## 1 Introduction  
As the development of construction technology and density of the underground environment, the surrounding soil behind a deep retaining wall is often constrained by many underground structures. For example, a large number of underground projects, including but not limited to deep foundation pits, slope protection, and underground pipe corridors, occur near existing large-rigid underground structures, such as an existing deep basement or an operating subway station, as shown in Figure 1. The force and deformation of soil behind the deep retaining structure cannot pass through the side wall of an adjacent existing large rigid underground structure. Thus, a soil mass with a limited width is formed. The stress and strain of the limited soil are significantly different from those of a semi-infinite soil [1; 2; 3; 4]. In this case, a reasonable soil of limited width between the newly constructed deep retaining wall and the existing underground structure must be scientifically considered. And the traditional Rankine method or Coulomb method is not suitable. It is still unclear whether the effect of the existing large-rigid underground structure on the clipped soil of limited width--and thus on the retaining structure--is positive or negative, especially when the soil condition is not simply dry sand. Many studies investigated the active earth pressure distribution behind a deep retaining wall with limited \(c\)-\(\varphi\) soil (cohesive soil) [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17] through multiple kinds of methods. Centrifuge model tests and field measurements [5; 6; 7; 8; 9; 10] have been conducted to analyze the active earth pressure distribution of different limited soil widths, providing a great deal of valuable data for the future analysis. Numerical solutions [11; 12; 13; 14] have also been implemented due to their comprehensiveness in considering multiple influencing factors if a suitable mechanical model and boundary conditions are considered carefully. Theoretical methods [15; 16; 17; 18; 19] have been introduced to ensure that the results of the field measurements, model tests, and numerical simulations are scientifically expressed and conveniently applied. Scholars [7; 15; 17; 18; 19] have incorporated the limited soil width into their calculations of the active earth pressure. The active earth pressure gradually increases with increasing of limited soil width. However, these methods do not consider the influence of soil cohesion [20; 21; 22; 23; 24; 25; 26] and ground surface overload [27; 28; 29], which are important factors in the design of a retaining structure. Further, these methods did not specify the explicit range of limited soil width. A proper analytical method is needed for calculating the active earth pressure acting on the retaining wall. The limit equilibrium method of a horizontal soil strip [30; 31; 32; 33; 34; 35] has been widely used because of its convenience and briefness in practical applications. And the methods for calculating the lateral coefficient of active earth pressure include moment equilibrium method [36; 37] and the soil arching method [38; 39; 40]. The moment equilibrium method does not consider the force transmitted by the soil arching, which leads to the calculation results being inconsistent with the actual values. The soil arching effect is a common phenomenon caused by the transfer of internal stress, such as the earth pressure decreasing in the yield region and increasing in the adjacent region [41]. And it is more notable in the soil of limited width between two wall faces. Thus, the soil arching method is used and improved in the limit equilibrium method of a horizontal soil strip for calculating the active earth pressure of limited soil in this study. Two main key problems about stress redistribution in the soil arching method include the stress direction deflection and the stress transfer path [39], and they are all reflected by the uneven deformation.