# Calculation of Nonlimit Active Earth Pressure against Rigid Retaining Wall Rotating about Base
## 4 Derivation of the Analytical Solution for Active Earth Pressure
### Definition of Small Principal Stress Traces Parameters
#### 4.2.2 Definition of Arc Trace Radius  
Principal stress traces typically take the form of parabolas, catenaries, circular arcs, and so on. However, the choice of principal stress traces has little effect on the analytical solution of earth pressure, so for calculation convenience, this paper utilized circular principal stress traces [15]. The method for determining \(R_{DE}\) and \(R_{D^{\prime}E^{\prime}}\) involves first connecting points \(D\) and \(E\), drawing the middle vertical line of line \(DE\), and finally drawing a vertical line through point \(D\). This normal intersects the mid-pipeline at point \(O\). The radius of the arc \(DE\) is represented by lines \(OD\) and \(OE\). The radius \(R_{D^{\prime}E^{\prime}}\) of arc \(D^{\prime}E^{\prime}\) is determined using the method presented in Figure 2b.  
Figure 2b depicts the geometric features of the minor principal stress trace, assuming that the minimal principal stress trace is a circular curve. The length of the arc at the upper interface of the unit is \(l_{\overline{DE}}\), the center of the circle is \(O\), and the depth of burial at point \(D\) is y. Although the length of the straight line \(OD\) is \(R_{DE}\), the length of line \(O^{\prime}D\) is \(R_{D^{\prime}E^{\prime}}\)  
\(\theta_{a}\) and \(\theta_{b}\) are the angles of \(OD\) and \(OE\) to the horizontal, and \(\theta_{1}\) is the angle between the vertical and line \(DE\), which can be derived from the geometric relationships in Figure 2b. The following can be derived from the angle relationship of \(\Delta ODE\):  
\[2(\theta_{a}+\gamma)+\theta_{b}-\theta_{a}=180^{\circ} \tag{16}\]  
\[\theta_{1}+\gamma=90^{\circ} \tag{17}\]  
where \(\gamma\) is the included angle between segment \(DE\) and the horizontal direction.  
The simplification of Equations (16) and (17) yields  
\[\theta_{1}=\frac{\theta_{a}+\theta_{b}}{2} \tag{18}\]  
The length of line \(DE\), denoted as \(l_{DE}\), can be determined from the geometry of \(\Delta DEB\) by using the sine theorem as follows:  
\[\frac{H-y}{\angle DEB}=\frac{l_{DE}}{\angle DBE} \tag{19}\]  
\[\angle DBE=90^{\circ}-\alpha_{s} \tag{20}\]  
\[\angle DEB=90^{\circ}+\alpha_{s}-\theta_{1} \tag{21}\]  
Substituting Equations (20) and (21) into Equation (19) to simplify it yields  
\[l_{DE}=\frac{(H-y)\cos\alpha_{s}}{\cos(\theta_{1}-\alpha_{s})} \tag{22}\]  
Equation (22) incorporates the geometric relations in the \(\Delta ODE\) and \(\Delta OD^{\prime}E^{\prime}\) to derive the magnitude of \(R_{DE}\) and \(R_{D^{\prime}E^{\prime}}\).  
\[R_{DE}=\frac{\frac{1}{2}l_{DE}}{\sin(\frac{\theta_{b}-\theta_{s}}{2})}=\frac{ (H-y)a_{2}}{2} \tag{23}\]  
\[R_{D^{\prime}E^{\prime}}=\frac{\frac{1}{2}l_{D^{\prime}E^{\prime}}}{\sin( \frac{\theta_{b}-\theta_{s}}{2})}=\frac{(H-y-dy)a_{2}}{2} \tag{24}\]  
\[a_{2}=\frac{\cos\alpha_{s}}{\sin(\frac{\theta_{b}-\theta_{s}}{2})\cos(\theta_ {1}-\alpha_{s})} \tag{25}\]  
where \(H\) is the retaining wall height, \(y\) is the depth of burial at point \(D\), and \(dy\) is the length of line \(DD^{\prime}\).