Geotechnical Engineering-II [Lec #2: Mohr-Coulomb Failure Criteria]

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

6
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
Y
v
h
h
GL
As loading progresses, Mohr
circle becomes larger…
.. and finally failure occurs
when Mohr circle touches
the envelope

v

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
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
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
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
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

Geotechnical Engineering-II [Lec #2: Mohr-Coulomb Failure Criteria]

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