## 1 Introduction  
Keeping the tunnel-face support pressure, soil-layer groundwater pressure, and soil pressure in a balanced state is the key to ensuring the smooth construction of mud-water-balanced pipe jacking and reducing the impact of disturbance on the surrounding soil. In a pipe-jacking project, if the tunnel-face support pressure is not properly controlled, tunnel-face instability problems will be caused. When the tunnel-face support pressure is too low, the front soil is unloaded. And there is a large influx of soil into the mud and water silo, causing over-excavation, surface subsidence, and pipe-jacking obstruction. Whenthe support pressure on the tunnel face is too high, the soil in front is compressed due to loading, causing uplift and damage in a certain area on the surface [1]. Therefore, in order to avoid problems such as surface subsidence with too much support pressure or uplift damage with too little support pressure, reasonable and accurate control of the support pressure on the tunnel face is key in mud-water-balanced pipe-jacking construction [2]. Research on the ultimate support pressure currently uses indoor physical model tests, the stability coefficient method, the limit analysis method, the limit equilibrium method, and numerical calculations [3; 4]. However, physical model tests are costly and difficult to generalize to complex real-world projects [5; 6; 7]. The stability coefficient method was first proposed by [PERSON] [8], who, in 1967, established a mathematical relationship between the support force and the surface load, soil gravity, overburden burial depth, and clay undrained strength and defined the ratio of each parameter as the stability coefficient N. Based on a large amount of surface settlement data, [PERSON] [9] proposed that N values of 5 to 7 lead to destabilization of the excavation surface and induce surface collapse, and they explored the relationship between the stability coefficients and the overburden burial depth, the unsupported length, and variations in settlement. However, it was found in practice that the prediction error of the stability coefficient method was large, and it was very easy to misudge an unstable condition as a stable condition. At the same time, the stabilization coefficient method can only be applied to clay formations (i.e., with cohesion \(\
eq 0\)) and cannot be applied to sandy soil formations. Although the stabilization factor method has received a series of improvements to diversify the form of expression, one point that has never been taken into account is that the effect of permeability on the excavated surface is suitable for calculations in groundwater-free construction conditions. In addition, the arc effect overestimates the destructive power. In contrast, the limit analysis [10; 11; 12; 13] method is based on the kinematic permissive velocity field and the static permissive stress field. It establishes the equilibrium conditions and damage conditions in the critical state of the soil body and deduces the excavation surface support force based on the principle of virtual work [14; 15]. The limit equilibrium method [16; 17] assumes that damage to the soil body in front of the excavation produces a sliding body, and it analyzes the equilibrium of forces in the soil body at the moment of damage to find the solution to the problem. Then, the complex static indeterminate problem of soil damage is transformed into a simpler static deterministic solvable problem. The theory of the limit equilibrium method [18; 19; 20; 21] is rigorous, the mechanical mechanism is clear, and the calculation formula is concise and refined. After decades of improvement and enhancement, it has been developed to a fairly perfect degree and is widely used in practical engineering [22; 23]. Based on the two-dimensional limit equilibrium model and silo theory [24], [PERSON] et al. [25] established a three-dimensional wedge-shaped model for the stability of a tunnel face; the model can determine the magnitude of the ultimate support pressure.