Abstract
In the present study, based on previous research results, a finite element method that considered the grouting pressure and displacement control of the tube-soil side friction coefficients was established for the purpose of estimating the jacking forces of large sections of rectangular pipe jacking. Furthermore, the pipe jacking project of Zhong-Zhou Avenue was taken as an example in this study, in which the rectangular pipe jacking models A1 and B1 under silty clay geological conditions were established. The two estimation models were verified using the pipe jacking cases A2 and B2, respectively. The estimation model can effectively estimate the jacking force, and the rectangular jacking force is distributed as a logarithmic function with the jacking distance. The shallow buried rectangular pipe jacking has some common characteristics in buried depth, grouting pressure, the length-width ratio of outer diameter, construction geological conditions, and so on. The main independent factors that affect the jacking force are the buried depth and the outer perimeter of the jacking pipe. Based on the numerical model of case A1 and case B1, the logarithmic functions of jacking force of case A1 and case B1 with jacking distance were obtained by changing the buried depth. The calculation formula of the jacking force can reflect the variation law of the jacking force to some extent.
1. Introduction
The accurate estimations of jacking forces are important problems in jacking pipe construction and design projects. Numerical analysis methods are commonly applied in the simulations of construction processes. It has been found that, through numerical simulations, the contact mechanical behaviors between the surrounding soil and the outer walls of the jacking pipes can be quickly determined. This information is very favorable for the accurate estimation of jacking forces. There are generally two main control methods used in the numerical simulations of engineering mechanical behaviors. The first is a stress control method, and the second is a displacement control method. Regarding to the stress control method, the side friction resistance levels of the pipes and the surrounding soil are obtained through various empirical and theoretical equations. Also, the resistance levels of the end faces of the jacking heads can be obtained using a stress control method. Then, as the input data of the numerical model, these values can be applied to the corresponding outer walls of the jacking pipes, along with the end faces of the jacking heads, respectively [1–3].
However, it has been observed that due to the uncertainties of the applied empirical and theoretical calculation formulae, the results of the jacking force estimations are uncertain. For example, excessively high jacking forces will lead to the destruction of the pipe segment structures, whereas excessively low jacking forces will lead to jacking failures. This is known to be a deficiency when stress control methods are utilized. In the designs of jacking pipes, the jacking directions and jacking displacements are clear. This effectively opens the door for finite element methods of displacement control to accurately simulate pipe jacking processes. Based on the numerical simulation method, Zhao [4] simulated the jacking process of pipe jacking through the displacement control method, to explore the stress and strain variation rule of surface and surrounding pipe jacking under different conditions. Fan et al. [5], taking the actual engineering as the research background, adopted the finite element method based on displacement control method to analyze the influence law of pipe jacking on the expressway. Yen and Shou [6] first proposed that jacking forces could be accurately estimated using finite element methods of displacement control for jacking pipes. The jacking forces at different jacking positions were first calculated by displacement control finite element methods. Then, the curve fitting relationships between the jacking forces and jacking distances in the middle sections of jacking pipes were successfully obtained through data fitting. The jacking forces were determined to be related to the construction conditions and jacking distances. For example, in the cases of circular jacking pipes, the calculations of jacking forces were required to consider the contact ranges and friction coefficients between the pipes and the soil. In fact, there are many factors that are known to affect the jacking forces, such as the shapes of the jacking pipes and the grouting pressure levels. For shallow rectangular pipe jacking, the grouting pressure levels are often minimal, so as to prevent ground uplift. Therefore, in the overcutting ranges, the slurry tends to interact with the soil under the action of pressure to form substances, which are similar to soft soil distributed on the outer walls of the jacking pipes. As a result, there will be tangential friction resistance levels on the entire outer wall sections of the jacking pipes. The existence of slurry will only reduce the side friction coefficients of the outer walls of the jacking pipes but not the contact ranges between the outer walls of the jacking pipes and the surrounding soil.
At present, there have been few research studies conducted regarding the effects of the grouting pressure levels on the jacking forces. In this study, jacking forces at different jacking positions were obtained by applying given displacements to sections of the jacking pipes in a launch shaft using the numerical analysis software ABAQUS [7]. Then, the contact properties and grouting pressure levels between the pipes and the surrounding soil during the jacking processes were determined. The longitudinal stress components that had been exerted on the pipe sections were used to back-calculate the jacking forces.
2. Jacking Force Estimation Models of Rectangular Pipe Jacking
There have been various studies conducted to investigate the jacking forces of circular pipes using theoretical derivations. The jacking forces include the end resistance levels of the jacking heads and the side friction between the pipes and surrounding soil. The end resistance levels of the jacking heads can be calculated while accounting for the overburden pressure levels on the pipes. For example, in a particular jacking pipe project, the end resistance of the jacking head will usually be constant. The side friction between the pipes and soil is known to be related to the pipe-soil contact areas and grouting pressure levels, as well as the friction coefficients of the contact surfaces between the pipes and soil. Also, the side friction will tend to increase linearly as the jacking distance increases. The lateral friction between the pipes and soil plays a controlling role in the magnitude of the jacking forces. Therefore, the pipe-soil contact pressure levels and the pipe-soil friction coefficients are the main parameters used to determine the jacking force. At present, no theoretical estimation formulae of the jacking forces of rectangular pipes exist. The jacking forces of rectangular pipes are usually calculated by referencing the circular pipe calculation formula. However, varying geological conditions and complicated construction scenarios are often found in real project situations. Therefore, it is quite difficult to assume that the current theoretical derivations accurately match the actual scenarios.
It has been observed that the presence of the grouting pressure causes a certain range of mud jacketing to form between the pipes and the surrounding soil during pipe jacking projects. The contact surfaces between the pipes and the soil become partially replaced by the contact surfaces between the pipes and the mud. Additionally, during the calculations of jacking forces, the friction characteristics of the two contact surfaces will need to be considered [8, 9]. The friction coefficients between the pipes and the soil have been observed to be greatly reduced. At the same time, the friction coefficients between pipes and the mud have been determined to be much smaller than the friction coefficients between the pipes and the soil. However, it has been found that the grouting pressure is usually greater than the overburden pressure, which tends to increase the side frictional resistance between the pipes and the contact objects to some extent. The reductions in the friction coefficients of the contact surfaces due to the presence of mud are only considered in the existing theoretical formulae. However, the contribution of the grouting pressure to the side friction has not been previously considered.
At present, there are two types of jacking force calculation models that are commonly used in China and internationally. In accordance with the stability of tunnels, there are two types of pipe outer wall and soil partial contact models and pipe outer wall and soil full-contact models. In previous related studies, Milligan and Norris [10] proposed that jacking forces were equal to the products of pipe self-weight and the pipe-soil friction coefficients. Haslem [11] held that jacking pipes only had elastic contact and relative sliding with the bottoms of the hole walls. An elastic contact model introduced by Hertz was adopted to obtain the contact widths between the pipes and surrounding soil, and the jacking forces were calculated using the product of the cohesive forces between the pipes and soil, and the contact areas. The elastic contact model proposed by Hertz was also utilized by Khazae et al. [8] to calculate the contact widths between the outer walls of jacking pipes and the surrounding soil; consideration was given to the friction between the outer walls of the pipes and the surrounding mud. O’Reilly and Rogers [12] believed that the outer walls of the jacking pipes were in full contact with the pore walls of the soil, and the jacking forces were equal to overcoming the friction resistance caused by the soil pressure on the pipes. In another related study, Sofianos et al. [13] made comparative analyses of the results of a full-contact model and a partial contact model. At this point, outer wall-soil full-contact pipe models are used to calculate the jacking forces in the majority of the relevant design codes in China and most other countries. However, in Japan, a soil pressure model presented by Terzaghi [14] has been used to calculate jacking forces. Zhang et al. [15] proposed the use of a coordination surface person contact model to obtain a jacking force calculation formula, which would consider the influences of pipe friction. In 2007, Cheng et al. [16] proposed that displacement data could easily be obtained in practical geotechnical engineering processes. Therefore, in the simulation models of geotechnical engineering construction processes, the selection of the displacement boundary conditions could simplify the problems that were encountered. In a specific simulation model, such displacement boundary conditions were referred to as displacement loads by Zhang et al. [17]. In the present study, this numerical simulation method is referred to as a displacement control approach. Numerical models based on displacement control approaches have often been used to solve the problems of the reverse analysis of the parameters in geotechnical engineering processes. These parameters included soil-structure interactions and the properties of the material. During the displacement control approaches, the pulling-up processes of the plate anchors were simulated using a finite element method. Then, simulation models of the installations of the pile foundations were established using a displacement control numerical method. Furthermore, the installations of the pipe roof support in tunneling systems were successfully simulated by Zhang et al. [18]. In the aforementioned studies, displacement control methods were also applied in order to simulate the contact surfaces or the specified points. However, displacement control approaches have been seldom used in the estimations of jacking forces in rectangular pipe jacking. Yen and Shou [6] used a coupling finite element method, as well as a displacement control method, to estimate the required jacking forces in circular pipe jacking. The research results provided important guidance for a displacement control element method for estimating the jacking forces of pipe jacking. However, there are still many problems to be solved in order to accurately estimate the jacking forces using finite element methods with displacement control. It has been determined that the fitting precision of the estimated results will be related to the grouting pressure, the number of calculation models, distributions of the friction coefficients of the side walls, contact areas of the pipe and soil, and other factors. Therefore, further research is required regarding the influencing factors on the accuracy of the estimations, which was a motivation for the present study. The different models and their respective advantages and disadvantages are summarized in Table 1.
3. Numerical Analysis Methods
3.1. Cases for Numerical Models
This study involved four cases in the Zhong-Zhou Avenue subway station of Zhengzhou Metro Line 4. The four cases were part of a subway station project composed of a cross-street underground passage pipe jacking design. The four cases were designated the numbers A1, A2, B1, and B2, respectively. The spatial position distribution of the four rectangular pipe jacking cases is shown in Figures 1(a)–1(c). The numerical software ABAQUS 17.1 was utilized as the finite element software in this study. The analyses of the study results focused on the two construction cases A1 and B1, whereas cases A2 and B2 were used as the verification cases. The detailed parameters were the same as those in cases A1 and B1, respectively. The A1 and B1 case results are described and illustrated in Table 2. The analysis and verification cases were all constructed via rectangular pipe jacking machines equipped with jacking force and grouting pressure monitoring systems. Images of the jacking head and pipe segment of case A1 are shown in Figures 2(a)–2(e). The jacking force and pressure histories were collected and processed, and the averages of the values with the jacking times for each pipe section were obtained.
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In accordance with the detailed survey reports of the engineering geology in the field, the pipe jacking had passed through the silty clay layers and above the fine sand layers, which belonged to the three-level alluvial terrace of the Yellow River. The stratum distribution is shown in Figure 1(d). The physical and mechanical properties of each layer are shown in Table 3. The dry strength and toughness of the silty clay layers containing iron-manganese oxide and shell fragments were determined to be low. Also, the states of silty clay layers, which possessed a high internal friction angle and cohesion, were found to range from plastic to hard plastic. The material parameters of the soil and pipes are summarized in Tables 4 to 5, respectively. The groundwater level was approximately 11 m below the surface, which was much higher than the underground excavation. It was observed that the presence of the groundwater had only barely influenced the pipe jacking constructions.
3.2. Finite Element Model Design
3.2.1. Finite Element Meshes
The purpose of the current study was to obtain the fitting function of the jacking forces and the jacking distances based on the stresses exerted on the pipe sections by employing a displacement control method. It was determined that, provided that the numerical model was reasonably established, the fitting function of the jacking forces with the jacking distances on rectangular pipes under the same conditions should be consistent. Furthermore, if the entire process of the jacking was simulated, then the calculation time would be much longer, and the soil elements of the contact surfaces could be easily distorted and destroyed, resulting in the termination of the calculations. Therefore, at different positions on the jacking line, the lengths of various pipe sections were selected in order to establish the jacking numerical model. Then, the jacking forces at those different positions along the line were calculated, and the fitting curves and functional relationships between jacking forces and jacking distances were obtained. In the present study, the pipe elements were focused on, rather than the soil elements. It was determined that the mesh of the pipe segments and nearby soil elements required proper encryption, and the mesh of the other positions of the model was required to be appropriately sparse.
In the current study, in accordance with the St. Venant Principle, the influences of the boundary effects were ignored when the model sizes were above 3D. The buried depths of case A1 and case B1 were both 3.5 m, and the heights of case A1 and case B1 were 5.4 m and 7.25 m, respectively. Therefore, the ground boundary effects of the model were required to be considered. The boundary effects of the side vertical and bottom horizontal boundaries were evaluated. Meanwhile, the boundary effects of the side horizontal vertical and bottom horizontal boundaries were avoided. The case A1 model was 105 m long, 47.75 m wide, and 20.3 m high. Meanwhile, the case B1 model was 105 m long, 47.75 m wide, and 27.75 m high. The rationality of grid elements, along with the elimination degrees of the boundary effects, were tested using a series of experimental analyses. The grid element divisions of the jacking pipes and the soil on their side walls were found to be relatively fine and evenly distributed along the jacking direction, as shown in Figure 3. It was determined that these conditions would lead to long-running times of approximately two hours for each model. However, it was believed that the calculation accuracy could potentially be improved. In the current study, for case A1, 65,520 soil elements and 2,826 pipe elements were used, as well as 71,788 soil nodes and 5,556 pipe nodes. Meanwhile, in case B1, 72,644 soil elements and 3,204 pipe elements were used, along with 79,104 soil nodes and 6,300 pipe nodes. The grid elements were encrypted in the areas around the pipes and jacking heads, as well as the surrounding soil of the piping tunnels. Then, a displacement load method was used to apply displacements to the end cross sections of the pipes in order to achieve the jacking of each segment. In cases A1 and B1, the proposed jacking distances were both 105 m. Therefore, in this research study, six models with various jacking distances (12 m, 27 m, 42 m, 57 m, 69 m, and 84 m) were used in the jacking force calculations, respectively.
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3.2.2. Boundary Conditions for Finite Element Models
In regard to the boundary conditions in the numerical models, the ground was considered to be a free surface without any surface forces or constraints, and the displacements in the x, y, and z directions of the bottom horizontal boundary of the model were fixed by roller. The displacements in the x direction of the x-axis surface vertical boundary were fixed by connecting rods. Finally, the displacements in the y direction of the y-axis surface vertical boundary were fixed by connecting rods. Then, a three-dimensional hexahedron element (C3D8R) with eight nodes was used in this study’s simulation.
3.2.3. Contact Properties
In previous studies, in order to obtain jacking force results that were closer to the actual scenarios using displacement control simulations, Yen and Shou proposed that the soil and pipes would not be in direct contact with each other in the range of the overcuts of the jacking pipes, in which the contact properties could be set as frictionless. Therefore, in the current study, the friction contact properties were only set between the pipes and the soil in the nonovercut range. However, it has been shown from previous engineering experience that for rectangular jacking pipes, the friction between the pipes and soil cannot be eliminated by the overcutting of the machine head. It has been observed that even with the presence of lubricating slurry, tangential friction will still exist within the overcut range. However, the coefficients of the friction will tend to be smaller than in the nonovercut range. Therefore, in order to characterize the differences between the two types, the contact friction coefficients in the overcut range was set as 0.2, and the friction coefficients in the nonovercut range was set as 0.25. In regard to the shallow buried tunnels, the development of the jacking forces was determined to be influenced by the grouting pressure to a large extent. In the current study, in order to examine the influences of the grouting pressure on the jacking forces, the grouting pressure was set between the jacking pipes and the soil in the overcut range, namely, the top edge and two sides of the rectangular jacking pipes, as shown in Figure 4. The distribution of grouting pressure on the side wall of pipe jacking is shown in Figure 4.
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3.3. Numerical Simulations
This study’s numerical simulations of the pipe jacking processes included the following steps: (1) The initial settlement in the soil mass under self-heavy stress conditions was omitted in the method of the initial ground stress balance. (2) The elements within the tunnel section along the jacking direction, which simulated the soil excavation, were eliminated using a pipe jacking machine. (3) The contact pairs between the soil and the pipes or the slurry and the pipes were defined, and two different friction coefficients were set, respectively, on the pipe-slurry and pipe-soil contact surfaces according to the lubrication and overcut conditions. (4) The grouting pressures at the interfaces between the slurry and the pipes under different programs and perpendicular to the outer walls of the pipes were applied. (5) The grouting pressures and pipe self-weight stress conditions were enforced in order to create friction on the interfaces between the soil and pipes or the slurry and pipes. (6) The pipes were displaced to the designated positions in order to obtain the stress fields of the soil and pipes.
The pipes in the launching shaft were jacked forward one step in the displacement control finite element method. The distance for one step was 1.5 m in both the A1 and B1 cases. The displacement boundary conditions in the numerical models were employed in the pipe sections of the launching shaft in order to cause the pipes to be jacked forward one step, namely, 1.5 m. In such a scenario, the pipes and the neighboring soil were influenced during the process of the pipe jacking. In regard to the simulations of the overcut areas, Yen and Shou considered that the interfaces would be frictionless in the overcut ranges. Therefore, from that perspective, the lubricating slurry had fully filled the gaps between the pipes and the surrounding soil under the grouting pressure, and the surrounding soil had not directly contacted the jacking pipes. Meanwhile, the lubricating slurry was able to generate only compressive stress on the outer walls of the pipes but not shear stress. Also, in accordance with the stress boundary conditions, the tangential friction forces on the outer walls of the jacking pipes should not exist. In the current study, it was observed that previous engineering experiences had shown that the lubrication slurry had reduced the side friction coefficients between pipes and soil masses, instead of eliminating the side friction resistance of the pipes in the overcutting range. Therefore, in this study, it was considered that the lubricating slurry had interacted with the surrounding soil under the grouting pressure to form a material that was similar to soft soil, and the friction coefficients had decreased. At the same time, the outer walls were subjected to grouting pressure. The overcutting ranges of the rectangular pipes include the outer walls of the top edges and both sides, and the friction coefficient of the outer walls was determined to be 0.2. The grouting pressure and friction coefficients had jointly determined the side friction resistance within that range. The nonovercutting range included the outer walls of the bottom edges of the rectangular pipes. It was determined that as a lubricant, the slurry had also reduced the friction coefficient of the outer walls of the bottoms of the pipes to a certain extent, and the friction coefficient was determined to be 0.25. Therefore, the grouting pressure, pipe weights, and friction coefficients were found to have determined the side friction resistance.
In the cases of A1 and B1, the jacking pipe operation cycle lengths were both 1.5 m. In the present study, six numerical models were established for cases A1 and B1 along the pipe directions ranging from 12 m to 13.5 m; 27 m to 28.5 m; 42 m to 42.5 m; 57 m to 59.5 m; 72 m to 73.5 m; and 87 m to 88.5 m, as shown in Figure 5, respectively. During jacking, grouting pressure and contact are activated to form lateral friction resistance.
3.4. Displacement Control Finite Element Method for Estimating Jacking Force
In order to better estimate the jacking forces, a displacement control finite element method was proposed by Yen and Shou. The stress distributions of the neighboring soil bodies and pipes were obtained by means of numerical analysis methods. Taking case A1 as an example, under the action of grouting pressure in program 3, when the jacking was completed, the longitudinal stress distribution of the first section before the pipe jacking machine head of each jacking position was as shown in Figure 6. The longitudinal stress components of the elements were replaced by the longitudinal stress components at each junction of the pipes, and the average values were obtained according to the total number of elements, which was shown as Figure 7 for case A1. Then, the jacking forces were obtained by integrating the sectioned areas. For example, in case A1, when the jacking position was 73.5 m and the grouting pressure was 0.13 MPa, the longitudinal stress at each node of the first section before the jacking head was as shown in Table 6, and the calculated value of the jacking force was 18,818.21 kN.
Theoretically, the jacking forces could be obtained by calculating that the longitudinal stress components of any pipe section were equal, which was somewhat similar to the section methods in which the internal forces were determined using continuum mechanics. However, in reality, the jacking forces calculated by each section will differ, due to continuous discretization. Moreover, through comparative calculation, the jacking forces that are calculated by the initial section of the first pipe piece in front of the head will be the largest. Therefore, this study selected the aforementioned section for the jacking force estimations. The flow chart of the applied displacement control method is shown in Figure 8.
Using the above method, we can calculate the jacking force at any jacking position of the pipe and then draw the fitting curve and corresponding fitting function of the jacking force with the jacking distance according to the size of the jacking force at each position. Such a fitting function of jacking force with jacking distance is of general significance for shallow buried pipe jacking under the same construction parameters and geological conditions.
4. Results of the Numerical Simulations
4.1. Grouting Pressure Programs of the Models
The jacking forces of the rectangular jacking pipes consisted of the resistance at the ends of the jacking heads and the resistance at the sides of the outer walls of the pipes. The lateral resistance was found to be related to the grouting pressure of the mud sleeves and the lateral friction coefficients of the outer surfaces of the pipes. During the jacking processes, the pipe segments were subjected to grouting pressure on the top, left, and right sides of the jacking pipes, as shown in Figure 4. It was observed that the friction coefficients between the outer walls of the jacking pipes and the soil could be greatly reduced by the mud sleeves formed by the bentonite plasticizer, which were usually 0.2. Then, by applying the displacements of the pipe joint lengths in the jacking direction (i.e. 1.5 m), the initial sections of the first pipe joints in front of the jacking heads could be selected in order to search the stress components of each element node in the examined sections along the jacking direction, which were essentially the longitudinal stress components. Therefore, this study only focused on the distributions of the longitudinal stress σyy.
Thixotropic mud pressure had an impact on the jacking force. For shallow buried rectangular pipe jacking, the grouting pressure of thixotropic mud is usually no higher than 0.2 MPa. In the initial stage of pipe jacking, the grouting pressure is large, and the grouting pressure of the middle and end sections will decrease. During the whole jacking process, the monitoring value of grouting pressure is not stable and has a jumping property, which brings difficulties to the selection of grouting pressure parameters in the calculation model. In specific projects, we must choose the corresponding grouting pressure scheme according to the engineering experience. In this study, according to engineering experience, we selected several groups of grouting pressure schemes, estimated the jacking force of each model, and obtained the jacking force at different jacking positions under different grouting pressure schemes. By comparing with the measured monitoring curve of jacking force, the optimal fitting curve of jacking force and the corresponding grouting pressure scheme are obtained. By comparing the grouting pressure scheme with the actual monitoring value of grouting pressure, the rationality of the selection of the optimal grouting pressure scheme is determined.
For case A1, four grouting pressure application programs were adopted, as shown in Table 7. The grouting pressure was not considered for all of the models included in Program 1. However, the grouting pressure was set as 0.1 MPa for all of the models in Program 2. The grouting pressure was set as 0.13 MPa, for all of the models in Program 3, and for Program 4, the grouting pressure of the 12 m to 13.5 m segment model was set as 0.12 MPa. The grouting pressures of the other five models were set as 0.13 MPa. The jacking forces and jacking distances obtained by the four sets of models were found to satisfy the logarithmic function, and the correlation coefficients ranged between 0.9821 and 0.9907, as shown in Figure 9. The jacking forces, which were calculated in Program 1, were found to be much smaller than the monitoring data. In Programs 2 and 3, the deviations were continuously reduced by increasing the grouting pressure. In regard to Program 4, the fitting deviations were observed to be further reduced by adjusting the grouting pressure of the first model of Program 3. Therefore, it was confirmed that Program 4 was the best fitting scheme in this study’s simulations. Also, under the scenario in Program 4, the deviations in the jacking force results, which had been calculated by the six models, had been maintained within approximately 5% when compared with the actual monitored data, as shown in Table 8.
For case B1, three grouting pressure application programs were adopted, as detailed in Table 9. The grouting pressure was not considered in all of the models in Program 1. The grouting pressure was set as 0.1 MPa for all of the models in Program 2. For Program 3, the grouting pressure of the 12 m to 13.5 m segment model was set as 0.12 MPa, and those of the other five models were set as 0.1 MPa. It was found that Program 3 had displayed the best fitting results, as shown in Figure 10. It was observed that under the optimal program, the jacking deviations, which were calculated by the six models, could be maintained within 10% when compared with the actual monitored data, as shown in Table 10.
4.2. Comparison of the Jacking Forces
At the present time, the formulae that are commonly used for calculating the jacking forces of rectangular jacking pipes refer to those used for circular jacking pipes. However, it is known that under different geological conditions, the jacking force calculation formulae of circular jacking pipes that are used in construction processes utilize the empirical formula proposed by Staheli [19], along with the empirical formula proposed by the Japan Micro Tunnelling Association [20]. The empirical formula of jacking forces for the partial contact of circular jacking pipes was obtained by Yen and Shou [6] in a displacement control finite element method. The estimation formula of jacking forces provided by the China Code was basically the same as that proposed by the Japan Micro Tunnelling Association [20]. In the aforementioned formulae, the jacking forces of jacking pipes consisted of the side resistance and end resistance, with the end resistance fixed, and side resistance increasing linearly with the jacking distances. In the present study, displacement control finite element models under different grouting programs were selected to estimate the jacking forces at different jacking distances. As a result, the fitting curves of the jacking forces with the jacking distances under different grouting programs were successfully drawn. The relationships between the jacking forces and jacking distances under different grouting programs in cases A1 and B1 are illustrated in Figures 9 and 10, respectively.
It was observed in this study that in both cases A1 and case B1, the jacking forces and jacking distances under different grouting programs were more in line with a logarithmic function relationship than a linear relationship. The correlation coefficient R2 of the logarithmic function fitting was determined to range between 0.9675 (case B1, grouting pressure Program 3) and 0.9979 (case B1, grouting pressure Program 1), as shown in Table 11. Meanwhile, the correlation coefficient R2 of the linear fitting was determined to range between 0.7825 (case A1, grouting pressure Program 1) and 0.9892 (case B1, grouting pressure Program 3). It was observed that the same rule was also reflected in the fitting of the jacking force monitoring values. In regard to the A1 and A2 cases, the logarithmic function fitting correlation coefficient R2 had ranged between 0.8712 and 0.9875, respectively. Meanwhile, the linear fitting correlation coefficient R2 had ranged between 0.7825 and 0.7552. Furthermore, for the B1 and B2 cases, the logarithmic function fitting correlation coefficient R2 had ranged from 0.7851 to 0.8397, and the linear fitting correlation coefficient R2 had ranged from 0.8939 to 0.9639, respectively. The fitting degree of the logarithmic function was observed to be usually higher than that of the linear function in both the numerical estimations and monitoring values. However, there were exceptions observed, such as the monitoring fitting values of case B2.
In the current study, the grouting pressure configuration programs were found to have major influences on the estimation results of the jacking forces. For example, in both case A1 and case B1, the estimated jacking forces were far smaller than the monitoring values when the grouting pressure was not considered, as shown in Figures 9 and 10. In the full-contact simulations, with the assumptions of the jacking pipes and surrounding soil, it was necessary to consider the grouting pressures when calculating the jacking forces using the proposed finite element method of displacement control. In this study, the grouting pressure of the four jacking pipes (cases A1, A2, B1, and B2) had ranged between 0.1 MPa and 0.2 MPa. The monitoring point layout of in-situ pipe jacking grouting pressure was shown in Figure 11. It was determined through on-site monitoring that the grouting pressure had fluctuated over time for each jacking return, as detailed in Figures 12 and 13. As can be seen in Figures 12 and 13, both the optimal grouting pressure programs of case A1 and case B1 were in good agreement with the grouting pressure monitoring, which further indicated that the influences of grouting pressure on the jacking forces were real. Therefore, it was determined in this study that for large-sectioned rectangular jacking pipes, a finite element method of displacement control, which considered the grouting pressure, was reasonable in a sense under the assumption of full contact.
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4.3. Applications of the Jacking Force Estimation Formula in Cases A2 and B2
Currently, there is no theoretical formula for accurately estimating the jacking forces of rectangular jacking pipes. The existing jacking force estimation formula generally refers to that used for circular jacking pipes. In accordance with the “Code for Construction and Acceptance of Water Supply and Drainage Pipeline Engineering GB50268-2008,” the formula for estimating the jacking forces of rectangular jacking pipes is as follows:
Therefore, the jacking force at different positions of pipe case A2 and case B2 could be calculated according to formula (1). The jacking force at different positions of case A2 and case B2 could also be obtained by actual monitoring. At the same time, the jacking force at different positions of case A2 and case B2 could be obtained by fitting the jacking force formula of case A1 and case B1 based on the displacement control finite element method. Through the above three methods, the functional relationship curves of jacking force and jacking distance of pipe A2 and pipe B2 are obtained, as shown in Figures 14 and 15, respectively. The deviation analysis results of the jacking force A2 and B2 obtained by the three methods are shown in Tables 12 and 13.
It was found in this study that when compared with the measured jacking force fitting values of case A2, the deviation in the predicted jacking force values by the estimation formula of case A1 had ranged between −52.80% and −16.46%, and by theoretical formula, it had ranged between −54.87% and 18.73%. Therefore, it could be seen that the two error results were basically the same. Moreover, when compared with the measured jacking force fitting values of case B2, the deviation in the predicted jacking force values by the estimation formula of case B1 had ranged between −24.29% and 4.10%, and by theoretical formula, it was between −24.95% and 7.4%. Once again, it could be seen that the two error results were basically the same. Therefore, in a certain sense, the correctness of the jacking force prediction method had been successfully proven.
The predicted values of the jacking forces in individual positions were found to have large deviations. For cases A2 and B2, which were placed at distances of 15 m, the prediction deviations had ranged between −52.80% and −24.29%, respectively. It was determined that this may have been due to some accidental working conditions, which had been encountered during the pipe jacking process. Overall, the average deviations of the jacking forces for the A2 and B2 estimation formulae obtained by the displacement control method were −26.69% and −10.37%, respectively. Also, the estimated average deviations of A2 and B2 by the theoretical estimation formula were −31.43% and −17.17%, respectively. It was found that the former was significantly lower than the latter. Therefore, it was shown in this study that the displacement control method, which had considered the construction processes, was more advantageous than the traditional theoretical estimation formula in regard to the accurate predictions of the jacking forces on the rectangular jacking pipes examined in this study.
4.4. Discussion of the Results
In the current research study, although the jacking forces of cases A2 and B2 under the same parameter conditions had been successfully predicted by the numerical simulation fitting results of cases A1 and B1, respectively; the deviations were obvious. The jacking forces of cases A1 and B1 were predicted by their individual numerical simulation fitting results, respectively, and the deviations were small. It could be seen in the comparison results and arrangement of data, as well as the trends of fitting functions, that the jacking forces in the middle sections of jacking pipes could be effectively predicted using a finite element method of displacement control, which considered the grouting pressure of the mud sleeves on the outer walls of jacking pipes. In other words, satisfactory prediction accuracy could be obtained by adjusting the grouting pressure of each set of models.
In accordance with to the numerical simulation and monitoring results, the respective logarithmic fitting regression lines were obtained, as shown in Figures 9 and 10. As can be seen in the figure, the functional relationships between the jacking forces and jacking distances of the rectangular jacking pipes had satisfied the logarithmic relationships, which was observed to be different from the linear relationships of the traditional theoretical formulae. Furthermore, as can be seen from the logarithmic relationships, the jacking forces were also composed of two parts. It was found that part of the jacking force included the contact resistance between the jacking head-end faces and the surrounding soil. However, under the same parameter conditions, the jacking head resistance was a constant.
The other part of the jacking force included the contact side resistance between the outer walls of the jacking pipes and the surrounding rock. This part had increased with the jacking distances, not as a linear function, but as a logarithmic function. It was observed that there were many factors that had influenced the jacking forces, and the deviations of the traditional theoretical formula were often large. As can be seen in Figures 9 and 10, the grouting pressure had sometimes played a critical role in jacking force estimations. During the jacking processes of the pipe jacking, the grouting pressures had fluctuated between 0.1 and 0.2 MPa and were not a fixed value. It could be seen that for shallow buried jacking pipe, the grouting pressure was much higher than that of the overlying soil layer. In a sense, the grouting pressure and friction coefficients had jointly determined the variable parts of the jacking force characteristics. The fluctuations in the grouting pressure during the jacking processes had caused the jacking forces to rise with the jacking distances. This law could be seen in the fitting curves of the measured jacking forces of the four jacking pipes examined in this study.
It was observed in this study that when compared with circular jacking pipes, it was much simpler to consider the contact areas between the outer walls of jacking pipes and the surrounding soil in the rectangular jacking pipes. Generally speaking, in accordance with previous engineering experience, the outer walls of the bottom edges of the rectangular jacking pipes will be in full contact with the surrounding soil. Meanwhile, the top edges and outer walls on both sides will be in contact with the surrounding mud. It has been determined that under stable slurry jacket conditions, the side friction coefficients of the same slurry can generally be considered as a constant. Therefore, the influences of the side friction coefficients on the jacking force estimations were not considered in this study.
It was previously determined that regardless of whether the traditional theoretical formula or other empirical formulae were implemented, the jacking forces were directly proportional to the jacking distances. However, in actual pipe jacking operations, although the jacking forces increase with the jacking distances, the increases will tend to fluctuate. This may be due to the fact that the jacking forces may be influenced by accidental factors, such as human and mechanical deviations, geological changes, and so on. The empirical formula that was used in this study was in the form of a logarithmic function. It was observed that in both the simulated fitting curves or the actual monitoring curves, the correlation coefficients of the fitting curves were higher than those of the linear curves. It was found that, to some extent, the logarithmic function curves were able to better reflect the actual situations of the pipe jacking construction in the middle sections, as well as some accidental factors that could not be reflected by the linear curves. Therefore, the logarithmic curve fitting was determined to be more reasonable.
During the processes of long-distance pipe jacking projects, the top edges and outer walls on both sides of rectangular jacking pipes will form a hyper cutting space, which is often filled with mud. As a result, the surrounding soil will not directly contact the outer walls of the jacking pipes. This presents large deformation problems in the couplings between Euler bodies and Lagrange bodies. In this study, only the effects of mud on the reductions in the friction coefficients were considered, and the top edges and outer walls on both sides of the rectangular jacking pipes were assumed to still be in direct contact with the surrounding soil. This was also determined to be an important cause of the deviations in the jacking force estimations.
According to the engineering experiences, the large-size rectangular section pipe jacking project in Chinese cities can be summarized as follows:(1)Large-size rectangular section pipe jacking project is usually used for shallow buried street passage under the city. The strata through which the jacking pipe passes are usually river terraces, and they are usually silty clay, sandy clay, clay, and other stratum that is not easy to leak mud. This type of formation tends to form a stable mud jacket, greatly reducing the jacking force.(2)The buried depth of large-size rectangular section pipe jacking usually does not exceed 10 m. If it is more than 10 m, the construction is difficult because the jacking force is too large.(3)The length-width ratio of underground street crossings in cities is usually from 1.35 to 1.7. Therefore, although the section size of the pipe jacking may be large or small, the length-width ratio of the rectangular pipe jacking is generally about 1.35 to 1.7. The length-width ratio of case A1 and B1 is 1.39.(4)The grouting pressure used for reducing side friction of rectangular pipe jacking is usually no more than 0.2 MPa. The grouting pressure is related to the section size and the buried depth of pipe jacking.
Through the above analysis, the main factors of the large-size rectangular section pipe jacking force in Chinese cities are the buried depth and the section size of pipe jacking. If the relationship between the logarithmic function coefficient of the jacking force and the buried depth of the jacking pipe and the section size can be concluded, the research results in this study have more practical significance. By changing the buried depth and section size of the numerical model, some statistical laws of the logarithmic function coefficients of the jacking force of the rectangular pipe jacking can be obtained. For case A1 and B1, other model parameters were fixed, and the buried depth of pipe jacking was changed to obtain the fitting relation curves of jacking force and jacking distance at different buried depths. The logarithmic function coefficients a and b of the jacking force are summarized as shown in Table 14. For the shallow buried rectangular pipe jacking with length-width ratio range from 1.35 to 1.7, the jacking force can be estimated according to Table 14. The outer wall perimeter of case A1 is 25.8 m. The outer wall perimeter of case B1 is 34.7 m. The jacking force of other outer wall perimeter is calculated by interpolation of outer wall perimeter, and the jacking force of other buried pipes is also calculated by depth interpolation.
5. Conclusions and Suggestions
The results of this study showed that a pipe-soil total contact displacement control model, which considered the grouting pressure, could be used to estimate the jacking forces of large cross-sectioned rectangular jacking pipes. The influences of the grouting pressure and side friction coefficients on the jacking forces were considered in this study’s models. In order to ensure that the simulations were accurate, this study established the jacking force estimation models of cases A1 and B1.
According to the numerical results, the conclusions are summarized as follows:(1)During the middle jacking stage, the applied displacement control finite element method, with consideration given to the grouting pressure and side friction coefficients, was able to produce accurate jacking force estimations.(2)A regression equation could be applied to the construction of the rectangular pipe jacking with the same section size and buried depth under the conditions of silty clay layers.(3)The applied finite element method for estimating the jacking forces based on displacement control was also found to be applicable to the cases of rectangular pipe jacking under other conditions.(4)In the empirical formula of the jacking force estimations, it was determined that the jacking forces had increased with the jacking distances. However, the increases had not been linear. In real scenarios, the jacking forces tend to be large and jump in the transition section of the shield TBM launching shafts and receiving wells.(5)The empirical formula established in this study was limited to the middle stage of the jacking process.
Because the constructions of long-distance and large cross-sectioned rectangular pipe jacking have become popular, it is potentially worthwhile to apply finite element methods in which grouting pressure and pipe-soil total contact displacement control are considered when simulating future pipe jacking construction projects. It has been observed that under allowable jacking forces, reasonable grouting pressure and side friction coefficients can be calculated in reverse, after which the construction parameters of the pipe jacking can be effectively optimized. In addition, segmental displacement control methods can be used to analyze the jacking forces of curved jacking pipes, which will potentially open up new techniques for estimating the jacking forces of curved pipe jacking, as well as the optimization of construction parameters.
Notations
| γt: | Unit weight of the soil (kN/m3) |
| P0: | Earth pressure at the top of the pipe (kN/m2) |
| φ: | Friction angle (°) |
| PW: | Hydrostatic head pressure of the jacking pipe (kN/m2) |
| ψ: | Dilation angle (°) |
| K0: | Coefficient of earth pressure on the side of pipe jacking |
| σc: | Yield strength (kPa) |
| : | Unit weight of the water (kN/m3) |
| E: | Young’s modulus (kN/m2) |
| H: | Pipe jacking buried depth (m) |
| V: | Poisson’s ratio |
| H1: | Pipe jacking height (m) |
| Fp: | Jacking force (kN) |
| A: | Pipe jacking width (m) |
| l0: | Socket circumference (m) |
| fk: | The average friction resistance per unit area between the pipe wall and the soil (kN/m2) |
| l: | Pipe jacking distance (m) |
| NF: | End resistance of pipe jacking machine (kN) |
| S: | Pipe jacking machine excavation area (m2). |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.