Let's calculate the moment of inertia for a wide flange beam. In this case I have
chosen a beam that is known as a W12 X 87. The W stands for wide flange. The twelve means
that this dimension here is nominally about 12 inches tall and the third variable is 87,
that stands for pounds per foot. So this would give you the weight of the beam. So each foot
of the beam weights 87 pounds. There are all sorts of these W12 beams. They range all the
way from W12 X14 all the way up to W12 X 136. So this means that each beam, of the different
sizes, could range anywhere from 14 lb/ft all the way up to 136 lb/ft. And all of them,
nominally, have this dimension of about 12 inches. And although this dimension stays
at about 12 inches, what changes to change the weight is mainly the width of the flange,
so a lighter beam would tend to have a narrower flange, and a heavier beam tends to have a
wider, thicker flange. Let's start by defining a coordinate axes. I'll run the x axis down
the length of the beam, the y axis runs vertically and the z axis, sticking out in the other
direction. When we look at the section on the right, this dot in the middle represents
the centroid, and from the direction, y axis upward, z axis to the left, and the x axis
comes straight out at us. Typically for a wide flange beam, a load would be applied
downward to the upper flange and the beam would tend to bend about the z axis. And consequently
we want to calculate the area moment of inertia about the z axis and that is equal to the
integral of y^2 dA where y is the vertical distance from the centroid. Note that how
you typically see this reported in reference manuals, you will typically see this being
the x axis and this being a vertical axis. I am keeping it, this z axis, to be consistent
with the isometric view. But just keep that in mind. We are calculating what I am calling
Iz you will typically look up and it is called Ix. To evaluate this integral, let's break
the cross section into three parts. The first will be the webb, the vertical component,
it is a rectangle, I used lines here to delineate the rectangle. The second part is the upper
flange, part 2, and the third part is the lower flange, part 3. There are a few other
dimensions that we need to make the integral a little easier. The first one here is this
dimension is 6.0625. Simply 12.125 inches divided by 2. The other dimension that we
want is this dimension. The distance from the centroid to the bottom of the upper flange.
And this is 5. 455 inches. And the other dimension that we will want is the distance from the
centroid to the top of the upper flange. And this is 6.265 inches. Here is the integral
from the webb, and I am integrating with respect to z, the left side of the web to the right
side to the web. And for y I am integrating negative 5.455, so down here, up to a positive
value of 5.455 at the top of the web. And then I will evaluate that y^2 dy dz. I can
do something similar for the upper flange. I am integrating again from the left, -6.0625
to the right of the upper flange, and I am integrating from the bottom of the upper flange
to the top of the upper flange. And I will do something real similar for the lower flange.
These integrals are straightforward to evaluate. Here is the, for the web, for the upper flange,
and zone 3, the lower flange. When I do the arithmetic, I come up with Iz equal to 731.3
inches to the forth. When you look this beam up in a handbook, you will find a value Iz
reported to be 740 in^4 and I think that the difference that we are seeing here between,
this very small difference between the two is a fact that in a real beam there will be
wells at the top and bottom of that, and I think that that difference accounts for the
wells. What is interesting is just how little of a difference the web actually makes in
terms of the area moment of inertia. For example if we forget about the web all together, delete
that first term, I compute an area moment of inertia of only about 676 in^4. What this
means is that the presence of the web, although its area is nearly a third of the entire cross
section of the beam, it only increases the are moment inertia by about 8%. It really
doesn't make much difference at all in terms of stiffness about the z axis. The high moment
of inertia about the z axis is due primarily to the flanges. And it is due primarily to
the flanges because they are spaced so far away from the centroid. For example, here
is a simulation that I ran for a beam that supports a downward load across its flange.
And the two sides are both pinned. So it is a simply supports beam with a downward force.
And what we observe is the top of the beam is under compression, the material at the
top of the beam is trying to be squeezed together. And the material at the bottom of the beam
is being pulled apart as that beam is being separated. So the bottom of it is under tension.
When we look at the cross section, I have cut a cross section down the center of the
beam where the bending moment is the greatest. And what we observe is that, again the majority
of the normal stress is supported but the top and the bottom flange. Areas of white
means there is no normal stress at all. We will see an area of which down the neutral
axis of the beam. So if you want to make something as stiff as possible about a particular axis
it is good to move as much material as you can away from the central axis. And this is
why wide flange beams are designed the way they are. they are designed to support loads
acting on the top flange. Another possibility is to load the beam from the side. So here
I have applied a force along the right side of the beam. And if we look at the cross section
we will see that the right side of the cross section is under compression and the left
side of the cross section is under tension. In this configuration the web itself supports
very little, if any bending, or normal stress. And the upper and the lower flange, a significant
portion of the flange, doesn't support any normal stress either. In this configuration
you would be looking up Iy, and when I look it up I get a value of 241 in^4. In this configuration
the flange is only about 1/3 as stiff.