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# Probability of Failure

The definition of Probability of Failure in SWedge (6.0) is:

PF = Number of Failed Wedges / Number of Samples

where:

This definition of the Probability of Failure is considered representative of the overall probability of wedge failure for a given set of probabilistic input data.

NOTE: The default value of Design Factor of Safety = 1, however this value can be changed in the Project Settings dialog.

## Probability of Sliding

Another measure of probabilistic failure for wedges is the Probability of Sliding. This is defined as:

Probability of Sliding = Number of Failed Wedges / Number of Valid Wedges

where:

Number of Valid Wedges = Number of Samples - Number of Invalid Wedges

The Probability of Sliding is the probability that a kinematically valid wedge will fail (slide).

The generation of Valid or Invalid wedges is related to the random sampling of orientation distributions (joint, slope or tension crack). Depending on your geometry input, it is possible for the random sampling of the input data to generate invalid wedge geometries.

• A Valid Wedge is a kinematically feasible wedge, which conforms to the wedge plane definitions used in SWedge, and which can be removed from the slope.
• An Invalid Wedge is a wedge which does NOT conform to the wedge plane definitions used in SWedge, or which cannot be removed from the slope.
• An Invalid Wedge can also be defined by the settings in the Failure Mode Filter dialog (e.g. if you exclude certain failure modes then these become invalid wedges).

The Number of Valid Wedges is also equal to the Number of Failed Wedges + Number of Stable Wedges.

The Probability of Sliding is listed in the Info Viewer, or you can calculate the value from the probabilistic analysis summary in the Sidebar (i.e. Number of Failed / Number of Valid).

If the statistical sampling of orientations does NOT produce any Invalid Wedges (i.e. Number of Valid = Number of Samples), then the Probability of Sliding = Probability of Failure.

Note that the Probability of Failure can also be written as:

Probability of Failure = (Probability of Kinematically Valid Wedges) x (Probability of Sliding)

= (Number Valid/Number Samples) x (Number Failed/Number Valid)

= Number Failed / Number Samples

## SWedge Version 4.0 Definition of Probability of Failure

It should be noted that in previous versions of SWedge (prior to version 5.0), the Probability of Failure which was reported, was in fact the Probability of Sliding (as defined above).

That is, in SWedge version 4.0:

Probability of Failure = Number of Failed Wedges / Number of Valid Wedges

As of SWedge version 5.0, the definition of Probability of Failure was revised to the current definition, because the earlier version (where Probability of Sliding was reported as the Probability of Failure), had a tendency to overestimate the Probability of Failure in some cases, which could lead to overly conservative slope designs.

For an interesting discussion of Probability of Failure in rock slopes, click here to read a discussion posted on the Rocscience website.