# Anti-Plane Dynamics Analysis of a Circular Lined Tunnel in the Ground under Covering Layer

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## Abstract

**:**

## 1. Introduction

## 2. Model and Analysis

- The density, the shear velocity, and the shear modulus of the lower soil layer: ${\rho}_{1}$, ${c}_{1}$, ${\mu}_{1}$.
- The density, the shear velocity, and the shear modulus of the covering layer: ${\rho}_{2}$, ${c}_{2}$, ${\mu}_{2}$.
- The density, the shear velocity, and the shear modulus of the lined tunnel: ${\rho}_{3}$, ${c}_{3}$, ${\mu}_{3}$.
- The inner radius and outer radius of the lined tunnel: $a$, $b$.
- The radius of the upper and lower boundary of the covering layer: ${R}_{U}$, ${R}_{D}$.
- The thickness of the covering layer: $h$.
- The distance from the center of the circular lining to the lower boundary: $d$.

## 3. Results and Discussion

- A: (The covering layer is “softer”) Domain II is coal of density ${\rho}_{2}=1500\mathrm{kg}/{\mathrm{m}}^{3}$ and shear velocity ${c}_{2}=1000\mathrm{m}/\mathrm{s}$, then ${\rho}^{*}=0.6$, ${k}^{*}=2.4$.
- B: (The covering layer is “stiffer”) Domain II is dense limestone of density ${\rho}_{2}=2900\mathrm{kg}/{\mathrm{m}}^{3}$ and shear velocity ${c}_{2}=3200\mathrm{m}/\mathrm{s}$, then ${\rho}^{*}=1.2$, ${k}^{*}=0.75$.

_{D}. Accordingly, the problem studied in this paper is reduced to the problem of scattering of SH waves by a circular lining in a single medium half-space, which was solved in previous work by using the wave function expansion method combined with the image method [9]. Figure 2 shows the DSCFs around the inner surface of the circular lining for ${R}_{d}=100a$, $h=0.5a$, $d=1.5a$, ${\mu}^{\#}={\mu}_{3}/{\mu}_{1}=3.2$, ${k}^{\#}=0.7$ and $b/a=1.1$ when ${k}_{1}a=0.1\text{},\text{}1.0,\text{}2.0$, respectively. Through careful comparison, the results are basically consistent with the previous results.

## 4. Conclusions

- The parameters of the different soil layer mediums, the frequency of the incident waves, and the lining thickness all affect the dynamic stress concentration factor of the inner and outer surfaces of the lining. Therefore, engineering designs should consider the influence of various factors in combination with different geological conditions.
- When the SH wave incidence is of a low frequency, the soft covering layer has a significant amplification effect on $DSCF{\sigma}_{\theta z\mathrm{max}}^{*}$, while the stiff cover layer has a shielding effect on the SH wave. When the SH wave incidence frequency is high, this effect is not obvious. Compared with the SH wave scattering problem of lining in half-space, the combination of the soil layer and lining medium parameters in the covering layer is more complicated. The presence of a softer covering layer makes the dynamic response of the lining most sensitive to frequencies less than the presence of a stiffer covering layer.
- When the SH wave incidence frequency is low, increasing the thickness of the lining is effective to reduce the dynamic stress concentration on the outer surface of the lining, but it has little effect on the inner wall. Only when the SH wave incidence frequency is higher, increasing the thickness is meaningful for reducing the dynamic stress concentration of the inner surface. This effect should be considered in the project, and different strengthening measures should be taken for the inner and outer surfaces.
- It is also worth noting that although only the SH-wave disturbance is considered in this article, the large-circle hypothesis method is adopted, so based on this method, the scattering of similar models under p-wave and SV-wave disturbances can be further studied.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The circular lined tunnel under the covering layer disturbed by shear horizontal (SH) wave.

**Figure 2.**$DSCF{\sigma}_{\theta z}^{*}$ of the inner surface when degenerated into a circular lining in half-space.

**Figure 4.**Variation of the $DSCF{\sigma}_{\theta z\mathrm{max}}^{*}$ around outer surface with ${k}_{1}a$ for geological combination A.

**Figure 5.**Variation of the $DSCF{\sigma}_{\theta z\mathrm{max}}^{*}$ around outer surface with ${k}_{1}a$ for geological combination B.

**Figure 6.**$DSCF{\sigma}_{\theta z}^{*}$of the surface with the geological combination A when ${k}_{1}a=0.1,0.3,1.2$.

**Figure 7.**$DSCF{\sigma}_{\theta z}^{*}$ of the surface with the geological combination B when ${k}_{1}a=0.1,0.5,1.2$.

**Figure 8.**Variation of the $DSCF{\sigma}_{\theta z\mathrm{max}}^{*}$ around outer surface with $d/a$ for geological combination A.

**Figure 9.**Variation of the $DSCF{\sigma}_{\theta z\mathrm{max}}^{*}$ around outer surface with $d/a$ for geological combination B.

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**MDPI and ACS Style**

Qi, H.; Chu, F.; Zhang, Y.; Wu, G.; Guo, J.
Anti-Plane Dynamics Analysis of a Circular Lined Tunnel in the Ground under Covering Layer. *Symmetry* **2021**, *13*, 246.
https://doi.org/10.3390/sym13020246

**AMA Style**

Qi H, Chu F, Zhang Y, Wu G, Guo J.
Anti-Plane Dynamics Analysis of a Circular Lined Tunnel in the Ground under Covering Layer. *Symmetry*. 2021; 13(2):246.
https://doi.org/10.3390/sym13020246

**Chicago/Turabian Style**

Qi, Hui, Fuqing Chu, Yang Zhang, Guohui Wu, and Jing Guo.
2021. "Anti-Plane Dynamics Analysis of a Circular Lined Tunnel in the Ground under Covering Layer" *Symmetry* 13, no. 2: 246.
https://doi.org/10.3390/sym13020246