This document provides a summary of Lecture 2 on the Mohr-Coulomb failure criteria in geotechnical engineering. It introduces the relationship between normal and shear stresses on a failure plane using the Mohr-Coulomb equation. It then discusses the graphical representation of this relationship using Mohr's circle, showing how the major and minor principal stresses and maximum shear stress can be determined. It demonstrates how the Mohr circle expands under increasing loads until it contacts the failure envelope, indicating failure. The criteria is shown for both total and effective stresses, accounting for pore water pressure. Steps to derive the failure condition directly from the geometry of the Mohr circle are also presented.

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 # 2
8-Sep-2017

2. 2
The relationship between normal and shear stress on the failure
plane
 tan cf
f = shear strength
c = cohesion
 = normal stress
Φ = angle of internal friction
)( f
σ1
σ3

Friction angle
f
 

Graphical
representation
Cohesion
c
MOHR-COULOMB FAILURE CRITERIA

3. 3
MOHR-COULOMB FAILURE CRITERIA
  tanff c
’f
f
’

’
c’ c’
’f tan ’ frictional
component
Shear strength consists of ____ components:
• c and  are measures of
shear strength.
• Higher the values,
higher the shear strength
• Cohesive and Frictional
two

4. 4
MOHR CIRCLE OF STRESS

’
2
'
3
'
1' 


m
2
'
3
'
1  
'
3 '
1
PP = Pole w.r.t. plane
q
Soil element q
’1
’1
’3 ’3
'
Major principal stress: σ1
Minor principal stress: σ3
Mean principal stress: σm
Maximum shear stress: τmax
max (’, )
2
31
2
312
22





 





 



Equation of Mohr Circle
Compare it’s formulation with equation of circle

5. 5
MOHR CIRCLE AND FAILURE ENVALOPE
Soil elements at different locations
Failure surface
X X
X ~ failure
Y
Y
Y ~ stable

’
  tancf

6. 6
MOHR CIRCLE AND FAILURE ENVALOPE
Y
v
h
h
Mohr circle before
external loading

v+

Soil element does not fail if the Mohr circle is
contained within the envelope
GL
v

7. 7
MOHR CIRCLE AND FAILURE ENVALOPE
Y
v
h
h
GL
As loading progresses, Mohr
circle becomes larger…
.. and finally failure occurs
when Mohr circle touches
the envelope

v

8. 8
MOHR CIRCLE IN TERMS OF TOTAL AND
EFFECTIVE STRESSES
= X
v’
h’
X
u
u
+
h’
Effective Stresses
v’
u
vh
X
v
h
Total Stresses

 or ’

9. 9
MOHR CIRCLE IN TERMS OF TOTAL AND
EFFECTIVE STRESSES
= X
v’
h’
X
u
u
+
v’h’
Effective Stresses
u
vh
X
v
h
Total Stresses

 or ’
If X is on
failure
c

Failure envelope in
terms of total stresses
’
c’
Failure envelope in terms
of effective stresses

10. 10
MOHR-COULOMB FAILURE CRITERION
WITH MOHR CIRCLE OF STRESS
X
’v = ’1
’h = ’3
X is on failure ’1’3

’
’ c’
Failure envelope in terms of
effective stresses
c’ Cot ’ (’1+ ’3)/2
(’1 - ’3)/2





 











 

2
'
2
''
'
3
'
1
'
3
'
1 


 SinCotc
Therefore,

11. 11
MOHR-COULOMB FAILURE CRITERION
WITH MOHR CIRCLE OF STRESS





 











 

2
'
2
''
'
3
'
1
'
3
'
1 


 SinCotc
( ) ( ) ''2''
3
'
1
'
3
'
1  CoscSin 
( ) ( ) ''2'1'1 '
3
'
1  CoscSinSin 
( )
( ) ( )'1
'
'2
'1
'1'
3
'
1





Sin
Cos
c
Sin
Sin


















2
'
45'2
2
'
452'
3
'
1

 TancTan

12. 12
CONCLUDED
REFERENCE MATERIAL
Principles of Geotechnical Engineering – (7th Edition)
Braja M. Das
Chapter #12
Geotechnical Engineering – Principles and Practices – (2nd Edition)
Coduto, Yueng, and Kitch
Chapter #12