Consider a beam,
straight, prismatic,
born of one stuff,
formed isotropic.
Now enforce
some constraint
and simply support
the beam in place.
A cut anywhere along its length,
if cut plane will stay the same.
Let us restrict our ordered space
to just the bounds of this page.
A force P applied along this beam,
normal to length and within the page,
will left-and-right produce in an instant
reactions determined by their distance.
With these forces in balance, we reach quasi-stasis
(between those reacting to the one applied).
Note how no moments are placed at the ends
but the beam sectioned will show how it bends.
*A quick aside:
as is the norm
to keep this right
we must prescribe
the beam’s deform
and bend is only slight!*
But a bending moment acts!—it’s
on the neutral axis.
From there going up, you’ll find it compress,
developing normal negative stress.
If rather going down, it’s another case:
the tensile fibers are stretched in place.
How does it vary? Not right-to-left?
It varies linear through the depth.
If instead
your interest is
at our cut’s centroid
This normal stress
does not compress
or pull, it’s just devoid.
Back to the cut and what appeared,
there’s also the force we know as shear.
That force which acts along a plane,
across the section, contorts in shape.
As a stress, at bottom and top,
you’ll find that shear stress simply stops.
How does it vary? Verily observe:
from mid to end, quadratic curve.
Stop considering; let loading adjourn.
Lest it had yielded, it’s shape will return.
Unconstrain all nodes, unequilibrate,
save your force P for some other date.
No more “Consider…”
No more “Suppose...”
Let your pupils retire.
As for what’s next
I will leave what is left
for you as an exercise.