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# Failure probability associated with a given factor of safety3

## Failure probability associated with a given factor of safety

(OP)
Does anyone know of any good references or research papers on failure probability vs factor of safety. I'm looking for something like 'slopes with a factor of safety of 1 have an annual failure probability of 1/5, slopes with a factor of safety of 1.5 have an annual failure probability of 1/100,000'.

I have the idea in my head that T William Lambe might have presented something to this effect in a distinguished speaker lecture (Terzhagi or Peck ?) but I can't find it - although it may have been for earth dams.

### RE: Failure probability associated with a given factor of safety

I remember many years ago there was a paper in ASCE Geotechnical Journal - the professor was from Worchester in Massachusetts - the name escapes me. I think the paper was in the late 70s (?). Something that struck me was an example where a FoS of 1.5 had a probability of failure of 20% (from memory).

### RE: Failure probability associated with a given factor of safety

John Christian produced this very handy plot. This is from 2013, but I expect that there are earlier versions as well.

### RE: Failure probability associated with a given factor of safety

I've read a couple of different papers that discuss this topic, unfortunately I do not have access right now. I'll look for them next chance I get.

My recollection is that there is no one probability of failure related to a slope's factor of safety. There are too many elements that go into the factor of safety calculation. For a given slope, there are ways of calculating the probability of failure, but it requires that the stability calculations be performed with various input parameters and that the variation of the input parameters be statically/probability established.

Always thought it would be interesting to go through the process, but never had the project that would pay for the effort. Some of the commercial software packages may allow you to do this automatically now days. You might look into the GeoStudio Suite (Slope/w).

### RE: Failure probability associated with a given factor of safety

Hi guys, I haven't been dealing with probabilities for a while now, but from a quick look these are my thoughts.

Since Pf=P[FS<1], the probability of failure will be a function of the statistical distribution used to model FS (for example, normal, lognormal, beta...) and the parameters of such distribution (mean and standard deviation for example).
This concept is clear in the plot from Christian posted above by ATSE, where the smaller the COV (coefficient of variability), the higher the probability of failure, since the probability mass is less concentrated around the mean or median values. A higher COV means a higher variability due to higher uncertainty in the parameters used for the slope stability analysis (phi, c', density and so on).

Of course, the probability of failure of a slope with a deterministic FS=1 (average FS=1) should always be 50% if the distribution is assumed to be normal since the average is equal to the median which by definition is the 50% percentile of the distribution and 50% of the values lie below it. If FS is lognormal, then the average value is greater than the median value hence Pf should be >50%.
One clear practical example of a rock slope with average FS=1.34 and P(f)=0.064 or 6.4% is given in Hoek's rocscience handouts

https://www.rocscience.com/assets/resources/learni...

### RE: Failure probability associated with a given factor of safety

(OP)
Thanks McCoy, really interesting input

To give an idea of what I'm trying to do: I'm assessing on a regional (well not really regional, but a large subdivision proposed on a slope) landslide susceptibility, hazard, and risk. In New Zealand and in the terrain I'm dealing with landslides are largely triggered by rainfall and seismic events, so what I'm attempting to go is calculate: probability that a triggering event occurs * probability that the slope actually fails given that triggering event * (various other probabilities that represent run out distance / hazard to people and buildings).

my current line of thought is to calculate porewater pressure changes in response to rainfall using a software program called TRIGRS published by the USGS, and then calculate a DEM of factors of safety using Scoops3D. Based on those factors of safety I will than calculate the probability of failure. Of course - there are alot assumptions involved (most importantly that I largely have to assume the probability distribution of the various parameters based on published ranges (eg. JM Duncan) calibrated against my site specific CPT data to calculate the probability of failure for a given FS ?)

It's a very interesting and challenging topic; the council authorities involved have high (almost absurd) standards for the risk assessment they want but the published methods are all inadequate. For instance, using 'the past as the key to the future' isn't very helpful because the development completely changes the distribution of groundwater infiltration (decreasing it over most of the site but increasing it in the pond areas). Also the soils involved have some unique aspects that make it quite challenging (in my opinion - pumiceous soils)

### RE: Failure probability associated with a given factor of safety

#### Quote (geotechguy)

Based on those factors of safety I will than calculate the probability of failure. Of course - there are a lot of assumptions involved (most importantly that I largely have to assume the probability distribution of the various parameters based on published ranges (eg. JM Duncan) calibrated against my site-specific CPT data to calculate the probability of failure for a given FS ?)

I don't know the way 3Dscoop calculates Factors of safety and if they are time-dependent (annualized) or not. But to transform them into probabilities of failure, I would need to calibrate the 3Dscoop Fs with a FS calculated probabilistically with suitable software (like rocscience SLIDE) or by a calc sheet. That is, on a small number of sample slopes, calculate both and see if they are more or less the same, and then see what's the corresponding Pf.

Unless somewhere in the literature there is a reliable study on the variability of FS in various conditions (like we know for example that the COV of sands is on average 5%), then you might apply the plot from Christian above posted. It would remain an imprecise estimate though.

From the parameters you collected, you could study how their variability propagates to the FS variability but, again, you would need software with probabilistic capabilities.

The most brutally imprecise estimate would be to adopt a range of technically plausible COVs and use Christian's plot to derive a range of plausible probabilities of failure. Obviously, that would not be very acceptable in the design of your hazard risk map.

One final consideration: if you have CPT data, assuming the soil has not a totally frictional behaviour, then you don't have a reliable value of c', which governs FS substantially, as has been suggested in the article posted by BigH. Then pairs of phi-c' values from lab tests would be needed.

### RE: Failure probability associated with a given factor of safety

(OP)
Thanks for the input MCcoy

I did look into using Rocscience Slide3 but this is one of the few software packages that I don't have access to; I do have Plaxis LE (rebranded soilvision 3d) but it's not quite as good as Slide3 in this context because it's not as good at producing a plot of lowest FS in each cell; also both software packages are extremely computationally expensive / inefficient compared to Scoops3D and TRIGRS.

### RE: Failure probability associated with a given factor of safety

Geotechguy, I tried a quick search but wasn't able to come up with articles related to the typical COV of slope FS. At this point, based on the investigations you have and the inherent variability of soil in the area you are studying, you might propose a gross estimate of Pf values based on uncertainty propagation reasoning, like for example the following article, which yield confidence intervals for FS related to the numerosity of samples. This variability (which is rigorously valid in the specific example described in the article) may be extrapolated to your area, using the Christian plot. A pretty Imprecise estimate, but I can think of nothing better. You can always propose that estimate, with an abundance of caveats, since objections might legitimately ensue. You might even propose lower and upper bound intervals of Pf maps, underlining the fact that you contemplated the variability , the uncertainty and the subsequent imprecision of the final results.

https://papers.acg.uwa.edu.au/p/2025_51_Valderrama...

### RE: Failure probability associated with a given factor of safety

I've been pondering about the apparently high probabilities of failures related to FoS's.

One reason might be the wrong choice of the representative distribution: sample distribution or distribution of the average value of data?

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