Presented by: Civil Engineering Academy

Lateral Earth PressurePresented by: Civil Engineering Academy

Earth pressure is the force per unit area exerted by soil. The

ratio of horizontal to vertical stress (effective stresses) is called

coefficient of lateral earth pressure (K)

K=𝜎ℎ

𝜎𝑣=

𝜎′ℎ

𝜎′𝑣

Earth pressure is at-rest, active, or passive (CERM Ch. 37):

𝜎′ℎ(𝑎𝑐𝑡𝑖𝑣𝑒) or

𝜎′ℎ (𝑝𝑎𝑠𝑠𝑖𝑣𝑒)

𝜎′ℎ(𝑎𝑡−𝑟𝑒𝑠𝑡)

At-rest: Ko = 1-sinΦ

Active: Ka = tan²(45-Φ/2) or 1/ tan²(45+Φ/2)

Passive: Kp = tan²(45+Φ/2)

You have 2 ways of solving for K. Rankine or Coulomb

method. If you have no backfill and the wall is vertical then

use Rankine, it’s the most popular method.

Use the Coulomb equation to calculate K when given the batter

angle and the friction angle between the backfill soil and the

wall surface is given. Use CERM Eq. 37.5 for active earth pressure,

CERM Eq. 37.14 for passive earth pressure, and CERM Eq. 37.21

for at-rest soil pressure.

λ rake angle of retaining wall face

angle of internal friction

angle of external friction

slope of backfill

Equations to remember:

Rankine Active

Ka = tan²(45-Φ/2) or 1/ tan²(45+Φ/2)

𝑃𝑎=Ka𝜎′𝑣-2c(Ka)1/2 (If cohesion isn’t given then that term goes to 0)

Ra=1

2𝑃𝑎H =

1

2𝐾𝑎𝛾H²

Rankine Passive

Kp = tan²(45+Φ/2)𝑃𝑝=Kp𝜎′𝑣 +2c(Kp)1/2 (If cohesion isn’t given then that term goes to 0)

Rp=1

2𝑃𝑝H =

1

2𝐾𝑝 𝛾 H²

Rankine At-Rest

Ko = 1-sinΦ

𝑃𝑜=Ko𝑃𝑣 (If cohesion isn’t given then that term goes to 0)

Ro=1

2𝑃𝑜H =

1

2𝐾𝑜𝛾H²

A basement wall is shown. Plot the distribution of total active earth

pressure acting on the wall. Also, compute the active horizontal

thrust acting on the wall due to the soil.

3 ft

9 ft

C=0, Φ=40°, γ=125

pcf

C=0, Φ=34°, γ=130

pcf

5 ft

Solution:

1) Solve for Ka for each layer.

1) Top 3’: Ka=tan²(45-40/2)= 0.217

2) 3’ to 12’: Ka=tan²(45-34/2)=0.283

2) Solve for vertical stresses as you move from top to

bottom. Be sure to account for the water table.

3) Solve for active pressure acting (𝑝𝑎) on the wall for each layer.

𝑃𝑎=Ka𝜎′𝑣-2c(Ka)1/2

4) Solve for the areas of each distribution (resultant force) and sum them all

up to find the total active horizontal force acting on the wall.

= 0.5(B)(H) or area of triangle

= 0.5(B)(H) + (B)(H) or area of triangle + square

What if you have a surcharge at the top of the wall?

More later earth pressure problems! Yay!

Next Topic: Soil Consolidation