## 1 Introduction
However, subsequent studies [17; 18; 19] have shown that the Haar-Karman assumption tends to overestimate the contribution of circumferential stress, as confirmed through extensive analytical and experimental investigations. For example, [PERSON] [17] introduced the concept of a circumferential stress coefficient \(\lambda\), defined as the ratio of circumferential stress to the major principal stress, and proposed that it should be taken as the static earth pressure coefficient \(K_{0}\). Using the limit equilibrium method, he derived a formula for calculating the active earth pressure while accounting for the circumferential arching effect. The calculated results showed good agreement with experimental data in shallow soils but deviated significantly at greater depths. Furthermore, some researchers have considered both vertical and circumferential arching effects simultaneously. For instance, [PERSON] and [PERSON] [20] and [PERSON] et al. [21] incorporated the spatial arching effect and developed active earth pressure formulations for circular retaining walls based on the limit equilibrium approach. [PERSON] et al. [22; 23] developed a calculation model for active earth pressure under various shaft wall displacement modes based on the horizontal layer analysis method, incorporating the influence of the spatial arching effect. [PERSON] et al. [24] adopted the Coulomb failure mechanism and derived a formulation for active earth pressure in circular shafts using the energy method, accounting for energy dissipation induced by circumferential stress. Recognizing that shaft wall displacement in practical applications is often insufficient to mobilize the active limit state, [PERSON] [25] proposed a non-limit active earth pressure model that explicitly considers shaft wall displacement effects. Most of the aforementioned studies assume that the circumferential stress coefficient \(\lambda\) remains constant. However, its magnitude is, in fact, strongly dependent on the spatial location of the soil element [26; 27]. Based on discrete element simulations, [PERSON] and [PERSON] [28] observed that the circumferential arching effect diminishes progressively with increasing distance from the wall, leading to a corresponding decrease in the circumferential stress coefficient \(\lambda\). Similarly, [PERSON] et al. [29; 30] analytically derived the circumferential stress under axisymmetric active limit conditions and concluded that the influence of the circumferential arching effect decreases with radial distance. Therefore, assuming a constant circumferential stress coefficient inevitably introduces significant discrepancies between calculated and actual earth pressure distributions [31]. In addition, due to the presence of soil arching in the vertical plane, the principal stress direction undergoes rotation, which induces horizontal shear stresses. These shear stresses significantly affect both the magnitude of the active earth pressure and the location of its resultant force [32; 33]. However, this effect has not been adequately addressed in existing studies on shaft earth pressure. Based on the above analysis, this study investigates the sliding soil mass behind the shaft wall, taking into account the radial variation of the circumferential stress coefficient. An analytical expression for the inclination angle of the sliding surface under active limit state conditions is derived. Incorporating the effects of spatial soil arching and interlayer shear, a calculation method for active earth pressure is established using a horizontal layer analysis approach combined with vertical and radial static equilibrium conditions. The proposed equation is validated through comparison with existing theoretical models and experimental results. Furthermore, an internal force calculation method for the shaft lining is developed based on elastic theory, considering the spatial effects of soil arching and interlayer shear. The proposed method is verified against field monitoring data, demonstrating its accuracy and practical applicability.