Design Moment Strength Calculation of a Reinforced Concrete T-beam Video Lecture - Civil Engineering (CE)

hi what's up everyone welcome back and
in this video we're going to do some
more reinforced concrete beam analysis
and in particular we're going to do an
example problem where we have to
calculate the design flexural strength
of a reinforced concrete tea beam that's
given here and I have some dimensions
the flange width is 36 inches the total
height of the section is 30 inches and
the depth to steel from the extreme
compression side is 27 inches and what
this cross-section implies is that there
is an applied or an ultimate moment here
mu causing compression at the top
tension at the bottom and the material
properties I have FC prime of 4 ksi F Y
or grade 60 steel with F y equals 60 ksi
and I have three number ten bars to
represent my area of steel in that area
of Steel this a s is just three times
one point two seven inches squared which
is going to be three point eight one
inches squared from the given
information what I know is or released
from the drawing here I know that the
width of the flange is 36 inches the
width of the web here BW is 16 inches in
the depth to steel from the extreme
compression fiber to the center of the
steel here is 27 inches my approach to
this problem is going to be the same as
if I were analyzing any cross-section
for the nominal or the nominal moment
strength and the design flexural
strength it's going to involve the
following step and this is how and this
is how we're going to do it we're going
to draw the strain and stress profiles
first at ultimate and I look at that as
drawing a Freebody diagram or schematic
and if you can draw this you can you
don't have to memorize any equations for
this and for any beam analysis really
and the next thing we do is use apply
force equilibrium to calculate the
neutral axis depth then verify some of
the assumptions that we used in this
force equilibrium step and then
determine the strength reduction factor
this fee variable here so that we can
calculate the design moment strain then
we'll will calculate the nominal moment
using moment equilibrium about some
point on the cross section and then just
multiply that number M sub n by fee to
get the design moment strength so the
first thing that we want to do is draw
the strain and stress pro
files at ultimate for the reinforced
concrete section and so for the tea beam
I've kind of read on it right here and
the thing I like to do is draw a
horizontal line to the top and the
bottom and and have this vertical lines
here for the first vertical line will
represent my strain profile and then the
second one will be my stress kind of
slash force profile but the thing to
remember here is that you are drawing
this the strain and stress profile at
ultimate and what that means this word
ultimate implies a specific strain value
for the cup for the compressive strain
and in this case the concrete is
crushing and so here that means the
compressive strain in concrete reaches
the ultimate compressive strain and if
you're using the American concrete
Institute standard this is point zero
zero three in compression and here I
will have a linear strain profile and
that linear strain profile is always
true and here we're going when we draw
this line now for the strain profile
we're going to assume for now that the
neutral axis more like the depth of the
equivalent stress block that we're going
to use for the stress profile is within
the flange so what we're going to do is
assume that the depth of the equivalent
stress block is within the flange and
here is what the strain profile might
look like like that here and this where
is zero is equal to is where my neutral
axis location is here this would be C
and a and what the depth and my steel
strain would be here this epsilon s this
would be my strain in the tensile steel
right here and that's something that we
need to figure out and the stress
profile looks like this if we're using
the equivalent stress block here or the
Whitney stress block I'm going to have a
stress block that goes here down to a
distance a here and here this has a
stress intensity of 0.85 FC Prime and my
steel represented by the layer of steel
location here will have a stress over
here of FS ok and you know what this
debt a means the depth of the equivalent
stress block is at my that my neutral
axis is shallow enough if you will that
my equivalent stress block that stays
within the flange region right here and
so this I'm assuming right now is my
area in compression
this has a resultant of I'll call that C
sub C a resultant for C sub C and the
steel has a resultant force I'll call
that T sub B s in tension here and there
will be a centroid of this compression
block which will be this distance here
we'll call that Y bar and the distance
between the two or the couple really the
T the tension and the compression force
is I'll call Z and that's just another
symbol for the arm right there and
really this comes out to D minus y bar
here I would like to emphasize to you
that this drawing here assumes that the
depth of the equivalent stress block a
is less than the height of the flange or
this four inches and so we have this
nice rectangular compression zone if
this turns out to be not true we're
going to have to redo the formulation
and probably modify this drawing a
little bit to reflect reality or reflect
what's really going on all right so the
next thing I want to do is utilize my
force equilibrium equation to calculate
or determine the neutral axis step the
CNA the way I'm going to do that is just
take some of the forces the horizontal
here equal to zero and that just tells
me that t s is equal to C sub C and if I
assume assume that the strain in my
tension steel has yielded then my
relationship here I can substitute and I
can just have that this ts is a sfy is
equal to 0.85 times FC prime times the
area in compression and so here I can
solve for the area and compression and
this area and compression is going to be
equal to a SF why go over 0.85 FC Prime
and if I substitute some numbers that's
just going to tell me I have what three
point eight one inches squared times 60
ksi divided by 0.85
times five ksi and that tells me that
the area in compression is 53.78
eight inches squared and as long as this
area and compression that I need is less
than the area of the flange which I'll
put in purple here here is the geometry
of the flange of this t beam as long as
that's true then this assumption is true
right here and so if I - if I were to
calculate the area of the flange area
flange that would just be 36 times four
inches which is 144 inches squared and
that means that AC is less than a flange
so my stress or my Aryan compression is
this red zone it still stays as a
rectangular block and so I can say
therefore because the area of the
compression is less than the area of the
flange the area and compression I can
now say is AC is equal to a times B
where a is the depth of the Whitney
stress block B is the width of the
flange and I can solve for a here
directly now so going over here to the
left here and let me scroll down a
little bit
this a is equal to AC / B which is 53.78
8 inches squared divided by the flange
with 36 inches and that tells me a the
depth of the equivalent stress block is
one point four nine four inches and then
I can back calculate because I know a is
equal to beta 1 times CNA I can solve
for CNA the neutral axis depth which is
going to be a over beta 1 and this will
be one point four nine four inches
divided by beta one and in this case
here because I have FC prime equal to
five ksi beta 1 is equal to zero point
eight zero and this gives me a neutral
axis depth of one point eight six eight
inches now the next thing I want to do
is verify my assumptions you know I went
ahead and I verify that the depth of the
equivalent stress block is less than
this four inches which was the height of
the flange but I have this other
assumption here that I have yet to
verify and that's that detect that's the
tension steel has yielded and so in
order for me to do that I can just go
ahead and use similar triangles from the
strain profile which if you look over
here to calculate this strain right here
I would just have that this distance
right here is d minus CNA and by using
similar triangles I can come up a
relationship for the strain here as a
function of the geometry and the strain
in compression so that relationship by
similar triangles it's epsilon s over D
minus C na is equal to epsilon C over C
na and this whole thing just becomes
that epsilon s is d minus C na over c na
times zero point zero zero three and if
I substitute some numbers I have 27
inches minus one point eight six eight
inches divided by one point eight six
eight inches times zero point zero zero
three and that tells me epsilon s is e
- 0.04 now this number is much greater
than epsilon Y and this number is
greater than 0.005 which according to
the ACI code means this is a tension
controlled beam and by that fee is equal
to 0.9 and this is an ACI chapter 9 this
is from ACI chapter 9 if you're
interested in looking that up and you'll
see there a diagram that looks like this
and there will be a fee value and
epsilon T which is just another way to
say the extreme tensile fiber and here
this will be 0.05 and we are in this
zone which is a tension control zone and
this is 0.9 right here okay but anyways
you can see some other videos and
whatever that'll be all there you know
and this is something you got to know
right just kind of it again yeah you
know the reality is you've got to be
able to look it up alright don't be weak
so we verified that the steel yielded
and now the next thing that we can do is
calculate the nominal moment and to do
that there's no memorizing equations
here you know you just go back and look
at that stress profile again right here
to calculate the nominal moment I look
over here at the stress profile I
realized that my area and compression
stays within the flange so it's a nice
little rectangle so that means the depth
of my equivalent stress block the result
dents are the depth of the resultant C
sub C this is just a over two and so if
I sum moments and I like to draw this
little MN here if I sum moments about
any point and it's conventional to take
moments about where the C sub C is then
I can solve for this M sub n right here
so let's let me show you what that
equation might look like here if I sum
moments about C sub C equal to zero and
I say this is positive I'll get an
equation that actually looks like this
minus MN plus TS times Z equals zero and
that just tells me that MN is TS times Z
and because the steel is yielded this is
a sfy z is d minus y bar and because my
rectangle
stress block and the area and
compression is still a rectangle this Y
bar is just a over two and now all I got
to do is substitute here a s was three
point eight one inches squared this
yield strength of steel in this problem
with 60 ksi D was 27 inches minus a over
two which is one point four nine four
inches divided by two and ice plug and
chug oh let's convert this into kip feet
so one foot every 12 inches and this
number becomes 500 point one two kip
feet right and it ain't no thing man you
just multiply by fie MN to get the
design moment strain this point nine
times this 500 point one two Kip feet
and here this is 450 point one one kip
feet and gosh you know I think this is
as deep or as challenging as the FE exam
might get and you know you can do it
alright so hopefully you enjoyed take it
easy