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

Repeated Failure ExperimentsRepeated Failure Experiments

Stress Requirements for Rock Stress Requirements for Rock

FailureFailure

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

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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.

ReferencesReferences

Slide 15

Twiss, R. J. and E. M. Moores, Structural Geology, W. H. Freeman & Co., New York, 532 p., 1992.

Slides 16-19Davis. G. H. and S. J. Reynolds, Structural Geology of Rocks and Regions, 2nd ed., John Wiley & Sons, New York, 776 p., 1996.