Class notes of Geotechnical Engineering course I used to teach at UET Lahore. Feel free to download the slide show. Anyone looking to modify these files and use them for their own teaching purposes can contact me directly to get hold of editable version.

1. 1
Geotechnical Engineering–II [CE-321]
BSc Civil Engineering – 5th Semester
by
Dr. Muhammad Irfan
Assistant Professor
Civil Engg. Dept. – UET Lahore
Email: mirfan1@msn.com
Lecture Handouts: https://groups.google.com/d/forum/geotech-ii_2015session
Lecture # 25
8-Dec-2017

2. 2
COULOMB’S ACTIVE EARTH PRESSURE
a
b
q
180-a-q
a+b
A
B
C
D
q-b
W
W = Weight of soil wedge ABC
𝑊 = 1
2∙𝐴𝐶∙𝐵𝐷∙1∙𝛾 ⋯⋯⋯(1)
𝐴𝐶 =
𝐻
sin 𝛼 ∙ sin(𝜃 − 𝛽)
∙ sin(𝛼 + 𝛽)
𝐵𝐷 = sin(𝛼 + 𝜃) ∙
𝐻
sin 𝛼
𝑊 = 1
2 ∙
𝛾𝐻2
𝑠𝑖𝑛2 𝛼
∙
sin(𝛼 + 𝛽) ∙ sin(𝛼 + 𝜃)
sin(𝜃 − 𝛽)
Eq. 1 →
H

3. 3
COULOMB’S ACTIVE EARTH PRESSURE
a
b
q
180-a-q
a+b
A
B
C
D
q-b
W
W
f
R = Resultant of shear and normal forces
acting on failure plane
R
(q-f)
(a-d)180-(a-d+q-f)
d
Pa
𝛿 = 2
3 𝜙
Our Goal:
Determine active force (Pa) on the wall.
 Draw force polygon of the system.
Pa
d = angle of wall friction
(𝑅𝑒𝑠𝑜𝑛𝑎𝑏𝑙𝑒 𝑎𝑠𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛)

4. 4
COULOMB’S ACTIVE EARTH PRESSURE
a
b
q
180-a-q
a+b
A
B
C
D
q-b
W
W
f
R = Resultant of shear and normal forces
acting on failure plane
R
(q-f)
(a-d)180-(a-d+q-f)
d
Pa
𝛿 = 2
3 𝜙
Our Goal:
Determine active force (Pa) on the wall.
 Draw force polygon of the system.
Pa
d = angle of wall friction
(𝑅𝑒𝑠𝑜𝑛𝑎𝑏𝑙𝑒 𝑎𝑠𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛)

5. 5
COULOMB’S ACTIVE EARTH PRESSURE
a
b
q
180-a-q
a+b
A
B
C
D
q-b
W
W
R
(q-f)
(a-d)180-(a-d+q-f)
d
Pa
𝑃𝑎
sin(𝜃 − 𝜙)
=
𝑊
sin[180 − 𝛼 − 𝛿 + 𝜃 − 𝜙 ]
Applying sine law on force polygon
Pa
𝑃𝑎
sin(𝜃 − 𝜙)
=
𝑊
sin(𝛼 − 𝛿 + 𝜃 − 𝜙)
Replacing value of ‘W’
𝑃𝑎 =
1
2
∙
𝛾𝐻2
𝑠𝑖𝑛2 𝛼
∙
sin(𝛼 + 𝛽) ∙ sin(𝛼 + 𝜃) ∙ sin(𝜃 − 𝜙)
sin(𝜃 − 𝛽) ∙ sin(𝛼 − 𝛿 + 𝜃 − 𝜙)
f

6. 6
COULOMB’S ACTIVE EARTH PRESSURE
a
b
q
180-a-q
a+b
A
B
C
D
q-b
W
W
R
(q-f)
(a-d)180-(a-d+q-f)
d
Pa
As designers, we want to determine max. value of Pa
Pa
𝑃𝑎 =
1
2
𝛾𝐻2 ∙
𝑠𝑖𝑛2(𝛼 + 𝜙)
𝑠𝑖𝑛2 𝛼 ∙ sin(𝛼 − 𝛿) 1 +
sin(𝜙 + 𝛿) ∙ sin(𝜙 − 𝛽)
sin(𝛼 − 𝛿) ∙ sin(𝛼 + 𝛽)
2
To determine critical value of b for max. Pa, we have
𝑑𝑃𝑎
𝑑𝛽
= 0
𝑃𝑎 =
1
2
∙
𝛾𝐻2
𝑠𝑖𝑛2 𝛼
∙
sin(𝛼 + 𝛽) ∙ sin(𝛼 + 𝜃) ∙ sin(𝜃 − 𝜙)
sin(𝜃 − 𝛽) ∙ sin(𝛼 − 𝛿 + 𝜃 − 𝜙)
f

7. 7
COULOMB’S ACTIVE EARTH PRESSURE
a
b
q
180-a-q
a+b
A
B
C
D
q-b
W
d
Pa
𝐾 𝑎 =
𝑠𝑖𝑛2(𝛼 + 𝜑)
𝑠𝑖𝑛2 𝛼 ∙ sin(𝛼 − 𝛿) 1 +
sin(𝜙 + 𝛿) ∙ sin(𝜙 − 𝛽)
sin(𝛼 − 𝛿) ∙ sin(𝛼 + 𝛽)
2
Since,
𝑃𝑎 =
1
2
∙ 𝛾𝐻2
∙ 𝐾 𝑎
𝑃𝑎 =
1
2
𝛾𝐻2
∙
𝑠𝑖𝑛2
(𝛼 + 𝜙)
𝑠𝑖𝑛2 𝛼 ∙ sin(𝛼 − 𝛿) 1 +
sin(𝜙 + 𝛿) ∙ sin(𝜙 − 𝛽)
sin(𝛼 − 𝛿) ∙ sin(𝛼 + 𝛽)
2
f

8. 8
COULOMB’S ACTIVE EARTH PRESSURE
a
b
q
180-a-q
a+b
A
B
C
D
q-b
W
d
Pa
𝑃𝑎 =
1
2
𝛾𝐻2
∙
𝑠𝑖𝑛2
(𝛼 + 𝜙)
𝑠𝑖𝑛2 𝛼 ∙ sin(𝛼 − 𝛿) 1 +
sin(𝜙 + 𝛿) ∙ sin(𝜙 − 𝛽)
sin(𝛼 − 𝛿) ∙ sin(𝛼 + 𝛽)
2
For a vertical wall face and horizontal
levelled ground
𝛼 = 90° , 𝑎𝑛𝑑 𝛽 = 0°
𝑃𝑎 =
1
2
𝛾𝐻2 ∙
1 − sin 𝜙
1 + sin 𝜙
Above equation is reduced to
i.e. same as Renkine’s Solution
f

9. 9
COULOMB’S PASSIVE EARTH PRESSURE
Failure wedge and forces acting
on it for passive earth pressure
Force triangle to establish Pp
𝑃𝑝 =
1
2
𝛾𝐻2 ∙
𝑠𝑖𝑛2(𝛼 − 𝜙)
𝑠𝑖𝑛2 𝛼 ∙ sin(𝛼 + 𝛿) 1 −
sin(𝜙 + 𝛿) ∙ sin(𝜙 + 𝛽)
sin(𝛼 + 𝛿) ∙ sin(𝛼 + 𝛽)
2

10. 10
Practice Problem #7
f’ = 35°
 = 18.5 kN/m3
d = (2/3) f6 m
Using Coulomb’s theory, find out Pa and its point of
application.
a = 90°
b = 20°
𝑃𝑎 =
1
2
𝛾𝐻2
∙
𝑠𝑖𝑛2
(𝛼 + 𝜙)
𝑠𝑖𝑛2 𝛼 ∙ sin(𝛼 − 𝛿) 1 +
sin(𝜙 + 𝛿) ∙ sin(𝜙 − 𝛽)
sin(𝛼 − 𝛿) ∙ sin(𝛼 + 𝛽)
2

11. 11
Practice Problem #8
A counterfort retaining wall of 10 m height retains sand. The
void ratio and angle of internal friction of the sand respectively
are 0.70 and 30° in loose state and 0.40 and 40° in dense state.
Compute and compare active and passive earth pressure in both
cases. Take Gs = 2.66

12. 12
Practice Problem #9
A vertical wall 10 m high retains sand having specific gravity
of 2.65 and void ratio of 0.54. The WT is 4 m below the ground
surface. The surface of the backfill is horizontal and carries a
uniform surcharge of 25 kPa. The sand above the WT has an
average degree of saturation of 70% and a f-value of 35°. The
sand below the WT has a f-value of 30°. Determine the active
thrust on the back of wall and its point of application.

13. 13
Practice Problem #10
Determine the magnitude and location of Pa for several
possible alternatives that are produced due to tension crack.
c’ = 20 kPa
f’ = 10°
 = 17.5 kN/m3 4 m
f’ = 15°
 = 20.5 kN/m3
2 m
WT
f
ff


+
-





 
-



sin1
sin1
2
45tan2
o
a
aK
aaa KczK - 2
a
c
K
c
z



2

14. 14
Practice Problem #11
f
ff


+
-





 
-



sin1
sin1
2
45tan2
o
a
aK
aaa KczK - 2
c’ = 13.5 kPa
 = 18 kN/m3
WT
f’ = 30°
sat = 20 kN/m3
2m
6m
For a retaining wall shown in figure below, compute;
a. Total active earth pressure per meter width of the wall
b. Location of active thrust from the base of wall
c. The total thrust behind the wall if WT is lowered to the base of wall
q = 30 kPa
CLAY
SAND

15. 23
CONCLUDED
REFERENCE MATERIAL
Principles of Geotechnical Engineering – (7th Edition)
Braja M. Das
Chapter #13
Essentials of Soil Mechanics and Foundations (7th Edition)
David F. McCarthy
Chapter #17
Geotechnical Engineering – Principles and Practices – (2nd Edition)
Coduto, Yueng, and Kitch
Chapter #17