# Investigation of Steep Waste Dump Slope Stability of Iron Ore Mine--A Case Study
## 2 Materials and Methods
### Slope Stability Analysis
#### 2.4.1 Morgenstern-Price Method  
The Morgenstern-Price method is a sophisticated approach to limit equilibrium theory that is famous for its accuracy and comprehensiveness in slope stability analysis. The method is particularly adept at accommodating slip surfaces of arbitrary shape, which is a crucial feature for the realistic modeling of natural slopes. This requires that the setting of the unknown variables fully match the mechanical properties of the soil and rock, ensuring that the analysis is based on realistic physical properties. Similar to many other limit equilibrium-based approaches, the Morgenstern-Price method does not employ [PERSON]'s modulus (\(E\)) in assessing the stability of the soil slope. This approach is mainly based on the effective parameters of the soil (effective cohesion (\(c^{\prime}\)), effective internal friction angle (\(\varphi\)), pore pressure (\(u\))), as well as geometrical data of the slope. This method considers both normal and tangential equilibrium along with the moment equilibrium for each slice and is applicable to both circular and noncircular slip surfaces. The main core of this approach is to establish differential equations of static equilibrium and equations of moment equilibrium for individual slices of the slope. This segmented approach allows for a detailed and accurate understanding of slope stability under different conditions. As depicted in Figure 3, it is essential that the shear stresses in the vertical plane along both sides of the divided soil strip do not exceed the shear capacity available in this plane. This constraint ensures that the analysis accurately represents the soil's ability to resist the shear under applied loads. Through this method, we are able to systematically describe slope stability and provide a comprehensive and reliable assessment of potential risks and failure mechanisms. The detailed structural system and implementation of the Morgenstern-Price approach, as adopted in our investigation, are illustrated in Figure 3. To this end, consider a slice with an infinitesimal length of \(\Delta x\), and the slope angle of \(\alpha\). All exerted forces on the presented slice have been presented and defined in the following. In the presented subfigures, \(E\) and \(X\) in order stand for the normal and tangential interslice forces, \(G\) represents the resultant force of these two interslice forces with angle \(\beta\) with respect to the horizontal, \(\Delta W\) is the self-weight of the soil strip, \(N\) and \(T\) in order are the normal force and tangential resistance force at the bottom of the soil slice, \(U\) is the normal component of the pore pressure force (=\(\Delta x\)\(r_{u}\) sec(\(\alpha\)), where \(r_{u}\) is the pore pressure at the bottom of the slice), and \(h_{t}\) is the height of the slice (=\(z(x)-y(x)\)). The static equilibrium differential equation is given as follows:  
\[\cos\bigl{(}\varphi^{\prime}_{e}-\alpha+\beta\bigr{)}\frac{\mathrm{d}G}{\mathrm{d }x}-\sin\bigl{(}\varphi^{\prime}_{e}-\alpha+\beta\bigr{)}\frac{\mathrm{d}\beta}{ \mathrm{d}x}G=-P(x) \tag{3}\]  
\[ \begin{split} P(x)=&\biggl{(}\frac{\mathrm{d}W}{ \mathrm{d}x}+q\biggr{)}\sin\bigl{(}\varphi^{\prime}_{e}-\alpha\bigr{)}-r_{u} \frac{\mathrm{d}W}{\mathrm{d}x}\sec\alpha\sin\varphi^{\prime}_{e}+C^{\prime}_ {e}\mathrm{se}\\ &\cos\varphi_{e}t-\eta\frac{\mathrm{d}W}{\mathrm{d}x}\cos\bigl{(} \varphi^{\prime}_{e}-\alpha\bigr{)}\end{split}  \tag{4}\]  
\[C^{\prime}_{e}=\frac{C^{\prime}}{F} \tag{5}\]  
Figure 3: Schematic representation of a portion of the slope profile, and an infinitesimal slice element of length \(\Delta x\) with the exerted forces used in the Morgenstern–Price approach. \[\varphi_{e}^{\prime}=\text{arctg}\bigg{(}\text{tg}\frac{\varphi^{\prime}}{F}\bigg{)} \tag{6}\]  
The moment equilibrium equation reads:  
\[G\sin\beta=-y\frac{\text{d}}{\text{d}x}(G\cos\beta)+\frac{\text{d}}{\text{d}x}( y_{t}G\cos\beta)+\eta\frac{\text{d}W}{\text{d}x}h_{t} \tag{7}\]  
and the boundary conditions of the problem are taken as follows:  
\[