reduction relying on main vibration duration superposition exhibits strong spatial dependency—the same millisecond delay time may yield significantly varied effects across different locations. Consequently, by utilizing site-specific blast seismic wave parameters (e.g., Δt_m and T at protected structures), optimal millisecond delay times can be selected to achieve targeted superposition mechanisms. This approach concurrently minimizes vibration impact while maintaining construction efficiency.

4. Parameter Optimization of Millisecond Delay Blasting for Vibration Control in Classified Surrounding Rock

During propagation, blasting seismic waves undergo attenuation effects from the surrounding rock medium. The attenuation characteristics vary across different rock mass classes, ultimately causing divergence in seismic wave superposition outcomes. Consequently, the optimal delay time becomes contingent on site-specific geological conditions.

4.1. Rock Mass Classification Effects on Blasting Wave Superposition

To investigate blasting seismic wave effects in various rock masses, this study simulates single-hole initiation and progressive hole-by-hole initiation (2 ms delay) of cutting holes in Class III, IV, and V surrounding rock. Rock parameters for each class are listed in

Table 8. Parameters of the different class rock masses [35].

Class	Density (g/cm−3)	Elastic Modulus (GPa)	Poisson’s Ratio	Yield Stress (GPa)
III	2.4	13	0.25	8 × 10−4
IV	2.2	4.6	0.3	4.1 × 10−4
V	1.8	1.6	0.35	1.6 × 10−4

. Figure 9, Figure 10 and Figure 11 present the temporal evolution of stress distributions under distinct cases.

To analyze the evolution law of seismic waves generated on the surface after single-hole detonation, the surface stress evolution during this process is illustrated in Figure 9a. As shown, at t = 1.9 ms, changes appear in the stress contour map, indicating the seismic wave reaches the surface. Peak stress occurs at t = 5.4 ms. Subsequently, the seismic wave enters the attenuation phase, and due to the absence of further detonation, the stress ultimately attenuates, dissipating essentially by t = 7.4 ms. Figure 9b presents the surface stress evolution under a 2 ms delay time, where the detonation time of the second wave differs from the first by 2 ms. The second wave arrives at the surface at 3.9 ms, peaking at 7.4 ms, while the third wave arrives at 5.9 ms and peaks at 9.4 ms. Crucially, at t = 7.4 ms (when the second wave peaks), the stress from the first wave at the surface has essentially dissipated, as evidenced in Figure 9a. Consequently, the second wave experiences minimal impact from the first wave. In summary, when applying millisecond delay blasting (MS Delay Blasting) with a 2 ms interval in Class III surrounding rock, the stress waves generated by sequential blasts do not produce significant superposition effects on the surface.

Similarly, Figure 10 presents the stress evolution during single-hole and millisecond delay blasting in Class IV surrounding rock. Figure 10a illustrates the single-hole blasting stress contour, revealing seismic wave arrival at the surface at t = 3.0 ms, peak stress occurrence at t = 7.1 ms, and subsequent stress decay with near-complete dissipation (central zone) at t = 9.1 ms, culminating in full dissipation by t = 12.1 ms. Figure 10b depicts stress contours under 2 ms millisecond delay blasting, with the second wave arriving at 5.0 ms (peaking at 9.1 ms) and the third wave at 7.0 ms (peaking at 12.1 ms). Comparative analysis with Figure 9 demonstrates that the superposition effect between the first and second waves in Class IV is more significant. Similarly, the third and second waves also superimpose. This results in a significant stress enhancement zone above the tunnel face at 7.1 ms, 9.1 ms, and 12.1 ms. Consequently, stress wave superposition intensifies markedly in Class IV compared to Class III.

Figure 11 examines the stress evolution in Class V surrounding rock under single-hole and 2 ms millisecond delay blasting. An analysis of Figure 11a,b indicates that: the first wave arrives at 4.0 ms (peaking at 10.2 ms), followed by the second wave at 6.0 ms (peaking at 12.2 ms) and the third wave at 8.0 ms. Crucially, when the second wave peaks at t = 12.2 ms, substantial residual stress from the first wave persists on the surface, resulting in significant superposition between the first and second waves.

The preceding analysis indicates surface stress wave evolution across different surrounding rock classes. For Classes III and IV, superposition occurs between initial and subsequent waves but with negligible effects. In Class V surrounding rock, due to the slowed attenuation of blast-induced seismic waves, preceding and subsequent waves overlap. When the stress of the seismic wave generated by the first detonation reaches its maximum value, the seismic wave from the third detonation has already arrived at the ground surface. Moreover, when the stress of the second wave sequence peaks, the energy of the first wave sequence has not yet dissipated. Consequently, the stress wave superposition effect is relatively significant in Class V rock conditions.

Table 9. Result analysis of the different class rock masses.

Class	Maximum Surface Stress (kPa)	First Arrival Time (ms)	Duration (ms)
III	4.0	1.9	4.2
IV	3.1	3.0	5.4
V	2.0	4.0	9.7

records the Maximum Surface Stress, first arrival time, and duration of stress waves under different classes of surrounding rock. As can be seen from the table, as rock quality declines from Class III to V, surface arrival times increase progressively (1.9 ms for III, 3.0 ms for IV, 4.0 ms for V) and peak moments delay correspondingly (5.4 ms → 9.1 ms → 10.2 ms) due to altered rock mechanics. This is because the reduced mechanical parameters enhance intra-rock stress wave absorption, attenuating high-frequency components and decreasing propagation velocity. Additionally, maximum surface stress diminishes from 4.0 kPa (III) to 3.1 kPa (IV) and 2.0 kPa (V). This demonstrates that the damping effect across rock classes causes fundamentally distinct blast wave propagation and superposition behaviors, resulting in different superposition effects for the same millisecond delay durations in different rock classes.

4.2. Optimal Delay Time Determination for Various Classes of Surrounding Rock

To investigate wave superposition in distinct rock classes under varying millisecond delay conditions, single-hole blasting simulation were conducted on Class III, IV, and V surrounding rock. The results obtained the main vibration phase duration and main period at different monitoring points. The obtained parameters are summarized in Figure 12.

As shown in the figure, both the main vibration phase duration and the main period increase with declining rock quality. When the rock quality decreases from Class III to IV, variations in the main period across monitoring nodes remain comparatively minor, whereas variations in the main vibration phase duration approximately double. Compared with Class III and IV, significant changes occur in blasting seismic wave parameters for Class V surrounding rock.

When determining the optimal delay time, the fundamental principle is to reduce the vibration effect of tunnel blasting. The vibration reduction rate can directly reflect the vibration reduction effect of the different delay times. Let us select the point at the surface directly above the tunnel surface as the coordinate origin O point. Take the excavated direction in front of the tunnel as the positive direction of the coordinate axis. Select a node at the surface every 2.5 m along the axis of the excavated tunnel. Select the millisecond delay time with the largest vibration reduction rate of each point as the optimal delay time. To meet the needs of engineering practice, the rock broking effect of the tunnel face should be optimized and the construction efficiency improved. If the vibration reduction rate values of multiple millisecond times are similar, let us further screen the smallest delay time. To determine the optimal delay time for different locations, this study carries out numerical simulations of different classes of surrounding rock with different delay blasting. Figure 13 shows the optimal delay time for different classes of surrounding rock.

For Class III conditions, optimal vibration reduction is achieved with 3 ms delay blasting within the ranges −40 m to 0 m and 10 m to 40 m. Delay blasting of 1 ms yields a vibration reduction within the ranges −50 m to −47.5 m, 0 m to 10 m, and 45 m to 50 m. In the range of −40 m~40 m, when 1.5 ms < Δt_m < 3.1 ms, the vibration reduction through waveform separation can be achieved by using 3 ms millisecond delay time blasting. In the range of 10 m in front of the tunnel face, the optimal delay time is 1 ms. In this case, the vibration reduction through main-period superposition is better than waveform separation. At 40~50 m ahead/behind the tunnel face, the main period of blasting seismic waves increases with distance. Within this range, 1 ms millisecond delay time blasting can achieve vibration reduction through main-period superposition with better efficacy than other delay times.

For Class IV conditions, optimal vibration reduction is achieved with 1 ms delay blasting within the ranges −50 m~−35 m. A 3 ms millisecond blasting vibration reduction effect is most effective in the range of −35 m~−7.5 m. The 2 ms delay provides optimal vibration reduction in −7.5 m to 17.5 m. The 7 ms millisecond blasting vibration reduction effect is most effective in the range of 17.5 m~30 m, and the 5 ms vibration reduction effect is most effective in the range of 30 m~50 m. Compared to Class III, the main period T changes minimally for Class IV, but the main vibration phase duration Δt_m increases. This makes Class IV more conducive to vibration reduction through main vibration phase superimposition. When using 2 ms, 3 ms, or 5 ms delays within their respective ranges, vibration reduction through waveform interference outperforms separation waveform. However, vibration reduction through main vibration phase superimposition exhibits high randomness, with effects varying significantly by location.

For Class V conditions, optimal vibration reduction is achieved with 1 ms delay blasting within the ranges −50 m~25 m. A 4 ms delay blasting yields vibration reduction within the ranges 25 m~50 m. Behind the tunnel face at 40 m, when 1.6 ms < T < 2.1 ms, 1 ms delay blasting achieves vibration reduction through main-period superposition with better efficacy than other delay times. However, at 25 m to 50 m in front of the tunnel face, 4 ms millisecond blasting provides the best vibration reduction effect. However, ahead of the tunnel face from 25 m to 50 m, 4 ms delay blasting provides the optimal vibration reduction effect. In this range, the vibration reduction effect of main-period superposition weakens, whereas that of main vibration phase superposition strengthens. At this time, vibration reduction through main vibration phase superimposition has a better vibration reduction effect.

Table 10. Optimal delay time of the different class rock masses.

Class	Range (m)	Optimal Delay Time (ms)
III	−50~−47.5, 0~10, 45~50	1
	−40~0, 10~40	3
IV	−50~−35	1
	−7.5~17.5	2
	−35~−7.5	3
	30~50	5
	17.5~30	7
V	−50~25	1
	25~50	4

summarizes the optimal delay time for different grades of surrounding rock in different ranges. In summary, for Class III conditions, when protected objects lie within 40 m before/after the tunnel face, using waveform separation provides optimal vibration reduction. Beyond 40 m, we should consider main-period superposition. For Class IV, waveform interference based on proximity for near-face protected objects should be selected. For distant objects, shorter delays should be considered to achieve main-period superposition. For Class V, due to the increase in the main period, vibration reduction through main period superposition can be realized according to the protected object.

5. Conclusions

Taking Zhongliangshan Tunnel of Chongqing as the engineering background, this study carries out numerical simulation for the controlled blasting of tunnel blasting construction crossing under existing railways. Through the numerical simulation results, the vibration reduction through waveform interference of millisecond blasting is summarized. It also analyzes the corresponding law of the surface stress in the different classes of the surrounding rock conditions. The selection basis of optimal millisecond time for tunnel cutting hole blasting is proposed. The main conclusions are as follows:

(1)

According to the superposition mechanism of blasting seismic waves, the vibration reduction through waveform interference can be further divided into two types of vibration reduction: superposition of the main vibration phase and superposition of the main period.

(2)

The millisecond delay time Δt is less than the main period T of the blasting seismic wave, and, the blasting seismic waves weaken each other. At this point, it realizes the vibration reduction through the main-period superposition with the best vibration reduction effect.

(3)

With the deterioration in the surrounding rock quality, the dissipation of low-frequency waves in the formation increases. The first arrival time and peak time of stress waves arriving at the surface are delayed. This increases the degree of wave superposition between the blasting seismic waves.

(4)

Surface wave arrival times progressively increase (from 1.9 ms in Class III to 3.0 ms in Class IV and 4.0 ms in Class V). The maximum surface stress exhibits a progressive decline across different classes of surrounding rock: 4.0 kPa for Class III, decreasing to 3.1 kPa for Class IV, and further reducing to 2.0 kPa for Class V.

(5)

For cut-blasting in tunnels crossing under existing railways, millisecond delay blasting can be used to reduce vibration effects. Class III adopts a 3 m delay time to achieve the vibration reduction through waveform separation. A delay time of 1 ms can be used for Class V to achieve vibration reduction through waveform interference. By combining on-site blasting monitoring data and determining the relevant parameters of blasting seismic waves, this method can be used to determine the optimal delay time.

Limitations: In practical engineering applications, rock mechanics parameters and geological media exhibit complexity, containing features such as faults. Although this study included different classes of surrounding rocks, the mechanism of seismic wave superposition caused by explosions under complex geological conditions is still unclear. Meanwhile, the vibration reduction effect of delayed blasting is also limited by specific parameters of the surrounding rock and the accuracy of electronic detonators. These aspects will be extensively studied in future research. Optimizing delay time in complex geological environments is a key direction for future research.