# Passive Earth Pressure and Soil Arch Shape: A Two-Dimensional Analysis
## 1 Introduction  
The issue of earth pressure on retaining walls is one of the core research topics in geotechnical engineering, with significant implications for the design optimization and stability assessment of retaining structures. While the classical Coulomb, Rankine, and Caquot theories [1] are widely applied in practice due to their simplicity and computational efficiency, their assumption of linear earth pressure distribution conflicts with the nonlinear patterns observed in field measurements and model tests [2, 3]. This discrepancy has prompted extensive theoretical studies on nonlinear earth pressure distributions, primarily through the application of limit equilibrium methods and limit analysis approaches. The majority of earth-retaining structures are primarily subjected to active earth pressure, leading to extensive research on active earth pressure theory by scholars worldwide. Following the introduction of a nonlinear earth pressure calculation method based on horizontal layer analysis, this approach has been systematically refined and extended to analyze active earth pressure under complex conditions [4, 5]. Moreover, substantial experimental evidence has demonstrated the ubiquity of soil arching effects within soil masses. [PERSON] [6] attributed the nonlinear distribution of earth pressure to soil arching, characterizing the arch shape as the minor principal stress trajectory, and subsequentlydeveloped a curved distribution calculation for active earth pressure by integrating this concept with horizontal layer analysis. Thereafter, calculation methods that integrate the horizontal layer analysis with soil arching effects have gained increasing traction among scholars, yielding substantial advancements. For instance, [PERSON] et al. [7] derived analytical solutions for the lateral earth pressure coefficient and active earth pressure under seismic conditions, considering both soil arching and vertical cracks within the framework of the unified strength theory under plane strain conditions. To better represent soil failure mechanisms and improve computational accuracy, analytical solutions incorporating curved sliding surfaces have been developed for active earth pressure, addressing sliding surface morphology [8; 9]. Additionally, limit analysis methods, which served as an alternative approach, have also been effectively employed to study active earth pressure under complex conditions. [PERSON] et al. [10], based on limit analysis theory, introduced the concept of critical width to distinguish between finite-width and semi-infinite soil masses in scenarios like retaining walls near rock or existing structures, and formulated computational equations for active earth pressure in finite soil masses, providing a theoretical foundation for relevant engineering designs. Compared to the extensively studied problem of active earth pressure, theoretical investigations on passive earth pressure remain relatively limited. Nevertheless, passive earth pressure plays a critical role in the safety and stability of various geotechnical structures, such as excavation support systems, bridge pier foundations, retaining walls, and pile cap foundations. Therefore, a thorough theoretical understanding of passive earth pressure is of significant importance for both design optimization and reliable performance in engineering practice. The horizontal layer method (see Figure 1a), combined with major principal stress trajectories, is also widely employed to investigate the distribution patterns of passive earth pressure [11; 12; 13]. Variations in existing studies primarily stem from differing assumptions regarding the shape of the sliding surface and the soil arch (trajectory of the major principal stress). These assumptions directly impact the calculated magnitude and distribution characteristics of passive earth pressure. Extending this approach, researchers have adapted it to complex conditions, including cohesive and multilayered backfill, seismic loading, seepage, and narrow excavations [14; 15; 16; 17].