Introduction to Unsymmetric Bending Video Lecture - Civil Engineering (CE)

all right what up everybody welcome back
to structure fruit learning and in this
video we're going to continue our
mechanics of material sequence and talk
about unsymmetric bending and in
particular what I hope to do is describe
unsymmetric bending to you as it relates
to an asymmetrical cross section or
biaxial bending and hopefully within
that we'll talk about how to identify
the principal axes and then determine
the orientation of the neutral axis
alright so before we get all crazier to
unsymmetric bending let's think about
our original bending formula because
that's kind of the background that you
need before you get into this and this
flexure formula I affectionately called
it the most selfish equation in this
world this negative my over I and it was
based on a transversely loaded beam and
the loading was in-plane the cross
section the when it was derived had one
axis of symmetry and so that when we
looked at a cross-section if I just take
took some generic rectangular
cross-section and let's say that this
internal moment were applied positively
so that was causing compression at the
top this thing had a width B at a height
H and in particular I could take a look
at this cross section locate the
centroid determine where the neutral
axis was you can calculate the moment of
inertia so I could calculate the normal
stress or convert this moment and
determine the normal stress caused by
this bending moment on the cross section
and so here if this is the neutral axis
location we had a stress profile that
varied linearly from top to bottom and
we define this positive Y direction from
zy equals zero at the neutral axis
pointing upwards plus y this top was in
compression and the bottom was in
tension and everything was great
assuming that the cross section had at
least one axis of symmetry and the
loading was in-plane so that the bending
happened in plane so the question we're
going to try to answer here is what
happens if the cross section isn't
symmetric what if it's just some random
looking potato looking cross section or
what if the loading and the bending
don't happen in
how do I calculate the normal stress
Ismail it is my stress profile still
linear now before I get really into it
and try to answer these questions for us
you know one of the things I could do is
I could I can mess around with this
cross section drawing I could assign a
coordinate system that matches this
positive Y and at the face of the cut I
could say that my positive Y goes from
the centroid of the geometry upwards
this way the x-axis would come through
the axial direction of the book of the
beam and it'd be coming out of the
screen if you will and my positive Z
would be in this direction here towards
the left on this face of this cut and
this moment could be expressed as a
vector like this so here using the right
hand rule you put your thumb your right
thumb in the direction of that
double-headed arrow and the moment would
curl causing compression at the top and
tension at the bottom this moment would
equal em Z or better yet I could say MZ
has a magnitude of M so that's one way I
could describe this with the coordinate
system but we'll talk more about that
I'll repeat this again so that you can
see it more than once you know saying
sometimes you got to look at things more
than once just to get a feel for what
it's like the question is is is this
flexure formula is this linear stress
profile are these things still valid if
I have an asymmetrical cross section so
let's say I do instead of with this
moment applied let's say my cross
section is some random looking potato
deal some asymmetric section so here's
my cross section let's say it has some
centroid right here this is the
geometric centroid we'll define a
coordinate system from this geometric
centroid will say up Y is upwards here
we'll say Z is to the left and X is
coming out of the screen and my moment
is applied as before the only difference
is that my cross section now is some
random arbitrary thing all right so
let's consider this asymmetrical cross
section with this moment applied and
let's say that this moment is applied on
the z axis just rotating about the z
axis and the z axis also represents the
neutral
axis so that when I look at this I have
compression above the z axis or this
neutral axis and tension below it and
because Z is a neutral axis we can go
ahead and say that this stress variation
the stress here along the z axis is zero
and it's going to vary linearly as I go
away from this neutral axis so my stress
profile in 2d might look like this so
here because I've assumed Z is my
neutral axis again I know that my my
normal stress varies linearly as I go a
distance away and I have zero normal
stress at Z here and if you recall from
how the flexure formula was derived it's
essentially was based on equilibrium and
so if I had let's say the X direction
and I look at this entire cross-section
I sub forces in the X which in this case
is coming out of the page right here I
would say that the ink that if I could
integrate over all the forces here I
need the compression force resultant to
equal the tension force resultant so
that my sum of the forces in the X is
equal to zero in the way that this was
expressed was that you would take an
infinitesimal area this little da an
increment of area at some location let's
say here this would be some distance Z
in some distance Y and we would say
let's sum the forces and the force on
this little increment was D Sigma times
da if I want the entire cross section I
have to integrate and notice here on
this location that I've chosen this da
at a positive Y and a positive Z that is
in compression so technically that would
be a negative here and I would integrate
over the entire area and this would
equal zero and what this showed at least
previously was that the neutral axis has
to pass through the geometric centroid
in order for this to be true so now I'm
going to look at moments about Z and if
I take moments about Z this should equal
this M this moment resultant on the
inside or at the face of that cut and if
I say a pause
moment is this way and if I again if I
want to take if I have a force resultant
Sigma da on this element and I've got to
multiply that by the arm which for
moments about Z would be the Y arm Y
Sigma da and if I integrate let's see
this would be this would be this Y Sigma
da would be a positive moment so I'm
have a positive sign and this thing when
I solve this this should be equal to M
and this if you recall from substituting
let's say the linear relationship of the
stress profile so if I had some random
stress at some distance Y here right
here I knew that the normal stress Sigma
over Y is equal to Sigma max over C and
because this normal stress here was in
compression we gave it a sigit of sine
as in like the magnitude is a negative
compressive stress we gave it a negative
sign because the magnitude with Sigma
max but it was causing a compressive
stress at the top and so here if I
substitute for Sigma here we got this
relationship which ended up becoming the
flexure formula but now since we have an
asymmetric cross section we don't have
symmetry about the vertical axis like we
did before we've got to check moments
about the Y and that has to equal zero
because I only have single axis bending
I have bending only about one axis so I
need to make sure I have no moment
applied in the Y direction so that
moment resultant should equal zero when
I integrate this entire stress component
and so here this would be a positive Y
moment in this case here so this again
this increment of for Sigma da right
here with an arm of Z is causing a
negative moment about the y axis
integrated over the y is equal to zero
and again if I substitute this
relationship over here into there and in
order for this thing to be true I need
this term here to equal zero and this
and this term is called the product of
and if I want everything that I've said
to be true that assumption that this
moment in the Z is the neutral axis and
I have linear variation
I need this product of inertia and in
equilibrium to be true and I need this
iy z to be equal zero