This is what you should've come up with.

On one end you can see that it's pinned,

on the other end we have a roller support,

and that we have a load,

a concentrated load down at the center of the beam.

So now that we have this model,

what I want you to do is to solve for

the reactions at the ends of the beam.

If you do that, you can see those reaction forces here,

and you'll recall from module 14 in

my applications in engineering mechanics course that

we came up with

this relationship which said that the negative value of

the load at a point equals

the slope or rate of change of the shear diagram.

So as a review,

I'd like you to go back and do

a shear diagram for this loading condition on this beam.

This is what your sheer diagram should look like.

Then if you also recall back to module

16 of applications in engineering mechanics,

we came up with this relationship,

which said that the change of bending moment between

two points equals the area under the shear force curve.

So here is our shear force diagram,

and we can now find our bending moment diagram.

I like you to do that on your

own as review and then come on back.

So this is what you should have come up

with for the bending moment diagram.

This is our equation now,

our differential equation for the bending moment.

It's equal to EI d squared y dx squared.

So what we can do is since

the shear is equal to the derivative of the moment,

we can take this derivative and we now

have an expression for our shear.

In this case, I've assumed that

the flexural rigidity EI is a constant.

Then I can take the derivative again of the shear,

to come up with an expression for the load.

So we now have some interesting relationships,

and we can go in

the other direction by

integrating to find first the slope.

Here we have an expression for d squared y dx squared.

If we integrate that, we'll come up with

a slope which is dy dx,

and that's shown here.

So what I want to do now is I want to come

up with what we're going to call the slope diagram.

The slope diagram is going

to be the area under the moment diagram,

will be the change in slope.

So you can see the area from this point to the center,

under the moment curve is PL

squared over 16 because its a triangle.

The slope of the slope diagram starts off at zero.

We also see if we look at

our loading condition that the slope

at the center has to be zero,

because we're down at the bottom of the deflection.

So we know that the slope at the center is zero,

we know that the slope of the slope diagram is

zero at the left-hand side,

but then it increases because of the area

under the moment curve to PL squared over 16,

so we start off with minus PL squared over 16 EI,

and we go to a slope of zero.

Then we have a positive slope beyond that,

going down to a slope of

our slope diagram zeroing out at the end with again,

the change in the slope

being the area under the moment curve,

which is PL squared over 16 EI.

Then we can do the same thing for the deflection diagram.

The area under the slope

will be the change in deflection.

Here's our equation, we've integrated again.