In pipe jacking engineering, accurate prediction of jacking force is the key to pipe jacking design. Based on a project of the Beijing Daxing Airport Line, the influence of the advance jacked pipes on the jacking force of the subsequent pipe is carried out in the present work. First, the verified numerical model of practical engineering was established, and the jacking force and radial stress of different pipes were analyzed. Then, the two pipes were taken as research object, and the parameters of spacing, angle, buried depth, and pipe diameter were investigated, respectively. The results show that in the actual project, the advance jacked pipes have amplification and superposition effects on friction resistance of the subsequent pipe, and the maximum growth rate is 37.2%. The friction resistance of the subsequent pipe presents a trend of first increasing and then decreasing with the change of the layout angle of advance jacked pipe from 0° to 180°. With the increase of buried depth and pipe diameter, the absolute value of incremental friction resistance of the subsequent pipe increases gradually, but the growth rate remains constant. Finally, the empirical formulas for predicting the friction resistance growth rate of subsequent pipes under different angles are proposed. The research results can provide some reference for the design of pipe jacking.

In recent years, rapid urbanization resulted in a great development in underground space. The pipe jacking method has been widely used in energy transportation engineering, water conservancy, and tunnel engineering because of its advantages of fast construction speed, little disturbance to the surrounding environment, and easy-to-control jacking accuracy [1-12]. Besides, pipe jacking is also used in pipe roof engineering. For example, the Xinle Ruin station of Shenyang subway line 2 was built with the pipe roof method. The pipe roof structure consists of 19 steel pipes with a diameter of 2000 mm and 2 steel pipes with a diameter of 2300 mm [13]. The Gongbei tunnel was built by 36 steel pipes with a diameter of 1620 mm [14].

The core problem of pipe jacking is still the calculation of friction resistance. The magnitude of the jacking force is directly determined by the friction resistance, and the radial stress of the pipe–soil interface determines the distribution and magnitude of friction resistance stress. At present, the existing literatures have done a lot of research on the friction resistance. The numerical simulation and laboratory test methods have been adopted to study the pipe–soil interaction under different types of lubricants and their combinations by Shou et al. [15]. They concluded that the reduction of the jacking force is closely related to the decrease of friction coefficient, and the effect of lubrication is slightly more significant in the case of curved pipe than in the case of linear pipe. Yen and Shou [16] took the obtained average friction coefficient as the input parameter of the finite element software, and found that the numerical results were consistent with the measured data in jacking force. It is shown that the jacking force can be well predicted by finite element simulation. Ong and Choo [17, 18] and Choo and Ong [19, 20] proposed a method to determine the soil–pipe interface parameters in the finite element model by reconstituting and subsequently shearing scalped tunneling rock spoil. It is proved that this method can accurately predict the jacking force in highly weathered geological formations. Based on the background of Guanjingkou water conservation project in Chongqing, Li et al. [21] revealed the variation law of the friction resistance between concrete pipe and the surrounding rock during the construction of micro-shield pipe jacking, and successfully predicted the jacking force in the jacking stage by using three-dimensional numerical method. Based on the continuum and discontinuum numerical analyses, Barla et al. [22] explained the boring machine was stuck during pipe jacking in the limestone formation, and pointed that the design phase is crucial for micro-tunneling in a rock mass.

Nowadays, a large number of crossing projects and shallow buried projects have appeared in the urban underground space. In order to minimize the disturbance of new projects on the surrounding environment and improve the utilization rate of urban underground space, the pipe roof method is usually used for advanced support in these projects [13, 14, 23-25]. Based on trenchless technology, the pipe roof structure is jacked into the formation by the jacking force applied by the pipe jacking machine. Usually, the pipe roof structure is composed of steel pipes with lock joint, and the distance between adjacent steel pipes is relatively close. For example, in the Xinle Ruin station project mentioned above, the distance between the steel pipes with 2000 mm diameter is only 350 mm [13]. In the Gongbei tunnel project, the distance between the steel pipes with 1620 mm diameter is only 355–358 mm [14]. There are various layout forms of pipe roof structure, including horizontal layout, arch layout, door shape layout, and rectangular layout, as shown in Figure 1. Tao et al. [26] studied the interaction of jacking force in horizontally arranged pipe by field measurement and numerical method and proposed the group effect of jacking force in horizontally arranged pipe. Apparently, the jacking force of the subsequent pipe will be inevitably affected by the advance jacked pipes. However, many scholars still focus on predicting the jacking force in single pipeline, while ignoring the interaction of jacking force between multiple pipes [17-20, 27-35]. Therefore, the influence of the advance jacked pipes on the subsequent pipe needs to be further studied, and it is of great practical significance for the design of the pipe roof method.

In this work, the influence of the advance jacked pipe on the subsequent pipe is analyzed in the pipe jacking stage, and the factors include pipe jacking spacing, buried depth, layout angle, and pipe diameter. The results of the jacking force and radial stress of the pipe–soil interface under different influence factors are expounded in detail. This work is organized as follows: The pipe–soil contact model and the pipe jacking technology without “overcut” are introduced in Section 2. Based on the pipe jacking project of the New Airport Line of Beijing, the detailed numerical model of pipe jacking is established in Section 3. Based on the validated numerical model, the results of the jacking force and radial stress of different pipes are analyzed in Section 4. The parametric analysis model and the cases of the advance jacked pipe affecting the subsequent pipe are presented in Section 5. The results including the friction resistance and radial stress of the subsequent pipe are analyzed in Section 6. The fitting function between the friction resistance growth rate of the subsequent pipe and spacing is proposed under different angles in Section 7. Finally, conclusions are summarized in Section 8.

2.1. Pipe–Soil Contact Model

In general, in order to reduce the friction between the pipe–soil interface and facilitate the pipe jacking construction, micro shield construction technology is used for pipe jacking. Under this construction technology, the outer diameter of the micro shield is slightly larger than that of the pipes following it. This “overcut” is typically 10–30 mm in diameter. Milligan and Norris [36] pointed that the excavation of reasonably stiff and stable soils (such as cohesive soil) will stand stable temporarily and the pipe simply slides along the bottom of the bore. This pipe–soil interface is regarded as “partial contact,” as shown in Figure 2(a). However, when pipe jacking is carried out in cohesionless soil or very soft clay, the bore will not be stable, and the pipe–soil interface is regarded as “full contact,” as shown in Figure 2(b). When the bore is stable, the lubricant (such as bentonite slurry) can be used to partially or completely fill the section of overcut to reduce the friction resistance.

2.2. Pipe Jacking Technology

During the construction process of pipe jacking with pipe roof method, due to the complex surrounding environment, the “overcut” is generally not allowed in order to prevent large disturbances to the surrounding strata. Therefore, the underbreak pipe jacking technology is adopted in the construction of the pipe roof method in Beijing, China. During the pipe jacking, the spiral drill is installed inside of the steel pipe. The soil of the excavation face is continuously squeezed and cut by the spiral drill, and the pipe-jacking construction is carried out step by step (Figure 3). In addition, it is worth noting that lubricant is not used at the pipe–soil interface due to the underbreak pipe jacking technology.

In the project of the Beijing Metro Line 8 under crossing existing shield tunnel of Line 10, the deep hole displacement sensors were used by our research team to evaluate the disturbance effect of underbreak pipe jacking technology in the sandy gravel [37]. The outer diameter of the shield tunnel in Line 10 is 6 m, the inner diameter is 5.4 m, the concrete strength of the pipe segment is C50, and the burial depth of the arch crown is 11.7 m. The newly built Line 8 is a horseshoe-shaped tunnel constructed using the shallow underground excavation method. The width and height of the newly built tunnel are 6.3 and 6.6 m, respectively, and the clear spacing between the left and right lines is 10.7 m. The distance between the new tunnel arch crown and the existing shield tunnel invert is only 2.5 m. From top to bottom, the soil layers are backfill, silty sand, round pebble, silty clay, and sandy gravel. In order to ensure the normal operation of Line 10 during the construction of Line 8, it is proposed to adopt the techniques of “large-diameter pipe roof + deep hole grouting” to strictly control the deformation of the existing tunnel, as shown in Figure 4. The pipe roof is composed of 29 steel pipes with a diameter of 299 mm and a center distance of 350 mm, and is constructed above the arch crown of the new tunnel. a test steel pipe was set up for on-site testing, and six deep hole displacement sensors were arranged above the test pipe to monitor the deformation of the soil during the pipe jacking process, as shown in Figure 5. The monitoring results during the jacking process of the test pipe are shown in Figure 6. It indicates that the soil at different measuring points shows the tendency of settlement. With the increase of the vertical distance between the monitoring point and the pipe crown, the soil settlement decreases nonlinearly. When the jacking length of steel pipe is 17 m, the settling rate of each sensor significantly increases. When the jacking length of the steel pipe exceeds 25 m, the monitoring data of each sensor shows a stable trend. It should be noted that the monitoring results of sensors at points B and E show a slow upward trend. It may be caused by signal transmission issues with the sensor. After pipe jacking was completed, the settlement of point A was the largest, only −0.14 mm, which proves that this pipe jacking technology has an obvious positive effect on reducing the disturbance and deformation of surrounding soil in pipe jacking stage. All in all, there is no “overcut” between pipe and soil, and the pipe–soil interface type is “full contact” with this pipe jacking technology.

3.1. Engineering Background

The newly built Beijing Daxing Airport Line needs to cross the existing shield line 10 and Zhenguosi North Street at the same time. The underground excavation method was adopted to construct the new tunnel. The layout and cross-section of the project are shown in Figures 7 and 8 , respectively. The construction site can be classified into three layers of soil, including sandy gravel, silty sand, and miscellaneous fill from bottom to top. The maximum excavation width and height were 14.8 and 9.3 m, respectively. The minimum distance between the new tunnel and the existing shield tunnel was only 0.85 m and the crown of the new tunnel was 3.9 m below the ground surface (Figure 8). In order to ensure the normal operation of the existing shield tunnel, the pipe roof structure was constructed under the floor of the new tunnel. The pipe roof is composed of 26 steel pipes. The diameter of the steel pipe is 402 mm and the wall thickness is 16 mm. The clear spacing between adjacent steel pipes is 48 mm. The steel pipes are connected by lock joint so as to form a pipe roof structure to protect the shield tunnel. In order to minimize the disturbance of pipe jacking to surrounding strata, the underbreak pipe jacking technology introduced in Section 2 is adopted. The completed pipe roof structure and the sequence of pipe jacking are shown in Figures 9 and 10, respectively.

3.2. the Numerical Model of Pipe Jacking

The ABAQUS was used in this study. It should be noted that before the pipe jacking of pipe roof structure, the soil of part I in the new structure has been excavated (Figure 8). Therefore, the buried depth of pipe jacking is assumed to be 8.6 m in the numerical model. In addition, researchers have pointed out that the soil pressure acting on the steel pipe is mainly determined by the overlying soil, while the underlying soil only provides a reaction force to the steel pipe [27, 29, 38-44]. At the same time, the interaction of jacking force between pipes is mainly concerned in this study. Therefore, the influence of the shield tunnel on jacking force is ignored in the numerical model to simplify the calculation. The numerical model with a width of 15 m, a height of 12 m, and a length of 50 m is established for avoiding the boundary effect (Figure 11). For the boundary condition, the lateral boundaries are set as normal constraint, and the bottom boundary is set as fixed constraint. The four numerical models with different jacking lengths (10, 20, 30, and 40 m) were set to simulate the whole process of pipe jacking [16].

3.3. The Contact Properties and Model Parameters

The soil is described using the Mohr–Coulomb yield criterion. The linear elastic model is chosen for the steel pipe. Table 1 provides the model parameters. According to the pipe jacking technology of non “overcut”, the “full contact” is set at the soil–pipe interface. The normal direction of the pipe–soil interface is set as “hard contact,” and the tangential direction is set as “penalty friction.” According to the suggestion by Shou et al. [14], a simple laboratory test was carried out to determine the friction coefficient of the soil-pipe interface. The experimental device and test results are shown in Figure 12. During the experiment, shake the runner slowly to make the steel block move forward slowly at a uniform speed, and record the data displayed in the sensor. Three groups of cases were tested, and the results were 23.2, 24.1, and 23.7 N, respectively. Through measurement, the weight of the steel block is 7.9 kg, and the calculated friction coefficients are 0.29, 0.3, and 0.3, respectively. Therefore, the friction coefficient is taken as 0.3.

3.4. The Process of Numerical Simulation

The displacement control method is used to simulate the pipe jacking process in the numerical model [15, 19]. A coupling point is set at the center of the steel pipe tail, and this coupling point is coupled with the section of the steel pipe tail for displacement. The pipe jacking construction is simulated by applying z-direction displacement to this coupling point. After the calculation is completed, the node reaction force along the z-direction of the coupling point is extracted as the jacking force.

The numerical simulation process of pipe jacking is as follows: (1) geostress balance, (2) remove the elements of soil to simulate the excavation, and (3) apply displacement load to the coupling point. According to the field investigation, the distance from the spiral drill to the head of the pipe is 0.05–0.1 m. In this model, the displacement load of 0.05 m is applied at the coupling point. According to the actual construction sequence of pipe jacking, the numerical calculation is carried out for pipes of 1–13 (Figure 11).

4.1. Model Validation of Jacking Force

The numerical results of the jacking force in pipes 1, 2, 3, 5, 6, 10, and 13 are compared with the field measured results, as shown in Figure 13. It can be seen:

  1. Both numerical and measured results show that with the increase of jacking length, the numerical results of the jacking force increase linearly, and the numerical results are consistent with the field measured results, which proves the effectiveness of the numerical model.

  2. The design of the pipe jacking machine can provide a jacking force of 1600 kN. Pipes 1 and 2 were successfully jacked into the formation and reached the design position. The maximum jacking force is 1550 kN in pipes 1 and 2, which was close to the designed jacking force limit of the pipe jacking machine. However, in the subsequent pipe jacking process, when pipe 3 was jacked to around 30 m, the recorded jacking force approached the limit jacking force of the pipe jacking machine, causing the subsequent pipe to be stuck. In order to solve this problem, another pipe jacking machine was added temporarily on-site for pipe jacking, but the jacking force could not be recorded by this machine.

  3. According to the sequence of pipe jacking, pipes 1 and 2 are far away from each other, and it can be considered that there is no mutual interference. Starting with pipe 3, due to the influence of the advance jacked pipes, both the numerical and the measured results indicate that the jacking force increases gradually with the increase of the number of advance jacked pipes. For example, the friction resistance of pipes 1 and 2 is about 38.2 kN/m, while that of pipes 3, 5, 6, 10, and 13 is about 44.5, 47.8, 48.5, 49.5, and 52.4 kN/m, respectively and the growth rates of friction resistance are 16.5%, 25%, 27%, 29.6%, and 37.2%, respectively. The results show that the advance jacked pipes have the effect of amplification and superposition on the friction resistance of the subsequent pipe.

4.2. Radial Stress

In the pipe jacking construction, the jacking force is the macroscopic manifestation of the friction stress in the pipe–soil interface, and the friction stress is determined by the radial stress and the friction coefficient. When the friction coefficient remains constant, the change of jacking force can be explained essentially by the variation law of radial stress. Figure 14 shows the radial stress of pipes 1(2), 3, 5, 7, 9, 11, and 13 during pipe jacking. It can be seen that the radial stress distribution of pipes 1 and 2 are symmetrical, and the radial stress at the horizontal position (90° and 270°) is the largest due to the influence of “elliptization” deformation of the pipe. Starting with pipe 3, affected by the advance jacked pipes, the radial stress distribution of the subsequent pipe is asymmetric, and the radial stress at different positions is greater than that of the single pipe (pipes 1 and 2). In addition, it is obvious that the increase in radial stress near the advance jacked pipes (the position of 90°) is the largest, as shown in Figure 15. Combined with the research results of the jacking force in Section 4.1, it is shown that the radial stress of subsequent pipes is amplified by the advance jacked pipe, which leads to the increase of the jacking force of subsequent pipe.

In this section, in order to quantify the influence of the advance jacked pipe on the subsequent pipe, the two pipes are taken as research objects, and the numerical models considering the influence of parameters such as spacing, buried depth, angle, and pipe diameter are established.

5.1. Numerical Model

As the influence of the advance jacked pipe on the friction resistance of the subsequent pipe is focused on in this paper, the friction resistance is only considered in the finite element model. Thus, the pipe “passes-through” the soil layer, as shown in Figure 16. The lengths of soil and steel pipe in the numerical model are set to 10 and 12 m, respectively. The width and height of the model are dynamically adjusted considering the influence of different cases and boundary effects. Since the face resistance is not considered, the displacement of 0.5 m along the z-direction is applied at the coupling point. Only the soil of silty sand is selected for simulation, and the model parameters are shown in Table 1.

5.2. Cases Setting

On the premise of soil properties unchanged, the interaction between pipes is mainly affected by pipe jacking parameters and layout. The parameter diagram of spacing, buried depth, angle, and pipe diameter is shown in Figure 17. The detailed analysis of cases is given in Table 2.

In this section, based on the parameters of spacing, buried depth, angle, and pipe diameter, the influence of advance jacked pipe on the friction resistance and the radial stress of the subsequent pipe are analyzed. It should be noted that the value of increment of friction resistance ΔFs mentioned later is defined as:

ΔFs=Fs`Fs
(1)

The growth rate of friction resistance is defined as:

ω=(FsFs)/Fs
(2)

where the Fs is the friction resistance of the subsequent pipe and Fs is the friction resistance of the single pipe.

The single pipe refers to that there is no advance jacked pipe around the jacking pipe. It is to ensure that the buried depth of the single pipe is consistent with that of the subsequent pipe.

6.1. Influence of Layout Angle

Taking the calculation results of a pipe diameter of 400 mm and buried depth of 8 m, Figure 18 depicts the curve of the incremental friction resistance of the subsequent pipe under different angles and spacing. The positive value of the increment represents the increase of the friction resistance, while the negative value represents the decrease of the friction resistance. It can be seen that the curve of the increment of the friction resistance is similar to “normal distribution curve,” and the distribution is approximately symmetrical along the axis of 90°. In general, in the range of 60°–120°, the friction resistance of the subsequent pipe is greater than that of the single pipe. With the increase of angle, the increment of friction resistance increases at first and then decreases, and reaches the maximum value at 90°. However, in the range of 0°–60° and 120°–180°, the friction resistance of the subsequent pipe is less than that of the single pipe. In a word, compared with the friction resistance of the single pipe, the friction resistance of the subsequent pipe presents a “decrease–increase–decrease” trend with the change of the layout angle of advance jacked pipe from 0°to 180°.

In order to explain the causes of the above phenomena, the radial stress of the subsequent pipe is compared with that of the single pipe under different angles (Figure 19). The radial stress of the subsequent pipe is changed because of the influence of the advance jacked pipe. Due to the high symmetry of the model, the radial stress distribution of the subsequent pipe is similar to that of the single pipe under the cases of 0°, 90°, and 180°, and due to the asymmetry of the model, the radial stress distribution of the subsequent pipe is disorderly under the cases of 30°, 60°, 120°, and 150°. Overall, the radial stress of the subsequent pipe is less than that of the single pipe under cases of 0°, 30°, 150°, and 180°, while the radial stress of the subsequent pipe is greater than that of the single pipe under cases of 60°, 90°, and 120°. The reasons for these results are as follows:

(1) In the stratum where the vertical earth pressure is greater than the horizontal lateral pressure, the “elliptization” deformation of vertical compression and horizontal tension will occur at the pipe. When the advance jacked pipe is located at the position of 0°, 30°, 150°, and 180° of the subsequent pipe, the advance jacked pipe bears part of the load on the subsequent pipe. That is, the advance jacked pipe has an unloading effect on the subsequent pipe, which leads to the decrease of radial stress on the subsequent pipe.

(2) Under the cases of 60°, 90°, and 120°, the unloading effect of the advance jacked pipe disappears, and the advance jacked pipe shows the loading effect on the subsequent pipe. Under the influence of the “elliptization” deformation, the advance jacked pipe and the subsequent pipe squeeze each other, which leads to the increase of radial stress on the subsequent pipe.

It can be found in Figure 19 that the minimum radial stress of the subsequent pipe is shown in cases of 30° and 150°, and the maximum radial stress of the subsequent pipe is shown in cases of 60°, 90°, and 120°. This is caused by the comprehensive interaction between the advance jacked pipe and the subsequent pipe. Taking the cases of 30°, 60°, and 90° as example, the reason for the maximum or minimum values of radial stress is elaborated in detail, as shown in Figure 20. In the case of 30° (Figure 20(a)), the middle soil is located at the invert of the advance jacked pipe. Due to the influence of the deformation of the advance jacked pipe, the middle soil produces upward displacement. The “elliptization” deformation of the subsequent pipe has little influence on the displacement of the middle soil, and the middle soil is easy to separate from the subsequent pipe, resulting in the minimum radial stress. In the case of 60° (Figure 20(b)), due to the influence of the “elliptization” deformation of the advance jacked pipe, the middle soil at the position of maximum radial stress is squeezed by the advance jacked pipe, resulting in the maximum radial stress of the subsequent pipe appears at this position. In the case of 90° (Figure 20(c)), under the combined effect of the “elliptization” deformation of the advance jacked pipe and the subsequent pipe, the middle soil is squeezed by the advance jacked pipe and the subsequent pipe. As a result, the maximum radial stress appears at the 90° position of the subsequent pipe, which is greater than the other cases.

All in all, the different effect is reflected when the advance jacked pipe is located at different positions of the subsequent pipe. When the advance jacked pipe is located at the range of 0°–60° and 120°–180° of the subsequent pipe, the advance jacked pipe shows the unloading effect, and the friction resistance of the subsequent pipe is less than that of the single pipe. When the advance jacked pipe is located at the range of 60°–120°of the subsequent pipe, the advance jacked pipe shows the loading effect, and the friction resistance of the subsequent pipe is greater than that of the single pipe.

6.2. Influence of Spacing

The incremental friction resistance of the subsequent pipe with different spacing and angle under the cases of 400 mm pipe diameter and 8 m buried depth can be observed in Figure 21. In Section 6.1, it has been shown that the incremental friction resistance of the subsequent pipe under different angles is approximately symmetrically distributed along the axis of 90°. Therefore, the simulation results of 0°, 30°, 60°, and 90° cases are given in Figure 21. It can be seen that the absolute value of incremental friction resistance of the subsequent pipe progressively decreased and converged with the spacing increased. The decreased speed of absolute value of incremental friction resistance of 0° case is greater than that of 30° case, which indicates that the unloading effect of the advance jacked pipe in 0° case is more sensitive to the influence of spacing than that in 30° case. The growth rate of friction resistance is also shown in Figure 21. When the pipe jacking spacing d > 3D, the effect of the advance jacked pipe on friction resistance of the subsequent pipe is less than 5% in the case of 90°. In other cases, when the pipe jacking spacing d > 2D, the effect of the advance jacked pipe on friction resistance of the subsequent pipe is less than 5%. It is shown that the horizontal position of the pipe has the greatest influence. The interaction between the advance jacked pipe and the subsequent pipe can be ignored when the pipe jacking spacing d > 3D, which is consistent with the research results by Shou et al. [14]. Therefore, in the design of large-scale pipe jacking, the influence of pipe jacking spacing should be considered. It is necessary to revise the prediction of the jacking force of single according to the actual situation to obtain the jacking force of the subsequent pipe.

6.3. Influence of Buried Depth

Taking the pipe diameter of 400 mm, the results of increment and friction resistance growth rate with different spacing, buried depths, and angles are shown in Figures 22 and 23, respectively. Similarly, the calculation results are only given for cases of 0°, 30°, 60°, and 90° in this section. It is shown that with the buried depth of the pipe increased, the absolute value of incremental friction resistance gradually increased. It is remarkable that buried depth has no noticeable effect on the friction resistance growth rate. It can be considered that the friction resistance growth rate remains constant with the increase of buried depth. When the pipe jacking spacing is 1.5D, 2D, 3D, and 4D, the friction resistance growth rate in 0° case is −6.67%, −4.24%, −2.46%, and −1.69%, respectively. The friction resistance growth rate in 30° case is −4.81%, −4.59%, −3.88%, and −3.36%, respectively. The friction resistance growth rate in 60° case is 6.75%, 4.22%, 2.7%, and 1.3%, respectively. And the friction resistance growth rate in 90° case is 12.55%, 8.22%, 5.12%, and 3.59%, respectively. Therefore, the buried depth has an impact on the incremental friction resistance of the subsequent pipe, but has no effect on the friction resistance growth rate of the subsequent pipe.

6.4. Influence of Pipe Diameter

Taking the buried depth of 8 m, Figures 24 and 25 present the curve of incremental and growth rate of friction resistance with different spacing, pipe diameters, and angles. It is shown that the absolute value of incremental friction resistance increased with the pipe diameter increased. Under the same spacing and angle, the growth rate of friction resistance in different pipe diameters is almost unchanged. When the pipe jacking spacing is 1.5 D, 2 D, 3 D, and 4 D, the friction resistance growth rate in 0° case is −6.9%, −3.88%, −2.43%, and −1.55%, respectively. The friction resistance growth rate in 30° case is −4.79%, −4.59%, −3.89%, and −3.31%, respectively. The friction resistance growth rate in 60° case is 7.3%, 4.99%, 2.77%, and 1.3%, respectively. And the friction resistance growth rate in 90° case is 12.5%, 8.88%, 5.44%, and 3.96%, respectively. These results are close to those mentioned in Section 6.3. Generally speaking, the spacing and angle between the advance jacked pipe and the subsequent pipe have obvious influence on the growth rate of the friction resistance of the subsequent pipe, while the buried depth and pipe diameter have no influence on the friction resistance growth rate of the subsequent pipe. Therefore, the friction resistance of the subsequent pipe can be predicted according to the friction resistance of the single pipe when the pipe jacking spacing and angle are determined.

In Section 6, the parameters of spacing, buried depth, angle, and pipe diameter are analyzed in detail, and some meaningful conclusions that can be used in pipe jacking design of pipe roof method are obtained. With the increase of the buried depth and pipe diameter, the absolute value of incremental friction resistance of the subsequent pipe gradually increases. However, the growth rate of friction resistance fluctuates in a small range and can be regarded as a constant. The friction resistance of the subsequent pipe can be expressed as follows:

Fs`=(1+ω)Fs
(3)

where ω is the growth rate of friction resistance of the subsequent pipe and is the function of the spacing and angle, which can be expressed as follows:

ω=ω(d,θ)
(4)

In this section, the function fitting of the relationship between the growth rate of friction resistance of the subsequent pipe and the spacing under different angles is proposed, as shown in Figure 26. The symmetry of the calculated results from different angles is also considered, and only the fitting functions of 0°, 30°, 60°, and 90° are given in Figure 26. At the same time, in order to consider the limited influence range of pipe jacking, it is assumed that when d>10 D, there is no interaction between the advance jacked pipe and the subsequent pipe, that is, the growth rate of friction resistance is 0%. The function fitting results of 0°, 30°, 60°, and 90° are consistent with the calculated results, and the calculated correlation coefficient R2 is 0.98, 0.85, 0.98, and 0.98, respectively.

In the project introduced in Section 3, the pipe diameter is 402 mm and the spacing is 450 mm (1.1 D). According to the fitting function in Figure 26 (d), the growth rate of friction resistance of pipe 3 is 17.5%, while the growth rate of measured and numerical calculation is 16.5%. The two results are close to each other. It is proved that this fitting function can predict the growth rate of friction resistance of the subsequent pipe. Therefore, in the case of determining the angle and spacing of the subsequent pipe, the growth rate of the friction resistance of the subsequent pipe can be estimated by this fitting function, and the friction resistance of the subsequent pipe can be calculated according to friction resistance of the single pipe.

In this work, based on the pipe jacking project of Beijing Daxing Airport Line, the pipe-soil “full contact” numerical model of pipe jacking is established. Combined with the field-measured data of jacking force, the influence of advance jacked pipes on subsequent pipes is studied through the jacking force and radial stress. The two pipes are taken as research object, and the parameters of pipe jacking spacing, buried depth, angle, and pipe diameter are investigated successively. In addition, based on the results of friction resistance of the single pipe, the empirical expression for the predicting growth rate of friction resistance of the subsequent pipe at different angles is proposed in silty sand. The key findings are elaborated as follows:

  1. The radial stress of the subsequent pipe is affected by the advance jacked pipes, which leads to change of friction resistance of the subsequent pipe. In practical engineering, the advance jacked pipes have amplification and superposition effects on the friction resistance and the radial stress of the subsequent pipe, and the maximum growth rate of friction resistance of the subsequent pipe is 35.9%. The subsequent pipe was stuck due to the insufficient jacking force of the pipe jacking machine.

  2. When the advance jacked pipe is located at different angles of the subsequent pipe, the advance jacked pipe shows different effects. In the range of 0°–60° and 120°–180°, the advance jacked pipe has the unloading effect on the subsequent pipe, and the friction resistance of the subsequent pipe is less than that of the single pipe. In the range of 60°–120°, the advance jacked pipe has a loading effect on the subsequent pipe, and the friction resistance of the subsequent pipe is greater than that of the single pipe. This loading effect reaches the maximum at the 90° position. With the change of the setting angle of the advance jacked pipe (0°–180°), the value of incremental friction resistance of the subsequent pipe approximately presents a normal distribution curve with a symmetry axis of 90°.

  3. The interaction between the advance jacked pipe and the subsequent pipe is affected by the pipe jacking spacing, buried depth, and pipe diameter. With the increase of pipe jacking spacing, the absolute value of increment and growth rate of friction resistance of the subsequent pipe gradually decreased. With the increase of buried depth and pipe diameter, the absolute value of incremental friction resistance of the subsequent pipe progressively increased, but the growth rate of friction resistance of the subsequent pipe remains unchanged.

  4. The growth rate of the friction resistance of the subsequent pipe is only related to the arrangement angle and spacing, and has no relationship with the buried depth and diameter of the pipe. Based on the numerical results, the empirical expression for modifying the growth rate of friction resistance of the subsequent pipe is proposed.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare no conflicts of interest.

The National Natural Science Foundation of China (42072308), the National Key R&D Program (2019YFC1509704), Young Teachers’ Research Ability Improvement Plan of Beijing University of Civil Engineering and Architecture (Grant No. X23005) and Beijing Municipal Engineering Research Institute funded this research. We are very grateful for these financial supports.

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