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Article

Research on Analytical Solution of Stress Fields in Adjacent Tunnel Surrounding Rock Under Blasting and Verification Analysis

1
School of Zijin Geology and Mining, Fuzhou University, Fuzhou 350100, China
2
School of Resource Engineering, Longyan University, Longyan 364012, China
3
Zhongchang Nuclear Engineering (Fujian) Construction and Development Co., Ltd., Nanping 353200, China
*
Author to whom correspondence should be addressed.
Appl. Sci. 2025, 15(15), 8688; https://doi.org/10.3390/app15158688
Submission received: 20 May 2025 / Revised: 29 July 2025 / Accepted: 30 July 2025 / Published: 6 August 2025
(This article belongs to the Special Issue Recent Advances in Rock Mass Engineering)

Abstract

In tunnel blasting, an analytical solution for dynamic stress in the surrounding rock of adjacent tunnels is critical for dynamic response analysis, mechanical evaluations, and crack propagation control. Previous studies on stress field analytical solutions primarily modeled rock as a linear elastic material, focusing mainly on the P-wave effects from instantaneous detonation. Based on Heelan’s short cylindrical cavity model, this paper derives an analytical solution for blast-induced dynamic stresses in adjacent tunnel rock, incorporating both induced SV-waves and a rock mass damage factor through rigorous theoretical analysis. Numerical case studies and field measurements were used to analyze stress propagation during tunnel blasting, and theoretical results were compared with measured data. The key findings were as follows: Radial stress > axial stress > hoop stress. All three stresses decay with increasing distance and damage factor, following an inversely proportional relationship with distance. Radial stress decays faster than axial and hoop stresses. Stress also decays exponentially over time, with the peak occurring after the transverse wave arrival. The theoretical results show approximately 10% deviation from the existing empirical formulas, while field measurements closely match the theoretical model, showing consistent stress trends and an average error of 7.02% (radial), 7.56% (axial) and 7.05% (hoop), confirming the reliability of the proposed analytical solution.

1. Introduction

Tunnel engineering, a key technology for traversing complex terrains and reducing transportation distances, is widely applied in expressways, railways, and urban transit systems. Among construction methods, drilling and blasting methods remain predominant due to their advantages in speed, cost-effectiveness, and adaptability to varying geological conditions [1]. However, the transient stress waves generated during tunnel blasting operations can significantly affect existing tunnels [2], with potential damage requiring careful evaluation. In closely spaced adjacent tunnels, dynamic loading from subsequent blasts can pose serious safety risks to neighboring tunnel structures, including lining failures. As the primary source of dynamic loading, blast-induced stress waves critically impact surrounding rock masses and support systems. Understanding their response and control has become a key technical challenge in safe tunnel construction. Therefore, obtaining analytical solutions for dynamic stresses in surrounding rocks under blasting loads is essential for understanding stress propagation, controlling blast vibrations, and ensuring the integrity of existing tunnel support structures.
Current research challenges in analytical solutions for blast-induced stress mainly fall into two areas: The first is the complexity of stress superposition from transient P- and S-wave interactions, compounded by stress propagation through heterogeneous and discontinuous materials [3,4]. The second is difficulties in validation due to limitations in in situ stress measurement embedding sensors in rock, which is technically challenging, and field data are often affected by interference from air shock waves and construction equipment during signal acquisition [5]. Analytical solutions for dynamic stresses in surrounding rocks of adjacent tunnels are crucial for understanding stress evolution, calculating dynamic responses, analyzing mechanical behavior, and optimizing blast control parameters.
As a form of transient energy excitation, blast-induced stress waves have long been a central topic in explosion theory and its applications. Based on charge geometry, blasting can be classified into spherical charges and cylindrical charges. Tunnel excavation blasting typically uses cylindrical charges, and extensive international research has focused on their associated stress field characteristics.
Theoretical developments have led to three mainstream models for calculating blast-induced stress fields: the Starfield superposition method, the ideal fluid medium model, and Heelan’s short-cylinder solution. Starfield [6] proposed decomposing cylindrical charges into multiple equivalent spherical charges. Experimental studies on stress wave propagation, showed that this superposition method yields acceptable agreement with measured data in the medium-to-far fields. Based on this framework, Liu [7] derived computational formulas for stress fields generated by cylindrical charges. Neiman et al. [8] treated rock media as incompressible ideal fluids under transient blast loading, where particle velocity fields satisfy Laplace’s equations. Using boundary mapping methods, they developed solutions for stress fields around cylindrical charges under various borehole boundary conditions. Heelan [9] established a classical analytical solution for cylindrical charge blasting using elastodynamic theory. This model assumes an infinite, isotropic rock mass and provides an expression for radial stress ( σ r ) distribution due to single hole blasting:
σ r = P 0 r / r 0 α e β ( r r 0 )
where
  • P 0 is the initial blast pressure (MPa);
  • r 0 is the charge radius (m);
  • α and β are rock mass attenuation coefficients.
Because the mechanical properties of the rock mass will change after blasting [10], this model accounts for rock mass damage or jointed fissures, leading to prediction errors exceeding 40% in weak surrounding rock. To address this, Henrych [11] introduced an attenuation factor and proposed a modified stress wave propagation equation for heterogeneous rock masses. The relationship between the attenuation coefficient β and the rock mass integrity coefficient K v is expressed as follows:
β = 0.85 × ( 1 K v ) × ln r r 0
This model has shown good accuracy in hard rock tunnels but lacks adaptability in fractured zones. To address this, Zhang et al. [12] introduced a damage factor to revise the stress wave equation, establishing a nonlinear attenuation model that accounts for fracture propagation. Their results showed that stress attenuation in jointed rock masses exceeds that in intact rock by over 50%. Ma et al. [13] discretized cylindrical charges into multiple linked spherical charges and derived the dynamic stress field using vector superposition methods. Lei et al. [14] developed analytical solutions for stress fields under different detonation transmission directions using an equivalent unit spherical charge superposition method, incorporating directional effects. Favreau [15], Graff [16], and Achenbach [17] analyzed stress wave fields induced by cylindrical charges based on spherical cavity expansion models. Blair [18] extended Heelan’s short-cylinder solution by integrating finite detonation velocity and charge length, advancing a superposition model for long charges. Gao et al. [19] further partitioned elongated cylindrical charges into a series of instantaneously detonated short charges, deriving vibration velocity formulas for arbitrary points using time-delayed superposition of detonation wave propagation.
In experimental and field-testing research, advanced techniques such as dynamic caustics and ultra-dynamic strain measurement have been widely adopted. Yang et al. [20] systematically investigated stress–strain fields in centrally initiated cylindrical charges and the evolution of local stress fields near blast-induced crack tips using dynamic caustics. Their study revealed significant principal stress differences across orientations and pronounced end effects. Ding [21] conducted scaled model experiments to analyze stress wave superposition between adjacent boreholes. Qiu [22] used physical modeling to simulate blast-induced disturbances in underground tunnels, mapping surface strain distributions at various blast locations. Yang [23] employed a high-speed camera-based digital image correlation system in dual-borehole blast tests, showing that peak stress at the midpoint between two charges reached 2.4–2.7 times that of single-hole blasting. Liu et al. [24] developed an embedded strain bar system incorporating MEMS sensors and wireless transmission, enabling real-time monitoring of dynamic stress fields during construction of the Qingdao Jiaozhou Bay Tunnel. Their signal processing algorithm combining wavelet denoising with Kalman filtering improved the signal-to-noise ratios from 15 dB to 28 dB. Zhang [25] optimized field blast monitoring protocols to capture dynamic strain on tunnel primary lining (TSL) surfaces. Luo et al. [26,27,28] performed in situ dynamic strain measurements on existing tunnel surfaces, quantitatively confirming strain amplitudes on blast-facing sides.
In a numerical simulation, Gao et al. [19] used 3D dynamic finite element software to model cylindrical charge blasting in infinite rock masses. By developing a quarter-symmetry finite element model, they systematically examined the effects of initiation positions on explosive energy transmission and stress field distribution. Liang et al. [29] applied numerical simulation to analyze rock fragmentation under stress wave superposition from multi-source charge detonations, confirming that stress wave interactions from spherical charges align with theoretical rock-breaking mechanics. Xiang et al. [30] proposed a numerical analysis model for strip charge stress fields by integrating the Starfield superposition method with dynamic finite element analysis, providing a comprehensive understanding of the formation and evolution mechanisms of blast-induced stress fields under diverse initiation modes.
In summary, despite substantial progress in studying stress fields from cylindrical charge blasting, three critical limitations remain:
  • Methodological limitations: The widespread use of equivalent spherical charge models—which inherently omit S-wave generation—fails to reflect actual field conditions, where cylindrical charges produce both P- and S-waves.
  • Dynamic modeling oversights: Current superposition approaches for elongated charges, based on short-cylinder solutions often overlook two key factors: (a) axial pressure effects from instantaneous detonation of short cylindrical cavities, and (b) the evolution of rock mass damage in media treated as continuous, homogeneous materials.
  • Validation constraints: Most existing measurements focus on surface particles, lacking in situ dynamic stress data from within the rock mass, which limits the verification of theoretical formulations models.
To address these gaps, this paper builds on explosion stress wave propagation theory and Heelan’s short-cylinder solution [9], incorporating the axial pressure generated by the instantaneous detonation of short cylindrical cavities (Figure 1) and the effects of rock mass damage. Based on the phase delay effect and stress wave propagation intervals, a porous superposition method is used to derive an analytical solution for the stress in the surrounding rock of adjacent tunnels under blasting. Theoretical formulas are employed to calculate stress magnitude and propagation patterns of radial particles in adjacent tunnels. These results are then validated through field stress measurements and numerical simulations. The findings provide a theoretical basis for calculating dynamic stress in surrounding rock, understanding the propagation of internal cracks and fissures, and improving surrounding rock reinforcement and blast control.

2. Theoretical Derivation of Analytical Solutions for Blasting Stress Fields in Adjacent Tunnels

During tunnel face blasting, both surface waves (Rayleigh and Love) and body waves are generated and propagate through the surrounding rock mass. Body waves include P-waves and S-waves, with the latter divided into SV and SH components. P-waves travel faster than S-waves. The combined action of P-, SV-, and SH-waves induces three-dimensional dynamic stresses in the surrounding rock of adjacent tunnels: radial stress ( σ r ), axial stress ( τ z ), and hoop stress ( τ θ ). The mechanical model for an adjacent tunnel’s surrounding rock under coupled wave actions is shown in Figure 2.

2.1. Blasting Stress Field of a Single-Hole Cylindrical Charge Based on Heelan’s Short-Cylinder Solution

2.1.1. Stress Field of a Short-Cylinder Charge Incorporating Instantaneous Detonation Axial Pressure and Rock Mass Damage

Heelan modeled the stress induced by the instantaneous detonation of a short cylindrical cavity charge as a transient pressure applied to the boundary of a short cylindrical cavity (Figure 1), from which displacement solutions were derived. The results show that detonation generates both P-waves and S-waves, with the latter further decomposing into SH-waves (parallel to the horizontal plane) and SV-waves (perpendicular to it). Figure 3 shows the particle vibration direction as the angle between the wave propagation direction and the short cylindrical cavity wall varies. The displacement solutions for these three wave types are given by [9]:
u P ω P = F 1 ( ϕ ) R d d t p ( t R C p ) + G 1 ( ϕ ) R q ( t R C p ) sin ϕ cos ϕ
u S V ω S V = F 2 ( ϕ ) R d d t p ( t R C s ) + G 2 ( ϕ ) R q ( t R C s ) cos ϕ sin ϕ
v S H = K ( ϕ ) R d d t s ( t R C s )
where u P and u S V are the horizontal displacements of the P-wave and SV-wave (m); ω P and ω S V are their vertical displacements (m); and v S H is the tangential displacement of the SH-wave (m). In tunnel face blasting, u , w and v represent the radial, axial and hoop displacements, respectively (m); C p and C s are the propagation velocities of the P-wave and S-wave (m/s); p ( t ) , q ( t ) , s ( t ) , ϕ and R are defined as in Figure 1; F 1 ( ϕ ) and G 1 ( ϕ ) are the source functions for the P-wave; F 2 ( ϕ ) and G 2 ( ϕ ) are for the SV-wave; and K ( ϕ ) is for the SH-wave.
Source function definitions [9]:
F 1 ( ϕ ) = Δ 4 π G C p 1 2 C s 2 c o s 2 ϕ C p 2 G 1 ( ϕ ) = A 4 π G C p 2 C s 2 c o s ϕ F 2 ( ϕ ) = Δ 4 π G C s sin 2 ϕ G 2 ( ϕ ) = A 4 π G sin ϕ K ( ϕ ) = Δ 4 π G C s sin ϕ
where Δ is the volume of the short-cylinder cavity (m3), A is its lateral surface area (m2), and G is the shear modulus of the surrounding rock mass (GPa). The rock damage factor D is defined as follows [31]:
D = 1 ( v p , p o s t v p , p r e ) 2
where v p , p r e and v p , p o s t are the P-wave velocities of the rock before and after blasting, respectively (m/s), measured on-site using acoustic testing. The uncertainty in the damage factor D is ±0.01. Since the P-wave velocity in rock mass after blasting is generally lower than that before blasting, the damage factor D ranges from 0 to 1, with higher values indicating more severe damage to the surrounding rock. Based on the existing formula [32], and incorporating the damage factor, p ( t ) , q ( t ) and s ( t ) can be expressed as follows:
p ( t ) = P m e ( a t + b ) ( 1 D ) q ( t ) = μ d 1 λ p ( t ) s ( t ) = λ p ( t ) P m = 1 8 n ρ e D e 2 d c d b 3
where P m is the maximum pressure accounting for the combined effects of explosive stress waves and detonation product expansion (MPa); a and b are detonation decay coefficients of the explosion [33]; μ d is the dynamic Poisson’s ratio of the rock ( μ d = 0.8 μ ); λ is the lateral pressure coefficient, defined as λ = μ d /(1 − μ d ); n is the pressure amplification coefficient of the detonation products; ρ e is the explosive density (kg/m3); D e is the explosive detonation velocity (m/s); d c is the charge diameter (m); and d b is the borehole diameter (m).
During cylindrical charge detonation, the rock mass within the blast volume is subjected first to the shock front and then to explosive gas pressure. This process creates ‘‘crushed’’ and ‘‘fractured’’ zones, characterized by radial cracks, circumferential cracks, and non-radial fractures [34,35]. While blasting can effectively break rock within the intended excavation zone, it may also damage surrounding rock that must be preserved, compromising its mechanical properties, bearing capacity, and stability [36]. To assess such effects, Yu and Vongpaisal developed the Blast Damage Index (BDI) for mining applications [37].
BDI = IS DR = VdC KT
where IS is the induced stress; DR is the damage resistance; V is the vector sum of PPV (mm/s); d is the specific gravity of the rock mass (kg/m3); C is the compressional wave velocity of the rock mass (mm/s); K is the site quality constant; and T is the dynamic tensile strength of the rock mass (N/m2).
This relationship accounts for both the rock mechanics and wave propagation effects, but it depends on numerous site-specific parameters. To simplify the representation of the rock mass, this paper introduces the damage factor D (0 < D < 1). By incorporating D into the exponent of the stress attenuation term (Equation (8)), the model quantitatively captures the influence of rock degradation on stress wave propagation. Physically, increasing D reflects progressively damaged material, such as the formation and growth of micro-cracks, voids, or other defects. These features act as energy dissipation mechanisms by (i) reflecting, refracting, and scattering incident stress waves, which reduces the measured wave amplitude; (ii) converting mechanical wave energy into heat and other dissipated forms through crack friction and plastic deformation; and (iii) reducing the rock mass’s overall load bearing capacity.
The expression e−(at+b)/(1−D) models this behavior. The term 1/(1 − D) amplifies the decay rate: as D increases towards 1 (severely damaged rock), (1 − D) decreases, making the exponent more negative and leading to faster stress amplitude decay of the stress amplitude P m . This matches physical observations—higher damage leads to stronger attenuation of stress waves. Thus, 1/(1 − D) effectively quantifies how damage progressively accelerates wave energy dissipation and scattering.
During tunnel face blasting, the radial, axial, and hoop displacements of the surrounding rock are expressed as follows:
u = u P + u S V = F 1 ( ϕ ) R d d t p ( t R C p ) + G 1 ( ϕ ) R q ( t R C p ) sin ϕ + F 2 ( ϕ ) R d d t p ( t R C s ) + G 2 ( ϕ ) R q ( t R C s ) c o s ϕ
ω = ω P + ω S V = c o s ϕ F 1 ( ϕ ) R d d t p ( t R C p ) + G 1 ( ϕ ) R q ( t R C p ) + F 2 ( ϕ ) R d d t p ( t R C s ) + G 2 ( ϕ ) R q ( t R C s ) sin ϕ
v = v SH = K ( ϕ ) R d d t s ( t R c s )
By differentiating the displacement field with respect to time, the vibration velocity field of the short-cylinder charge is derived as follows:
V r = u t = F 1 ( ϕ ) R d 2 d t 2 p ( t R c p ) + G 1 ( ϕ ) R d d t q ( t R c p ) sin ϕ + F 2 ( ϕ ) R d 2 d t 2 p ( t R c s ) + G 2 ( ϕ ) R d d t q ( t R c s ) c o s ϕ
V z = ω t = c o s ϕ F 1 ( ϕ ) R d 2 d t 2 p ( t R C p ) + G 1 ( ϕ ) R d d t q ( t R C p ) + F 2 ( ϕ ) R d 2 d t 2 p ( t R C s ) + G 2 ( ϕ ) R d d t q ( t R c s ) sin ϕ
V θ = v t = K ( ϕ ) R d 2 d t 2 s ( t R C s )
where V r , V z and V θ represent the radial, axial, and hoop vibration velocities, respectively, in units of m/s.
During blasting operations, the relationships between the stresses ( σ r , τ z , τ θ ) generated by P-waves, SV-waves, and SH-waves and the corresponding particle vibration velocities ( V r , V z , V θ ) can be expressed as follows [38]:
σ r = V r ρ C p τ z = V z ρ C s τ θ = V θ ρ C s
where ρ is the rock density, kg/m3.
By synthesizing Equations (13)–(16), the stress field induced by a short-cylinder charge blasting can be derived as follows:
σ r = F 1 ( ϕ ) R d 2 d t 2 p ( t R C p ) + G 1 ( ϕ ) R d d t q ( t R C p ) ρ C p sin ϕ + F 2 ( ϕ ) R d 2 d t 2 p ( t R C s ) + G 2 ( ϕ ) R d d t q ( t R C s ) ρ C p c o s ϕ
τ Z = F 1 ( ϕ ) R d 2 d t 2 p ( t R C p ) + G 1 ( ϕ ) R d d t q ( t R C p ) ρ C s c o s ϕ + F 2 ( ϕ ) R d 2 d t 2 p ( t R C s ) + G 2 ( ϕ ) R d d t q ( t R C s ) ρ C s sin ϕ
τ θ = ρ C s K ( ϕ ) R d 2 d t 2 s ( t R C s )

2.1.2. Superposition of Blasting Stress Fields Considering Phase Delay Effects

The detonation of an elongated cylindrical charge can be modeled as the superposition of instantaneous detonations from multiple short-cylinder segments that detonate sequentially. Accounting for time delays due to detonation wave propagation, the stress field for cylindrical charge blasting (e.g., with bottom initiation) can be derived by combining the superposition model in Figure 4 with Equations (17)–(19):
Based on this criterion, elongated cylindrical charges can be divided and calculated. By substituting q(t) = µd (1 − λ)p(t), and s(t) = −λp(t) (from Equation (8)), the stress field of the columnar charges in matrix form is as follows:
σ i = σ r i τ z i τ θ i = ρ C F 2 ( ϕ i ) R i d 2 d t 2 + μ d ( 1 λ ) G 2 ( ϕ i ) R i d d t p t ( i 1 ) L m D e R i C p F 1 ( ϕ i ) R i d 2 d t 2 + μ d ( 1 λ ) G 1 ( ϕ i ) R i d d t p t ( i 1 ) L m D e R i C s λ K ( ϕ i ) R i d 2 d t 2 p t ( i 1 ) L m D e R i C s R θ
C = C p 0 0 0 C s 0 0 0 C s , F i = F 2 ϕ i R i d 2 d t 2 + μ d 1 λ G 2 ϕ i R i d d t p t i 1 L m D e R i C p F 1 ϕ i R i d 2 d t 2 + μ d 1 λ G 1 ϕ i R i d d t p t i 1 L m D e R i C s λ K ϕ i R i d 2 d t 2 p t i 1 L m D e R i C s , R θ = cos θ sin θ 0 sin θ cos θ 0 0 0 1
σ i = ρ C F i R θ , σ = i = 1 m σ i
In existing studies, the analytical solutions for derived blasting stress fields based on short cylindrical charges are formulated as follows [19]:
σ r i τ i = ρ C p 0 0 C s s i n ϕ i c o s ϕ i F 1 ( ϕ i ) R i d 2 d t 2 p ( t ( i 1 ) L m D e R i C p ) + ρ C p 0 0 C s c o s ϕ i s i n ϕ i F 2 ( ϕ i ) R i d 2 d t 2 p ( t ( i 1 ) L m D e R i C s )
σ r τ = i = 1 m σ r i τ i
Compared to the existing research Formulas (23) and (24), the Equations (20)–(22) derived in this study introduce two key advancements: the inclusion of instantaneous detonation axial pressure q ( t ) , and the rock mass damage factor D.

2.2. Multi-Borehole Stress Field Superposition Based on Stress Wave Propagation Time Intervals

In multi-borehole blasting, stress field superposition must account for time interval effects caused by differences in stress wave propagation paths (see Figure 5). These delays result from variations in propagation distances and anisotropic wave velocities in the rock mass, necessitating precise phase corrections for accurate dynamic stress predictions.
Based on the single-hole stress field derived above, the total stress field at a given observation point under multi-borehole blasting is obtained by superposition:
σ r τ Z τ θ = j = 1 n σ r j ( t t j ) τ Z j ( t t j ) τ θ j ( t t j )
where n is the number of boreholes; t j is the time required for stress waves generated from the j-th borehole to reach the observation point, given by t j = R j / C p , s ; R j is the distance from the j-th borehole to the observation point (m); and C is the corresponding P-wave or S-wave propagation velocity (m/s).

3. Application Examples and Comparative Analysis

3.1. Selection of Relevant Parameters and Calculation Points

A tunnel in Fujian Province is a double-track highway tunnel with a 10 m net spacing that was constructed using the upper- and lower-step method, with simultaneous construction of the left and right tunnels. The surrounding rock is weathered granite with well-developed joint fissures; its physical and mechanical properties are listed in Table 1. Engineering practice indicates that slotted holes in a single section carry a high charge, producing the greatest stress and vibration. Therefore, as an example, eight secondary slotted holes were detonated simultaneously on the upper steps. Each borehole (dot) has a diameter of 40 mm, an explosive charge diameter of 32 mm, a hole depth of 3 m, a charge of 2.4 kg per hole, and bottom initiation. The layout is shown in Figure 6. The explosion has a detonation velocity of 3200–3800 m/s, a density of 1200 kg/m3 and a detonation product pressure coefficient (n) of 10 [31]. Due to the extensive jointing in the rock mass, the rock damage factor is taken as D = 0.2.
According to existing studies [39], the blast-facing sidewall of the adjacent tunnel is a critical risk area. To evaluate the stress magnitude and distribution in the rock mass at this location, six calculation points (Points A–F) are placed at 1-m intervals along a horizontal line within the sidewall (see Figure 7). The first point (Point A) is located 11.42 m from the center of the cut holes. According to reference [17], a single cylindrical charge with an aspect ratio of 10 was divided into 15 short cylindrical charge units (m = 15), each 16 cm long with a base radius of 1.6 cm. These parameters were then used in Equations (20)–(22) and (25) for calculation.

3.2. Calculation Results and Comparative Analysis

The peak stress points A~F are shown in Table 2. The uncertainty in the peak stress is ±0.02. The results indicate that during blasting at the excavation face, the peak stresses in the rock mass follow the order: radial stress > axial stress > hoop stress.
To further examine stress evolution, Figure 8 shows the triaxial stress variations of radial points A–F along the tunnel axis, based on the data in Table 2. The results indicate that radial stress decays inversely with distance and attenuates more rapidly than axial stress. Hoop stress remains consistently lower than both radial and axial stresses. Based on Table 2, the time history curve of radial stress at Point A is plotted in Figure 9. This curve shows that radial stress decreases over time and exhibits a double peak. The peak occurs after the arrival of the S-wave, as the superposition of P- and S-waves causes a sudden increase in stress, indicating their combined effect on the surrounding rock particles. According to Table 2, radial, axial, and hoop stresses decay with distance following the power law σ = σ0dα, where σ0 is the stress at d = 0, d is the distance from the cut hole center, and α = −2.004 ± 0.01063 is the common power coefficient, identical for all stresses (see Figure 7). The initial stresses are σr0 = 215.427 ± 5.89566 MPa, σa0 = 123.68921 ± 3.36189 MPa, and σh0 = 102.10915 ± 2.85727 MPa.
To compare the stress results, parameters from the engineering case were applied to Equations (20)–(22) and (25) from this study—which account for the instantaneous axial pressure from short-cylinder charges and the rock mass damage factor D—and to Equations (23) and (24) from previous studies. The computed stress values are shown in Figure 10 and Figure 11.
Comparative analysis of instantaneous detonation axial pressure ( q ( t ) ): As shown in Figure 10, radial stress is higher when instantaneous detonation axial pressure is considered. The maximum difference reaches 12.42%, with an average difference of 11.25% across all calculation points. Similarly, Figure 10 shows that axial stress is also higher with this factor included, with a maximum difference of 10.36%, and an average difference of 9.68%.
Comparative analysis of rock mass damage (D): As illustrated in Figure 11, incorporating the rock mass damage factor D reduces all the stress components. Stress attenuation follows a nonlinear trend as D increases. At D = 0.4, radial stress decreases by 44.58%, axial stress by 29.55%, and hoop stress by 27.18%, compared to the undamaged case (D = 0).
Based on the results in Table 2 and the comparative analysis in Figure 10 and Figure 11, the formulas derived in this study (Equations (20)–(22) and (25))—which account for instantaneous detonation axial pressure and rock mass damage—demonstrate greater accuracy and rationality than existing models.

4. Field Monitoring and Stress Analysis

4.1. Engineering Situations

In a twin-tunnel highway project in Fujian Province, the left and right tunnels were excavated simultaneously, with the left tunnel advancing more rapidly. The upper bench excavation area of the active tunnel is approximately 77 m2. The blasting operations use digital electronic detonators, with 40 mm diameter boreholes and No. 2 rock emulsion explosive cartridges (32 mm diameter). The initiation sequence follows a ring-by-ring (row-by-row) pattern with a 50 ms inter-row delay, achieving a 3.0 m cyclic advance. Detailed blasting parameters are provided in Table 3, and the blasting network is shown in Figure 12.

4.2. Monitoring Test Setup, Materials, and Methods

The purpose of dynamic strain monitoring is to measure stress in the surrounding rock of the adjacent tunnel during blasting, verify the accuracy of the previously derived analytical stress solutions, and analyze stress variation patterns induced by blasting. An ultra-dynamic strain measurement system (Figure 13) was used to monitor dynamic strain in the tunnel surrounding rock. The testing procedure, shown in Figure 14, operates as follows: voltage signals were collected via embedded sensors, converted into strain values using Equation (26), and dynamic stress peaks are then calculated based on Hooke’s law (Equation (27)) [40].
ε = 4 × U / K × U 0 × n × G
σ = E ε
where U is the peak output voltage (V), K is the sensitivity coefficient of the strain gauge, U 0 is the bridge voltage of the strain amplifier (V), n is the number of effective bridge arms, G is the gain of the strain amplifier, and E is the elastic modulus of the rock mass (GPa). The dynamic strain test uses a half-bridge circuit, with the instrument parameters listed in Table 4.
The operating principle of the triggered voltage signal recording system is illustrated in Figure 15. When blast loads act on the strain bar, it undergoes minor deformation, causing the bonded strain gauges to change resistance. The resistance change produces a microvolt-level output via a Wheatstone bridge. The signal is then amplified by an ultra-dynamic strain amplifier, triggering a pulse that converts the microscopic deformation into a millivolt-level voltage signal recorded by the system.
The measurement point layout is shown in Figure 16. Horizontal boreholes are drilled into the sidewall of the adjacent existing tunnel, with five strain measurement points spaced at 1 m intervals (Point 1 is 1 m from the sidewall). Prefabricated strain bars, matched to the physical and mechanical properties of the tunnel surrounding rock, are used. Triaxial resistance strain gauges are mounted on the bars to measure strain via electrical resistance (Figure 17). The instrumented bars are inserted into the boreholes and fixed with grout. A schematic diagram of the strain bar inserted into the borehole is shown in Figure 18, where the specifications of the PVC pipe in the diagram are consistent with those of the on-site drilling. To determine the rock mass damage factor D at the same locations, a YL-LCT single-transmitter dual-receiver sonic probe (Figure 19) is used to measure P-wave velocities. Sonic testing boreholes are drilled 0.5 m from the strain measurement holes (Figure 20), ensuring spatial alignment for accurate comparative analysis.
During on-site dynamic strain testing, intense blasting vibrations caused collisions between data cables and surrounding structures, while mechanical equipment and welding operations introduced significant signal interference. To ensure measurement accuracy, the following measures were implemented before testing: data cables at borehole exits were wrapped with foam sleeves, and the remaining segments were enclosed in stainless steel junction boxes to prevent impact damage. Testing instruments were grounded using steel rebars to eliminate electromagnetic interference. All mechanical equipment was relocated, and welding operations were suspended during testing. Instrument connections were established and activated only after confirming the absence of interference sources. Sonic testing was completed prior to blasting. The ultra-dynamic strain amplifier was activated just before the blast and deactivated afterward. The field-testing configuration is shown in Figure 21. Each blasting round required approximately 15 min for complete monitoring, and the procedure was repeated over five consecutive rounds.

4.3. Monitoring Results

The wave velocity test results of the surrounding rock prior to the first blasting cycle are shown in Figure 22. The extracted results of measuring points 1–5 are shown in Table 5. Substituting the data into Equation (7), the calculated damage factors for monitoring Points 1 to 5 are 0.20, 0.18, 0.14, 0.10, and 0.09, respectively. Taking the radial strain test at monitoring Point 1 during the first blasting cycle as an example, the measured voltage signal is shown in Figure 23. The stress peak occurs at approximately 55 ms, closely aligning with the 50 ms delay associated with the highest charge, a single time-delay interval of the cut holes, confirming the reliability of the dynamic strain measurements. The wave velocities and dynamic strain data from the remaining four points were substituted into Equations (7), (26) and (27) to calculate the corresponding damage factors and dynamic stresses. The results are presented in Table 6, where “—” indicates that dynamic strain data were not obtained at certain points due field complexity and signal interference. The uncertainty in the peak stress and peak voltage is ±0.02.
As shown in Table 6, at each monitoring point, the stresses hierarchy is as follows: radial stress > axial stress > hoop stress. Additionally, all three stresses decrease with the components of increasing rock mass damage and with greater distance from the blast source.

4.4. Stress Propagation Law and Comparative Analysis

Based on the data in Table 6, the average damage factors and stress values at monitoring points 1–5 over five blasting cycles are summarized Table 7. To further examine the stress propagation behavior in the surrounding rock of the adjacent tunnel with respect to distance from the blast center, Figure 24 was generated using the data in Table 7. As shown in Figure 24, radial, axial, and hoop stresses all decrease with increasing distance, with radial stress showing the fastest attenuation rate, followed by axial and hoop stresses.
The mean relative error (MRE) between measured and theoretical stress values is defined as follows:
MRE ( i ) = σ ¯ measured ( i ) σ theoretical ( i ) σ theoretical ( i ) × 100 %
σ ¯ measured ( i ) = 1 n j = 1 n σ i j
where σ ¯ measured ( i ) is the mean measured stress at the i-th monitoring point, σ theoretical ( i ) is the theoretical stress at the i-th monitoring point, n is the number of blast events, and σ i j is the measured stress at the i-th monitoring point during the j-th blast.
And the average error (AE) of all measuring points is defined as follows:
AE = 1 k i = 1 k MRE ( i )
where k is the number of measuring points.
A comparison between measured and theoretical stress values is presented in Figure 25. The results show that the measured stresses are slightly lower than theoretical predictions, with average errors of 7.02% for radial stress, 7.56% for axial stress, and 7.05% for hoop stress.
Analysis of discrepancies: According to Reference [41], borehole stemming can improve explosive energy utilization by approximately 7.5%. In this test, however, the cut holes were unstemmed, leading to energy loss. Additionally, signal attenuation occurred due to excessive cable lengths at the signal receivers. According to reference [40], the voltage drop ( Δ U ) along the cable is calculated as follows [42]:
Δ U = Δ U 0 % × I × L
where Δ U 0 % is the percentage voltage drop per kilometer of cable, I is the load current (A), and L is the cable length (km). Based on data from reference [42], Δ U 0 % was taken as 0.766%. With a measured load current of I = 0.52 A and cable length L = 0.1 km, the voltage drop calculated using Equation (31) is 0.3983 mV. This voltage drop reduced the peak voltage during measurements, resulting in values lower than theoretical predictions and contributing to the previously noted error, further lowering the reduced measured stress values. These factors collectively explain the observed discrepancies.
Despite these minor deviations, the measured and theoretical values align well, with errors within acceptable limits (≤8%), confirming the reliability of the analytical stress field solutions developed in this study. In tunnel engineering, the MRE between theoretical and measured stress values typically remains below 10%, a range acceptable for meeting project reliability requirements [43,44].

5. Conclusions

Based on Heelan’s short cylindrical cavity solution, this study derives an analytical model for dynamic stresses in the surrounding rock of adjacent tunnels by incorporating the axial pressure q ( t ) generated by instantaneous detonation and the rock mass damage factor D. A case study was conducted to investigate stress magnitudes and propagation behavior, with validation through field measurements. The theoretical stress values derived from the model show consistent trends and minor numerical deviations when compared to measured data.
(1)
Stress magnitude and propagation laws: Both theoretical and measured results exhibit consistent triaxial stress trends in the following order: radial stress > axial stress > hoop stress. All three stresses decrease with increasing distance and damage factor, showing an inverse relationship with distance. Radial stress decays at the highest rates, followed by axial and hoop stresses. Stress also decays exponentially over time, with the peak occurring after the arrival of the transverse S-wave.
(2)
Advantages of proposed stress model: Incorporating instantaneous detonation pressure significantly improves accuracy compared to conventional models that neglect this factor, reducing average errors by 11.25% in radial stress and 9.68% in axial stress. Including the damage factor D reveals nonlinear stress attenuation with increasing damage. At D = 0.4, compared to the intact rock condition (D = 0), stress reductions reach 44.58% (radial), 29.55% (axial), and 27.18% (hoop), confirming the importance of accounting for detonation pressure and rock mass damage in stress prediction.
(3)
Validation through field measurements: Measured stresses show small deviations from theoretical values, with average errors of 7.02% (radial), 7.56% (axial), and 7.05% (hoop). This close agreement validates the reliability of the proposed analytical solution.
Note: The primary limitations of the stress calculation model in this study are its initial assumption of heterogeneous, anisotropic rock masses as homogeneous media, with stress computed through the introduction of a damage factor. This elasticity-based model simplifies reality and does not fully capture the complex behavior of anisotropic rock. In practice, stress wave propagation in such media is nonlinear, and influenced by transmission, refraction, and reflection at rock mass discontinuities, as well as strain-softening effects in tunnel rock [45]. During multi-borehole stress superposition, the rock between the cut holes and the central pillar is treated as a continuous medium. However, auxiliary (relief) and perimeter pilot holes cause additional stress attenuation due to wave scattering at these cavities. To address these limitations, future research will focus on the following: (1) Stress wave propagation in anisotropic and fractured rock masses using a true triaxial Hopkinson bar system to simulate in situ stress conditions, and ultrasonic transmission techniques to monitor P-wave velocity evolution under varying stress paths [46]. (2) Discrete element modeling (DEM) to simulate contact mechanics across rock discontinuities. (3) Numerical simulation tools to study stress wave behavior in rock masses containing pre-existing cavities. These approaches aim to improve theoretical accuracy and better reflect real-world conditions.

Author Contributions

Methodology, T.L., X.L. and X.H.; resources, Y.H. and Y.X.; data curation, Y.W., J.Z., Y.H. and Y.X.; writing—original draft, Y.W. and J.Z.; writing—review and editing, T.L., X.L. and X.H.; funding acquisition, T.L. All authors have read and agreed to the published version of the manuscript.

Funding

This work is supported by the Xiaoming Lou: Natural Science Foundation of China (No. 52109124).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

Authors Yelong Xie and Yan Hu were employed by the company Zhongchang Nuclear Engineering (Fujian) Construction and Development Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

References

  1. Liu, F.-X. Technological advancements and prospects of drill and blast tunnel construction equipment. Mod. Tunn. Technol. 2024, 61, 190–202. [Google Scholar]
  2. Shi, J.-J.; Xu, W.-X.; Zhang, H.; Ma, X.-Y.; An, H.-M. Dynamic response and failure mechanism of deep-buried tunnel with small net distance under blasting load. Buildings 2023, 13, 711. [Google Scholar] [CrossRef]
  3. You, S.; Yang, R.-S.; Xiao, C.-L.; Ding, C.-X.; Li, C.-X.; Yang, Z.; Li, J. Study on the superposition effect of stress waves and crack propagation law between blastholes at different angles. Opt. Lasers Eng. 2024, 178, 108193. [Google Scholar] [CrossRef]
  4. Ding, X.-H.; Yang, Y.-Q.; Zhou, W.; An, W.; Li, J.-Y.; Ebelia, M. The law of blast stress wave propagation and fracture development in soft and hard composite rock. Sci. Rep. 2022, 12, 17120. [Google Scholar] [CrossRef]
  5. Luo, H.-H.; Li, X.-L.; Wang, J.-G.; Li, G.-T.; Li, Z.-L. Study on peak overpressure prediction of underground blasting shock waves on straight-through roadways. J. Chin. Saf. Sci. 2019, 29, 57–62. [Google Scholar]
  6. Starfield, A.M.; Pugliese, J.M. Compression waves generated in rock by cylindrical explosive charges: A comparison between a computer model and field measurements. J. Rock. Mech. Min. Sci. 1968, 1, 65–77. [Google Scholar] [CrossRef]
  7. Liu, J.-C.; Gao, W.-X.; Zhang, S.-H.; Hu, Y. Study on initial stress and damage law of hole wall in water interval blasting. J. Arab. Geosci. 2022, 15, 96. [Google Scholar] [CrossRef]
  8. Neiman, I.-B. Modeling the explosion of a system of borehole charges in a scarp. Sov. Min. 1986, 22, 108–113. [Google Scholar] [CrossRef]
  9. Heelan, P.-A. Radiation from a cylindrical source of finite length. Geophysics 1953, 18, 685–696. [Google Scholar] [CrossRef]
  10. Yue, G.-W.; Li, M.; Wang, L.; Liang, W.-M. Optimal layout of blasting holes in structural anisotropic coal seam. PLoS ONE 2019, 14, e0218105. [Google Scholar] [CrossRef]
  11. Henrych, J. The dynamics of explosion and its use. J. Appl. Mech. 1980, 47, 218. [Google Scholar] [CrossRef]
  12. Zhang, M.-W.; Shimada, H.; Sasaoka, T.; Matsui, K.; Dou, L.-M. Evolution and effect of the stress concentration and rock failure in the deep multi-seam coal mining. Environ. Earth Sci. 2014, 72, 629–643. [Google Scholar] [CrossRef]
  13. Ma, J.-J. Theoretical Calculation and Verification of the Range of Damage Caused by Columnar Charge Blasting. Master’s Thesis, Kunming University of Science and Technology, Kunming, China, 2021. [Google Scholar]
  14. Lei, T.; Kang, P.-L.; Ye, H.-W.; Li, L.; Wang, Q.-Z. Study on the direction effect of stress wave superposition and fracturedistribution in rock mass during cylindrical charge blasting. J. Rock. Mech. Eng. 2024, 43, 399–411. [Google Scholar]
  15. Favreau, R.-F. Generation of strain waves in rock by an explosion in a spherical cavity. J. Geophys. Geophys. Res. 1969, 74, 4267–4280. [Google Scholar] [CrossRef]
  16. Graff, K.-F. Wave Motion in Elastic Solid; The Ohio State University Press: Columbus, OH, USA, 1991. [Google Scholar]
  17. Achenbach, J.-D. Wave propagation in elastic solids. J. Appl. Mech. 1980, 41, 544. [Google Scholar] [CrossRef]
  18. Blair, D. Seismic radiation from an explosive column. Geophysics 2010, 75, 55–65. [Google Scholar] [CrossRef]
  19. Gao, Q.-D.; Jin, J.; Wang, Y.-Q.; Lu, W.-B.; Leng, Z.-D.; Chen, M. Acting law of in-hole initiation position on distribution of blast vibration field. Explos. Shock. 2021, 41, 138–152. [Google Scholar]
  20. Yang, R.-S.; Guo, Y.; Li, Q.; Xu, P.; Chen, C. Evolution law on explosive stress and strain field of column charges at middle detonation position. J. Chin. Coal. Soc. 2019, 44, 3423–3431. [Google Scholar]
  21. Ding, C.-X.; Yang, R.-S.; Feng, C. Stress wave superposition effect and crack initiation mechanism between two adjacent boreholes. J. Rock. Mech. Min. Sci. 2021, 138, 104622. [Google Scholar] [CrossRef]
  22. Qiu, J.; Li, X.; Hu, C.; Zhao, Y.; Li, D.; Liang, L. Physical Model Test on the Deformation Behavior of an Underground Tunnel Under Blasting Disturbance. Rock. Mech. Rock. Eng. 2020, 54, 91–108. [Google Scholar] [CrossRef]
  23. Yang, L.-Y.; Ding, C.-X.; Yang, R.-S.; Yang, Q.-C. Experimental and Theoretical Analysis of Stress Superposition in Double-Hole Blasts. J. Test. Eval. 2020, 48, 201800932. [Google Scholar] [CrossRef]
  24. Liu, G.-Y.; Liu, X.-M.; Chen, S.-H. Influence of delay time on superposition effect of surface vibration. Eng. Blast. 2022, 28, 63–70. [Google Scholar]
  25. Zhang, Z.; Shi, X.; Qiu, X.; Ouyang, J. On-site tests investigating the effects of blasting vibration amplitude on the dynamic response of thin spray-on liner. Tunn. Undergr. Space Technol. 2024, 145, 105620. [Google Scholar] [CrossRef]
  26. Luo, C.; Yang, X.-A.; Li, K.; Wang, B.; Xiao, C.-Y. Dynamic strain test and measurement of a tunnel in nearby blasting vibration. Vib. Shock. 2020, 39, 262–268. [Google Scholar]
  27. Zhang, C.-H. Analysis of Velocity Maximum of the Linging of Existing Tunnel by the Blasting of the Tunnel Near-By. Master’s Thesis, Lanzhou University, Lanzhou, China, 2009. [Google Scholar]
  28. Liang, Q.-G.; Li, D.-W.; Zhu, Y.; Xie, F.-H. Vibration Control Technology for Blasting Construction Near Tunnels; Science Press: Beijing, China, 2015. [Google Scholar]
  29. Liang, R.; Zhu, M.; Zhou, W.-H.; Huang, X.-B.; Du, K.-F.; Zhou, Y.-T. Rock-breaking Characteristics of stress wave superposition between spherical charges. J. Yangtze. River. Sci. Res. Inst. 2020, 37, 67–72. [Google Scholar]
  30. Xiang, W.-F.; Shu, D.-Q.; Zhu, C.-B. Stress field analysis of strip charge blasting based on starfield superposition method. Explos. Shock. 2004, 05, 437–442. [Google Scholar]
  31. Zhang, B. Damage Characteristics of Surrounding Rock for Large-section Tunnel Excavated by Bench Blasting. Blasting 2023, 40, 69–76. [Google Scholar]
  32. Dai, J. Dynamic Behaviors and Blasting Theory of Rock; Beijing Metallurgical Industry Press: Beijing, China, 2013. [Google Scholar]
  33. Lu, W.-B. Study on the Propagation and Effects of Stress Waves in Rock Blasting. Ph.D. Thesis, Wuhan University, Wuhan, China, 1994. [Google Scholar]
  34. Onederra, A.-I.; Furtney, K.-J.; Sellers, E.; Iverson, S. Modelling blast induced damage from a fully coupled explosive charge. J. Int. Rock. Mech. Min. Sci. 2013, 58, 73–84. [Google Scholar] [CrossRef]
  35. Song, W.-H.; Jiao, H.-C.; Wang, Y.-W. Crack closure effect during the impact coal seam with high-pressure air blasting and the influence of gas drainage efficiency. Front. Earth Sci. 2023, 11, 1131386. [Google Scholar] [CrossRef]
  36. Costamagna, E.; Oggeri, C.; Segarra, P.; Castedo, R.; Navarro, J. Assessment of contour profile quality in D&B tunnelling. Tunn. Undergr. Space Technol. 2018, 75, 67–80. [Google Scholar] [CrossRef]
  37. Yu, T.-R.; Vongpaisal, S. New blasting damage criteria for underground blasting. CIM Bull. 1996, 89, 139–145. [Google Scholar]
  38. Cao, F.; Ling, T.-H.; Zhang, S. Safety threshold of blasting vibration velocity of highway tunnel considering influence of stress wave transmission. Vib. Shock. 2020, 39, 154–159. [Google Scholar]
  39. Cao, M.-X.; Yan, S.-H.; Zheng, Y.-Q.; Shao, C.; Wang, N.-X.; Du, J.-X. Analysis of blasting vibration response of adjacent tunnel. Eng. Blasting 2025, 31, 130–138. [Google Scholar]
  40. Lou, X.-M.; Lin, J.; Wang, X.-L.; Liu, H.-Y.; Wang, G. Characteristics of axial distribution of borehole wall pressure along theborehole of coupled charges with different borehole diameters under spherical detonation wave. J. Harbin. Inst. Technol. 2025, 57, 148–159. [Google Scholar]
  41. Luo, Y.; Shen, Z.-W. Investigation on length of stemming material and its effect in hole-charged blasting. Mech. Pract. 2006, 02, 48–52. [Google Scholar]
  42. Wang, G.-Z.; Pan, Q.-W. Voltage loss and voltage drop of low-voltage cables. Wire. Cable. 2014, 1, 17–19. [Google Scholar]
  43. Wang, B.; He, C.; Wu, D.-X.; Geng, P. Inverse analysis of in-situ stress field of Cangling super-long highway tunnel. Rock. Mech. 2012, 33, 628–634. [Google Scholar]
  44. Wang, T. Multi-Scale Analysis of In-Situ Stress Field of He-Jian No. 1 Tunnel Area of He-Jian Expressway in Yunnan Province. Master’s Thesis, Chang’an University, Xi’an, China, 2022. [Google Scholar]
  45. Cui, L.; Sheng, Q.; Zhang, J.; Dong, Y.-K.; Guo, Z.-S. Evaluation of input geological parameters and tunnel strain for strain-softening rock mass based on GSI. Sci. Rep. 2022, 12, 20575. [Google Scholar] [CrossRef]
  46. Tian, X.-C.; Tao, T.-J.; Liu, X.; Jia, J.; Xie, C.-J.; Lou, Q.-X.; Chen, Q.-Z.; Zhao, Z.-H. Calculation of hole spacing and surrounding rock damage analysis under the action of in situ stress and joints. Sci. Rep. 2022, 12, 22331. [Google Scholar] [CrossRef]
Figure 1. The instantaneous detonation pressure acts on the short-cylinder cavity: p ( t ) , q ( t ) and s ( t ) represent the radial, axial, and tangential pressures, respectively, which are generated by P-waves, SV waves, and SH waves (MPa); R is the distance from the charge center to the measurement point (m); ϕ is the angle from the vertical direction; 2a is the cylinder diameter (m); and l is the cylinder height (m).
Figure 1. The instantaneous detonation pressure acts on the short-cylinder cavity: p ( t ) , q ( t ) and s ( t ) represent the radial, axial, and tangential pressures, respectively, which are generated by P-waves, SV waves, and SH waves (MPa); R is the distance from the charge center to the measurement point (m); ϕ is the angle from the vertical direction; 2a is the cylinder diameter (m); and l is the cylinder height (m).
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Figure 2. Schematic diagram of the stress distribution on internal material particles in surrounding rock mass of adjacent tunnels.
Figure 2. Schematic diagram of the stress distribution on internal material particles in surrounding rock mass of adjacent tunnels.
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Figure 3. The propagation direction of three types of waves and the vibration direction of particles.
Figure 3. The propagation direction of three types of waves and the vibration direction of particles.
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Figure 4. Calculation model for superposition of the stress field in columnar charge blasting: L is the total length of the cylindrical charge (m); and m is the number of segmented short-cylinder charges into which the elongated charge is divided. According to reference [17], explosive charges with a length-to-diameter ratio (L/D) of less than 10 are classified as short cylindrical charges.
Figure 4. Calculation model for superposition of the stress field in columnar charge blasting: L is the total length of the cylindrical charge (m); and m is the number of segmented short-cylinder charges into which the elongated charge is divided. According to reference [17], explosive charges with a length-to-diameter ratio (L/D) of less than 10 are classified as short cylindrical charges.
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Figure 5. Schematic diagram of simultaneous explosion of multiple blastholes at the same point.
Figure 5. Schematic diagram of simultaneous explosion of multiple blastholes at the same point.
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Figure 6. Schematic diagram of cutting hole.
Figure 6. Schematic diagram of cutting hole.
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Figure 7. Schematic diagram of stress calculation points.
Figure 7. Schematic diagram of stress calculation points.
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Figure 8. Stresses decrease as a power law of the distance to the cut hole center.
Figure 8. Stresses decrease as a power law of the distance to the cut hole center.
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Figure 9. The radial stress–time history curve of point A.
Figure 9. The radial stress–time history curve of point A.
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Figure 10. Comparison of radial and axial stresses with and without considering relevant factors q(t) and D.
Figure 10. Comparison of radial and axial stresses with and without considering relevant factors q(t) and D.
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Figure 11. Variation patterns of triaxial stresses at calculation point A with different damage factors.
Figure 11. Variation patterns of triaxial stresses at calculation point A with different damage factors.
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Figure 12. Blasting network diagram.
Figure 12. Blasting network diagram.
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Figure 13. Ultra-dynamic strain measurement system.
Figure 13. Ultra-dynamic strain measurement system.
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Figure 14. Flowchart of the testing system.
Figure 14. Flowchart of the testing system.
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Figure 15. Schematic diagram of the voltage signal recording mechanism.
Figure 15. Schematic diagram of the voltage signal recording mechanism.
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Figure 16. Layout of monitoring points.
Figure 16. Layout of monitoring points.
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Figure 17. Strain bar, strain gauge and resistance testing.
Figure 17. Strain bar, strain gauge and resistance testing.
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Figure 18. Schematic diagram of strain bar installation into borehole.
Figure 18. Schematic diagram of strain bar installation into borehole.
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Figure 19. YL-LCT single-transmitter dual-receiver sonic probe.
Figure 19. YL-LCT single-transmitter dual-receiver sonic probe.
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Figure 20. Layout of wave velocity testing boreholes.
Figure 20. Layout of wave velocity testing boreholes.
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Figure 21. On-site Monitoring Layout.
Figure 21. On-site Monitoring Layout.
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Figure 22. Rock wave velocity test results (first blasting).
Figure 22. Rock wave velocity test results (first blasting).
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Figure 23. Measured voltage signal (radial direction at monitoring point 1 during the first blasting).
Figure 23. Measured voltage signal (radial direction at monitoring point 1 during the first blasting).
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Figure 24. Propagation law of measured stress with the distance from the blast center.
Figure 24. Propagation law of measured stress with the distance from the blast center.
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Figure 25. Comparison between measured and theoretical values of stress.
Figure 25. Comparison between measured and theoretical values of stress.
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Table 1. Physical and mechanical parameters of tunnel surrounding rock.
Table 1. Physical and mechanical parameters of tunnel surrounding rock.
Density (kg·m−3)Compressive Strength (MPa)Tensile Strength (MPa)Elastic Modulus (GPa)P-Wave Velocity (m/s)S-Wave Velocity (m/s)Poisson’s Ratio
265054.507.5030.70303617040.23
Table 2. Stress calculation results.
Table 2. Stress calculation results.
Calculation PointDistance to the Cut Hole Center (m)Peak Stress (MPa)
RadialAxialHoop
A11.421.600.930.76
B12.381.370.780.65
C13.351.180.680.56
D14.321.020.590.49
E15.290.900.520.43
F16.260.800.460.38
Table 3. Blasting parameters of upper bench.
Table 3. Blasting parameters of upper bench.
Borehole TypeNumber of BoreholesCharge per Hole (kg)Total Charge (kg)Delay Segment
Cut Holes61.59.0Segment 1 (0 ms)
82.419.2Segment 2 (50 ms)
Auxiliary Holes111.819.8Segment 3 (100 ms)
1323.4Segment 4 (150 ms)
1628.8Segment 5 (200 ms)
2443.2Segment 6 (250 ms)
Contour Holes181.527.0Segment 7 (300 ms)
1827.0Segment 8 (350 ms)
Floor Holes161.828.8Segment 9 (400 ms)
Total130 226.2
Table 4. Key parameters of the instrument.
Table 4. Key parameters of the instrument.
KU0/VnG
2.04.01100
Table 5. Sonic wave test results for measuring points 1–5.
Table 5. Sonic wave test results for measuring points 1–5.
Point 1Point 2Point 3Point 4Point 5
hole depth (m)1.02.03.04.05.0
original rock wave velocity (km/s)3.0363.0363.0363.0363.036
wave velocity before the first blasting (km/s)2.7162.7502.8162.8812.897
D0.200.180.140.100.09
Table 6. Stress measurement results.
Table 6. Stress measurement results.
Blasting TimesDamage Factor DMeasure PointPeak Voltage (mV)Peak Stress (MPa)
RadialAxialHoopRadialAxialHoop
10.2015.773.192.720.890.490.42
0.1826.273.572.940.960.550.46
0.1436.763.201.040.50
0.1047.174.203.411.100.650.52
0.0954.793.890.740.60
20.2215.703.142.700.880.480.42
0.2026.193.550.950.54
0.1536.733.763.181.030.580.49
0.1046.994.103.431.070.630.51
0.1057.974.693.781.220.720.59
30.2315.573.022.700.870.480.41
0.2126.083.490.930.54
0.1636.683.743.171.030.580.49
0.1444.043.320.620.50
0.1057.834.613.711.200.710.57
40.2315.733.142.700.870.480.41
0.2125.853.342.810.900.510.43
0.1736.533.653.091.000.560.47
0.1546.663.903.161.020.600.49
0.1257.564.453.601.160.680.55
50.2515.683.122.630.850.460.40
0.2325.852.780.900.43
0.1536.623.693.121.020.570.48
0.1246.894.033.251.060.620.50
0.1057.814.581.200.70
Table 7. The average value of measured damage factor and stress.
Table 7. The average value of measured damage factor and stress.
Measure PointDamage Factor DAverage Stress (MPa)Corresponding Theoretical Calculation PointsTheoretical Stress (MPa)
RadialAxialHoopRadialAxialHoop
10.230.880.480.41F0.800.460.38
20.210.930.540.44E0.900.520.43
30.151.030.580.49D1.020.590.49
40.121.060.620.50C1.180.680.56
50.101.200.710.57B1.370.780.65
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Luo, T.; Wei, Y.; Zhao, J.; Xie, Y.; Hu, Y.; Lou, X.; Huo, X. Research on Analytical Solution of Stress Fields in Adjacent Tunnel Surrounding Rock Under Blasting and Verification Analysis. Appl. Sci. 2025, 15, 8688. https://doi.org/10.3390/app15158688

AMA Style

Luo T, Wei Y, Zhao J, Xie Y, Hu Y, Lou X, Huo X. Research on Analytical Solution of Stress Fields in Adjacent Tunnel Surrounding Rock Under Blasting and Verification Analysis. Applied Sciences. 2025; 15(15):8688. https://doi.org/10.3390/app15158688

Chicago/Turabian Style

Luo, Tao, Yong Wei, Junbo Zhao, Yelong Xie, Yan Hu, Xiaoming Lou, and Xiaofeng Huo. 2025. "Research on Analytical Solution of Stress Fields in Adjacent Tunnel Surrounding Rock Under Blasting and Verification Analysis" Applied Sciences 15, no. 15: 8688. https://doi.org/10.3390/app15158688

APA Style

Luo, T., Wei, Y., Zhao, J., Xie, Y., Hu, Y., Lou, X., & Huo, X. (2025). Research on Analytical Solution of Stress Fields in Adjacent Tunnel Surrounding Rock Under Blasting and Verification Analysis. Applied Sciences, 15(15), 8688. https://doi.org/10.3390/app15158688

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