156
11 Dam Stability Failures
• there is uncontrolled erosion of the downstream slopes.
Consequences: road failure, personnel safety hazards, mining infrastructure damage.
Table 11.1 shows a summary of the FoS obtained by the engineers.
Let’s now apply Method 1 to the results of two cross-sections displayed in
Table 11.2.
If we look at the first column F.S. (FoS) it is easy to evaluate “experimental values”
of the FoS population as follows:
minimum = 0.998;
maximum = 1.350;
average = 1.168;
standard deviation = 0.115.
This leads to a coefficient of variation (COV) = standard deviation/average (%) of
roughly 10%. Let’s define the “tails” of the distribution of FoS as maximum-average
and average-minimum and express them in number of standard deviations (sigma
bounds).
As the “tails” are respectively only 1.48 sigma bounds and 1.58 sigma bounds the
distribution would be a bathtub-shaped one, i.e., a U-shaped distribution. That may
mean that we are sampling two different “families” of slope in that first column. It
seems this corresponds to a geological/geotechnical/physical reality as at least one
area of the North slope is less stable than the rest.
Let’s suppose now, for the sake of the discussion, that we would come to the
conclusion the last statement is not correct. Then we would conclude that the analyses
have actually not uncovered the “true” max, min values of the FoS, and could “play
around” with those extreme values to understand what happens to the distribution of
FoS.
We would notice that using an empirical Beta distribution requires us to lower
the lower bound (min) to 0.85 to get a pseudo-triangular distribution, which would
lead to p f (stability) = 0.06 = 6 × 10
−2 (approx. 6%). Raising the upper bound at
1.55 would lead p f = 0.13 = 1.3 × 10
−1 (approx. 13%) and a pseudo-bell-shaped
distribution. At this point we could build an ETA to complete the analysis of the case.
So, at the end of the day, by using a blended approach we can swiftly frame the
problem at hand and allow a rational discussion related to the “homogeneity” of the
considered slope.
Also, the 6–13% range evaluated with Method 1 sits well within our previous
experience on similar cases. In Chap. 15, and particularly in Sects. 15.5.4 and 15.5.5
we will see that this theoretical range may even be a low estimate as it does not
include defects and uncertainties that became known in the course of the study. That
being said, any dam with a probability of failure above 1% (0.01) can be considered
a very poor structure (see Table 10.1) and a candidate to failure. Incidentally, Mount
Polley and Fundao Dam, evaluated with the data available before their respective
failures had probabilities between 8 and 17% (see Sect. 11.2.2).
11 Dam Stability Failures
• there is uncontrolled erosion of the downstream slopes.
Consequences: road failure, personnel safety hazards, mining infrastructure damage.
Table 11.1 shows a summary of the FoS obtained by the engineers.
Let’s now apply Method 1 to the results of two cross-sections displayed in
Table 11.2.
If we look at the first column F.S. (FoS) it is easy to evaluate “experimental values”
of the FoS population as follows:
minimum = 0.998;
maximum = 1.350;
average = 1.168;
standard deviation = 0.115.
This leads to a coefficient of variation (COV) = standard deviation/average (%) of
roughly 10%. Let’s define the “tails” of the distribution of FoS as maximum-average
and average-minimum and express them in number of standard deviations (sigma
bounds).
As the “tails” are respectively only 1.48 sigma bounds and 1.58 sigma bounds the
distribution would be a bathtub-shaped one, i.e., a U-shaped distribution. That may
mean that we are sampling two different “families” of slope in that first column. It
seems this corresponds to a geological/geotechnical/physical reality as at least one
area of the North slope is less stable than the rest.
Let’s suppose now, for the sake of the discussion, that we would come to the
conclusion the last statement is not correct. Then we would conclude that the analyses
have actually not uncovered the “true” max, min values of the FoS, and could “play
around” with those extreme values to understand what happens to the distribution of
FoS.
We would notice that using an empirical Beta distribution requires us to lower
the lower bound (min) to 0.85 to get a pseudo-triangular distribution, which would
lead to p f (stability) = 0.06 = 6 × 10
−2 (approx. 6%). Raising the upper bound at
1.55 would lead p f = 0.13 = 1.3 × 10
−1 (approx. 13%) and a pseudo-bell-shaped
distribution. At this point we could build an ETA to complete the analysis of the case.
So, at the end of the day, by using a blended approach we can swiftly frame the
problem at hand and allow a rational discussion related to the “homogeneity” of the
considered slope.
Also, the 6–13% range evaluated with Method 1 sits well within our previous
experience on similar cases. In Chap. 15, and particularly in Sects. 15.5.4 and 15.5.5
we will see that this theoretical range may even be a low estimate as it does not
include defects and uncertainties that became known in the course of the study. That
being said, any dam with a probability of failure above 1% (0.01) can be considered
a very poor structure (see Table 10.1) and a candidate to failure. Incidentally, Mount
Polley and Fundao Dam, evaluated with the data available before their respective
failures had probabilities between 8 and 17% (see Sect. 11.2.2).