Hi, welcome to module 16 of applications in engineering mechanics.

Today we're going to continue on and do a bending moment diagram,

for a multiforce member. So here is my generic beam again.

And my differential element that I've taken out

of the beam and drawn my free body diagram.

Now we are going to do the, the moment equilibrium equation

on this free body diagram to come up with a moment relationship.

So, I am going to call this right hand side point A.

I am going to sum moments about point A.

I will choose my sign convention for assembling the equation as counter clock,

clockwise positive and I end up with this is clockwise so that's minus M,

minus V times its moment arm which is dx. Plus I have

a q times dx force down, so that's going

to be positive in accordance with my sign convention.

Its moment arm is going to be dx over two.

And then I have plus M dM.

This shear force on the right-hand side actually goes through

point A, and so it will not cause a moment.

Now we can neglect this higher order term, because dx is infinitesimally small.

And when you have a very small value,

the square of that value or, if you

multiply that value together, it's orders of magnitude smaller.

And you're making it', it's a very

negligible compared to the rest of the terms.

And so by doing that, I can then cancel M and M, and I come up with a relationship

that the value of the shear at any point is

equal to the change of moment in the x direction.

So in words,

the value of the Shear Force equals the slope or the change,