## 8 The Mobilized Shear Strength of Soil
### The Mobilized Shear Strength of Soil in the Active and Passive State  
As known, the mobilized shear strength of soils under static conditions is equal to the peak strength. However, only part of the shear strength of soil is mobilized under dynamic conditions. For example,  
Figure 4: Mohr’s diagram for the geometrical derivation of the mobilized shear strength of soil in the state at rest.  
in the active state it stands that \(K_{ae}>K_{a}\). Thus, a Mohr circle representing the active state of stress under dynamic conditions with \(\sigma_{3}=K_{ae}\gamma z(1-k_{v})\) and \(\sigma_{1}=\gamma z(1-k_{v})\) lies inside the respective one representing static conditions with \(\sigma_{3}=K_{a}\gamma z\) and \(\sigma_{1}=\gamma z\). The latter, as known, corresponds to state of failure touching the [PERSON]'s failure criterion, whilst the Mohr circle representing dynamic condition does not. The same also stands, of course, for the passive state because \(K_{pe}<K_{p}\). In the example above, \(k_{v}\) was considered positive; however, a negative \(k_{v}\) does not change the general observation.  
As it will be shown below, however, an earthquake excitation causes the reduction of the shear strength of soil mobilized during an earthquake event. Thus, with smaller (mobilized) shear strength, the soil exerts greater earth pressure on the retaining structure. The amount of the resisting forces (or moments) in relation to the destabilizing forces (or moments) defines, finally, whether the earth retaining structure will be stable or not. If the retaining structure is allowed to be displaced enough so that failure occurs, the [PERSON]'s theory will give again the minimum or maximum earth pressure in the active and passive state, respectively.  
For the active and passive state, the mobilized shear strength is calculated following the steps described for the state at rest. In this respect, geometrically, from Figure 4 (after replacing \(K_{oe}^{c-\varphi}\) with \(K_{ae}^{c-\varphi}\)), the mobilized friction angle for the active state is given by the following equation:  
\[\sin\varphi_{m}=\frac{\frac{\pm\left(\frac{(1-k_{v})\gamma z-K_{ae}^{c-\varphi }(1-k_{v})\gamma z}{2}\right)}{\frac{c^{\prime}}{\tan\varphi^{\prime}}\ +\ \frac{(1-k_{v})\gamma z+K_{ae}^{c-\varphi}(1-k_{v})\gamma z}{2}}. \tag{40}\]  
The plus sign, in the equation above, stands for the more probable case where \(K_{ae}^{c-\varphi}<1\), although, an extreme earthquake event in combination with a weak soil could cause \(K_{ae}^{c-\varphi}\) to exceed unity. A parametric study carried out by the author, however, showed that precondition for this unusual case to occur is that the shear strength of soil is very small and simultaneously that the seismic event is rather unrealistically extreme. Thus, since this condition is rather impossible to occur in practice, it is not further discussed here.  
By substituting Equation (19) into Equation (40) with the term \(2c_{m}/\gamma z\) inside \(K_{ae}^{c-\varphi}\) to have been replaced by the term \(2c^{\prime}/(\gamma zf_{m})\) (recall Equation (34)), Equation (36) is again obtained, but with  
\[e_{1}=\frac{(1-k_{v})\left(1-\frac{1-\sin\varphi^{\prime}}{1+\sin\varphi^{ \prime}}\left(1+2\frac{k_{v}}{1-k_{v}}\tan\varphi^{\prime}\right)\right)}{ \frac{2\sigma^{\prime}}{\gamma z}\tan\left(45^{\circ}-\frac{\varphi^{\prime}} {2}\right)}, \tag{41}\]  
and  
\[e_{2}=\frac{\tan\left(45^{\circ}+\frac{\varphi^{\prime}}{2}\right)}{\tan \varphi^{\prime}}+\frac{(1-k_{v})\left(1+\frac{1-\sin\varphi^{\prime}}{1+\sin \varphi^{\prime}}\left(1+2\frac{k_{v}}{1-k_{v}}\tan\varphi^{\prime}\right) \right)}{\frac{2\sigma^{\prime}}{\gamma z}\tan\left(45^{\circ}-\frac{\varphi^ {\prime}}{2}\right)}. \tag{42}\]  
In a similar way, the \(e_{1}\) and \(e_{2}\) parameters for the passive state are  
\[e_{1}=-\left(\frac{(1-k_{v})\left(1-\frac{1+\sin\varphi^{\prime}}{1-\sin \varphi^{\prime}}\left(1-2\frac{k_{v}}{1-k_{v}}