WEBVTT
00:00:10.707 --> 00:00:12.540
STUDENT: Let's talk
about the Mohr's Circle,
00:00:12.540 --> 00:00:14.430
an important tool
in materials science
00:00:14.430 --> 00:00:17.860
and mechanical engineering.
00:00:17.860 --> 00:00:20.812
What is Mohr's Circle,
and what does it do?
00:00:20.812 --> 00:00:22.270
Assume you have a
rod, and you want
00:00:22.270 --> 00:00:23.960
to break it with your hands.
00:00:23.960 --> 00:00:26.330
What can you do about it?
00:00:26.330 --> 00:00:29.830
You can pull it,
compress it, or twist it.
00:00:29.830 --> 00:00:31.590
Each of these will
induce a stress
00:00:31.590 --> 00:00:34.190
in the rod in a different way.
00:00:34.190 --> 00:00:37.390
And the Mohr's Circle
helps you analyze this.
00:00:37.390 --> 00:00:40.120
Mohr's Circle is a
graphical tool, a visual way
00:00:40.120 --> 00:00:44.530
to see how to go from one
state of stress to another.
00:00:44.530 --> 00:00:46.420
Usually you are given
a state of stress
00:00:46.420 --> 00:00:49.580
that you calculated from
using stress equations.
00:00:49.580 --> 00:00:51.520
Then you want to find
a state of stress
00:00:51.520 --> 00:00:54.700
at a new angle of
orientation, or principal
00:00:54.700 --> 00:00:58.750
stresses at that orientation,
or the maximum in-plane shear
00:00:58.750 --> 00:01:00.400
stresses.
00:01:00.400 --> 00:01:02.200
And Mohr's Circle
allows you to draw
00:01:02.200 --> 00:01:04.750
this representative
element to visualize
00:01:04.750 --> 00:01:06.010
the calculations for you.
00:01:10.920 --> 00:01:13.530
Now, let's derive
the Mohr's Circle.
00:01:13.530 --> 00:01:15.830
So now we take a look
at the piece of material
00:01:15.830 --> 00:01:20.690
that we cut from the rod
that we're trying to break.
00:01:20.690 --> 00:01:25.460
And originally we had
this coordinate system.
00:01:25.460 --> 00:01:28.340
And we have some stresses,
where the signals
00:01:28.340 --> 00:01:31.010
represent the normal
stress and the tiles
00:01:31.010 --> 00:01:33.090
represent the shear stress.
00:01:33.090 --> 00:01:35.330
So we draw this
representative pair
00:01:35.330 --> 00:01:38.060
of stresses for convenience.
00:01:43.570 --> 00:01:48.550
And then we want to
analyze the stresses
00:01:48.550 --> 00:01:51.560
of this cross-section area.
00:01:51.560 --> 00:01:56.560
So what we do is, we need to
rotate the coordinate system
00:01:56.560 --> 00:02:01.660
so that x-axis would line up
with the unit normal vector
00:02:01.660 --> 00:02:06.280
to this cross-sectional area.
00:02:06.280 --> 00:02:10.600
And then the y-axis would be
rotated to this direction,
00:02:10.600 --> 00:02:16.042
and would be lined up with the
n prime unit vector shown here.
00:02:20.470 --> 00:02:24.410
So this is done by rotating this
coordinate system by an angle
00:02:24.410 --> 00:02:28.170
theta counterclockwise.
00:02:28.170 --> 00:02:33.230
Then we can obtain the unit
normal vector in matrix form.
00:02:33.230 --> 00:02:36.350
And that is just cosine
theta and sine theta.
00:02:36.350 --> 00:02:40.490
And similarly, the n prime
vector, that's just negative
00:02:40.490 --> 00:02:43.340
sine theta and cosine theta.
00:02:43.340 --> 00:02:47.570
So we can now get
the stress tensor
00:02:47.570 --> 00:02:51.440
by taking the dot
product of the stresses
00:02:51.440 --> 00:02:55.660
in the x-direction with the
unit normal vector, and the dot
00:02:55.660 --> 00:02:59.780
project of stresses in the
y-direction with the unit
00:02:59.780 --> 00:03:01.860
normal vector.
00:03:01.860 --> 00:03:05.000
So now we can write
out the normal stress
00:03:05.000 --> 00:03:07.700
in the new coordinate
system by just
00:03:07.700 --> 00:03:11.480
taking the dot product
of the stress tensor
00:03:11.480 --> 00:03:13.850
with the unit normal vector.
00:03:13.850 --> 00:03:18.590
And similarly, we can
find the shear stress
00:03:18.590 --> 00:03:21.290
in the new coordinate
system by taking
00:03:21.290 --> 00:03:25.055
the dot product of the stress
tensor with the n prime.
00:03:27.740 --> 00:03:30.890
And we can do these
complicated calculations
00:03:30.890 --> 00:03:33.530
on Mathematica, which I
will show you right now.
00:03:36.520 --> 00:03:40.290
So I've already typed in these
vectors into Mathematica.
00:03:40.290 --> 00:03:46.440
And n represents the unit normal
vector, and np is just n prime,
00:03:46.440 --> 00:03:49.830
and t is the stress tensor.
00:03:49.830 --> 00:03:53.250
So we want to find
the normal stress
00:03:53.250 --> 00:03:55.770
in the new coordinate
system, sigma prime.
00:03:55.770 --> 00:04:00.160
And that is just
dotting t with n.
00:04:00.160 --> 00:04:04.530
And we get this
simplified version.
00:04:04.530 --> 00:04:05.385
That.
00:04:05.385 --> 00:04:07.930
So that's our sigma prime.
00:04:07.930 --> 00:04:11.430
Then we want to
find our tau prime.
00:04:11.430 --> 00:04:18.029
And that is just
t dotted with np.
00:04:18.029 --> 00:04:21.089
Again, we want to find
the simplified version.
00:04:21.089 --> 00:04:22.560
So that's that.
00:04:22.560 --> 00:04:25.710
And don't worry about
this thetas and 2 thetas.
00:04:25.710 --> 00:04:29.260
We'll use trig identities
to change them.
00:04:29.260 --> 00:04:31.480
So here's our trig identities.
00:04:31.480 --> 00:04:33.170
And we just plug them back in.
00:04:33.170 --> 00:04:39.130
And we obtain these two
equations for our new stresses.
00:04:39.130 --> 00:04:41.820
Then we move all
the terms containing
00:04:41.820 --> 00:04:44.190
2 theta to the right
side of the equations,
00:04:44.190 --> 00:04:47.220
and all the rest of the terms to
the left side of the equations.
00:04:47.220 --> 00:04:50.190
And we square both
sides of the equations
00:04:50.190 --> 00:04:53.850
to get equation 1 and
equation 2, as shown here.
00:04:57.700 --> 00:05:03.330
We add equation 1 and equation
2, and some terms cancel.
00:05:03.330 --> 00:05:08.140
Finally, we get this really
clean equation shown here.
00:05:08.140 --> 00:05:10.470
Now look closely
to this equation.
00:05:10.470 --> 00:05:12.570
What form of equation
does it look like?
00:05:15.990 --> 00:05:17.490
You might have
guessed that it looks
00:05:17.490 --> 00:05:21.970
like the equation of a circle,
and you're absolutely right.
00:05:21.970 --> 00:05:24.270
This is exactly how
Mohr's Circle is derived.
00:05:26.820 --> 00:05:29.820
We can compare the equation
with the equation of a circle,
00:05:29.820 --> 00:05:33.060
and we see that Mohr's Circle
is centered at the coordinates
00:05:33.060 --> 00:05:36.960
sigma x plus sigma y over 2, 0.
00:05:36.960 --> 00:05:41.190
And sigma x plus sigma y over
2 is the average normal stress
00:05:41.190 --> 00:05:43.570
in the old coordinate system.
00:05:43.570 --> 00:05:46.010
We can also obtain the
radius of Mohr's Circle.
00:05:50.160 --> 00:05:53.500
Now let me draw this
Mohr's Circle for you.
00:05:53.500 --> 00:05:56.230
So we go back to our
old coordinate system
00:05:56.230 --> 00:06:00.310
with some calculated stresses.
00:06:00.310 --> 00:06:06.820
And then we need to construct
this new coordinate system
00:06:06.820 --> 00:06:11.080
to draw this Mohr's Circle on.
00:06:11.080 --> 00:06:17.260
I take the positive
x-direction as the plus sigma
00:06:17.260 --> 00:06:23.890
and the negative y-direction as
plus tau, just by convention.
00:06:23.890 --> 00:06:25.630
Then, in order to
draw the circle,
00:06:25.630 --> 00:06:30.220
we take a pair of normal
stress and shear stress
00:06:30.220 --> 00:06:35.470
and we plot their values
as the coordinates
00:06:35.470 --> 00:06:38.950
in our coordinate system.
00:06:38.950 --> 00:06:41.590
So we find a dot,
and it's represented
00:06:41.590 --> 00:06:45.490
by sigma x and tau xy.
00:06:45.490 --> 00:06:49.000
Then, by our previous
equations that we derived,
00:06:49.000 --> 00:06:52.480
we can find the center
of the circle, which is
00:06:52.480 --> 00:06:58.180
located at sigma average and 0.
00:06:58.180 --> 00:07:05.890
And sigma average is just
sigma x plus sigma y over 2.
00:07:05.890 --> 00:07:11.920
Then we find its values, and
we plot that on our sigma axes.
00:07:14.630 --> 00:07:17.190
That's just our
center of the circle.
00:07:17.190 --> 00:07:20.690
Then we connect these
two dots that we drew,
00:07:20.690 --> 00:07:24.170
and that's just the
radius of the circle.
00:07:24.170 --> 00:07:27.380
Now we can draw this circle,
since we know its center
00:07:27.380 --> 00:07:31.130
and a dot on its circumference.
00:07:31.130 --> 00:07:35.570
We can now confirm the equation
for the radius of the circle
00:07:35.570 --> 00:07:39.800
by doing some geometry here.
00:07:39.800 --> 00:07:46.510
So, this length is
just sigma average.
00:07:46.510 --> 00:07:50.470
And then we have another
length that's just sigma x.
00:07:50.470 --> 00:07:52.450
And the difference
between these lengths
00:07:52.450 --> 00:07:59.540
is just sigma x
minus sigma average.
00:07:59.540 --> 00:08:02.600
Then this line right
here is simply tau xy.
00:08:06.270 --> 00:08:09.720
And by looking at
this right triangle
00:08:09.720 --> 00:08:12.060
and using Pythagoras'
Theorem, we
00:08:12.060 --> 00:08:17.460
can obtain the radius of the
circle, which is written here
00:08:17.460 --> 00:08:18.870
under the square root.
00:08:21.380 --> 00:08:24.570
Actually, it confirms
our original equation
00:08:24.570 --> 00:08:28.750
for obtaining the radius
r of Mohr's Circle.
00:08:28.750 --> 00:08:33.330
So Mohr's Circle is very useful
for visualizing the stresses
00:08:33.330 --> 00:08:36.179
on the material.
00:08:36.179 --> 00:08:39.150
So we can, just by
doing simple geometry,
00:08:39.150 --> 00:08:43.210
find the values for
its principal stresses.
00:08:43.210 --> 00:08:45.920
So in green, you can
see on the circle,
00:08:45.920 --> 00:08:48.470
there is the maximum
normal stress
00:08:48.470 --> 00:08:49.710
and minimum normal stress.
00:08:49.710 --> 00:08:53.370
And now, in red, I'm
labeling the minimum
00:08:53.370 --> 00:08:57.200
and maximum shear stresses.