1

Lateral Earth Pressure

• We can calculate σv’

• Now, calculate σh’ which is the horizontal stress

• σh‘/ σv‘ = K• Therefore, σh‘ = Kσv‘ (σV‘ is what?)

σv’

σh’H

2

Lateral Earth Pressure

• There are 3 states of lateral earth pressureKo = At Rest

Ka = Active Earth Pressure (wall moves away from soil)

Kp = Passive Earth Pressure (wall moves into soil)

Passive is more like a resistance

σv

σh

z

H

3

At Rest Earth PressureAt rest earth pressure occur when there is no wall rotation such as in a braced wall (basement wall for example)

Ko can be calculated as follows:

Ko = 1 – sin φ for coarse grained

soilsKo = .44 + .42 [PI / 100] for NC soils

Ko (oc) = Ko (NC) (OCR)1/2 for OC soils

σv

σh

z

H

4

Active Earth PressureActive earth pressure occurs when the wall tilts away from the soil

(a typical free standing retaining wall)

5

Active Earth PressureActive earth pressure occurs when the wall tilts away from the soil

(a typical free standing retaining wall)

6

Active Earth PressureActive earth pressure occurs when the wall tilts away from the soil

(a typical free standing retaining wall)

7

Active Earth PressureActive earth pressure occurs when the wall tilts away from the soil (a typical free standing retaining wall)

Ka can be calculated as follows:

Ka = tan2 (45 – φ/2)

thus: σa‘ = Ka σv’ – 2 c (Ka)1/2

45 + φ/2

Failure wedge

H

Soil sliding down pushing the wall

8

Passive Earth PressurePassive earth pressure occurs when the wall is pushed into the soil (typically a seismic load pushing the wall into the soil or a foundation pushing into the soil)

Kp can be calculated as follows:

Kp = tan2 (45 + φ/2)

thus: σp‘ = Kp σv’ + 2 c (Kp)1/2

45 - φ/2

Failure wedgeH

Soil being pushed up the slope

9

Active Stress Distribution (c = 0)

γc = 0Φdry soil

H

σa‘ = Ka σv’ – 2 c (Ka)1/2

σa‘ = Ka σv’0

σa‘ is the stress distribution Pa is the force on the wall (per foot of wall)How is Pa found?

Pa = ?

? - What is this value

10

Passive Stress Distribution (c = 0)

γc = 0Φdry soil

H

0

σp‘ is the stress distribution Pp is the force on the wall (per foot of wall)How is Pp found?

Pp = ?

? - What is this value

σp‘ = Kp σv’ – 2 c (Kp)1/2

σp‘ = Kp σv’

11

Stress Distribution - Water Table (c = 0)

H1

Ka γ H1

or

Pa = Σ areas = ½ Ka γH12 + Ka γH1H2 + ½ Ka γ’H2

2 + 1/2γwH22

H2

Ka γ H1 Ka γ’ H2

Ka (γ H1 + γ’ H 2)

Effective Stress Pore Water Pressure

Pa

γw H2

12

Stress Distribution With Water Table

H1

Ka γ H1

or

H2

0

Ka γ H1 Ka γ’ H2

Ka (γ H1 + γ’ H 2)

Effective Stress Pore Water Pressure

Pa

γw H2

Why is the water pressure considered separately? (K)

13

Active Stress Distribution (c ≠ 0)

-

γc ≠ 0Φdry soil

H

Ka γH

zo

2 c (Ka)1/2 Ka γH – 2 c (Ka)1/2

=

Find zo:

Ka γzo – 2 c (Ka)1/2 = 0

Zo = 2c / γ (Ka)1/2

Pa = ?

_

14

Passive Stress Distribution (c ≠ 0)

-

γc ≠ 0Φdry soil

H

Kp γH 2 c (Kp)1/2 Kp γH + 2 c (Kp)1/2

=

Pp = ?

+

15

Practice Problems

Chapter 12 – Problems:13

15 (part c is using equation 12.47 and part d uses equation 12.49)

25

13 and 25 are the same problem just one is active calcs and the other is passive calcs