Rankine Earth Pressures

[MacGruber22]

Does anyone have a reference for reducing active/at-rest pressures for a retaining wall for a soil that slopes downward behind the wall? My texts only have equations for when the slope on the retaining side goes upward.

Thanks!

 

[BigH]

Use a graphical method . . .

 

[civeng80]

Would a negative angle work in the equations?

 

[MacGruber22]

Well, a negative angle of, say -30, is equal to 330, and the Rankine (as well as Coulomb) equations are only good for that angle less than the soil's internal friction angle. Since the cosines of -30 and 330 are the same, it doesn't work. I think BigH's idea may be the right direction.

 

[PEinc]

Coulomb earth pressure coefficient equations allow sloping behind the back of a wall and in front of the bottom of the wall.
Many geotech books also have graphs or tables for these coefficients with respect to slope angles.
As mentioned above, the angles for the equations need to be less than the phi angle of the soil.
You should check out various geotech books or call your geotech consultant.

http://www.PeirceEngineering.com

 

[MacGruber22]

PEinc - you are telling me that Coulomb's method does allow for a negative beta angle? If, so I need to check out other texts as you suggest.

 

[ukbridge]

Check out Eurocode 7 Part 1 - Annex C, which should give you what you need.

 

[BigH]

Bowles 5th Edition - page 597 (Chapter 11 on Lateral Earth Pressures) - does -10deg and -5 deg.

 

[MacGruber22]

ukbridge - I didn't see any negative beta angles in that Eurocode appendix.

BigH - Thanks a lot - I think I may have a pdf of Bowles somewhere. I can at least see how the negative beta was derived so I can rationalize a reduction for more than 10 deg (mine is about 25).

 

[MacGruber22]

One should not use the Rankine method for Kp when beta > 0, since an inspection of Table 11-4 shows that it decreases with increasing beta. This is clearly not correct-Ka does properly increase. Note also that one can use a (-) beta in the Rankine equations, but the computed coefficients are those of (+) beta. The Coulomb equations are valid for both (+) and (-) beta. That is, Kp increases with increasing beta and decreases with (-) beta values.

 

[PEinc]

An interesting discussion on earth pressure theory.

http://nptel.ac.in/courses/105101083/download/lec7.pdf

http://www.PeirceEngineering.com

 

[MacGruber22]

Rankine's theory is not intended to be used for determining earth pressures directly against a wall (friction angle does not appear in equations above). The theory is intended to be used
for determining earth pressures on a vertical plane within a mass of soil.

This statement competes with the Coduto and Bowles texts I have. Both texts use Rankine as one of many methods for determining horizontal pressures on retaining walls.

 

[dgillette]

MacGruber - Does that last quote refer to the friction angle between fill and wall, rather than internal friction within the soil? The original Rankine model assumed (in effect) that there is no friction on the wall, so the major and minor effective stresses are vertical and horizontal. That's not exactly how WJM Rankine set up his formulation (as I recall), but the effect is the same.

(You can derive Rankine yourself very easily, beginning with a Mohr circle touching the MC envelope, as if the soil is in a state of failure. Assume sigma-1 is vertical and sigma-3 is horizontal, and solve for sigma-3 as a fn. of sigma-1 and phi'. There is of course no shear stress on the planes of principal stress. Reverse -1 and -3 for passive pressure. If you have more patience than I do, you can derive Coulomb for any beta, negative or positive, by a sliding block.)

 

[BigH]

After all this - I still think a Culmann graphical method, easy enough to do, would put all this to bed . . .

 

[MacGruber22]

dgillette,

I am not sure. Good question - you are probably correct.

BigH - no worries, as that is what I have decided to go with.

Thanks