An Analytical Method Evaluating the Evolution of Group Effect for Vertically Loaded Pile Groups Subjected to Tunnel Excavation

Abstract

Tunnel excavations near existing vertically loaded pile groups are frequently encountered in urban areas, and most available studies have focused on the additional deformation and stress induced in the pile group, lacking in consideration of the variation in the pile-soil interaction (PSI) in the process, which plays an important role in pile group behavior with close pile spacing. In addition, for vertically loaded pile groups subjected to excavations in their proximity, the combined actions of the overlapping stress and shielding effect can lead to complicated variations in the group effect, which in turn results in difficulty in evaluating the PSI relationship and the pile group response. Thus, the modified Poulos method is extended in this paper to account for the effects of tunneling, and the variations in the group effect, load redistributions and pile settlements are also investigated. The validity of the proposed method is firstly verified by an available centrifuge model test with tunneling near a 2 × 2 fixed-head pile group. Further, the influences of the pile spacing and relative distance between the pile group and the tunnel are analyzed, and the most unfavorable working condition could be a tunnel excavated near the soil surface, close to a pile group with a small pile spacing.

1. Introduction

Tunnel excavations adjacent to existing piled foundations are frequently encountered in urban areas. Ground loss induced by tunneling can lead to nonnegligible soil settlements around a tunnel and, in turn, additional stressing and deformation on adjacent piled foundations, usually in the form of a pile group for high rise buildings, which are originally designed to withstand the vertical loads transmitted from the upper structure.

When a pile group is only subjected to vertical loads on the pile head, the pile–soil interaction (PSI) is in an active mode [1], and a close pile spacing less than 6 times the pile diameter usually requires considering the group effect in the vertical direction due to overlapping stress in engineering practices [2,3]. Simplified analysis methods to evaluate the response of an active vertically loaded pile group have extensive applications and can usually be classified into three categories: (1) the elastic analysis method, assuming the soil as an infinite elastic space loaded beneath the surface [4,5], and limited to analysis of pile-soil interaction in the elastic range; (2) the nonlinear foundation column (NFC) model, which needs to be combined with other methods to incorporate the pile-soil-pile interaction [6], for example, the shear displacement method [7,8]; (3) and numerical simulation [9,10,11].

The modified Poulos method [12] avoids the disadvantages of the NFC and elastic analysis methods by combining the above two methods and was recommended by the American Petroleum Institute [13] to evaluate the group effect of active pile groups in ocean engineering practices. The pile head settlement of the pile group is assumed to consist of two components: the nonlinear head settlement of an equivalent isolated pile and the additional settlement induced by pile group interactions. The head settlement of the isolated pile is calculated using the NFC model combined with t−z/Q−z curves, while the additional settlement is determined based on the elastic analysis method suggested by Poulos and Davis [4], which is proportional to the pile head loads. Furthermore, modified t−z/Q−z data are developed by multiplying the Z values of the original t−z/Q−z data by a Z multiplier larger than unity. The Z multiplier is determined by trial calculations so that the isolated pile head settlement equals the pile group head settlement predicted above. The modified t−z/Q−z curves can account for the pile group effect, and the Z multiplier is considered the parameter that represents the group effect of the active pile group.

With tunneling taken into consideration, although a single pile response can be easily considered by either the NFC model or elastic analysis method, analysis of the pile group response will suffer from the difficulties in load redistribution among the group piles and variations in group effect during tunneling, and the addition pile head settlement is no longer proportional to the pile head loads.

In terms of computation, the additional pile head settlement cannot be directly determined based on previous research for an active pile group, leading to difficulties in group effect evaluation. Finite element modelling of both the tunneling and the existing pile group loaded at the pile head is also frequently adopted as a workover in present practices because of both the group effect and complex boundary conditions, but it entails a heavy computational burden, not to mention a complicated parameter determination process [9,10,11]. Simplified methods have been proposed to include the additional pile head settlement by introducing a shield settlement based on the shear displacement method [7], and it was proven that neglecting the working loads can lead to an unconservative prediction of pile response [1,14]. However, neither the pile head load redistribution nor the group effect is taken into consideration, which is not consistent with the design practices of active pile groups. Franza et al. [15] considered the pile-soil interaction and the pile-to-pile interaction by the elastic analysis method and incorporated the effects of superstructure stiffness in their analysis. The importance of considering the vertical loads before tunneling and load redistributions among piles was also highlighted, while the influence of the group effect on pile-soil interaction was not considered, which may lead to an overestimated pile group capacity with small pile spacing.

Thus, an analytical method to predict the effects of tunneling on a vertically loaded pile group in the vertical direction is proposed by extending the modified Poulos method in this paper. The method is then verified by comparison with an available centrifuge model test conducted by Loganathan et al. [16]. Finally, parametric studies are performed to further investigate the evolution mechanism of the group effect during tunneling.

2. Methodology

The modified Poulos method proposed by Focht et al. [12] is a combination of the NFC model and the Poulos elastic analysis, with both methods modified to use their features applicable to active pile groups considering both nonlinear pile-soil interaction and pile group effect.

For a group of m piles only subjected to the pile head load, Focht et al. [12] rationally predicted the settlement of a pile group using Equation (1), known as the modified Poulos method. It is assumed that the settlement $z_{kG}^1$ of group pile k consists of settlement $z_k^1$ of an equivalent isolated pile considering the nonlinear pile-soil interaction and the sum of elastic additional settlements $\stackrel{-1}{z_{ke}}{\displaystyle \sum _{i=1,j\ne k}^m{\alpha }_{ki}{P}_i}$ induced by the other piles in the pile group. The superscripts hereafter represent the node number along the pile shaft, and 1 means the first node at the pile head.

$$z_{kG}^1=z_k^1+\stackrel{-1}{z_{ke}}{\displaystyle \sum _{i=1,j\ne k}^m{\alpha }_{ki}{P}_i}$$

(1)

where $\stackrel{-1}{z_{ke}}$ is the elastic pile settlement for an isolated pile applied by a unit pile head load and α_ki is the interaction factor, obtained by the additional response of pile k due to the existence of group pile i divided by the elastic settlement of an equivalent isolated pile k.

The NFC method used to calculate $z_k^1$ assumes that the soil be discretized into a series of soil springs along the pile shaft and the pile interacts with the soil by discontinuous nonlinear springs. The Poulos elastic analysis considers the soil as continuous semi-space, and can thus be used to predict both isolated single piles and pile groups, and is adopted to calculate $\stackrel{-1}{z_{ke}}$ and α_ki in Equation (1). It can be indicated that the elastic additional settlement is proportional to the pile head load P_i on pile i in this equation.

If P_i is known for each group pile, the group pile settlement $z_{kG}^1$ can be easily calculated by Equation (1). For a pile group connected by a cap, and the distributional pile head load P_i is unknown, $z_{kG}^1$ can be obtained by incorporating Equations (2) and (3).

$$P={\displaystyle \sum _{i=1}^n{P}_i}$$

(2)

$$\begin{array}{cc}{z}_{kG}^1=z_{iG}^1& i=1,2\dots n,i\ne k\end{array}$$

(3)

where P is the total pile head load for the pile group and Equation (3) means that the settlement for every group pile is equal at the pile head.

When the vertically-loaded pile group is simultaneously subjected to tunneling, however, the elastic additional settlement is not proportional to the pile head load, and the modified Poulos method Equation (1) should be rewritten as Equation (4) to account for the effects of tunneling-induced soil settlements along the pile shaft.

$$z_{kG}^1=z_k^1+\Delta z_{ke}^1$$

(4)

where $\Delta z_{ke}^1$ is the additional elastic settlement of group pile k due to the existence of other group piles, when the pile group is subjected to a combination of the pile head load and tunnel excavation, as determined by Poulos elastic analysis.

Therefore, the prediction of $z_{kG}^1$ consists of two parts: calculation of the nonlinear isolated pile head settlement $z_k^1$ by the NFC method and evaluation of the elastic additional pile head settlement $\Delta z_{ke}^1$ through Poulos elastic analysis.

2.1. Calculation of $z_k^{\mathit{1}}$

The vertical equilibrium equation of the isolated pile can be obtained as Equation (5) if the pile can be regarded as a column connected by a series of nonlinear soil springs in the vertical direction. To describe the nonlinear pile-soil interaction along the piles, proper pile-soil interaction relationships in the vertical direction were assumed as follows: the relationships of skin friction vs. vertical pile-soil relative displacement at a depth of y below the surface (see Equation (6)), and the relationship of end bearing capacity vs. vertical pile-soil relative displacement at the pile tip (see Equation (7)).

$$E_p{A}_p\frac{{\mathrm{d}}^{2}z(y)}{\mathrm{d}{y}^2}-t(y)=0$$

(5)

$$t(y)=k_t(y)\cdot \left[z_s(y)-z(y)\right]$$

(6)

$$Q_b=k_b\cdot (z_{sb}-z_b)$$

(7)

where t(y) and Q_b are the peripheral skin friction and the bearing force at the pile tip, k_t(y) and k_b are the stiffness of skin friction and end bearing, and with nonlinear PSI relationships (t−z/Q−z curves) taken into consideration, k_t(y) and k_b vary with pile-soil relative displacement, z_s(y) and z_sb are the vertical free-field soil settlements along depth and at the depth of pile tip, and z(y) and z_b are the pile settlements along the pile shaft and at the pile tip, respectively.

Given that the PSI relationships vary with depth, Equation (5) is solved by the finite difference method, where the isolated pile with L in length is first discretized uniformly into n segments, with each segment length equal to δ, and two virtual nodes above the top and below the tip of the pile (see Figure 1).

Equation (5) is then transformed into Equation set (8), where there are n equations and n + 2 unknown variables. Therefore, two additional boundaries are added to solve the equation set. Equations (9) and (10) indicate that a concentrated load P is applied at the pile head and the pile tip is connected to a nonlinear soil spring obeying the Q−z relationship, respectively, are adopted to form the (n + 2) × (n + 2) Equation set (8), whose solution is the pile settlement vector {z}. The superscripts hereafter also represent the node number of the pile.

$$z^{i-1}-(2+k_t^i{\delta }^2/E_p{A}_p)z^i+z^{i+1}+k_t^i{\delta }^2{z}_s^i/E_p{A}_p=0\hspace{1em}i=2,3,\dots n+2$$

(8)

$$E_p{A}_p(z^3-z^1)=-2\delta P$$

(9)

$$E_p{A}_p(z^{n+3}-z^{n+1})=-2\delta k_b(z^{n+2}-z_{{}^s}^{n+2})$$

(10)

The initial tangents of both the t−z and Q−z curves can be adopted as k_t and k_b to calculate the pile settlement, which in turn returns new k_t and k_b values according to the t−z and Q−z curves, leading to another solution of the pile settlement. Repletion of the above procedure with updated k_t and k_b values will be stopped if the relative error of the pile settlements between two iterations is less than a given tolerance of 1 × 10⁻⁵.

2.2. Calculation of $\Delta z_{ke}^{\mathit{1}}$

Considering that the group effect has been proven to be elastic [17], it is reasonable to calculate the additional pile response by Poulos elastic analysis. According to Equation (4), $\Delta z_{ke}^1$, the additional elastic settlement of pile k due to the existence of other piles in the group, should be solved by Equation (11) in turn to predict the settlement of the pile group.

$$\Delta z_{ke}^1=z_{kge}^1-z_{ke}^1$$

(11)

where $z_{kge}^1$ and $z_{ke}^1$ are the elastic head settlements of the group pile k and the isolated pile equivalent to pile k, respectively, both calculated by Poulos elastic analysis.

In the Poulos elastic analysis, the soil is assumed to be an isotropic, elastic semi-space, and pile-soil interaction is analyzed by Mindlin’s solution of the displacement in the soil due to a vertical load applied below the soil surface. As depicted in Figure 2, an isolated pile with length L is discretized into n uniform segments with each segment length of δ. The skin friction $t^i$ of element i along the pile shaft is simplified to be evenly distributed on the element. The pile end bearing stress $t_b$ is also considered to be evenly distributed on an enlarged pile tip with a diameter of $d_b$. As a result of n skin friction stresses along n pile segments and an end bearing stress, the PSI function for an isolated pile can be written as Equation (12).

$$\left\{z_r\right\}=\frac{d}{E_s}\left[I_s\right]\left\{t\right\}$$

(12)

where z_r is the relative pile-soil displacement, including the midpoint displacements of n pile segments and a pile tip displacement; d is the pile diameter; E_s is the soil modulus; [I_s] is the pile-soil interaction matrix; and t is the pile-soil interaction force, including n skin friction stresses $t^i$ and a tip bearing stress $t_b$. The element I_sij of such a matrix signifies the displacement of the midpoint of element i induced by a unit load of skin friction or the tip bearing stress and can be obtained by integration of Mindlin’s solution [4].

When taking tunneling into consideration, as a result of pile-soil displacement compatibility, z_r equals the free-field soil settlement {z_ks} minus pile settlement {z_ke} for the group pile k, and Equation (12) can be rewritten as Equation (13).

$$\left\{z_{ke}\right\}=\frac{d}{E_s}\left[I_s\right]\left\{t\right\}+\left\{z_{ks}\right\}$$

(13)

Given that the pile is applied with a vertical head load of P, the relationship between the displacement and stress along the pile shaft in its differential form is given in Equation (14). By substituting Equation (13) into Equation (14), Equation (15) can be obtained to calculate {t}, and in turn, $z_{ke}^1$ of the isolated pile subjected to both tunneling and the vertical pile head load can be easily calculated.

$$\left\{t\right\}=\frac{d}{4{\delta }^2}{E}_p{A}_p\left[I_p\right]\left\{z_{ke}\right\}+\left\{Y\right\}$$

(14)

where E_p is the pile modulus, A_p is the pile sectional area, [I_p] is the (n + 1) × (n + 1) pile action matrix, {Y} is a (n + 1) column vector, and $\{Y\}={\{\frac{nP}{\pi d^2\delta },0,0,\dots 0\}}^{-1}$ (Poulos and Davis, 1980).

$$\left([I]-\frac{d^2{E}_p{A}_p}{4E_s{\delta }^2}\left[I_p\right]\left[I_s\right]\right)\left\{t\right\}=\frac{d{E}_p{A}_p}{4{\delta }^2}[I_p]\left\{z_{ks}\right\}+\{Y\}$$

(15)

For a pile group consisting of m group piles, the group effect of group pile i on the target pile k can be considered by superposition of the displacements of segments on pile k induced by the skin friction and end bearing loads on pile i. With tunneling further taken into consideration, the pile-soil interaction Equation (13) can be modified by adding the contribution of {t_i} along the group pile i, and the pile-soil interaction equation for group pile k can be given as Equation (16).

$$\left([I]-\frac{d^2{E}_p{A}_p}{4E_s{\delta }^2}\left[I_{p1}\right]\left[I_{skk}\right]\right)\left\{t_k\right\}+{\displaystyle \sum _{i=1,i\ne k}^m\frac{d^2{E}_p{A}_p}{4E_s{\delta }^2}\left[I_{p1}\right]\left[I_{ski}\right]}\left\{t_i\right\}=\frac{d{E}_p{A}_p}{4{\delta }^2}[I_{p1}]\left\{z_{sk}\right\}\hspace{1em}k=1,2\dots m,i=1,2,\dots m,i\ne k$$

(16)

where [I_skk] and [I_ski] are (n + 1) × (n + 1) pile-soil interaction matrices. The elements I_skk,_ij and I_ski,ij signify the displacements of the midpoint of element i induced by a unit load of skin friction or the tip bearing stress on the jth element of pile k and pile i, respectively. The calculations of [I_skk] and [I_ski] are the same as those of [I_s]. {Y_k} is the (n + 1) column vector, and $\{Y_k\}={\{\frac{n{P}_k}{\pi d^2\delta },0,0,\dots 0\}}^{-1}$, where P_k is the head load of pile k. Considering that the distributed pile head load for each group pile is unknown, [I_p1], a n × (n + 1) pile action matrix is used instead of [I_p], and [I_p1] =

$$\left[\begin{array}{ccccccc}1& -2& 1& 0& 0& \dots & 0\\ 0& 1& 2& 1& 0& \dots & 0\\ & & & \begin{array}{l}\\ \end{array}& \dots & & \\ 0& \dots & & 0& 1& -2& 1\\ 0& \dots & & 2& 2& -5& 3.2\\ 0& \dots & & 0& -4f/3& 12f& -32f/3\end{array}\right]$$

where $f=\delta /n$. Pile group configuration and relative position between the pile group and tunneling can be easily considered by different free-field soil settlements and pile-soil interaction matrices [I_ski].

If the distributed pile head load P_k is known for every group pile, Equation (17) can be obtained for m group piles based on the force equilibrium, and the elastic pile group settlement {z_kge} can be calculated by combining Equation (17) and Equation set (16). On the condition that a fixed-head pile group withstands a total pile head load P and the settlements of group piles are equal, Equations (17) and (18) can be obtained. The elastic pile group settlement {z_kge} can be calculated after solving {t_k} based on Equation set (16), Equations (17) and (18).

$${\displaystyle \sum _{k=1}^m\{E_k\}\{t_k\}}=P$$

(17)

$$\{G_k\}\{t_k\}+{\displaystyle \sum _{i=1,i\ne k}^m\{F_{ki}\}\{t_i\}}+z_{ks}^1=z_{kge}^1$$

(18)

where $\{E_k\}=\{\begin{array}{ccccc}\pi d\delta & \pi d\delta & ..& \pi d\delta & \pi d_b^2\end{array}\}$, $\{G_k\}=\frac{d}{E_s}\{\begin{array}{ccccc}{I}_{skk,11}& I_{skk,12}& \dots & I_{skk,1n}& I_{skk,1n+1}\end{array}\}$, and $\{F_{ki}\}=\frac{d}{E_s}\{\begin{array}{ccccc}{I}_{ski,11}& I_{ski,12}& \dots & I_{ski,1n}& I_{ski,1n+1}\end{array}\}$.

2.3. Calculation Procedure

A procedure to evaluate the influence of tunneling on the adjacent pile group is then given as follows:

The free-field soil settlements {z_sk} at the group pile locations due to tunnel excavation are first evaluated. The soil settlement profiles can be obtained from either field measurements or available empirical equations [18,19];

The elastic pile group head settlement $z_{kge}^1$ is then calculated based on the pile head fixity conditions and the soil settlements by the elastic analysis method;

If the group piles are individually loaded and the pile head loads are known, the elastic isolated pile head settlement $z_{ke}^1$ for each group pile can be readily calculated combined with the free field soil displacement {z_sk} at its location by Poulos elastic analysis. For fixed-head group piles, however, trial distributed pile head loads are assumed in the calculation of $z_{ke}^1$, satisfying the condition that the sum of the distributed loads is equal to the total group pile head load P. The additional settlement $\Delta z_{ke}^1$ is then obtained based on Equation (11).

For individually loaded piles, an NFC analysis of an isolated pile subjected to both the pile head load and tunnel excavation is conducted, and the pile group head settlement $z_{kG}^1$ can also be readily obtained combined with $\Delta z_{ke}^1$ calculated in step (3). For fixed-head group piles, the trial pile head loads in step (3) are adopted in the NFC analysis of isolated group piles to determine $z_k^1$ and $z_{kG}^1$ and if the calculated pile group head settlement $z_{kG}^1$ is not equal for the group piles, return to Step (3), where the trial distributed pile head loads should be adjusted.

NFC analyzes for every group pile are conducted again by adjusting the PSI relationships by Z multipliers no less than 1.0.

A flow chart of the above calculation procedure is given in Figure 3. Variation of the pile group settlement combined with the Z multipliers is deemed to give an overall evaluation of the group effect due to tunnel excavation from the aspects of pile deformation and pile-soil interaction.

3. Verification

Loganathan et al. [16] conducted a series of centrifuge model tests to investigate the response of a vertically loaded 2 × 2 fixed-head pile group subjected to tunnel excavation at different depths. Although the test series were not designed to investigate the variation of group effect due to tunneling, the measured pile group response can serve to verify the validity of the proposed method.

The case with 18 m excavation depth was analyzed (Figure 4). The soil sample for the test was measured to vary in shear strength linearly from 50 kPa at the surface to 75 kPa at the pile tip, and the bulk modulus was 16.5 kN/m³. A 2 × 2 pile group consisting of 4 18-m long circular concrete piles with a diameter of 0.8 m and an elastic modulus of 30 GPa in the prototype was replicated. The pile spacing was 3.125 times the pile diameter (represented by d), and the pile head was designed to be connected by a rigid pile cap, applied with a vertical load of 4.46 MN before tunneling. The tunnel with a diameter of 6 m was excavated with a centerline located 18 m below the surface, and the center-to-center distance between the tunnel and the front pile row was 5.5 m. The ground loss ratio ε₀ was designed as 1%.

Tunneling-induced free-field soil settlement has been thoroughly investigated by researchers, and various empirical prediction formulas have been proposed [18,19]. Considering the nonlinear effects of soil deformation, [19] proposed a closed-form analytical solution of subsurface soil settlement due to tunnel excavation, see Equation (19). The free-field soil settlement profiles z_sk(y) at the group pile locations adopted in the following analysis are predicted by the formula.

$$\begin{array}{l}{z}_{sk}(y)={\epsilon }_0{R}^2\left(-\frac{y-H}{x^2+{(y-H)}^2}+(3-4\mu )\frac{y+H}{x^2+{(y+H)}^2}-\frac{2y\left[x^2-{(y+H)}^2\right]}{{\left[x^2+{(y+H)}^2\right]}^2}\right)\exp \\ \left[-\frac{1.38x^2}{{(H+R)}^2}-\frac{0.69y^2}{H^2}\right]\end{array}$$

(19)

where R is the tunnel radius, H is the excavation depth of the tunnel center, y is the depth below the surface, and x is the distance from the tunnel centerline.

In NFC analyzes, considering that the hyperbola relationship is simple and extensively used in pile response analysis [1,20], the t−z relationship is assumed to follow a hyperbola in the following NFC analyzes, see Equation (20).

$$t(y)=\frac{z(y)}{\frac{1}{k_{s,ini}(y)}+\frac{t_u(y)}{z(y)}}$$

(20)

where k_s,ini(y) and t_u(y) are the initial stiffness and ultimate skin friction of the soil spring in the vertical direction at a depth of y below the surface and can be determined based on Equations (21) and (22), respectively [7,21].

$$k_{s,ini}=\frac{E_i}{2(1+\mu )r_0\ln (r_m/r_0)}$$

(21)

$$t_u=K_0\gamma z\tan \phi +c$$

(22)

where r₀ and r_m represent the pile radius and an empirical distance, respectively, and r_m is typically taken as 2.5r₀. E_i is the initial modulus of the soil, and E_i = 2E₅₀ for a hyperbola relationship, where E₅₀ is the soil modulus at 50% of the ultimate shear strength measured in an unconfined compression test. Based on the relationship between the shear strength and E₅₀ recommended by Code JTS167-4-2012 [2], E_i can be easily obtained as 18 MPa.

The Q−z relationship is also described by a hyperbola [7]; see Equation (23).

$$Q_b=\frac{z_b}{\frac{1}{k_{bs,ini}}+\frac{Q_u}{z_b}}$$

(23)

where b denotes the pile tip and k_bs,ini and Q_u are the initial stiffness and end bearing capacity at the pile tip, respectively, as determined by Equation (24), proposed by Randolph and Wroth [7], and Equation (25), the Vesic bearing capacity equation [22].

$$k_{bs,ini}=2E_i{r}_0/(1+2\mu )(1-\mu )$$

(24)

$$Q_u=c{N}_c+(1+2K_0\gamma z)N_q/3$$

(25)

where N_c and N_q are the bearing capacity factors.

In the elastic analyzes, the soil modulus is also taken as 18 MPa, consistent with that in the NFC analyzes, considering that the pile group effect is purely elastic behavior.

Additional pile head settlement and the axial force distributions along the shaft of the front-row piles induced by excavation were measured in the centrifuge model test and compared with the predicted results to verify the validity of the proposed method.

The pile group responses before and after tunneling are both predicted by the method, and the tunneling-induced settlement can be obtained as 10.9 mm, which is acceptable when compared with 12.2 mm measured in the test. The predicted tunneling-induced axial force distribution with and without considering the Z multiplier are also given in Figure 5. For the front-row piles, the predicted results show good consistency with the measured ones. It can also be observed that if the group effect is not taken into account, stronger negative skin friction will be induced and lead to overestimated additional response. Besides, the pile head load is redistributed among the group piles, and the load discrepancy can be approximately 6% of the original value. If the tunnel is excavated nearer to the pile group or with a higher ground loss ratio, the load discrepancy can be larger and should be considered in the evaluation of the pile group response.

In addition, the variation in the Z multipliers for group piles is given in

Table 1. Variation in the Z multipliers before and after tunneling for a 2 × 2 pile group.

Pile Row	Before Tunneling	After Tunneling
Front	1.52	1.45
Rear	1.52	1.74

. It can be observed that the group effect decreases for the front row piles and increases for the rear row piles, indicating that reanalysis of the Z multipliers is necessary when tunneling is taken into consideration. The PSI relationship may be either stiffer or softer due to tunneling because the variation of the Z multiplier is subjected to a combined effect of load redistribution and stress overlapping in this stage, different from that of an active pile group.

4. Discussion

Stiff clay with a shear strength changing from 50 kPa to 75 kPa and dry sand were adopted in the above centrifuge model tests, and both can be classified as favourable soil conditions. In addition, the pile–tunnel interaction can be affected by both the vertical and lateral clearances according to available studies, but mostly from the perspective of load redistribution and pile group settlement during tunneling. [9,10,23].

Thus, variations in the group effect during tunnel excavation with varying tunnel–pile relative position and pile spacing are investigated by the proposed model by considering the following working conditions, as shown in

Table 2. Details of working conditions.

Case No.	s	C1/R	C2/L
1	2	1.5	0.25
2	2.5	1.5	0.25
3	5	1.5	0.25
4	2.5	0	−0.25
5	2.5	1.5	−0.25
6	2.5	2	−0.25
7	2.5	1.5	0

. The influence of the tunnel-pile relative position is analyzed by varying the lateral distance C₁ between the tunnel center and the front pile and the vertical distance between the tunnel center and the pile tip, represented by C₂ (see Figure 6), where a positive C₁ and C₂ indicate that the tunnel is excavated on the left side of the front pile and above the pile tip (tunnel drawn as a dashed line). Relatively soft soils are included to consider unfavorable tunnel excavation conditions, and a uniform shear strength along the pile shaft is taken as 40 kPa in the following analyzes. The pile group dimensions and material parameters are kept the same as those in Section 3, except that the influence of pile spacing in pile-tunneling interaction is also investigated with changing pile spacing. Considering the relatively soft soil condition, the total load of the vertically loaded fixed-head pile group is determined to be 4.0 MN based on analysis of a single pile, with a safety factor of 2.0.

For each case, both the pile group settlement, axial force, and the variation in the Z multipliers of the group piles are examined to evaluate the influence of tunneling.

4.1. Influence of Pile Spacing

To investigate the influence of pile spacing on the response of the pile group, Cases 1, 2 and 3 with pile spacings of 2, 2.5 and 5 times the pile diameter are analyzed by the proposed method, and both the variations in pile head settlement, axial force distributions, and Z multipliers are compared in Figure 7, Figure 8 and Figure 9.

Figure 8 shows that with increasing pile spacing, pile settlement is significantly decreased due to the group effect both before and after tunneling. In addition, with the settlements before tunneling taken as the baseline, the tunneling-induced settlement also decreases with larger spacing, ranging from 0.0308 m for Case 1 to 0.0232 m for Case 3, which can be attributed to the stronger support provided by the rear-row pile subjected to smaller tunneling-induced soil settlements. However, discrepancy in the pile head axial force can also be induced with larger pile spacing, ranging from 30% of the maximum axial force before tunneling for Case 1 to 40% for Case 3, indicating increasing load redistribution with larger spacing and, in turn, nonnegligible shear force in the pile cap.

The variation in the group effect before and after tunneling is also investigated for Cases 1, 2 and 3, and the group effect in the vertical direction, manifested by the Z multipliers larger than 1.0, also decreases with increasing pile spacing. However, group piles are shown to behave differently due to unsymmetrical loading conditions after tunneling. The Z-multipliers for the front row piles increase after tunneling, while those for the rear row piles decrease, but the Z multipliers for the rear row piles are still larger than 1.0, meaning that the group effect cannot be neglected in the analysis of the pile group response. In addition, the discrepancy between the front row and rear row piles is also shown to decrease with increasing pile spacing.

4.2. Influence of Lateral Clearance between the Front Row Piles and Tunneling

The influence of lateral clearance (C₁) between the front row piles and tunneling is a key factor for tunnel–pile group interaction. However, the variation in the group effect before and after tunneling and load redistribution among group piles have not been thoroughly investigated. To consider the close tunnel–pile group lateral distance, Cases 4, 5 and 6 with tunnels excavated below the pile tip (C₂ = −0.25 L) and lateral distances of 0, 1.5 and 2 times the tunnel radius are selected, and the predicted pile settlements are first given in Figure 10.

Figure 10 shows that with the tunnel center approaching the center of the pile group, the piles are subjected to increasing soil settlements, resulting in increasing tunneling-induced pile settlement. For cases with the same tunnel excavation depth, a tunnel excavated directly below the pile group can lead to the largest additional settlement.

Tunneling-induced axial forces for Cases 3 to 6 are given in Figure 11, and the influence of C₁ can be explored by considering the axial force distribution before tunneling for the pile group with 2.5 d pile spacing as the baseline. It can be observed that the pile head load is not significantly redistributed for the three cases, and the influence of lateral distance on pile head load redistribution is not monotonic. However, the maximum axial force along the pile shaft increases with decreasing lateral distance, meaning that the tunnel excavated directly below the pile group is also the most unfavorable working condition from the perspective of load redistribution.

The variation in the group effect is also examined for the three cases and shown in Figure 12. For Case 4 with almost identical loading conditions for the group piles, the Z multipliers for all the group piles decrease after tunneling. However, for Cases 5 and 6, the Z multipliers for the rear row piles increase after tunneling, while those for the rear row piles decrease, but the group effect should also be considered in pile group analysis due to the Z multipliers larger than 1.0.

4.3. The Influence of Vertical Distance between the Pile Tip and Tunneling

Cases in Section 4.1 and Section 4.2 include tunnels excavated both below and above the pile tip, and a significant discrepancy in load redistribution is found in excavation above the pile tip, while there is a subtle difference in settlement induced by tunneling. Thus, Case 7 is added and compared with Cases 2 and 5, and the corresponding results are given in Figure 13, Figure 14 and Figure 15.

Pile settlements before and after tunneling are shown in Figure 13, and it can be indicated that with the same lateral distance between the front-row piles and the tunnel center, the tunneling-induced pile head settlement is almost the same. However, load redistribution among group piles is shown to be largely affected by the vertical distance between the pile tip and the tunnel center. Figure 14 shows that with increasing tunnel excavation depth, the discrepancy in the pile head load significantly decreases from 60% of the pile head load before tunneling to less than 5%. The front-row piles are subjected to a larger pile head load when the tunnel is excavated above the pile tip, while they withstand a smaller load with excavation below the pile tip.

Variations in the group effect with different vertical distances are shown in Figure 15. The Z multipliers for the front row piles can be observed to decrease with deeper excavation, and the group effect for the rear row piles shows opposite changing patterns. For the three cases, the difference in the Z multipliers is minimal when the excavation depth is at the level of the pile tip due to the most likely loading condition in this case. From the perspective of the group effect, the most unfavorable condition may occur with tunneling above the pile tip.

5. Conclusions

The modified Poulos method for evaluation of active pile group response was extended in this paper to estimate the variation in group effect for an existing vertically loaded pile group due to tunnel excavation. The main refinement is taking into account the effect of tunneling-induced soil displacements when forming the pile-soil interaction function in prediction of the additional elastic pile head settlement. Therefore, this simplified method can evaluate the effects of tunneling on the pile group, where the tunneling-induced group settlement and load redistribution and the variation in the Z multipliers can be predicted, provided that the free-field soil settlements due to tunneling are given. The validity of the proposed method was demonstrated by an available centrifuge model test with a 2 × 2 fixed-head pile group. A parametric study was conducted and it shows that the influence of the lateral tunnel-pile group distance is significant on the pile group settlement but very subtle on load redistribution and the group effect, manifested by the Z multipliers. In the vertical direction, however, the tunneling-induced pile settlement, the Z multipliers and the extent of load redistribution can be largely decreased with deeper excavation depth.

Similar to an active vertically loaded pile group, both the pile group settlement and the Z multipliers can increase with closer pile spacing. However, load redistribution can occur due to large pile spacing, which may lead to unfavorable loading conditions when the tunnel is excavated at a shallow depth and close to the pile group.

Further studies using field measurements or centrifuge prototype may be useful to investigate the pile group effect evolution during tunneling, and can also be used to prove the feasibility of the proposed method with varying pile group configurations, softer ground soils and closer relative positions between the pile group and the tunnel.