LATERAL EARTH PRESSURE
A. Normally Consolidated Soil
𝐾𝑜 = 1 -sin ∅
𝐾𝑜 = coefficient of at rest earth pressure
(Normally consolidated soil)
∅= angle of international friction
B. Over Consolidated Soil
𝐾𝑜 = (1 - sin ∅)√𝑂𝐶𝑅
𝐾𝑜 = coefficient of at rest earth pressure
(over consolidated soil)
OCR = present over consolidated ratio
C. For Normally consolidated Clays, whose plasticity index is known
(Brook and Ireland’s Formula)
₁. For plasticity between 0 and 40
𝐾𝑜 = 0.4 + 0.007 (P.I.)
₂. For plasticity between (40 and 80)
𝐾𝑜 = 0.64 + 0.001 (P.I.)
At Rest Pressure on the Wall
1. Without Surcharge
𝜎 = 𝛾𝑠 h 𝑘𝑜
𝛾𝑠 = unit wt. of soil
𝛾 ℎ2 𝐾𝑜
F= 𝑠 2
2. With Surcharge
q = surcharge (KPa)
𝜎 = 𝐾𝑜 (q + 𝛾𝑠 h)
F = q 𝐾𝑜 h
𝛾
2 𝐾𝑜
𝐹2 = 𝑠 ℎ2
3. With Surcharge and Water Table
𝛾1 =𝛾𝑠𝑎𝑡 - 𝛾𝑤
𝜎 = 𝑘𝑜 [ q + 𝑦𝑠 ℎ1 + 𝛾ℎ2 ] +𝛾𝑤 ℎ2
𝛾1 = 𝛾𝑠𝑎𝑡 - 𝛾𝑤
(𝛾𝑠𝑎𝑡 - 𝛾𝑤 ) ℎ2 = 𝛾ℎ2
Rankine Active Earth Pressure with a Horizontal Backfill
Rankine Active Pressure coefficient
1. Non cohesive Soil
1 − sin ∅
𝐾𝑎 = 1+ sin ∅
∅
𝐾𝑎 = 𝑡𝑎𝑛2 ( 45 - 2 )
ACTIVE PRESSURE
𝜎 =𝛾𝑠 h 𝐾𝑎
ACTIVE FORCE
𝛾
2
F = 𝑠2ℎ 𝐾𝑎
2. Cohesive Soil
A. Before tensile crack occurs
ACTIVE PRESSURE
𝜎 = 𝛾𝑠 h 𝐾𝑎 -2𝑐 ℎ √𝐾𝑎
ACTIVE FORCE BEFORE TENSILE CRACK OCCURS
1
F = 2 𝛾𝑠 ℎ2 𝐾𝑎 - 2 c h√𝐾𝑎
B. After tensile crack occurs
DEPTH OF CRACK
Z=𝛾
2𝑐
𝑠 √𝐾𝑎
ACTIVE FORCE AFTER TENSILE CRACK OCCURS
F = (𝛾𝑠 h 𝐾𝑎 – 2c √𝐾𝑎 )
(ℎ−𝑧)(1)
2
Rankine Passive Earth Pressure with horizontal Backfill
Rankine passive earth pressure coefficient
𝜎
𝐾𝑝 = 𝑡𝑎𝑛2 (45 + 2 )
1+ sin ∅
𝐾𝑝 = 1− sin ∅
Passive force per unit length of wall
1
F = 2 𝛾𝑠 ℎ2 𝐾𝑝 + 2𝑐 ℎ √𝐾𝑝
Rankine active Pressure for Inclined Backfill
A. For Cohesionless Soil
Rankine Active Earth Pressure Coefficient
𝐾𝑎 = Cos 𝛼
𝐶𝑜𝑠 𝛼 . √𝐶𝑜𝑠 2 𝛼 . 𝐶𝑜𝑠 2 ∅
𝐶𝑜𝑠 𝛼 + √𝐶𝑜𝑠 2 𝛼 . 𝐶𝑜𝑠 2 ∅
𝛼 = angle that the backfill makes with horizontal
∅ = angle of friction of soil
Rankine active force for Inclined Backfill
1
𝑃𝑎 = 2 𝛾 ℎ2 𝐾 𝑝
B. For Cohesive Soil
1. Z =Depth of tensile crack
2𝑐
1 + sin ∅
Z = 𝛾 √ 1 . sin ∅
c = cohesion of backfill in KPa
𝛾 = unit wt. of cohesive backfill in kN/𝑚3
∅ = angle of friction of cohesive backfill
2. 𝜎𝑎 = active pressure at depth H
𝜎𝑎 = 𝛾𝐻 𝐾𝑎 𝐶𝑜𝑠𝛼
3. Resultant active Force after crack occur
1
𝑃𝑎 = 2 𝛾 𝐻 𝐾𝑎1 𝐶𝑜𝑠 𝑎 h
h=H–Z
𝐾𝑎1 = active pressure coefficient for cohesive backfill
𝑃𝑎 = active force after crack occurs
Rankine Passive Earth Pressure for Inclined Backfill
A. For Cohesionless Soil
Rankine Passive Earth Pressure Coefficient:
𝐾𝑝 = 𝐶𝑜𝑠 𝛼
𝐶𝑜𝑠 𝛼+√𝐶𝑜𝑠2 𝛼 .𝐶𝑜𝑠2 ∅
𝐶𝑜𝑠 𝛼 .√𝐶𝑜𝑠2 𝛼 .𝐶𝑜𝑠2 ∅
Rankine Passive Force on Wall with Inclined Backfill
𝑃𝑝 =
1
𝛾 ℎ2 𝐾 1 𝑝
2
B. For Cohesive Soil
𝐾 1 𝑝 = passive earth pressure coefficient
𝛼 = angle the backfill makes with the horizontal
∅ = angle of friction of soil
c =cohesion of soil
1
𝑃𝑝 = 2 𝛾 ℎ2 𝐾 1 𝑝
𝜎𝑎 = 𝛾𝑧 𝐾 1 𝑝 𝐶𝑜𝑠 𝛼
Coulomb’s Active Force on a Horizontal Backfill
𝛼 = 0°
𝛽 = 90
𝛾
2
𝑃𝑎 = 𝑠 ℎ 𝐾𝑎
2
𝐾𝑝 =
𝑠𝑖𝑛2 (𝛽+ ∅)
2
𝑆𝑖𝑛 (∅+𝛿) 𝑆𝑖𝑛(∅−𝛼)
𝑠𝑖𝑛2 𝛽 sin( 𝛽− 𝛿)[1+√
]
𝑆𝑖𝑛(0𝛽−𝛿) 𝑆𝑖𝑛(𝛼+𝛽
𝛿 = wall friction angle
∅ = angle of Internal friction of soil
Coulomb’s Passive Force on a Horizontal backfill
𝛼 = 0°
𝛽 = 90
𝛾
2
𝑃𝑝 = 𝑠 ℎ2 𝐾𝑝
𝐾𝑝 =
𝑆𝑖𝑛2 (𝛽− ∅)
2
𝑆𝑖𝑛(∅+𝛿)𝑆𝑖𝑛(∅+𝛼)
𝑆𝑖𝑛2 𝛽 𝑆𝑖𝑛(𝛽+𝛿)[1√𝑆𝑖𝑛 (𝛽+𝛿)𝑆𝑖𝑛(𝛼+𝛽)]
Coulomb’s Active Earth Pressure for Inclined Backfill
Coulomb’s Active Earth Pressure coefficient
𝐾𝑝 =
𝑆𝑖𝑛2 (𝛽+ ∅)
2
𝑆𝑖𝑛(∅+𝛿)𝑆𝑖𝑛(∅∙𝛼)
]
𝑆𝑖𝑛 (𝛽∙𝛿)𝑆𝑖𝑛(𝛼+𝛽)
𝑆𝑖𝑛2 𝛽 𝑆𝑖𝑛(𝛽∙ 𝛿)[1+√
𝛽 = angle the back face is inclined with the horizontal
𝛿 =wall friction angle
∅ = angle of internal friction
Coulomb’s Active force:
1
𝑃𝑎 = 2 𝛾𝑠 ℎ2 𝑘𝑎
Coulomb’s Passive Earth Pressure for inclined Backfill
Coulomb’s Passive Pressure Coefficient
𝐾𝑝 =
𝑆𝑖𝑛2 (𝛽+ ∅)
Coulomb’s Passive Force:
1
𝑃𝑎 = 2 𝛾𝑠 ℎ2 𝑘𝑝
2
𝑆𝑖𝑛(∅+𝛿)𝑆𝑖𝑛(∅∙𝛼)
]
𝑆𝑖𝑛 (𝛽+𝛿)𝑆𝑖𝑛(𝛼+𝛽)
𝑆𝑖𝑛2 𝛽 𝑆𝑖𝑛(𝛽− 𝛿)[1∙√
FAILURE WEDGE Of RETAING WALL
A. Active Pressure:
D. Passive Pressure:
∅ = angle of friction
B. Passive Pressure:
E. Active Pressure:
C. Active Pressure:
F. Passive Pressure: