# The Study for Longitudinal Deformation of Adjacent Shield Tunnel Due to Foundation Pit Excavation with Consideration of the Retaining Structure Deformation

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## Abstract

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## 1. Introduction

## 2. Study on the Sidewall Unloading of the Foundation Pit

#### 2.1. The Deficiency of the Existing Foundation Pit Unloading Model

#### 2.2. An Unloading Model Considering the Deformation of Retaining Structure

_{e}is the excavation depth. λ is the horizontal distance between any point on the retaining structure and the foundation pit corner. η is the buried depth at any point on the sidewall of the retaining structure. One could set the displacement of any point on the retaining structure of the side wall toward the inside of the foundation pit as v(λ, η).

_{i}(η, H

_{ei}) is the deformation increment of the sidewall retaining structure at the depth η, caused by the excavation of the ith layer in the foundation pit, while H

_{ei}is the depth of the excavation face after the excavation of the ith layer. δ

_{max}

_{i}is the maximum deformation increment of the sidewall retaining structure, caused by the excavation of ith layer. The applications of engineering generally use the accumulated maximum deformation of the retaining structure as the control index. This paper selected the ratio of accumulated maximum deformation of the retaining structure to the excavation depth as the control parameter of the retaining structure’s deformation during design and calculation. The accumulated deformation satisfies the control values when excavating each soil layer, so the maximum deformation increment of the ith layer excavation can be represented as

_{max}is the accumulated maximum deformation of the retaining structure caused by the excavation of the foundation pit; H

_{e}is the excavation depth of the foundation pit; δ

_{j}(H

_{ei}, H

_{ej}) is the deformation increment of the sidewall retaining structure at the depth H

_{ei}, caused by the excavation of the jth layer in the foundation pit; and H

_{ej}is the depth of the excavation face after the excavation of the jth layer.

_{e}ratio is 4.00, and the PSR near the corner is 0.72. By substituting this into the fitting formula, the following equation can be obtained:

_{e}, the cumulative deformation distribution of the retaining structure is as follows:

_{acr}is the displacement required when the soil is in the active limit state, generally taking v

_{acr}= 0.001~0.003 H [32]. The unloading of the foundation pit sidewall can be considered as the difference between the static earth pressure at the initial state and the retaining structure’s lateral load after excavation. Therefore, the unloading of the sidewall can be expressed as

## 3. Additional Stress Distribution Caused by the Excavation of the Foundation Pit

#### 3.1. Numerical Model and the Area Division of the Unloading Impact

_{e}. The horizontal distance between the axis of the tunnel and the center of the foundation pit is a, the outer diameter of the shield tunnel is D, the buried depth of the tunnel’s axis is h, and the minimum clearance between the retaining structure and the tunnel is s (s = a − B/2 − D/2). The height of the retaining structure within the scope of deformation is H. Therefore, it can be ascertained that the coordinate of any point on the tunnel axis is (a, l, h), where l is the horizontal distance between the calculation point on the tunnel along the y-axis and the excavation center of the foundation pit.

#### 3.2. Calculation of Additional Stress Caused by Sidewall Unloading of the Foundation Pit

_{c}(L/2 − ζ, η)dζdη. Thus, if the equation about the unloading effect in the distribution area of sidewall ① is integrated, then the horizontal additional stress at any point (a, l, h) on the tunnel axis caused by the unloading of sidewall ① is taken as

_{c}(B/2 − ξ, η)dξdη. Thus, by integrating the equation about the unloading effect in the distribution area of the sidewall (②), the horizontal additional stress at any point (a, l, h) on the tunnel axis caused by the unloading of the sidewall (②) is taken as

## 4. Longitudinal Deformation of the Shield Tunnel by the Side of the Foundation Pit

#### 4.1. Collaborative Deformation Mode of the Shield Tunnel with Rotation and Dislocation

_{x}, the relative horizontal displacement induced by the rigid rotation of the segment rings was δ

_{x1}, the relative horizontal displacement generated by the dislocation of the segment rings was δ

_{x2}and the rotation angle between the segment rings was θ

_{x}. This satisfied the equation δ

_{x}= δ

_{x1}+ δ

_{x2}. Let us set δ

_{x1}= j

_{x}δ

_{x}, where j

_{x}is the proportionality coefficient for the rotation effects of the segment rings with the horizontal displacement of the tunnel axis (i.e., the ratio of the relative horizontal displacement generated by the rotation between adjacent segment rings to the total relative horizontal displacement). When j

_{x}= 0, there is no relative rotation between adjacent segment rings, which is the deformation mode of complete shear dislocation. This is consistent with the deformation model of dislocation proposed by Zhou et al. [22]. When j

_{x}= 1, the horizontal displacement is completely caused by the rotation of adjacent segments, which is the deformation mode of complete rotation. According to the research in previous studies [34], the value of the proportionality coefficient j

_{x}is 0.1~0.3, and it can be assumed that j

_{x}= 0.2 in this study.

_{t}(l) is the horizontal displacement of the soil around the tunnel along the tunnel axis.

_{t}is the width of the segment ring.

_{x}between the segment rings is low, it can be assumed that sin (θ

_{x}) = θ

_{x}and cos (θ

_{x}) = 1. Therefore, the calculation formula for the horizontal displacement caused by rotation between the segment rings is

_{x1}= j

_{x}δ

_{x}into Equation (14), it is easy to get

_{sl}and k

_{t}are the shear stiffness and tensile stiffness between the rings of the tunnel, respectively, and the value method can be referred to in [35]. In Equation (18), Vesic’s formula [36] is used to calculate k, which is the foundation reaction coefficient. $k=\frac{0.65{E}_{0}}{(1-{\mu}^{2})D}\sqrt[12]{\frac{{E}_{0}{D}^{4}}{{(EI)}_{\mathrm{eq}}}}$, where μ is the Poisson ratio of the soil. E

_{0}is the deformation modulus of the foundation soil. ${E}_{0}=\frac{(1+\mu )(1-2\mu )}{(1-\mu )}{E}_{\mathrm{s}}$, where E

_{s}is the compression modulus of soil. (EI)

_{eq}is the equivalent flexural stiffness of the tunnel. According to the research of Ye et al. [37], the equivalent stiffness of the shield tunnel can be calculated according to the following formula:

_{c}is the elastic modulus of the segment; n is the number of longitudinal bolts; D

_{t}is the width of the segment rings; K

_{b}is the average linear stiffness of the joint bolt; and A

_{s}is the cross-sectional area of the tunnel. When the horizontal diameter and vertical diameter of the tunnel are not considered to change, ${\lambda}_{1}=tD({A}_{1}+{A}_{2}-{A}_{3}-{A}_{4}-{A}_{5})$; ${\lambda}_{2}=tD({A}_{1}+{A}_{2}+{A}_{3}+{A}_{4}+{A}_{5})$; ${A}_{1}=\pi {D}^{2}/16$; ${A}_{2}=\pi {D}^{2}{\mathrm{sin}}^{2}\psi /8$; ${A}_{3}={D}^{2}\psi /8$; ${A}_{4}={D}^{2}\psi {\mathrm{sin}}^{2}\psi /4$; and ${A}_{5}=3{D}^{2}\mathrm{sin}\psi \mathrm{cos}\psi /8$. Among this, ψ is the location parameter of the segment ring’s central axis, which assumes ψ = 30° in this study, and the determination methods can be referenced in [37]. t is the thickness of the segment, and D is the outside diameter of the tunnel.

#### 4.2. The Total Potential Energy of the Shield Tunnel during Deformation

- (1)
- Work done by the horizontal additional load caused by the excavation of the foundation pit:$${W}_{\mathrm{L}}={\displaystyle \sum _{m=-N}^{N-1}{\displaystyle {\int}_{m{D}_{\mathrm{t}}}^{(m+1){D}_{\mathrm{t}}}u(l)}}{P}_{\mathrm{ax}}(l)\mathrm{d}l={\displaystyle {\int}_{-N{D}_{\mathrm{t}}}^{N{D}_{\mathrm{t}}}u(l)}{P}_{\mathrm{ax}}(l)\mathrm{d}l$$
_{ax}(l) is the horizontal additional load caused by foundation pit excavation along the axis of the adjacent shield tunnel, which can be obtained with Equation (11), and N is the number of shield tunnel segment rings within the calculation range. The value taken in this paper’s calculation should be related to the range affected by the excavation of the foundation pit by the side of the shield tunnel. Theoretically, the greater the value of N is, the higher the calculation accuracy will be. However, the corresponding calculation amount will also increase, and the calculation efficiency will be affected. In order to ensure the accuracy of the calculation, the length of the tunnel included in the calculation range should not be less than the scope of the tunnel affected by the foundation pit. According to previous research [25], many factors influence the length of the impact region, and the excavation size of the foundation pit parallel to tunnel (L) is the main factor, and the impact region is about 2~3 times the size of L. Therefore, in the initial calculation, the value of N should not be less than 1.5 L/D_{t}, where D_{t}is the width of the segment ring. According to the method proposed in previous studies [34], if it is found that the influence range is large, the value of N should be increased appropriately until the calculated value tends to be stable. - (2)
- Work done by overcoming the soil layer resistance:$${W}_{\mathrm{R}}=-{\displaystyle \sum _{m=-N}^{N-1}{\displaystyle {\int}_{m{D}_{\mathrm{t}}}^{(m+1){D}_{\mathrm{t}}}\frac{1}{2}u(l)}}kDu(l)\mathrm{d}l=-{\displaystyle {\int}_{-N{D}_{\mathrm{t}}}^{N{D}_{\mathrm{t}}}\frac{1}{2}}kD{\left[u(l)\right]}^{2}\mathrm{d}l$$
- (3)
- Work done by overcoming the shear forces between the rings:$${W}_{\mathrm{S}}=-{\displaystyle \sum _{m=-N}^{N-1}\frac{1}{2}{Q}_{\mathrm{x}}{\delta}_{\mathrm{x}2}}=-{\displaystyle \sum _{m=-N}^{N-1}\frac{1}{2}{k}_{\mathrm{sl}}(1-{j}_{\mathrm{x}}{)}^{2}{\delta}_{\mathrm{x}}^{2}}$$
- (4)
- Work done by overcoming the tension caused by rotation between the rings:$${W}_{\mathrm{T}}=-{\displaystyle \sum _{m=-N}^{N-1}\left({\displaystyle {\int}_{r=0}^{r=D}\frac{1}{2}\frac{{k}_{\mathrm{t}}}{D}{\theta}_{\mathrm{x}}^{2}{r}^{2}\mathrm{d}r}\right)}=-{\displaystyle \sum _{m=-N}^{N-1}\frac{{k}_{\mathrm{t}}{\theta}_{\mathrm{x}}{}^{2}{D}^{2}}{6}}$$

#### 4.3. Fourier Expansion of the Curve Function of the Shield Tunnel’s Horizontal Displacement

_{n}is the Fourier expanding coefficients for each term; ${T}_{n}(l)=(1\text{}\mathrm{cos}\frac{\pi l}{N{D}_{\mathrm{t}}}\text{}\mathrm{cos}\frac{2\pi l}{N{D}_{\mathrm{t}}}\text{}\cdots \text{}\mathrm{cos}\frac{n\pi l}{N{D}_{\mathrm{t}}})$; and $A={({a}_{0}\text{}{a}_{1}\text{}{a}_{2}\text{}\cdots \text{}{a}_{n})}^{T}$.

#### 4.4. Solve the Variational Control Equation

_{p}takes the extremum of each undetermined coefficient. In other words,

_{i}is the ith element in matrix A (i.e., the coefficient of the polynomial of the tunnel deformation curve function).

_{r}]A

^{T}is the interaction effect between tunnel rings:

_{s}]A

^{T}is the effect of the soil resistance:

_{ax}]

^{T}represents the effect of additional load on the tunnel lining:

^{T}can be obtained:

^{T}back into Equation (26) reveals the distribution function of the horizontal displacement along the tunnel axis beside the foundation pit:

_{x}between adjacent segments of the shield tunnel is

## 5. Engineering Case Analysis

#### 5.1. Case History 1

_{e}= 15.8 m, and the underground diaphragm wall was 37.2 m below the ground. The minimum clearance from the sideline of the foundation pit’s retaining structure to the tunnel was s = 9.5 m [38]. The outer diameter of the shield tunnel was D = 6.2 m, which adopted C50 concrete segments. The thickness of the segments was t = 0.35 m, and the ring width was D

_{t}= 1.2 m. Sixteen M30 longitudinal bolts were used to connect the adjacent segment rings. According to the calculation, k

_{sl}= 2.23 × 10

^{6}kN/m, k

_{t}= 9.39 × 10

^{5}kN/m and (EI)

_{eq}= 1.1 × 10

^{8}kN·m

^{2}. The tunnel was buried 14.3 m deep in the silty silt and sandy silt layer. In the excavation scope of the foundation pit, the main distribution of the soil was miscellaneous fill, silty clay, sandy silty soil and silty sand with silty soil. According to the actual engineering geological conditions, the value of the soil weight chosen weighted a mean of γ = 18.4 kN/m

^{3}, and the Poisson’s ratio of the soil was μ = 0.4. In the process of the foundation pit’s excavation, this case was divided into four soil layers and excavated layer by layer, with the thickness of each layer from top to bottom being 1.6 m, 4.9 m, 4.8 m and 4.5 m, respectively.

_{max}/H

_{e}= 0.2% (i.e., the accumulated maximum deformation after the excavation of each soil layer was 0.2% of the excavation depth under such condition). The accumulated maximum deformation of the sidewall retaining structure in this case was within 31.6 mm. At this time, the rotation effects proportionality coefficient of the shield tunnel was j

_{x}= 0.2 (i.e., the horizontal deformation caused by the relative rotation of the segment rings accounted for 20%, while the horizontal deformation caused by the dislocation of the segment rings accounted for 80%). It can be seen from the figure that the value of horizontal displacement was mainly negative, indicating the displacement of the shield tunnel was along the negative direction of x-axis (toward the side of the foundation pit). The maximum calculated value of the horizontal displacement was 12.20 mm, whereas the maximum measured value of the horizontal displacement was 11.90 mm. The horizontal displacement of the tunnel was normally distributed, and the horizontal displacement was large near the center of foundation pit, while the two sides decreased successively.

^{−5}degrees. Both of them appear near the two inflexion points of the displacement curve in Figure 11, with a 36 m horizontal distance to the center of the foundation excavation.

#### 5.2. Case History 2

_{t}= 1 m and the segment thickness was t = 0.35 m. Sixteen M30 longitudinal bolts were used to connect the segment rings. According to the calculations, k

_{sl}= 2.23 × 10

^{6}kN/m and k

_{t}= 9.39 × 10

^{5}kN/m. The tunnel axis was buried 10.1 m deep and was located in the sandy silt and muddy clay layer. The soil layers in the excavation scope of the foundation pit mainly included filling soil, silty clay and sandy silty soil. According to the engineering geological conditions, the value of the soil weight took on a weighted mean γ = 18.2 kN/m

^{3}, and the Poisson’s ratio of the soil was μ = 0.35.

_{x}= 0.2 (i.e., the horizontal deformation caused by the rotation of the segment rings accounted for 20%, while the horizontal deformation caused by the dislocation of the segment rings accounted for 80%). In this case’s calculations, the control parameter of the retaining structure deformation was v

_{max}/H

_{e}= 0.15% (i.e., the accumulated maximum deformation after excavating each layer in the foundation pit was 0.15% of the excavation depth under this working condition). The accumulated maximum deformation of the sidewall retaining structure in this case was within 15 mm. It can be seen from the figure that the maximum horizontal displacement shown by the measured data was 3.66 mm. In contrast, the maximum horizontal displacement of the side shield tunnel calculated by the method in this paper was 4.04 mm. The horizontal displacement of the tunnel presented a normal distribution, and the tunnel’s horizontal displacement at the center of the excavation was large, while the two sides decreased successively.

^{−5}degrees, respectively. Figure 15 shows the distribution curve of the calculated shear force between the adjacent shield tunnel segment rings. As shown in the figure, the shear force between the adjacent segment rings was the largest at a horizontal distance of 33 m from the excavation center, with a maximum value of 176.66 kN.

## 6. Analyzing the Influence Factors of the Adjacent Tunnel’s Longitudinal Deformation

#### 6.1. Deformation Control Parameters of the Foundation Pit Retaining Structure v_{max}/H_{e}

_{max}/H

_{e}) of the underground diaphragm wall in Suzhou is between 0.05% and 0.40%, with an average value of 0.20%. According to the statistics of Wang et al. [41], the variation range of the control parameter (v

_{max}/H

_{e}) of the underground diaphragm wall in Hangzhou is between 0.09% and 0.32%, and the average value is 0.26%. Xu et al. [42] analyzed the measured data of 93 foundation pits in Shanghai which use underground diaphragm walls as retaining structures and found that the v

_{max}/H

_{e}was between 0.1% and 1.0%, with an average value of 0.42%.

_{max}/H

_{e}) of the foundation pit’s retaining structure under the condition that the other parameters remained constant. Figure 16 shows the curve of the maximum calculated value of the horizontal displacement of the shield tunnel by the side of the foundation pit changing with the deformation control parameter (v

_{max}/H

_{e}) of the retaining structure. As shown in the figure, when v

_{max}/H

_{e}= 0.25%, the maximum horizontal displacement of the shield tunnel adjacent to the foundation pit reached 13.37 mm. When v

_{max}/H

_{e}increased to 0.30%, the deformation of the retaining structure in this numerical example was controlled within 47.4 mm. At this time, the horizontal displacement distribution of the shield tunnel already changed a little, with a maximum increment of horizontal displacement of about 0.5 mm. This was because the soil mass at the maximum deformation of the retaining structure was close to the limit state. The deformation of the retaining structure continued to increase, while the increase of the sidewall unloading was not obvious. Figure 17 shows the longitudinal distribution curve of the calculated horizontal displacement value of the shield tunnel by the side of the foundation pit when the deformation control parameter v

_{max}/H

_{e}was taken as 0.05%, 0.10%, 0.15%, 0.20%, 0.25% and 0.30%, respectively. As shown in the figure, when v

_{max}/H

_{e}= 0.05%, the deformation of the retaining structure in this calculation was controlled within 7.9 mm. Meanwhile, the horizontal displacement of the shield tunnel by the side of the foundation pit was small, with a maximum horizontal displacement of 3.74 mm. With the increase of the deformation control parameter (v

_{max}/H

_{e}) of the retaining structure, the shield tunnel’s horizontal displacement value and the influence range of longitudinal deformation also increased.

#### 6.2. The Clearance s between the Foundation Pit and the Tunnel

_{e}. Figure 19 shows the longitudinal distribution curve of the calculated horizontal displacement value of the shield tunnel by the side of the foundation pit when clearance s equaled 1/3 H

_{e}, 2/3 H

_{e}, H

_{e}, 4/3 H

_{e}, 5/3 H

_{e}, respectively. As shown in Figure 18 and Figure 19, with the increase of the clearance s, the adjacent shield tunnel’s horizontal displacement decreased. The main influence range was within 100 m from the center of the excavation.

#### 6.3. The Buried Depth h of the Tunnel

_{e}, 2/3 H

_{e}, H

_{e}, 4/3 H

_{e}, and 5/3 H

_{e}, respectively. As shown in Figure 20 and Figure 21, the excavation of the foundation pit had a great impact on the adjacent shield tunnel with a shallow burial depth. In this calculation, the horizontal displacement of the adjacent tunnel was the largest when the depth of the tunnel axis was about two-thirds of the excavation depth. At this time, the excavation face was about 2 m below the bottom of the tunnel. Subsequently, with the increase of the adjacent shield tunnel’s buried depth, the horizontal displacement of the tunnel caused by the excavation of the foundation pit obviously decreased. Meanwhile, the influence range of the longitudinal deformation of the shield tunnel also decreased.

## 7. Conclusions

- (1)
- The calculated results of the method in this paper are in good agreement with the measured values, and the horizontal displacement of the shield tunnel axis by the side of the foundation pit presents a normal distribution. The longitudinal deformation of the adjacent tunnel is mainly caused by the unloading effect of the foundation pit sidewall, which is parallel and close to the tunnel. When the excavation depth is in the soil layer near the buried depth of the adjacent tunnel, the longitudinal deformation of the shield tunnel by the side of the foundation pit will increase sharply. Thus, more attention should be paid to the control of the tunnel’s deformation in this working condition.
- (2)
- The horizontal displacement near the excavation center of the foundation pit is relatively large, but the values of the dislocation and rotation angle between the segment rings are relatively small. The maximum rotation angle between the segment rings, the maximum dislocation and the maximum shear force between the segment rings all appear in the vicinity of the two inflexion points of the horizontal displacement curve of the tunnel.
- (3)
- With the increase of the retaining structure’s deformation, the longitudinal deformation of the adjacent shield tunnel and its influence range also increase. The longitudinal deformation of the adjacent shield tunnel decreases with the increase of the clearance between the foundation pit and the tunnel. The foundation pit’s excavation has a great influence on the adjacent shield tunnel at a shallow burial depth. Moreover, the impacts on the tunnel caused by the excavation decrease with the increase of the burial depth of the adjacent tunnel.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic diagram of the foundation pit unloading model (K

_{0}is the coefficient of static earth pressure, γ is the unit weight of soil for calculation, z is the depth of the calculation point and β is the stress loss rate).

**Figure 4.**The schematic diagram of the position relationship and influence between the foundation pit and the adjacent shield tunnel.

**Figure 5.**The schematic diagram about the partition of the region affected by the unloading of the sidewall.

**Figure 7.**Calculation model for the collaborative deformation of rotation and dislocation between the shield tunnel segment rings.

**Figure 8.**Comparison of the calculated values and measured values of the horizontal displacement of the shield tunnel adjacent to the foundation pit.

**Figure 9.**Horizontal displacement curve of the tunnel, caused by the unloading of each sidewall of the foundation pit.

**Figure 10.**Horizontal displacement curve of the tunnel during the process of excavation, layer by layer.

**Figure 11.**Longitudinal distribution of dislocation and rotation between the segment rings of the shield tunnel.

**Figure 12.**Longitudinal distribution of shear forces between the segment rings in the shield tunnel.

**Figure 13.**Comparison of the calculated value and measured value of the horizontal displacement of the tunnel beside the foundation pit.

**Figure 16.**The variation curve of the maximum calculated value of the shield tunnel’s horizontal displacement with different values of v

_{max}/H

_{e}.

**Figure 17.**The longitudinal distribution curve of the adjacent tunnel’s horizontal displacement when v

_{max}/H

_{e}takes different values.

**Figure 18.**The calculated curve of the maximum horizontal displacement of the shield tunnel by the side of the foundation pit changing with s/H

_{e}.

**Figure 19.**The longitudinal distribution curve of the horizontal displacement of the shield tunnel by the side of the foundation pit under different s/H

_{e}conditions.

**Figure 20.**The changing curve of the maximum horizontal displacement (calculated) of the adjacent shield tunnel with different h/H

_{e}values.

**Figure 21.**The longitudinal distribution curve of the horizontal displacement of the shield tunnel by the side of the foundation pit under different h/H

_{e}conditions.

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## Share and Cite

**MDPI and ACS Style**

Zhang, X.; Wei, G.; Jiang, C.
The Study for Longitudinal Deformation of Adjacent Shield Tunnel Due to Foundation Pit Excavation with Consideration of the Retaining Structure Deformation. *Symmetry* **2020**, *12*, 2103.
https://doi.org/10.3390/sym12122103

**AMA Style**

Zhang X, Wei G, Jiang C.
The Study for Longitudinal Deformation of Adjacent Shield Tunnel Due to Foundation Pit Excavation with Consideration of the Retaining Structure Deformation. *Symmetry*. 2020; 12(12):2103.
https://doi.org/10.3390/sym12122103

**Chicago/Turabian Style**

Zhang, Xinhai, Gang Wei, and Chengwu Jiang.
2020. "The Study for Longitudinal Deformation of Adjacent Shield Tunnel Due to Foundation Pit Excavation with Consideration of the Retaining Structure Deformation" *Symmetry* 12, no. 12: 2103.
https://doi.org/10.3390/sym12122103