# Calculation of Displacement-Dependent Active Earth Pressure for Deep Excavations in Soft Soil
## 1 Introduction  
With the exploitation of underground space, many deep excavations occur. In order to improve the safety of foundation pit engineering, which can reduce potential harm to the environment, there are strict control requirements for the deformation of the supporting structure. For key foundation pits, lateral wall deformations are controlled within 0.18% and 0.2\(\sim\)0.5% of the excavation depth in Shanghai (DG/T) 08-6l-2018) [1] and Zhejiang province (DB33/T 1096-2014) [2], respectively. Meanwhile, the excavation deformation in clay at full active state is 1-2% of the excavation depth [3, 4] ([PERSON] and [PERSON], 1991; [PERSON] and [PERSON], 2004); additionally, measurements performed using a centrifuge test in a paper by [PERSON] and [PERSON] [5] determined that the active limiting deformation in clay is 0.9-1% of the excavation depth. Therefore, the support does not show sufficient lateral movement to achieve an active condition. The Rankine and Coulomb earth pressure theories are commonly used in design and are only valid for the limiting condition; therefore, these theories are not useful in calculating the magnitude and distribution of displacement-dependent earth pressure in nonlimiting states. Many field monitoring [6, 7] and experimental results [8, 9, 10] have shown that earth pressure is closely related to the deformation of the retaining walls. The research methods of these studies focus on the relationship between earth pressure and the deformation of a rigid retaining wall, which can be summarized in three aspects. First, the classical equation of active earth pressures adopts the mobilized shear strength value instead of the limiting strength value. [PERSON] [11] assumed a linear variation in the mobilized angle of internal friction along the entire depth of the wall. [PERSON] [12] set a linear relationship between the mobilized shearing resistance represented by the tangent and the corresponding walldisplacement. According to the Mohr circle of stress under the situation of an intermediate active state and the unloading stress-strain relationship, [PERSON] and [PERSON] [13, 14] established the relationship between the mobilized internal frictional angle and the wall movement. Using the same approach, [PERSON] et al. [15] provided the mobilized shear strength for clay. In addition, the strength reduction method [16] and the disturbed state concept [17] have also been applied to calculate the mobilized shear strength. Second, various empirical functions have been proposed to fit the earth pressure-displacement relationship. On the basis of two centrifugal test results, the hyperbolic function model was derived by [PERSON]. Based on the well-known curve [18] of the earth pressure-displacement relationship in practice, as shown in Figure 1, various functions, including the sigmoid functions [19, 20], the exponential function [21], the sine function [22], and the hyperbolic function [23, 24], can be assumed to fit the relationship. Third, earth pressure can be determined on the soil strain state. [PERSON] et al. [25] extended the Rankine and Coulomb theories depending on the strain increment ratio. With a new insight, [PERSON] et al. [26] proposed a method of taking the isotropic stress state as the turning point, where the lateral strain has zero value. [PERSON] et al. [27] developed a model based on the hyperbolic stress-strain relationship to predict earth pressure with different wall movements. Current methods of predicting earth pressure that account for wall deformation focus on rigid retaining walls. However, the supporting structure of a deep foundation pit is thinner than a rigid retaining wall, the so-called flexible retaining wall, which can cause obvious deflection. A series of model tests [28, 29, 30, 31] were carried out to investigate the nonlinear earth pressure distribution of the cantilever retaining walls, while the effects of wall displacement on the earth pressure were not considered. Based on the progressive rupture mechanism that occurs in backfill, [PERSON] et al.