# On Two Approaches to Slope Stability Reliability Assessments Using the Random Finite Element Method
## 4 Quantitative Consequence Assessment of Slope Failures  
### Radiating Scan Method  
As the sliding volumes will be used later to explain the differences of the two approaches, a procedure called the radiating scan method is proposed and briefly introduced here. To quantify the failure volume, an estimation of slip area (\(A\)) has to be made for the problem. This is done by locating the rupture surface first, then smoothing this surface and at last, finding the arc length (\(L_{a}\)) and \(A\) by integration.  
The technique used in [21] has been adopted to define the rupture surface based on the shear strain invariant,  
\[\bar{\gamma}=\sqrt{\frac{2}{3}(\epsilon_{x}^{2}-\epsilon_{x}\epsilon_{y}+ \epsilon_{y}^{2})+\frac{1}{2}\gamma_{xy}^{2}} \tag{3}\]  
in which \(\epsilon_{x}\), \(\epsilon_{y}\) and \(\gamma_{xy}\) are the normal strains in the \(x\) and \(y\) directions and shear strain, respectively.  
First, the maximum shear strain invariant was found along the line just below the toe and the line just below the slope face. Next, an imaginary point in space, above the centre of the slope face, was chosen. Then, a series of lines radiating out from this point (across the slope mesh, clockwise) was considered (the first radiating line is the line connecting the point located in the first step and the imaginary point). The maximum value of shear strain invariant along each line (i.e., clockwise) was detected and, by connecting the locus of maxima from all the lines, the rupture surface was defined.  
Figure 4: Postprocessing an isotropic random field to produce an anisotropic field by squashing and stretching (\(\bar{\zeta}=6\)) (52 cells after Crop2 to be able to interpolate between the last two columns of cells).  
This is shown in Figure 5 schematically. Note that the shear strain invariant within an element is interpolated using the shape functions.  
However, no precise information about the length of the rupture surface is available by adopting the above-mentioned Radiation Scan method only. A procedure is implemented to smooth the curve by polynomial curve fitting; that is, by finding the coefficients of a polynomial \(p(x)\) of degree \(n\) that fits the data, \(p(x(i))\) to \(z(i)\), in a least squares sense, \(((x(i),z(i))\) are the coordinate pair defining the  
Figure 5: A Schematic illustration of the radiating scan method for locating the failure path in 2D.  
Figure 6: Polynomial curve fitting of the scanned rupture path in 2D.  
rupture surface). By doing so, the rippled curve shown in Figure 6 is smoothed to the curve shown in red. Next, the arc length is found by numerically integrating \(dL_{a}\) between 0.5 and 10:  
\[dL_{a}=\sqrt{1+\left(\frac{dp}{dx}\right)^{2}}dx \tag{4}\]  
Figure 6 shows that the slip area is found by computing the integral of \(p\) with respect to \(x\), using repeated trapezium integration and then subtracting the area of the triangle above the slope face. For the present slope geometry and failure mechanism, the integrated length of the rupture surface is \(L_{a}=12.0\) m and the slip area is \(A=23.0\) m\({}^{2}\).