1. The document discusses slope stability analysis using the Swedish slip circle method for analyzing finite slopes made of cohesive soils. 2. It describes the assumptions of the method and calculates the factors of safety for circular failure surfaces with and without tension cracks. 3. The document also covers other methods like the ordinary method of slices for c-f soils and discusses locating the critical slip circle using empirical relationships.

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 # 28
20-Dec-2017

2. 2
SLOPE STABILITY ANALYSIS
Finite Slope (Swedish Slip Circle Method)
Assumptions:
1. Material of the slope is homogeneous.
2. Soil is purely cohesive in nature i.e. f = 0.
3. Failure surface has a curved/circular or spoon like surface.
4. Shear strength of the soil is uniformly distributed along
failure plane. (only possible if f = 0)

3. 3
SLOPE STABILITY ANALYSIS
Swedish Slip Circle Method (Cohesive soils (f=0))
NSLC
A
B
𝐹𝑂𝑆 =
𝑅𝑒𝑠𝑖𝑠𝑡𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡 (𝑀 𝑅)
𝐷𝑖𝑠𝑡𝑢𝑟𝑏𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡 (𝑀 𝐷)

4. 4
NSLC
A
B
𝐹𝑂𝑆 =
𝑅𝑒𝑠𝑖𝑠𝑡𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡 (𝑀 𝑅)
𝐷𝑖𝑠𝑡𝑢𝑟𝑏𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡 (𝑀 𝐷)
𝐷𝑖𝑠𝑡𝑢𝑟𝑏𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡
𝑀 𝐷 = 𝑊 ∙ 𝑥
tr = c + sn tan f
For saturated clay under
undrained loading; f=0
 tr = c = su
𝑅𝑒𝑠𝑖𝑠𝑡𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡
𝑀 𝑅 = 𝜏 𝑟 ∙ 𝐴𝐵 ∙ 𝑅
𝑀 𝑅 = 𝑐 ∙ (𝑅 ∙ 𝜃) ∙ 𝑅
𝑀 𝑅 = 𝑐 ∙ 𝜃 ∙ 𝑅2
𝐹𝑂𝑆 =
𝑀 𝑅
𝑀 𝐷
𝐹𝑂𝑆 =
𝑐 ∙ 𝜃 ∙ 𝑅2
𝑊 ∙ 𝑥
SLOPE STABILITY ANALYSIS
Swedish Slip Circle Method (Cohesive soils (f=0))
Case-I: No Tension Crack
W
x
q
R
𝑊 = (𝐴𝑟𝑒𝑎 𝑜𝑓 𝐴𝐵𝐶𝐴 × 1) × 𝛾
→ q in radians

5. 5
W
x
q
R
NSLC
A
B𝐹𝑂𝑆 =
𝑅𝑒𝑠𝑖𝑠𝑡𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡 (𝑀 𝑅)
𝐷𝑖𝑠𝑡𝑢𝑟𝑏𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡 (𝑀 𝐷)
𝑀 𝐷 = 𝑊 ∙ 𝑥
tr = c + sn tan f
For saturated clay under
undrained loading; f=0
 tr = c = su
𝑀 𝑅 = 𝑐 ∙ 𝜃2 ∙ 𝑅2
𝐹𝑂𝑆 =
𝑀 𝑅
𝑀 𝐷
𝐹𝑂𝑆 =
𝑐 ∙ 𝜃2 ∙ 𝑅2
𝑊 ∙ 𝑥
SLOPE STABILITY ANALYSIS
Swedish Slip Circle Method (Cohesive soils (f=0))
Case-II: Development of Tension Crack
ℎ 𝑡 =
2𝑐
𝛾 𝐾𝑎
q2
FOS will reduce after development of tension crack [∵ q2 < q]
𝐷𝑖𝑠𝑡𝑢𝑟𝑏𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡
𝑅𝑒𝑠𝑖𝑠𝑡𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡

6. 6
W
x
R
C
A
tr = c = su
SLOPE STABILITY ANALYSIS
Swedish Slip Circle Method (Cohesive soils (f=0))
Case-III: Tension Crack filled with water
ℎ 𝑡 =
2𝑐
𝛾 𝐾𝑎
q2
𝛾wht
PW
h
2
3
ℎ 𝑡
𝑃 𝑊 =
1
2
𝛾 𝑤 ∙ ℎ 𝑡
2
𝑀 𝐷 = 𝑊 ∙ 𝑥 +
𝑀 𝑅 = 𝑐 ∙ 𝜃2 ∙ 𝑅2
𝐹𝑂𝑆 =
𝑐 ∙ 𝜃2 ∙ 𝑅2
𝑊 ∙ 𝑥 +
1
2
𝛾 𝑤 ∙ ℎ 𝑡
2
ℎ +
2
3
ℎ 𝑡
FOS will reduce further when tension crack is filled with water
𝐹𝑂𝑆 =
𝑅𝑒𝑠𝑖𝑠𝑡𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡 (𝑀 𝑅)
𝐷𝑖𝑠𝑡𝑢𝑟𝑏𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡 (𝑀 𝐷)
𝐷𝑖𝑠𝑡𝑢𝑟𝑏𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡
𝑅𝑒𝑠𝑖𝑠𝑡𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡
1
2
𝛾 𝑤 ∙ ℎ 𝑡
2
ℎ +
2
3
ℎ 𝑡

7. 7
Practice Problem #3
Determine the factor of safety of the cohesive slope shown in
the figure for the following two cases;
A. No tension crack
B. 2m deep tension crack filled with water (q1 = 38°)
NSLC
A
B
gb = 17.75 kN/m3
Cu above line AD = 21.5 kPa
Cu below line AD = 33.5 kPa
W
3.1m
q1=
40°
R
q2=
35°
D
D’
4m
2m
Area of ABCA
= 90m2

8. 8
SLOPE STABILITY ANALYSIS
Ordinary Method of Slices (OMS) (c-f soils)
 For c-f soils, normal stress would change along slip circle
 Different normal stress means, shear resistance would also be
different (∵ M-C equation)
 Failing slope divided into slices
TR
Guidelines for Slice Selection
 Slices do not have to be of equal
width
 For convenience, base arc of each
slice should pass through one soil
type only
 Slice width should be limited
(curved base approximated as
straight line)

9. 9
SLOPE STABILITY ANALYSIS
Ordinary Method of Slices (OMS) (c-f soils)
Wt
a
a
TR
l
b
𝑙 =
𝑏
cos 𝛼
𝑙 = 𝑏 sec 𝛼
𝐹𝑂𝑆 =
𝑀 𝑅
𝑀 𝐷 TR = Total shear resistance force acting on slice
𝑇𝑅 = 𝜏 𝑅 × (𝑙 ∙ 1)
𝑇𝑅 = (𝑐′
+ 𝜎 𝑛′ tan 𝜙) × 𝑙
TR
𝑅𝑒𝑠𝑖𝑠𝑡𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡
tR = Shear resistance (stress) offered by soil
𝑀 𝑅 = 𝑇𝑅 × 𝑟 … (𝐸𝑞. 1)

10. 10
SLOPE STABILITY ANALYSIS
Ordinary Method of Slices (OMS) (c-f soils)
where,
𝜎 𝑛′ =
𝑊𝑡 cos 𝛼
𝑙 × 1
𝑇𝑅 = 𝑐′
𝑙 +
𝑊𝑡 cos 𝛼
𝑙 × 1
𝑙 tan 𝜙
𝑇𝑅 = 𝑐′ 𝑙 + 𝑊𝑡 cos 𝛼 tan 𝜙
𝑀 𝑅 = 𝑟 (𝑐′
𝑏 sec 𝛼 + 𝑊𝑡 cos 𝛼 tan 𝜙)
Wt
a
a
a
TR
l
b
𝑙 =
𝑏
cos 𝛼
𝑙 = 𝑏 sec 𝛼
Wt cos a
Wt sin a
𝑀 𝑅 = 𝑇𝑅 × 𝑟 … (𝐸𝑞. 1)
𝑇𝑅 = 𝜏 𝑅 × (𝑙 ∙ 1)
𝑇𝑅 = (𝑐′ + 𝜎 𝑛′ tan 𝜙) × 𝑙
𝑅𝑒𝑠𝑖𝑠𝑡𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡
𝐸𝑞. 1

11. 11
SLOPE STABILITY ANALYSIS
Ordinary Method of Slices (OMS) (c-f soils)
Wt
a
a
a
TR
l
b
𝑙 =
𝑏
cos 𝛼
𝑙 = 𝑏 sec 𝛼
Wt cos a
Components of disturbing force (Wt)
1. Wt cos a  Passes through center of rotation, i.e. zero
moment
2. Wt sin a  Tangential component; causing sliding
𝑀 𝐷 = 𝑟 (𝑊𝑡 sin 𝛼)
Wt sin a
𝐷𝑖𝑠𝑡𝑢𝑟𝑏𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡
TR

12. 12
SLOPE STABILITY ANALYSIS
Ordinary Method of Slices (OMS) (c-f soils)
Wt
a
a
a
TR
l
b
𝑙 =
𝑏
cos 𝛼
𝑙 = 𝑏 sec 𝛼
Wt cos a
𝐹𝑂𝑆 =
(𝑐′ 𝑏 sec 𝛼 + 𝑊𝑡 cos 𝛼 tan 𝜙)
(𝑊𝑡 sin 𝛼)
Wt sin a
𝐹𝑂𝑆 =
𝑀 𝑅
𝑀 𝐷
𝑀 𝑅 = 𝑟 (𝑐′
𝑏 sec 𝛼 + 𝑊𝑡 cos 𝛼 tan 𝜙)
𝑀 𝐷 = 𝑟 (𝑊𝑡 sin 𝛼)

13. 13
CRITICAL SLIP CIRCLE
NSL
A
B
Many slip circles are possible on any slope
Slip circle having minimum FOS  Critical Slip Circle /
Critical Failure Plane
Design has to satisfy safety against critical slip circle
FOS = 2.4
1.20
1.79
1.55

14. 14
LOCATION OF CRITICAL SLIP CIRCLE
In Cohesive soils (f=0)
NSL
A
B
 Plot the configuration according to
scale
 Draw two lines from point A and B
at angles as has been shown in
figure.
 Taking radius equal to OA, draw a
circle passing through the slope
Case-I: Toe Failure
(Fellenius Method)
r
q1
q2
Slope Slope Angle q1 q2
1V : 0.5H 60° 29° 40°
1 : 1 45° 28° 38°
1 : 1.5 34° 26° 35°
1 : 2 27° 25° 35°
1 : 3 19° 25° 35°
Empirical values of q1 and q2

15. 15
LOCATION OF CRITICAL SLIP CIRCLE
In Cohesive soils (f=0)
NSL
A
B
 Plot the configuration according to
scale
 Draw a vertical line at the mid-point
of slope (the center of critical slip
circle always lies on a vertical line
passing through the mid-point of
slope)
 Determine the center by hit and trial
method by comparing the FOS
 Circle with minimum FOS is the
critical circle
Case-II: Base Failure
(Fellenius Method)
𝐻
2
𝐻
2
FOS = 1.65
FOS = 1.10
133.5°
 Angle made by critical circle at the
center is about 133.5°. (Fellenius)

16. 16
NSL
A
B
H
H
 Plot the configuration according to
scale
 Find the intersection point of ‘4.5H’
horizontal and ‘H’ distance vertical
downward from A
 Draw the direction angle q1 and q2
 Join the points of intersection O and
C
 Locate OC by hit and trial. For this
try O1, O2,…… and make circles.
 The circle giving minimum FOS is
the critical circle
q2
q1
O1
O2
O3
Omin = Ocr
LOCATION OF CRITICAL SLIP CIRCLE
In c-f soils
O4
4.5H
C

17. 17
SHORT TERM AND LONG TERM
STABILITY
Clay Core
Construction of dam core
- Clay material
- Very low permeability
- Construction in layers with compaction
at OMC
SHORT TERM STABILITY
Stability of slope immediately after construction
 Undrained conditions
 Undrained parameters (cu and fu) to be used for slope stability analysis
 Obtained from UU or CU triaxial tests
 Total unit weight of soil (gb) to be used
 Called as TOTAL STRESS ANALYSIS
 Change in pore water pressure totally dependent upon stress change

18. 18
SHORT TERM AND LONG TERM
STABILITY
Clay Core
Construction of dam core
- Clay material
- Very low permeability
- Construction in layers with compaction
at OMC
LONG TERM STABILITY
Stability of slope long time after construction
 Drained conditions
 Drained parameters (cd (or c’) and fd (f’)) to be used for slope stability analysis
 Obtained from CD triaxial tests, or CU tests with PWP measurements
 Effective unit weight of soil (gsub (or g’)) to be used
 Called as EFFECTIVE STRESS ANALYSIS
 Change in pore water pressure independent of stress change

19. 19
THE END
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
In fact, this is just the beginning!