1 Hello this is Professor Kitch.
Welcome to this webcast on section 9.3 which covers Mohr circle analysis
2 When you finish this module you should be
able to plot a Mohr�s circle given stresses applied to a soil element
Once you�ve properly constructed the Mohr circle, you should be able to determine both
the magnitude and orientations of the principle stress as well as the magnitude of the maximum
shear stress and orientation of the planes on which it acts.
Finally, you should be able to use the Mohr circle to compute the stresses applied to
any arbitrary plane using the pole method. 3
To explain why the Mohr circle can be important to geotechnical analysis, let�s consider
a point A below a slope as shown here. Initially, we might know the stresses acting
on horizontal and vertical surface at point A as shown here
However, it might be important to know the stress acting on some other plane at point
A. For example, we might be analyzing a potential sliding surface such at this, and want to
know the stress acting tangent to the sliding surface
If we know the stresses acting in the original x- and y- directions, can determine the stresses
acting in the new a- b- directions, so long as we also know the angle theta between these
two orientations. We will use the Mohr circle analysis to do the computation.
4 The Mohr circle is a graphical representation
of the stress at a point in a solid. It�s drawn on a graph such as this where the x-axis
represents normal stress and the y-axis represents shear stress.
In geotechnical engineering we define compression to be positive, because soils are generally
in compression. This is opposite of the sign convention in most other disciplines of engineering
which assume tension is positive�just one of the many ways in which geotechnical engineers
are special. first we will use the know state of stress
in the x- z- directions to construct the Mohr circle
The process is pretty simple. first we plot the stresses acting on the x-plane, show here
as point X. the x-coordinate represents the normal stress acting on the x-plane, sigma-z,
and the y-coordinate represents the shear stresses acting on the x-plane tau-xz. note
that the shear stress on the x-plane tends to rotate the element counter clockwise which
indicates that it is a positive shear stress. Next we plot point Z, representing the stresses
acting on the Z plane, sigma-z and tau-zx We connect the points X and Z. This line represents
a diameter of the Mohr circle with its center on the x-axis and we draw the circle.
The center of the circle represents the mean normal stress, sigma-m, which we will define
later. Using Pythagoras, we can compute the diameter
of the circle as the square root of sigma-z minus sigma-x squared plus twice tau-zx. This
represents twice the maximum shear stress at point A.
Every pair of opposite points on the circle represent stress on two orthogonal plane in
the original x-y space. so all of the these pairs of points represent
stress on planes at point A but with different orientations in x-y space. And the coordinates
of the pairs of points represent normal and shear stresses on these planes.
It�s important to point out that the Mohr circle plot is in sigma tau space, not x-
a-space. In the Mohr circle plot the horizontal axis represents normal stress and the vertical
axis represents shear stress. They are not the x and z directions in the physical space.
5 There is a relationship between the location
of points on the Mohr circle and x- z-space. That relationship is best understood by the
Pole point, or origin of planes The pole point is a special point on the Mohr
circle that can be used to determine the physical orientation of stresses acting at a point.
In its formal definition: The pole point, or origin of planes, is a
point, P, on the Mohr circle located such that any line drawn from that point will intersect
the circle at a point, D, which represents the stresses acting on a plane parallel to
the line connecting P and D Or said another way
A line drawn from a point D on the circle through the pole, P, will be parallel to plane
on which the stresses represented by point D act
These definitions of the pole point, while completely accurate, are hard to visualize.
The concept is best illustrated by an example. 6
Here is the Mohr circle we just constructed. The point X on the circle represents stresses
acting on the vertical X-plane, and the opposite point Z, represents stresses acting on the
horizontal Z-plane. If I start at X and draw a line parallel to
plane on which those stresses act, that is a vertical line, it will intersect the circle
at this point. Likewise the Z-point represents stresses acting
on a horizontal plane. If I draw a horizontal line from Z, it will intersect the circle
at the same point. That point is the Pole point or the origin
of planes. Any line drawn from P will intersect the circle
at a point which represents the shear and normal stresses acting on a plane parallel
to line draw. One what to illustrate the Pole point concept
is to extent one of the lines beyond the circle and then draw your soil element with one side
along the extended line. This soil element will now be oriented correctly in the x- z-
coordinate system As shown here, this element has one side parallel
to the line extended through the pole. And it is oriented with sigma-z acting on a horizontal
plane and sigma-x acting on a vertical plane. 7
We can use the Mohr circle to determine the magnitude of the principle stress and the
orientation of the principle planes We know from our engineering mechanics classes
that the principle planes are those where there is no shear stress. They are also the
planes where the normal stresses are at maximum and minimum values.
There are only two places on the circle where the shear stresses are zero. They are the
points where the circle intersects the horizontal axis.
The point on the right represents the major principle stress, sigma-1, with zero shear
stress. The point on the left represents the minor
principle stress, sigma-3. Again with zero shear stress.
To determine the orientation of the principle planes, we start at the pole point, P, and
draw a line through the point of maximum principle stress. The line is parallel to the plane
of maximum principle stress and forms an angle, beta, with respect to the horizontal.
If we draw a square soil element along this line it will be oriented with the principle
planes and the major on minor principle stresses will act in the directions shown.
If we want to measure the direction, we simply place a protractor on the Mohr circle plot
and read off the angle, which is in this case 18 degrees.
Since the Mohr circle is a graphical method, it is important to understand that it must
be drawn to scale. Today it is easy to draw a Mohr circle with CAD software
8 We can also use the Mohr circle to find the
shear and normal stresses on any arbitrary plane
in this case we wanted to know the stresses on the A and B planes.
To find these we simple draw a line parallel to one of the planes we want, in this case
the A plane Which forms and angle of 31 degrees to the
horizontal We then draw a line from the pole point at
the same angle That line will intersect the circle at point
A and the stresses at point A will represent
those acting on the A plane similarly, if we draw a line from the Pole
normal to the first line, it will be parallel to the B-plane and intersect the circle at
B which will represent the stresses acting on the B-plane.
A line connecting A and B will be a diameter of the circle and pass through the center,
indicating that these two points represent stresses on orthogonal planes.
9 You can also use formulas to compute the principle
stresses and the stresses on any arbitrary plane
These formulas come from your textbooks as show by the equation numbers on this slide.
Both the formulas and the Mohr circle graphical solution should give the same results
10 Finally, I�d like to briefly discuss the
concept of a state of stress and stress invariants Here we see the point A under our slope and
the two different orientations that we have been discussing. These two orientations are
also shown on this Mohr circle diagram. Now the soil at point A is unconcerned about
our choice of axis system. It is under a certain state of stress�a combination of normal
and shear stress. It doesn�t know the x- z-system from the a- b-system. Those are arbitrary
orientations chosen by us. The soil doesn�t know or care what axis system we use to define
shear and normal stresses. It is under a single state of stress that is the same regardless
of our chosen axis system. It follows then, that we should be able to
describe the state of stress of the soil at A by some values that don�t change as we
change axis systems. We call these special values stress invariants,
because they are invariant to our selection of axis system.
so let�s think for a moment. What�s not changing in our Mohr circle as we rotate from
the x- z-axis system to the a- b-axis system? If we think about it, it obvious that the
Mohr circle itself isn�t changing. The points we�re selecting on the circle are changing,
but the circle itself isn�t changing. That�s what�s invariant, the Mohr circle.
Specifically that means the location of the center of the circle doesn�t change. This
point represents the mean normal stress, sigma-m and is equal to the average of the principle
stresses. This is the average compressive stress that the soil is subject to.
The other value that�s not changing is the radius of the circle. This represents the
maximum shear stress that the soil is subject to. And is equal to one half of the difference
between the major and minor principle stress. The quantity is also called the deviator or
deviatoric stress. So these are our stress invariants, the mean
normal stress and the deviatoric stress. No matter what location on the Mohr circle we
pick, of what axis system we use, these don�t change.
11 You should now be able to draw a Mohr circle
and compute stresses on any planes. Here�s a practice problem you should complete before
class. Given the stresses on the x- and z-planes
shown Draw the Mohr circle for this state of stress
and then determine the following values. You should use both the Mohr circle and equations
given and check the results to you get from each method. You can also your solution against
Example 9.1 in your text. Do this example before class
That�s all for this webcast.