A 3D sliced-soil–beam model for settlement prediction of tunnelling using the pipe roofing method in soft ground

Abstract

The pipe roofing method is widely used in tunnel construction because it can realize a flexible section shape and a large section area of the tunnel, especially under good ground conditions. However, the pipe roofing method has rarely been applied in soft ground, where the prediction and control of the ground settlement play important roles. This study proposes a sliced-soil–beam (SSB) model to predict the settlement of ground due to tunnelling using the pipe roofing method in soft ground. The model comprises a sliced-soil module based on the virtual work principle and a beam module based on structural mechanics. As part of this work, the Peck formula was modified for a square-section tunnel and adopted to construct a deformation mechanism of soft ground. The pipe roofing system was simplified to a three-dimensional Winkler beam to consider the interaction between the soil and pipe roofing. The model was verified in a case study conducted in Shanghai, China, in which it provided the efficient and accurate prediction of settlement. Finally, the parameters affecting the ground settlement were analyzed. It was clarified that the stiffness of the excavated soil and the steel support are the key factors in reducing ground settlement.

Graphical abstract

Keywords

pipe roofing method / soft ground / numerical simulation / settlement prediction / simplified calculation / parametric analysis

1 Introduction

The pipe roofing method is widely used in tunnel excavation. The method is considered an efficient way of controlling ground settlement induced at the surface [1,2] and restraining tunnel excavation deformation [3–5]. Owing these advantages, the pipe roofing method is often adopted in projects with complex geological conditions and high requirements for environmental deformation control [6–10]. Therefore, the pipe roofing method has great application potential in urban tunnel engineering, especially in soft ground. There are many simplified calculation methods for predicting ground settlement due to tunnel construction in soft ground. Several aspects of current prediction methods should be noted. 1) Many prediction methods focus on the modified Peck formula [11–15]. Cheng et al. [11] used the formula i = KZ₀ to evaluate the settlement trough generated by the shield excavation of the Beijing large-scale EPB and determined the width parameter K of the surface settlement trough. Fang et al. [12] proposed a simple settlement prediction method that considered various types of shield, ground conditions, and tunnel depth and diameter parameters. The advantages of the modified Peck formula are its simplicity and convenience of use. However, the correction coefficient of the modified Peck formula depends on the geological and construction conditions. 2) Most surface settlement prediction methods are suitable for tunnels with a circular section [16–19]. Fang et al. [16] completed a series of model tests in sandy soil, discussed the surface settlement according to the depth of a circular tunnel and ground volume loss and established an equation with which to predict the distribution of surface settlement. Mirhabibi and Soroush [17] established a numerical model using ABAQUS to study the surface settlement and ground building settlement caused by the excavation of circular twin tunnels and evaluated the effects of the tunnel depth, center distance and other factors on the settlement trough. Ocak [18] used field test data and modified the Peck formula to estimate the surface settlement caused by circular twin tunnels on the basis of considering the interaction of double tracks. However, these methods cannot be adopted directly for pipe roofing tunnels with square sections. Hence, researching the surface settlement due to square-section pipe roofing is of considerable importance. 3) In many methods, it is assumed that the surface settlement is due to the gap between the tunnel lining structure and the surrounding soil [20–25]. Fang et al. [21] studied the surface settlement deformation during the construction of fully overlapping tunnels (with a tunnel boring machine (TBM)) and improved the Peck formula to take into account changes in the depth of the tunnel axis and the stratum loss ratio. An et al. [24] used FLAC3D to build a numerical model of a TBM tunnel and studied the effects of the ground stiffness, face pressure, tail void grouting pressure and additional volume of muck discharge, on surface settlement. Using monitoring data of a Tianjin project site, Hou et al. [25] statistically fitted the recommended stratum loss ratio during tunnel excavation and modified the Peck formula. All the aforementioned studies focused on the surface settlement due to the pipe roofing and neglected the section shrinkage of the tunnel resulting from the deformation of the lining structure. This is a reasonable approach for TBM construction. However, the deformation of the pipe roofing is critical to the surface settlement and cannot be omitted. Hence, the validity of the aforementioned studies in the prediction of ground surface settlement for pipe roofing remains limited.

In summary, existing methods for predicting ground deformation cannot be directly applied to pipe roofing tunnels with square sections, and the impact of pipe roofing deformation on ground deformation has not been considered. In this study, Peck’s formula [26] was modified by introducing a correction coefficient of the settlement trough width for soft ground at a Shanghai location. A soil deformation mechanism of a tunnel construction with a square section based on the modified Peck formula was then established. Furthermore, the interaction between the soil and pipe roofing system was considered. Finally, a three-dimensional (3D) sliced-soil–beam (SSB) model for predicting surface settlement which incorporates the soil loss and the steel pipe deformation, when adopting the pipe roofing method, was constructed. Using the SSB model, parametric studies were conducted to investigate the factors of surface settlement.

2 Deformation mechanism of soil

2.1 Numerical simulations

In investigating the mechanism of soil deformation surrounding a tunnel constructed by the pipe roofing method, FLAC3D was used to establish a series of square-section tunnel models with various normalized tunnel depths. In these models, the shrinkage ratio was 0.5% of the tunnel section; this is a value typical for the pipe roofing method in soft ground.

In the numerical simulations, the plastic-hardening (PH) model was used to simulate the soft ground [27]. The PH model is a shear and volumetric hardening constitutive model for the simulation of soil behavior. Additionally, the PH model considers the attenuation of the shear modulus for small values of strain. PH model parameters of soft ground typical for Shanghai [28–35] were selected as given in Tab.1.

As shown in Fig.1, the models had a width of 100 m and different depths (28, 32, 36, 40, and 44 m). The width of the models was sufficiently large, > 6L, where L is the width of the square section of the tunnel.to avoid the boundary effect [36]. The height H and width L of the square section of the tunnel were both 8 m. The length of the models along the length of the tunnel was set to be 2 m. The central depth of the tunnel Z_0 was 8, 12, 16, 20, and 24 m in the models., and the normalized tunnel depths {\displaystyle \frac{Z_0}{H}} were 1, 1.5, 2, 2.5, 3, respectively.

The bottom of each model was fixed, whereas the left and right sides were constrained by rollers. The top surface of the model was left free.

The settlement trough width (i_{\mathit{\text{z}}}), which is the horizontal distance from the tunnel centerline to the point of inflexion on the surface settlement trough, is an important parameter for settlement prediction:

(1)i_{\mathit{\text{z}}}=K_{\mathit{\text{z}}}{Z}_0,

where

(2)K_{\mathit{\text{z}}}=K_{\mathit{\text{z}}=0}{\left(1-\frac{\mathit{\text{z}}}{Z_{\mathrm{m}}}\right)}^{\alpha },

where Z_{\mathrm{m}} is the maximum depth of the mechanism and \alpha is a constant.

In the conventional Peck formula for a circular-section tunnel, the surface settlement trough coefficient is a constant K_{\mathit{\text{z}}=0}=0.5 [37,38]. Cheng et al. [11], Fang et al. [21], and Fattah et al. [13] have previously conducted much work on the value of K_{\mathit{\text{z}}} under different geological conditions.

This paper discusses the method of determining the parameter K_{\mathit{\text{z}}=0} in Peck’s formula. Fig.1 shows that there is a linear relationship between K_{\mathit{\text{z}}=0} and the normalized tunnel depth {\displaystyle \frac{Z_0}{H}}. The regression equation is

(3)K_{\mathit{\text{z}}=0}=-0.028\frac{Z_0}{H}+1.042.

The parameter K_{\mathit{\text{z}}} in the Peck’s formula for a square-section tunnel is modified as

(4)K_{\mathit{\text{z}}}=\left(-0.028\frac{Z_0}{H}+1.042\right){\left(1-\frac{\mathit{\text{z}}}{Z_{\mathrm{m}}}\right)}^{\alpha }.

2.2 Displacement field

Mair [39] proposed the deformation field of a circular-section tunnel \left({\displaystyle \frac{Z_0}{D}}=2.17\right) obtained in centrifuge tests whereas Fig.2(b) presents the deformation field for a square-section tunnel \left({\displaystyle \frac{Z_0}{H}}=2.00\right) obtained by numerical simulations in this study. It is seen that for the tunnels with similar normalized depths ({\displaystyle \frac{Z_0}{H}} or {\displaystyle \frac{Z_0}{D}}), the patterns of the soil deformation field were similar even though the tunnel section shapes were different. This implies that the same family of field functions (i.e., the same displacement mechanism) may be applied to approximately describe the soil deformation surrounding tunnels having different section shapes, of course using different K_{\mathit{\text{z}}}, as discussed in Subsection 2.1.

In this study, on the basis of the similarity of the soil mechanism as mentioned above, a two-step method of determining the displacement field of the soil surrounding a square-section tunnel was proposed.

The first step was an equivalency step. A displacement field generated by a circular-section tunnel was determined. This circular-section tunnel had the same section area and depth as the square-section tunnel. As shown in Fig.5, the displacement field proposed by Osman et al. [40,41] was used. We have:

(5)v=\frac{A{v}_{\mathrm{m}}}{2K_{\mathit{\text{z}}}}\left\{\exp \left[-\frac{1}{2}{\left(\frac{x}{i_{\mathit{\text{z}}}}\right)}^2\right]-\exp \left(-\frac{B^2}{2}\right)\right\},

(6)u=-\frac{\alpha A{v}_{\mathrm{m}}x}{2K_{\mathit{\text{z}}}\left(Z_{\mathrm{m}}-\mathit{\text{z}}\right)}\left\{\exp \left[-\frac{1}{2}{\left(\frac{x}{i_{\mathit{\text{z}}}}\right)}^2\right]-\exp \left(-\frac{B^2}{2}\right)\right\},

(7)v_{\text{m}} =\frac{V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{s}}}{\sqrt{2\text{π}}{i}_{\mathit{\text{z}}}}\left(\frac{\text{π}{D}_{\mathrm{t}}^2}{4}\right),

where A and B are parameters that control the shape of the deformation mechanism, v_{\mathrm{m}} is the maximum ground settlement, V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{s}} is the soil volume loss. The equivalent tunnel diameter D_{\mathrm{t}} is assumed according to Fig.3:

(8)\text{π} \frac{D_{\mathrm{t}}^2}{4}= HL,

where H is the height of the square-section tunnel and L is the width of the square-section tunnel as shown in Fig.3.

Using Eqs. (5) and (6), the vertical and horizontal displacements v and u can be determined at any point (x, z). More details can be found in Osman’s studies [40,41].

The second step was a modification step. The displacement field obtained in the first step should be modified to reflect the effect of the square section. On the basis of the numerical simulations in Subsection 2.1, the modified settlement trough width for the shallow buried ({\displaystyle \frac{Z_0}{H}}\le 3) tunnel with a square section is:

(9)i_{\mathit{\text{z}}}=\left(-0.028\frac{Z_0}{H}+1.042\right){\left(1-\frac{\mathit{\text{z}}}{Z_{\mathrm{m}}}\right)}^{\alpha }{Z}_0.

The settlement trough width (i_{\mathit{\text{z}}}) in Eqs. (5) and (6) are replaced by the expression in Eq. (9) to get the final displacement field.

3 Pipe roofing system

3.1 Simplified pipe roofing model

The bending stiffness between the steel pipes may be neglected, as illustrated in Fig.4, because the steel pipes are linked by a CT joint rather than welding to create pipe roofing. Therefore, from the cross-section of the tunnel, the pipe roofing can be regarded as a series of separated beams.

As shown in Fig.5, when the pipe roofing in soft ground is subjected to earth pressure, a 3D Winkler beam model can be used to analyze the deformation of each steel pipe separately. The earth pressure acting on the pipe roofing is assumed to be uniform in the transverse direction. Fig.5 shows the distribution of earth pressure along the tunnel’s longitudinal direction. For the excavation part, the earth pressure is a constant supporting pressure, which is determined by the interaction of the pipe roofing system and soil. For the un-excavation part, the earth pressure decreases from the constant supporting pressure at the excavation face to zero at a distance 6H from the excavation face. The distance 6H is assumed to be influencing range of the excavation.

During the excavation of the pipe roofing tunnel, the two ends of the pipes are inserted into the diaphragm walls of the launching shaft and arriving shaft. Therefore, both ends of the Winkler beam are fixed.

3.2 Excavation procedure

In practice, to ensure the stability of the tunnel excavation face in soft ground and to control the deformation resulting from excavation, the soil inside the pipe roofing is generally improved to have higher strength and stiffness. As a result, the foundation reaction coefficient (k_{\mathrm{s}}) of the excavated soil in the pipe roofing increases.

As shown in Fig.5, the excavated soil and the pipes are divided into n sections to simulate staged excavation. When the soil is excavated to node i of the beam, i-1 nodes of the beam have been supported by an inner steel support, which is simplified as a spring with stiffness k_{\mathrm{t}}. The unexcavated n-i nodes of the beam are supported by the improved soil with a foundation coefficient stiffness of k_{\mathrm{s}}. At the excavation face (i.e., node i of the beam), half of the improved soil is excavated and it is thus considered that the foundation reaction coefficient k_{\mathrm{s}} of the soil is {\displaystyle \frac{k_{\mathrm{s}}}{2}}.

The excavation procedure is simulated by changing the spring stiffness of each node from k_{\mathrm{s}} to k_{\mathrm{t}}. At the same time, the earth pressure applied to the beam should be changed correspondingly. The pressure acting on the excavated and unexcavated parts of the beam was stated in Subsection 3.1.

3.3 Principle of virtual work

On the basis of the principle of virtual work, the work done by the supporting pressure, the increased internal energy due to shearing, and with the work done by gravity should satisfy the equation

(10)P_{\mathrm{p}\mathrm{s}}{V}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{s}} +{\int }_A\tau {\gamma }_{\mathrm{s}}\mathrm{d}A ={\int }_A\rho v \mathrm{d}A,

where τ is the shear strength, v_{\mathit{\text{z}}=0} is the surface settlement, and \rho is the unit weight of soil, P_ps is the equivalent supporting pressure from the pipe roofing, γ_s is the shear strain. In the “work done by supporting pressure” part, the work done by the supporting pressure from the pipe roofing is P_{\mathrm{p}\mathrm{s}}{V}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{s}}.

In the “increased internal energy due to shearing” contribution, because the soft ground is usually regarded as being in an undrained state during tunnelling, an increase in the soil internal energy can only result from the development of shear stress (τ). The increased internal energy of the deformed soil ({\int }_A\tau {\gamma }_{\mathrm{s}}\mathrm{d}A) may then be calculated through integration of the soil strain–stress curve (see Subsection 5.2 for the specific calculation process).

In the “work done by gravity” contribution, {\int }_A\rho v\mathrm{d}A can be calculated using Eq. (5).

Such methodology was first proposed by Osman et al. [40,41] and known as mobilizable strength design.

The SSB model can be used to predict the soil deformation field (v) according to Eq. (5) and then calculate the equivalent supporting pressure from the pipe roofing (P_{\mathrm{p}\mathrm{s}}) according to Eq. (10).

3.4 Deformation of the pipe roofing system

The pipe roofing system may be regarded as a sequence of 3D beams according to the foregoing discussion on the simplification of the pipe roofing system and tunnel excavation process.

The displacement of each beam \{w_i\} is solved using the equation:

(11)\left[A_{\mathrm{v}}\right] \left\{w_i\right\}+d_i\left[k\right] \left\{w_i\right\}=\left\{q_i\right\},

where \left[A_{\mathrm{v}}\right] is the general stiffness matrix of the beam, d_i is the pipe diameter, [k] is the foundation stiffness matrix, which is a combination of k_{\mathrm{t}} and k_{\mathrm{s}}, and \left\{q_i\right\} is the force acting on node i, which depends on the earth pressure P_{\mathrm{s}\mathrm{p}} and the position (excavated part or unexcavated part). The earth pressure acting on the pipe roofing (P_{\mathrm{s}\mathrm{p}}) equals the supporting pressure from the pipe roofing (P_{\mathrm{p}\mathrm{s}}) discussed in Subsection 3.3.

Finally, the tunnel section shrinkage (V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{p}}) due to the displacement of the steel pipes (w_i) under earth pressure acting on the pipe roofing (P_{\mathrm{s}\mathrm{p}}) is calculated as

(12)V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{p}}=\frac{H\sum _m{\left\{w_i\right\}}_m}{{\displaystyle \frac{\text{π}{D}_{\mathrm{t}}^2}{4}}},

where m is the number of pipes on the top, left and right sides of the pipe roofing. As shown in Fig.4, the surface settlement is mainly induced by deformation of the top and sides of the pipe roofing. Hence, the deformation of the pipe roofing bottom is omitted.

In this section, the deformation of the pipe roofing \{w_i\} is calculated using Eq. (11) and the tunnel shrinkage (V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{p}}) is then obtained using Eq. (12). Subsequently, V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{p}} and V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{s}} are equalized to begin the iterative procedure.

4 Framework of the sliced-soil–beam model

Based on the theoretical derivation mentioned earlier, Fig.6 lists the calculation process of the SSB model. The SSB model comprises two parts, namely the sliced-soil model (i.e., the red part on the left-hand side) for soil deformation surrounding the tunnel in the transverse direction and the beam model (i.e., the blue part on the right-hand side) for the pipe roofing deformation in the longitudinal direction.

In the sliced-soil model, once the soil volume loss (V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{s}}) is obtained, the soil calculation module predicts the soil deformation field using a modified Peck’s formula and then calculates the equivalent supporting pressure from the pipe roofing (P_{\mathrm{p}\mathrm{s}}) using the principle of virtual work.

In the beam model, after obtaining the earth pressure acting on the pipe roofing (P_{\mathrm{s}\mathrm{p}}), the deformation of the pipe roofing is calculated using the 3D Winkler beam model and the tunnel shrinkage (V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{p}}) is then obtained.

Under the fully undrained condition, it is obvious that the soil volume loss (V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{s}}) is equal to the tunnel shrinkage (V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{p}}) owing to soil–pipe roofing deformation compatibility, whereas the supporting pressure from the pipe roofing (P_{\mathrm{p}\mathrm{s}}) equals the earth pressure acting on the pipe roofing, also written as (P_{\mathrm{s}\mathrm{p}}). Therefore, the sliced-soil module and the beam module are connected by the soil–pipe roofing interaction and form an iteration loop, through which the mentioned deformations and pressures of both the soil and pipe roofing are determined.

Fig.7 is a flow chart showing the following four main steps.

1) Input of the trial soil volume loss (V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{s}}), tunnel height (width), tunnel depth and soil mechanical parameters into the soil calculation module to obtain P_{\mathrm{p}\mathrm{s}}=P_{\mathrm{s}\mathrm{p}}.

2) Input of P_{\mathrm{s}\mathrm{p}}, the mechanical parameters of the pipe roofing, and of improved soil and inner supports into the structural calculation module to obtain the tunnel section shrinkage (V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{p}}).

3) Calculation of the error {\displaystyle \frac{\left|V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{s}}-V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{p}}\right|}{V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{s}}}} ; if the value is greater than 1%, then V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{s}} =V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{p}} can be set and the above steps can be repeated until the error is less than 1%.

4) Input of the final V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{s}} into the soil calculation module to obtain the surface settlement distribution.

The above calculations are programmed and calculated using MATLAB^®. Through this calculation, the surface settlement of any section can be calculated. Then, by accumulating the settlements of each slice of soil along the longitudinal direction, the 3D settlement of the ground surface is obtained.

5 Verification

This section considers a tunnel project [42], constructed using the pipe roofing method in Shanghai, to compare the field measurements of the ground settlement with the prediction results and verify the SSB model.

5.1 Introduction to the Shanghai project

The project adopts the pipe roofing method to excavate a shallow tunnel with a rectangular section in soft ground in Shanghai. The excavation of the project can be regarded as being performed under a fully undrained condition. The dimensions of the tunnel section are 22.8 m × 7.6 m, the total length of the tunnel is 100 m, and the buried depth C is 5.4 m. Fig.8 shows the geological profile and the sectional view of the tunnel.

As shown in Fig.8, the excavated tunnel is divided into three cells of the same size. The pipe roofing is made using two kinds of steel pipe. Thick steel pipes are used at the bottom and both sides of the tunnel. The parameters of the steel pipes are given in Tab.2.

Tab.3 gives the mechanical parameters of the soils.

The Metro Jet System (MJS) [43–47] method is an all-around ground reinforcing and consolidating method used for ground improvement. The MJS method offers advantages such as active pressure adjustment and grout injection in any direction. The jet grouting process has an impact on the ground disturbances. It has been shown that the generated spoil during the horizontal jet grouting process is effective in mitigate ground disturbance [48].

Two types of soil improvement, namely latticed soil improvement and layered soil improvement, are adopted to avoid the instability of the excavation face and to reduce the surface settlement. As shown in Fig.9, the latticed soil improvement results in grouting reinforcement horizontally and vertically whereas the layered soil improvement results in grouting reinforcement horizontally. The latticed improvement is of high stiffness and the layered improvement is of low stiffness.

For the first 30 m of excavation, only latticed soil improvement is adopted. For the other 70 m of excavation, the latticed soil improvement and layered soil improvement are staggered at intervals of 1.5 m to reduce costs.

5.2 Calculation using the sliced-soil–beam model

There is no need to consider the celled excavation because it is only necessary to predict the ground surface settlement once the tunnel excavation is complete. The tunnel excavation surface is simplified into three equal parts A, B, and C. The final settlement distribution of the surface is the superposition of each settlement due to the independent excavation of the three cells A, B, and C, as shown in Fig.10.

The stress–strain curves for the clay in this project are expressed uniformly [49,50] by:

(13)\frac{\tau }{c_{\mathrm{u}}}= {\left(\frac{{\gamma }_{\mathrm{s}}}{{\gamma }_{\mathrm{s},\mathrm{f}}}\right)}^{\beta } ,

where {\gamma }_{\mathrm{s},\mathrm{f}} and \beta are material constants for different shear states. The limiting strength is c_{\mathrm{u}}, which is discussed later, and the corresponding strain is {\gamma }_{\mathrm{s},\mathrm{f}}. For \beta = 1, a perfectly elasto-plastic model was used to describe the shear stress–strain curve of the soil.

Vane tests are carried out on the undisturbed soil layer at the project site to obtain the undrained shear strength of the soft ground at different depths. The test results are shown in Fig.11. In this study, a fitting line is used to represent the undrained shear strength of the soft ground at different burial depths in the project:

(14)c_{\mathrm{u}}=3.153 \mathit{\text{z}}.

The foundation reaction coefficients of the latticed and layered improvement zones were 90 and 50 MPa/m, respectively. The linear stiffness of the inner support is 500 MPa/m. The mechanical parameters of the SSB model are summarized in Tab.4.

Fig.12 shows the error versus the number of cycles. The error decreased from 1597.4% to 0.46% in only five cycles. The SSB model thus has a high convergence speed. After iteration, the final soil volume loss V_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}^{\mathrm{s}} was 0.48%, which is a typical value when applying the pipe roofing method in soft ground.

As shown in Fig.13, another powerful function of the SSB model is that the model can be used to obtain the 3D surface settlement distribution when the tunnel is in the process of being excavated, to any stage. Fig.13(a)–Fig.13(c) show the 3D surface settlements when the tunnel is excavated to 25, 50, and 75 m respectively. As the tunnel is continuously excavated, the range of ground settlement expands and the maximum settlement increases from 12.55 to 26.30 mm. It is observed that the maximum surface settlement point is near to above the excavation face when the excavation distances are 25 and 50 m and located in the middle of the tunnel, rather than at the excavation surface when the excavation distance is 75 m. Further details of the surface settlement are discussed in Subsection 5.3.

5.3 Comparison of measurements and calculations

Fig.14(a) shows the final measured and predicted longitudinal settlements of the pipe roofing, whereas Fig.14(b) presents the final measured and predicted longitudinal settlement distributions of the ground surface. Notably, the measured and predicted results are in good agreement. Additionally, the measured settlements of Cells A, B, and C are similar to one another. It is reasonable that the pipe roofing was simplified as a series of independent beams because the settlement of Cell C would have been greater than those of Cell A or Cell B if there had been transverse bending stiffness between the pipes. Moreover, there is a notable increase in settlement after 30 m. This is because in the first 30 m of the excavation range, the soil underwent latticed improvement and had a high stiffness, whereas the soil in last 70 m underwent layered improvement and had a relatively low stiffness.

Fig.15 presents the final measured and predicted transverse settlement distributions of the ground surface at 40 and 50 m from the starting point of the tunnel. Again, the prediction is consistent with the measurement. Therefore, the SSB model was verified in that both the longitudinal and transverse results match the measurements well. Additionally, it is observed that the measured settlement of the surface outside the tunnel is much smaller than that beyond the tunnel at 40 m, whereas the measured settlements of the ground above Cells A and B are similar to each other. The assumptions of the SSB model are thus reasonable.

The trend of the calculated displacement along the longitudinal and transverse direction is consistent with the measured values, which indicates that the theoretical model and the estimated method presented in this study exhibits satisfactory performance.

6 Parametric studies on factors affecting surface settlement

As shown in Fig.2, the calculation process shows that the ground settlement is derived from the settlement of the tunnel pipe roofing. Furthermore, Fig.7 reveals that the settlement of the pipe roofing is affected by the stiffness of the inner supports and the improved soil inside the pipe roofing. Therefore, the inner supports and the improved soil are the two key factors controlling the surface settlement for a pipe roofing system.

6.1 Stiffness of the improved soil

The geometric and mechanical parameters in the Shanghai project were used to conduct parametric studies on the stiffness of the improved soil. Cases with different foundation reaction coefficients of the improved soil were simulated as listed in Tab.5.

Fig.16 shows that the settlement of the pipe roof and that of the ground surface decrease with increasing stiffness of the improved soil. However, the efficiency of the settlement reduction becomes low when the foundation reaction coefficient exceeds 90 MPa/m, which was the actual value for the improved soil in the project, as shown in Fig.17. The stiffness of the improved soil was 2–3 times that of the original soil. Therefore, this is a recommended stiffness for the improvement of soil, in terms of not only efficiently reducing the settlement but also controlling the cost of improvement.

6.2 Stiffness of the inner supports

A series of cases with different stiffness of the inner supports is investigated. The corresponding parameters are given in Tab.6.

Fig.18 shows that the settlement of the pipe roof and that of the ground surface decrease with increasing stiffness of the improved soil. The pipe roofing deformation and surface settlement are not sensitive to the increasing stiffness of the inner supports, as Fig.19 shows. The maximum settlement variation is less than 10 mm when the stiffness of the inner supports is increased by a factor of 4 (from 200 to 800 MPa/m). The reason for the small variation may be that the stiffness of the inner supports was already high and there was thus only a marginal gain in increasing the stiffness. The inner supports applied in the Shanghai project can thus be optimized to a lower stiffness of 200 MPa/m.

7 Conclusions

An SSB model was proposed to predict the settlement of ground due to tunnelling using the pipe roofing method in soft ground. The model comprises a sliced-soil module based on the virtual work principle and a beam module based on structural mechanics. A deformation mechanism of soft ground around the square-section tunnel was constructed using an equivalence method. The pipe roofing system was simplified as a 3D Winkler beam to consider the interaction between the soil and pipe roofing. The following conclusions are drawn from the results of the study.

1) The SSB model accurately predicts the 3D ground settlement and pipe roofing deformation due to the excavation of a shallow buried rectangular pipe roofing tunnel in soft ground.

2) The settlement of the pipe roof and that of the ground surface decrease with increasing stiffness of the improved soil. However, as long as the foundation reaction coefficient exceeds 90 MPa/m, the efficiency of settlement reduction decreases. It is recommended that the stiffness of the improved soil is 2–3 times that of the original soil.

3) The pipe roofing deformation and surface settlement have low sensitivity to an increase in the stiffness of the inner supports. The suggested stiffness of the inner supports is 200 MPa/m.

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