## 7 The \"Intermediate\" State of Earth Pressure  
Often the wall does not translate or rotate enough so that the active or passive state can be fully mobilized. Thus, the pressure near the base is larger than predicted by theory. These pressures of intermediate state could be calculated using Equation (8) along with the proper \(m\) value. It is noted that for \(m=1\) and \(\infty\), Equation (8) leads to the coefficients of earth pressure for the state at rest and the active state respectively given earlier. In addition, due to the known \"symmetry\" between the active and the passive earth pressure coefficient, Equation (8) can be written in the following form:  
\[ \begin{array}{ll}K_{xe}^{c-\varphi}&=\left\{\begin{array}{l}\text{active ``side'';}\\ \text{from }K_{oe}^{c-\varphi}\;(m=1)\text{ to }K_{oe}^{c-\varphi}\;(m=+\infty); \;\xi_{1}=\xi_{2}=1\\ \text{passive ``side'';}\\ \text{from }K_{oe}^{c-\varphi}\;(m=1)\text{ to }K_{pe}^{c-\varphi}\;(m=+\infty); \;\xi_{1}=1+\xi;\;\xi_{2}=\frac{2}{m}-1\\ \end{array}\right\}\\ &=\left(\frac{1\;\pi\;\sin\varphi^{\prime}}{1\;\pm\;\sin\varphi^{\prime}} \right)^{\xi_{1}}\!\!\left((1\;\mp\xi\sin\varphi^{\prime})\mp\xi_{2}\frac{k_{ 0}}{1-k_{0}}\tan\varphi^{\prime}(2+\xi(1\mp\sin\varphi^{\prime}))\right)\\ &\mp\frac{1}{1-k_{0}}\frac{2c_{m}}{\gamma^{2}}\!\!\left(\frac{\tan\left(45^{ \prime}\mp\frac{\varphi}{2}\right)}{\tan\left(45^{\prime}\pm\frac{\varphi}{2} \right)}\right)^{\xi_{1}}\tan\!\left(\!45^{\prime}\pm\frac{\varphi^{\prime}}{2 }\right)\end{array} \right\}\end{array}, \tag{27}\]  
where, \(\xi_{1}\) and \(\xi_{2}\) are parameters related to the transition from the soil wedge of the state at rest to the soil wedge of the passive state (i.e., inclination angle of soil wedge changing from \(45+\varphi^{\prime}/2\) to \(45\)- \(\varphi^{\prime}/2\)). \(m\) is always positive meaning that, for the passive \"side\" \(m\) ranges between \(1\) (state at rest) and \(+\infty\) (passive state). The symmetry stands for smooth, not battered wall with zero backslope angle. For any \(m\) value between these extremes (i.e., \(1\) and \(+\infty\)), Equation (27) gives the coefficient of earth pressure at rest; however, \(m\) must be defined with respect to the wall movement. In this respect, [PERSON] [31] gave analytical expressions for the pressure behind smooth and rough translating wall which moves outwards at a distance \(\Delta x\). He also gave expressions for the case of rotating wall as for the top. According to [PERSON], the lateral earth pressure at depth \(z\) on a smooth wall that has been displaced laterally by \(\Delta x\) is:  
\[\sigma_{h,Finn}=\frac{\mu}{1-\mu}\gamma z-\frac{E_{Young}}{\left(1-\mu^{2} \right)\pi}\frac{4H^{2}z}{\left(z+H\right)^{3}\!\left(H-z\right)}\Delta x, \tag{28}\]  
for smooth translating wall and  
\[ \begin{array}{ll}\sigma_{h,Finn}&=\frac{\mu}{1-\mu}\gamma z-\frac{4E_{Young }H}{\pi(3-\mu-4\mu^{2})(1+\mu)}.\\ &\left(\frac{1-\mu^{2}}{H^{2}-z^{2}}+\frac{\mu(1+\mu)H-\left(1-\mu^{2}\right) z}{\left(H+z\right)^{3}}\right)\Delta x\end{array} , \tag{29}\]  
for rough translating wall. Note: \"Equation 107\" in Finn [31] for \(\sigma_{h,Finn}\) for the case of rough translating wall has a typographical error. This error has been corrected in Equation (29). Additionally, some signs in Equation (29) have been inverted because herein \(\Delta x\) is always positive quantity; in Finn [31], when the wall moves away of soil, this distance in considered negative. Re-evaluating the above expression, the procedure for the active case will have as follows: first, \(\sigma_{h,Finn}\) is replaced by Equation (10) and the term \((\mu/(1-\mu)))\gamma z\) (representing the static earth pressure at rest in \(\mu\) terms) by Equation (16); then, solving as for \(\Delta x\), the required displacement for the active state to fully develop at depth \(z\) will be:  
\[\Delta x_{\max}=\frac{\pi}{4}\frac{\left(1-\mu^{2}\right)}{E_{Young}}\frac{ \left(H+z\right)^{3}(H-z)}{H^{2}z}\Delta K(1-k_{v})\gamma z, \tag{30}\]  
for smooth translating wall and  
\[