#### Abstract

To evaluate the tunnel deformation law and soil stress distribution between foundation pit excavation and tunnel in different locations, numerical analyses using the hypoplastic model are conducted based on reported centrifuge model tests. Two cases are designed to investigate the effects of the foundation pit excavation on the deformation of existing tunnels. In case C, an existing tunnel is directly located underneath the foundation pit; in case S, the tunnel is located at one side of the foundation pit. Three-dimensional tunnel deformation mechanisms along the tunnel axis are observed through the variation of stresses change in the soil circumambient tunnel. It is found that compared with case C, there are minor Earth pressure changes in case S. The different Earth pressure changes around the tunnel lead to different modes of tunnel deformation. The maximum additional tunnel bending strain appears at the crown and invert, while the minimum value appears at the spring lines in case C. In case S, the maximum values appear in the right shoulder and left knee, while the minimum values appear on the left shoulder and right knee. The additional tunnel bending strain and stress reduction at different tunnel cross sections in case S is much smaller than those in case C. In case C, after the excavation, lateral Earth pressure coefficient changes Δ*K*_{xz} and Δ*K*_{yz} rise to 70% and 150%, respectively, of their initial value at shoulders, while little changes can be found at spring lines. However, in case S, the maximum absolute value of both Δ*K*_{xz}/(*K*_{xz})_{0} and (Δ*K*_{yz}/(*K*_{yz})_{0}) is no more than 10%.

#### 1. Introduction

Rapid increment in urban development occasionally needs the excavation of building near or aligned above the existing tunnels. The excavations, sometimes unsymmetrical and highly distortive, impose changes in stresses and deformation on the tunnel lining, not only in the cross-sectional direction but also in the longitudinal direction.

Many excavations have been carried out next to running tunnels in various cities. To examine the response of the tunnels subjected to a nearby basement excavation, numerous studies have been performed using field tests [1–3], centrifuge model tests [4–6], and numerical analytical methods [7–27]. The main research objects of these studies are as follows: geometry of the pit foundation and tunnel [13, 15, 20]; the relative position of tunnel and excavation [5, 9, 11]; reinforcement method [10, 23]; different soil constitutive models [14]; and soil properties such as soil density [6].

Ng et al. [4] carried out centrifuge model tests to examine the effect of a foundation pit digging on the distortion of existing tunnels in the sand. Based on this centrifuge model test, several numerical analyses were carried out using the hypoplastic model for sand. What was found is that the hypoplastic model can effectively capture the soil behavior more compared to the previously reported centrifuge model test [6, 14, 15, 20].

Although the effect of excavation on the existing tunnels has been mainly investigated by field and model tests, numerical analysis, and analytical analyses, the three-dimensional behavior of the existing tunnels in different locations caused by foundation pit excavation is not fully understood, which is strongly detected in the tunnel deformation and soil stress distribution during the interaction between foundation pit excavation and the existing tunnel in different locations.

In this study, three-dimensional numerical analyses using an advanced constitutive model named the hypoplastic model were carried out to evaluate tunnel deformation and the stress-strain distribution during the excavation-soil-tunnel interaction process. Based on two centrifuge model tests carried out by Ng et al. [4], two cases were designed to investigate the effect of a foundation pit excavation on the deformation of existing tunnels. In case C, an existing tunnel is located directly underneath the foundation pit, while in case S, the tunnel is located at one side of the foundation pit. The purposes of these analyses are (i) to improve the fundamental understanding of tunnel response in bending strain and stress distribution due to foundation pit excavation and (ii) to reveal the different tunnel deformation law caused by foundation pit excavation for case C and case S.

#### 2. Three-Dimensional Numerical Analysis

##### 2.1. Finite Element Mesh and Boundary Conditions

Numerical simulations of centrifuge tests were conducted to study the tunnel deformation and stress distribution around the tunnel due to the foundation pit excavation. ABAQUS (ABAQUS Inc., 2017) [28] known as the finite element was adopted in this study.

Figure 1 demonstrates the 3D finite element mesh used in this analysis based on two centrifuge model tests carried out by Ng et al. [4]. The tunnel in test C was located directly beneath the basement, while the tunnel in test S was located at one side of the basement with a clear distance of 25 mm (1.5 m in prototype). A square excavation (on plan) was carried out with a side length of 300 mm, corresponding to 18 m in the prototype. In subsequent discussions, all parameters and results were presented in prototype scale except for special clarification. The mesh was 72 m long, 45 m high, and 59.4 m wide. It constituted 27,400 elements and 31,171 nodes. The sand was simulated by eight-node brick elements, and both tunnel and diaphragm were simulated by four nodes, respectively. The stress components exerted on each three-dimensional element were also demonstrated in Figure 1(c).

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Pins were designed to the base of the mesh, whereas rollers were designed to all the vertical sides of the mesh. Thus, the normal movements in all directions (*x*, *y,* and *z* direction) at the base of the mesh and the normal movement to all vertical sides (*x* or *y* direction) of the mesh were restrained. Although the influence of the stress on the boundary was not taken into consideration in the setting of boundary conditions, the parameter analysis shows that it has little influence on the calculation results and can improve the calculation efficiency.

##### 2.2. Parameter Model and Constitutive Model

A user-defined hypoplastic soil model for granular material proposed by Von Wolffersdorff [29] was adopted in this study. Mašín [30] developed a hypoplastic model to predict clay behavior. The model can reflect the very fundamental feature of sand behavior, which is important for correct modeling of sand-structure interaction such as the influence of porosity and mean stress on peak strength and dilatancy, nonlinear of soil behavior, the dependency of stiffness on distortive history, and stress level.

From the overview of the previous investigations, 8 material parameters (*φ*_{c}, *h*_{s}, *n*, *e*_{d0}, *e*_{c0}, *e*_{i0}, *α*, and *β*) and five sand parameters (*m*_{T}, *m*_{R}, *R*, *β*_{r}, and *χ*) are needed to capture stress path dependency of sand stiffness and effect of strain [31].

With Toyoura sand, parameters for the reference model can be obtained from Herle and Gudehus [32], while the stiffness degradation curve of Toyoura sand can calibrate five parameters for intergranular strain [33]. The lateral coordinate of Earth pressure at the initial state (*K*_{0}) is considered to be 0.5 based on the empirical formula *K*_{0} = 1-sin*φ*, where *φ* is soil internal friction angle. Table 1 summarizes the related sand parameters.

The diaphragm and tunnel lining wall were redesigned as materials with linear elasticity. Seventy GPa and 0.2 were taken to as Young’s modulus and Poisson’s ratio, respectively, for the tunnel lining. The unit weight of the tunnel and the wall was 27 kN/m^{3}. Seventy GPa was assumed to be Young’s modulus of the diaphragm wall. The model parameters for the tunnel and diaphragm wall are summarized in Table 2.

##### 2.3. Numerical Modeling Process

The centrifuge test process is as same as the numerical modeling. The difference between the two approaches is that in numerical modeling, excavation was simulated by directly removing soil while excavation in centrifuge model tests was activated by draining heavy fluid (ZnCl_{2}). The excavation step was divided into three parts, and the detailed activation steps are as follows:(1)The initial tension conditions are established using *K*_{0} = 0.5(2)The gravitational acceleration of the whole model is incrementally increased to 60 g(3)The basement in 3 m rounds is excavated until the deepest depth of the excavation is 9 m

#### 3. Verification of Numerical Analyses

Figure 2 demonstrates a reasonable match between measured and computed additional bending strain at the tunnel transverse section under the center of excavation (C-C section). The measured results are reported by Ng et al. [4]. The positive value denotes that bending strain in tunnel lining is tensile, and the negative value means compressive. It shows that the additional bending strain measured is in good agreement with those computed after the first step of excavation (3 m). With the progressive excavation, the differences between them are larger, especially at the invert of the tunnel, which probably results from the different simulating methods used in centrifuge tests and numerical analysis discussed before. However, what should be noted is that the results obtained from the two approaches are consistent, at least in trend. The comparisons between the measured and computed results show that the finite element analyses adopted in this study are reasonable.

#### 4. Results and Analyses

##### 4.1. Deformation of Tunnel Lining

Figure 3 shows tunnel radius change in section C-C for case C and case S. For case C, the maximum increase in the tunnel radius is about 0.128% *R* in a vertical direction, and the maximum reduction in the tunnel radius is approximately 0.125% *R* in a horizontal direction. However, in case S, the maximum increase in the tunnel radius is about 0.055% *R* in a subvertical direction, and the maximum reduction in the tunnel radius is approximately 0.052% *R* in a subhorizontal direction. By comparing the two cases, it can be found that the excavation of the foundation pit causes the vertical elongation and horizontal compression of the tunnel lining for case C, while the tunnel lining extends toward the foundation pit for case S. Foundation pit excavation causes significant deformation of tunnel lining underneath the foundation pit but has relatively little impact on tunnel lining at the side of the foundation pit.

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##### 4.2. Bending Strain in Tunnel Lining

Figure 4 demonstrates the distribution of extra tunnel bending strain at tunnel transverse section and along the tunnel axis for two cases.

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As seen in Figure 4(a) that the distribution of the tunnel bending strain is symmetrical with the decrease in the horizontal direction and the increase in the vertical direction for case C. The absolute value of the decrease is slightly larger than that of the increase. It depends on the excavation dimension and the coefficient of lateral Earth pressure. However, for case S, the distribution of the tunnel bending strain inclines toward the excavation side to some degree, which is determined by the relative location between the tunnel and the excavation.

Figure 4(b) demonstrates the distribution of additional bending strain along the tunnel axis for two cases. In case C, the maximum magnitude of the additional bending strain (70 *με*) occurs in the excavation center of the basement. The value decreases with the increase in distance from the center section and then reaches a minimum value (about −35 *με*) at the edge of the basement. Outside of the excavation, the absolute values decrease along the tunnel axis and little changes are found at the section *x*/(*L*/2) = 3. A similar distribution trend of the additional tunnel bending strain along the tunnel axis is found in case S. What should be noted is that additional tunnel bending strain in case C is much larger than that in case S.

In summary, at the tunnel transverse section (C-C section), the maximum additional tunnel bending strain occurs at the invert and crown, while the minimum value happens at the spring lines for case C. However, for case S, the maximum values appear at the right shoulder and left knee, while the minimum values appear at the left shoulder and right knee. In the longitudinal direction, the magnitude of the additional tunnel bending strain in case C is much larger than that in case S. The distribution trend is consistent. In conclusion, the maximum value of additional tunnel bending strain occurs at the center of excavation, then decreases with increased distance from the center section, and reaches zero at the edge of the basement.

##### 4.3. Bending Moment in Tunnel Lining

Figure 5 shows the additional bending moments in the tunnel lining due to basement excavation. The additional bending moments of the lining are shown for two cases (i.e., case C and case S) and two sections (i.e., C-C section at the center of the excavation and D-D section at the edge of the excavation). What should be noted is that the actual properties of the shotcrete, such as stiffness, are time dependent and are not considered in this study. Hence, only the computed values of bending moments are taken for mechanism analysis.

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It can be seen from Figures 5(a) and 5(b) that the distributions of the bending moments are similar in both sections in case C. The difference is that the magnitude of additional bending moment in section C-C is greater than that in section D-D, which suggests that the tunnel under the C-C section carries a larger portion of the load than the D-D section. The maximum and minimum additional bending moments occur at the invert and the spring lines of the tunnel, respectively. The symmetric nature of the distribution can be clearly seen in Figures 5(a) and 5(b).

Figures 5(c) and 5(d) show the distributions of the bending moments in sections C-C and D-D in case S. Not only the distribution trend but also the magnitude of the bending moments in these two figures is nearly the same, although the magnitude of additional bending moment in section C-C is slightly greater than that in section D-D. The reason is that for case S, the shielding effect by the diaphragm wall is significant. It also should be noted that the maximum additional bending moments occur at the right shoulder or left knee, while the minimum additional bending moments occur at the left shoulder or right knee of the tunnel, respectively.

##### 4.4. Ground Earth Pressures around Tunnel

Figure 6 shows the comparison of normalized Earth pressure change (Δ*P*/*P*_{0} (%), where *P*_{0} is Earth pressure around tunnel before excavation and Δ*P* denotes Earth pressure change) around the tunnel at C-C sections for cases C and S. Negative sign means the decrease in Earth pressure changes around the tunnel. Figure 6(a) shows that Earth pressures in the soil around the tunnel decrease by 50% at the crown and 20% at the spring lines and shows little change at the invert. Figure 6(b) shows that the maximum changes in Earth pressure decrease by 7% nearby the crown, while little changes can be found at the left knee in the soil element. As a whole, the change in the Earth pressure in case S is much less than that in case C. It can be deduced that when the stiffness of the retaining wall is large enough, the influence of foundation pit excavation on tunnel deformation underneath the pit is much greater than that of the tunnel outside the pit. The different Earth pressure changes around the tunnel lead to different modes of tunnel deformation, as shown in Figure 3.

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##### 4.5. Changes in Normalized Vertical Stresses

Figure 7 shows the changes in normalized total vertical (*z* direction) stresses with respect to their initial values (Δ*σ*_{zz}/(*σ*_{zz})_{0}) for three soil elements (C, SP, and I) at section C-C in case C and case S.

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It can be seen from Figure 7(a) that within the excavation zone, stress reduction for each soil element is almost uniform in case C. The maximum stress reduction and increase in *σ*_{zz} are approximately 75% and 40% of the initial stress value, respectively. No further stress change takes place at a distance of approximately 1.5 times the excavation length away from the center section of excavation. It can be seen from Figure 7(b) that unnoticeable stress changes at the three selected elements take place along the tunnel axis compared with case C. Within the excavation zone, the absolute stress reduction gradually decreases at all three elements and reaches zero from the C-C section to some distance (about 3 L/2) away. Foundation pit excavation only has a little influence (approximately 10% in vertical stress) on the existing tunnel in case S.

##### 4.6. Changes in Shear Stresses

Figure 8 shows shear stresses (Δ*τ*_{xy}, Δ*τ*_{xz}, and Δ*τ*_{yz}) induced at the monitored soil element along tunnel axis as the excavation advance in case C.

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As seen from Figure 8(b), the shear stress distribution along the tunnel axis is antisymmetric. The maximum absolute shear stresses Δ*τ*_{xz} take place at the edge of the basement of these three monitored elements. The maximum Δ*τ*_{xz} is the smallest at the invert and the largest at the crown. The Δ*τ*_{xz} induced at all elements first increases from zero to maximum at the edge of the excavation and then decreases to zero along the tunnel axis at the distance of about 3/2*L* away from the center of the basement.

Figures 8(a) and 8(c) show no shear stresses (Δ*τ*_{xy} and Δ*τ*_{yz}) induced at the crown and invert of the sand element. This is because both sand elements C and I were located in a symmetric axis. Besides, the magnitude of Δ*τ*_{xy} is very small compared with the large relief of lateral Earth pressure and can be ignored. It should be noted that the increase in *τ*_{yz} is almost uniform within the excavation zone but with a sharp reduction at the edge of the excavation. Then, the absolute magnitude of Δ*τ*_{yz} decreased to small at an interval of 3 *L*/2 parted from the center section of the tunnel.

By comparing Figures 8(a)–8(c), one can see that the pattern of stress distributions of Δ*τ*_{xy} and Δ*τ*_{xz} is antisymmetric, whereas the distribution of Δ*τ*_{yz} is symmetric about excavation center section. Special attention should be paid to the edge of excavation where the absolute magnitude of stress change is largest.

As the case for the normal stresses (see Figure 7), the shear stress distribution can be identified from the significance of the influence zone shown in Figure 8. It can be identified that significant shear stress is induced at the edge of the basement and the influence of shearing vanishes at 3/2*L* away from the center of the basement.

Figure 9 shows shear stresses (Δ*τ*_{xy}, Δ*τ*_{xz}, and Δ*τ*_{yz}) induced at the monitored soil element along tunnel axis as the excavation advance in case S.

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The Δ*τ*_{xy} is induced only at the spring line with no changes in other soil elements monitored as shown in Figure 9(a). The absolute magnitude of Δ*τ*_{xy} increases from zero to maximum at the edge of the excavation and then decreases to minor along tunnel axis at a distance of about 3 *L*/2 away from the center of the basement, which is consistent with the above findings.

Figure 9(b) shows the antisymmetric distribution of Δ*τ*_{xz} along the tunnel axis after excavation. It can be found that the absolute magnitude of Δ*τ*_{xz} mobilizes to their respective maximum values at the edge of basement for the spring line, at 3 *L*/4 from the center of the basement for the crown, and at the boundary of the model for the invert, which indicates that stresses transfer and redistribute in the sand, around the stress relief as a result of excavation.

Figure 9(c) shows the symmetric distribution of Δ*τ*_{yz} along the tunnel axis after excavation. The magnitude of Δ*τ*_{yz} in three soil elements occurs at the spring line with the largest at the crown and with the smallest at section A-A, respectively. The magnitude of Δ*τ*_{yz} decreases from maximum to minimum at a distance of about 3 *L*/2 from section A-A.

In summary, distribution of the shear stresses induced by excavation is antisymmetric for Δ*τ*_{xy} and Δ*τ*_{xz} but symmetric for Δ*τ*_{yz}. Shear stress in soil element at the spring line nearby the excavation is more significant than other elements. Therefore, stress changes in the spring line should be paid attention to in the side excavation case. Considering comprehensively, the influence zone can be detected at 3 *L*/2 parted from the center of excavation.

##### 4.7. Variations in *K*_{xz} and *K*_{yz}

Figure 10 shows the changes in normalized lateral Earth pressure coefficients *K*_{xz} and *K*_{yz} with respect to their starting point as the excavation advances.

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It can be found from Figure 10(a) that when the basement was excavated to 3 m, no significant change in *K*_{xz} on the *x*-*z* plane occurs. As the excavation reached 9 m depth, *K*_{xz} increases by 70% at shoulders and 35% at knees. However, little changes have been found at spring lines and the invert. At the end of the excavation, change in the lateral Earth pressure coefficients in *x*-axis direction Δ*K*_{xz} rises at all elements of the initial value of (*K*_{xz})_{0} due to substantial stress relief, which indicates that all these soil elements are in extension mode in the *x*-*z* plane. Attention must be paid to the shoulders of the sand elements because of the maximum increment of *K*_{xz} in the *x*-*z* plane.

Figure 10(b) shows the changes in the normalized coefficient of Earth pressure with respect to their initial values. A similar distribution form of Δ*K*_{yz}/(*K*_{yz})_{0} can be found when compared with that of Δ*K*_{xz}/(*K*_{xz})_{0}. The values of Δ*K*_{yz}/(*K*_{yz})_{0} significantly increase at the shoulders of soil element with 150% but 70% both at crown and knees, respectively. It means that these soil elements are in extension mode. It should be noted that little changes in the values of Δ*K*_{yz}/(*K*_{yz})_{0} could be found at the invert but a minor decrease at the spring lines, which indicates that the sand elements are in compression form.

Figure 11 demonstrates the changes in *K*_{xz} and *K*_{yz} for various sand elements at the excavation center sector around the tunnel for case S.

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It can be seen from Figure 11(a) that the maximum absolute value of Δ*K*_{xz}/(*K*_{xz})_{0} is no more than 5%. So, it can be concluded that no significant changes in Δ*K*_{xz}/(*K*_{xz})_{0} can be found in the soil around the tunnel. That is, the effect of the side excavation on the existing tunnel is little compared with the center excavation case. However, it may be interesting to investigate the distribution of the normalized Earth pressure coefficient when the tunnel is located at the side of excavation especially in the case of a flexible diaphragm wall. At the left spring line, little changes can be found in the lateral Earth pressure ratio (Δ*K*_{xz}/(*K*_{xz})_{0}). The maximum absolute value occurs both at the left shoulder and the right knee. Irregular changes can be found around the right spring line.

Figure 11(b) shows the distribution of lateral Earth pressure coefficient ratio (Δ*K*_{yz}/(*K*_{yz})_{0}) during excavation. The more the excavation of the basement, the more the changes in the ratio at the shoulders and knees. The decrease in Δ*K*_{yz} due to excavation at shoulders and knees indicates that the soil elements are all in compression mode. Besides, little changes can be found at the crown, invert, and spring lines. Because of the larger reduction in the ratio in the right knee and left shoulder and that in the left knee and right shoulder, pattern of the deformation is obliquely elliptic as shown in Figure 3.

The complex nature of the coefficients (Δ*K*_{xz}/(*K*_{xz})_{0}) and (Δ*K*_{yz}/(*K*_{yz})_{0}) can be obtained by three-dimensional finite element analyses. With the advance of excavation, the stress redistribution effect can be easily captured by three-dimensional finite element analyses rather than a two-dimensional plane-strain analysis.

#### 5. Conclusions

The findings that are useful to engineering practice are highlighted as follows:(a)For the tunnel directly located below the foundation pit (case C), the distribution of the tunnel lining deformation, additional bending strain, and bending moment are all symmetrical between the crown and invert. For the tunnel located outside the retaining wall (case S), the distribution of the tunnel lining deformation, additional bending strain, and bending moment are all symmetrical between the right shoulder and left knee.(b)Compared with the changes in Earth pressure in case C, the changes are more minor in case S. It is deduced that when the stiffness of the retaining wall is large enough, the foundation pit excavation has a greater impact on tunnel deformation underneath the pit than that of the tunnel outside the pit. The different Earth pressure changes around the tunnel lead to varying modes of tunnel deformation.(c)In case C, the reduction in normal stress (maximum 65%) is almost the same within the excavation zone according to all monitored elements. While in case S, foundation pit excavation only has a minor influence (approximately 10% in vertical stress) on the existing tunnel. Shear stress in soil elements at the spring line nearby the excavation is more significant than other elements.(d)Changes in the coefficient of lateral Earth pressure at rest (Δ*K*_{xz}/(*K*_{xz})_{0}) and (Δ*K*_{yz}/(*K*_{yz})_{0}) also illustrate three-dimensional stress transfer mechanisms in the soil around the tunnel. In case C, after the excavation, Δ*K*_{xz} and Δ*K*_{yz} rise to 70% and 150%, respectively, of their initial value at shoulders, while minor changes have been found at spring lines. However, in case S, the maximum absolute value of both Δ*K*_{xz}/(*K*_{xz})_{0} and (Δ*K*_{yz}/(*K*_{yz})_{0}) is no more than 10%, which suggests that little changes in *K*_{xz} and *K*_{xz} have been found.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest.

#### Acknowledgments

This research was supported by the National Natural Science Foundation of China (project nos. 51904112 and 51904113), the College Student Innovation and Entrepreneurship Training Program (202111049037Y), the Scientific and Technological Guidance Project of Jiangsu Construction System (project nos. 2018ZD268 and 2017ZD246), and the Jiangsu Provincial Project of Industry and School and Research Institution (project no. BY2020254).