# A Multiobjective Decision-Making Model for Risk-Based Maintenance Scheduling of Railway Earthworks
## 2 Stability Assessment of Earthworks
### Slope Stability  
The earthworks assets considered in this paper include cuttings; where the line is excavated below the existing ground level in either soil or rock (Figure 1a); and embankments where the line is carried above existing ground on man-made fill, usually soil (Figure 1b). Common problems causing instability of slopes are shallow translational landslides caused by high rainfall (Figure 1a,b), rock falls caused by freeze-thaw effects (Figure 1c), and deep-seated rotational failures caused by the construction of embankments over weak sub-soils (Figure 1d). The safety of earthworks is typically determined in a slope stability analysis.  
Figure 1: Examples of earthworks failures: (**a**) shallow translational failure in soil cutting, (**b**) shallow translational failure in an embankment, (**c**) deep rotational failure and, (**d**) rockfall.  
Shallow translational landslides are usually triggered by near-surface changes in moisture content. In dry periods negative pore water pressure (suctions) develop in the near-surface soils, which contribute significantly to their overall stability. This suction transient and reduce during periods of rainfall or snowmelt. Assuming an infinite slope failure model, the factor of safety (\(FOS=\text{capacity}/\text{demand}\)) can be calculated as follows:  
\[FOS=\frac{C+\gamma\ h\ \cos^{2}\ \alpha\ \tan\phi}{\gamma\ h\ \cos\alpha\sin\alpha} \tag{1}\]  
where \(\gamma\) is the unit weight of soil, \(h\) the wetting from depth (depth of infiltrated water), the friction angle of the soil \(\phi\) and \(\alpha\), the slope angle, See [PERSON] and [PERSON] [29]. In this simplified approach, the effective cohesion and matric suction are combined into the total cohesion, \(C\).  
The primary mechanism for deep rotational slip failures on railway networks is the presence of soft ground underlying an embankment. As a result, these failures tend to be geographically well-defined (e.g., in peatlands and delta regions) and are not significantly affected by climate hazards. As the depth of this failure mechanism is typically much larger than rainfall-induced failures, the volume of soil involved is much larger. The factor of safety is typically determined assuming a circular failure surface in which the failure surface is divided into a number of vertical slices, and moment equilibrium is considered [30]. The key inputs for the analysis are the slope geometry and shear strength parameters of the soil. The factor of safety of a slope subject to a rotational failure is as follows:  
\[FOS=\frac{\sum_{i=1}^{n}[c_{i}\Delta x_{i}+(W_{i}-u_{i}\Delta x_{i})\tan\phi_{ i}]\frac{\sec\alpha_{i}}{1+\tan\phi_{i}\tan\alpha_{i}/FOS}}{\sum_{i=1}^{n}W_{i} \sin\alpha_{i}} \tag{2}\]  
where: \(W_{i}\) is the weight (kN) of the _ith_ slice, \(\alpha_{i}\) is the inclination angle of the base of the _ith_ slice, \(\Delta x_{i}\) (m) is the _ith_ slice width, \(c_{i}\) (kPa) is the cohesion of the soil on the base of the _ith_ slice, \(u_{i}\) (kPa) is the pore water pressure at the base of the _ith_ slice, and \(\phi_{i}\) is the friction angle of the soil at the base of the _ith_ slice, See Figure 2.  
The stability of a rock slope depends on a number of internal (slope angle, geology, fracturing, etc.) and external parameters (weathering, tectonic events, human anthropogenic factors, etc.). The impact of freeze-thaw cycles on rockfalls is well documented, as water infiltrates into existing fractures, freezes and causes expansion of the fracture that may eventually lead to a rock wedge being dislodged from the mass. In this paper, a 2D rock wedge failure mode is considered, where the geometry of the critical failure surface is dependent on the slope geometry [31].  
Figure 2: Definition of terms used in rotational failure analyses.