Dynamic Response of Twin Circular Unlined Tunnels Subjected to Blasting P Waves

The dynamic response of twin circular unlined tunnels is studied by indirect boundary integral equation method (IBIEM). The twin tunnels are assumed to lie in an unbounded elastic space subjected to blasting P waves. The influence of incident angle on the distribution of DSCF around the twin tunnels is analyzed. Besides, the influences of normalized wave number αa and normalized distance d* (ratio of spacing distance to tunnel radius) on the reference DSCF, that is, the peak DSCF for different incident angles with an assurance rate of 95%, are discussed. Results show that 1) the IBIEM is a high-precision method analyzing the DSCF around twin circular unlined tunnels induced by blasting P waves; 2) the incident angle θ₀, αa, and d* all have significant influences on the distribution of DSCF; 3) for a constant d*, the peak DSCF around the right tunnel is slightly greater than the left one; 4) the reference DSCF decreases exponentially with the increasing of αa or d*, and the corresponding fitting functions are proposed.

Introduction

In recent years, underground tunnels have been widely developed due to transportation problems induced by the dramatic growth of population (Li et al., 2017; Zhang et al., 2021). In many populous cities, twin tunnels are increasingly constructed to improve transportation capacity (Alielahi and Adampira, 2016). When the drilling and blasting method is adopted, blasting waves will impose a potential danger on the adjacent existing tunnels (Xia et al., 2013; Zhou et al., 2016; Dang et al., 2018; Xue et al., 2019; Peng et al., 2021). Evaluating the dynamic response and proposing a safety criterion of the existing tunnels subjected to blasting waves are of the utmost importance.

In the past years, extensive studies have been carried out in evaluating the dynamic response of underground tunnels caused by incident waves. These studies mainly focus on the dynamic response of the rock or the lining of a single tunnel. Yi et al. (2008) studied the dynamic response of an arch-with-vertical-wall lining, and concluded that the incidence angle of incident waves has a significant influence on the critical vibration velocity. Furthermore, Yi et al. (2016) discussed the dynamic response of a circular lined tunnel with an imperfectly bonded interface under plane P waves, and found that the interface stiffness has a great influence on the distribution of dynamic stress concentration factors (DSCF) of the tunnel. Pao and Mow (1973) investigated the dynamic stress concentration of lined and unlined circular tunnels subjected to plane P and S waves numerically. Lu et al. (2019a), Lu et al. (2019b), and Lu et al. (2021) studied the DSCF and peak particle velocity of a circular tunnel subjected to cylindrical P waves theoretically. Wang et al. (2014) studied the dynamic responses of the tunnel at various depths in double-layer rocks by finite element method. Li et al. (2018) firstly proposed the theoretical formula to evaluate the DSCF around a circular opening subjected to blasting wave and studied the dynamic response of deep-buried tunnels under triangular blasting loads. Liu et al. (2011) and Chen et al. (2012) discussed the dynamic response of a rock tunnel subjected to harmonic P, S, and R waves by numerical simulation method. Xu et al. (2014) investigated the effects of incident frequency, incident angle, and rock conditions on a circular lining tunnel subjected to incident plane P waves. Zhang et al. (2022) explored the stresses and vibrations of a single circular tunnel under the incidence of cylindrical P waves and proposed the PPV criterion. Qiu et al. (2022) studied the blasting dynamic behavior of deep buried tunnels and discussed their safety. Zlatanović et al. (2021) and Yuan et al. (2020) explored the influence of new structures on the dynamic responses of existing tunnels subjected to different incident waves. Although the study on different factors influencing dynamic response of underground tunnels subjected to incident waves has made great progress, the quantitative relationship between different factors and dynamic response of tunnels subjected to incident waves is very limited.

In this article, the dynamic response of twin circular unlined tunnels subjected to blasting P waves is studied by indirect boundary integral equation method (IBIEM). The influence of the incident angle on DSCF of the twin tunnels is analyzed. Furthermore, the influences of normalized wave number and normalized distance (ratio of spacing distance to tunnel radius) on the reference DSCF, that is, the peak DSCF for different incident angles with an assurance rate of 95%, are discussed. The fitting functions determining the reference DSCF are proposed.

Interaction of Blasting P Waves and Twin Circular Unlined Tunnels

Assume two circular unlined tunnels with a distance d lie in an unbounded space, centering at O₁ and O₂, respectively. The radii are denoted by a₁ on the left side and a₂ on the right side. A right-handed Cartesian coordinate system O₁xy is established, as shown in Figure 1. The virtual sources are represented by S₁ and S₂ with less radii b₁ and b₂ to simulate the outgoing waves and avoid singularity on the tunnel boundaries. The displacement potential function of incident P wave can be expressed as:

${\phi }^{\left(i\right)}={\phi }_0{\mathrm{e}}^{i\alpha \left(x\cos {\theta }_0+y\sin {\theta }_0\right)-i\omega t}(1)$

where ${\phi }_0$ is the amplitude of incident wave, α is the wave number of P waves and α=ω/c_p, ω is the circular frequency of incident wave, c_p is the phase velocity of incident wave, θ₀ is the incident angle, and i is the unit of complex number.

FIGURE 1

FIGURE 1. Incident wave and its interaction with two circular unlined tunnels.

Let φ denote the linear sum of the potentials of the incident and diffracted P waves, ψ denote the linear sum of the potential of the incident and diffracted S wave, then

$\begin{array}{l}\phi ={\phi }^{\left(i\right)}+{\displaystyle \sum _{i=1}^M{\phi }_i^{\left(d\right)}}+{\displaystyle \sum _{i=M+1}^{M+N}{\phi }_i^{\left(d\right)}}\\ \psi ={\displaystyle \sum _{i=1}^M{\psi }_i^{\left(d\right)}}+{\displaystyle \sum _{i=M+1}^{M+N}{\psi }_i^{\left(d\right)}}\end{array}(2)$

where M and N are the numbers of virtual sources in the left and right tunnels, ${\phi }_i^{\left(d\right)}$ and ${\psi }_i^{\left(d\right)}$ are the potential functions of diffracted P and SV waves generated by the i-th virtual source.

The diffracted waves generated by the i-th virtual source can be expressed in terms of 0-th Hankel functions as:

${\phi }_i^{\left(d\right)}=A_i{H}_0^{\left(1\right)}\left(\alpha R_i\right){\mathrm{e}}^{-i\omega t}(3)$

${\psi }_i^{\left(d\right)}=B_i{H}_0^{\left(1\right)}\left(\beta R_i\right){\mathrm{e}}^{-i\omega t}(4)$

where R_i is the distance between an arbitrary point in the surrounding rock and the i-th virtual source, defined by $R_i=\sqrt{{\left(x-x_i\right)}^2+{\left(y-y_i\right)}^2}$, β is the wave number of S waves.

According to the relations between stress and displacement, the stress can be expressed as:

$\{\begin{array}{l}\begin{array}{l}{u}_x=\frac{\partial \phi }{\partial x}+\frac{\partial \psi }{\partial y}\\ u_y=\frac{\partial \phi }{\partial y}-\frac{\partial \psi }{\partial x}\\ \frac{{\sigma }_{xx}}{\mu }=\left(\frac{{\beta }^2}{{\alpha }^2}-2\right){\nabla }^2\phi +2\left(\frac{{\partial }^2\phi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial x\partial y}\right)\end{array}\hfill \\ \frac{{\sigma }_{yy}}{\mu }=\left(\frac{{\beta }^2}{{\alpha }^2}-2\right){\nabla }^2\phi +2\left(\frac{{\partial }^2\phi }{\partial y^2}-\frac{{\partial }^2\psi }{\partial x\partial y}\right)\hfill \\ \frac{{\sigma }_{xy}}{\mu }=2\frac{{\partial }^2\phi }{\partial x\partial y}+\frac{{\partial }^2\psi }{\partial y^2}-\frac{{\partial }^2\psi }{\partial x^2}\hfill \end{array}(5)$

Then substituting Eqs 1–4 into Eq. 5, the displacements and stresses can be written as follows:

$u_x=i\alpha {\phi }_0\cos {\theta }_0{\mathrm{e}}^{i\alpha \left(x\cos {\theta }_0+y\sin {\theta }_0\right)}-{\displaystyle \sum _{i=1}^{M+N}\frac{\alpha A_i\left(x-x_i\right)H_1^{\left(1\right)}\left(\alpha R_i\right)}{R_i}}-{\displaystyle \sum _{i=1}^{M+N}\frac{\beta B_i\left(y-y_i\right)H_1^{\left(1\right)}\left(\beta R_i\right)}{R_i}}(\mathrm{6a})$

$u_y=i\alpha {\phi }_0\sin {\theta }_0{\mathrm{e}}^{i\alpha \left(x\cos {\theta }_0+y\sin {\theta }_0\right)}-{\displaystyle \sum _{i=1}^{M+N}\frac{\alpha A_i\left(y-y_i\right)H_1^{\left(1\right)}\left(\alpha R_i\right)}{R_i}}+{\displaystyle \sum _{i=1}^{M+N}\frac{\beta B_i\left(x-x_i\right)H_1^{\left(1\right)}\left(\beta R_i\right)}{R_i}}(\mathrm{6b})$

$\begin{array}{c}\frac{{\sigma }_{xx}}{\mu }=\left(2{\alpha }^2{\sin }^2{\theta }_0-{\beta }^2\right){\phi }_0{e}^{i\alpha \left(x\cos {\theta }_0+y\sin {\theta }_0\right)}\\ +{\displaystyle \sum _{i=1}^{M+N}{A}_i\left\{\left[2{\alpha }^2\frac{{\left(y-y_i\right)}^2}{R_i^2}-{\beta }^2\right]H_0^{\left(1\right)}\left(\alpha R_i\right)+2\alpha \left[\frac{2{\left(x-x_i\right)}^2}{R_i^3}-\frac{1}{R_i}\right]H_1^{\left(1\right)}\left(\alpha R_i\right)\right\}}+\\ {\displaystyle \sum _{i=1}^{M+N}{B}_i\left\{4\beta \frac{\left(x-x_i\right)\left(y-y_i\right)}{R_i^3}{H}_1^{\left(1\right)}\left(\beta R_i\right)-2{\beta }^2\frac{\left(x-x_i\right)\left(y-y_i\right)}{R_i^2}{H}_0^{\left(1\right)}\left(\beta R_i\right)\right\}}\end{array}(\mathrm{6c})$

$\begin{array}{c}\frac{{\sigma }_{yy}}{\mu }=\left(2{\alpha }^2{\cos }^2{\theta }_0-{\beta }^2\right){\phi }_0{e}^{i\alpha \left(x\cos {\theta }_0+y\sin {\theta }_0\right)}\\ +{\displaystyle \sum _{i=1}^{M+N}{A}_i\left\{\left[2{\alpha }^2\frac{{\left(x-x_i\right)}^2}{R_i^2}-{\beta }^2\right]H_0^{\left(1\right)}\left(\alpha R_i\right)+2\alpha \left[\frac{2{\left(y-y_i\right)}^2}{R_i^3}-\frac{1}{R_i}\right]H_1^{\left(1\right)}\left(\alpha R_i\right)\right\}}\\ -{\displaystyle \sum _{i=1}^{M+N}{B}_i\left\{4\beta \frac{\left(x-x_i\right)\left(y-y_i\right)}{R_i^3}{H}_1^{\left(1\right)}\left(\beta R_i\right)-2{\beta }^2\frac{\left(x-x_i\right)\left(y-y_i\right)}{R_i^2}{H}_0^{\left(1\right)}\left(\beta R_i\right)\right\}}\end{array}(\mathrm{6d})$

$\begin{array}{c}\frac{{\sigma }_{xy}}{\mu }=-{\alpha }^2\sin 2{\theta }_0{\phi }_0{e}^{i\alpha \left(x\cos {\theta }_0+y\sin {\theta }_0\right)}\\ +{\displaystyle \sum _{i=1}^{M+N}{A}_i\left\{4\alpha \frac{\left(x-x_i\right)\left(y-y_i\right)}{R_i^3}{H}_1^{\left(1\right)}\left(\alpha R_i\right)-2{\alpha }^2\frac{\left(x-x_i\right)\left(y-y_i\right)}{R_i^2}{H}_0^{\left(1\right)}\left(\alpha R_i\right)\right\}}\\ -{\displaystyle \sum _{i=1}^{M+N}{B}_i\left\{2\beta \frac{{\left(x-x_i\right)}^2-{\left(y-y_i\right)}^2}{R_i^3}{H}_1^{\left(1\right)}\left(\beta R_i\right)-{\beta }^2\frac{{\left(x-x_i\right)}^2-{\left(y-y_i\right)}^2}{R_i^2}{H}_0^{\left(1\right)}\left(\beta R_i\right)\right\}}\end{array}(6e)$

The stress components in Cartesian coordinate system can be transformed to those in polar coordinate system by the following equations.

$\{\begin{array}{c}{u}_r=u_x\cos \theta +u_y\sin \theta \\ u_{\theta }=-u_x\sin \theta +u_y\cos \theta \\ {\sigma }_{rr}={\sigma }_{xx}{\cos }^2\theta +{\sigma }_{yy}\mathrm{si}{\mathrm{n}}^2\theta +{\sigma }_{xy}\sin 2\theta \\ {\sigma }_{\theta \theta }={\sigma }_{xx}\mathrm{si}{\mathrm{n}}^2\theta +{\sigma }_{yy}\mathrm{co}{\mathrm{s}}^2\theta -{\sigma }_{xy}\sin 2\theta \\ {\sigma }_{r\theta }=\left({\sigma }_{yy}-{\sigma }_{xx}\right)\sin \theta \cos \theta +{\sigma }_{xy}\cos 2\theta \end{array}(7)$

The traction-free boundary conditions are given by

${\sigma }_{rr}={\sigma }_{r\theta }=0\text{for}\{\begin{array}{l}x=a_1\cos \theta \\ y=a_1\sin \theta \end{array}\text{and}\{\begin{array}{l}x=d+a_2\cos \theta \\ y=a_2\sin \theta \end{array}(8)$

Substituting Eqs 6a–6e and 7 into Eq. 8, the boundary conditions can be rewritten in the matrix form.

$HA=S(9)$

where H is the Green’s influence matrix, A is the undermined coefficients representing the source density, and S is the vector of stresses induced by the incident wave.

In general, Eq. 9 is overdetermined, and only the approximate solution can be obtained by the least square method.

$A={\left[H^{\prime }H\right]}^{-1}{H}^{\prime }S(10)$

The relative error is always determined as follows:

$Err=\left|HA-S\right|/\left|S\right|(11)$

where $\left|\right|$ is the Euclidean norm of a vector.

DSCF is an important parameter to evaluate structural stability (Fan et al., 2019; Ming et al., 2019; Jang et al., 2020; Li et al., 2020), which is defined as follows:

$\text{DSCF}=\left|\frac{{\sigma }_{\theta \theta }}{{\sigma }_0}\right|(12)$

where σ₀ is the stress intensity of the incident wave in the direction of propagation, defined by σ₀=μβ²φ₀, μ is the shear modulus.

In order to get some general results, we need to define the following dimensionless parameter: the normalized distance as d*=d/a, where a is the maximum of a₁ and a₂. αa is another quite important index, generally called the dimensionless wave number, which equals 2π times the ratio of a to the wave length of the incident wave.

Accuracy Analysis

In order to determine the calculation accuracy, we choose the case of d*=4 for analysis. Table 1 shows the relative errors determined by Eq. 11 for different θ₀ when αa = 0.1 and 2. The numbers of source points and observation points are generally different and the previous is less.

TABLE 1

TABLE 1. Relative errors for different numbers of source and observation points.

It is found that the relative error decreases with the increasing numbers of source and observation points. When the numbers of source and observation points equal to 60 and 80, the maximum relative errors are 1.014 × 10⁻⁴ for αa = 0.1 and 1.542 × 10⁻³ for αa = 2, which are sufficiently accurate. When the numbers of source and observation points equal to 80 and 100, the maximum relative errors are 2.433 × 10⁻⁸ for αa = 0.1 and 2.328 × 10⁻⁶ for αa = 2, which become far smaller. Therefore, we believe that 80 source points and 100 observation points are suitable in the following analysis.

Results and Discussions

The distribution of DSCF around the twin tunnels under different αa and θ₀ is shown in Figure 2, where d* = 4.

FIGURE 2

FIGURE 2. Distribution of DSCF around twin tunnels. (A) θ₀ = 0°; (B) θ₀ = 30°; (C) θ₀ = 60°; (D) θ₀ = 90°.

The DSCF of the tunnels is approximately symmetrically distributed about the incident direction for αa = 0.1, and it becomes more complex with the increase of αa. It is obvious that the distribution of DSCF around the twin tunnels changes obviously with the variation of θ_0. For the left tunnel, the peak DSCF around the left tunnel at θ₀ = 0° is smaller than that at others for a fixed αa. For the right tunnel, the peak DSCF around the right tunnel occurs at θ₀ = 0° for αa = 0.1, while at others for αa = 0.5, 1, and 2. This suggests that the incident angle θ₀ of blasting P waves has a significant influence on the distribution of DSCF.

For a fixed θ₀, the peak DSCF at αa = 0.1 is obviously greater than other cases, but the values at αa = 0.5, 1, and 2 do not change apparently, which indicates that the peak DSCF decreases quickly at first, then varies slower and slower, eventually tends to a steady value.

Although the left tunnel is located at the incident side, the DSCF of the right tunnel is larger under the same αa, which may result from that the left tunnel deflects and focuses stress on, just like a convex lens.

According to the above analysis, the DSCF of tunnels changes with θ₀. In a practical project, θ₀ may be unknown and the upper bound of DSCF for different θ₀ has a critical reference value for engineers. Figure 3 and Figure 4 show the variation of the reference DSCF (i.e., 95% of peak DSCF values are less than the given value) with αa and d* ,respectively.

FIGURE 3

FIGURE 3. Variation of the reference DSCF with αa. (A) d* = 4; (B) d* = 10; (C) d* = 20.

FIGURE 4

FIGURE 4. Variation of DSCF with d*. (A) αa = 0.1; (B) αa = 1; (C) αa = 2.

It is found from Figure 3 that the peak DSCF around the left tunnel is more discrete than the right tunnel and its decreasing rate with αa for θ₀ = 0^o is obviously greater. The discreteness reduces with the increasing value of d*. By using the least square method, we found the variation of the reference DSCF can be well fitted exponential functions. According to the fitting functions, the attenuation of peak DSCF around the left tunnel is faster than the right tunnel for a same d*.

The summation of fitting functions in Figure 3 is listed in Table 2.

TABLE 2

TABLE 2. Summation of fitting functions with respect to αa.

It is observed from Figure 4 that the peak DSCF for the left tunnel fluctuates obviously at αa = 0.1, the variation tendency becomes clearer at αa = 1 and 2. Besides, the smallest DSCF occurs at θ₀ = 90° for αa =0.1 while at θ₀ = 0° for αa = 1 and 2. For the right tunnel, the peak DSCF shows a wave-like change at αa = 0.1. When αa = 1 and 2, the variation trends of DSCF with d* are similar, the peak value of DSCF occurs at θ₀ = 30° or 45°. Compared with the left tunnel, the distribution of peak DSCF of the right tunnel is more concentrated.

It is also found that the variation of the reference DSCF can be well fitted exponential functions. According to the fitting functions, the attenuation of peak DSCF around the right tunnel is faster for a constant αa while slower for a constant d* compared with that around the left tunnel.

The summation of fitting functions in Figure 4 is listed in Table 3.

TABLE 3

TABLE 3. Summation of fitting functions with respect to d*.

Based on the above analysis, the attenuation of the reference DSCF with αa or d* can be well expressed by exponential functions. Considering the joint impact of αa and d*, the DSCF changing with both αa and d* of the twin tunnels is shown in Figure 5.

FIGURE 5

FIGURE 5. DSCF fitting surfaces. (A) Left tunnel; (B) right tunnel.

The fitting surfaces for both tunnels are expressed in Eq. 13.

$\text{DSCF}=\{\begin{array}{cc}1.5902\exp \left(0.6324/d^{\ast }+0.0533/\alpha a\right)& \text{for the left tunnel}\\ 1.7241\exp \left(1.0277/d^{\ast }+0.0423/\alpha a\right)& \text{for the right tunnel}\end{array}(13)$

It is found that the fitting function can well reflect the previous results and αa has a greater influence on DSCF than d*. Eq. 13 can be proposed as the reference function and engineers can quickly obtain the critical value to prevent damage to such underground structures.

Conclusion

In this article, the dynamic response of twin circular unlined tunnels in an unbounded space subjected to blasting P waves is studied. According to the research results, the following conclusions can be obtained:

(1) The DSCF around twin circular unlined tunnels subjected to blasting P waves is calculated by indirect boundary integral equation method, which is of high precision.

(2) The incident angle θ₀, normalized wave number αa, and normalized distance d* all have a significant influence on the distribution of DSCF around the twin tunnels.

(3) For a constant d*, the peak DSCF around the right tunnel is slightly greater than the left one.

(4) The reference DSCF decreases exponentially with the increase of αa or d*, and the corresponding fitting functions are proposed. Compared with d*, the influence of αa is more distinct.

Data Availability Statement

The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found in the article/Supplementary Material.

Author Contributions

Conceptualization: SL, YY; Methodology: SL and YJ; Formal analysis and investigation: JS, SL, and ZZ; Draft preparation: SL and LJ.

Funding

The study was sponsored by the Hubei Provincial Natural Science Foundation (Grant No. 2019CFB224), Hubei Provincial Department of Education (Grant No. Q20191308), and Open Research Fund of Hubei Key Laboratory of Blasting Engineering (Grant No. HKLBEF202011).

Conflict of Interest

The 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.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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