# Geo Struct Sparks LLC

## When Mononobe-Okabe Equation Doesn't Work

### The Original (Mononabe-Okabe Equations)

This site gets enough visitors searching Mononobe-Okabe that we might as well have a resource for what they are looking for.  The following excerpt is from the AASHTO (1998) Standard Specifications for Highway Bridges, 16th Edition (1998).  Both the active and passive equations are provided below.

As graphed on our home page, the Mononobe-Okabe active pressure equation will result in imaginary numbers when the term inside the "psi" square root is negative, which will only occur for the sine term (phi(friction angle) - omega(earthquake excitation ratio) - i (slope inclination)) is less than zero. An interpretation is that as the seismic forces get stronger for steeper uphill slopes, the critical slip plane becomes rotated shallower than the ground slope, and the wedge becomes infinitely long.  The only way to overcome this limitation is to use a non-infinite slope, as we do in the WASP program.

### Nay-Sayers

Validity of Mechanics

There is plenty of evidence that Mononobe-Okabe and wedge equilibrium methods (including our program WASP) are at best an overly-simplistic engineering analog for dynamic seismic behavior.  Whitman (1990) was among the first to present the results of shake table or centrifuge studies which showed that the maximum seismic earth pressure does not occur while the soil behind the retaining wall is moving towards the wall.  Instead, the maximum force on the wall and the maximum bending moment occur as the wall and ground is accelerating towards the backfill (uphill side). This is further reinforced by a more recent paper by Atik and Sitar (2010) where they did the same experiment in a centrifuge.

This is merely be a question of relative acceleration.  The conventional view is incorrect that the active earthquake pressure must occur while the ground is moving toward the downhill side of the wall (to the right on Figure CA10 above).    From a Newtonian perspective, just because the base is moving and the wedge is trying to remain stationary, there is still a relative acceleration between the two masses that will mobilize the wedge and the forces.  The wedge behind the wall NEVER has its own acceleration of say 0.4 g towards the right (at least until outright failure), it is being accelerated by the ground behind it.  Rather, the wedge may have RELATIVE acceleration to the right when the ground is accelerating in the opposite direction.  This is definitely true of permanent displacements; careful review of Newmark sliding block theory indicates that relative displacements occur NOT when the earthquake accelerations are moving down-slope, but when the base acceleration is moving uphill while the soil wedge is only slowing down or not moving as fast uphill, e.g. relative displacements occur similar to pulling a table cloth out from under dishes.

I agree that if wedge sliding is in the opposite direction, it will be relatively more complex, because the stiffness and mass of the particular wall will tend to influence the result in any case.   This stiffness and strength is not modeled in a simple equilibrium model in any case.  Therefore we maintain the wedge equilibrium methods are approximately correct, as they were before the most recent paper came out.

Slope Failure

I have heard arguments that the Mononobe-Okabe fails because the slope above is failing, and you therefore shouldn't design a wall if the slope is failing.  This is not credible, because all earthquake response at high accelerations will involve failure, however only for the 1/10 or 1/100 of a second during the maximum peak ground acceleration spikes, which does not result in significant displacement.  Note that such slope failures in granular soils are typically  shallow surface displacements (similar to infinite slope stability) which do not provide significant forces to the wall.  It is far more acceptable for some gravel or dust in the slope above a wall to displace a few inches than for a retaining wall (or other engineered structure) to displace or rotate the same amount.  Mononobe-Okabe also fails because of the assumption of an infinitely-long slope failure whereas the slope above and the forces on the wall will definitely not be infinite.

Magnitude of Results

Atik and Sitar (2010) found that there was no incremental addition to the active pressure in their centrifuge tests below about 0.4g acceleration (for Mononobe-Okabe and WASP, presumably kh = 0.2 g).  They also tried predicting the correct earthquake active earth pressure using numerical methods, and found the results not accurate.  Therefore they did not recommend using numerical methods to predict active seismic earth pressures unless it can be backed up by physical modeling (e.g. in their paper, the centrifuge studies).

Atik and Sitar (2010) also found the bending moment in their model walls corresponded to a wedge-equilibrium (e.g. Mononobe-Okabe or WASP) force that was computed based on 0.65 of the peak ground acceleration.  This appears to be similar to the 0.5 x PGA that is conventionally used.

It should be noted that the Atik and Sitar model was a U-shaped structure, i.e. although the top of wall was free to rotate, the base of the wall was restrained against  displacement by the slab between the two retaining walls on either side of their model.

NCHRP Report 611 (Transportation Research Board 1998) recommends that the seismic condition generally does not control wall design below ground acceleration level of 0.15 g or less, and therefore determination of the seismic active pressure is probably not necessary.  At the same time, pseudo-static force equilibrium, e.g. the Mononobe-Okabe equations, are still required in many design codes.  Therefore, given its predominance in the design codes, determination of the active pressure by these methods including the WASP program is still necessary, at least for the time being.

References are at the end of the "Parameter Guide" page.