13.3 Reliability Analysis
A probabilistic approach, on the other hand, allows for the systematic analysis of uncertainties and for their inclusion in evaluating slope performance. Important geotechnical parameters such as shear strength parameters and pore water pressures may be used as random variables, each with a probability distribution, rather than deterministic values. Consequently, the factor of safety F of a slope under specified conditions must also be regarded as a random variable with a probability distribution.
If one could compute factors of safety with absolute precision, a value of F=1.1 or even 1.01 would be acceptable. However, the uncertainties involved in computing factors of safety, the computed values of factor of safety are never absolutely precise.
The reliability of a slope (R) is an alternative measure of stability that considers explicitly the uncertainties involved in stability analyses. The reliability of a slope is the computed probability that a slope will not fail and is 1.0 minus the probability of failure:
Where the probability of failure and R is is the reliability or probability of no failure. Factors of safety are more widely used than R or to characterize slope stability. Although R and are equally logical measures of stability.
There are two common methods of calculating the reliability of slope.
1. Margin of safety method
2. Monte Carlo method
The margin of safety is the difference between the resisting and displacing forces, with the slope being unstable if the margin of safety is negative. If the resisting and displacing forces are mathematically defined probability distributions and respectively then it is possible to calculate a third probability distribution for the margin of safety. If the lower limit of the resisting force distribution fD(r) is less than the upper limit of the displacing force distribution fD(d). The shaded area in Figure 5 being proportional to the area of the shaded zone. The method of calculating the area of the shaded zone is to calculate the probability density function of the margin of safety: the area of the negative portion of this function is the probability of failure (Figure 5). If the resisting and displacing forces are defined by normal distributions, the margin of safety is also a normal distribution, the mean and standard deviation of which are calculated as follows
Mean, margin of safety =
Standard deviation, margin of safety=
where fD(r) and fD(d) are the mean values, and σr and σd are the standard deviations of the distributions of the resisting and displacing forces respectively.
Figure 5: Calculation of probability of failure using normal distributions: (a) probability density functions of the resisting force and the displacing force in a slope; and (b) probability density function of difference between resisting and displacing force distributions
Monte Carlo analysis is an alternative method of calculating the probability of failure which is more versatile than the margin of safety method. Monte Carlo analysis avoids the integration operations that can become quite complex, and in the case of the beta distribution cannot be solved explicitly. The particular value of Monte Carlo analysis is the ability to work with any mixture of distribution types, and any number of variables, which may or may not be independent of each other .
The principal characteristic of the Monte Carlo method is that of generation a large quantity of random numbers varying between 0 and 1. The examined problem variables are generated by these random numbers in such a way as to respect the assumed probability distribution curves. The component input parameters in a slope stability analysis are modeled as random variables and are used to estimate the PDF of the factor of safety. This PDF is characterized by its mean value, , and standard deviation, . Although the PDF of the FOS, can take on any shape, it is usually assumed to be either normally or log-normally distributed. The total area under the curve for both the normal and lognormal PDF distributions is always equal to 1.0. If one considers FOS values less than one to represent failure, the shaded areas of these distributions will represent the probability of failure,.
Figure 6: Definition of Probability of failure for lognormal and normal PDFs
Lognormal distribution will always take on values greater than zero, and as this the same for the factor of safety, this distribution is often considered to be more appropriable. For a lognormal PDF of F, the reliability index, , is defined by (Wolff, 1996)
Where the critical factor of safety corresponding to unsatisfactory performance or 1,0 for the case of failure.
(standard deviation of a lognormally distributed FOS)
(coefficient of variation of FOS)
For the special case of the probability of failure, 1.0, and Equation 6.91 simplifies
Therdiability index, , describes safety as it is a measure of the number of standard imparating the best estimate of F, and the prescribed critical value of the
In the PDF of F is presumed to be normal, the reliability index is defined by
In the critical factor of safety corresponding to unsatisfactory performance, or 1.0 for the case of failure.
(mean or expected value of a normally distributed FOS)
(standard deviation of a normally distributed FOS)
The Monte Carlo method simulation can be carried out following these steps:
1. A deterministic method, such as the limit equilibrium method, is chosen and used to calculate the safety factor or the safety margin as being dependent on parameters of the problem which should be modelled probabilistically.(Example shear strength desility etc).
2. The probability density distribution obtained from experimental data measurements can be constructed in a histogram from.
3. The cumulative distribution is constructed for each random variable probability density distribution. This cumulative curve can be drawn, by dividing, for instance, the variation field of each probabilistic parameter into ten intervals, in such a way that an increasing value between 0 and 1 corresponds to each central value of these intervals.
4. Random values are generated and the correspondent values of the random variables are determined. In the case of the example, random values varying in the 0-1 field are generated and for each generation, the correspondent dip angle is determined.
5. The random variable values obtained by the random generation are used as input data for the determination of the correspondent safety factor with the chosen deterministic method.
6. The operation described in points 4 and 5 are repeated until a stable distribution of the safety factor or safety margin probability density is obtained. A reliable number of generations should be as much as to allow the distribution to be stable according to the designed approximation degree. The distribution stability can be controlled by means of the comparison between two probability density distributions obtained with different generation numbers. The test can be used for this purpose.
Monte Carlo simulation produces distributions of possible outcome values;
The number of trial calculation is given by
Where n=minimum number of trials for the Monte Carlo simulation
PF =probability of unsatisfactory performance (or failure)
E=relative percent error in estimating the probability, pF
m=number of component random variables
d=normal standard deviate according to the following confidence levels:
Confidence Normal Standard
Level Deviate, d
The relative % error can be calculated as
With E computed, the resulting Pf vale can be considered to be accurate to within a range of 0.01E × Pf , with the associated level of confidence.
Next, random values of component variable are generated and FOS is calculate using
performance function. Uniformly distributed values R1, R2,….., RN is generated between
0 and 1. With these, random values for a standardized normal distribution (μ=0, σ=1)
can be generated using Box and Muller method:
Where, Xi= value taken by the normally distributed random variable
For dependent random variable, one of the pair of independent random numbers, N1
or N2, are modified:
For each set of component variable , FOS is calculated using performance function.
For a given critical FOS value, probability of unsatisfactory performance is reported as:
There are two slope A and Slope B, and the PDFs are presumed to be lognormally distributed for F. Slope A has a mean FOS and slope B, has a higher mean FOS, In a deterministic approach. Sense, it would appear that Slope B is safer than Slope A. However, the FOS PDF for Slope A has a much smaller standard deviation in comparison to Slope B, which leads to reliability index values of for the critical F=1.0 assumption. As Slope A has a greater reliability index, Slope A can be considered safer than Slope B as there is less uncertainty associated with Slope A (Figure 7).
Consider another example of a slope with a conventional deterministic factor of safety F = 1.1 and compare with a reliability approach based on a mean value F = 1.1 and a standard deviation of σF = 0.1. F = 1.1 implies a conventional safety margin of 10% which does not reflect the uncertainties that might be associated with slope performance. The value of F is based on a constant value of each of the parameters on which it depends. The variability and thus the overall uncertainty associated with the value of each parameter is thus ignored in calculating the value of F. Therefore it will not represent the reliability of the slope accurately. Consider now a reliability approach within a probabilistic framework. Consider F as a random variable or stochastic parameter with mean value F = 1.1 and a standard deviation of 0.1 (this implies a coefficient of variation (c.o.v.) of F, VF = 9.1%). Then the reliability index from Equation (10.1) above is β = 1. With a mean F of 1.2 and a standard deviation of 10% (note that now this value implies a c.o.v., VF = 8.3%), the reliability index β = 2, and the probability of failure pF is about 2.2 %. Thus we note that the reliability index has doubled and the probability of failure has decreased by a factor of about 7.2 (about 86%) although the mean factor of safety increased by only about 9.1%.
Figure 7: Comparison of factor of safety for two slopes where Slope A has a higher reliability index due to its smaller standard deviation value.