Retaining Wall – Lateral Earth Pressure

Retaining Wall – Lateral Earth Pressure

Update: For spreadsheets and more examples of calculating active and passive pressures see Lateral Earth Pressure II

We will briefly discuss lateral earth pressure caused by soil weight and ground water effects. I’m not going to go through all the derivations just the results and how they are typically used in practice. More of the ‘I don’t want to hear about the labor just show me the baby’ technique.

See this post for a broader overview of earth retention design:

General Earth Retention Design

Rankine and Coulomb Methods

The most common theories for determining lateral pressure due to soil are Rankine and Coulomb methods. Both methods use an idealized failure plane where the soil ‘shears’ itself and causes the soil mass to move toward the wall. The Rankine method assumes that the soil is cohesionless, the wall is frictionless, the soil-wall interface is vertical, the failure surface on which the soil moves is planar, and the resultant force is angled parallel to the backfill surface. The Coulomb method accounts for friction between the wall and the soil and also a a non-vertical soil-wall interface (battered wall). Earth pressures may also be found in geotechnical reports as Equivalent Fluid (or Lateral) Pressures (EFP or ELP). Which are given in units of lbs per sq ft. per ft of depth or pcf. All this represents is a lateral earth coefficient already multiplied by the soil density. So if you find your active pressure coefficient using one of the formulas below say Ka=0.33 and multiply this by the soil density say 120 pcf you get about 40pcf. Because earth loads are applied as uniformly increasing loads (triangular distribution against the back of wall). The equivalent lateral pressure is 40psf / ft of depth.

Basic Geometry Sketches

Lateral Earth Pressure – Soil Basic 1

Lateral Earth Pressure – Soil Basic 1 Geometry Sketch

Rankine equations for Active and Passive pressure (more on that below):

φ = (phi, html format looks slightly different than image) effective friction angle of the soil

β = Angle of backslope from the horizontal

 K_a = \cos\beta \frac{\cos \beta - \left(\cos ^2 \beta - \cos ^2 \phi \right)^{1/2}}{\cos \beta + \left(\cos ^2 \beta - \cos ^2 \phi \right)^{1/2}}
 K_p = \cos\beta \frac{\cos \beta + \left(\cos ^2 \beta - \cos ^2 \phi \right)^{1/2}}{\cos \beta - \left(\cos ^2 \beta - \cos ^2 \phi \right)^{1/2}}

For the case where β is 0, the above equations simplify to

 K_a = \tan ^2 \left( 45 - \frac{\phi}{2} \right) \
 K_p = \tan ^2 \left( 45 + \frac{\phi}{2} \right) \

 Coulomb equations:

φ = (phi, html format looks slightly different than image) effective friction angle of the soil

β = Angle of backslope from the horizontal

δ = effective friction angle between the two planes being evaluated. Usually between wall and soil with typical values being 2/3*φ or between two soil surfaces (i.e. for segmental retaining walls – reinforced zone soil and retained soil)

θ  = batter or angle of wall from the horizontal (you may see some coulomb eqns which use values from the horizontal so don’t be confused)

 K_a = \frac{ \cos ^2 \left( \phi - \theta \right)}{\cos ^2 \theta \cos \left( \delta + \theta \right) \left( 1 + \sqrt{ \frac{ \sin \left( \delta + \phi \right) \sin \left( \phi - \beta \right)}{\cos \left( \delta + \theta \right) \cos \left( \beta - \theta \right)}} \ \right) ^2}
 K_p = \frac{ \cos ^2 \left( \phi + \theta \right)}{\cos ^2 \theta \cos \left( \delta - \theta \right) \left( 1 - \sqrt{ \frac{ \sin \left( \delta + \phi \right) \sin \left( \phi + \beta \right)}{\cos \left( \delta - \theta \right) \cos \left( \beta - \theta \right)}} \ \right) ^2}
To account for wall batter the hoizonatal and vertical component of the active pressure are:
Kah=cos(δ+θ)
Kav=sin(δ+θ)

The Active state referes to pressures where the soil is sliding toward the wall or the wall is giving. The Passive state refers to soil pressures where the soil is being compressed such as soil at the low side of a sheet pile wall. Passive pressures will be higher than active as you can imagine that the soil will ‘push back’ when it is being pushed. The soil may also be ‘at-rest’. You may wish to use at rest pressures when designing concrete basement walls which do not allow much movement or other type retaining walls where minimal movement is wanted.

At rest pressure coeffcient:

K0= 1 − sin (φ)

References

Reference for wall movement under to ‘engage’ active pressure:

In Winterkorn and Fang, “Foundation Engineering Handbook” Table 12.1
Sand:
Active Pressure: Parallel to Wall .001H
Active Pressure: Rotation About Base .001H
Passive Pressure: Parallel to Wall .05H
Passive Pressure: Rotation About Base >.1H
Clay:
Active Pressure: Parallel to Wall .004H
Active Pressure: Rotation About Base .001H
Passive (No values given) however NAVDAC DM2.2 states that the required strain or wall movement required to mobilize the passive soil is about 2x the movement required for active pressure.
A great reference, but I’m sure it’s been out of print for a while.  My copy is dated 1975.

 

Water Pressure

Water pressure can be greatly reduced by providing drainage aggregate and drain pipe directly behind the wall. The density of water is much less than most soils (64 pcf) however its lateral pressure coeffcient is = 1.0 so the Equivalent Fluid or Lateral pressure is 64.5 psf/ft which is higher than most soils in an active pressure case. Therefore water pressure can have a serious impact on the lateral load applied to wall and should be given proper attention if the water is not given a relief source as mentioned above.

 

Thoughts? Comments? Questions?