# Passive Earth Pressure and Soil Arch Shape: A Two-Dimensional Analysis
## 4 Parametric Analysis
### Soil-Wall Interface Friction Angle  
Figure 10a-c presents the effects of the soil-wall interface friction angle \(\delta\) on the passive earth pressure distribution, the passive earth pressure coefficient \(K_{p}\), and the normalized height of the resultant force application point \(h_{p}/H\), respectively, for a constant internal friction angle (\(\varphi=30^{\circ}\)).  
The variation in passive earth pressure distribution with \(\delta\) is illustrated in Figure 10a. As \(\delta\) increases, the distribution transitions from a linear to a curvilinear profile, with the degree of curvature intensifying at higher \(\delta\) values. At \(\delta=0\), a linear distribution is observed, consistent with [PERSON]'s passive earth pressure theory, reflecting no shear mobilization at the wall interface. With increasing \(\delta\), the pressure intensity diminishes in  
Figure 10: Effect of soil–wall friction angle on passive earth pressure: **(a)** passive earth pressure distribution; **(b)** passive earth pressure coefficient; **(c)** the height of the application point.  
the mid-to-upper wall region relative to [PERSON]'s prediction, while it exceeds [PERSON]'s values near the wall base. The curvilinear profile suggests a redistribution of stresses, likely driven by enhanced soil arching, which concentrates load transfer toward the lower wall section as wall friction strengthens.  
Figure 10b shows a clear non-monotonic relationship between the passive earth pressure coefficient and the soil-wall friction angle. Starting from the Rankine condition at \(\delta=0\), \(K_{p}\) increases with \(\delta\), reaching a peak value at an intermediate soil-wall friction angle, and then decreases as \(\delta\) approaches the internal friction angle \(\varphi\). This trend reflects the interaction between two competing mechanisms: increasing shear resistance along the wall interface tends to increase passive resistance, while the associated changes in stress state and excessive arching effects may become less efficient for mobilizing overall resistance beyond an optimal \(\delta\). The existence of this peak implies that maximizing soil-wall friction does not necessarily yield the highest passive resistance. Additionally, it is observed that, while a higher surface surcharge \(q\) consistently increases the magnitude of \(K_{p}\), the location (\(\delta\) value) of the peak \(K_{p}\) remains relatively unaffected by changes in surcharge.  
The influence of \(\delta\) on the normalized height of the resultant force application point is shown in Figure 10c. Similar to the trend observed for \(K_{p}\), this relationship is non-monotonic. Starting from \(h_{p}=H/3\) at \(\delta=0\), the application point initially shifts downward as \(\delta\) increases from zero. This initial downward shift is consistent with the pressure redistribution observed in Figure 10a, where soil-wall friction causes pressure to concentrate lower on the wall. Beyond a certain value of \(\delta\), \(h_{p}\) begins to increase again. With increasing \(q\), the curve retains its shape, but \(h_{p}\) shifts upward, with the increment diminishing at higher loads.