Three-Dimensional Ground Settlement Induced by Metro Tunnel Excavation Considering the Influence of Group Piles

Considering the influence of group piles, a prediction model for three-dimensional ground surface settlement induced by circular metro tunnels excavation in incompressible rock masses is proposed based on the stochastic medium theory and the shear displacement method. The surface settlement caused by the metro tunnel opening is divided into two parts. One part is soil mass settlement caused by the metro tunnel opening and calculated by the stochastic medium theory. The other part is the settlement induced by the friction force between the group piles and the soil mass around the metro tunnel cross section and calculated by the shear displacement method. The three-dimensional prediction of the ground surface settlement is obtained by the linear superposition of the two parts. The validation of the proposed prediction approach is proved by comparing with the measured data and the numerical model of the double tunnels under the Puyuan overpass where metro tunnels undercrossed group piles. The effects of buried depth, radial convergences, center distance of double tunnels, position and size of piles, and group piles are analyzed and discussed. The improved prediction approach can be applied to calculate the three-dimensional ground settlement, especially for the metro tunnels crossing through group piles.


Introduction
The ground surface settlement caused by metro tunnel excavation is an important factor to evaluate the impact of the subway tunnel construction on adjacent structures.It is necessary to control the subsidence deformation of soil mass strictly and reduce the damage induced by ground movement in the process of tunnel excavation, especially the metro tunnel crossing group piles.Some studies have been conducted to predict the ground displacement induced by excavation and solved many practical engineering problems [1][2][3][4][5][6].For example, Peck [7] developed a chart method (Gaussian distribution curve) to predict the surface settlement caused by tunnel opening based on the measured settlement data from lots of metro tunnel constructions.Eisenstein et al. [8] and New and O'Reilly [9] pointed out the ineffectiveness of Peck's method [7] in solving practical problems due to its deficiencies in theory.Sagaseta [10] proposed a virtual image technique to calculate the soil mass deformation in an isotropic and homogeneous incompressible soil mass due to the near-surface ground loss.Verruijt and Booker [11] presented an approximate analytical solution of the surface settlement for tunneling in an elastic half space.In addition, the equivalent ground loss parameters are redefined and the modified analytical formulas of vertical displacement are proposed to predict the tunneling-induced undrained ground movements in a soft ground by Loganathan and Poulos [12].However, these approaches make it difficult to predict ground surface settlement accurately because of a number of complex parameters.
The stochastic medium (SM) theory which is used for mining and studying the movement law of rock was proposed by Litwiniszyn [13,14].The SM theory was developed to study the ground surface movement due to underground excavation by Baochen and Dezhang [15] and Liu [16].Tunnelinginduced ground surface movements were modeled and the back analyses of 18 ground settlement profiles were performed using the SM theory by Yang et al. [17].Based on the SM theory, the simplified procedure to predict the ground surface settlement and the impact of different tunnel depth and angle on settlement were obtained by Yang and Wang [18].However, the influence of group piles foundation on surface settlement in subway excavation is ignored in those results.Moreover, those reported publications for studying the ground settlement are limited to two-dimensional (2D) calculation.
According to the shear displacement method [19] and the SM theory (Baochen and Dezhang [15] and Liu [16]), threedimensional (3D) models for predicting ground surface settlement induced by single and double metro tunnels excavations considering the influence of group piles are proposed, respectively.The interaction coefficient of group piles is introduced to calculate the settlement when tunnels cross through group piles.Moreover, the engineering example of metro tunnel which was located at Changsha in China is presented.

Methodology
2.1.Approach.The influence of group piles near the metro tunnels on the ground surface settlement cannot be neglected during the tunnels excavation.The ground surface settlement is predicted by linear superposition of settlements caused by metro tunnels excavation and the group piles.The approach followed was as follows: (1) Based on stochastic medium theory, the ground settlement caused by metro tunnel excavation is calculated.
(2) The ground settlement induced by the friction force between the group piles and soil mass is obtained by using the shear displacement method.
(3) A linear superposition of steps (1) and ( 2) is adopted, and the ground surface settlement which is caused by the metro tunnel excavation considering the effect of group piles is obtained.

Assumptions.
To predict the ground surface settlement induced by tunnel excavation considering the influence of group piles, the following assumptions are adopted: (1) The soil mass is incompressible, and it is ideally adapted to the stochastic medium theory.
(2) The piles bodies remain motionless when the tunnels are excavated (i.e., piles will not sink with tunnel excavation).
(3) There is no relative displacement between pile and adjacent soil mass.

Calculation Method of the Stochastic Medium Theory.
A three-dimensional Cartesian coordinate system is established in the space with -axis for the vertical direction.According to the stochastic medium theory, the vertical movement of soil mass is represented by the function (, , ) as follows [20]: where  11 ,  12 ,  22 ,  1 ,  2 , and  are parameters that depend on the structural characteristics of soil mass medium.
Based on the stochastic medium theory, the process of tunnel excavation is divided into numerous infinitesimal excavation elements, and the total surface displacements are obtained by the superposing displacements induced by elemental excavations.Dual systems are adopted in this study, and one is for the global coordinate (, , ) and the other is for the local coordinate (, , ).The calculation model of the ground settlement induced by the elemental excavation with the buried depth of  is shown in Figure 1.
According to the boundary conditions of elemental excavation, the elemental displacement function (  ) can be solved by the following equations: where  is a differential operator and  =  11 ( 2 / 2  ) +  12 ( 2 /    ) + ⋅ ⋅ ⋅ .(  ,   ) is a pulse function.
According to the assumption that the soil mass is incompressible (i.e., the volume of soil mass remains constant) during metro tunnels excavation, the horizontal displacements (  and   ) and the vertical displacement (  ) are satisfied with Combining ( 1), (2), and (3), soil displacements caused by the metro tunnels excavation can be obtained by where  1 (  ) and  2 (  ) are the influence radii of the ground surface settlement caused by elemental excavation that are related to  11 and  12 .(  ) is the offset distance of the ground surface settlement.In this paper, vertical settlement caused by metro tunnel excavation incorporating the influence of group piles is studied.According to (4), the calculation formula of the ground surface settlement at a point (, , 0) induced by elemental excavation can be simplified as follows: where () is the influence radius of the ground surface settlement caused by elemental excavation.

Ground Surface Settlement Induced by Single Tunnel
Excavation.A circular metro tunnel with radius  is excavated, where the distance from the center of excavation section to surface is .The three-dimensional calculation model of surface settlement caused by single tunnel excavation is shown in Figure 2. The center axial line of the tunnel is parallel to -axis, and any cross section that is perpendicular to the center line can be regarded as the  plane.The calculation of the ground surface settlement induced by tunneling can be regarded as a plane problem.Thus, the settlement calculation of elemental excavation () is simplified as a twodimensional problem, and the simplified model is shown as Figure 3.
The settlement of the ground surface induced by elemental excavation () at the point (, 0) of - coordinates is dependent on According to Liu [16], the main influence zone angle () of the settlement is introduced to determine the influence radius (()): where  is the depth of excavation.
The relationship between the main influence zone angle and the internal friction angle () of soil mass can be expressed as The ground surface movement is dependent on the radial convergence of tunnel's cross section.According to the stochastic medium theory, the surface settlement equals the settlement difference caused by the original excavation section Ω and section Λ after shrinking as follows: As shown in Figure 4, the initial radius of circular tunnel section Ω is ; after uniformly shrinking, radius becomes  − Δ.The surface settlement in 2D induced by tunneling can be expressed as follows: where Similarly, the three-dimensional surface settlement of the point (, , 0) induced by the uniform radial convergence from  to  − Δ is obtained as follows:

Ground Surface Settlement Induced by Double Tunnel
Excavation.As shown in Figure 5, the double circular tunnels with initial radius  are excavated at depth .The center distance between the two tunnels is  ( > 2).The radial convergences of the double circular tunnels are Δ 1 and Δ 2 , respectively.
According to the stochastic medium theory, the surface settlement caused by the double tunnels excavation can be considered as a linear superposition of settlement induced by two single tunnel excavations.Therefore, the ground settlement calculation in 3D can be simplified as a plane problem and shown in Figure 6.
The surface settlements caused by two single tunnel excavations in 2D are expressed as follows, respectively: where and ℎ 2 are the integration parameters of the double tunnels.These variables are expressed as follows, respectively: According to superposition principle, the total settlement of the ground surface caused by the double tunnels excavation at the point (, 0) of - coordinates in 2D can be expressed by Therefore, the three-dimensional surface settlements induced by the uniform radial convergences (Δ 1 and Δ 2 , resp.) are presented as follows: The three-dimensional total settlement of ground surface at the point (, , 0) induced by double circular tunnels excavations is obtained by

Ground Surface Settlement
Effected by the Pile.The interaction between the pile and the soil mass is incorporated, and the displacement by the friction resistance of pile must be considered [21].In the paper, the shear displacement method is used to calculate the ground displacement by the friction resistance of pile.The pile with radius  0 and the influence radius   of the friction resistance are shown in Figure 7.In Figure 7,  is the vertical displacement of the soil mass element while  is the distance between the soil mass element and the pile center.Horizontal surface displacement of soil mass element is neglected.Thus, the shear strain of the soil mass element is defined as Therefore, shear stress is expressed as where  is shear modulus.An annular soil mass element around the pile with the height being , the inner diameter being , and the outer diameter being  + , is defined.Then, the equilibrium equation of the submerged soil mass element and the pile is expressed as where  0 is radius of the pile. 0 is the friction resistance.By substituting ( 20) into (19), the differential equation is derived: The displacement of soil mass around the pile is presented by integrating (21) as follows: 2.6.Effect of Group Piles.According to the load transfer mechanism, the lateral load resistance of pile foundations is critically important due to pile-soil-pile interaction when lateral soils move down (Rollins et al. [22,23]).The interaction effect between the group piles and soil masses should be taken into account in the calculation of surface displacement.As an example of group piles as shown in Figure 8, the interactions between pile C and other piles are present.
According to the group piles tests and the back analysis, formulations have been developed to compute the interaction coefficients of group piles (  ) by Rollins et al. [22,23] and the expressions for each case are shown as follows: First (lead) row piles: Second row piles: Third or higher row piles: where  is center-to-center spacing between piles in the direction of loading and  0 is width or outside diameter of the pile.The values of   are limited to 1.0.The interaction between the pile group is neglected when the spacing between piles is larger than 6.5 0 .

Total Settlement of the Ground Surface
3.1.Single Tunnel.In Figure 9, a circular metro tunnel with initial radius  is excavated underground in incompressible soil mass.The central axis of tunnel is parallel to -axis in the Cartesian coordinate system (, , ).A pile with radius  0 is located at the area, and the center of pile is (, , 0).The approach expressed above is adopted to determine the total surface settlement caused by single metro tunnel excavation with the consideration of the pile effect.
The pile has a negative effect on the settlement because of the frictions force between pile and soil mass.The maximum friction force is obtained by the largest relative displacement only induced by the metro tunnel excavation between pile and soil mass.However, the friction force is impossible to reach its limits and 0.8 times of the maximum friction force is adopted [20].According to (22), the displacement induced by the impact of pile is obtained as follows: where  and  are the coordinates of the center of the pile for  and , respectively.The total three-dimensional settlement of the ground surface caused by single metro tunnel excavation considering the influence of pile is obtained by   (, ) =  1 (, ) +  2 (, ) . (25) However, the metro tunnel often crosses the area with group piles.Incorporating the effect of group piles, total settlement of the ground surface is predicted by superposing the settlement caused by tunnel excavation and the displacement effected by the group piles: where  and  represent the order number of piles and the total number of group piles, respectively.

Double Tunnels.
As shown in Figure 9, the double circular metro tunnels have been excavated with initial radius , the same depth , and center distance  ( > 2).A pile with radius  0 remaining motionless is located near the tunnels, with the coordinate of pile center (, , 0).According to the superposition principle, the ground surface settlement for the double tunnels excavation considering the influence of single pile is computed by The three-dimensional prediction of ground surface settlement considering the influence of group piles is presented as follows (Figure 10): (28)

Comparison with Measured Data.
In order to validate the effectiveness and accuracy of the proposed approach, the predicted settlements induced by tunneling with the influence of   In Figure 12, the direction which is approximately parallel to tunnel centerlines is taken as -axis and the vertical direction of -axis is -axis.Due to  2,11 = 17.91 m > 6.5 0 = 13 m, the piles 1-10 and piles 11-20 are regarded as the first and second group piles, respectively.It shows that both  1,2 and  11,12 are 5 m from the field data.According to the calculation by ((23a), (23b), and (23c)), the interaction coefficients (  ) are both 0.467.Geotechnical parameters were obtained by tests, and the local test parameters are shown in Table 1.
Th e B e ij in g -G a n g z h o u r a il w a y   595 points are set to measure the actual displacements of ground surface which are induced by the metro tunnels excavation in the segment from Xujiachong station to Railway College station.A part of monitoring data and the predicted results by the proposed approach in the area of 100 m×100 m is shown in Figure 13.
The comparison between the measured data of six points and the predicted results by the proposed approach are carried out, and the predicted settlements of the ground surfaces for points 1, 2, 4, and 6 are larger than those of the field data as shown in Figure 13.The results by the proposed approach are smaller than the measured data of points 3 and 5 which are located at the influence zone of group (29) The calculation results depict that the surface settlement reaches its maximum value on the centerline of tunnels.The surface settlement decreases gradually with the increasing of horizontal distance and at infinity  → 0. The settlements near piles are smaller than those far from them.

Numerical Validation.
For further validation of the prediction approach in this paper, an analysis for the ground surface settlement was presented by numerical software.The basic geotechnical parameters and engineering parameters are the same as the engineering example above.The results of numerical simulation are shown in Figure 14, with a negative value indicating settling down.
It is evident from the results in Figure 14 that the ground surface settlements around piles are smaller than those far away from the piles in the same  coordinates.Comparing the results in Figures 13 and 14, the numerical results are well agreeable with those of the proposed approach.

Analysis and Discussion
5.1.Effect of the Buried Depth.In order to illustrate the influence of parameters for the proposed approach on the surface displacement, the analyses and discussions for buried depth of the tunnel, center distance of double tunnels, size and position of piles, and convergence are performed.To determine the effect of the buried depth of tunnel on the surface settlement, the radial convergences presented by Yang and Wang [18] and local test geotechnical parameters in Table 1 are  As shown in Figure 15, the surface displacements which are induced by tunneling reach the maximum values on the center lines of tunnel.The greater the depth of the tunnel is, the smaller the surface settlement is.In comparison of the four examples, the frictional resistance between the pile and soil mass shows a countereffect on soil mass movement in the influence zone of piles.The countereffect is particularly evident when the tunnels buried depth is 50 m.The scope of ground movement will broaden, and the interaction effect of double tunnels on settlement and the pile effect are more significant with the buried depth increasing.

Effect of the Radial Convergences.
In order to study the effect of the radial convergences of tunnels on the surface settlement, the convergences presented by Yang and Wang [18] are adopted in Table 2  are shown in Table 2.The surface settlements considering different convergences are predicted by the proposed approach as shown in Figure 16.
The effects of radial convergences on the surface settlement are significant.For example, with the Δ 1 and Δ 2 increasing from 0.0105 to 0.0310 m, the surface settlement is almost three times as examples 1 and 3 shown in Figure 16.Moreover, the significant differences of settlement caused by different tunnel convergences after working can be seen from example 4. The ground surface settlements induced by radical convergence of 0.0310 m are smaller than those by radical convergence of 0.0489 m.In Figure 17, the surface settlements induced by tunneling with  = 10 m are larger than those in examples 2, 3, and 4 (i.e.,  = 20, 40, and 60 m, resp.)under the condition of same pile's coordinates.The smaller the center distance is, the more obvious the interaction effect of double tunnels on surface settlement is.The effect of piles on settlement will enhance with decreasing of center distance.3. The results are shown in Figure 18.

Effect of the Center
As it is shown from the results depicted in Figure 18, the effect of pile on settlement is greatest when pile is located at the neutral axis of tunnel.The effect will decrease with the increase of distance between pile and the neutral axis of tunnel.In the light of examples 3 and 4 in Figure 18, the scope

Figure 1 :
Figure 1: Calculation model for elemental excavation based on the stochastic medium theory.

Figure 2 :Figure 3 :
Figure 2: Ground surface settlement caused by single tunnel excavation in 3D.

Figure 4 :
Figure 4: Cross section of circular tunnel.

Figure 5 :
Figure 5: Ground surface settlement caused by double tunnels excavation in 3D.

Figure 6 :
Figure 6: Plane calculation model of surface settlement induced by double tunnels excavation.

Figure 7 :
Figure 7: Calculation model of ground surface displacement effected by pile in 3D.

Figure 10 :
Figure 10: Double tunnels excavation with the effect of piles in 3D.

Figure 13 :
Figure 13: Three-dimensional surface settlement induced by double tunnels excavation considering the influence of group piles and measured data.
piles.The comparison shows that the three-dimensional surface settlements induced by double tunnel excavations considering the influence of group piles are close to the monitoring data, with an average error of approximately 4.28%: 0.071 + 0.009 + 0.0295 + 0.0925 + 0

Figure 14 :
Figure 14: Numerical simulation of ground displacement in 3D.

Table 3 : 4 :
The coordinates of the piles center on the ground and radii of piles.The parameters of pile group.Examples  0 (m)   (m) Coordinates  0 (m)  (m) Distance.To investigate the effect of center distance on ground settlement, four different examples are analyzed using the proposed approach with the data as follows:  0 = 1 m,   = 20 m,  =  = 0,  = 3 m, Δ 1 = Δ 2 = 0.0156 m,  = 20 m, and  = 10, 20, 40, and 60 m, respectively.The results of settlement for different center distances are shown in Figure 17.

5. 4 .
Effect of the Position and Size of Piles.Substituting the parameters  = 3 m, Δ 1 = Δ 2 = 0.0156 m,  = 40 m, and  = 20 m, the coordinates and radii of piles are shown in Table