# Calculation Method of Earth Pressure Considering Wall Displacement and Axial Stress Variations
## 2 Theoretical Analysis
### Special Earth Pressure Points
\[\varepsilon_{xp}=\Big{[}\big{(}K_{p}-K_{0}\big{)}\sigma_{z}+2c\sqrt{K_{p}} \Big{]}\frac{(1-\mu^{2})}{E_{tp}} \tag{15}\]  
\(E_{tp}\) is the tangent modulus in the passive limit state, where \(\sigma_{x}\) is the major principal stress. The direction of the minor principal stress is determined from the presence of the minor principal stress second inflection point. If the point exists, the minor principal stress is \(\sigma_{z}\), otherwise the minor principal stress is \(\sigma_{y}\). The specific conditions for the determination are described in Section 2.5. Substituting Equation (15) into Equation (9) gives the \(F_{\Sigma x}\) acting on the back of the retaining wall as shown in Equation (16). \[F_{\Sigma xa}=\frac{1}{2}K_{p}\gamma h^{2}+2 ch\sqrt{K_{p}} \tag{16}\]  
(2) Static earth pressure point  
At the static earth pressure, both horizontal strains in the soil are zero, noted as \(\varepsilon_{0}=0\). The two horizontal stresses are equal and equal to the static earth pressure, as shown in the following equation. \[\sigma_{y}=\sigma_{x}=K_{0}\sigma_{z} \tag{17}\]  
It is clear from Equations (8) and (10) that \(\sigma_{x}<\sigma_{y}\) in the active earth pressure zone and \(\sigma_{x}>\sigma_{y}\) in the passive earth pressure zone. Therefore, the static earth pressure point is the minor principal stress inflection point. When \(\varepsilon_{x}=0\), the retaining wall is subjected to static earth pressure, where \(\sigma_{x}<\sigma_{z}\). As the strain \(\varepsilon_{x}\) increases, there exists a point within the interval (0, \(\varepsilon_{xp}\)) where \(\sigma_{x}=\sigma_{z}\). Before this point, \(\sigma_{x}>\sigma_{z}\), and after this point, \(\sigma_{x}<\sigma_{z}\). Based on Equation (3) and the range of values of \(\mu\), it is known that \(\sigma_{x}(\sigma_{z})>\sigma_{y}\) at this point. Therefore, this point is called the major principal stress inflection point. Let \(\sigma_{x}=\sigma_{z}\), and the strain \(\varepsilon_{x1r}\) at this point can be obtained from Equation (8) as indicated in Equation (18). \[\varepsilon_{x1r}=\frac{(1+\mu)(1-2\mu)\sigma_{z}}{\Big{(}1-R_{f}S\Big{)}^{2} E_{i}} \tag{18}\]  
(4) Minor principal stress second inflection point  
Within the interval \((\varepsilon_{x1r},\varepsilon_{xp})\), \(\sigma_{x}\) is the major principal stress. When \(\varepsilon_{x}=\varepsilon_{x1r}\), \(\sigma_{y}\) is the minor principal stress. As the soil strain \(\varepsilon_{x}\) gradually approaches \(\varepsilon_{xp}\), according to Equation (10), \(\sigma_{y}\) continuously increases. There may be a point where \(\sigma_{y}=\sigma_{z}\), before which \(\sigma_{y}>\sigma_{z}\), and after which \(\sigma_{y}<\sigma_{z}\). This point is called the minor principal stress second inflection point. Substituting \(\sigma_{y}=\sigma_{z}\) into Equation (10) gives the strain at this point (\(\varepsilon_{x3r}\)) as shown in Equation (19). \[\epsilon_{x3r}=\frac{(1+\mu)(1-2\mu)\sigma_{z}}{\left(1-R_{f}S\right)^{2}E_{i}\mu} \tag{19}\]  
By substituting Equation (19) into Equation (8), we obtain the stress \(\sigma_{x3r}\) at this point, as depicted in Equation (20). \[\sigma_{x3r}=K_{0}\sigma_{z}+\frac{(1-2\mu)}{\mu(1-\mu)}\sigma_{z}=\frac{1}{K_ {0}}\sigma_{z} \tag{20}\]  
(5) Discussion of the order of the special points  
The special points mentioned above consist of the active limit earth pressure point, the static earth pressure point, the major principal stress inflection point, the minor principal stress second inflection point, and the passive limit earth pressure point. The corresponding sequence of soil strain and earth pressure acting on the retaining wall at these points, arranged from small to large, is presented in Equations (21) and (22).