# Field Study on Earth Pressure of Finite Soil Considering Soil Displacement
## 4 Theoretical Model  
The relationship between displacement and earth pressure is in accordance with hyperbolic curves in the small deformation stage, as shown in Equation (1) [25]:  
\[k_{0}\sigma_{z}-\sigma_{x}=\frac{u}{a_{0}+b_{0}u} \tag{1}\]  
where \(k_{0}\) is the coefficient of static earth pressure; \(\sigma_{z}=yz\) is the vertical pressure; \(\gamma\) is the weight of soil; \(z\) is the burial depth of soil; \(k_{0}\sigma_{z}\) is the earth pressure without deformation; \(\sigma_{x}\) is the earth pressure; \(u\) is the horizontal displacement of the enclosure structure; \(a_{0}\) and \(b_{0}\) are the undetermined coefficient. It can be obtained from Equation (1), when \(u\rightarrow\infty\), there are:  
\[k_{0}\sigma_{z}-\sigma_{x,\text{lim}}=\frac{1}{b_{0}} \tag{2}\]  
where \(\sigma_{x,\text{lim}}\) is the soil pressure at the limit state. Since the lateral displacement of the soil cannot be infinite, it will become the active earth pressure state after reaching a certain value, so the parameters \(b_{0}\) are:  
\[b_{0}=\frac{1}{k_{0}\gamma z-\epsilon_{0}} \tag{3}\]  
The radial stress and radial strain have the following hyperbolic relationship:  
\[\sigma_{r0}-\sigma_{r}=\frac{\epsilon_{r}}{a_{1}+b_{1}\epsilon_{r}} \tag{4}\]  
where \(\sigma_{r0}\) is the initial radial pressure; \(\sigma_{r}\) is the radial pressure; \(\epsilon_{0}\) is the active earth pressure; \(\epsilon_{r}\) is the radial strain. \(a_{1}\) and \(b_{1}\) are the test coefficients, where:  
\[a_{1}=\left(\frac{\epsilon_{r}}{\sigma_{r0}-\sigma_{0}}\right)=\frac{1}{E_{i}} \tag{5}\]  
where \(E_{i}\) is the initial tangential elastic modulus of the initial soil in the test under the condition of lateral unloading. [PERSON] divides the fill behind the enclosure into partitions by analyzing the soil deformation state [26], as shown in Figure 5. The horizontal strain in the plastic equilibrium zone is equal, and the horizontal strain in the elastic equilibrium zone decreases linearly with the increase in \(x\), namely:  
\[{\epsilon_{x}}^{\prime}(z)=\epsilon_{x}(z)\bigg{[}1-\bigg{(}x-\frac{H-z}{ \tan\theta}\bigg{)}\bigg{/}\bigg{(}b-\frac{H-z}{\tan\theta}\bigg{)}\bigg{]} \tag{6}\]where \(x\) is the distance between the soil mass and the enclosure; \(H\) is the height of enclosure structure; \(b\) is the width of finite soil; \(\theta=45^{\circ}+\varphi/2\); \(\varphi\) is the friction angle in the soil behind the enclosure. If \(b<\frac{H}{\tan\theta}\), then it constitutes a limited soil. That is, the soil width is less than this value. For limited soil to meet the deformation control requirements under normal working conditions, the lateral displacement of the enclosure structure is very small, which is far less than the displacement required to reach the active state so it is difficult to form the ultimate earth pressure mode. At this time, the earth pressure on the enclosure structure should be between the static earth pressure and the active earth pressure, which forms the so-called quasi active earth pressure. Based on the boundary conditions, the lateral displacement of the existing building is 0, and the relationship between the lateral deformation and strain is obtained:  
\[\left\{ \begin{array}{l}u=\varepsilon_{x}b\,\ 0\leq z\leq\beta\\ u=\varepsilon_{x}[b+(H-z)/\tan\theta]/2\,\ \ \beta\leq z\leq b\end{array} \right. \tag{7}\]  
where \(\beta=H-b\tan\theta\) is the height of the rectangular soil divided by the limit state of the finite soil behind the wall. Substitute Equation (7) into Equation (4) and compare it with Equation (1) to obtain the expression for the parameter \(a_{0}\):  
\[\left\{ \begin{array}{l}a_{0}=b/E_{i}\,\ 0\leq z\leq\beta\\ a_{0}=[b+(H-z)/\tan\theta]/2E_{i}\,\ \beta\leq z\leq b\end{array} \right. \tag{8}\]  
Substitute Equations (8) and (3) into Equation (1) to obtain the formula for the non-limit state earth pressure of finite soil:  
\[\left\{