# Calculation Method of Earth Pressure Considering Wall Displacement and Axial Stress Variations
## 2 Theoretical Analysis
### Special Earth Pressure Points  
In this paper, it is assumed that the soil behind the retaining wall is in a plane strain state, namely the \(y\) direction being the principal stress direction. The retaining wall is assumed to be vertical, and the friction angle between the wall and the soil is assumed to be zero. This means that both the \(x\) and \(z\) directions are also principal stress directions. Based on Mohr-Coulomb strength theory, the soil behind the retaining wall is considered to be in limit equilibrium state when the Mohr circle formed by the major and minor principal stresses is tangent to the Coulomb line. This is illustrated in Figure 2.  
In the figure, \(\sigma_{o}\) represents the static earth pressure. As the retaining wall is moved away from the soil, the soil behind the wall produces active strain, causing the earth pressure behind the wall to decrease. The minimum value of the earth pressure is the active limit earth pressure \(\sigma_{xa}\), and this interval is known as the active earth pressure zone. If the retaining wall has no displacement, it experiences static earth pressure and at this point \(\sigma_{x}=\sigma_{y}\). The direction of the minor principal stress changes after this point, making it a special earth pressure point. As the retaining wall is displaced toward the soil, the soil behind the wall produces passive strain, resulting in an increase in earth pressure behind the wall. As \(\sigma_{x}\) increases and becomes equal to \(\sigma_{z}\), the major principal stress inflection point \(\sigma_{x1r}\) occurs. Similarly, when \(\sigma_{y}\) increases and becomes equal to \(\sigma_{z}\), the minor principal stress inflection point \(\sigma_{x3r}\) occurs. The maximum value of the earth pressure is the passive limit earth pressure \(\sigma_{xp}\), and this interval is known as the passive earth pressure zone. The stress-strain calculation formulas for these earth pressure points are described below. (1) Limit earth pressure point  
Based on the above assumptions, the soil behind the retaining wall satisfies the Rankine earth pressure condition in the limit state. In the active limit state, the \(x\) direction earth pressure is minor principal stress. The limit equilibrium condition gives the active limit earth pressure \(\sigma_{xa}\) shown in Equation (11). \[ \begin{array}{l}\sigma_{xa}=\sigma_{z}\tan^{2}(45^{\circ}-\frac{\sigma}{2})- 2c\tan(45^{\circ}-\frac{\sigma}{2})\\ =\sigma_{z}K_{a}-2c\sqrt{K_{a}}\end{array}  \tag{11}\]  
By assigning \(\sigma_{x}=\sigma_{xa}\), the active limit strain \(\epsilon_{xa}\) can be determined using Equation (8), as shown in Equation (12). \[\epsilon_{xa}=\left[(K_{a}-K_{0})\sigma_{z}-2c\sqrt{K_{a}}\right]\frac{(1-\mu ^{2})}{E_{ta}} \tag{12}\]  
Figure 2: Earth pressure zoning according to the magnitude of the earth pressure. \(E_{ta}\) is the tangent modulus in the active limit state, where \(\sigma_{z}\) is the major principal stress and \(\sigma_{x}\) is the minor principal stress. Substituting Equation (12) into Equation (9) gives the \(F_{\Sigma x}\) acting on the back of the retaining wall as shown in Equation (13). \[ \begin{array}{ll}F_{\Sigma xa}&=\frac{1}{2}K_{0}\gamma h^{2}+\int_{0}^{h} \big{[}(\sigma_{z}K_{a}-2c\sqrt{K_{a}})-K_{0}\sigma_{z}\big{]}dz\\ &=\frac{1}{2}K_{a}\gamma h^{2}-2 ch\sqrt{K_{a}}\end{array}  \tag{13}\]  
In the passive limit state, the \(\sigma_{x}\) corresponds to the major principal stress. The passive limit earth pressure \(\sigma_{xp}\) is given by Equation (14). \[ \begin{array}{ll}\sigma_{xp}&=\sigma_{z}\tan^{2}(45^{\circ}+\frac{\varphi} {2})+2c\tan(45^{\circ}+\frac{\varphi}{2})\\ &=\sigma_{z}K_{p}+2c\sqrt{K_{p}}\end{array}  \tag{14}\]  
Similarly, let \(\sigma_{x}=\sigma_{xp}\), the passive limit strain \(\varepsilon_{xp}\) can be obtained using Equation (8), as depicted in Equation (15).