Mohr-Coulomb Model
MOHR-COULOMB MODEL
Computational Geotechnics
Course ‘Computational Geotechnics’
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Mohr-Coulomb Model
Contents
Idealized and Real Stress-Strain Behavior of Soils
Basic Concepts of Mohr-Coulomb Model
Mohr-Coulomb Model Modeling
Friction Angle
Influence of Intermediate Principal Stress on Friction
Angle
Drained Simple Shear Test
How to Understand 
Drained Triaxial Test
Reference
Appendix A
Appendix B
Course ‘Computational Geotechnics’
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Mohr-Coulomb Model
Idealized and Real Stress-Strain Behavior of Soils
Course ‘Computational Geotechnics’
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Mohr-Coulomb Model
Basic Concepts of Mohr-Coulomb Model
Bilinear approximation of triaxial test
Basic law:
 i   ie   ip
i = x, y, etc.
 ie  reversible elastic strain
 ip  irreversible plastic strain

p
i
 0
for f < 0
f  f ( x ,  y ,  z ,  xy ,  yz ,  zx )
f = yield function
Course ‘Computational Geotechnics’
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Mohr-Coulomb Model
Mohr-Coulomb Soil Modeling
Course ‘Computational Geotechnics’
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Mohr-Coulomb Model
Mohr-Coulomb Soil Modeling

p
i
 
Flow rule for plastic strains:
This means:

g
 
 x

p
x
,

p
y
 
g
 i
g
 y
, etc.
a multiplier that determines the magnitude of plastic strains
g

 i
determines the direction of plastic strains
Classical associated plasticity: g = f
General non-associated plasticity: g  f
M-C model: f  r – s sin – c cos
: yield function
g  r – s sin – c cos : plastic potential function
Course ‘Computational Geotechnics’
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Mohr-Coulomb Model
Mohr-Coulomb Soil Modeling
Nonlinear Failure Envelope Representation
Friction Angle Definitions
(Kulhawy and Mayne, 1990)
Course ‘Computational Geotechnics’
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Mohr-Coulomb Model
Strength Envelopes for a Range of Soil Types
(Bishop, A.W., 1966)
 tc versus Relative Density and Unit Weight
(NAVFAC, “Soil Mechanics Design Manual”, DM-7)
Course ‘Computational Geotechnics’
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Mohr-Coulomb Model
Friction Angle
Dilatancy Angle Relationships
(Bolton, M.D., 1986)
cv  36  1  2  3  4   5
in which:
o
1
=
2
=
3
=
4
=
5
=
Correction for particle shape
For high sphericity and subrounded shape
1 = -6o
o
For low sphericity and angular shape
1 = +2
Correction for particle size (effective size, d10)
For d10 > 2.0 mm (gravel)
2 = -11o
o
For 2.0 > d10 >0.6 mm (coarse sand)
2 = -9
o
For 0.6 > d10 >0.2 mm (medium sand)
2 = -4
For 0.2 > d10 >0.06 mm (fine sand)
2 = 0
Correction for graduation (uniformity coefficient, Cu)
For Cu > 2.0 (well-graded)
3 = -2o
o
For Cu = 2.0 (medium graded)
3 = -1
For Cu < 2.0 (poorly graded)
3 = 0
Correction for relative density (Dr)
For 0 < Dr <0.5 (loose)
4 = -1o
For 0.5 < Dr < 0.75 (intermediate)
4 = 0
For 0.75 < Dr < 1.00 (dense)
4 = +4o
Correction for type of mineral
For quartz
5 = 0
o
For feldspar, calcite, chlorite
5 = +4
o
For muscovite mica
5 = +6
(Kulhawy and Mayne, 1990)

Course ‘Computational Geotechnics’
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Mohr-Coulomb Model
cv for NC Clays vs. PI
(Mitchell, 1993)
max  cv  0.8
(Bolton, 1986)
Course ‘Computational Geotechnics’
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Mohr-Coulomb Model
Influence of Intermediate Principal Stress on Friction Angle
Introduction:
Models and soil mechanics
Special conditions
Name of test
Diagram
a  b  c
True triaxial
b c r
Cylindrical compression
The “triaxial” test
b  0
Plane strain or biaxial
b  0
Plane stress
b  c  r  0
One-dimensional compression
The oedometer test
b  c  r  0
Uniaxial compression or
Unconfined compression
a  b  c  0
Isotropic compression
(Ladd et al., 1977)
Course ‘Computational Geotechnics’
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Mohr-Coulomb Model
Drained Simple Shear Test
Course ‘Computational Geotechnics’
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Mohr-Coulomb Model
How to Understand 
 =  + i
or
 =  - i
i = inter particle angle of friction
quartz sand:    – 30o
Course ‘Computational Geotechnics’
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Mohr-Coulomb Model
Drained Triaxial Test
Now
 v   x   y   z  2 x   y
tan  
2sin
1  sin
Course ‘Computational Geotechnics’
(identical to biaxial test)
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Mohr-Coulomb Model
References
Bishop, A.W. (1966), “The Strength of Soils as Engineering Materials”, Geotechnique, Vol. 16,
No. 2, pp. 89-130.
Bolton, M.D. (1986), “The Strength and Dilatancy of Sands”, Geotechnique, Vol. 36, No. 1, pp.
65-78.
Duncan, J.M and Buchignani, A.L. (1976), “An Engineering Manual for Settlement Studies”,
Report, Department of Civil Engineering, University of California, Berkeley, 94 pages.
Janbu, N. (1963), “Soil Compressibility as Determined by Oedometer and Triaxial Tests”, Proc.
3rd European Conference on Soil Mechanics and Foundation Engineering, Wiesbaden, Germany,
Vol. 1, pp. 19-25.
Janbu, N. (1985), “Soil Models in Offshore Engineering”, Geotechnique, Vol. 35, No. 3, pp.
241-281.
Kulhawy, F.H. and Mayne, P.W. (1990), “Manual on Estimating Soil Properties for Foundation
Design”, Report EL-6800/1493-6, Electric Power Research Institute.
Ladd, C.C., Foott, R., Ishihara, K., Schlosser, F., and Poulos, H.G. (1977), “Stress-Deformation
and Strength Characteristics”, Proc. 9th Int. Conference on Soil Mechanics and Foundation
Engineering, Tokyo, Japan, Vol. 2, pp. 421-494.
Mitchell, J.K. (1993), “Fundamentals of Soil Behavior”, John Wiley & Sons, Inc., New York, 2nd
Edition.
Naval Facilities Engineering Command, NAVFAC, “Soil Mechanics Design Manual”, DM-7,
Alexandria, Virginia, 1982, 355 pages.
Pestana, J. (2001), PLAXIS Lecture Notes, Department of Civil and Environmental Engineering,
University of California, Berkeley.
Wroth, C.P. and Wood, D.M. (1978), “The Correlation of Index Properties with Some Basic
Engineering Properties of Soils”, Canadian Geotechnical Journal, Vol. 15, No. 2, pp. 137-145.
Course ‘Computational Geotechnics’
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Mohr-Coulomb Model
Appendix A
 y
Determination of
from plasticity theory in drained simple shear test:
 xy


x  0 z  0

 y
 xy


x   y  z   y
Rate of volume change:
We will proof:



y

 xy
 tan
At failure only plastic strains!

g 1   x   y
x    
 2
 sin   0
 x
 2r



p
x


p
y
y    

 xy  

p
xy
  y  x

g
 12  
 sin    sin
 y
 2r


 xy
g

 cos
 xy
r

Hence:
y

 xy
Course ‘Computational Geotechnics’
 tan
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Mohr-Coulomb Model
Appendix B
Drained Biaxial Test
Determination of  in drained biaxial test:
tan   

p
v

p
y



g 1   x   y
 
 2
 sin 
 x
 2r


p
x

g 1   y   x
 
 2
 sin 
 y
 2r


p
y
g
 
0
 z

p
z
+

p
v
   sin
 xy  0 
 x  y
2r
1

p
y
  12   1  sin 
tan  
2sin
1  sin
Course ‘Computational Geotechnics’
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Mohr-Coulomb Model
Appendix B
Drained Biaxial Test
Difference with triaxial test:
Course ‘Computational Geotechnics’
Plane strain
z  0
z x
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Mohr-Coulomb Model
Appendix B
Drained Biaxial Test
v   x   y   z   x   y
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