This is a continuation of the previous blog entry - where the client was running into a numerically ridiculous result (basically unstable) because the horizontal earthquake coefficient selected was really high (0.75 g, or 1.25 x PGA, rather than the typical 0.5 x PGA) and the soil friction angle was low.  WASP does not produce a meaningful result, because the resulting linear failure envelope is at 1 degree from horizontal - basically an infinite failure length on level ground. (It would be even flatter or zero, but I only have WASP calculate at 1 degree slip surface increments)  Some slightly better success would be to analyze the horizontal coefficient from a circular slip surface from slope stability - by definition, a circular slip has to intersect a level ground surface somewhere.

The example is using SLIDE by Rocscience, a slope stability program which has really nice standard feaatures.  The problem setup is shown below, which models the previous example my client provided - a 4.05 m high vertical excavation with soil with a friction angle of 30 degrees and a pseudo-static horizontal coefficient of 0.75g (This combination results in the Mononobe-Okabe equation in imaginary numbers, and in WASP also nearly imaginary Kae > 400,000). 

In most current slope stability programs, there is a method to confine the search - in SLIDE this can be done under "Refine Search - Add Point."  The cross hair symbol at the base of the wall shows the selected point that all slope trial surfaces must pass through for this example.  I have added a load of 300 kN/m where the wall will be, which I will vary below in order to get the active earthquake pressure.

I used the Bishop method since it only considers force equilibrium, not moment equilibrium, similar to the M-O equation and WASP. I have left a large area on the uphill side of the retaining wall to allow for a very long failure surface.  An example of the output from this model is shown below.  Factor of safety is indicated by the color represented at the axis of rotation.  Yup, that's an earthquake sliding surface for a 4m high face that extends 30 m horizontally (This is showing what the theory says, perhaps not what occurs in real life?):

Second, SLIDE has a nice feature that you can enable sensitivity analysis.  First you have to enable sensitivity analysis under the Analysis menu, Project Setup tab.  Then you have to select the parameter to vary under the Statistics menu.  The original value of the horizontal load of 300 kN/m was allowed to vary from 0 to 600 kN/m.  After running the analysis, a plot of the horizontal load versus the slope stability factor of safety can be obtained:

A factor of safety of 1.0 corresponds to a horizontal load of 450 kN/m - this is Pae, the total active earthquake force.  We take this result and back-calculate:

Ka = 2*Pa/(gamma * H^2) = 2*450 (kN/m) /20 (kN/m^3) *(4.05 m)^2  or Kae = 2.74.

Yes, this is still ridiculously high, but if you want to create "non-moving at-rest" vaults in Japan, this still might be the appropriate. Or, as noted in the accompanying blog entry, you might pick a better choice of parameters.

One note is that for a "near infinite" slope stability solution, the length of the model and the height of the circular slip surface will change your result, since your longest slope model is always the controlling one.  If we rerun the model above with an increased length and wider selection of rotational points, we get lower FOS (to 0.79) and a slipping surface extending to 10 times the height of the excavation (to around 50 m):

You will see also that the sensitivity analysis gets less sensitive - 0 to 600 kN/m results in only a variation from FOS = 0.74 to 0.87. 

I didn't calculate it, but the resulting Kae would be well above 4.  One thing you might consider with this method is selecting the Kae by deciding what the maximum practical slip surface length might be.  If you decide a reasonable slip surface is at most 5 times the retained depth, you might come up with an active earth pressure coefficient this way that you might be able to live with.

This goes to show that even using slope stability analysis, you get a pretty crazy estimate of the active earthquake pressure, if you have poor input values.