# Numerical Simulation of Tunnel Face Support Pressure in Layered Soft Ground
## 1 Introduction  
With the rapid pace of urbanization both domestically and internationally, the frequency of underground space development is increasing significantly, making the 21 st century an era dedicated to the utilization and advancement of underground spaces, which has become a widely accepted consensus [1]. Shield tunnel construction boasts remarkable benefits such as accelerated excavation speed and high automation levels, rendering it the primary method employed for constructing urban subterranean tunnels. A key aspect of this technique involves providing appropriate support pressure at the tunnel excavation face to ensure ground stability [2]. The support pressure exerted on the excavation face plays a critical role in minimizing ground settlement during tunnel construction. Insufficient supporting pressure can lead to active failure, while excessive supporting pressure can lead to passive failure [3; 4; 5]. Many researchers and engineers have conducted theoretical and analytical studies on the stability of the excavation face during shield construction, and the specific research methods can be roughly divided into theoretical analysis, model tests, and numerical analysis. Among the theoretical analysis methods, the limit equilibrium method is extensively employed. In the early 1960s, [PERSON] [6] proposed an initial limit equilibrium failure mechanism, assuming that instability failure of a shield excavation face occurs on a rigid plastic soil adhering to the Mohr-Coulomb failure criterion. The support pressure required on the front wedge surface is calculated through force-based limit equilibrium analysis. Due to its straightforward concept and simple calculation, many researchers [7; 8] have widely embraced and continuously improved the wedge slider model to accommodate stability analysis of the shield excavation face and support pressure under varying geological conditions. [PERSON] [7] used an enhanced wedge-shaped silo model to assess tunnel face stability for earth-pressure-balanced shields in water-free conditions. [PERSON] [9] refined the wedge stability model by investigating how soil heterogeneity influences theminimum surface support pressure. [PERSON] [10] proposed an improved 3D working face stability analysis model based on the limit equilibrium method, which took into account the advanced support effect of a small conduit. Limit analysis represents a significant approach for addressing stability issues of excavation faces. [PERSON] et al. [11] were pioneers in utilizing the upper-bound plastic limit analysis method to study the limiting support pressure of tunnel excavation faces and instability failure mechanisms while evaluating unlined tunnel face stability within pure cohesive soils under undrained water assumptions. [PERSON] and [PERSON] [12] proposed a three-dimensional mechanism based on the motion of two rigid conical blocks within the friction material, from which they derived the upper and lower limits of support pressure for maximum and minimum excavation faces in a shield roadway. Building upon this work, [PERSON] et al. [13] presented an upper-bound solution to tunnel excavation face instability in layered soil. While most studies estimate the limiting surface support pressure when an upper layer (single or multiple layers) overlays the lower strata during tunnel excavation, it is important to re-evaluate this situation as multiple strata crossings prior to tunnel excavation are common in reality. [PERSON] [14] established a simplified model of the three-dimensional failure mechanism of a roadway face in multi-layered surrounding rock, obtained a theoretical model for upper limit analysis of support pressure, and found that the stability of the tunnel face could be improved by improving its formation properties or optimizing the tunnel section design. The second type of analysis method is the model test method. There are two types of model tests: the 1-g model test [15; 16] and centrifuge tests under the n-g model test [17; 18].