# Passive Earth Pressure and Soil Arch Shape: A Two-Dimensional Analysis
## 2 Derivation of Equilibrium Differential Equations and Solution for Passive Earth Pressure
### Calculation of Passive Earth Pressure  
The intensity of the horizontal component of passive earth pressure acting on the wall face at depth \(z\), denoted \(\sigma_{\rm wx}\), is obtained by setting \(x=0\) in Equation (16):  
\[\sigma_{\rm wx}(z)=\frac{K_{\rm wp}\gamma H}{1+m_{\rm p}}\Big{[}\Big{(}\frac{z }{H}\Big{)}^{-m_{\rm p}}-\frac{z}{H}\Big{]}+K_{\rm wp}q\Big{(}\frac{z}{H} \Big{)}^{-m_{\rm p}}. \tag{17}\]  
The resultant horizontal passive force \(P_{\rm ph}\) acting on a wall of height \(H\) is obtained by integrating the pressure intensity over the depth of the wall. When \(0<m_{\rm p}<1\), the integration yields the following:  
\[P_{\rm ph}=\int_{0}^{H}\sigma_{\rm wx}(z)\mathrm{d}z=\frac{K_{\rm wp}\gamma H^ {2}}{1-m_{\rm p}}(\frac{1}{2}+\frac{q}{\gamma H}) \tag{18}\]  
It is noted that, if \(m_{\rm p}\geq 1\), the integral in Equation (18) diverges, suggesting an infinite horizontal force, which is physically unrealistic. However, as illustrated in Figure 3, contour plots of \(m_{\rm p}\) indicate that \(m_{\rm p}\) typically reaches a maximum value of approximately 0.5 under the condition \(\delta\leq\varphi\). Therefore, the condition \(m_{\rm p}\geq 1\) is not encountered, ensuring that Equation (18) yields a finite resultant horizontal force within this theoretical framework. The resultant vertical force acting on the wall is the integral of the shear stress \(\tau_{\rm w}\) along the wall height:  
\[P_{\rm pv}=\int_{0}^{H}\tau_{\rm w}(z)\mathrm{d}z=P_{\rm ph}\tan\delta. \tag{19}\]  
The total resultant passive earth force is then given by the following:  
\[P_{\rm p}=\sqrt{P_{\rm ph}^{2}+P_{\rm pv}^{2}}=\frac{P_{\rm ph}}{\cos\delta}. \tag{20}\]  
The moment exerted by the horizontal earth pressure about the bottom of the wall (\(z\) = 0) is calculated as follows:  
\[M_{\rm p}=\int_{0}^{H}\sigma_{\rm wx}(z)z\mathrm{d}z=\frac{K_{\rm wp}\gamma H^ {3}}{2-m}(\frac{1}{3}+\frac{q}{\gamma H}), \tag{21}\]  
and the height of application \(h_{\rm p}\) of the horizontal resultant force is then given by the following:  
\[h_{\rm p}=\frac{M_{\rm p}}{P_{\rm ph}}=\frac{2}{3}\frac{1-m_{\rm p}\ \gamma H+3q}{3-m_{\rm p}\ \gamma H+2q}H. \tag{22}\]  
Figure 3: Contour plot of the parameter \(m_{\rm p}\). Finally, the passive earth pressure coefficient, defined as \(K_{\rm P}=2P_{\rm P}/\gamma H^{2}\), can be derived as follows:  
\[K_{\rm P}=\frac{K_{\rm wp}}{(1-m_{\rm P})cos\delta}(1+\frac{2q}{\gamma H}). \tag{23}\]  
Figure 4a-d illustrates the distributions of normalized vertical stress, shear stress, horizontal stress, and the relative principal stress ratio within the soil wedge. These results were obtained using the following parameters: internal friction angle \(\varphi=20^{\circ}\), soil-wall friction angle \(\delta=0.2\varphi\), and applied surcharge \(q=0.5\gamma H\) (with wall height \(H=1\) m assumed for scaling). The term \(K_{\rm pr}\) in Figure 4d represents the Rankine passive earth pressure coefficient, defined as \(K_{\rm pr}=\tan^{2}(\pi/4+\varphi/2)\). Analysis of these distributions (Figure 4) reveals the following: (1) The vertical normal stress shows a clear layered distribution and increases nonlinearly with depth. In particular, within the middle and lower regions of the wedge, the vertical stress significantly exceeds the nominal overburden pressure \((\gamma 2+q)\). This phenomenon is attributed to the soil arching effect, in which the frictional resistance along the wall back and the sliding surface restricts vertical displacement, leading to stress redistribution and concentration. (2) The shear stress \(\tau_{xx}\) is observed to decrease progressively with increasing horizontal distance \(x\) from the wall. However, its variation with vertical depth \(z\) appears non-monotonic, generally increasing initially before subsequently decreasing. (3) The horizontal normal stress \(\sigma_{x}\) exhibits a continuous increase with depth \(z\) at any given \(x\). Conversely, along a horizontal plane (constant \(z\)), \(\sigma_{x}\) varies non-monotonically with horizontal distance.