This document contains lecture notes on slope stability analysis in geotechnical engineering, focusing on infinite slopes and their factors of safety (fos) under various conditions, including dry, fully saturated, and cohesive soils. It includes mathematical formulas for calculating shear stress and factors of safety, as well as practice problems for students to solve. The material references notable geotechnical engineering textbooks for further study.

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 # 27
15-Dec-2017

2. 2
SLOPE STABILITY ANALYSIS
Infinite Slope
Applied Shear Stress (t)
𝜏 =
𝑇
( 𝐿 cos 𝛼)(1)
𝜏 =
𝛾𝐻𝐿 sin 𝛼
( 𝐿 cos 𝛼)(1)
𝜏 = 𝛾𝐻 sin 𝛼 cos 𝛼
𝐹𝑂𝑆 =
𝑐′ + 𝛾𝐻𝑐𝑜𝑠2 𝛼 ∙ tan 𝜙′
𝛾𝐻 sin 𝛼 cos 𝛼
Factor of Safety (FOS)
Case-A: c-f Soil (c>0; f>0) L
H h
a
W
a
b
c
d
T
N
a
tr
𝑭𝑶𝑺 =
𝑨𝒗𝒂𝒊𝒍𝒂𝒃𝒍𝒆 𝑺𝒉𝒆𝒂𝒓 𝑺𝒕𝒓𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝑺𝒐𝒊𝒍
𝑨𝒑𝒑𝒍𝒊𝒆𝒅 𝑺𝒉𝒆𝒂𝒓 𝑺𝒕𝒓𝒆𝒔𝒔
Available Shear Strength of Soil (tr)
𝜏 𝑟 = 𝑐′ + 𝛾𝐻𝑐𝑜𝑠2 𝛼 ∙ tan 𝜙′
Without considering the effect of WT

3. 3
SLOPE STABILITY ANALYSIS
Infinite Slope
𝐹𝑂𝑆 =
𝑐′ + 𝜎 𝑛′ ∙ tan 𝜙′
𝛾𝐻 sin 𝛼 cos 𝛼
Case-A: c-f Soil (c>0; f>0)
 c’ & f’; effective strength parameters
 Obtained through drained triaxial test
𝜎 𝑛 = 𝜎 𝑛
′
=
𝐴𝐵 = ℎ
𝐴𝐶 =
L
H
h
a
W
a
b
c
d
T
N
a
tr
A
C
B
D
a
𝐴𝐷 = ℎ 𝑤 =
𝑢 =
ℎ 𝑤 =
𝑢 = 𝛾 𝑤ℎ 𝑐𝑜𝑠2 𝛼
𝐹𝑂𝑆 =
𝑐′ + (𝛾𝐻 − 𝛾 𝑤ℎ) ∙ 𝑐𝑜𝑠2 𝛼 tan 𝜙′
𝛾𝐻 sin 𝛼 cos 𝛼
𝛾𝐻𝑐𝑜𝑠2
𝛼 𝜎 𝑛 − 𝑢
ℎ cos 𝛼
𝐴𝐶 cos 𝛼 = ℎ 𝑐𝑜𝑠2 𝛼
𝛾 𝑤ℎ 𝑤
𝐴𝐷 = ℎ 𝑐𝑜𝑠2
𝛼
Pore water pressure, u
Considering
presence of WT

4. 4
SLOPE STABILITY ANALYSIS
Infinite Slope
Case-B: Cohesive soil (f=0)
L
H h
a
W
a
b
c
d
T
N
a
tr
𝐹𝑂𝑆 =
𝑐′
𝛾𝐻 sin 𝛼 cos 𝛼
Case-C: Granular soil (c=0)
𝐹𝑂𝑆 =
(𝛾𝐻 − 𝛾 𝑤ℎ) ∙ 𝑐𝑜𝑠2
𝛼 ∙ tan 𝜙′
𝛾𝐻 sin 𝛼 cos 𝛼
𝐹𝑂𝑆 = 1 −
𝛾 𝑤ℎ
𝛾𝐻
tan 𝜙′
tan 𝛼
𝐹𝑂𝑆 =
𝑐′
+ (𝛾𝐻 − 𝛾 𝑤ℎ) ∙ 𝑐𝑜𝑠2
𝛼 tan 𝜙′
𝛾𝐻 sin 𝛼 cos 𝛼
(For c-f soil)
𝐹𝑂𝑆 = 𝛾𝐻 1 −
𝛾 𝑤ℎ
𝛾𝐻
𝑐𝑜𝑠2
𝛼 ∙ tan 𝜙′
𝛾𝐻 sin 𝛼 cos 𝛼

5. 5
SLOPE STABILITY ANALYSIS
Infinite Slope
Case-C: Granular soil (c=0)
For critical case; FOS = 1
𝐹𝑂𝑆 =
tan 𝜙
tan 𝛼
tan 𝛼 = tan 𝜙′
i.e. the slope is only stable up to the angle of friction of soil.
𝐹𝑂𝑆 = 1 −
𝛾 𝑤ℎ
𝛾𝐻
tan 𝜙′
tan 𝛼
For dry slope, h=0
𝛼 = 𝜙′
L
H h
a
W
a
b
c
d
T
N
a
tr

6. 6
SLOPE STABILITY ANALYSIS
Infinite Slope
Case-C: Granular soil (c=0)
For fully saturated slope, h=H
𝐹𝑂𝑆 = 1 −
𝛾 𝑤ℎ
𝛾𝐻
tan 𝜙′
tan 𝛼
𝐹𝑂𝑆 =
𝛾 − 𝛾 𝑤
𝛾
tan 𝜙′
tan 𝛼
𝐹𝑂𝑆 = 0.5
tan 𝜙′
tan 𝛼
For critical case; FOS = 1
1 = 0.5
tan 𝜙′
tan 𝛼
tan 𝛼 = 0.5 ∙ tan 𝜙′
i.e. a fully saturated slope will be stable only up to half of its
friction angle
𝛼 = 0.5 ∙ 𝜙’
L
H h
a
W
a
b
c
d
T
N
a
tr

7. 7
Practice Problem #1
Calculate the maximum permissible slope angle for a sandy soil
having gsat = 20 kN/m3, f’ =30°, to give a FOS of 1.5 when;
a. the slope is dry
b. the slope is fully saturated with water table at the ground
surface

8. 8
Practice Problem #2
An infinite slope is shown in the figure. The position of water
table is 2m below the ground surface. Determine the change in
FOS of the slope if WT is brought to a depth of 4m below the
surface.
gsat = 18 kN/m3
c’= 10 kPa
f’ = 20°
H = 6m
a = 15°

9. 9
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