Torsion Formula Video Lecture - Civil Engineering (CE)

all right what's up everybody welcome
back
got my afternoon cup of coffee and about
to make this video on torsion and
hopefully by the time this video is over
you will be able to describe the
deformation of a circular shaft with an
internal torque in it and then you'll
know where the torsion formula comes
from which will be another very
important equation that for your
introductory course the mechanics of
materials and and so we'll start off
maybe you've all experienced torsion in
some way or another maybe twisting your
back right you've done it some you've
created an internal torque in your
spinal cord or spinal column so that you
can have some muscles that are twisting
right or taking a piece of chalk and
twisted it till failure and seeing the
plane of failure that occurs but in any
case let's take a look at here a solid
circular shaft that's subjected to
twisting or this external torque here or
this actually let's make it seem like if
I made a cut at this distance L I would
see this internal torque on the inside
of that that would be what's going on on
the inside and as always this double
arrow indicates the direction in which
your thumb should go and then the the
rest of your fingers will curl around in
that direction right there okay so here
that's the indicating the direction of
the torque and and just to make sure
just in case there are any questions
torque is really just another moment
it's a moment that causes twist moment
causing a twisting action causing twist
right there and you know it's no
different than any other moment except
you know the moments that you've looked
at in 2d are about the out-of-plane axis
in 2d and here in this case we're
talking about a moment about the x axis
which happens to be running through the
centroid of the circular shaft and so
when we apply an external torque the
things you want to know is that the the
circular cross-sections the circular
section so here let me see if I can draw
a circular section right here BAM just
using my little actually have a little
tool that I use or it's actually not
it's not a tool it's just inside of a
tape but here BAM
to draw my
circular sections right here these
circular sections when you twist
something circles remain secure so
there's no deformation remain circular
in terms of the cross-section there's no
warping out of plane or anything like
that but the thing that does change the
thing that does is that these lines that
are longitudinal at the start line so
here let me draw a longitudinal line
that's parallel to all this business BAM
that these longitudinal lines twist or
they deform so here what would happen is
if that line right there if that gray
line was my first line or before at my
undeformed before bloating then I apply
a load to it and I end up with right
here some sorts whisking action so this
gray line turns and rotates and twists
right there and as you can imagine so
what what the actual deformation here
because we have a small deformation
assumption the the deformation that we
would look at or measure is this angle
change right here that angle change and
as you may have noticed from the strain
videos an angle change angle change is
associated with ding ding ding shear
strain which means that this torque
internal torque induces shear stress yes
shear stress and we will write here this
gamma in fact this angle right there is
a shear strain that's going on and and
another thing to note is that if I look
at the cross section of this so if I
look at my cross section again let's
start with this gray being my undeformed
line right here oh gotta get that
through the middle right there okay and
if that gray right there oh okay that
gray
line here right through the center of my
cross-section but that dot back there
let's say that's the middle right there
that deforms to here this position right
here man and this angle change right
here which one I use green just to let
you know that's different than the shear
strain this angle change right here
we'll call that fee that fee right here
is called the angle of twist and in
particular in this one this is the angle
of twist at L if this you know this is
zero right here along my x fill my x
axis this is my angle of twist at L and
if you you know you can tell from the
side view of this this line right here
this angle twist decreases as I get
closer and closer to my fixed condition
right here so it's a function of this
function of the length or where we are
along the length of my circular shaft
using our reasoning of the angle change
here we said that this is associated
with shear strain and by you know
knowing that a shear strain we know that
this twisting induces a shear stress and
that shear stress is on the surface of
the circular shaft here and so if I look
at let's say here let's say again from
this line right here
I see let me draw for ourselves here let
me draw this line all the way across
this blue line right here BAM like that
right there
okay and so we notice that we have no
deformation right here at the center and
then we as we go out our deformation but
the amount of or distance from the
original to the the deform line if you
will
increases which would tell us that hey
you know what since our deformation
right here compared to this line versus
this line is increasing linearly I would
imagine that the stress also increases
linearly and and the fact that's true
okay and so here if I have for instance
where I have the most deformation I
would anticipate
on this little if I could zoom in and I
see a little rectangular area on the
surface here this will experience this
location right here will experience the
largest shear stress this will be my
towel Max and then and this would be
some incremental area so I could say in
between here along the length or along
the radius of this right here I have
some shear stress which I will denote as
tau acting on this little square area
which I will call da right here and here
if this if the radius of my shaft here
let me take the radius here let's say
the radius of the shaft we'll call that
a lot of your books use the symbol C so
Casey is the radius of the shaft right
here okay maybe you like to use are not
Rho but are okay and will say Rho is the
distance to some specific location some
specific don't all right here we go row
right there BAM that's row and I don't
even but hopefully you can okay zoom in
on your monitor or whatever right and in
here you've got this this relationship
going on so we know that this the stress
here because the deformation or the ink
the deformation going along the radius
from here or zero deformation at the
center line to the outer edge where it's
the maximum deformation we know that's
varying linearly so we know we have this
intuition or feeling that that hey the
shear stress along the face here is also
probably varying linearly okay so let's
take a closer look at this cross section
here and so if I go there I kind of I
just redrew it here if I look down on
this right here if i zoom in again just
because here at the outer edges where I
experienced my largest affirmation I
anticipate having here and I'll use a
towel for stress or use red right here
the shear stress on this little
incremental area right here I anticipate
it's saying it's at the very outer edge
towel max right here at the very outer
edge a distance see from the from the
center here a radial distance see I
expect the towel max or Maxima shear
stress and I anticipate this all the way
around the entire cross-section okay on
the outer edge is my maximum shear
stress and then here if I take an
arbitrary location or arbitrary radial
distance from the center here I know
that I'm going to have some sort of
linear relationship okay because my
deformation is also linear I know my
stress is also linear because Hookes law
is linear okay and so here I have BAM
some stress some shear stress on this
increment area some arbitrary radial
distance row away so here I'll call that
tau and I know that the two I can relate
the two by similar triangles by similar
triangles this relationship would be tau
max over C is equal to the shear stress
at any arbitrary location divided by the
radial distance to that location from
the center and it doesn't matter which
line on the cross section I look at so
even if I were looking at some arbitrary
line at this angle right here my stress
profile my shear stress profile is the
same right because it's also again no
matter what line I'm looking at I have a
linear distribution or linear
deformation going from the center out
towards the outer radius and I'm going
to experience here we'll call this tau
max and then as well like this and it
doesn't matter what line I'm going to
have if this is my some location right
here and this is my distance Rho as well
this would be again an arbitrary
distance row away from my Center right
here intermediate if you will acting on
this da my incremental area here and we
know that the the resultant of all the
shear stresses that if we can if you
will the resultant of the shear stresses
applied over this area
the torque that we sum up here has to
equal to that internal applied torque
that we that internal torque that exists
at this cross-section so what that means
is that the torque this internal torque
has to be equal to the sum of all the
shear stresses da times Rho because we
have this is a torque right force times
it so tau times da it's going to give us
a force and then we need an arm or
distance also have this Rho and then if
I and then this da just to make sure
it's really a double integral in maybe
like polar coordinates or something that
that you have to to resolve if you want
to calculate this without the help of
some other things that we're going to
have but here let's for instance let's
substitute now this definition of the
shear stress tau which is tau is equal
to Rho times tau max divided by C and if
I substitute that for tau over here then
I'll get that the torque is equal to
let's see I'll group the rows together
so I'll say tau max over C times Rho
squared da
and in this case tau max is a specific
value at the very end or at the outer
radius or outer edge of my shaft and see
it's also the outer edge it's just the
radius these are all constants and so
this becomes this tau max over C is Rho
squared da and this right here probably
looks like a moment of inertia
yeah from statics anyway right here this
is also it the polar moment of inertia
because this right here we're talking
about circular sections and and radial
and polar coordinates so this is our
polar moment of inertia which is defined
as the resistance to twisting of my of
my of my cross sectional area the
resistance to twist okay just like a
moment of inertia area moment of inertia
was a resistance to bending this polar
moment of inertia is a resistance to
twisting and the larger my polar moment
of inertia the harder it is to twist
something and so this ends up in our
very popular result T
and the symbol use for this is J right
here so this ends up with our t max over
C times J the polar moment of inertia
and here if this is often rearranged so
that the maximum shear stress in a
circular shaft with an internal torque T
is equal to times the radius divided by
the polar moment of inertia so this is
specifically for the maximum and at any
other location tau the shear stress at
some location Rho a radial distance away
from the center divided by the polar
moment of inertia gives us our shear
stress and this is the shear stress due
to torsion or an internal torque okay I
have one word left for you done or in
the famous words I guess I have more
than one word the famous words of our of
our former California Governor I'll be
back later later