# Passive Earth Pressure and Soil Arch Shape: A Two-Dimensional Analysis
## 2 Derivation of Equilibrium Differential Equations and Solution for Passive Earth Pressure
### Solution of the Equilibrium Equations  
The traditional horizontal layer method assumes that a uniform average vertical stress acts across the entire soil layer at a given depth. To further account for soil arching effects within this framework, later studies generally introduced additional assumptions. Specifically, it is commonly assumed that, at the same depth, the magnitudes of the principal stresses remain constant while their orientations vary, and that all soil elements are in a state of limit equilibrium. Nonetheless, the soil experiences intricate stress-strain evolutions throughout its interaction with the retaining wall as it approaches the limit equilibrium state, which makes it challenging to strictly fulfill these assumptions. An alternative, influential simplification originates from [PERSON]'s [24] foundational analysis of silo pressures, wherein the vertical normal stress \(\sigma_{z}\) is assumed to remain constant along any horizontal plane (i.e., \(\partial\sigma_{z}/\partial x=0\)). While this is an idealization, its effectiveness for analyzing granular media has been supported by multiple lines of evidence: (i) its successful application in silo pressure and arching theories [25; 26]; (ii) numerical simulations of retaining wall problems reveal only minor horizontal variations in \(\sigma_{z}\)[21]; (iii) its incorporation into analytical derivations of active earth pressure has yielded results consistent with experimental observations [9]. Considering this theoretical development and supporting evidence, the present analysis of passive earth pressure adopts [PERSON]'s assumption, positing that the vertical normal stress \(\sigma_{z}\) is invariant with the horizontal coordinate \(x\) at any given depth \(z\). Applying this condition to the vertical equilibrium equation employed in this study (Equation (2)), and subsequently differentiating with respect to \(x\), leads to the following:  
\[\frac{\partial^{2}\tau_{xz}}{\partial x^{2}}=0. \tag{9}\]  
The general solution to Equation (9) indicates a linear distribution of shear stress with \(x\) as follows:  
\[\tau_{xz}(x,z)=C_{1}(z)x+C_{2}(z), \tag{10}\]  
where \(C_{1}(z)\) and \(C_{2}(z)\) are integration constants determined by boundary conditions. Applying boundary conditions--specifically, at \(x=0\), \(\tau_{\rm w}(z)=A_{\rm p}\sigma_{\rm wz}(z)\); and at \(x=z\cot\phi\), \(\tau_{xx}(z,x)=0\)--yields the specific expression for the shear stress:  
\[\tau_{zx}(z,x)=(A_{\rm p}-\frac{A_{\rm p}}{z\cot\phi}x)\sigma_{\rm wz}(z)=(A_{ \rm p}-m_{\rm p}\frac{x}{z})\sigma_{z}(z), \tag{11}\]  
where \(m_{\rm p}=A_{\rm p}\tan\phi\). Substituting this expression for shear stress (Equation (11)) into the vertical equilibrium equation (Equation (2)) results in a first-order linear non-homogeneous ordinary differential equation governing the vertical stress \(\sigma_{z}\):  
\[\frac{\partial\sigma_{z}}{\partial z}+m_{\rm p}\frac{\sigma_{z}}{z}=-\gamma. \tag{12}\]  
The general solution to Equation (12) takes the following form:  
\[\sigma_{z}(z)=\left\{ \begin{array}{ll}C_{3}z-\gamma z\ln(z)&\qquad m_{\rm p }=-1\\ C_{4}z^{-m_{\rm p}}-\frac{\gamma z}{m_{\rm p}+1}&\qquad m_{\rm p}\
eq-1\end{array}  \right., \tag{13}\]  
where \(C_{3}\) and \(C_{4}\) are integration constants determined by the boundary conditions. Given that \(m_{\rm p}=A_{\rm p}\tan\phi\geq 0\) under considered conditions, the case \(m_{\rm p}=-1\) is excluded.