I want to talk for a minute about the steps you need
to draw a shear and bending-moment diagram.
The first one is to draw a free-body diagram of the entire structure.
So in this problem we have a roller that gives me one force,
and I have a pin and that gives me two forces.
Once I have that, I can go back,
and I can write the equations of equilibrium for the entire structure
(It's just a rigid body like anything else.) and solve for the external loads.
Now to draw the shear and bending-moment diagram,
what I want to do is cut the beam at each change in loading.
So I'm going to imagine that I'm a little ant,
and I'm going to crawl down the beam.
Every time something changes, I need to draw another free-body diagram.
So as I'm going through here, I need to cut this three times.
Take three slices. Or draw three free-body diagrams,
however you want to think about it.
As my little ant is crawling,
the free-body diagram for any place between A and C will be the same.
For any x in between 0 and 12, I'll have the same free-body diagram.
I'll have a bit of this, I'll have that, and I'll have my two internal loads.
So it doesn't matter where in between A and C I cut this,
I'll have the same free-body diagram.
Similarly it doesn't matter where between 12 and 18 I cut this,
somewhere in here, and somewhere in here after those loads.
And we're going to talk for a minute about how this works with these loads,
and being an equivalent system to what we had before.
But let's talk about the first cut, anywhere between 0 and 12.
Specifically what I want to do is make this at x,
so that any x between 0 and 12 will work.
Include any portion of the distributed load that acts on that piece of beam.
Not the whole thing. Only this bit acts on this beam,
so my equivalent point load will be 40*x, not 40*12.
Make sure that you're using the right sign conventions for your internal loads.
If you don't, your beams, your graphs are going to be incorrect.
And we're going to note that N is often neglected
because generally it's not an extension load that breaks a beam.
Write your equations of equilibrium for this portion of your beam and solve.
So V(x) and M(x) are given here.
They are functions of x,
because I have down here a load that includes the distance, the 40 times x,
and I have a distance that is simply x,
so these are going to be functions of x that will let us graph them.
It's the first piece of my piecewise function.
The second chunk is about the same.
I'm going to cut [the beam] somewhere after this load ends and this one begins.
So somewhere in here.
I don't know where the somewhere is; I'm going to call that x.
Now I have my entire distributed load chunk, so I will have in fact all of 40*12.
And I can write my equations of equilibrium and solve for V(x) and M(x).
That's the second part of my piecewise function.
The third chunk is the same;
I have to come after this load here in the middle,
after the entire distributed load, but before I get to the end of the beam.
Now I want to say for just a second, this is the picture we had originally
where the beam was loaded with a 40-pound load at this elbow piece.
What I'm looking at is whether this beam will break.
So I need to move that 400-pound load up to D where it acts on the beam itself.
The equivalent system there,
-- Remember an equivalent system is the same, if the sum of the forces is the same,
and the sum of the moments is the same. --
that gives me these: a force and a moment acting at that point.
But the shear bending-moment steps are the same.
At the end of the day, draw the free-body diagram,
put your loads on, solve.
That gives me my third piece of the piecewise function.
And now I can take all pieces 1, 2, and 3 and graph them.
One note about this is that you need to label your graphs.
You need to label this axis. You need to label your points.
So if you have a maximum, it's 515 pounds. What are the values? What are these values?
I need units on every number you give me, please, or on the axes.
All of these stuff, information is essential.
Also any place it crosses an axis or has an inflection point needs to be labeled.