We came up in the last course, my part three course,

with a moment curvature relationship and this is what it was.

Kappa was called the curvature and we found it was proportional to the moment.

It was also equal to 1 over the radius of curvature, which is one over rho.

We said that EI was the flexural rigidity or

the resistance of the beam to bending for a given curvature.

So we made some assumptions.

We were operating in the linear elastic region.

We were working with pure bending or

flexure under constant bending moment and we have no shear force.

We actually find that the beam deflection due to shearing can be negligible.

So, here again is our Moment-curvature relationship where one

over the radius of curvature is equal to M over EI.

The Curvature equation, which you can find in any standard calculus

textbook is shown here where y is in the transverse direction or

the direction of the deflection and x is in the direction along the beam.

And we're looking at small deformations and so

dy/dx is a small value and if we square that like we do in the numerator, that's

much much less than 1 since it's a small number squared and we can Neglect it.

And so our curvature, 1 over the radius of curvature becomes d squared y dx squared.

And then we can substitute that into the top equation for

the moment curvature relationship.

And we get a differential equation for the elastic curve of the beam shown here.

So we have this differential equation for the elastic curve of a beam.

If we have an equation for the moment along the beam now,

we can find the deflections by integrating twice and

using boundary conditions to find the constants of integration.

And so what we're looking for is why.

Our sign convention will be that the bending moment is positive,

if we have a smiley face here.

This is the same as we've used in my previous courses where negative is

shown on the right.

So, if it's a positive bending moment, d squared y dx squared is positive.

If it's a negative bending moment, then d squared y dx squared is negative.

And so for horizontal beams, I will always select

deflection y as being positive upward for beam deflection problems.

And we'll get started with looking at that in the next module.