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Geology 3120 Geology 3120 Powerpoint notes available online at:Powerpoint notes available online at:
http://www.colorado/edu/geolsci/courses/GEOL3120http://www.colorado/edu/geolsci/courses/GEOL3120
Geology 3120 - Geology 3120 - The Mohr Stress DiagramThe Mohr Stress Diagram
nn
Stress SpaceStress Space
00
ss
OutlineOutline
• Setting up the Problem
• The Mohr Stress Diagram
• Mohr-Coulomb Law of Failure
• Exercises
Setting up the ProblemSetting up the Problem
is defined as the angle between the plane and the is defined as the angle between the plane and the
force vector. Clockwise is positive (+).force vector. Clockwise is positive (+).
Decomposing StressesDecomposing Stresses
After several trigonometric and algebraic simplifications, After several trigonometric and algebraic simplifications,
the two equations left are ……the two equations left are ……
n = (1 + 3) - (1 - 3)cos 22222
s = (1 - 3)sin 222
Fundamental Stress EquationsFundamental Stress Equations
Normal StressNormal Stress
Shear StressShear Stress
Physical and Stress SpacePhysical and Stress Space
11
11
3333 nn
ss
Physical SpacePhysical Space Stress SpaceStress Space
00
Conventions - Physical Conventions - Physical
SpaceSpace
33
is defined as the is defined as the
angle between the angle between the
plane and the plane and the 11
stress direction. A stress direction. A
clockwise direction is clockwise direction is
positive (+).positive (+).
11
11
33
Conventions - Stress SpaceConventions - Stress Space
nn
ss
00 33 11
n n = compression= compressionn n = tension= tension
Conventions - Stress SpaceConventions - Stress Space
nn
ss
00 33 11
nn(p)(p), , ss
(p)(p)
n(p) = (1 + 3) - (1 - 3)cos 2
2222
sp
= (1 - 3)sin 222
ss(p)(p)
nn(p)(p)
Conventions - Stress SpaceConventions - Stress Space
nn
ss
00 33 11
(1 - 3)sin 222
nn(p)(p), , ss
(p)(p)
Conventions - Stress SpaceConventions - Stress Space
nn
ss
00 33 11
(1 + 3)22
nn(p)(p), , ss
(p)(p)
Mean Stress - Mean Stress -
center of circlecenter of circle
Conventions - Stress SpaceConventions - Stress Space
nn
ss
00 33 11
(1 - 3)22
nn(p)(p), , ss
(p)(p)
Deviatoric Stress - Deviatoric Stress -
radius of circleradius of circle
Conventions - Stress SpaceConventions - Stress Space
nn
ss
00 33 11
(1 - 3)
nn(p)(p), , ss
(p)(p)
Differential Stress - Differential Stress -
diameter of circlediameter of circle
nn
ss
00 33 11
(1 - 3)cos 222
nn(p)(p), , ss
(p)(p)
Difference between mean stress Difference between mean stress
and normal stress on planeand normal stress on plane
Conventions - Stress SpaceConventions - Stress Space
Laboratory Experiments in Rock Laboratory Experiments in Rock
DeformationDeformation
Deformed marble rock cylindersDeformed marble rock cylinders
Mohr-Coulomb Law of FailureMohr-Coulomb Law of Failure
n
c
= angle of internal friction= angle of internal frictiontan tan = coefficient of internal friction [slope; m]= coefficient of internal friction [slope; m]nn = normal stress [X] = normal stress [X]cc = critical shear stress required for faulting [Y] = critical shear stress required for faulting [Y]0 0 = cohesive strength [y-intercept; b]= cohesive strength [y-intercept; b]
n
Y = mX + b
Y = mX + b
(
(
Influence of Pore Fluid PressureInfluence of Pore Fluid Pressure
Applied Stress
Effective Stress
pf
Pore fluid pressure decreases normal stresses by the fluid pressure Pore fluid pressure decreases normal stresses by the fluid pressure
amount.amount.
Rock can then fail under the Mohr-Coulomb Law.Rock can then fail under the Mohr-Coulomb Law.
Problems 1 Problems 1
& 2& 2
1. For the maximum and minimum principal stresses of 600 and 200 MPa oriented as a vertical vector and a horizontal, E-W striking vector, respectively, determine the normal and shear stress on a plane oriented N0°E, 45°E.
2. For the stress state in the problem above determine the deviatoric stress and mean stress.
Problems 1 & 2Problems 1 & 2
1. For the maximum and minimum principal stresses of 600 and 200 MPa oriented as a vertical vector and a horizontal, E-W striking vector, respectively, determine the normal and shear stress on a plane oriented N0°E, 45°E. n = 400 Mpa, s=200 MPa
2. For the stress state in the problem above determine the deviatoric stress and mean stress. Deviatoric Stress = 200 MPa, Mean Stress = 400 MPa
WW EE
11
33
Problem 3Problem 3
3. Given two planes P1 and P2 oriented where equals 90° (P1) and 45° (P2), P1 has a normal stress of 500 MPa and P2 has a normal stress of 300 MPa and a shear stress of 200 MPa, determine the magnitudes of the principal stresses, the deviatoric stress and the mean stress. Is this stress state more or less likely to produce failure as that in Problem 1?
Problem 3Problem 3
3. Given two planes P1 and P2 oriented where equals 90 (P1) and 45 (P2) degrees, P1 has a normal stress of 500 MPa and P2 has a normal stress of 300 MPa and a shear stress of 200 MPa, determine the magnitudes of the principal stresses, the deviatoric stress and the mean stress.
1 = 500 MPa and 3 = 100 MPaDeviatoric Stress = 200 MPaMean Stress = 300 MPa
Problem 3Problem 3
3. Is this stress state more or less likely to produce failure as that in Problem 1?
The stress state of Problem 3 is more likely to produce failure than in Problem 1 since the Mohr circle is closer to the failure envelope.