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.

very beautifully explained

nice presentation. thanx