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NICHOLAS BURNAND: Hello and
welcome to this short video

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on Mohr's circle.

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The goal of this
video is to show you

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how Mohr's circle works, how
they could be useful for you,

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and to make some interesting
observations as well.

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So first of all, Mohr's
circle is a graphical way

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to find the principle
normal stresses

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and the maximal shear stress
of a given stress state, which

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could be the one right
here, where you have

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some normal forces acting
on sigma x, y, and z,

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and also some different
shear stresses--

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the tau's you have here.

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Your stress state could also
be shown using a tensor--

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stress tensor-- which is shown
right here by this symmetric 3

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by 3 matrix.

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It has to be
symmetric because we

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don't want to have any movement
acting on our stress element

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to prevent any rotation.

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And also of course, we're
using an isotropic material.

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So I'm going to start with
the 2D Mohr's circle--

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the simplest one.

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So now obviously, my
stress element is a square.

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We have sigma x and sigma y
as normal forces acting on it.

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And the shear stress tau x,y.

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So now, let's move to
the Mohr's circle itself

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and I'll explain to
you how to draw it.

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So first of all, I have
to put my two stresses

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on my graph right here.

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I have normal axis right here
and a shear axis right here.

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So I'm going to take sigma
x, which is [INAUDIBLE] here,

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and tau x,y, which is going to
give me my first point right

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here.

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And then sigma y and tau y,x--

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which is the
opposite of tau x,y--

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and this is going to give me
my second point right here.

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Now, I'm going to say that this
is the diameter of my Mohr's

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circle.

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And this allows me to draw the
outermost circle right here,

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as you can see.

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Now, I see that this
Mohr's circle crosses

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the normal axis on two points--

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the two endpoints that
you see right here.

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And B, these are my
principal normal stresses

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acting on my stress element.

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And I can actually also
get on the Mohr's circle,

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the angle how much I have
to turn my stress element

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to get this principal
normal stresses.

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And it's important to
remember that the angle I see

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on the Mohr's circle--
so this angle here--

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is actually twice
the real angle I

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have to turn my stress element
by, which is shown right here.

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And then I have a
blue point here,

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which is my maximal
shear stress.

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So its maximal stresses,
as you can see,

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is the radius of
my Mohr's circle.

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And again, I have an angle
of maximal shear stress,

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which is displayed right here.

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And again, the angle
of my Mohr's circle

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is twice the real angle.

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Now, some interesting
observations

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we can do right here is that
even if I don't apply any shear

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stress to my stress
element, I will still

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have some maximal shear stress.

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And as you can see, the angle
of the maximal shear stress

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is, well, 90 degree on my Mohr's
circle, but in reality, 45

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degrees.

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And as you probably already
know, if I'm doing for example

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a tensile stress, then
my maximum shear stress

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is going to be on
a 45 degrees plane.

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Now, the only way to avoid
having any shear stresses

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in my stress element
is actually to have

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sigma x being equal to sigma y.

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And that's the only way I
can avoid any shear stresses.

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Now, the other way to
get my principal stresses

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would be to get the eigenvalues
of my two by two stress tensor.

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And that's what we're going
to see later for the 3D case.

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So in 3D now, what
I have to do is

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get this principal normal
stresses by, as I've told you,

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find the eigenvalues
of my stress

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tensor, which is right here.

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That's the random
one I've chosen.

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It's symmetric,
as I've explained

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to you in the beginning.

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And finding the
eigenvalues gives me

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my actual principal
normal stresses

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sigma 1, sigma 2, and sigma 3.

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So now, let's move to
the 3D Mohr's circles.

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So it's the same
principle, but now I

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have three principal
stresses instead of two.

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So I also have three Mohr's
circle instead of one

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because I have one Mohr's circle
between sigma 1 and sigma 2,

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another Mohr's circle
between sigma 2

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and sigma 3, and then bigger
one between sigma 1 and sigma 3.

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So now, the orange
zone right here--

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same at the bottom--

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is the allowed pairs
of stresses sigma n--

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the principal normal force-- and
tau, which is my shear force.

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And the only [INAUDIBLE] pairs
of normal and shear stresses

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are in this two orange
zones because it

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has to be inside the bigger
circle and outside of the two

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smaller circles.

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Now, the observation
we can do here

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is that first of all,
the maximal shear stress

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only depends on the difference
between the smallest

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and the largest
principal stresses.

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So as you can see, if I
move my sigma 2, which

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is the middle one, well,
my maximal shear stress

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doesn't change at all.

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And the other
observation we can make

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is that if I equal sigma 3
to 0, as it is right now,

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and this would be the same
actually-- or almost the same

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if we look in 2D, as the
example I've shown you before.

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But actually, the fact that I'm
now in 3D and that even if I

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don't apply a force
on the z-axis--

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any normal force--
well, this still

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has an influence on the maximal
shear stress because in 2D,

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my maximal shear stress
would lie right here

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at the top of the first
gray Mohr's circle.

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But now, because I am in
3D, my maximal shear stress

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is actually a bit higher.

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And now, actually another way
to get this principal normal

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stresses using Mohr's circles--

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so instead of as
I've done it here,

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get the eigenvalues
of my stress tensor,

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I can actually use projections.

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So I have my 3D
stress tensor here.

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And what I'm going to do is
project it all along my x,y,

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and z-axis.

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So that's for example,
projection on the z-axis.

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And what I get is
a 2D stress tensor.

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So doing this, what
I get again of course

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is my 2D Mohr's circle,
as we've seen it

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in the beginning of this video.

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And that's for example, the
projection along the x-axis.

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And I again, get my principal
normal stresses, two endpoints,

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and my maximal shear stress
blue point right here.

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And what's more
interesting is to see

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for example, the angle of
principal stresses, which

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is minus 19 right
here and to half

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the angle between the red
line here and the normal axis.

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And doing three different
projections along the x,y,

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and z-axis, i get
three different angles.

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So here, that's another one
and that's the third one.

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And now if I actually
apply this angle,

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so if I actually rotate
my stress element

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by minus 19.9539 degrees
around the x-axis

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and do the same around
the y-axis and the same

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again around the
z-axis, then I actually

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get my new set of
coordinates where

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I get my principal normal
stresses acting on my stress

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element.

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So I would get again,
these three values up here.

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Just by applying
three rotation, I've

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found by projecting
my stress state

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on my three different axis.

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So that's it.

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I hope you enjoyed the video
and actually learned something

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on Mohr's circles.

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Thanks for watching.