## 1 Introduction  
As an emerging shaft construction technique, the mechanized shaft method integrates undrained excavation, overall suspended sinking, mechanical cutting using a milling drum, and prefabricated shaft assembly. This method exhibits strong adaptability to weak, disturbed, or otherwise unstable ground conditions. It significantly mitigates the impact of shaft construction on the surrounding strata, while reducing construction time and overall cost. Owing to these advantages, it has been increasingly adopted in modern shaft engineering practice. Similar to other underground structures, mechanized shafts are inevitably subjected to earth and rock pressures from the surrounding strata, resulting in stress and deformation of the support system. Therefore, accurate estimation of the earth pressure acting on the support structure is essential in the structural design process. The classical Rankine and Coulomb earth pressure theories are widely applied in engineering practice due to their simplicity. However, their assumption of a linear earth pressure distribution often deviates from actual field conditions, leading to discrepancies between the theoretical and measured resultant earth pressures. Numerous laboratory tests have demonstrated that the active earth pressure behind plane retaining walls exhibits a nonlinear distribution [1,2]. Model tests on circular shafts have further confirmed this nonlinearity, revealing significant deviations from the classical Rankine earth pressure predictions [3]. In recent years, extensive research has been carried out by both domestic and international scholars on the distribution of earth pressure behind retaining walls. Current analytical approaches for evaluating earth pressure on plane retaining walls primarily include the limit equilibrium method, slip-line theory, and limit analysis. Among these, the limit equilibrium method based on horizontal layer analysis has been widely adopted and has yielded numerous valuable results. For example, [PERSON] [4], [PERSON] and [PERSON] [5], as well as [PERSON] and [PERSON] [6], incorporated the soil arching effect in the backfill and, based on the horizontal layer analysis method, derived the nonlinear distribution of active earth pressure in sandy soils. Since then, the earth pressure calculation method combining horizontal layer analysis with the soil arching effect has been further extended to various conditions, including seismic loading, limited-width backfills, seepage effects, and curved failure surfaces [7; 8; 9; 10; 11; 12]. These developments have significantly advanced the understanding of earth pressure on plane retaining walls. Compared with the extensive research on plane retaining walls, studies on the calculation of earth pressure acting on curved retaining structures (e.g., circular shafts) remain relatively limited. [PERSON] [13] was among the first to consider the effects of soil friction angle and the height-to-diameter ratio. Following an approach similar to [PERSON]'s earth pressure theory, he derived an expression for the earth pressure on circular shafts based on the overall static equilibrium of the sliding soil mass. However, numerous studies have shown that under limit equilibrium conditions, stress redistribution in the soil behind the shaft gives rise to a vertical arching effect, similar to that observed in plane retaining walls. Additionally, a horizontal circular arching effect also develops. These two mechanisms, collectively referred to as the spatial arching effect, significantly influence both the magnitude and distribution of earth pressure [14; 15]. To account for the influence of the circumferential arching effect, [PERSON] [16] adopted the Haar-Karman assumption, which assumes that the circumferential stress \(\sigma_{\theta}\) is always equal to the major principal stress \(\sigma_{1}\), and applied slip-line theory to derive the active earth pressure acting on circular shafts.