## 2 Mobilized Friction Angle Under RB Active Earth Pressure Mode  
### Mobilized Wall-Soil Friction Angle Under RB Active Earth Pressure Mode  
The magnitude of earth pressure on retaining walls is directly influenced by their displacement modes. Retaining walls exhibit three fundamental displacement modes: translational movement (T mode), rotation about the top (RT mode), and rotation about the base (RB mode). As shown in Figure 1, the RB displacement mode schematic illustrates a common displacement pattern for gravity retaining walls in engineering practice. Under the RB mode, for a rigid retaining wall with a horizontal displacement \(S\) at the wall top at a given moment, the horizontal displacement \(S(z)\) at depth \(z\) can be expressed as follows:  
\[S(z)=(1-z/H)S \tag{1}\]  
As shown in Figure 2, soil failure is observed to reach a depth of approximately \(y=0.4\), while interfacial failure between the wall and soil occurs only within the range from the soil surface to \(y=0.2\) when the rotating angle is 0.7 mrad. This phenomenon correlates with the distribution of earth pressure: within the range \(y\in[0,0.2]\), the active earth pressure state is achieved. In the deeper range \(y\in[0.2,0.5]\), the earth pressure remains slightly higher than the active earth pressure value.  
Within this deeper zone, the retaining wall provides partial support to the soil, preventing sliding along the wall-soil interface. Consequently, the weight of the failed soil mass is partially redistributed through interactive arching mechanisms to both the wall and the adjacent non-failed soil that has not undergone yielding [36].  
As illustrated in Figure 3, based on PLAXIS finite element numerical simulations, the critical horizontal displacement corresponding to the wall-soil interface friction angle reaching its ultimate state was extracted. The results demonstrate that when the friction angle between the retaining wall and soil attains its maximum value \(\delta_{m}\), the critical horizontal displacement \(S_{\delta}\) required to reach the ultimate friction angle exhibits an approximately linear relationship with depth \(z\), observed consistently under RB, RT, and T modes. Conse  
Figure 1: Displacement schematic diagram of retaining wall under RB mode.  
quently, an expression for the mobilized friction angle at varying depths under non-limit states is proposed:  
\[\delta_{\text{m}}=\begin{cases}\delta_{0}+\frac{S(z)}{S_{d}(z)}(\delta-\delta_{0} ),\ S(z)<S_{\delta}(z)\\ \delta,\ \ S(z)\geq S_{\delta}(z)\end{cases} \tag{2}\]  
where \(\delta\) is the ultimate wall-soil friction angle at the limit state, \(\delta_{0}\) is the initial internal friction angle of the wall-soil interface, \(S_{\delta}(z)\) is the critical displacement required for the interface friction angle to reach the ultimate state.  
For the selection of the initial wall-soil friction angle \(\delta_{0}\), [PERSON] [37] recommends that it be determined based on actual engineering conditions: for embedded retaining walls, it may be taken as 0; for backfilled retaining walls, when exact values are unavailable, it can be conservatively estimated as \(\delta_{0}=\varphi/3\).  
According to Equation (2), by selecting \(\delta_{0}=5^{\circ}\), \(\delta=15^{\circ}\), \(S_{\delta}(z)=0.1\%\)\(z\), the mobilized friction angle between the wall and soil under RB mode is illustrated in Figure 4 when horizontal displacements at the top of the retaining wall reach different values.  
Figure 3: Relationship between critical horizontal displacement and retaining wall height.  
Figure 2: The variation in wall–soil interface contact pressure, sliding friction angle, and shear plastic strain increment with the rotation angle of the retaining wall from Reference [36].