Mononobe Equation

[Geola]

Anyone knows how to deal the case with the negative magnitude inside the square root of Mononobe-Okabe Equation. Can I assume zero when it becomes negative? Thanks!

 

[dgillette]

No, do not assume zero. What you are seeing is where Mononobe-Okabe just blows up because the assumptions of the method are violated. We've run into this problem with very severe earthquakes on spillway walls that retain dam embankment.

M-O is just Coulomb active theory extended to have a pseudostatic horizontal load in addition to the gravity load. There is an angle (called theta) in the paper I have) that increases with increasing PHA. Inside the square root, there is a term sin(phi-beta-theta) that goes to zero, causing the term inside the square root to have a zero in the denominator, and kaboom, the equation blows up. I believe that causes the base angle of the slide mass, called alpha, to approach zero, which makes the mass of the slide approach infinity. With level backfill, that happens when kh=tan(phi), e.g., 0.577 if phi=30.

I have refs for a couple of papers that purport to improve on that. Can't vouch for them, however. I won't type the whole list out unless you really want them and have access to "Geotechnique" and "Soils and Foundations" and "Geotechnical and Geological Engineering."

 

[Geola]

Actually the problem is not the denominator being zero, but the term sin(phi-beta-theta) becomes negative when phi<beta+theta which leads to a negative value inside the square root. I have a case with friction angle of phi=35,a backfill slope of 2/1(H/V), then kh cannot be larger than 0.1g, otherwise no solution. Looks like M-O equation is too limited.

 

[dgillette]

Sorry about that. I should have said that I was talking about the equation for [alpha-beta], which does have it in the denominator inside the square root but, regardless, you are correct that M-O doesn't work there. I don't know whether/how well these "improved" versions work.

The inclination of the critical base plane (alpha) starts at 45+phi/2 at zero acceleration, then flattens out as PHA increases. In the equation for (alpha-beta), cot(alpha-beta) becomes infinite as (phi-beta-theta) approaches zero, which means alpha=beta and the base plane of the critical surface becomes parallel with the surface of the slope, making the slide mass infinite. I'm pretty sure going past there hoses up the geometry (slide going over the top of the wall as well as the load on the wall) so it violates the assumptions of the M-O equation.

Regards,
DRG