A Segmented Calculation Method for Friction Force in Long-Distance Box Jacking Considering the Effect of Lubricant

Abstract

In box jacking, injecting lubricant around the box is an essential method to reduce excessive friction forces caused by the interaction between the box and soil. This method introduces complexity to factors controlling the friction forces, such as the pipe-soil contact state, earth pressure, and friction coefficient. In particular, during long-distance construction, different lubricant conditions come into play. These intricate scenarios hinder the accurate estimation and control of friction force throughout the entire construction period. This study analyzed the variation patterns of frictional resistance based on monitoring data from two actual cases. The lubricant condition changes during the long-distance jacking process were categorized, the effect of lubricant actions on factors controlling friction force in each segment was discussed, and a new method for calculating friction forces by partitioning the long-distance box jacking was proposed. This approach aims to enhance the prediction accuracy and was compared with the results obtained from existing models. The rationality of the new model was further validated by combining numerical simulation results with field data. The results indicate that the proposed segmented calculation model demonstrates better prediction accuracy when facing variations in actual construction conditions. It can serve as a reference for the process design and construction control of long-distance box jacking.

1. Introduction

The box jacking technique has the advantages of being green, economical, and environmentally friendly and safe, and the space utilization rate is higher than that of circular sections, which have been widely used in urban underground structure construction, such as subway tunnels, utility tunnels, and pedestrian underpasses [1,2,3]. As shown in Figure 1a, the technique forms a tunnel using a boring machine, and the hydrocylinder provides a jacking force to synchronously push the box pipe to realize the trenchless construction of underground structures. In order to maximize the use of shallow underground spaces to meet the needs of urban underground facilities, long-distance projects have gradually increased in recent years, resulting in higher accuracy requirements for the evaluation of jacking forces. However, in practice, it has been found that the resistance calculations based on circular pipe jacking have obvious errors, which increases the cost of auxiliary facilities and creates construction challenges [4,5].

The magnitude of the jacking force is mainly determined by the cutterhead resistance of the boring machine and the friction force around the box. The cutterhead resistance is usually controlled at a constant value to prevent excavation face collapse, while the friction resistance is obtained by multiplying the earth pressure, friction coefficient, and active area. With the increase in jacking length, the area of frictional resistance becomes larger and larger, which consumes a major part of the jacking force. Therefore, an accurate calculation of friction forces is essential for project design and construction.

In long-distance pipe jacking projects, lubricants are effective and necessary to reduce the friction forces [6,7]. As shown in Figure 1b, the lubricant consists of bentonite, carboxymethyl cellulose (CMC), sodium hydroxide (NaOH), hydrolyzed polyacrylamide (PHP), and water. The proportion of each component should be optimized based on the factors of filtration loss, viscosity, thixotropy, economy, and pumping capacity so that the lubricant viscosity should be greater than 45 s and the filtration loss should be less than 15 mL/30 min. The lubricant injection pressure is usually controlled at 1.1~1.3 times the earth pressure.

Figure 1. Box jacking technical principle.

Figure 1. Box jacking technical principle.

The over-cutting space around the box is produced by the size difference between the tunnel and the box, providing a space for lubricant injection. However, this also changes the factors controlling the friction forces, such as the box-soil interaction, earth pressure, and friction coefficient.

Numerous studies have been conducted on the friction force for box jacking. Assuming full circumference pipe-soil interaction and neglecting lubricant action, the jacking force formulas are derived for various burial depths with different earth pressure models, such as the soil column model, the Protodyakonov load-transmitting arch model, and the Terzaghi model [8]. For circular pipe jacking, evidence suggests that the actual force requirements are much less than the calculated value based on the above assumptions. Therefore, researchers have suggested partial pipe-lubricant interaction assumptions [6,9,10], which are referenced to solve box-jacking friction calculation models. Jiao et al. [11] used numerical simulations to predict the jacking force of box jacking, considering different box-soil contact areas. Xue [12] established the jacking force formula for box jacking using the Bill Bowman theory and additionally considering the friction resistance caused by the boring machine shell. Wen et al. [13] calculated earth pressure based on the soil column model and the Protodyakonov model and established the calculation formula for friction resistance for five kinds of box-soil interaction models. Ma et al. [14] and Zhang et al. [15] proved that the assumption of partial box-soil contact is reasonable and proposed a calculation method for box-jacking friction forces by considering the soil arching effect.

However, the synchronous influence of lubricant on the three control factors is not considered completely in the above calculation model. More importantly, there is a general lack of discussion on the variation in lubricant conditions during long-distance jacking, and the entire construction process is assumed to be an idealized single working condition, ignoring the interference of uncertain factors such as changes in lubricant properties and slurry fluctuations.

In this study, the control factors of friction were analyzed in detail via theoretical deduction and shear tests. The lubricant effect on each control factor in the whole jacking length is discussed according to the measured data of two long-distance box jacking cases. On this basis, the commonly existing friction prediction models are summarized, and a new friction force calculation model that is segmented according to the effect of lubricant is proposed. Moreover, the accuracy of the new model was verified using theoretical calculation and numerical simulation.

2. Factors Controlling Friction Forces

2.1. Box-Soil Interaction State

Table 1. Dimensions, buoyancy, and gravity per unit length of common box.

H (m)	B (m)	b (m)	Box Gravity F1 (kN)	lubricant Buoyancy F2 (kN)	F2 − F1 (kN)
3.5	3.8	0.4	130	139.65	9.65
4	6	0.45	204.75	252	47.25
4.2	6.9	0.5	252.5	304.29	51.79
5.5	9.1	0.7	462	525.525	63.525

shows the dimensions of the common box proposed in China. It can be found that the lubricant buoyancy is greater than the box gravity due to the large internal clearance of the box, and the lubricant tends to gather at the bottom of the box under the action of gravity. Therefore, it can be deduced that the box pipe has a tendency to float upward when the over-cutting space is filled with lubricant.

As shown in Figure 2, under the lubricant effect, there are four contact status evolutions during long-distance jacking.

(1) Full box-soil interaction: As shown in Figure 2a, when no lubricant is injected, the soil around the tunnel cannot be stabilized after the excavation using the boring machine, and there is a 20–50 mm over-cutting space between the hole and the subsequent box pipe, resulting in radial soil collapse toward the box’s outside surface.

(2) Three or two side box-soil interaction: As shown in Figure 2b, after the first injection of lubricant, the tunnel cannot be immediately stabilized. However, the lubricant permeates into the surrounding soil, gradually reducing the soil permeability and leading to the retention of some lubricant in the overcut space between the box and the soil. Due to the accumulation of lubricant on the sidewalls and bottom of the box under the action of gravity, the external earth pressure is isolated, and the box forms three or two sides of contact with the soil.

(3) One side box-soil interaction: As shown in Figure 2c, with multiple injections of lubricant, an impervious clay layer, that is, a mud cake, is formed, and the over-cutting space is filled with lubricant. The soil around the tunnel remains stable under the action of the lubricant pressure. Due to the buoyancy of the lubricant, the box floats and tightly adheres to the tunnel roof, causing direct contact between the top of the box and the soil.

Figure 2. The box-soil contact status evolutions.

Figure 2. The box-soil contact status evolutions.

2.2. Earth Pressure around the Box

The over-cutting space causes the soil to loosen, creating a new stress state. The Terzaghi loose soil pressure theory considers the restraining effect of the soil column around the tunnel on the overlying soil mass, known as the soil arching effect. This concept has been adopted by circular jacking specifications in various countries [14,15].

As shown in Figure 3, the Terzaghi model assumes that part of the soil pressure in the shear zone is balanced by shear resistance. The earth pressure on the box can be calculated according to Equation (1).

$$\begin{array}{c}{\sigma }_{\mathrm{z}}=\frac{1-{\mathrm{e}}^{-2K\tan \phi \frac{Z}{B_1}}}{2K\tan \phi \frac{Z}{B_1}}\left(\gamma -\frac{2c}{B_1}\right)Z+q_0{\mathrm{e}}^{-2K\tan \phi \frac{Z}{B_1}}\\ B_1=B+2H\tan \left(45°-\phi /2\right)\end{array}$$

(1)

where σ_z—earth pressure (kPa);

c—the cohesion (kPa);

φ—the internal friction angle (°);

B₁—the the shear zone width (m);

B—the box width (m);

H—the box height (m);

q₀—the additional load on the ground (kPa);

γ—the unit weight of the soil (kN/m³);

Z—the box buried depth (m);

K—the earth pressure coefficient, which numerous experiments and simulation studies suggest setting it to 1.

Figure 3. Terzaghi loose soil pressure calculation model.

Figure 3. Terzaghi loose soil pressure calculation model.

Equation (1) is applicable for calculating the soil pressure under conditions without lubricants. This paper provides a correction considering the effect of the lubricant. When the gate moves downward relative to the stationary soil, shear stress is generated upward at the shear interface, known as positive soil arching. When the gate moves upward relative to the stationary soil, shear stress is generated downward at the shear interface, known as negative soil arching [16]. As shown in Figure 4, during the repeated injection and loss of lubricant, the box continuously floats up and sinks. Therefore, the soil pressure at the top of the box is controlled between positive and negative soil arching. Meanwhile, Evans [15] introduced the non-associated plastic flow rule into the soil shear strain problem, as shown in Equation (2):

$$\tau =\sigma \frac{\cos \psi \sin \phi }{1-\sin \psi \sin \phi }$$

(2)

Using on-site measurements, numerical simulations, and PIV experiments, research on the parameters of the dilation angle (ψ) has demonstrated that it can be set to 0° [17,18]. Therefore, Equation (2) can be simplified to τ = σsinφ.

Figure 4. Effect of lubricant on soil arching direction.

Figure 4. Effect of lubricant on soil arching direction.

(1) Earth pressure in the positive soil arching state.

As shown in Figure 4a, as the lubricant gradually dissipates, the soil within the shear zone slides downward relative to the soil on both sides. Shear stress acts upward to inhibit its sliding tendency, presenting a positive soil arching state, and the earth pressure reaches a minimum. Considering the force transmission via particle compression between layered soils, the bottom earth pressure of the upper layer is defined as the additional load on the top of the lower layer, as shown in Equation (3):

$$\begin{array}{c}{\sigma }_{zn}=\frac{1-e^{-2\sin \phi \frac{Z_n}{B_1}}}{2\sin \phi \frac{Z_n}{B_1}}\left({\gamma }_n-\frac{2c}{B_1}\right)Z_n+{\sigma }_{zn-1}{e}^{-2\sin \phi \frac{Z_n}{B_1}}\\ B_1=B+2H\tan \left(45°-\phi /2\right)\end{array}$$

(3)

where σ_zn—the earth pressure in the nth layer (kPa);

Z_n—the soil thickness in the nth layer (m).

(2) Earth pressure in the negative soil arching state.

As shown in Figure 4b, the lubricant volume in the over-cutting space gradually increases during the lubricant injection. The formed mud cake seals the filtration channel, causing the lubricant pressure to act directly on the box and the soil around the tunnel. When the lubricant pressure exceeds the static soil pressure, there will be an upward shear tendency in the overlying soil. The soil within the shear zone slides upward relative to the surrounding soil, and the shear stress is downward, indicating a negative soil arching state. The earth pressure reaches a maximum, as shown in Equation (4):

$$\begin{array}{c}{\sigma }_{zn}=\frac{e^{2\sin \phi \frac{Z_n}{B_1}}-1}{2\sin \phi \frac{Z_n}{B_1}}\left({\gamma }_n-\frac{2c}{B_1}\right)Z_n+{\sigma }_{zn-1}{e}^{2\sin \phi \frac{Z_n}{B_1}}\\ B_1=B+2H\tan \left(45°-\phi /2\right)\end{array}$$

(4)

2.3. Box-Soil Friction Coefficient

As shown in Table 2, the friction coefficient (μ) can be calculated using back analysis based on field monitoring data, a method that has been adopted in various friction solutions [19,20]. Additionally, some standards reduce the friction coefficient based on the internal friction angle of the soil, which fails to reflect the lubricant effect.

In practical engineering, the friction coefficient (μ) is related to soil type and lubricant properties, making it challenging to accurately evaluate using the mentioned methods. Therefore, conducting contact shear tests for box-soil-lubricant interactions is a more feasible method for obtaining the friction coefficient [21,22].

Table 2. Friction coefficients μ proposed in existing studies.

Category	Soil Type	Friction Coefficient
		Without Lubricant	With Lubricant
Stein [19]	Sand	0.3~0.4	0.1~0.3
	Clay	0.2~0.3
Pellet-beaucour [20]	Sand	0.2~0.4	0.07~0.1
	Clay	0.1~0.3

Table 2. Friction coefficients μ proposed in existing studies.

Category	Soil Type	Friction Coefficient
		Without Lubricant	With Lubricant
Stein [19]	Sand	0.3~0.4	0.1~0.3
	Clay	0.2~0.3
Pellet-beaucour [20]	Sand	0.2~0.4	0.07~0.1
	Clay	0.1~0.3

In this study, shear friction tests on box-soil interactions were conducted, measuring the friction coefficient under different lubrication conditions and soil particle sizes. As shown in Figure 5, the upper shear box was filled with soil, and the lower shear box was placed with concrete blocks simulating the outer surface of the box. Constant normal stress is applied to the soil-concrete interface by a vertical motor, and simultaneously, the upper shear box was fixed while the lower shear box was driven horizontally along a straight guide rail at a constant speed using a horizontal motor. The friction force exerted on the driven lower shear box was measured. In addition, the test considered the lubrication conditions at the soil-concrete interface. As the over-cutting space in practical engineering can accommodate a 2 cm thick layer of lubricant, a 2 cm thick layer of bentonite lubricant was poured onto the concrete block. The average surface roughness of the concrete block was 20.23 μm, while the actual box jacking pipe section had an average surface roughness of 15.45 μm. The surface roughness difference of within 5 μm meets the test requirements. The experimental lubricant used a material and ratio commonly employed in sand-layered box jacking projects (bentonite:water = 0.06:1). The funnel viscosity was 69 s, and the fluid loss was 12 mL/30 min, meeting the pumping capacity and anti-filtration requirements of practical engineering.

As shown in

Table 3. The friction coefficients f by shear test.

	Soil Type	Gravelly Sand	Coarse Sand	Medium Sand	Fine Sand	Silty Sand	Clay
Lubrication Condition
Without lubricant (water-free)		0.586	0.594	0.629	0.619	0.620	0.782
Without lubricant (Saturation)		0.559	0.603	0.581	0.606	0.603	/
With slurry lubricant		0.166	0.164	0.167	0.175	0.207	0.062
Reduced proportion (fwater-free − fwith slurry lubricant)/fwater-free		71.7%	72.4%	73.4%	71.7%	66.6%	92.1%

, the friction coefficients for various soils were obtained using shear tests. Compared with Table 2, the classification of soil types and lubrication conditions is more detailed, providing a reference for other studies. It can be observed that the friction coefficients under saturated and dry soil conditions were relatively close, indicating limited lubrication effects from pore water. Under lubrication conditions, the friction coefficient of sand decreased by 66.6~73.4%, and that of clay even decreased by 92.1%. Field data also show that the friction force can be reduced by 70~90% using a large amount of lubricant, and the friction coefficient no longer depends on the soil type but on the liquid limit of the lubricant [23,24].

2.4. Box-Lubricant Friction Force

The box-lubricant friction force is the adhesive shear action of flowing liquid on a solid interface. It differs in mechanism from the formation of solid–solid shear friction. Therefore, it is quantitatively characterized using the flat-plate fluid model and the Herschel–Bulkley rheological model.

The flat-plate fluid model for box jacking is shown in Figure 6. The shear velocity (U) of the lubricant is given by:

$$U=\frac{V}{d}y$$

(5)

where V—the box moving speed;

d—the lubricant thickness.

Figure 6. The flat-plate fluid model.

Figure 6. The flat-plate fluid model.

The shear velocity of the lubricant between the plates varies along the y coordinate, with the soil-lubricant interface at 0 and the box-lubricant interface at v. As the lubricant is a thixotropic non-Newtonian fluid, a modified Herschel–Bulkley rheological model was adopted, as shown in Equation (6):

$$\tau ={\tau }_y+K{\left(\frac{dU}{dy}\right)}^n$$

(6)

where τ—the shear stress (Pa);

τ_y—the yield shear stress (Pa);

K—the consistency coefficient (Pa·sⁿ);

n—the flow behavior index (s⁻¹).

By substituting Equation (5) into Equation (6), the formula for calculating the box-lubricant friction force can be obtained:

$$f_2={\displaystyle \sum \tau S_2}={\displaystyle \sum \left({\tau }_y+K{\left(\frac{V}{d}\right)}^n\right)S_2}$$

(7)

where τ_y, K, and n can be calculated using Equation (8); θ₆₀₀, θ₃₀₀, and θ₃ are measured using a rotational viscometer; and S₂ is the contact area between the box and the lubricant.

$$\left\{\begin{array}{l}{\tau }_y=0.511{\theta }_3\\ K=0.511\left({\theta }_{300}-{\theta }_3\right)/{511}^n\\ n=3.322\lg \left[\left({\theta }_{600}-{\theta }_3\right)/\left({\theta }_{300}-{\theta }_3\right)\right]\end{array}\right.$$

(8)

3. Friction Force Variation Characteristics

3.1. Project Introduction

Two cases in China, Case A and Case B, were used in this study. As shown in Figure 7, Case A is located in Jiangsu City. The box’s external dimensions are 9.1 m in width, 5.5 m in height, and 1.5 m in length, with a wall thickness of 0.65 m. The total jacking length is 233.6 m, and the average buried depth is 9 m. During construction, lubricant is injected into over-cutting space through 10 injection holes. The stratum of the box jacking is silty sand with sily soil ➅-1 and silty sand ➅-2; the soil properties are shown in Table 4. As shown in Figure 8, Case B is located in Shanghai City. The box dimensions are 9.8 m in width, 6.3 m in height, and 1.5 m in length, with a wall thickness of 0.7 m. The total jacking length is 163 m, and the average buried depth is 12 m. Each box is also designed with 10 injection holes. The stratum of the box jacking is silty sand ➁-3, and the soil properties are detailed in Table 5.

Table 4. Soil properties in Case A.

Code	Stratum	Thickness (m)	Density (g/cm3)	Cohesion (kPa)	Friction Angle (°)
➀-5	Plain fill	4.3	1.92	27.9	16.8
➃	Clay	2.0	1.99	41.4	15.7
➄	Silty clay with silt	1.3	1.92	16.8	22.7
➅-1	Silty sand with silt	4.4	1.91	4.6	31.4
➅-2	Silty sand	4.0	1.94	3.8	33.4
➆	Silty clay	5.7	1.93	25.7	17.7

Table 4. Soil properties in Case A.

Code	Stratum	Thickness (m)	Density (g/cm3)	Cohesion (kPa)	Friction Angle (°)
➀-5	Plain fill	4.3	1.92	27.9	16.8
➃	Clay	2.0	1.99	41.4	15.7
➄	Silty clay with silt	1.3	1.92	16.8	22.7
➅-1	Silty sand with silt	4.4	1.91	4.6	31.4
➅-2	Silty sand	4.0	1.94	3.8	33.4
➆	Silty clay	5.7	1.93	25.7	17.7

Table 5. Soil properties in Case B.

Code	Stratum	Thickness (m)	Density (g/cm3)	Cohesion (kPa)	Friction Angle (°)
➁-3	Silty sand	18.5	1.85	6	30.5
➃	Mucky clay	7	1.69	14	11.5
➄	Silty clay	3.2	1.82	16	20.5

Table 5. Soil properties in Case B.

Code	Stratum	Thickness (m)	Density (g/cm3)	Cohesion (kPa)	Friction Angle (°)
➁-3	Silty sand	18.5	1.85	6	30.5
➃	Mucky clay	7	1.69	14	11.5
➄	Silty clay	3.2	1.82	16	20.5

Figure 7. Overview of Case A.

Figure 7. Overview of Case A.

Figure 8. Overview of Case B.

Figure 8. Overview of Case B.

3.2. Field Monitoring Results

The actual jacking forces during construction were obtained (see Figure 9). As shown in Figure 9a, during the initial 0–17 m of jacking length in Case A, no lubricant was injected, and the jacking force increased linearly with the jacking length. The jacking force decreased gradually after a jacking length of 17~80 m due to the significant friction reduction effect after lubricant injection. From 80 m to 240 m, the jacking force increased linearly again, but the slope was significantly smaller than that from 0–17 m. As shown in Figure 9b, the measured jacking force in Case B also showed the same variation pattern.

The cutterhead resistance is typically kept constant, and the slope of the jacking force curve can reflect the change in friction forces [10,14,15]. It was observed that the construction process includes three conditions: without lubrication, lubrication gradually coming into play, and lubrication fully in play, which should be divided into zones when calculating the friction force.

(1) Without lubrication zone (WL).

During the hoisting and installation of the box pipe, the hydrocylinder will be contracted, removing the jacking force acting on the already jacked box. After the soil in front of the excavation face loses the jacking force support, active earth pressure will be generated, which will cause the cutterhead and the already jacked box to have a tendency to retreat towards the launching shaft, and the acting direction of the friction force will be reversed. If the friction force cannot balance the active earth pressure, the slight retreat will induce instability of the excavation face and may even cause accidents such as collapse and ground subsidence.

Therefore, without injecting lubricant in the initial stage of jacking, a significant friction force can be achieved to prevent the instability of the excavation face when the hydrocylinder contracts. This is a characteristic method for long-distance box jacking.

(2) Lubrication gradually coming into play zone (GP).

After the first injection of lubricant, free water in the lubricant filters into the surrounding soil, causing a rapid decrease in lubricant pressure, and it is difficult to maintain stability in the soil around the tunnel. As the solid particles of the bentonite fill the soil pores via flocculation and bridging, a mud cake is formed in the over-cutting space, reducing the box-soil friction coefficient. Since the complete filling of soil pores and the continuous stability of the surrounding soil require repeated actions of the lubricant, the box-soil contact state can not be changed from full box-soil interaction to one-side box-soil interaction immediately; instead, it gradually transitions to a state where three sides or two sides of the pipe are in contact.

(3) Lubrication fully in play zone (FP).

After multiple injections of lubricant, a complete mud cake is formed around most box sections, stabilizing the surrounding soil. The contact state develops into a one-side box-soil interaction, and the unit area friction force significantly decreases. The earth pressure follows the evolution cycle of positive soil arching and negative soil arching as the lubricant is injected and filtrated. Although the jacking force still increases with the jacking length, the required jacking force per unit length (i.e., the slope) is significantly lower than that in the case of jacking without lubricant.

Figure 9. Actual jacking forces during construction.

Figure 9. Actual jacking forces during construction.

4. Theoretical Model for Friction Force

4.1. Existing Friction Force Models

Some countries, including the Japan Society of Trenchless Technology (JSTT) (Tokyo, Japan) and the China Society of Trenchless Technology (CSTT) (Beijing, China), have formulated technical standards for the calculation of friction force in box jacking projects. However, there is no calculation model for box jacking specified by the British Pipe Jacking Association (PJA 1995), the American Society of Civil Engineers (ASCE 27), the French Society of Trenchless Technology (FSTT 2006), the German Trenchless Association (ATV A-161), and do not stipulate a separate calculation model.

Table 6 inventories the calculation formulas for the friction force in the above standards. These formulas can be classified into two patterns: (1) theoretical derivation based on different earth pressure models, including Japan’s standard BMDN and China’s standard TSP, and (2) data statistics based on engineering experience, such as China’s standard TCB and TSR.

TCB and TSR have the same formula, calculating the overall friction force by considering the outer area of the box and the friction force per unit area. The selection of the friction force unit is based on extensive engineering data, as presented in Table 7. However, data statistics are limited due to the lack of mechanical explanation, which leads to prediction deviations. In addition, BMDN and TSP primarily focus on precise calculations of normal earth pressure, differing only in consideration of the soil arching effects. Although the above models do not fully consider the lubricant effect or use segmented calculations, these models have been widely used in practical engineering, and the prediction accuracy will be discussed below.

Table 6. Review of the friction force for box jacking.

Technical Standard	Calculation Model Ff	Symbols and Notes
BMDN: Box jacking method design needle in Japan	$F_f=\mu L\beta \left(\left(B_c{\sigma }_z+G\right)+B_{c}c\right)$	μ-tan(φ/2); L-jacking length; β-reduction factor (0.45 for sand soil); Bc-circumference of box; σz-normal force from Terzaghi model; G-the weight of box pipe; c-cohesion. α-assurance factor, 1.2; σ1-normal force from soil column model; μ1,2,3-the friction coefficients on the top, bottom, and side walls of the box pipe, respectively; Ka-active soil pressure ratio. fu-friction force per unit area.
TSP: Technical specification for pipe jacking of water supply and sewerage engineering (T/CECS 246-2020) [25]	$F_f=\alpha L\left(B{\sigma }_1{\mu }_1+B\left({\sigma }_1+G\right){\mu }_2+2K_{a}H{\sigma }_1{\mu }_3\right)$
TCB: Technical code for box jacking construction of utility tunnel (DB32/T 2020) [26]	$F_f=2\left(B+H\right)L{f}_u$
TSR: Technical specification for rectangular pipe jacking engineering (DBJ/T 15-229-2021) [27]

Table 6. Review of the friction force for box jacking.

Technical Standard	Calculation Model Ff	Symbols and Notes
BMDN: Box jacking method design needle in Japan	$F_f=\mu L\beta \left(\left(B_c{\sigma }_z+G\right)+B_{c}c\right)$	μ-tan(φ/2); L-jacking length; β-reduction factor (0.45 for sand soil); Bc-circumference of box; σz-normal force from Terzaghi model; G-the weight of box pipe; c-cohesion. α-assurance factor, 1.2; σ1-normal force from soil column model; μ1,2,3-the friction coefficients on the top, bottom, and side walls of the box pipe, respectively; Ka-active soil pressure ratio. fu-friction force per unit area.
TSP: Technical specification for pipe jacking of water supply and sewerage engineering (T/CECS 246-2020) [25]	$F_f=\alpha L\left(B{\sigma }_1{\mu }_1+B\left({\sigma }_1+G\right){\mu }_2+2K_{a}H{\sigma }_1{\mu }_3\right)$
TCB: Technical code for box jacking construction of utility tunnel (DB32/T 2020) [26]	$F_f=2\left(B+H\right)L{f}_u$
TSR: Technical specification for rectangular pipe jacking engineering (DBJ/T 15-229-2021) [27]

Table 7. Recommended friction force per unit area values from TCB and TSR.

Soil Type	Recommended Value (kN/m2)
Clay	3~5
Fine sand	5~8
Silty sand	8~11
Medium coarse sand	11~16

Table 7. Recommended friction force per unit area values from TCB and TSR.

Soil Type	Recommended Value (kN/m2)
Clay	3~5
Fine sand	5~8
Silty sand	8~11
Medium coarse sand	11~16

4.2. Segmented Calculation Model

(1) WL

The friction force is entirely composed of box-soil frictional resistance in this condition. The top and side walls of the box are subject to active earth pressure, which is calculated using the traditional Terzaghi model, and the bottom is subject to foundation reaction force. The friction coefficient μ is selected from the shear test value without lubrication condition in Table 3, and the calculation formula under the condition without lubrication is as follows:

$$F_f=(2B{\sigma }_z+2H{\sigma }_s+BG)L\mu$$

(9)

(2) GP

After the lubricant is injected, the full box-soil contact gradually changes into three or two side box-soil contact. Meanwhile, the lubricant action has a weak effect on the direction of soil arching, and the earth pressure is calculated according to the positive soil arching state. The box-soil interface develops from a soil-concrete interface to a lubricant-soil-concrete mixed interface. The friction coefficient μ is selected from the shear test value of the lubrication condition in Table 3, and the calculation formula is obtained:

$$\begin{array}{l}Three\text{-}side:F_f=\left(B{\sigma }_z\mu +2H{\sigma }_s\mu +B\left({\tau }_y+K{\left(\frac{V}{d}\right)}^n\right)\right)L\\ Two\text{-}side:F_f=\left(B{\sigma }_z\mu +H{\sigma }_s\mu +\left(H+B\right)\left({\tau }_y+K{\left(\frac{V}{d}\right)}^n\right)\right)L\end{array}$$

(10)

(3) FP

When the lubricant is fully utilized, the contact state changes from three or two-side box-soil contact to one-side box-soil contact. Considering the effect of the lubricant on the earth pressure, the pressure should be calculated according to the modified Terzaghi’s model for both the maximum and minimum values. The friction coefficient is selected considering the change in contact interface characteristics. The calculation formula is as follows:

$$F_f=\left(B{\sigma }_{zn}\mu +\left(2H+B\right)\left({\tau }_y+K{\left(\frac{V}{d}\right)}^n\right)\right)L$$

(11)

4.3. Comparison of Friction Force Solutions

The calculated values of the existing models and the segmented calculation model were contraposed with the measurements. The calculation parameters are provided in Table 8. According to the results of the shear tests, the friction coefficients for Cases A and B were 0.603 during the WL stage and 0.207 during the GP stage. In the FP stage, an additional high-viscosity bentonite slurry was injected at the box top in Case A, resulting in a clay-concrete contact interface at the top of the box, and a friction coefficient of 0.062 was used. For Case B, the average of the friction coefficients for clay and sand was used, resulting in a value of 0.1345. Additionally, to compare the applicability of the existing models, TSP’s friction coefficient was assumed to be the same as that of the FP stage, BMDN’s friction coefficient was calculated using the specified formula, and the unit area friction force for TCB/TSR was selected to be within the recommended range of 8 to 11 kN/m².

As shown in Figure 10, the segmented calculation model proposed in this paper demonstrated good predictive accuracy in the three segments of Case A and Case B. In contrast, the existing models failed to predict the friction force for different lubrication conditions during long-distance jacking, especially in the WL and FP stages. The friction force was significantly underestimated or overestimated, posing challenges to cost savings and construction safety.

In particular, during the 0~17 m in Case A and 0~11 m in Case B without lubrication, both the existing model and the segmented calculation model adopted the assumption of full box-soil interaction. However, the segmented model considered the actual contact interface of this condition, resulting in calculated values closer to the measured values, while the existing model’s assumption about lubrication condition friction coefficients was clearly inconsistent with the actual situation. In the GP stage, the friction force gradually decreased with increasing jacking length. Due to the unstable changes in the contact state and friction coefficients, the overall results showed fluctuations around the calculated values of the three-side and two-side box-soil interaction models. The FP stages of 80~233.6 m in Case A and 31~163 m in Case B, where slurry lubrication played a predominant role, accounted for the largest proportion of the entire jacking length. The proposed model considered the effect of earth pressure on the friction force during the lubricant injection-filtration process. The predicted curve of the maximum and minimum values for these stages essentially covered the measured values, with the actual friction forces fluctuating between the curves.

Table 8. The calculation parameters for friction force.

Calculation Models		Interaction State	Friction Coefficient μ
			Case A	Case B
Existing models	TSP	Full box-soil interaction	0.062	0.1345
	BMDN		0.2905	0.2726
	TCB/TSR		fu = 8~11 kN/m2
Models in this paper	WL	Full box-soil interaction	0.603
	GP	Three/two side box-soil interaction	0.207
	FP	One side box-soil interaction	0.062	0.1345

Table 8. The calculation parameters for friction force.

Calculation Models		Interaction State	Friction Coefficient μ
			Case A	Case B
Existing models	TSP	Full box-soil interaction	0.062	0.1345
	BMDN		0.2905	0.2726
	TCB/TSR		fu = 8~11 kN/m2
Models in this paper	WL	Full box-soil interaction	0.603
	GP	Three/two side box-soil interaction	0.207
	FP	One side box-soil interaction	0.062	0.1345

The errors between the calculated values and measured values for the maximum friction force are presented in

Table 9. Error comparison of maximum friction force.

Maximum Friction Force (kN)/Error (%)	TSP	BMDN	TCB/TSR	Segmented Calculation Model		Measured Value
				Minimum	Maximum
Case A	68,897	102,727	54,569	14,528	24,164	23,110
	198.1%	344.5%	136.1%	−37.1%	4.6%
Case B	134,879	95,590	37,959/52,194	32,841	56,060	47,055
	288.9%	103.1%	−19.3%/10.9%	−30.2%	19.1%

Notes: Error% = (Calculated value − Measured value)/Measured value.

. It can be observed that, despite considering the lubrication-reducing effect, the existing models still overestimate the maximum friction force. In Case A, the errors of the minimum and maximum friction force calculated by the proposed model were −37.1% and 4.6%, respectively. However, although TSP adopts the same friction coefficient as the proposed model, the error in the maximum jacking force still reached 198.1%. BMDN simplified the effect of the box-soil contact area and friction coefficient using the reduction factor β, but the calculation result was still conservative, and the error reached 344.5%. TCB/TSR, which used the minimum unit area friction force, also exhibits an error of 136.1%. In Case B, the errors of the minimum and maximum friction forces calculated by the proposed model were −30.2% and 19.1%, respectively. The errors for TSP and BMDN reached 288.9% and 103.1%, respectively. Based on the recommended range of unit area friction force, TCB/TSR’s calculation results showed relatively good accuracy, with errors of −19.3% and 10.9%, respectively. This suggests that the empirical values from TCB/TSR have practical utility. However, its calculation formula does not consider the burial depth, earth pressure, and specific lubrication processes, leading to unstable prediction accuracy and limiting its engineering applicability. Moreover, the phenomenon of overestimating the actual jacking force in Case A by TCB/TSR indicates that the lubrication technology in Case B can still be further optimized. Additional improvement can be realized by a high-viscosity bentonite slurry injection, improving the lubrication interface and reducing the friction coefficient, thereby controlling the friction forces within a lower range.

In summary, this paper proposes a segmented calculation method for friction forces that aligns better with actual working conditions. It provides a scientific basis for estimating and controlling friction forces during the design and construction phases.

Figure 10. Comparison of results from different calculation models.

Figure 10. Comparison of results from different calculation models.

5. Numerical Analysis

5.1. Numerical Model Establishment

To further validate the reliability of the segmented calculation model proposed in this paper, the finite element software Abaqus v2023 was employed in Case A to simulate the construction process of box jacking under different lubricant conditions. The model was simplified based on the following assumptions:

(1) Constitutive model: (i) Assume the soil is a homogeneous, continuous, isotropic elastic-plastic material, and adopt the Drucker-Prager elastic-plastic constitutive model with better convergence [14]. (ii) Neglect the effect of box joint connections. (iii) Surround the box with a 20 mm equivalent layer to simulate the lubricant, employing an elastic constitutive model.

(2) Construction process: (i) Use the birth and death element method to excavate the soil, simulating the forward jacking of the box via displacement control [28,29,30]. (ii) Neglect axial offset and box deflection during jacking, assuming the box jacks forward in a straight line. (iii) Neglect the time effects on soil consolidation settlement.

Figure 11 shows the three-dimensional finite element numerical model. The model parameters are shown in Table 10. The dimensions of the box model were 9.1 × 5.5 m, with a wall thickness of 0.65 m and a soil cover depth of 9 m. To eliminate boundary effects, the dimensions of the stratum model were 50 m in width, 30 m in height, and 253.6 m in length. The stratum model used a structured mesh, while the box and boring machine used a swept meshing command, and the element type of the box, boring machine, and soil is C3D8R (the solid hexahedron elements). The entire model consisted of 205,584 nodes and 179,492 elements. In the stratum model, the ground surface was unconstrained, and the bottom was constrained in the x, y, and z-direction displacements.

Table 10. Material parameters used in this model.

Material Type	Strata Thickness (m)	Density (g/cm3)	Young Modulus (MPa)	Poisson Ratio	Cohesion (kPa)	Friction Angle (°)
Plain fill	4.3	1.92	21.4	0.32	27.9	16.8
Clay	2	1.99	25.8	0.29	41.4	15.7
Silty clay with silt	1.3	1.92	22.9	0.32	16.8	22.7
Silty sand with silt	4.4	1.91	32.8	0.32	4.6	31.4
Silty sand	4	1.94	40	0.31	3.8	33.4
Silty clay	14	1.93	22.1	0.3	25.7	17.7
Equivalent layer	0.2	1.05	1	0.38	/	/
Box	/	2.4	3.34 × 104	0.2	/	/

Table 10. Material parameters used in this model.

Material Type	Strata Thickness (m)	Density (g/cm3)	Young Modulus (MPa)	Poisson Ratio	Cohesion (kPa)	Friction Angle (°)
Plain fill	4.3	1.92	21.4	0.32	27.9	16.8
Clay	2	1.99	25.8	0.29	41.4	15.7
Silty clay with silt	1.3	1.92	22.9	0.32	16.8	22.7
Silty sand with silt	4.4	1.91	32.8	0.32	4.6	31.4
Silty sand	4	1.94	40	0.31	3.8	33.4
Silty clay	14	1.93	22.1	0.3	25.7	17.7
Equivalent layer	0.2	1.05	1	0.38	/	/
Box	/	2.4	3.34 × 104	0.2	/	/

Figure 11. 3D finite element numerical model.

Figure 11. 3D finite element numerical model.

5.2. Condition Design and Calculation

The lubricant significantly influenced the controlling factors of the friction force; there was only one lubricate condition in the whole jacking length in previous numerical simulation studies [28,29,30], which does not reflect the changing lubrication condition in practical construction.

Based on different lubrication conditions, four models of box-soil interactions were established in this paper, including (1) the WL stage (the full box-soil interaction). (2) the GP stage (box-soil interaction occurs at the top and both side walls). (3) the GP stage (box-soil interaction occurring at the top and one side wall). (4) the FP stage (box-soil interaction only occurs at the top of the box).

The equivalent layer and surrounding soil adopted common node constraints and surface-to-surface contact with the box. The contact surface’s normal direction was considered rigid contact, and the contact surface’s tangential direction adopted a penalty function to define the friction coefficient, and its value is the same as that in Section 4.3. When simulating the friction force between the box and lubricant, the friction coefficient was defined as 0. The calculations proceeded in the following steps:

Step 1: Initial geostatic stress equilibrium.

Step 2: After excavating the soil elements, activate the equivalent layer and the box, simultaneously applying support forces.

Step 3: The displacement control method is used to push forward 1.5 m (length of a single box) and remove the support force of the excavation face.

Step 4: Repeat steps 2 and 3 until the construction is completed.

Step 5: Calculate the friction force via the integral of stress on the outer wall of the box, and export the stress nephogram.

5.3. Simulation Results Validation

As shown in Figure 12, the initial geostatic stress analysis observed a stress equilibrium, with a vertical displacement (the upward displacement is positive) less than 10-7 and the vertical stress (the downward stress is negative) exhibiting layered distribution, consistent with the initial stress conditions before the construction. The analysis focused on the stress distribution characteristics under different lubrication conditions to compare and verify the reliability of the segmented calculation model.

(1) Earth pressure

As shown in Figure 13, there was minimal difference in earth pressure under different contact states, overall presenting a pattern of bottom > top > left wall = right wall. The average normal pressures were extracted from Figure 13 and compared with calculated values (

Table 11. Comparison of calculated and simulated pressure.

Position	Simulated Average Value (kPa)				Theoretical Calculated Value (kPa)
	a	b	c	d	WL	FP
Box top	135	134	134	135	114	110/183
Box bottom	179	183	180	179	138	/
Left wall	72	69	67	67	67	/
Right wall	72	69	67	67	67	/

). Due to the small elastic modulus and thickness of the equivalent layers, the soil tended to contract towards the box, resembling active earth pressure (WL). However, the stress concentration occurred at both ends of the tunnel due to excavation, leading to a simulated earth pressure larger than the calculated values. Meanwhile, considering the effect of the lubricant, the simulated earth pressure at the box top fell between positive and negative soil arching pressures (FP). Overall, even though the numerical model could not simulate the actual injection and filtration of the lubricant, the simulation effect on the soil arching action is satisfactory.

(2) Friction force

As shown in Figure 14 and Table 12, there were significant differences in the friction force under different lubricant conditions. The magnitude of the friction force followed the order of full box-soil interaction > three-side box-soil interaction > two-side box-soil interaction > one-side box-soil interaction. As mentioned earlier, under different lubricant conditions, the earth pressure was generally consistent. The main reason for the differences in the friction force is the variation in contact area and friction coefficients. In particular, in Figure 14b–d, the friction force significantly decreased after the transition from box-soil friction to box-lubricant friction.

As shown in Figure 15, in the FP stage, the simulated values for full, three, and two-side box-soil interactions exceeded the measured jacking force. Since there were changes in the box-soil contact state, friction coefficients, and friction types, the simulated values for one side box-soil interaction were the closest to the measured values. This once again proves that assuming full box-soil interaction and constant friction coefficients in the existing calculation models is inconsistent with actual construction. Due to the lubricant effects on the controlling factors, the outer surface of the box in a certain range does not interact with the soil, and the interface friction characteristics are also different, which corresponds with the assumption of the segmented calculation model in this paper. This demonstrates that the segmented calculation is more practical for guiding long-distance box jacking.

Table 12. Simulation value of friction force per unit length.

Position	Simulated Value/(kN/m)
	a	b	c	d
Box top	729.8	489.1	252.4	75.7
Box bottom	969.6	0.003333	1	0.004
Left wall	179.7	132.8	65.9	2.9
Right wall	186.2	97.3	9.5	2.9
Total value	2065.3	719.2	328.8	81.5

Table 12. Simulation value of friction force per unit length.

Position	Simulated Value/(kN/m)
	a	b	c	d
Box top	729.8	489.1	252.4	75.7
Box bottom	969.6	0.003333	1	0.004
Left wall	179.7	132.8	65.9	2.9
Right wall	186.2	97.3	9.5	2.9
Total value	2065.3	719.2	328.8	81.5

Figure 15. Comparison of simulated and measured values.

Figure 15. Comparison of simulated and measured values.

6. Conclusions

In order to improve the prediction accuracy, a new segmented friction calculation model was proposed that takes into account the effect of lubricants on the contact state, earth pressure, friction coefficient, and other factors during construction. Combined with the field-measured values and numerical simulation results, the effectiveness of the new method was verified. The conclusions of this study are as follows:

(1) The measured jacking force data of two box jacking cases showed that there are differences in lubrication conditions during long-distance jacking, and the variation law of the friction force in each zone is obviously different. In the condition without lubrication, the full box-soil contact state occurs, the friction coefficient is large, and the friction force increases linearly with the jacking length. In the condition of gradual lubrication, the full box-soil contact gradually changes to three or two side contacts, the friction coefficient decreases gradually, and the friction force markedly decreases. In the condition of full lubrication, the friction coefficient decreases to the lowest values, and the friction increases linearly with the jacking length, but the increasing slope is obviously smaller than that without lubrication.

(2) Compared with the existing models, the segmented friction calculation model proposed in this study has a better prediction accuracy for the changes in the lubricant conditions during the whole construction length and provides technical support for long-distance box jacking. Meanwhile, the existing models cannot reflect the changes in lubrication conditions, which generally misjudge the actual friction force. Especially for TSP, the friction coefficient was the same as the proposed model, but the calculated value was still far greater than the measured value. Meanwhile, the empirical value is used for TCB and TSR without considering the buried depth of the box, the earth pressure, or other factors, which limits its engineering applicability.

(3) The numerical model could not simulate the actual injection and filtration of the lubricant, as there were some differences between the simulated results and the theoretical results for the earth pressure. However, by setting the interaction mode between the outer surface of the box and the soil, the numerical model can reflect the difference in lubrication conditions, and the simulation results prove the rationality of the theoretical hypothesis that this study considers the box-soil contact state, friction coefficient, and friction type.

(4) From the perspective of the influence mechanism, it is suggested to select a calculation model to estimate the actual friction force combined with the real-time injection of the lubricant. The lubricant technology widely used at present can basically meet the construction requirements in terms of reducing the friction coefficient, maintaining soil stability, and preventing lubricant leakage. On this basis, adding injection holes at the top of the box and injecting a proper amount of high-viscosity bentonite slurry can further reduce the jacking force.

It should be noted that due to the invisibility of underground engineering and the complexity of geological conditions, the transition points in each zone are still uncertain under actual conditions, and the changes in lubrication conditions need to be further studied via controllable and visual test work or more cases.