With the invention of the Bessemer process in the second half of the 19th century the
production of steel increased rapidly which revolutionized the way we design
or build structures. With its high strength and durability steel became the building block of
modern cities. Steel made tall buildings such as the Empire State and the Chrysler Building
become possible. If we pay closer attention to the steel frame of these buildings we can notice
a particular shape appearing in the steel beams and columns. Have you ever wondered
why we shape steel as an "I" ? Well today we are going to find out the secret behind the "I" shape
To determine why the "I" is the most common steel shape, let's analyze different types
of beams under load and see their behavior. We will compare a hollow square, an "I" shape and
a solid circle with dimensions as shown. All beams are made out of the same material and
with the same cross-sectional area. I prepared a simulation where a simply supported beam is
loaded with a single load at mid point. On the bottom left we can see a graph of the applied
force and midpoint deflection and on the right we can observe the yielding of the section. As
the load increases the deflection of the beam increases in a linear fashion. At around 37 kN,
the first yielding of the material occurs. Once the yield point is reached, the beam will continue
to take load however the deflection will no longer follow a linear line the beam will start to bend
a lot more while not taking much more load. This is because the material is undergoing
plastic deformations which are permanent and characterized with large strains. At around 49
kN the beam has gone fully plastic and has reached its ultimate capacity. At this point on,
the deflection will keep increasing while the load capacity will remain the same. Now, let's
plot the same graph for the three different shapes and see how they compare. Again, on the y-axis we
have the applied load and on the x-axis is the midpoint deflection. The circular shape reaches
its ultimate capacity at around 26 kN, the green line or the square shape as we saw earlier reached
its ultimate capacity at around 49 kN and lastly we have the "I" shape which more than double the
square shape and reach 97 kN. As we expected, the "I" shape performed much better than the square or
circular shape even though all of them use the exact same amount of material. Okay, so let's
load up a beam and look at what's happening on the inside and how the load is being carried. To load
our beam we will use a fake animated hydraulic press, the internal bending moment or torque at
mid span of the beam for a single load can be calculated as P times L over 4. If you look at
the loaded cross-section at mid-span we can notice that the top half of the beam is under compression
and the bottom half is under tension while at the midpoint, also known as the neutral axis,
the stress equals zero. Because these forces act in an opposite direction as you can see on the
diagram they induce an internal torque or bending moment. If this is not intuitive to you let's look
at a so-called foam beam. At first the lines drawn on the beam are vertical however when the beam is
bent we can see that the lines on the top got much closer together which means the material
is being compressed and on the bottom they got further apart which means the material is being
stretched. With that in mind let's measure the stress along the cross section of the beam. We
can notice that as the load is applied the stress is higher the further away we move from the center
of the beam. When the stress reaches 300 MPa or the yield stress of steel then we mark that
as the yield point. If we continue to load the beam then the yielding will propagate towards
the neutral axis. When the yielding reaches the neutral axis then the beam has gone fully plastic
or it's reached its ultimate capacity. Upon further load, the beam will excessively deflect
or rupture. This is the capacity that we use in design of steel members but in reality still has
additional reserve strength that we don't really rely on where the stress profile looks something
like the dashed red line. If we multiply the compressive area times the compression stress
we'll end up with an equivalent compression force acting at the centroid of the area. If we do that
for both the tension and compression side we will end up with two forces acting in opposite
directions of one another. These two forces are equal in magnitude and when multiplied
by the distance between them they represent the moment resistance of the beam. If we look at the
circular section we can see that again we have the same two forces, the magnitude is the same
as in the "I" shape however the difference is in the location at which they are acting. The
centroid of the compression and tension side are much closer towards the middle which represents
the moment arm and since the moment resistance of the beam depends on the distance between
these forces we end up with a much smaller moment capacity. If you could shift the centroid of the
area further away from the middle then we would increase the bending resistance of the beam and
the ability to carry more load. This is where the I-beam is superior since most of the area
is located away from the middle of the beam the parameter that captures the tendency of a beam
to resist angular acceleration is called moment of inertia but simply said moment of inertia is
a parameter that shows which section is harder to bend.
I hope this video helped you understand the
secret behind the "I" shape thank you for watching and don't forget to subscribe see you next time.