Geo Struct Sparks LLC

When Mononobe-Okabe Equation Doesn't Work

Diagram

WASP is a simplified wedge limit equilibrium solution with two slopes (one above the wall, and the second a horizontal bench).  The geometry closely matches that for a retaining wall below a road embankment.  The main trick to avoiding the limitations of the Mononobe-Okabe equation is to avoid an infinite slope. Cohesion may also provide a non-infinite and much more acceptable earthquake active coefficient (just as adding 50 to 100 psf cohesion may be useful for eliminating low FOS shallow failures in circular slope stability analysis - if the soil properties justify it).  Diagram a shows (mostly) the input parameters (plus Pa and alpha) and Diagram b shows the internally computed parameters.

 A full description including equations can be found in Pease and Menes (2010) (click here to download a pdf of the paper).

Settings

Units - metric or English.  The program does not change values (does not change unit weight from 20 kN/m3 to ~ 120 pcf, for example) when the user changes this option.  The correct units for entry are shown adjacent to each parameter.

Coulomb versus Rankine - The Coulomb condition is the general case for which Mononobe-Okabe was prepared.  For this condition, the wall angle (omega) and earth pressure rotation (delta) can be any value selected by the user - more below.

The Rankine condition describes the soil stress under level ground or a slope which is not in intimate contact with a frictional surface.  The wedge is assumed to have a vertical face (omega = 0) and the earth pressure rotation (delta) matches the slope of the ground surface.  From a retaining wall design perspective, this matches common assumptions for design of concrete cantilever "semi-gravity" walls, i.e. those with a heel extending back from the vertical wall element, and the rectangular or trapezoidal soil block behind the wall and above the heel is assumed to act as part of the gravity wall.

Wall Geometry

Height of Wall (Hw) - This is the height of the wall (vertically) as measured along the slip surface behind the wall, and so it also the height of the edge of the wedge.  For many walls, this will be the actual height of the exposed wall (minus stickup, plus embedment distance);  For concrete cantilever ("semi-gravity") walls, it is common to model with a vertical wedge surface extending upward from the buried heel of the wall, so that there is a rectangular soil block included above the concrete footing which is part of the wall mass (Note that the program can enforce this geometry with the "Rankine" setting).  If this is the case, the designer should consider the actual soil height of the slip surface, which could be higher than the wall if the upward slope starts immediately behind the wall. 

Slope Angle (beta) - Negative (downward) angle is permitted for the retained soil slope so that you can obtain all the possible solutions for the active case, however Mononobe-Okabe should work just fine for most downward slopes, this program does not add anything unless you wish to consider cohesion or surcharges. 

Maximum Slope Height (Hsmax) - This defines the maximum height and length of the uniform slope, beyond this point, the retained soil is flat.  Slope Length (Lsmax) is provided on the next line as a check so that what you can verify the slope and Hsmax result in the correct distance to the slope break.  This is most appropriate for embankments which have a maximum height, we hope to expand to provide a full variable slope solution at a later date. 

Wall Angle (omega) - Look at data entry diagram (note that negative slope is shown) or look at the graph of the problem geometry to make sure omega makes the wall lean the direction you intend.  Wall Height Above Heel (H) is a check value, which represents the vertical height of soil directly above the wall heel.  This value is the same as Hw for omega = 0.  We have included this value because the soil wedge mass is determined by two triangle defined the wall, the vertical line, and the ground surface in front of this line; and the triangle or trapezoid defined by this line, the slip surface, the vertical back-of-failure (including tension crack) behind this line.

Wedge Limit Length (Lwlimit) - This option allows you to define a horizontal distance beyond the back edge retaining wall which the failure wedge will not be terminate, due to a rigid vertical face at that distance.  This would potentially include a building wall or an uphill retaining wall provided that it does not add lateral load or vertical load to the lower wall.  This vertical surface is not a tension crack (apply tension crack and cohesion for that option). Instead there is an active pressure Pa (static, not earthquake) due to active earth pressure of the soil wedge pushing against this boundary.  For equilibrium, there is a reactive or reflected force from the rigid structure back to the soil wedge that must be considered as an external force on the wedge.  This value is not Kae because it the actual active pressure is acting opposite the direction of kh.  Technically, the active force would be due to -kh, but an earth pressure based on kh= 0 is conservative.  

Buildings above to a retaining wall are usually sufficiently-flexible structures for which causing settlement of the wall is a concern, unless the foundations are safely below the active wedge.  I had a project ten years ago with tiered retaining walls for which having the wedge limit length would have been useful, however that was before I started considering WASP.  For an uphill retaining wall which is not fully retained internally to that wall, you may need to (manually at this point) add a horizontal surcharge load for the downhill wall if the horizontal load exceeds the reflected Pa. 

 

 

 

Soil Parameters

Unit Weight (gamma) - WASP does not consider ground water level within the active wedge, which could cause the effective unit weight to change in the middle of the failure wedge.  For an undrained analyses (using undrained strength Su or total c and phi) the presence of water table would only affect the initial effective stress defining the soil strength.  Just as with Mohr-Coulomb and Mononobe-Okabe, separate values of Pa for total and submerged unit weight can be used after the coefficient is obtained.  As with conventional earth pressures, the active pressure below the water table can be reduced by multiplying by the submerged rather than total unit weight, but unless there is a way for water to drain rapidly from one side of the wall to the other, the hydrostatic force must be included.

Cohesion (c)- Unlike for static, drained long-term analyses, it is possible that cohesion is present for the short-term design case, particularly for granular soil with at least some fines.  Sources of cohesion may include dry cohesion, capillary rise above the groundwater table, or short term (undrained) cohesive strength.  A selected value of cohesion should generally be verified by the lower bound or reduction from average of several strength tests. 

A common "cheat" in earthquake slope stability, where a 2H:1V slopes does not have the required factor of safety under seismic conditions (FOS >= 1.1), is to add a small cohesion (50 to 100 psf, 2.5 to 5 kPa) to the near surface soil, which drives the critical failures slightly deeper and eliminates the shallow "infinite" failure surfaces less than 1 foot (0.3) m below the ground (the other cheat method common in slope stability programs is to specify a minimum depth of slip surface to eliminate the same circles).  In one project, we visualized what height we would expect a vertical cut to remain standing during construction for the materials we encountered.  A design cohesion was obtained by back-calculating from the estimated tension crack height, confirmed with the results of other testing.

Friction Angle (phi) -  Consult your geotechnical engineer.

Earth Pressure Rotation (delta)-  The earth pressure rotation depends not only on the interface friction angle or coefficient, reasonable limits on rotation, and also the relative amount of sliding between the soil and the wall. 

First, the interface friction angle is usually about 100% of the soil friction for rough (cast-in-place concrete), 80% for smooth concrete (formed or precast) and 60 to 70% for soil on steel.  However, more realistic values of earth pressure rotation for earth pressure analysis should be limited to 1/3 to 1/2 of the soil friction angle.  In particular, large positive rotation of delta with linear failure surface can over-estimate the earth pressure by as much as 30%, compared to the more accurate solution would be to use a log-spiral or circular/linear approximation.

For the Rankine case, the earth pressure is assumed to be parallel to the sloped ground surface.  I don't know if anyone has checked the validity of this assumption between earth pressures in Rankine versus the limitations above, but it is based on simple equilibrium theory.

Second, the orientation:  The earthquake case is slightly simpler than the static case.  For most retaining walls (static case), the backfill behind the wall is expected to settle downward relative to the wall, resulting in positive delta.  For the static case for certain compressible walls, such as gabion-faced walls, or if for some other reason (insufficient bearing capacity, but we won't design that hopefully) the wall could settle more than the backfill, resulting in an upward delta.  For the earthquake condition, one would expect that nearly all the time, the instantaneous deformation of the wall versus the soil wedge behind it would result in positive delta.  Always be on the lookout for exceptions, negative delta increases the active pressure and will make the wall (slightly) less stable.  Note that the horizontal component earth pressure coefficient (Kae)h changes more slowly than Kae with changing delta or omega.

Tension Crack Present - The occurrence of a tension crack, despite the decrease in the length of the failure surface for cohesion, generally increases Kae or Ka slightly compared to the case with cohesion but no tension crack.  Therefore for earthquake conditions, it is advisable to consider a tension crack.  The tension crack height is determined per NAVFAC (1986) as 2*c/gamma * tan(45+phi/2), and is provided in the next line for reference.  For cases where the tension crack height is greater than the wall height, Ka of zero is obtained. 

 Earthquake Parameters

Horizontal Acceleration Kh - This is the design acceleration coefficient for the wall, not the actual earthquake acceleration.  Most previous sources (published papers, text books) indicate using Kh that is half of the site peak ground acceleration, and looking at the displacement-based design method (below) indicates why.  If the horizontal coefficient is half of the site acceleration, the upper-bound displacement is generally less than 2 - 4 cm (1 - 1.5 inch).

AASHTO (LRFD Design Manual, 2006) specifies to use 0.5 of the peak acceleration for walls that are free to translate, which is assumed to indicate that the top of the wall is free to deflect at least 0.5 percent of the wall height.  For structures restrained against lateral movements, where there is doubt that the backfill can yield sufficiently to mobilize soil strengths, such as a wall or abutments with tiebacks or battered piles, AASHTO suggests the use of a factor of 1.5 times the peak ground acceleration.

Atik and Sitar (2010) indicate that the maximum bending moment and the total earth pressures on the wall correlate to Mononobe-Okabe (force equilibrium method) using 0.65 of the peak ground acceleration.  This appears to support the use of approximately half of the peak ground acceleration used above. As noted above, the difference between using 0.5 and 0.65 can be comfortably justified by the effects of displacement-based design

Vertical Acceleration Kv -  Previous papers have agreed that changing Kv will result in little change in the active earth pressure value, because the driving and resisting forces on the wedge are all largely proportional to the wedge weight.  If you are looking at a retained soil designed with significant cohesion (or surcharge loads, although most vertical surcharge loads would likely vary with Kv as well) one should evaluate whether this is still true.  This is because these resisting factors are not a function of the vertical mass of the wedge system, so do not respond proportionally. 

Displacement Based Design

Franklin and Chang (1977) developed a numerical estimation of seismic displacement originally for earth dams.  Their report basically further evaluated permanent ground displacements using Newmark's method (Newmark, 1965).  As reported by FHWA (1998), Richard and Elms (1979) extended the work of Franklin and Chang to gravity retaining walls.  From reading the article, it looks like Richards and Elms did not do any analyses to verify the validity of scaling Franklin and Chang's resuilts (presumably applicable to large dam embankment slopes based on their report's title) to typically much shorter retaining walls, or they just assumed the dam data by Franklin and Chang (1977) was applicable. My next task is to find a copy of Franklin and Chang, which appears to be far less accessible

The seismic displacement equations are used in this program as stated by FHWA (1998).  Its pretty simple, the equation basically curve fits a large cloud of data prepared by Duncan and Chang.  The program only uses the Richard and Elms equation for upper bound, and does not use the other equations which will estimate slightly lower values when very low N/A ratios occur.   

Both the WASP program and the Mononobe-Okabe spreadsheet include two of the displacement upper bound equations, as shown in solid lines.  The upper line takes over at N/A or Kh/PGA of approximately 0.35, i.e. where the design lateral force is 3 times less than the actual PGA for the site.

 

Wall Seismic Resistance Coefficient (N) - is the same as the transmittable block acceleration on the diagram above, in this program is the horizontal acceleration coefficient selected to determine Kae.  It is set on the Earthquake tab.

Peak Acceleration (A) - should be the local peak ground acceleration determined by building codes, charts, or other methods applicable to the project design.  In the U.S., this is typically the International Building Code (ICC, 2006) which includes maps of short period and 1-second period spectral accelerations for design, or these values can also be taken from the USGS web-based program.  As stated in the IBC, the peak ground acceleration (spectral value at 0 second period) can be determined as 40% of the short-period spectra times the soil class Multiplier (for local ground condtions), i.e. Sms= 0.40 * Ss *Fa.  Further reduction by 2/3 to use SDs (in our opinion) is not considered representative of the actual peak ground motion, and is an appropriate reduction for structural design only.  The use of a lower N already provides the reduction equivalent to that used by the structural engineer.

Peak Velocity Normalized by A (VoverA) - This is an approximation proposed by FHWA (1998), which allows the user to not have to worry about determining seismic peak velocity separately, and avoids the need to enter dimensional values.  FHWA (1998) does not state and we have not further researched the limitations of this approximation.  The stated value in mm/s is correct for either English or metric calculations. 

Peak Velocity (V) - This value is stated so that the user (one that is highly familiar with earthquake records, that is) to verify that the calculated peak velocity is reasonable. V computed using VoverA is stated in the design units (mm/s or in/s) 

Upper Bound Displacement and 50% Displacement (Dub and D50) - These are the upper bound equation by Richard and Elms, and an average trend through the middle of the shaded cloud (above) reported by Whitman (1990) as having been estimated by Wong and Whitman (1982).  If these levels of wall displacement are permissible by the project owner, it may be possible to decrease the design coefficient and decrease Kae for the wall design.

As far as can be determined, the displacement-based model makes no consideration of failure mode, i.e overstressing of wall reinforcement, sliding, or overturning.  Presumable, it would be preferable to have greater reserve capacity in the wall and overturning, and allow permissible displacement to occur in sliding.  This could be implemented by choices of seismic factor of safety for overturning versus sliding in the design codes, and may also occur due to either structural conservatism (reinforcing bar sizes or concrete widths rounded up to next common size) or material strengths in excess of design strengths for the wall strength (concrete strength increasing with age).

As noted in the various design methods, the displacement-based estimate is only appropriate for  soils which are not susceptible to liquefaction or other significant strength loss (Newmark suggests 85% degradation or less).   Greater displacement will occur for these cases, either because they fail outright, because the displacement methods does not consider deterioration of the seismic coefficient during the duration of ground shaking, or because the strength loss results in a much lower stiffness and the site fundamental period is closer to the acceleration spectra of the earthquake.

Surcharge Loading

Surcharge loads are provide in WASP primarily to be able to evaluate whether the presence of surcharge loads "draws" the active wedge flatter or restricts the wedge steeper than if the loads were not present.  We would suggest that final designs be done without surcharge, and add these in separately.

Note that the surface surcharge pressure can only be a negative (downward) value. 

The horizontal load is correct for a load to the right (such as a tieback or deadman pulling away from the wall face).  However, if the horizontal load is toward the wall, the horizontal pressure distributed in the soil will carry some or all of the load further towards the structure.  Therefore, the effect of surcharge will likely be felt well to the left of its point of application, whereas the wedge analysis only considers load at the point of application.  Use with care.

Results

For easy reference on the program screen, and also included in the printout version, results are shown in a line at the bottom of the page.  These results include:

Alpha - the failure wedge angle corresponding to the maximum Kae.  Note that in many cases, the distribution of Kae is gradual, such that minor changes is loading could probably shift a flatter or steeper.

Pae - The total earth force used to calculate Kae

Kae - The calculated Kae value, also shown on the graph.  This is 2 * Pae/H^2, it does not consider a tensile zone if there is a tension crack.  For earthquake loading, it would be more reasonable that the tension crack is at the back of the wedge, but the soil is bearing strongly on the upper portion of the retaining wall.  If you prefer the other way, compute Kae from Pae manually.

(Kae)h - I don't know how many times I have computed Kae to too much precision, then wrote the report with only the total force without accounting for the rotation angle, to a design audience that doesn't know the difference.  This is the horizontal component.

(EFP)h - the horizontal component of the equivalent fluid pressure (in English units, actually pounds per square foot per foot of wall height).  Not a big deal, but a slight time saver.

Other methods

NCHRP Report 611 (Transportation Research Board, 2008) shows that the same results as WASP can be obtained using standard slope stability programs (I have used SLIDE, it may be possible to use SLOPE/W).  We have used this method to back-check our program.  In our experience, this is critical if one wants to model the overall force for a complex slope, possibly with multiple soil layers and water table.  However, to come up with the same answer to get a "stand-alone" earth pressure coefficeint can be time consuming.  Specifically, the steps include:

1.  Model the geometry of your retaining wall and slope in the program and check it.  Apply the desired seismic coefficients.

2.  Find the control in the program which sets the failure slope to a single linear element, and the search method which fixes the exit of the failure surface at the toe of the wall, and allows for multiple searches for the failure surface entrance.

3.  Apply an arbitrary force to represent the wall supporting the active pressure, and run the slope stability program. Find the factor of safety.

4. Iterate (repeatedly) with different wall support force until the factor of safety is exactly 1.  This represents the equilibrium result for the active condition.

5. Back-calculate the active pressure coefficient from the active force (Kae = 2 * Pae/(gamma H^2) where Pa is the active earthquake force result from the slope stability runs, gamma is the unit weight, and H is the wall height.

This can easily take 1/4 to 1/2 hour per single result, in my experience.  Using WASP, entry of 20 parameters can take less than 5 minutes, and changes in wall height and geometry can be easily made with a single parameter change.

Position of active earthquake force

For static active pressures on a flexible retaining wall, the pressure distribution is a triangle with the maximum force at its base and the centroid of loading at 1/3 of the wall height.  

Older papers (Seed?, Whitman?) have suggested that the application of the seismic active force may trapezoidal or inverted triangular, with the centroid of loading either at mid-height or 2/3 of the wall height.  This sound good from a theoretical hand-waving approach (I have included this recommendation in many a soil report) but was apparently not supported by much data.  Atik and Sitar (2010), however, show this is not true - the active earthquake pressure is still a triangular distribution for which the centroid of loading would be at the lower 1/3 point. 

References

AASHTO, 2006, LRFD Bridge Design Manual.

Al Atik, L.  and N. Sitar, 2010, Seismic Earth Pressure s on Cantilever Retaining Structures, ASCE Journal of Geotechnical and Geoenvironmental Engineering, Vol. 136 No. 10.

Federal Highway Administration, 1998, Geotechnical Earthquake Engineering Reference Manual, NHI Training Course in Geotechnical and Foundation Engineering, NHI Course 13239- Module 9, Publication FHWA HI-99-012.

Federal Highway Administration, 2011, LRFD Seismic Analysis and Design of Transportation Geotechnical Features and Structural Foundations Reference Manual, FHWA-NHI-111-032.

Franklin A.G., and F.K. Chang, 1977, Earthquake Resistance of Earth and Rock-Fill Dams, Report 5: Permanent Displacement of Earth Embankments by Newmark Sliding Block Analysis, Misc. Paper 5-71-17, Soils and Pavements Laboratory, US Army Engineers Waterways Experiment Station, Vicksburg Mississippi.

International Codes Council, 2006, 2006 International Building Code

Transportation Reserch Board 2008, NCHRP Report 611

National Cooperative Highway Research Program (NCHRP) "NCHRP Project 12-70, Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes and Embankments, Recommended Specifications, Commentaries, and Example Problems,", NCHRP Report 611, Transportation Research Board, Washington DC. 

Newmark, N. M., 1965, Effects of Earthquakes on Dams and Embankments, Geotechnique, Vol 15, No. 2, pp 139 - 160.

Richards, R. Jr and D.G. Elms, 1979, Seismic Behavior of Gravity Walls, Journal of the Geotechnical Engineering Division, ASCE, Vol 105 No GT4, pp. 449 - 464. 

Transportation Research Board, Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankment, NCHRP Report 611, Washington D.C.

Whitman,R. V., 1990, Seismic Design of Gravity Retaining Walls, Proceedings, Design and Performance of Earth Retaining Structures, ASCE Geotechnical Special Publication No. 25, pp. 817 - 842.

 

 

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