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Geo Struct Sparks LLC

When Mononobe-Okabe Equation Doesn't Work


 This blog is an area to try out new text or content on the website and get feedback on it before adding it to the site.

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Well yes, you can "break" this program

Posted by geostructsparks on 16 September, 2015 at 19:00

After selling this program for five years, suddenly two separate clients came up with scenarios that "broke" WASP within the same month. By break, I don't mean destroy the software, but rather they managed to create scenarios with infinite results on level ground, for separate projects, one in Japan, and one in Panama.


The examples were both where the designers were in high seismic areas, were on level ground, wanted to simulate "non-moving (at-rest) Ko" conditions, and they were not using particularly strong soil materials. The example from the client below is for a 4.05m high vertical wall with soils with 30 degree friction angle. Following guidelines in the US FHWA manual on earthquake design, they figured that to get "Ko-earthquake" conditions, rather than designing for 0.5 times the PGA, they would design for 1.25 times the PGA - which resulted in 0.66 g x 1.25 = 0.75g.


Results are shown below. Basically they "broke" my promise that WASP would never come up with a infinite active pressure. This is because as the PGA was increased, their calculation actually brings the "infinite slope" scenario to flat ground (Mononobe-Okabe would also result in an imaginary result). In other words, the linear failure surface associated with M-O and the WASP code which provide the maximum active pressure dropped to, and slightly below, and 0 degree angle. At first it appears the graph is not working, but on closer examination Kae is in the range of 400,000, and this occurs only at a failure inclination of 1 degree from horizontal.



Valid options are:


  • Increase the soil strength, 30 degree friction angle would be representative of a only medium dense sand - which if saturated and 0.75g PGA, would liquefy, you've got bigger problems - or if an intermediate sand/silt/clay add cohesion. Increasing the friction angle to 37 degrees for this geometry started getting semi-realistic results.
  • The concept that to get a wall to move less in the earthquake scenario, simplify design for a higher ground acceleration, has its limits.
  • In M-O and WASP the failure occurs when the ground is accelerating away from the wall, and the wall simply cannot accelerate fast enough to keep up. In this case, the designer was developing loads for a buried underground vault, where the soil on the other side of the vault almost certainly will give the vault a push so that it does not "fall behind" the soil that is accelerating away from it at 0.75 g.
  • Be aware how big a "failure envelope" you are making - it basically extends to the horizon.
  • Despite our engineering tools, it may not be realistic to say that a structure "will not move" (the premise of Ko soil pressures) at 0.75g, You could try going to higher quality tools such as finite element programs, however I would be willing to get even these methods would have a hard time modeling 0.75 g without movement(much less being able to validate your results against a case study).

An alternate solution shown here is to increase the cohesion to 10 kPa (208 psf) with the friction angle of 30 degrees - might be appropriate for a silty sand. While for a simple problem such as the client was designing, you probably don't have enough testing, consider this implies that the soil is capable of standing in a vertical cut 1.7 m (5 feet high), which is roughly the maximum vertical cut for trenching allowed by US OSHA. If you have already seen construction on your site, you may have observed trench excavations for utilities this deep constructed vertical...


The next blog post shows how I can model the seismic active pressure using a slope stability program for the example above.

Modeling Kae with Slope Stability Programs

Posted by geostructsparks on 11 September, 2015 at 5:20

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.

Discussion Published in ASCE JGGE

Posted by geostructsparks on 7 September, 2012 at 5:50

My discussion on Atik and Sitar's paper was published in the ASCE JGGE August 2012 issue (Jonathan Pease, Discussion of  Al Atik, L.  and N. Sitar, 2010, Seismic Earth Pressures on Cantilever Retaining Structures, ASCE Journal of Geotechnical and Geoenvironmental Engineering).  Much of the discussion and questions I included in the discussion are included under the Nay Sayers section. 

Unfortunately, there is no response / closure by the authors in this issue that would answer my questions on their paper.  I will post a copy of the discussion in the next day or two on the site.  The only problem with a discussion on a very dense (generally good) paper is that its like reading a footnote discussion where you need the entire paper to understand what the issues are.

Mandatory Upgrade March 2012

Posted by geostructsparks on 28 March, 2012 at 8:30

In the first released version of the program in March 2010, we included a mandatory upgrade feature where the program, either in demo mode or with a license, stopped working 2 years after last compiled.  When the progam is opened, the "About" screen opens but there is a message in red "Suggesting" that the user go to this website and download a new version.  Rather than continue to the main program, the program closes after this screen.  This was to make certain that we could correct any bugs or make improvements if necessary and require everyone to upgrade to a new version with those changes.

This seemed like a good idea at the time (sorry!), but frankly I forgot about this until after the program lapsed this March.  (Meanwhile I had switched computers and did not have the compiler installed, so it took a week to catch up, etc...).  For users of either demo or licensed copies, thd fix is simple:  download the new version and install it over the old version.  I have found that it is not necessary to uninstall the old version before installing the upgrade.

The following changes were made to the new program version:

The license expiration counter (number of days left on license) counted up rather than down, although the license did correctly expire after a year;

We added a second equation to the Franklin and Chang upper bound displacement model applicable if the design Kh is much lower than the earthquake acceleration (see parameter guide);

The program will still suggest that you upgrade in two years time, but it will continue running.

Errata on Paper

Posted by geostructsparks on 24 July, 2010 at 9:26

There is an error in the version of the paper to be published online and in the paper proceedings version for the Toronto Conference.  Equation 10, which computes the mass or weight of the soil wedge, has four additive terms.  Each term should include gamma, or the unit weight; it is missing from one of the terms.  This is corrected in the version available for download from the "Parameter Guide" page as of the date of this entry.  This hopefully should be obvious to anyone using the paper since the units have to be consistent.

WASP Conference Paper Accepted

Posted by geostructsparks on 4 March, 2010 at 20:23

Our paper on "Active Wedge Analysis of Seismic Pressure for Retained Slopes" has been accepted for the 9th US National/10th Canadian Conference in Earthquake Engineering has been accepted.  This conference is in July 2010 in Toronto, Ontario.  It provides a summary of the methodology and equations, and benefits of using this alternate method.

A preconference copy of the paper can be reached by the link in the start of the parameter guide.

WASP 1.0.1 License Available

Posted by geostructsparks on 30 November, 2009 at 9:40

The WASP program is finally available for sale!  The last 2 to 3 weeks have been taken up with learning or trying to learn how to make the purchase work on the web.  There have been a number of obstacles thrown up, which I think we have finally overcome.  Its not elegant, but hopefully it works.  With this task completed, I intend to focus on writing up the online help.  I will have screen shots of the program up in the "About WASP" page in a day or two.


The odyssey started about 4 years ago, when I first prepared WASP on a spreadsheet.  I wrote WASP towards the end of a specific project (more later) because I could not find a wedge analysis in online sources or in literature that was reasonably cost effective to check the seismic active pressure (I did see a Culmann Analysis module available for $3000!).  While I have continued to search and not find anything particularly applicable, there may be some codes out there that do the same thing.  We hope that with the added bells and whistles (being able to observe on screen the predicted failure wedge, incorporating displacement-based design into the analysis) we have made this program a useful tool for design that you will find cost-effective to purchase.  When we started Geo-Struct Sparks 8 months ago, essentially to license this software, we determined that we would develop the spreadsheet into a standalone program.  Thanks to Chris Cortopassi provided excellent programming services.