diagram mohr
TRANSCRIPT
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The Mohr Stress Diagram
Edvard Munch as a young geologist
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Material in the chapter is covered
in Chapter 7 in Fossens text
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The Mohr Stress Diagram
A means by which two stresses acting on a plane of known
orientation can be plotted as the components of normal and
shear stresses (derived separately from each of the two stresses).
The Mohr circle is thus an elegant way to determine the shear and
normal stresses for a pair of stresses oriented obliquely to the plane
in question. The Mohr circle allows you to quickly read this for
planes of any orientation.
It also makes it easy to visualize mean stress and differences in
stress, or deviatoric stress and relate these to deformation.
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!n= (!1+ !3) + (!1- !3) cos 2"
2
2
!s= (
!1-!
3) sin 2"
2
Stress Equations
Normal Stress
Shear Stress
Two perpendicular stresses oriented at any
angle to a plane
Theta is the angle between the maximum stress and the pole
to the plane the stresses are acting upon.
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The Mohr Stress Diagram!n= (!1+ !3) + (!1- !3) cos 2"
22
!s= (!1- !3) sin 2"
2
Theta = angle between
The normal to the planeand the maximum
principle stress (see Fig
4.1 in Fossen)
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The Mohr Circle - (mean or average stress)
Mean Stress
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The Mohr Circle radius or deviatoric stress
Deviatoric Stress
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The Mohr Circle diameter or differential stress
Differential Stress
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Laboratory Experiments in Rock Deformation
Deformed marble rock cylinders
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Laboratory Experiments in Rock Deformation
Stress a rock sample until it fractures (or flows)
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In-class exercise, work in groups of 4, turn in write up of answers.
Given these samples, discuss how the magnitude of stresses likelyvaried relative to one another in these four experiments. Each
cylinder was deformed in a different experiment, each with itsown axial and radial load (which varied relative to one another).
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In-class problem
1. For maximum and minimum stresses of 600 and 200
mega-pascals (MPa) oriented as a vertical vector and ahorizontal, E-W striking vector (respectively), determinethe normal and shear stresses on a plane oriented North-South, 45 degrees East. It helps to first draw a block
diagram.
2. So max stress is oriented vertically and equal to 600 MPA
3. Min stress is horizontal, oriented east-west and = 200 MPa
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In-class problem
There are two ways to solve these problems.
Use the Mohr Stress Diagram or
Use the equations
(extra credit if you do both)
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Determine the normal and shear stresses on a plane oriented N-S, 45oE
Use the Mohr Stress Diagram
Maximum stress is oriented vertically and equal to 600 MPA
Minimum stress is horizontal, oriented east-west and = 200 MPa
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Use the Equations
!n= (!1+ !3) + (!1- !3) cos 2"
22
!s= (!1- !3) sin 2"
2
For the minimum and maximum principle stresses of 600 and 200
megapascals (MPa) oriented as a vertical vector and a horizontal,
E-W striking vector (respectively), determine the normal and shear
stresses on a plane oriented North-South, 45 degrees East
For maximum and minimum stresses of 600 and 200 (MPa) oriented asa vertical vector and a horizontal, E-W striking vector (respectively),
determine the normal and shear stresses on a plane oriented North-
South, 45 degrees East. It helps to first draw a block diagram.
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2 For the stress state in the previous problem, determine the differentialstress and mean stress. Start by plotting the solution for normal and
shear stresses on the Mohr Stress Diagram.
3 Discuss how a change in differential stress might affect whether arock might be more or less likely to break. It may help by arbitrarily
varying the stresses and looking at how they plot on the circle, or byimagining stress on a cube.
4 Now discuss whether increasing the mean stress would cause a rock
to break more readily. Would this be more or less likely withincreasing depth in the crust?
5 Draw the stress state where the minimum and maximum stresses are
both equal to 600 Mpa.
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Following set of slides should be copied and handed out to students for
exercises. Work in groups of 4, so for a class of 80 students, print out 20
sets.
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In-class exercise, work in groups of 4 & turn in a write up of your answers.
Problem 1.1 Given these samples, discuss how the mean stress likelyvaried relative to one another in these four samples. Each cylinder was
deformed in a different experiment, each with its own axial and radial
load (which varied relative to one another).
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In-class problem 1.2
For maximum and minimum stresses of 600 and 200 mega-pascals (MPa) oriented as a vertical vector and a horizontal, E-W striking vector (respectively), determine the normal and shearstresses on a plane oriented North-South, 45 degrees East. Ithelps to first draw a cross section.
So max stress is oriented vertically and equal to 600 MPA
Min stress is horizontal, oriented east-west and = 200 MPa
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Use the Equations
!n= (!1+ !3) + (!1- !3) cos 2"
22
!s= (!1- !3) sin 2"
2
For maximum and minimum stresses of 600 and 200 megapascals(MPa) oriented as a vertical vector and a horizontal, E-W striking
vector (respectively), determine the normal and shear stresses on a
plane oriented North-South, 45 degrees East. It helps to first draw across section.
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Or use the Mohr Circle
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For the stress state in the previous problem, determine the differentialstress and mean stress. Start by plotting the solution for normal and shear
stresses for planes of all orientations on the Mohr Stress Diagram.
Discuss how a change in differential stress might make the sample moreor less likely to break. It may help by arbitrarily varying the stresses and
looking at how they plot on the circle, or by imagining stress on a cube.
Now discuss whether increasing just the mean stress would cause a rockto break more readily (i.e. not deviatoric or differential stress).
Draw the stress state where the minimum and maximum stresses are both
equal to 600 MPa. What is the differential stress? Would you expect the
rock to deform under these conditions?
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In-class problem 2.2
1. For the stress state in the previous problem, determine thedifferential stress and mean stress. Start by plotting thesolution for normal and shear stresses for planes of allorientations on the Mohr Stress Diagram.
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In-class problem 2.4
Discuss how a change in differential stress might make thesample more or less likely to break. It may help byarbitrarily varying the stresses and looking at how theyplot on the circle, or by imagining stress on a cube.
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In-class problem 2.5
1. Now discuss whether increasing the mean stress would causea rock to break more readily (i.e. not deviatoric or differentialstress).
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In-class problem 2.6
1. Draw the stress state where the minimum and maximum
stresses are both equal to 600 Mpa. What is the differentialstress? Would you expect the rock to deform under theseconditions?