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# Lateral earth coefficient for phi<beta3

## Lateral earth coefficient for phi<beta

(OP)
Hi all,

I'm designing a gabbion wall for retaining a slope of 25° while the soil friction angle is only 20°. I need to calculate the lateral earth pressure coefficients and I have always adopted the Coulomb's formula, like the following for the active state https://wikimedia.org/api/rest_v1/media/math/rende.... This formula is not defined when the friction angle (phi) is lower than the slope angle. Why is this case not considered by the theory? Which value may I adopt for designing the wall?

thanks

### RE: Lateral earth coefficient for phi<beta

Make a search for "Culmann" in this forum. You can use the Culmann graphical method instead the Coulomb method when the math does not work (you are getting the sqrt of negative number when phi<beta). Bowles book has also a good example for the Culmann method.

### RE: Lateral earth coefficient for phi<beta

2
Let me see if I understand this correctly: I take it that the soil you are retaining has a friction angle of 20deg. The slope angle of the material "above your retaining wall" will be 25deg. You have not mentioned what material this is, i.e., cohesive or cohesionless.

Let's look at Cohesionless. Forget the gabion wall; do you think your slope of 25deg will stand up with a material friction value of 20deg? FS = 1 if friction angle = slope angle (dry material). If friction angle is less, it gives a FS < 1 - will not happen. So while the soil retained by the retaining wall (to the height of the wall) may/will be stable, the slope above will not. I think you really need to go back to the value of 20deg for friction angle and see if it is realistic.

Let's look at cohesive: Typically the undrained strength is use in the short term. Again, for long term, the friction angle would be operative (similar c,phi) analysis and the slope can be considered as per cohesionless. The slope of 25deg might stand in the short term (if the soil is overconsolidated) but will not in the long term. Again, this is a case of the retained soil above the retaining wall.

So, in my view, if I understand your issues, you really have a problem with the soil sloping up from the top of the retaining wall (that which is unsupported).

### RE: Lateral earth coefficient for phi<beta

(OP)
Hi All,

@Okiru: I'll check out Culmann's method
@BigH: The soil is cohesive, c'=43kN/m^3. The slope is rather short (less than 30m), so from the theory, the slope angle might be higher than the fricition angle in cohesive soil, right?

### RE: Lateral earth coefficient for phi<beta

You may need a taller wall (and probably not a gabion wall) so that you can reduce the slope behind the wall. And, as BigH said, Check if your soil properties are realistic. Is the ground surface in front of the wall flat or sloped? If the phi angles of the retained and/or foundation soils are so low, you should also check global stability.

### RE: Lateral earth coefficient for phi<beta

pietro82,

It doesn't matter how high or short the slope is. If the slope angle is steeper than the phi angle, it will eventually fail.

As PEinc pointed out, you need a taller wall and you need to check global stability.

Good luck.

Mike Lambert

### RE: Lateral earth coefficient for phi<beta

cohesive soil vulnerable to wetting/drying and freezing/thawing will loose cohesion over time. Such temporal changes will also affect the friction angle, rendering peak strength obsolete and reducing the strength to fully-softened. These are the risks, depending on the design life.

Otherwise, good luck.

f-d

ípapß gordo ainÆt no madre flaca!

### RE: Lateral earth coefficient for phi<beta

GeoPaveTraffic - "It doesn't matter how high or short the slope is. If the slope angle is steeper than the phi angle, it will eventually fail."

I trust you mean "It" = the slope above the wall in the case of a short slope. The wall will not necessarily fail, just the slope above.

The trial wedge procedure can determine a thrust from a short broken back slope with phi < backslope but if the slope was checked by itself, it would not solve.

### RE: Lateral earth coefficient for phi<beta

Doctormo, could you explain "...but if the slope was checked by itself, it would not solve. "? Sorry, I did not quite understand that part...thanks.

### RE: Lateral earth coefficient for phi<beta

Maybe my description of it "not solving" is not exactly correct. If you ran a slope stability analysis or tried to create a trial edge for the steep slope by itself, they would provide factors of safety less 1.0. The trial wedge would come up with the slope being the most unstable condition.

The factor of safety of a simple slope is defined as Tan(phi)/Tan(slope) as I recall from my Terzaghi book. Once the FS < 1.0, there is no equilibrium (driving forces exceed resisting forces). At this point, there is nothing left to do or solve for. Clear as mud?

### RE: Lateral earth coefficient for phi<beta

Doctormo,

You are correct, I was talking about the slope above the wall failing. But if that slope fails, then the wall is not performing a meaningful function.

With respect to the original post, the wall needs to be higher so the slope can be flatter. Beta cannot be greater than phi for the retained material or you will eventually have a failure of the slope above the wall resulting in a failed condition for the wall slope combination.

Mike Lambert

### RE: Lateral earth coefficient for phi<beta

Mike - I agree with raising the wall height to decrease the back slope which would be my first recommendation also. I would also want to know why anyone is working with phi = 20 deg soil for anything structural without remediating or disposing of it.

One could also put some lightweight reinforcement in the short slope above the wall to increase stability and design the wall accordingly since a short slope would not long enough to affect the overall stability of the wall structure that could not be accounted for even with the phi angle problem.

### RE: Lateral earth coefficient for phi<beta

(OP)
Hi all,

@Doctormo: that is correct for a cohesionless soil. In that case, Therzaghi says: FS=c′/γzcosβsenβ+tanϕ′/tanβ. So in case c'>0 and the slope length is limited, FS>1 even with ϕ′>β. Don't you use this formula for the slope stability?

### RE: Lateral earth coefficient for phi<beta

pietro82 - Yes, the discussion is about cohesionless soil. Keep in mind the discussion relates to earth pressure calculations for slopes steeper than the phi angle. I was relating the issue of trial wedge analysis to slope stability analysis and how neither will solve for the long or infinite slope condition when phi < slope.

Adding cohesion will still not allow a typical earth pressure equation to solve for phi < slope but will obviously benefit a trial wedge or slope stability analysis which does not rely on earth pressure equations. My point was that a short steep slope behind a retaining wall can be solved for a thrust when phi < slope using trial wedge methods. The slope itself may not be stable but that is different than calculating a lateral load on a wall. I have seen a few concrete walls with slope failures above the wall where the wall is fine and the slope is not.

Cohesion is a subject by itself as it is generally a time dependent strength condition so what works today may not work in the future. There are many discussions about the use of cohesion in earth pressure calculations.

### RE: Lateral earth coefficient for phi<beta

(OP)
Speaking about cohesion, how do engineers usually deal tensile cracks which occur in the active side according to Rankine's theory?
Thanks

### RE: Lateral earth coefficient for phi<beta

Refer to Foundation Analysis and Design by Bowles for earth pressure analysis with cohesion.

Most engineers ignore cohesion so they do not have to deal with tension cracks. Bowles discusses that also.

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