# Propagation of Nonplanar SH Waves Emanating from a Fault Source around a Lined Tunnel

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{l}and shear wave velocity c

_{l}, while the half-space is characterized by shear modulus μ

_{s}and shear wave velocity c

_{s}. The center of the tunnel is located at point O

_{1}with a distance d + b below the ground surface. The fault is considered an arch of circular O

_{3}with radius a

_{1}. In coordinate system O, the fault is located at r = r

_{f}, θ = −α

_{f}. In coordinate system O

_{3}, the fault is located at r

_{3}= a

_{1}, between the angles 2π − α

_{2}and 2π − α

_{1}. To facilitate the solution to the problem, an imaginary tunnel and an imaginary fault with respect to the ground surface are introduced herein. The definitions of the five Cartesian and five polar coordinate systems are shown in Figure 1.

_{n}is the complex coefficient to be determined, C

_{n}is the Bessel function with nth order, which can be either J

_{n}, Y

_{n}, H

_{n}

^{(1)}, H

_{n}

^{(2)}depending on the physical conditions of the problem.

_{3}, the wave field inside the circle O

_{3}can be written as:

_{3}can be written as:

- (1)
- Stress-free boundary conditions on the flat surface and the inner surface of the lining$$\frac{{\mu}_{s}}{r}\frac{\partial \left({u}_{s}+{u}_{o}\right)}{\partial \theta}=0,\theta =\pm \frac{\pi}{2}$$$${\mu}_{l}\frac{\partial {u}_{l}}{\partial {r}_{1}}=0,{r}_{1}=a$$
- (2)
- The continuity of both displacement and stress fields on the outer surfaces of the lining$${u}_{l}\left({r}_{1},{\theta}_{1}\right)={u}_{s}\left({r}_{1},{\theta}_{1}\right)+{u}_{o}\left({r}_{1},{\theta}_{1}\right),{r}_{1}=b$$$${\mu}_{l}\frac{\partial {u}_{l}}{\partial {r}_{1}}={\mu}_{s}\frac{\partial \left({u}_{s}+{u}_{o}\right)}{\partial {r}_{1}},{r}_{1}=b$$
- (3)
- The continuity of stress field on the circle O
_{3}$${u}_{i}\left({r}_{f},{\theta}_{f}\right)={u}_{s}\left({r}_{f},{\theta}_{f}\right)+{u}_{o}\left({r}_{f},{\theta}_{f}\right),{r}_{f}={a}_{1}$$$${\mu}_{s}\frac{\partial {u}_{i}}{\partial {r}_{f}}={\mu}_{s}\frac{\partial \left({u}_{s}+{u}_{o}\right)}{\partial {r}_{f}},{r}_{f}=b$$ - (4)
- Assuming there is a unit-amplitude dislocation with out of plane motion, the boundary condition on the fault can be written as:$${u}_{i}\left({r}_{f},{\theta}_{f}\right)-\left[{u}_{s}\left({r}_{f},{\theta}_{f}\right)+{u}_{o}\left({r}_{f},{\theta}_{f}\right)\right]=f\left({\theta}_{f}\right),{r}_{f}={a}_{1}$$
_{f}) is a function written as:$$f\left({\theta}_{f}\right)=H\left[\theta -\left(2\pi -{\alpha}_{1}\right)\left]+H\right[\theta -\left(2\pi -{\alpha}_{2}\right)\right]$$$$H\left(\theta \right)=\left\{\begin{array}{cc}0,& \theta <0\\ 1,& \theta \ge 0\end{array}\right.$$

_{f}) can be written as:

_{n}is the Fourier transform of f(θ

_{f}), which can be written as:

_{n}, B

_{n}, D

_{n}, E

_{n}and F

_{n}can be obtained by combining Equations (18)–(22) and truncating the infinite series in these equations to a finite number. Once the coefficients are known, the wave fields in the half-space are ready for numerical computation. A detailed flowchart for the summarization of the present method is shown in Figure 2.

## 3. Numerical Results and Discussion

_{s}is the wavelength of the incident SH waves.

_{s}/c

_{l}= 2/5, μ

_{s}/μ

_{l}= 8/75 [15].

_{s}/c

_{l}= 1, μ

_{s}/μ

_{l}= 1/3, α = 0°, φ = 90°. One can see that the present results are in excellent agreement with those by Lee & Trifunac (1979) [4] for large source-receiver distance.

_{1}/b = 8 for each case. Figure 5 represents the case for η = 1, and Figure 6 represents the case for η = 2.

_{θz}represents the shear stress in the lining.

_{1}/b = 8 and R

^{i}/b = 30. Figure 7 represents the case for η = 1, and Figure 8 represents the case for η = 2. Fault orientation plays an important role in DSCF. Comparing Figure 7a to Figure 7d, one can find that the location of the maximum DSCF changed from the lower part of the tunnel to the upper part of the tunnel as α changed from 90° to 0°. Similar trends can be seen by comparing Figure 7c to Figure 7f. However, the location of the maximum DSCF is almost unchanged compared to Figure 7b to Figure 7e. That means, different from the plane wave assumption, the fault orientation should be considered for the seismic design of tunnels.

_{i}= 0) in Figure 9a, the shaded area, where the displacement amplitudes reduce, appears on the ground surface right over the tunnel. What’s more, with the increase of dimensionless frequency η, the reduction of displacement amplitudes becomes more and more obvious. For cases of non-vertical incidence (α

_{i}≠ 0, Figure 9d–f), one can also see the illuminated area on the left ground and the shaded area on the right ground. The motions on the ground surface decrease quickly from the illuminated area to the shaded area. Comparing the ground motion for different fault orientations, one can find that there is almost no difference in the ground motion for horizontal fault orientation ((α

_{1}+ α

_{2})/2 = 90°) and vertical fault ((α

_{1}+ α

_{2})/2 = 0°). However, the ground motion for the oblique fault ((α

_{1}+ α

_{2})/2 = 60°) is obviously different from the vertical and horizontal cases. Moreover, the ground motion over the illuminated area for oblique faults is larger than that for vertical and horizontal faults. This means that the damage to the oblique fault is greater than the vertical and horizontal faults.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

a | inner radius of tunnel |

b | outer radius of tunnel |

d | burial depth of tunnel |

R^{i} | source-receiver distance |

μ_{l} | shear modulus of lining |

c_{l} | shear wave velocity of lining |

μ_{s} | shear modulus of half-space |

c_{s} | shear wave velocity of half-space |

k_{l} | shear wave number of lining |

k_{s} | shear wave number of half-space |

λ_{s} | wavelength of incident SH waves |

J_{n}, Y_{n}, H_{n}^{(1)}, H_{n}^{(2)} | Bessel function |

A_{n}, B_{n}, D_{n}, E_{n}, F_{n} | unknown coefficients |

η | dimensionless frequency |

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**Figure 4.**Comparison of ground motion between the present assumption and that of plane waves presented by Lee & Trifunac (1979) [4].

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**MDPI and ACS Style**

Zhang, N.; Zhang, Y.; Dai, D.; Zhang, Y.; Sun, B.; Chen, X.
Propagation of Nonplanar SH Waves Emanating from a Fault Source around a Lined Tunnel. *Sustainability* **2022**, *14*, 10127.
https://doi.org/10.3390/su141610127

**AMA Style**

Zhang N, Zhang Y, Dai D, Zhang Y, Sun B, Chen X.
Propagation of Nonplanar SH Waves Emanating from a Fault Source around a Lined Tunnel. *Sustainability*. 2022; 14(16):10127.
https://doi.org/10.3390/su141610127

**Chicago/Turabian Style**

Zhang, Ning, Yunfei Zhang, Denghui Dai, Yu Zhang, Baoyin Sun, and Xin Chen.
2022. "Propagation of Nonplanar SH Waves Emanating from a Fault Source around a Lined Tunnel" *Sustainability* 14, no. 16: 10127.
https://doi.org/10.3390/su141610127