# On Two Approaches to Slope Stability Reliability Assessments Using the Random Finite Element Method
## 2 Finite Element Strength Reduction Analysis  
As the strength reduction method comprises an integral part of the RFEM, it is now briefly introduced to facilitate the understanding of the following auto-search algorithms and the later comparison of the two approaches to slope reliability. In the strength reduction method [15], the factor of safety (FOS) of the slope is defined as the proportion by which \(\tan\phi\) and \(c\) must be reduced to cause failure with the gravity loading held constant. Gravity loads are generated first and applied to the slope in a single increment. A trial strength reduction factor (SRF) loop gradually scales down the soil strength until the algorithm fails to converge within a given iteration limit (e.g., 500). Each entry of this loop implements a different strength reduction factor SRF (usually increasing). The factored soil strength parameters that go into the elasto-plastic analysis are obtained from  
\[\phi_{f}=\arctan(\tan\phi/SRF) \tag{1a}\] \[c_{f}=c/SRF \tag{1b}\]  
where \(c\) is the cohesion and \(\phi\) is the friction angle. Progressively larger values of the SRF factor are attempted until the algorithm hits the iteration ceiling (which also indicates a sudden increase in the nodal displacement, as shown in Figure 1), at which point the SRF is then interpreted as the factor of safety FOS, therefore \(FOS=SRF_{limit}\).  
An automatic search routine (called AutoFOS) was written to find this limiting SRF to an accuracy of 0.01, taking advantage of the pattern of progression of the iteration count with SRF [16]. The AutoFOS search routine is initialised by first defining a starting SRF, \(SRFinitial\) and the step size to the next trial SRF value, \(\Delta_{SRF}^{initial}\) (where superscript \(initial\) refers to the initial step). Three other artificial iteration cut values (Cut1, Cut2 and Cut3) are used when the iteration counts are close to failure, at which point \(\Delta_{SRF}\) is truncated in a prescribed manner (i.e., a larger step was taken for smaller counts and a smaller step size for larger counts, as shown in Figure 1).  
The pseudocode for the AutoFOS procedure is as follows (\(I\) and \(C\) are the actual and maximum-allowed/ceiling plastic iteration counts):  
* 10 Generate a slope and shear strength profile with mean, \(\mu_{c}\)
* 20 Do While (\(I^{j-1}\)\(<\)C), \(j\) is loop number:
* 30 If (\(j=1\)) then \(SRF^{j-1}=SRFinitial\), \(\Delta_{SRF}^{j}=0\)
* 40 Else If (\(j=2\)) then \(\Delta_{SRF}^{j}=\Delta_{SRF}^{initial}\)
* 50 Else If (\(j\geq 3\)) then
* 60 If (\(I^{j-1}\)\(<\)Cut1) then \(\Delta_{SRF}^{j}=\Delta_{SRF}^{initial}-\left(\Delta_{SRF}^{initial}\sqrt{\frac {I^{j-1}}{C}}\right)\)  
Figure 1: Strength reduction factor versus maximum displacement and iteration numbers for a 45\({}^{\circ}\) slope with an undrained shear strength of 20 kPa.  
70 Else If (\(I^{j-1}\)\(>\)\(Cut1\)) then \(\Delta^{j}_{SRF}=0.05\)  
80 Else If (\(I^{j-1}\)\(>\)\(Cut2\)) then \(\Delta^{j}_{SRF}=0.02\)  
90 Else If (\(I^{j-1}\)\(>\)\(Cut3\)) then \(\Delta^{j}_{SRF}=0.01\)  
100 Else If (\(I^{j-1}\)\(>\)\(C\)) then Slope failed, finish  
110 End If  
120 End If  
130 Calculate new \(SRF\): \(SRF^{j}=SRF^{j-1}+\Delta^{j}_{SRF}\)  
140 Reduce the shear strength: \(\mu^{j}=\frac{\mu}{SRF}\)  
150 Finite element plastic analysis of the slope, recording the number of iterations (\(I^{j}\)) taken to converge  
160 End Do  
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In the above and in the following, \(\mu_{c}\) represents, in general, the mean of \(c\) and \(\phi\). For an undrained analysis, it represents the mean of the undrained shear strength \(\mu_{S_{u}}\).