Study on the Generalized Displacement Boundary and Its Analytical Prediction for Ground Movements Induced by Shield Tunneling

The process of shield tunnel excavation would inevitably cause surrounding ground movement, and excessive displacement in the soil could lead to large deformation and even collapse of the tunnel. The methods estimating convergence deformation around tunnel opening is summarized. Then, a universal pattern of displacement boundary condition around the tunnel cavity is originally introduced, which is solved as the combination of three fundamental deformation modes, namely, uniform convergence, vertical translation, and ovalization. The expression for the above-mentioned displacement boundary condition is derived, by imposing which the analytical solution for ground movements, based on the stress function method, is proposed. The reliability and applicability of this proposed solution are veriﬁed by comparing the observed data in terms of surface settlement, underground settlement, and horizontal displacement. Further parametric analyses indicate the following: (1) the maximum settlement increases linearly with the gap parameter and the tunnel radius, while it is negatively related to the tunnel depth; (2) the trough width parameter is independent of the gap parameter and the radius, while it is proportional to the tunnel depth. This study provides a new simple and reliable method for predicting ground movements induced by shield tunneling.


Introduction
Over the last decade, an increasing urbanization has led to the over-crowded ground transportation issue around the world [1][2][3]. In order to alleviate the surface traffic congestion, underground subway construction has gained rapid development, in which shield tunneling method is widely applied due to its advantages (e.g., automation, all-weather, and widespread applicability) [4][5][6]. However, the process of shield tunnel excavation would inevitably disturb the original stress state of surrounding soil and cause its horizontal displacement and subsidence, which could even damage neighboring surface structures [7][8][9]. erefore, it is of great significance to predict ground movements induced by tunneling, especially for the cases of tunneling under intense buildings, in order to assess the potential correlated damage of surface structures [1,[10][11][12]. e development of prediction methods to calculate tunnel-induced ground movement is among the hottest topics in tunnel engineering [13]. A large number of research efforts have been paid with respect to methods for predicting tunneling-induced ground movements. Generally, they can be classified into three categories: empirical [14][15][16][17], numerical [18][19][20][21][22], and analytical [23][24][25][26][27][28][29][30]. Empirical formulas are principally obtained based on field observations and intuitive deductions, such as the Peck formula [17], which is able to accurately reflect transverse settlement profile. However, these empirical methods are usually lacking rigorous theoretical derivations and thus result in very limited applicability. On the other hand, numerical approach has been greatly used for predicting ground movements, because it can accurately simulate tunnel excavation processes under various geological conditions. However, the numerical analyses often produce unsatisfied results, due to insufficient information of in situ conditions and unrealistic modeling of soil behavior. In addition, advanced computing equipment is required and a large amount of computation time is needed for such analysis. For the analytical method, it adopts assumptions concerning the calculation model and constitutive behavior of soil, satisfying principles of mechanics and rigorous mathematical deduction. Furthermore, extensive parametric analysis is much easier to be carried out using the analytical method.
Benefiting the aforementioned advantages, the analytical method has then become the primary approach to compute tunneling-induced ground movements. Generally, this method is conducted by developing mathematical equations satisfying soil behavior and solving them based on special boundary conditions. e analytical methods for predicting tunnel-induced ground movements cover four main categories: the virtual image technique [23][24][25]31], the complex variable method [26,[32][33][34], the stochastic medium theory [27,35,36], and the stress function method [28-30, 37, 38]. Among them, the stress function method may develop solutions lacking physical meaning, for a function ln r existing, which does not meet the common condition of U r(θ) | r(θ) ⟶ ∞ ⟶ 0, where U r(θ) refers to the ground movement in the radial or hoop direction. And thus mathematical correction must be taken, such as the boundary of zero vertical displacement introduced by Chou and Bobet [38] and modifying the expressions by using − U θ instead of U θ recommended by Park [28], which leads to considerably smaller settlement far away from the excavation face, larger ground uplift, and narrower settlement trough than the observed results. e boundary condition is imposed and then the closed form solution is developed. Two types of boundary conditions are classified: the far end and the near end (namely, around the tunnel opening). For the former case, the boundary condition far from the tunnel opening is physically based on the initial stress state. For the latter case, the boundary of the tunnel opening can be divided into two kinds: the stress boundary and the displacement boundary. ese two boundary conditions are developed based on ground-liner interaction mechanism and convergence deformation pattern around the tunnel section, respectively [39]. Usually the radial stress at the tunnel opening is assumed to be zero, while the stress at the ground-liner interface can be determined in case of the consideration of ground-liner interaction. However, the displacement boundary is ambiguous because of the varied convergence deformation patterns under different geological conditions, and construction techniques [40].
In order to describe the deformation at tunnel cavity, Sagaseta [23] suggested a uniform radial convergence pattern (Figure 1(a)); this is, only uniform radial ground movement is considered. However, the practical deformation around the tunnel opening is highly non-uniform due to soil pressure and construction quality; and it is noted that the simplified pattern results in wider settlement trough and larger horizontal displacement than the observed deformation shape. Verruijt and Booker [40] extended the model suggested by Sagaseta [23] and proposed an oval-shaped pattern, as shown in Figure 1(b). But this mode may lead to a wider settlement trough and smaller settlements. On the other hand, Rowe and Kack [42] stated that the radial ground movement around the tunnel opening is not uniform (Figure 1(c)), because the gap (tail void) around the tunnel is non-circular. Loganathan and Poulos [25] took into account this nonuniform radial movement by introducing an equivalent ground-loss parameter, which can be estimated with respect to the gap parameter proposed by Lee et al. [43]. is deformation pattern may calculate a smaller maximum settlement in some cases because the deformation of liner is neglected.
Further researches and numerous practices pointed out that the deformation at the tunnel cavity can be considered as the sum of three above-mentioned modes [40]. Based on this, Park [28] presented four boundary conditions of the empirically prescribed displacement around the tunnel. Tong et al. [30] and Zhang et al. [29] further summarized the deformation pattern around the tunnel and separately proposed a kind of displacement boundary condition by contrast analysis with three empirically appointed displacements at the tunnel opening. To sum up, as for analytical prediction for tunneling-induced ground movement, limitations still exist for both the stress function method and displacement boundary condition, and further researches are still needed to be conducted.
Looking at these challenges, this paper originally introduces a universal pattern of displacement boundary condition around the tunnel cavity (defined as the generalized displacement boundary herein) based on the three fundamental deformation modes. An optimal plane analysis model is chosen to avoid mathematical correction for the results.
en, an analytical method to compute ground movements induced by shield tunneling is proposed based on the stress function method by imposing the generalized displacement boundary condition. e analytical method is validated against a set of 20 field cases with monitoring data. e method associated with engineering properties is finally introduced to estimate the characteristic parameters of the generalized displacement boundary.

The Generalized Displacement Boundary
2.1. Basic Assumptions. Generally, the time is short between shield excavation and liner installation, so is the time for compatible deformation between soil and liner. At this point, the pore water pressure is slow during the dissipation process [29]. erefore, time-dependent behavior and drainage conditions are ignored in this study. To simplify the calculation, the following assumptions are adopted: (1) e soils are assumed to be ideally elastic materials, and the tunneling-induced movements are considered as plane strain problem (2) e tunnel opening is perfectly a circle, and the movement focus of soils is exactly the tunnel center (3) e soil is in close contact with the liner, and the distortion of liner causes no ground loss, and the thickness of liner is assumed to be zero 2 Advances in Civil Engineering

Convergence Deformation Patterns around Tunnel
Opening. e ground movements around the tunnel opening are solved as the combination of three fundamental deformation modes, as shown in Figure 2, with input parameters, u 0 , u v , and u e , corresponding to uniform convergence, vertical translation, and ovalization, respectively. Among them, the components u v and u e reflect the nonuniform convergence deformation.

Derivation of Expression for Generalized Displacement Boundary Condition.
e generalized displacement boundary refers to the displacement of soils along the circumference in the radial direction. e analysis model of generalized displacement boundary is illustrated in Figure 3, where rectangular coordinate and polar coordinate with the same origin are established, respectively. Based on the basic assumption and convergence deformation pattern above, the radial displacement U 0 along the circumference using polar coordinate can be expressed as where R and u 0 are tunnel radius and uniform convergence, respectively, and ρ(θ) refers to the transverse curve of liner, which can be expressed by an elliptic equation: where a � R + u e , b � R − u e , and c � u v . While in rectangular coordinate, the curve of liner can be obtained as By substituting equation (3) into equation (1), then the expression for generalized displacement boundary condition is obtained.

Analytical Solution for Ground Movements
Based on Generalized Displacement Boundary

Analytical Model and Boundary
Conditions. e initial stress state of soil depends on the analysis model of tunnel excavation, which covers three types, as shown in Figure 4. e major difference between the three models is the stress state far from the tunnel opening. e first model (Figure 4(a)) assumes that uniform pressure is exerted around the plane edge [44,45], which conforms with the practical condition, while the lateral pressure coefficient is considered in the second case [28], as shown in Figure 4(b), which is relatively consistent with the practical condition. e third further introduces the tunnel depth and radius [29,30], as shown in Figure 4(c), which can better estimate the soil stress of shallow buried tunnel, but otherwise may lead to complicated analytical solution or even violated physical laws in some cases. erefore, the second analytical model is chosen in this study.
e Airy stress function can be assessed by dividing the applied stresses into isotropic and deviatoric parts [28][29][30]: , and σ v and σ h are the vertical and horizontal stress, respectively. It is noted that the axial stress is ignored in the above definitions for deviator and mean effective stresses. Sheng et al. [46] stated that the errors due to the simplified definitions are negligible. Based on the relationship between Airy stress function and stresses [47], the stresses in the initial field can be derived as Excavation contour Deformed contour

Excavation contour Deformed contour
Excavation contour Deformed contour     Advances in Civil Engineering e boundary conditions far from and around the tunnel opening can be obtained, respectively, as where equation (7) is the generalized displacement boundary condition originally introduced in this paper.

Analytical Prediction for Ground
Movement. Using the coordinate system shown in Figure 3, Park [28] simplified the Airy stress function introduced by Timoshenko and Goodier [48] and obtained a general solution for the tunnel excavation problem. Substituting equations (6) and (7) into the general solution presented by Park [28], the analytical formula, which is able to compute tunneling-induced ground movements basing on generalized displacement boundary condition, can be obtained as Further, the ground surface and subsurface settlements U z and the lateral deformation U x can be estimated as follows: where r � ������ x 2 + y 2 , sin θ � (y/r), and cos θ � (x/r). Compared with available analytical solutions (Park [28]; Tong et al. [30]; Zhang et al. [29]), due to inclusion of the term ln r, equations (8) and (9) need no mathematical corrections; consequently, it eliminates the resultant errors.

Estimation of the Characteristic Parameter of the Generalized Displacement Boundary
As mentioned above, the ground movements around the tunnel opening are considered as a combination of three basic modes, namely, uniform convergence u 0 , vertical translation u v , and ovalization u e . erefore, in order to accurately assess the analytical solution presented in Section 3, the characteristic parameter (u 0 , u v , u e ) of these three modes must be first estimated. Unfortunately, rather few data on vertical translation and ovalization of tunnel are currently available [39], which mostly are estimated empirically [31,49] or by contrast analysis with several groups of prescribed displacement [28][29][30]. So these parameters are subjective and thus have limited applicability. In fact, the characteristic parameters u v and u e are related to numerous factors, such as geological conditions and construction technique [40]. A slight change in engineering conditions may produce significant effects in the degree of u v and u e .

Uniform Radial Convergence Parameter
e uniform radial convergence u 0 can be estimated with respect to the volume loss as shown in where R is tunnel radius and V l is volume loss, which can be estimated based on the gap parameter introduced by Lee et al. [43].

Nonuniform Convergence Parameter
In order to associate the nonuniform convergence parameter with engineering conditions and facilitate engineering application, u v and u e are expressed as multiples of the uniform radial convergence parameter u 0 in this study as shown in where m ∈ [0, 1], n ∈ [0, ∞). m � 0 and n � 0 refer to only uniform radial convergence, as shown in Figure 1(a), while m � 0, n ≠ 0 indicate that only ovalization occurs, as shown in Figure 1(b); m ≠ 0 and n � 0 suggest convergence patterns as shown in Figure 1(c), with only vertical translation. Table 1 presents the engineering information of the 20 case studies previously investigated by various authors. e optimal value of m and n in equation (11) can be obtained by applying the least square method to equation (9). Based on the observed data and the resulting solution, Figure 5 is prepared with the distance from center line of tunnel as the horizontal coordinate, and the surface settlement as the vertical coordinate, while the optimal m and n are plotted against the dimensionless h/R ratio in Figure 6. It can be seen that none of the parameters shows a tendency to vary with the relative depth h/R. Note that the optimal m are concentrated around 0, 0.5, and 1, which coincide well with the research findings reported by Bobet [37], Zhang et al. [29], Tong et al. [30], and Jiang and Zhao [49].

e Ovalization Coefficient n.
e effect of the tunnel ovalization can be further quantified by the dimensionless distortion coefficient with respect to u e and R as shown in Based on the 20 case studies listed in Table 1 and the corresponding optical m, Figure 7 shows the relationship between the dimensionless distortion coefficient and the volume loss. It can be seen that the relative ovalization coefficient increases overall linearly with the volume loss. erefore, in this study, the relationship between the Advances in Civil Engineering In the absence of measured tunnel ovalization, by combining equations (10)- (13), an empirical method for estimating the tunnel ovalization can be obtained as shown in the following: u e � 1.365 V l R + 0.0027R.
For many cases of tunneling in soft ground, the typical volume losses are generally around 1% under well controlled conditions [63], while they may be within 0.1-0.5 % by   Advances in Civil Engineering taking special measures [64]. Figure 7 indicates that a good agreement can be found from equation (13) when the volume loss is under 1%. erefore, the tunnel ovalization parameter can be rationally estimated by equation (14) in most cases.

e Vertical Translation Coefficient m.
e transverse settlement curve is characterized by the maximum settlement and the trough width parameter. In this study, taking the maximum settlement Z 0 where m � 0 as a reference, the maximum settlement Z m�i at m � i ∈ (0, 1] is defined as Similarly, based on C 0 where m � 0, the trough width parameter C m�i at m � i ∈ (0, 1] is defined as where the coefficients a i and b i are greater than zero. e greater the a i , the larger the maximum settlement; and the greater the b i , the wider the trough width parameter. Figure 6 shows that the coefficient m is concentrated around 0, 0.5, and 1. erefore i � 0.5 and i � 1 are selected as an example for further analysis in this section. Figures 8  and 9 show the relationship between the coefficients a i ,b i and the nonuniform convergence coefficient m, n, the gap parameter g, the tunnel depth h, and the tunnel radius R, respectively. e following findings are found: (1 ) e coefficient a i is independent of the gap parameter, the depth, and the radius, while it is positively and negatively related to the coefficients m and n, respectively. e coefficientn can be estimated within 3.27∼8.17 from equation (14) for most cases of tunneling where typical volume losses are within 0.1-1%. According to Figure 8(c), for most tunnel projects, the ratio of Z m�i and Z 0 falls in the quadrilateral cdef, with the maximum ratio of 1.23; that is, the maximum settlement is only 23% higher than the reference in the most unfavorable case.
(2) It is noted that the change of the gap parameter, the depth, and the radius produced negligible effects on the degree of the coefficient b i , where no more than 1% can be observed. erefore, it can be considered that the coefficient b i has no relation with the above parameter. Similarly, it is positively and negatively related to the coefficientsm and n. Figure 9(c) indicates that, for most tunnel projects, the ratio of C m�i and C 0 falls in the quadrilateral ghij, with the maximum ratio of 1.082; that is, the trough width parameter is only 8.2% higher than the reference in the most unfavorable case.
To sum up, the nonuniform convergence coefficient m is assumed to be 0 when estimating the tunneling-induced ground movement by the method presented in this paper. e final result can be obtained increasing by 0-23%, depending on construction conditions. For example, the increase by 0% may be adopted when taking special measures, while the increase by 23% may be adopted under poor grouting quality, where the result is relatively conservative so as to ensure the safety.

Comparisons with Field Data.
Tunnel excavation would inevitably cause vertical and horizontal ground movements. Tables 1-3 summarize the engineering information of previous various case studies concerning surface settlement, underground settlement, and horizontal displacement, respectively. Based on the case studies listed in Tables 1-3, we compare the analytical prediction with the same kind of other methods presented by [28][29][30] and then check the reliability and applicability of the proposed solutions in this paper. It should be noted that the above three other methods adopt the same deformation pattern as in this paper, but impose different displacement boundary condition, and their nonuniform convergence parameters are empirically prescribed.
(1) Surface settlement: comparison of analytical surface settlements and field observation is presented in Figure 10. Only some results of cases listed in Table 1   Advances in Civil Engineering are selected as examples to illustrate the reliability and applicability of the proposed solutions. From Figure 10, the following can be seen: (1) Generally, the surface settlements estimated by the analytical method in this paper agree well with the field observations. As for some cases, such as cases 18 and 19, the results increasing by 23% is preferable, for the segment liner to sink to the bottom of excavation opening due to not timely filling into the gap between tunnel opening and liner. (2) e methods proposed by Park [28], Tong et al. [30], and Zhang et al. [29] all have the risk of seriously overestimating the maximum settlement and underestimating the trough width, as shown in cases 18 and 19. e main reason may be that their displacement boundary condition is unable to fully reflect the convergence patterns    [59] Note: the value of gis obtained by inversion with respect to the maximum surface settlement.  (2) Underground settlement: comparison of analytical underground settlements and field observation is shown in Figure 11. To illustrate the applicability of the proposed method in this paper, results at different depths are compared. Figures 11(a)-11(d) present results at depths of 0.91 m, 2.9 m, 3.0 m, and 7.0 m, respectively. e following findings are found: (1) Compared with other method, the underground settlements calculated by the proposed method in this paper are more reasonable. Similarly, the results increasing by 23% is preferable for some cases as shown in Figures 11(b)-11(d).  smaller than the observed value. Comparatively speaking, Tong's method is also considerably reliable in the prediction of underground subsidence. It can be noted that Tong's method always predicted a larger value than Park's method and Zhang's method, because Tong's method doubled the value of characteristic parameters.
(3) Horizontal displacement: comparison of analytical horizontal displacements and field observation is shown in Figure 12. Unfortunately, an uncommon phenomenon appears where the horizontal displacement obtained by the proposed method is negative around tunnel depth; this is because the ovalization parameter obtained by the proposed empirical formula is so large that the soils at the excavation opening move back to the center of tunnel within a certain angle of the opening, thus squeezing the surrounding soils. In order to eliminate this uncommon phenomenon, it is suggested to ignore the influence of ovalization when predicting the horizontal displacement proposed in this paper. It is worth noting that this special approach has no theoretical support but is based on empirical analysis. Figure 13 presents comparison of analytical horizontal displacement and field observation when ovalization is neglected by the proposed method. e following can be seen: (1) e horizontal displacement calculated by the proposed method in this paper is reasonable. It can be noted that the position of maximum horizontal displacement obtained by the method in this paper is always close to the tunnel axis, which is quite consistent with the practical condition.  14 Advances in Civil Engineering

Parametric Analyses.
e primary parameters, including the tunnel geometry, the tunnel depth, and the gap parameter, have a significant influence on the ground settlements. e influence of those parameters on the ground settlements was investigated by parametric analyses in terms of the maximum settlement and the trough width parameter. e following basic parameters are set: the soil elastic modulus, E � 2.5 × 10 4 kPa, the soil Poisson's ratio, v � 0.5, the soil lateral pressure coefficient, k � 1, the calculated position of settlement,z � 0 m, namely, surface settlement. e analysis cases are summarized in Tables 4-6 . And the related results are shown in Figures 14-16 , respectively. e parametric analysis demonstrates the following: (1) e maximum settlement increases linearly with the gap parameter and the tunnel radius. is is because the ground movements are related to the deformation around the opening, and the larger the gap parameter and the larger the radius, the larger the displacement at the excavation surface, while the maximum settlement is negatively related to the tunnel depth. Due to the limited scope of stress unloading caused by tunnel excavation, the greater the depth, the less the impact on the surface settlement. It should be noted that when the depth is more than 20 m, the increase of this parameter slightly affects the maximum surface settlement. (2) e trough width parameter increases linearly with the gap parameter and the tunnel depth, while being negatively related to the tunnel radius. However, further analysis indicates that when the gap parameter increases by 2 times, the trough width parameter only increases by 1.35%, and when the radius increases by 1 time, the trough width parameter only increases by 0.47%. erefore, it can be considered that the trough width parameter is independent of the gap parameter and the radius. (3) To date, various expressions have been proposed for estimating the trough width parameteri. Generally, they could be divided into three categories: (1) i is regarded as a function of the friction angle of soil ϕ and tunnel depth H, namely, i � f(ϕ, H) [11]; (2) i is related to the tunnel radius R and tunnel depth H in the form of i � Ra(H/2R) n or i � a(bH + cR) [68]; (3) i depends only on the tunnel depth H with i � aH + b [69]. e parameters a, b, c, n are all pending constants fitted from field observation. According to the above findings, this paper considers that the third kind is more reasonable and suggests i � 0.66H + 0.45.  1  30  10  3  2  60  10  3  3  90  10  3  4  120  10  3  5  150  10  3  6 180 10 3  1  90  5  3  2  90  10  3  3  90  15  3  4  90  20  3  5  90  25  3  6 90 30 3 Table 6: Tunnel radius of six cases. Advances in Civil Engineering 15

Conclusion
In this study, the convergence deformation around tunnel opening is summarized, and then a universal pattern of displacement boundary condition around the tunnel cavity is originally introduced, which is solved as the combination of three fundamental deformation modes, namely, uniform convergence, vertical translation, and ovalization. e expression for the above-mentioned displacement boundary condition is derived and the stress boundary is obtained based on an optimal plane analysis model, by imposing which, the analytical solution for ground movements, based on the stress function method, is proposed.
(1) e analytical method is validated against a set of 20 field cases with monitoring data, based on which, the nonuniform convergence parameters of the generalized displacement boundary are summarized and then the empirical methods estimating the nonuniform convergence parameters are obtained as follows: ovalization parameter u e � 1.365V l R+ 0.0027R; vertical translation parameter u v � 0u 0 , and the final result can be obtained increasing by 0∼23%, depending on construction details. (2) e reliability and applicability of this proposed solution are verified by comparing the observed data in terms of surface settlement, underground settlement, and horizontal displacement. For some cases, the available predicted methods based on the stress method all have the risk of seriously overestimating the maximum settlement and underestimating the trough width. e difference in boundary conditions between Park's method and Zhang's method has little influence on predicting surface settlement but has obvious effect on the horizontal displacement. Among them, Tong's method always predicted a larger value. (3) Further parametric analyses indicated the following: (1) the maximum settlement increases linearly with the gap parameter and the tunnel radius, while it is negatively related to the tunnel depth and it should be noted that when the depth is more than 20 m, the increase of this parameter slightly affects the maximum surface settlement. (2) e trough width parameter is independent of the gap parameter and the radius, while it is proportional to the tunnel depth and it suggests i � 0.66H + 0.45.

Data Availability
All data are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.