Blasting safety criterion of existing high speed railway tunnel over tunnel

3.1. On-site monitoring data analysis

The existing high-speed railway tunnel and Science City tunnel are monitored and analyzed by using the measuring and control blasting vibration monitoring instrument of Chengdu China Science Center, and the on-site blasting vibration monitoring data are counted. The specific monitoring point layout diagram is shown in Fig. 3.

Data regression analysis is carried out on the two tunnels according to Sadowski formula, where: $\alpha$ is the earthquake safe velocity, cm/s; $\nu$ is the coefficient related to geological conditions; $Q$ is the maximum burst charge of a single stage, kg; $R$ is the distance from the detonation source, m; $k$ is the attenuation coefficient of blasting. The regression formula of vibration velocity of Science City tunnel is:

The vibration velocity regression formula of the existing high-speed railway tunnel is as follows:

Fig. 3Schematic diagram of vibration measurement point layout of Science City tunnel

According to the relevant provisions of the Blasting Safety Regulations [16,20-21] on the frequency, as shown in Table 1, combined with the analysis of the field monitoring data of the Science City tunnel and the existing high-speed railway tunnel, it can be seen that the blasting vibration frequency is mostly greater than 50 Hz, so the control threshold of the peak vibration speed is selected as 15 cm/s. Through the analysis of Table 2, it is not difficult to find that, the peak value of the maximum vibration tunnel in the tunnel is less than 2.32 cm/s, which is far less than the tunnel peak control threshold, indicating that the blasting construction scheme design of Science City tunnel is reasonable, safe and reliable, and the surrounding rock of the tunnel is stable.

3.2. Establishment of mathematical model of attenuation law of existing high-speed railway tunnel

According to the relevant field measurement experiments and numerical simulation research results of a large number of scholars on the attenuation law of blasting vibration in rock and soil mass, it can be seen that the propagation of blasting vibration in rock and soil mass is related to a variety of factors (Hastings M. C. 2005, Govoni J. J. 2008), and the main relevant parameters are shown in Table 3.

From Buckingham’s theorem of dimensional analysis ($\pi$ theorem), shock wave overpressure ($P$) can be expressed as:

From Buckingham’s theorem of dimensional analysis ($\pi$ theorem), Rock and soil mass point vibration peak velocity ($\upsilon ’$) can be expressed as:

Substituting Eq. (4) into Eq. (3) yields:

By recombining ${\pi }_3$, ${\pi }_4$, $\pi$, we get the dimensionless number ${\pi }_8$:

where: ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ are respectively the index of ${\pi }_2$, ${\pi }_3$, ${\pi }_4$.

For a certain site, $\rho$ and $c$ can be approximated as constants. Therefore, a functional relationship can be considered from Eq. (6):

To sum up, this functional relationship can be written as:

Then:

where: ${\alpha }_1$ and ${\beta }_1$ are the coefficients given in the process of function transformation respectively.

Then $\mathrm{l}\mathrm{n}{k}_1=\mathrm{l}\mathrm{n}{\alpha }_1$:

Eq. (10) is the prediction formula of blasting vibration velocity without the influence of elevation difference. By substituting Eq. (10) into Eq. (9), we can get:

Then Eq. (11) can become:

Then a mathematical model reflecting the attenuation law of blasting vibration velocity under flat terrain conditions under the influence of elevation difference is established as follows:

where: $j$ represents the site influence coefficient; ${\beta }_1$ is the attenuation coefficient of peak strength of vibration velocity; ${\beta }_2$ is the effect coefficient of the peak intensity of vibration speed elevation, ${\beta }_3$ is the effect coefficient of the horizontal distance of vibration speed.

By comparison with Eq. (10) and Eq. (13), it can be seen that during the propagation attenuation of the existing high-speed railway tunnel in the rock mass, the elevation between the measuring point and the explosion source is affected. In order to evaluate the rationality and accuracy of the prediction model of blasting vibration of the existing high-speed railway tunnel established by Eq. (13), it is compared with the traditional prediction formula according to the fitting coefficient of the fitting curve, as shown in Table 4.

4.1. Blasting loading method

Combined with the theory of explosive stress wave [1-2], the blasting load is loaded in the form of triangular load, and the initial pressure of the explosive is:

Type: $P_0$ is the explosive detonation pressure; ${\rho }_0$ is the explosive density; $D$ is the detonation speed of the explosive; $\gamma$ is the isentropic index of the explosive. The approximate value is $\gamma =$ 2-3 [3], in this paper, $d_c$ and $d_b$ are the diameter of the cartridge and the diameter of the hole.

In this paper, emulsified explosive of, is adopted, and the specific loading form is shown in Fig. 4.

Fig. 4Loading curve of equivalent explosion impact load

4.2. Numerical model and parameter selection

4.2.1. Model size and boundary conditions

Through Flac3D numerical simulation, the dynamic response of new tunnel construction to existing high-speed railway tunnels can be predicted and evaluated, which provides a scientific basis for the optimization of construction scheme and the reinforcement of tunnel structure. According to the research needs, a 3D numerical model with dimensions of 120 m×80 m×99.65 m was established. The distance between the bottom of the model and the tunnel is set to 3 times that of the tunnel to reduce the influence of boundary effects on the simulation results. Two main formation materials, calc mudstone and sandstone, are considered in the model.

The physical and mechanical parameters of these two materials are shown in Table 5, including density, elastic modulus, Poisson's ratio, internal friction Angle, cohesion force, etc. These parameters are the basis of simulation calculation. Non-reflective boundary (such as static boundary or viscous boundary) is used on the tunnel contour to reduce the influence of wave reflection on the simulation results. The remaining boundaries are free boundaries. The stress distribution and change of the rock mass around the existing tunnel are analyzed, and the influence on the stability of the tunnel structure is evaluated. Monitor the displacement and deformation of existing tunnels during construction, especially the deformation of tunnel linings, to assess their impact on the safety of high-speed rail operations. The propagation law of vibration wave caused by new tunnel construction in rock mass and the response characteristics of existing tunnel to vibration wave, including vibration frequency and amplitude, are studied. In the laboratory, it is usually necessary to use a series of precise test methods and equipment to measure physical and mechanical parameters such as rock density, elastic modulus, Poisson’s ratio and shear modulus. The relevant physical and mechanical parameters are shown in Table 5.

Fig. 5Tunnel numerical model

Table 5Selection table of relevant physical and mechanical parameters

Parameters	Units	Type
		Calcareous mudstone	Sandstone
Density	g/cm3	2.2	2.6
Modulus of elasticity	GPa	0.1	0.45
Shear modulus	Mpa	39.06	187.5
Poisson's ratio	-	0.28	0.2
Cohesion	Mpa	10	15
Friction Angle	°	22	45
Compressive strength	Mpa	130	150
Yield strength	Mpa	0.4	0.8

5.1. Particle vibration change rule when a new tunnel is straddling an existing high-speed railway tunnel

Seismic waves generated by blasting propagate from far to near, and gradually transform from body waves to surface waves. In order to understand the vibration change law of tunnel particles caused by blasting seismic waves at different tunnel sections, when the new tunnel is directly across the existing high-speed railway tunnel, the particle distribution law of the existing high-speed railway tunnel contour at the distance of 1 m, 5 m and 30 m from Zhangziplane is selected.

It is not difficult to see from Fig. 6 that in the near area of tunnel blasting, the peak vibration velocity of particles in the tunnel is greater in the vertical direction than in the radial direction of the tunnel than in the axial direction of the tunnel ($Y>X>Z$), indicating that the vibration in the vertical direction is the main direction of vibration propagation in the area closer to the face of the hand, which is due to the larger buried depth of the tunnel and the better grade of the surrounding rock of the tunnel. With the initiation of the V-shaped cut area, the seismic waves generated by tunnel blasting mainly include body waves and surface waves. Since there are free planes in both axial and radial directions of the tunnel, they all attenuate to a certain extent, resulting in a large peak vibration velocity in the vertical direction of the tunnel. At the same time, as the distance from the tunnel palm surface continues to extend, the vibration propagation speed of particles inside the tunnel gradually shows a slowing trend. Specifically, the attenuation in the $Y$ direction is the most significant, followed by the $X$ direction, and the attenuation in the $Z$ direction is relatively minimal. This phenomenon shows that the attenuation of particle velocity in the tunnel is mainly concentrated in the $Y$ direction, that is, as the distance away from the tunnel face increases, the vibration energy dissipation in the $Y$ direction is the most obvious.

From Fig. 6 (a), the propagation law of particles in the tunnel in the $X$ direction, it is not difficult to see that the vibration speed of surrounding rock is small at the spinner of the tunnel, while the vibration speed is larger at the vault, arch foot and floor of the tunnel. This is because the tunnel has an elliptical structure and stress concentration is prone to occur at the arch foot of the tunnel, resulting in excessive stress here. Due to the larger stress at the arch foot of the tunnel, the wave impedance here is larger, so the peak vibration velocity here is larger. At the same time, due to the high ground stress of the floor, blasting excavation leads to the redistribution of the overall stress in the tunnel, and the unloading rebound occurs, resulting in the propagation of stress waves here being amplified to a certain extent.

Fig. 6Peak vibration velocity distribution of tunnel excavation section

a)$X$ direction

b)$Y$ direction

c)$Z$ direction

From Fig. 6(b), the propagation law of the tunnel particle in the vertical direction, it is not difficult to see that the vibration speed of the tunnel floor and the tunnel side wall is large, and the arch foot is the smallest. Tunnel excavation leads to the free surface in the tunnel, which provides conditions for the further deformation of the tunnel. At the same time, the vertical vibration in the tunnel corresponds to the shear wave in the tunnel blasting seismic wave, that is, SH wave and SV wave. The shear wave is very easy to reflect and refract in the free plane and the rock mass interface, so the vertical vibration speed is larger at the tunnel top, side wall and bottom plate near the tunnel blasting area, and the arch foot and spinner due to the phenomenon of stress concentration and special site structure, resulting in the vibration speed caused by stress waves is small.

5.2. Calculation model of safety threshold of high-speed railway tunnel

Combined with the numerical simulation results, the stress and vibration velocity values at different positions of the existing high-speed railway tunnel were calculated, and the stress and vibration velocity of the lining of the high-speed railway tunnel were analyzed by regression according to the principle of least square method, as shown in Eq. (6). As the existing high-speed rail tunnel is made of C30 concrete, after the actual engineering situation, for safety considerations, the value of the existing high-speed rail tunnel is calculated according to C25, and its ultimate tensile strength is 1.46 MPa. Combining with Eq. (16), the threshold value of blasting vibration safety control of the existing high-speed rail tunnel can be obtained as 19.41 cm/s:

Stress waves are pressure fluctuations caused by the energy released instantaneously by blasting, which are characterized by high pressure, high speed propagation and high energy. When the blasting operation is carried out in the mountain directly above the tunnel, the shock wave generated by the explosion will propagate through the mountain, and part of the energy will be converted into stress wave and transferred to the tunnel structure. Such impact load can cause the structure to produce dynamic strain and stress, and affect the stability and strength of the structure. The dynamic stress may exceed the static bearing capacity of the structure, resulting in the destruction of the structure.

The impact and vibration effects in the process of stress wave propagation can make the tunnel structure vibrate. In the long run, vibration will cause fatigue damage to the structure and weaken the strength and stability of the structure.

The propagation speed of P wave and S wave is expressed by:

where: $C_P$ is the propagation speed of p-wave; $C_S$ is the propagation speed of shear wave; $E_d$ is the dynamic elastic modulus; $\mu$ is Poisson’s ratio; $\rho$ is density.

According to Snell’s rule, when light passes through the interface of two media, the relationship between the incidence Angle and the refraction Angle satisfies the following:

where the $\theta ₁$ is the incidence Angle and ${\theta }_2$ is the refraction angle.

On the free surface of concrete lining, when the incident wave is P-wave, the stress-strain generated by P-wave and S-wave contains the following relations:

where: $R_0$ is the reflection coefficient of the stress wave. ${\sigma }_{Pi}$, ${\tau }_{Si}$, ${\sigma }_{Pr}$, ${\tau }_{Pr}$ are the incident and reflected P-waves and S-waves.

Suppose there is a plane burst wave in the rock mass that travels with velocity $v$, and a displacement wave that travels with velocity $u$. According to stress wave theory, dynamic stress can be expressed as:

where: $v_P$ and $v_S$ are the vibration speed of the concrete particles of the tunnel lining caused by the longitudinal wave and the transverse wave. [${\sigma }_t$], [$\tau$] are the ultimate tensile stress strength and shear stress strength of tunnel lining concrete.

By combining Eqs. (17-23), the safety criterion model of blasting vibration of the existing high-speed railway tunnel under blasting stress can be obtained, as shown in Eq. (13):

where: $v_P\left(\sigma \right)$, $v_S\left(\sigma \right)$ is the safe vibration speed determined by the limit tensile stress strength criterion under the action of S wave.

Under the action of blasting vibration, the ultimate tensile strength of tunnel concrete lining may be improved to some extent. This is because the blasting vibration will produce severe dynamic load and vibration effect, and produce strong force on the concrete structure. This force can promote the friction between the particles inside the concrete to reduce, make the concrete more compact, improve its mechanical properties. Specifically, under the action of blasting vibration, the particles in the concrete will have relative displacement, and the friction between the particles will be reduced, thus reducing the internal friction resistance. In this way, under the action of external load, the concrete is more prone to displacement and deformation, making the internal stress distribution more uniform. Since the tensile strength of concrete is related to its internal stress distribution, the ultimate tensile strength of concrete may be improved when the stress distribution is more uniform.

The relationship between static tensile limit strength and dynamic tensile limit strength of lining material is shown in Eq. (24):

where: ${\left[\sigma \right]}_t$ is the dynamic tensile strength; ${\sigma }_{t0}$ is static tensile strength; ${\nu }_H$ is the loading rate, ${\nu }_H={\sigma }_H/{\sigma }_1$; ${\sigma }_H$ is any loading speed (${\sigma }_H\ge$1); ${\sigma }_1$ is the loading speed, ${\sigma }_1=$0.1 MPa/s; ${\stackrel{-}{K}}_D$ is the dynamic strength improvement coefficient. Generally, 1.32 is taken.

5.3. Safety threshold

The dynamic tensile strength of C25 concrete is obtained by synthesizing the dynamic strength improvement coefficient obtained from Eq. (26). According to the characteristics of concrete materials, the dynamic elastic modulus is generally 1.2 times of the static elastic modulus, and the dynamic elastic modulus of C25 concrete is calculated. The parameters used in C25 concrete material are shown in Table 7.

The experimental data clearly reveal the influence of different types of seismic waves (P-wave and S-wave) on the critical vibration velocity of tunnel lining structure at different incident angles. Specifically, when the longitudinal wave (P wave) acts on the tunnel lining at an incident Angle ranging from 0° to 40°, the critical vibration velocity required for structural damage increases correspondingly with the gradual increase of the incident Angle. This phenomenon indicates that the resistance of the lining structure to vibration is enhanced when the longitudinal wave is inclined. It is particularly significant that the minimum critical vibration velocity that the tunnel lining structure can withstand under the extreme case of vertical incidence of the P-wave (i.e. the incidence Angle is 0°) is determined to be 14.73 cm/s, which is an important threshold to ensure the safety of the structure under this condition.

On the other hand, the behavior of S-wave in the range of 0° to 40° is completely opposite, and the critical vibration velocity of blasting induced by S-wave decreases gradually with the increase of incident Angle. This means that the tunnel lining structure is more vulnerable to damage when the shear wave is oblique incident. Especially, under the condition of vertical incidence of shear wave, the minimum critical vibration velocity of the lining structure to resist damage is reduced to 32 cm/s, which is much lower than the value of vertical incidence of longitudinal wave, which highlights the potential threat of shear wave to the safety of tunnel structure.

Further analysis shows that the failure mechanism caused by different waveforms is also different: under the action of P-wave, the tunnel lining structure first occurs tensile failure, which is caused by the direct compression and tension of the P-wave medium, resulting in the internal stress of the material exceeding its tensile strength. Under the action of shear waves (S-waves), the structure is the first to experience shear failure, because the shear waves mainly cause the transverse vibration of the medium, resulting in the shear stress concentration and exceeding the shear strength of the material.

Fig. 7 intuitively shows the complex relationship between critical vibration velocity and incidence Angle under these two waveforms, providing important theoretical basis and reference data for predicting and controlling the impact of blasting vibration on tunnel lining structure in engineering practice. With a deeper understanding of these relationships, engineers can more accurately design blasting plans and take effective shock absorption measures to ensure structural safety and stability during tunnel construction and operation.

Fig. 7Relation between critical vibration velocity and incidence angle

The safety threshold of concrete blasting vibration velocity obtained by numerical simulation is 19.41 cm/s, which is larger than the result obtained by theoretical analysis. This is because although the two methods are based on the ultimate tensile stress criterion, the numerical simulation cannot completely simulate the actual engineering situation, resulting in errors in the results. Therefore, it is not recommended to adopt only a single method of numerical simulation when designing and studying similar projects. Finally, for the sake of safety, the blasting vibration safety threshold of the existing high-speed railway tunnel can be set as 14.73 cm/s, and the calculated results can be used as the safety criterion of the blasting vibration speed of the project, so as to guide the blasting operation on site efficiently and ensure the safety and stability of the tunnel. The theoretical analysis of safety criterion of blasting vibration velocity of tunnel lining based on ultimate strength criterion is reliable.