Beam Bending the Flexure Formula Video Lecture - Civil Engineering (CE)

okay team welcome back we're going to
talk today about the beam bending
flexure formula we're going to get a new
formula our formula sheet today
woohoo okay so what we're talking today
about is beam bending okay so if I have
my beam here's my beam again right like
nerf Doodle right and as I start bending
this beam what's happening to that beam
well if you can see the top of my beam I
don't know if you can see that or not
there's a little wrinkles on iyr there
wrinkles on the top because as I bend
this the top fibers of this beam are in
compression right whereas the bottom
ones down here are being stretched their
intention so somewhere in between the
top and the bottom there has to be
something one has to switch from
compression to tension right
what does then and what's called the
neutral action okay or the centroidal
axis of the part here's a beam I have
that I have a cross-section of right the
beam goes into the board like that and
I'm taking a cross-section on the end
though look at that cross-sectional view
instead of a t-shape okay so somewhere
on this thing this being is what's
called the neutral axis right the
neutral axis so if it's if it's
compression to top and tension at the
bottom let's go up the neutral axis
nothing so I get in your car put it in
neutral where do you go nowhere okay so
the equation for this this is what's
going to happen to your beam bending is
this Sigma equals MC over I how that is
a big time equation we're used about
million times in the end of the semester
so what does that mean what kind of
stress is Sigma remember that's normal
stress that's the stress fatigue makes
us be longer or shorter right a
compressive or tensile stress okay well
what's happening on the top we'll have
compression right and I'm being squeezed
together to the top there's stress do
those fibers being in compression on the
bottom there's stress
do those fibers being pulled apart in
tension okay so that's that's where this
comes in so on the top of this beam I'm
going to have compression I'm going to
have attention from the compression and
on the bottom from pinches getting
longer and remember this that
compression is given half Sigma is going
to have a negative side and then tension
what attention start with a positive
side right Sigma is going to be positive
right so those things that are in
tension or getting longer positive
stress if it's getting shorts and
compression negative stress okay so how
we solved it kind of problems okay we
remember always think what is ill Emma's
bending moment we don't want to solve
that right if we have to we can draw a
shear moment diagram we can tell you
what M is at any point of a beam on this
problem they say that you know here's
the beam they have this moment like that
right something is twisting it's 4 foot
tips so that that moment is twisting it
like so okay so we already know the
bending moment for this one would it see
well that is the distance from the
neutral axis to the point I'm interested
in and then this point they say find max
tension so where's maximum going to
happen well it's 0 here and it starts
increasing increasing increasing so the
max is going to be on the very outside
of the park so this is max or that is
max right this is max for compression
this is max for tension okay
so C is a distance okay and then of
course I we learned I in the statics
didn't we I is the area moment of
inertia right the boy okay and what was
the equation for that remember we had to
the parallel axis theorem the equation
was I equals 1/12 b h-cubed plus a d
squares scuse me so that was the
equation to find I for a rectangular
shape we're going to have to do that to
here okay so this is a calculated thing
we can just look it up this is given
that's kind of given except we don't
have it here so you know that neutral
axis words that occur it also occurs it
would call the centroidal axis so step
one for this problem we get by the
centroid of this w so we're going back
to statics here and always start from
the bottom apart I think that C's is if
you just use the bottom of the part of
the data and find the centroid from the
bottom so how far is it from the bottom
of the park up to there right and that
looks like it's in the Y direction so we
need to find y bar for the spars okay
how do we do that
we're going back to status here we go
we're talking about a teen cross section
this thing started point five inches
typical what is typical mean that means
this point by their one by their point
by there the whole fitness is point five
everywhere so what do you say we divide
this into oh I don't know four shapes
right let's define it here
and divide it there and we'll divide it
across there okay so we'll call you
shake one shape to shape three and shape
for okay so if you remember from
centroids we're going to use our little
table method and wait here's the same
lesbaz shape one two three four right
and then on the top up here or do I have
I have Y a my egg okay so all we have to
do is fill in this little table here
we'll be good to go
okay because what is what is the
equation for y bar y bar is equal to the
sum of the Y a divided by the sum of the
areas right okay make it so black pants
work better
there we go so let's giving Phil this is
where is what I've learned for shape
number one okay now let's just call this
zero down here let's just think of this
as our x-axis okay so this piece shape
number one is ten inches tall whereas Y
bar is right in the middle of that so Y
bar is at five inches and the area of
that is a half-inch wide by 10 inches
long so that's five inches squared right
okay and then shape number two where
shape number two T's way over there
where is y bar well this is three so the
whole thing is three and a half isn't it
this is three point five okay and how
far is it from the bottom to the top of
the part
well that's 10 plus 1/2 so that's ten
point five right so where is y bar for
this guy right here
well you got to go to ten point five and
then come back half a three point five
right so here we go when you look a
little a mistake on three point five
divided by two equals one point seven
five so ten point five minus one point
seven five is eight point seven five so
Joe guide and this one's also eight
point seven five now the area for those
is 3 point 5 divided by 2 right or three
point five times 1/2 Y so that's one
point seven five and finally shape
number four words the centroid is this
guy from the bottom it's ten plus 1/2 of
1/2 right so 10 plus 0.25 which is ten
point two five and the area of that guy
is three times point five under the
1/2 okay so we need to fill in one eye
times a so that's 25 look this one eight
point seven five times one point seven
five equals fifteen one three one and
this one is same and this one ten point
two five times one point five equals
fifteen point three seven five okay
think rolls there so we need to some of
the A's
well that's going to be right there
isn't it some of the A's is five plus
one point seven five plus 1.75 plus one
point five five equals ten and then we
need to some of the Y that's twenty five
plus fifteen point three one two five
times two plus fifteen point three seven
five is 71 okay so what is y bar it's 71
divided by 10 which is seven point one
and injuring inches are they so Y bar is
seven point one inches that neutral axis
is seven point one inches from the
bottom of the park which means that from
the top of the port right the whole
thing was ten point five so ten point
five - well no clear ten point five -
seven point one is three point four so
this right here is three point four okay
over here
seven two point one
that's step one right we don't even know
anything about that we know where the
neutral axis is we had to find the
neutral axis matter of fact for tensile
stress right the part that's in tension
which is the bottom what is see going to
be what is the distance 7.14 the neutral
axis to the bottom of 7.14 compressive
stress which is a top what see going to
be from the neutral X to the top that C
is going to be three point four so we
did kind of find something for our
equation there now the moment is already
given we're going to have to be careful
because that's in foot tips and all our
stuff is in inches so that's another
trick there I think the last thing we
need to do it let's calculate I put
definitely I think we're going to be
there okay
so I equals 1/12 bh cube plus ayby
squares so since we have this in four
parts let's just do this in four parts
okay we'll find I in four parts okay so
let's do these number one okay piece
number one so my is 1/12 the base what's
the base point five what's the height 10
plus the area of piece number one this
is nice already have that plus five
times here's where most people mess up
is what is B remember the D is the
distance from the neutral axis of the
whole thing to the centroid of the part
that I'm working with so I'm working
with this piece here so this piece here
is in the middle of this 10 right so
ignore the outside of four seven point
one and I come back five so that means
that the D here is two point one okay
that's piece number one okay so let's do
piece number 2 so plus 1/12
what's the based on piece number two
this little guy the base half
hi 3.5 plus the area area is 1.75 and
then what is D forever D is if I go from
the neutral axis to the outside of the
part that's three point four and then
come back half of that three point five
so 3 point 5 divided by 2 equals no 3
point 5 divided by two is one point
seven five and so three point four minus
one point seven five equals one point
six five okay now that's piece number
two there these are wonders piece number
two is piece number three the same is -
yep so let's just do this
let's two times that let's put a tool in
there times 2 and then we have one more
plug last piece is piece number four
through the top so we've got one 12 the
base is what three the height this time
is 0.5 b h-cubed b h-cubed eh open up a
cube both they're doing cubes that would
be bad plus the area which is 1.5 times
d well D is 3 point 4 minus 0.25 right
so 3 point 4 minus 0.25 equals three
point one five okay that's squared so
now we're going to do is put those in
our calculator okay and this is going to
give us what is it's going to give us
this is going to give us inches to the
four okay so I'm gonna put these in my
calculator piece one two one two three
right so here we go let's see if I can
do this point five times ten n cubed
equals divided by 12 equals 41.67 plus 5
times 2.1 squares equals ok so the first
part is sixty three point seven two
that's what that is number this piece
here this chunk here okay and that's
five times three point five cubes equals
divided by 12 equals plus one point
seven five times one point six five
squared equals six point five five and
then I'm going to multiply all that
times two left 13.10 that will is last
piece down here three times 0.5 cubed
equals divided by 12 equals plus one
point five times three point one five
squared equals fourteen point nine two
so I'll add all those together a plus
thirteen point one plus sixty three
point seven two equals ninety one point
seven four ninety one point seven four
inches to the fourth and that is what I
about bending around the x-axis is equal
to okay so that has a lot of work for
the moment of inertia okay this should
be a sentence review if you don't
understand how to do this go back to
static the very last three four videos
in my Fed series and review that stuff
okay so here we go
Sigma and upper tension is equal to four
tension tensions where the top of the
bottom the bottom right it equals four
I'm do this equation here in C over on
so for and that's foot kiss
remember Kip is a thousand pounds but we
don't want that in feet do we so one
foot is 12 inches right that way our
feet will get canceled out and what is
that going to give us x what M times C
and what's the distance C is for the
bottom part retention so C is going to
be seven point one inches divided by the
moment of inertia which is ninety one
point seven four and that's inches to
the fourth right but two of those are
going to cancel the weather with one of
them and there goes the other one
so let's go lettuce with Kip's per
square inch or K and I write so 4 times
12 times seven point one equals divided
by ninety one point seven four equals
three point seven one so Sigma in
tension equals three point seven one KS
i okay we do the same thing for
compression that's the top of the part
right where our wrinkles work so sigma
compression was going to be different
it's going to be four times 12 times
what about C it's C different yeah for
the top apart is 3.4 divided by same I
ninety one point seven four so when
compression I have 48 times three point
four equals divided by 91.7 41.78 so
Sigma for compression equals one point
seven eight KS I does this surprised me
the compression is less than tension it
shouldn't because you live in a change
was the distance and this is farther
away from the neutral axis than that is
right so it's going to be bigger on the
ball that's all there is to it okay so
we're going to get a new equation
our question sheet I'm going to add that
one over here and let's see if it gets
harder okay hang on