## 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. 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  
Figure 1: Box jacking technical principle. to solve box-jacking friction calculation models. [PERSON] et al. [11] used numerical simulations to predict the jacking force of box jacking, considering different box-soil contact areas. [PERSON] [12] established the jacking force formula for box jacking using the [PERSON] theory and additionally considering the friction resistance caused by the boring machine shell. [PERSON] 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. [PERSON] et al. [14] and [PERSON] 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.