Today we discuss the Factor of Safety and probability of failure of geostructures.

We presented this discussion at TMW 2020 and are delighted to share with you our talk.

This information is applicable to dams as well as dumps and pit slopes. However that requires the analyst pay attention to the specific details pertaining each type of structure.

Factor of safety and probability of failure of geostructures

The Factor of Safety (FoS) measures the ability of a structure to withstand its loadings. It is the ratio between resisting forces and driving forces acting on or within a structure. Both resistance and loadings are subject to uncertainties, thus are stochastic variables. The intrinsic variability of geomechanical parameters, water table position, external loadings, etc. influence resistance and loading.

Common practice is to consider those variabilities by reducing resistance and increasing loadings to “prudent” values leading to a single deterministic value of FoS. Simplified probabilistic analyses consider the probability of failure as the probability the FoS to be smaller or equal to one. FoS equal one means loadings equate resistance, thus the structure is metastable.

The reliability is equal to resistance minus loading. Thus also expresses the ability of a structure to withstand its loadings. The probability of failure is the probability that reliability assumes negative values. That means that loadings exceed resistance.

In the formulations above the reasoning is purely mechanical. No human factors, standard of care, maintenance intervene.

In recent years various authors have expressed the probability of failure as a combination of Key Performance Indicators (KPIs). They include human factors thereby altering the value of the purely mechanical approach. Indeed, they combine models and empirical approaches to include human factors and quality of studies, design, monitoring, maintenance.

Semi empirical approaches limits

Like usual there are detractors and excessively enthusiastic users.

Detractors consider these approaches as doubtful and unverifiable, however, without delivering a counter-proof. Excessively enthusiastic users apply these semi-empirical approaches without considering the specificity of the considered structures.

We showed in our book on Tailings Management how misleading that can be (Fig. 15.6, page 244). We have recently seen a major mining company using semi-empirical approaches in this manner and immediately wrote to warn them of the potential misjudgment.

Lessons learned

What we have learned is that the semi-empirical approaches help answering some perplexing questions geotechnical engineers have. The most important one is why are failures less frequent than our reliability studies predict?

In the paper and the presentation we discuss the answer to this question and show examples going back to 1984.

We also show how to view the geomechanical factor of safety and put it in perspective with a semi-empirical probabilistic approach.

Indeed, we carried out a number of trials. We used a probabilistic geomechanical approach and compared the results to a semi-empirical empirical methodology. The test cases were in cohesive and granular slopes of non-engineered materials (natural soils). In each case we found comparable results, within the same order of magnitude for the two approaches.

The probability of failure of real-life dams lies in a range of six orders of magnitude and world-wide historic tailings dams performance lies within one order of magnitude. Thus, finding a correspondence between a theoretical and empirical probability of failure within one order of magnitude is more than adequate.

Now, engineered materials have fewer uncertainties than natural non engineered ones. That is because they are selected and carefully compacted under controlled water content, reducing their variability. Proper management, monitoring, etc., further reduce uncertainties and thus enhance reliability of structures using them.

As a result, engineered slopes necessarily have lower probability of failure than non-engineered slopes. Accordingly, the empirical methods reflect the reduction of uncertainties leading to lower annualized probabilities.

Closing remarks

The ability to generate and compare Factor of Safety and probability of failure of geostructures is paramount for the formulation of risk informed design and to foster engineer’s confidence in probabilistic solutions.

We advice you to drop the FoS as a decision tool and only perform probabilistic analyses. Our preference is to rely on semi-empirical systemic approaches. They should couple with benchmarking, to ensure anchoring to reality.

Geomechanical probabilities of failures can be used for comparing design alternatives among themselves. However they are not valid to perform risk assessments, in particular at portfolio level. Indeed, their values are too high and correspond to non-engineered geotechnical structures.

One big merit of the purely geomechanical approaches is to show, for example, that the factor of safety should be different for slopes built with different materials. As a result, clayey slopes should have a larger factor of safety than granular slopes. That is because the variability of cohesion is way larger than the variability of friction.

Good discussion