Shear and bending-moment diagrams are the last topic that we will consider
in this semester's class.
It is essentially taking the internal loads
and finding them at all parts of the beam.
See, this is how it goes.
A beam will break when the internal loads get too big.
Now if you're looking at a beam, if it breaks anywhere, then it broke.
You can't break a little bit here and not a little bit there and be okay.
So to say that it's--,
I want a beam to not break,
I need to be able to say that exceeding a maximum value,
the maximum internal load, anywhere will make it break.
Thus each of my internal loads has to be below the max
at every point on the beam.
So what I need to know to determine whether or not my beam will break
is what these values are for the internal loads
at every point along the beam.
Everywhere. For all x.
So what I'm going to do is I'm going to get V(x), M(x), and N(x)
for all x and plot them.
And then I can look and see whether the maximum value of the graph
is below this maximum value that would make my beam break.
So if you're looking at a beam that's L long,
I want to find out what each of these is for every x between 0 and L.
And just as a point of convention, we will start x equals 0 on the left-hand side
and go this way as we go along.
So what do we do when we're doing that then?
Let's just look at this basic beam with a cantilever on the left-hand side,
P in the middle, and Q coming off the end as an axial load.
I'm going to take these in metric units
just so that we know what units we're dealing with.
Step 1 is to find the external loads on your beam. So I have a wall.
I've replaced it with two forces and a moment.
Solve the equations of equilibrium to find out what the loads are at the wall.
Once you have these, --
-- This is the principle of a shear and bending-moment diagram by definition. --
I want to draw the free-body diagram for every single position along the beam.
Clearly, I do not want to draw infinitely many free-body diagrams. But I don't really need to.
Because of algebra, I can say this is the free-body diagram for any x
as long as I haven't gotten to this P yet.
So if x is [equal to] L/8 or 3L/8 or anything less than L/2,
I will not have yet have gotten to this load.
So any slice I take between the wall and the mid-P gives me this free-body diagram.
And that means that I can solve these equations of equilibrium
to get N(x), V(x), and M(x)
for every value of x as long as I'm before L/2.
So as long as I'm between 0 and L/2, that's my free-body diagram,
and these are the values for my internal loads.
Similarly, every free-body diagram I would draw
for any x bigger than L/2 but less than L,
any slice I take in here, will give me this free-body diagram.
It will have the V, but it will not yet have the Q
because I haven't gotten to the end of the beam.
So this free-body diagram is valid for every x between L/2 and L.
So I'm going to solve the equations of equilibrium
to find out what my internal loads are for this point.
Now, be a little bit careful.
x still starts at the left-hand end. It's our coordinate. It has to start at the origin.
So the distance along this beam is x,
which means that when you're actually summing this moment,
you've got P * x, this is your one at the end.
This is the moment at the wall, and the vertical load at the wall.
And this is my P which is acting in the middle times my distance now is (x - L/2).
That's this little black distance right here.
This distance is how far away the break is from my mid-applied P.
Be careful with your algebra here. Lesson.
Now that all adds up to being 0 for both for both V and M.
Here I have then my internal loads as a function of x
for both the first part of the beam and the second part of the beam.
Step 3 just is plotting them.
I just rewrote those things again. These are the values we just found.
And then I'm going to plot these three things.
N is equal to Q all the way along because nothing changed in the middle of the beam.
So my axial load diagram just is N equals Q for all values of x between 0 and L.
A couple things about these plots.
They need to be labeled so it needs to have N and units in both places.
Values need to be labeled, and x values need to be labeled.
All of these things have to be actually labeled. It's not just a shape I'm looking for.
I'm looking to see whether that value exceeds the maximum allowable
at any point along the beam.
So I have to know what those values are.
The shear diagram is up here.
It started from x=0 at V=P for the first part of my beam,
and then it went to V=0 for the second part of my beam.
M started at -PL/2. That was the moment at the wall.
And it increases with a slope of P for the first part of the beam.
Now for the second part of the beam,
M was uniquely equal to 0.
These are the shear and bending-moment diagrams for that beam and the axial load diagram.
It's not atypical that we don't have one of these.
So often what you'll see is shear and bending-moment diagrams
because when we're talking about beams,
we are most likely talking about gravitational loads or things that go down.
A "beam" is what we call something that supports a transverse load.
So often times you will not have this graph, but it's the same principle.
We're graphing the internal loads.
Now, one last sort of thing I want to point out about this problem:
there's an alternate Step 1.
Remember that the internal loads are the same
whether you take the left-hand portion of the beam or the right-hand portion of the beam.
So, in fact, I don't even have to find in this problem what the external loads are.
Do be careful though: x=0 still starts at the wall
because it's our coordinate.
So if I took the right-hand side of the beam,
I'm using the right-hand side of the beam sign conventions,
then this length is L minus x. That's not a problem.
This now is any x between L/2 and L, and I get the same numbers I had before.
Between 0 and L/2 now I have the mid P again because I'm going this way.
And I get the same values I had before.
So you can do this in a smart way,
and you can sort of look at either side and use whatever is easiest.