This paper was presented in APSSIM 2016 (First Asia Pacific Slope Stability in Mining Conference). Probabilistic slope stability can be used to optimise the geotechnical studies.

1. Probabilistic slope stability analysis as a tool to optimise a
geotechnical site investigation program
Mahdi Zoorabadi
Ismet Canbulat
Marc Ruest

2. Objectives of geotechnical investigation
• Provide a base level of geotechnical data on which to conduct mine design.
• Quantify the geotechnical environment.
• Identify specific geotechnical hazards impacting on mine design.
• Provide spatial characterisation of the geotechnical domains to assist with
planning/budgeting and evaluating project NPV and risk

3. Uncertainty is inherent feature of geotechnical quantification
• Geological uncertainties
• Spatial variability
• Lack of sufficient data
• Human errors

4. Probabilistic slope stability analysis
Limitations of deterministic analysis :
• Variability of the input parameters
• “How stable is the slope?”
• Probability of failure
Eddleston et al. (2004)
Probabilistic slope stability analysis methods:
• Taylor series (First-order Second-moment expansion)
• Rosenblueth’s point estimation
• Monte Carlo simulation
• Fourier analysis
“Probability is part of the language of risk, much used
and understood by managers and non-engineers. Giving
them advice using risk language would therefore help
them reach the right decisions.” (Briddle, 2002)

5. First-Order Second-Moment Expansion of Taylor Series
𝐹𝑂𝑆 = 𝑔(𝑋1, 𝑋2, … , 𝑋 𝑛
𝑔(𝑋 is the performance function or FOS calculation function (Bishop’s method, …) and 𝑋𝑖 are the component of the input
parameters.
Input parameters are modelled as random variables:
𝐸 𝑋 =
𝑎𝑙𝑙 𝑐
𝑥𝑓(𝑥 𝑑𝑥 𝑉𝑎𝑟 𝑋 =
𝑎𝑙𝑙 𝑥
𝑥 − 𝜇 𝑥 𝑓 𝑥 𝑑𝑥 𝑓(𝑥 : porbabilistic distribution function
𝐸 𝐹𝑂𝑆 = 𝜇 𝑓𝑜𝑠 ≈ 𝑔(𝐸 𝑋1 , 𝐸 𝑋1 , … , 𝐸 𝑋 𝑛 +
𝑖=1
𝑘
𝑗=1
𝑘
𝜕2
𝐹𝑂𝑆
𝜕𝑋𝑖 𝜕𝑋𝑗
𝐶𝑜𝑣 𝑋𝑖, 𝑋𝑗 + 𝑒 𝑓𝑜𝑟 𝑖 < 𝑗
𝑉𝑎𝑟 𝐹𝑂𝑆 = 𝜎 𝐹𝑂𝑆
2
=
𝑖=1
𝑘
𝜕 𝐹𝑂𝑆
𝜕 𝑋𝑖
2
𝑉𝑎𝑟[𝑋𝑖] + 2
𝑖=1
𝑘
𝑗=1
𝑘
𝜕 𝐹𝑂𝑆
𝜕 𝑋𝑖
𝜕 𝐹𝑂𝑆
𝜕 𝑋𝑗
𝐶𝑜𝑣 [𝑋𝑖, 𝑋𝑗 + 𝑉𝑎𝑟 𝑒 𝑓𝑜𝑟 𝑖 < 𝐽
𝜕 𝐹𝑂𝑆
𝜕 𝑋1
=
𝑔 𝑋1 + 𝜎1, 𝑋2, 𝑋3, … , 𝑋 𝑛 − 𝑔 𝑋1 − 𝜎1, 𝑋2, 𝑋3, … , 𝑋 𝑛
2𝜎1
=
∆ 𝐹𝑂𝑆1
2𝜎1
𝜕 𝐹𝑂𝑆
𝜕 𝑋2
=
𝑔 𝑋1, 𝑋2 + 𝜎2, 𝑋3, … , 𝑋 𝑛 − 𝑔 𝑋1, 𝑋2 − 𝜎2, 𝑋3, … , 𝑋 𝑛
2𝜎2
=
∆ 𝐹𝑂𝑆2
2𝜎2
..
.
𝜕 𝐹𝑂𝑆
𝜕 𝑋 𝑛
=
𝑔 𝑋1, 𝑋2, 𝑋3, … , 𝑋 𝑛 + 𝜎 𝑛 − 𝑔 𝑋1, 𝑋2, 𝑋3, … , 𝑋 𝑛 − 𝜎 𝑛
2𝜎 𝑛
=
∆ 𝐹𝑂𝑆 𝑛
2𝜎 𝑛
𝜕 𝐹𝑂𝑆
𝜕 𝑋𝑖
2
𝑉𝑎𝑟 𝑋𝑖 =
1
4
[∆ 𝐹𝑂𝑆𝑖]2
𝑉𝑎𝑟 𝐹𝑂𝑆 = 𝜎 𝐹𝑂𝑆
2
≈
1
4
𝑖=1
𝑘
∆ 𝐹𝑂𝑆𝑖
2
+
1
2
𝑖=1
𝑘
𝑗=1
𝑘
∆ 𝐹𝑂𝑆𝑖 ∆ 𝐹𝑂𝑆𝑗 𝜌𝑖𝑗 + 𝑉𝑎𝑟 𝑒 𝑓𝑜𝑟 𝑖 < 𝑗
𝜌𝑖𝑗 shows the correlation coefficient between variables

6. Reliability Index
𝛽 =
ln (𝐹𝑂𝑆𝑐 − ln 𝜇 𝐹𝑂𝑆 +
1
2
(ln 1 +
𝜎 𝐹𝑂𝑆
𝜇 𝐹𝑂𝑆
2
ln 1 +
𝜎 𝐹𝑂𝑆
𝜇 𝐹𝑂𝑆
2
Lognormal PDF
Normal PDF
𝛽 =
𝐹𝑂𝑆𝑐 − 𝜇 𝐹𝑂𝑆
𝜎 𝐹𝑂𝑆
distance between average
calculated FOS and a critical FOS,
(for example FOCc=1 for stability)
normalised by the standard
deviation of the FOS Abramson et.al (2002)

7. Probability of Failure
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑢𝑛𝑠𝑎𝑡𝑖𝑠𝑓𝑎𝑐𝑡𝑜𝑟𝑦 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑎𝑛𝑐𝑒 = 𝑃(𝐹𝑂𝑆 < 𝐹𝑂𝑆𝑐 = Φ(−𝛽 𝐹𝑂𝑆
If 𝐹𝑂𝑆𝑐 > 1
If 𝐹𝑂𝑆𝑐 = 1
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 = 𝑃(𝐹𝑂𝑆 < 1 = Φ(−𝛽 𝐹𝑂𝑆=1
US Army Corps, 1997 Eddleston et al. (2004)
Φ :Normal Cumulative distribution function

8. 1. Determine the average FOS by using average values of random variables
2. From each material unit, the average value of its cohesion (𝝁 𝒄) is replaced by 𝝁 𝒄 + 𝝈 𝒄 (where 𝝈 𝒄
is the standard deviation) and FOS is determined for the previously obtained failure surface
(𝑭𝑶𝑺 𝒄
+
).
3. For each material unit, the above calculation is repeated by replacing of 𝝁 𝒄 with 𝝁 𝒄 − 𝝈 𝒄, the
results would be 𝑭𝑶𝑺 𝒄
−
.
4. Steps 2, 3 are repeated for the friction angle of each unit and 𝑭𝑶𝑺 𝝋
+
and 𝑭𝑶𝑺 𝝋
−
are calculated.
5. For a slope with n units, the FOS is calculated (4n+1) times, therefore it is possible to do this
analysis even using existing commercial software.
6. Now, ∆ 𝑭𝑶𝑺 𝒄 = 𝑭𝑶𝑺 𝒄
+
− 𝑭𝑶𝑺 𝒄
−
and ∆ 𝑭𝑶𝑺 𝝋 = 𝑭𝑶𝑺 𝝋
+
− 𝑭𝑶𝑺 𝝋
−
are calculated for each unit.
These two parameters represents the relative contribution of uncertainty projected by cohesion
and friction angle of each material units to the uncertainty of FOS.
Steps to calculate the reliability index (only cohesion and friction angle are random variables)

9. Demonstration of the Method

10. Demonstration of the Method

11. Demonstration of the Method
Uncertainties caused by friction angle
of the material unit of 1 have a
higher impact on the reliability of the
FOS. Then, geotechnical study
program should be redesigned to
reduce the standard deviation of
friction angle for material unit of 1

12. Conclusions
• Probabilistic slope stability analysis can be used to quantify the impact of uncertainties
of input parameters on the reliability of slope stability.
• First-Order Second-Moment expansion of Taylor series is easy to use.
• This method determines the relative contribution of uncertainty projected by each
component random variable.
• This capability can be used at early stages of geotechnical studies or during mining to
optimise the designs.