We came up in the last course,

my Part 3 course, with

a moment-curvature relationship. This is what it was.

Kappa was called the curvature and we

found that was proportional to the moment,

that was also equal to one 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.

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 one since that's

a small number squared and we can neglect it.

So our curvature, one over

the radius of curvature becomes d squared y dx squared.

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 a 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.

So what we're looking for is

why our sign convention will be

that the bending moment as

positive if we have a smiley face here,

this is the same as we've used in my previous courses,

our 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.

So for horizontal beams,

I will always select the deflection y as being

positive upward for beam deflection problems.

We'll get started with looking at

that in the next module.