1 lateral earth pressure we can calculate σ v ’ now, calculate σ h ’ which is the horizontal...
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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
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