# Calculation Method of Earth Pressure Considering Wall Displacement and Axial Stress Variations
## 2 Theoretical Analysis
### Calculation Method for Earth Pressure Considering Axial Stress  
According to the elastic theory, the relationship between the strain in the \(x\) direction and the stresses in the \(x,y\), and \(z\) directions is shown in Equation (1):  
\[\varepsilon_{x}=\frac{\sigma_{x}}{E}-\frac{\mu(\sigma_{y}+\sigma_{z})}{E} \tag{1}\]  
In the above equation, the stress in the \(z\) direction \(\sigma_{z}\), represents the self-weight stress of the soil, where \(\sigma_{z}=\gamma z\), with \(z\) is the depth of the soil.  
According to the plane strain assumption, the \(y\) direction strain \(\varepsilon_{y}=0\). Since the \(y\) direction stress \(\sigma_{y}\) satisfies the following formula.  
\[\varepsilon_{y}=\frac{\sigma_{y}}{E}-\frac{\mu(\sigma_{x}+\sigma_{z})}{E} \tag{2}\]  
Therefore, the relationship between \(\sigma_{y}\) and the stresses in other directions can be expressed as follows:  
\[\sigma_{y}=\mu(\sigma_{x}+\sigma_{z}) \tag{3}\]  
By substituting Equation (3) into Equation (1), the relationship between \(\sigma_{x}\) and \(\varepsilon_{x}\) can be obtained as expressed in Equation (4).  
\[\sigma_{x}=\frac{\mu}{1-\mu}\sigma_{z}+\frac{E\varepsilon_{x}}{1-\mu^{2}}=K_{ 0}\sigma_{z}+\frac{E\varepsilon_{x}}{1-\mu^{2}} \tag{4}\]  
The \(\sigma_{x}\) is the earth pressure on the retaining wall. It is a unified calculation formula of active and passive earth pressure expressed by strain, taking into account the change of axial stress of the soil along the retaining wall. Where \(K_{0}\) is the static earth pressure  
Figure 1: Direction diagram of earth pressure behind the wall.  
coefficient. The above equation indicates that the earth pressure behind the wall consists of two components: the static earth pressure when the retaining wall has no displacement, and the incremental earth pressure which varies with the direction and magnitude of the wall displacement.  
Substituting Equation (4) into Equation (3), the relationship between \(\sigma_{y}\) and \(\varepsilon_{x}\) strain is obtained as shown in Equation (5).  
\[\sigma_{y}=K_{0}\sigma_{z}+\frac{E\mu}{1-\mu^{2}}\varepsilon_{x} \tag{5}\]  
It can be seen that the \(\sigma_{y}\) is also a function of the \(\varepsilon_{x}\), that is, it varies with the direction and amount of displacement of the retaining wall, and it can be seen that [PERSON] [49] assumed that the \(\sigma_{y}\) in the soil behind the wall is the static earth pressure is not appropriate.  
The total earth pressure \(F_{\Sigma x}\) of earth pressure behind the wall is as follows:  
\[F_{\Sigma x}=\int_{0}^{h}\sigma_{x}\mathrm{d}z=\frac{1}{2}K_{0}\gamma h^{2}+ \frac{E}{1-\mu^{2}}\int_{0}^{h}\varepsilon_{x}\mathrm{d}z \tag{6}\]  
where \(h\) is the depth of the soil behind the wall. The above formula is derived from elastic theory and seems theoretically reasonable. However, comparison with the test results shows that while the results of this paper and the test results are in good agreement at the bottom of the retaining wall, the results of this paper are generally much greater than the test results at the top of the retaining wall, with a significant difference between the two. The problem lies in the modulus of the soil, which varies with changes in stress state. However, the above formula treats the modulus of the soil as a constant and ignores the difference between soil and other elastic materials. This leads to a deviation of the earth pressure from the actual value calculated by the method in this paper.