Beams – shear stress and bending stress

So hopefully, you had a chance to
experiment with the beam simulator to figure out what
factors affect the shear and moment along the length of a beam. Hopefully you found that the load and
the type of supports and the length all affect the shear
and moment in the beam. So as you go along the length of a
beam, the shear and moment change, and it’s going to be affected
by the type of supports, the loads and the length. You should have hopefully found
that the height, the material and the cross sectional shape did
not affect the shear and the moment, however. So if I put in a beam that is
made of wood versus one that’s made of plastic versus
one that’s made of steel, as long as they have the same supports
and they have the same load applied, they will have the same shear
force and moment along the length. That to me is somewhat
counter intuitive. It seems like a wood beam and a steel
beam shouldn’t have the same forces. They do end up having the same
forces, at least as long as they’re statically determinate beams. Statically indeterminate
beams would vary slightly, but that goes a little
beyond the course– but even those don’t
vary by all that much. What does change are
the stresses though. So a wood beam versus
a steel beam, as long as they have the same
support conditions, if they have different shapes
and different materials they’re still going to get
the same force and moment. They will have different stresses
though, and that’s the difference. So stresses are what an engineer will
compare with material properties. Steel will have a much higher
allowable stress in bending, like it did with
compression and tension. It’ll have a much higher allowable
bending stress than a wood beam, say, or a plastic beam, or whatever
type of beam we have in there. So in a beam, the two types
of stresses that we calculate are shear stress and bending stress. Shear stress is analogous to the
tension or compression stress that we calculated for columns
and for ropes and cables. So to get an average shear stress,
we can just take the shear force. So whatever that force was that we
calculate along the length of the beam and divide it by the
cross sectional area. So whatever this cross
sectional area is, that’s the area we’re going to be trying
to shear across so we can get a stress, then we can compare that to
allowable material properties. The other thing we calculate
with beams is a bending stress. So it’s a measure of the stress
along the length due to bending. The equation for that
is we take the moment and we multiply by a value called c,
and c is half the height, typically. So it’s the distance
from our neutral axis, or our bending axis– in
other words, a neutral axis. When we bent the beam
with the grid, we found that we had tension on one side
and compression on the other side. To go from tension to compression, we
necessarily need to go through a place where there’s zero stress,
or zero force induced, and that’s our neutral surface. For a symmetric cross
section, so a cross section that’s equal, if we put
a line through the middle that line will go right
through the center. And then my distance from that
neutral access to the outside edge is just half the height, and that
is typical for a symmetric section. So again, we have the
moment, whatever we calculate for the moment along the
length, and then half the height of the cross section, and then we
divide that by the moment of inertia. So what’s a moment of inertia? A moment of inertia– we talked
a little bit with columns, because that was a
function of the buckling. Moment of inertia is a quantity–
it’s the resistance to bending. So different shapes are going to
have different moments of inertia, different resistance to bending. The units are always going
to be length to the fourth. And it tends to be
that the more mass you have away from a central axis,
the more resistance to bending. So if we look at just
this rectangular section, so it’s a longer dimension in
one dimension, it’s easier for me to bend it this way. You could try this with a
ruler if you had it, too. It’s easier to bend it one
direction than the other. It’s much harder for me to bend it
when the beam is oriented vertically. So why is that happening? It’s happening because when I do it
vertically I have more of my mass away from that central axis–
so from that center line. As opposed to when I do it
horizontally, now my line is going through the
middle of that dimension and my height is not very high. So you can go ahead and try it. But if you have a taller
section, it’s going to tend to be stiffer and stronger. Just a note on stiffness and
strength– lots of people use those interchangeably. Stiffness is a measure of how
much it’s going to deflect. So as I try to bend this, one of them
is going to deflect more than the other. The one that deflects the least
is referred to as a stiffer beam. And then strength has to do
with when it’s going to fail. They’re related in that they
both use moment of inertia, but one has to do with a failure and
one has to do with a deformation. But in both cases the moment of inertia
is higher if I have a vertical section, and that is because
that height makes a much bigger difference in
my moment of inertia, and more mass is away from the center. If I were to calculate
the moment of inertia, I’d need calculus to
figure out what it is. But I could tell you what the
formula is for a rectangular section. A rectangular section, if you want
to compute the moment of inertia, is the base times the
height cubed over 12. That helps explain, again, why
vertically it’s going to be stronger. If that height is cubed, I want
my bigger height to be cubed, and that’s going to make a bigger
difference in my moment of inertia. And we can also look at
these three different beams. So this one is the
shorter one, so it’s going to have the least moment of inertia. It’s also got more of the
mass closer in to the axis, even if I do it vertically. If I take a hollow section–
I 3D printed all of them– they all have exactly the
same cross sectional area. So they all use exactly the
same amount of material, but they have different shapes. So this one is hollow, more of
the mass is away from the center, so it’s going to end up being stiffer
in both directions than all my mass closer to the central axis. So by taking that same cross
sectional area and moving it away, I tend to make stronger beams. Similarly with this I-beam, I-beam is a
pretty common beam for vertical loads. It’s bending about this
axis in the middle. And more of the mass
is away from that axis, so it tends to be a stronger, stiffer
system, has a higher moment of inertia. So shear versus bending–
when we talked about these, we talked about there being
possible sheer versus bending. So a beam has to resist both the
vertical forces, which causes a shearing and then a bending behavior. Most beams if they’re long and
slender will fail in bending, so they’ll reach their
tensile or compressive capacity before they shear apart. The exception would be
a very short, deep beam. So sometimes, say, in
parking garages when we get really deep concrete
beams that are fairly short, they could have issues with shear. But in this course we’ll focus
primarily on the bending behavior. But go ahead and experiment
with the beam simulator. Your goal is to try to determine how
the stress varies as the cross section changes. So if we use a rectangular section
versus a hollow section versus maybe an I-beam, which of those
tends to be most efficient? You can also try designing a paper
beam or a set of paper beams. So can you design a
set of paper beams that will support a book or multiple
books and span a certain distance.

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