# Stability Analysis of the Longitudinal Slope Linear Shield Tunnel Excavation Face
## 1 Introduction  
As urbanization continues to accelerate, major infrastructure projects such as urban rail transit and road tunnels are being constructed at an increasingly rapid pace. In densely developed urban core areas, subway tunnel networks are highly concentrated, and the shallow underground space has been largely occupied by various existing underground structures, imposing significant constraints on underground construction. To avoid existing tunnels, pile foundations, and other underground structures, many shield tunnels must be excavated under challenging conditions, including large longitudinal slopes [1]. For instance, the section between Sanyuan Station and Wuxi Railway Station features a maximum longitudinal slope of \(2.8\%\) on Wuxi Metro Line 3, and The Rogfast Tunnel, currently under construction in Rogaland County, southwestern Norway, features a maximum longitudinal slope of approximately \(5\%\). During shield tunneling in longitudinally sloped tunnels, the soil in front of the excavation face is subjected to uneven stress distributions and otherdestabilizing factors. Therefore, studying the instability and deformation mechanisms of the excavation face in shield tunnels with longitudinal slopes is of critical importance [2; 3; 4]. Both Chinese and international researchers have conducted extensive studies on the instability and deformation of soil at the excavation face of shield tunnels using various methodologies, including theoretical analysis, numerical simulations, and physical experiments. These studies have yielded significant insights, contributing to a deeper understanding of excavation face stability. [PERSON] et al. [5] introduced the Kinematical Element Method (KEM) as an analytical tool for tunnel face stability assessment. Compared to traditional limit analysis, KEM offers improved flexibility in representing 3D failure mechanisms and allows for automatic optimization of collapse surfaces. [PERSON] et al. [6] proposed an effective coupling of seepage force estimation and upper bound limit analysis, enabling realistic stability evaluation for tunnels driven under the water table. [PERSON] et al. [7] proposed an improved wedge prism model based on the limit equilibrium approach, along with an enhanced three-dimensional rotational failure model derived from the limit analysis method. [PERSON] et al. [8] proposed a novel instability mechanism for the shield excavation face based on the limit analysis method, considering the complete tunnel surface. Their approach enhanced the optimal available solution for collapse pressure. [PERSON] et al. [9] conducted three-dimensional finite element simulations to analyze tunnel face stability, demonstrating that numerical approaches can accurately capture progressive failure mechanisms and provide realistic estimates of minimum support pressures. [PERSON] et al. [10] employed the finite difference software FLAC3D to investigate the stability of the excavation face in deep-buried sandy tunnels, obtaining the ultimate support pressure under different internal friction angles and burial depths. [PERSON] et al. [11] investigated tunnel face collapse using Numerical Limit Analysis (NLA) and the Material Point Method (MPM), showing both methods accurately predict failure mechanisms and minimum support pressures, with MPM enabling detailed simulation of the post-collapse run-out process. [PERSON] et al. [12] applied the discrete element software PFC3D to analyze the instability and failure process of tunnel excavation faces. Their findings revealed that as the burial depth ratio (\(C/D\)) increased, the ultimate support pressure initially increased and then gradually stabilized. [PERSON] et al. [13] conducted large-scale model tests to investigate the stability of the excavation face in Earth Pressure Balance (EPB) shield tunneling within multi-layered foundations.