Bending stress in beams – cross-sectional shape


So what did you find this
time with the beam simulator? The first time we used
the beam simulator, we found that the cross-sectional
shape in the material, and the height of the cross-section
did not change the moment in the beam. So it was not a factor. But, hopefully, you found that the
cross-sectional shape and the height were a factor in the bending stress. So a cross section is
not a factor in bending moment, that the actual
force or moment, but it is a factor in the bending stress. And this is, actually, helpful to
engineers when we try to design beams. So how will we predict
when it’s going to fail? The first thing we have
to figure out is what the moment is going to be
along the length of the beam. In this class you can
use the beam simulator to help you calculate those moments. What an engineer will usually know,
they’ll have some idea of the length that they need to span. They’ll have some idea
of the loads that that are going to be applied, depending
on how it’s going to be used. And they’ll have some idea what the
supports are going to look like. So with that information,
they can then figure out what the moment is along the length. And once they know that
maximum moment, they can start to make some estimations
on what material will work and what heights of cross-section
and what shape they want to use. Makes it a little more
straightforward, though. If I can calculate that moment
without the shape affecting it, I can get that moment and then
I’ll be able to experiment with different shapes and heights. So where is it this beam likely to
fail, horizontally or along the length? It will fail where
the moment is maximum. And then where along the height,
that’s the other thing the engineer has to figure out, is where on the
height do they expect failure, so they can calculate the stress at
that location and make sure it’s OK. What typically happens, is we’ll
have it on the top or bottom edge. And I brought my flexible beam again,
so we could take a look at this grid again. So as I flex this beam, I’m
going to exaggerated it. But the bottom is in tension again
and the top is in compression. So the bottom is getting
longer, top is getting shorter. We have tension and compression. Somewhere along that height,
we’ll have a zero line. And it’s called a neutral line,
where there’s zero stress. If we go from tension to compression,
we have to go through a place where there’s no stress. The other thing you can look
at when I bend this grid is when the grid starts
with vertical lines. And then as I bend it, those line
stay linear, they’re still lines, but now they’re off at an angle. This is important, because it shows
that the deformation and the stress will have a linear distribution. So we’ll go from tension to
compression along the line. And stress and defamation
will be merit in that sense. But they will also tell
us that we’re going to get our maximum deformation
at the bottom or the top surface. So an engineer will, typically, look
for where’s the maximum tension, where’s the maximum compression. And, again, materials often behave
differently in tension or compression, and beams are included in that. What is one example that has
one allowable stress in tension and one in compression? It’s not unlike a truss,
except for now everything is happening in one single beam. So beams versus trusses,
is one of the questions an engineer will try to answer
when they’re designing a system. Is it better to use a beam or truss? And that’ll depend on different factors. Truss tends to be more efficient,
because the tension and compression or in those separate members. But it also tends to be a deeper system. So if you don’t have the space for
a truss, you might go to a beam. The equation we used to calculate
the bending stress in this beam is Mc over I again. And I will go through and
calculate those bending stresses for a couple examples
including the Eiffel Tower, which we can model as a beam. It’s a vertical beam, but it’s
got an excellent parabolic shape.

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