Curved Beams - Bending of Curved Beams, Advanced Solid Mechanics Video Lecture - Mechanical Engineering

in this lesson we're going to discuss
the effect of curvature on beams with
moments applied so first let's review
the assumptions the assumptions listed
are for straight beams so we're going to
consider only we're not going to sitter
consider the segments that are directly
next to or near the applied loads we're
going to assume that the beam is load
loaded in the plane of symmetry we're
going to assume that the cross-sections
that we take are in a plane
perpendicular to the neutral axis we're
going to assume the material is
homogeneous and obeys Hookes law we're
going to assume the stresses are not
long and not great enough to cause
elastic deformation and that the
deformations that we see will be small
excuse me the stresses are still in the
elastic region we won't see any plastic
deformation the segment has pure bending
and the only assumption we're going to a
lab relax is now we're not going to say
that the beam is initially straight so
let's take a look at a beam this
particular one has a rectangular
cross-section but we'll generalize it
shortly and what we're going to notice
is that the centroidal axis has to do
with the geometry what is in the center
of the geometry the neutral axis is
defined as to place along the axis in
which there are no stresses so for a
straight beam the neutral axis lines up
on the centroidal axis and everything is
ok but for a curved beam we notice that
the stress distribution now has it still
has a compressive component in it on the
top of the beam and a tensile component
on the bottom of the beam but the center
of the beam is no longer at zero and
there exists this new this neutral axis
where the stress is zero we should also
notice that the stress is no longer in a
straight line
it now has
a hyperbolic shape rather than a linear
shape and that the stress at the inner
surface appears to be greater than the
stress at the outer surface and that's
going to be typical of curved beams so
there's a couple definitions we need to
see the the biggest one is the distance
that we're going to call e that the
neutral axis has moved from the
centroidal axis and the the strict
definition of that is RC is the distance
from the radius of curvature of this of
the centroid okay so this is how curved
the beam is that's what RC is and we're
going to subtract the area over the
integral of DA over R and that's going
to tell us what the what the actual
distance is well this is obviously quite
mathematical but for common shapes
jiggly's machine design book lists
several RC and R in values so now it's
just a simple matter of calculating what
E is from this table so a couple things
to notice and we'll notice on the back
going back code diagram as well RI is
the radius from the right from the
centre of curvature to the inside of the
beam or o is to the outside of the beam
RC is to the centroid and then you have
these values CI is the distance from the
neutral axis to the outside Co is the
distance from the neutral axis to the
outside and again E is the distance
between the centroidal axis and the
neutral axis RN is the distance from the
center to the neutral axis that should
cover all of our definitions so now
using this table as a guide for when we
have things like eye beams or circles or
rectangles or whatever shape that we
have
we can consider a curved beam so in this
case being shown the the curved beam
only experiences a moment okay so
there's a moment that's twisting the
edges of the beam causing a pure moment
stress with inside the curved part of
the beam and what we're going to do now
is we're going to say that the inside
stress is equal to M times e we know how
to get calculate e a we know what the
cross sectional area is R is the radius
to the inside and then CI is the
distance from the neutral axis to the
outside okay so this this becomes our
new equation for finding stress and
finding the maximum stress so we we
should expect that Sigma I will be the
greatest of the stresses the beam
experiences and we should express that
the outside will be in compression
there's a minus sign there there's a lot
of cases where instead of a bending
moment being applied we can actually
have two forces for instance a hook or
something like that so this is a fairly
common analysis and looking at this
analysis we're going to it's now become
a combined loading so we have a moment
that's induced if you were to do a free
body diagram across the interior right
here where this break occurs you would
recognize there has to be a moment that
counteracts the moment that's created by
this force times the distance away that
this this is but there's also an axial
load and so here's the free body diagram
of it broken in half with the internal
moment we can calculate what that moment
is again it's going to be this force
times the distance away that the force
is and now we have the same term term
for the stress so this M is still
producing a stress caused by the moment
but we also have F over a as well so
this is again a combined loading
situation
in this particular loading situation
there is no transverse shear forces and
there's no twisting forces that would be
rotating the hook the other direction so
out of the into the page or maybe around
the F there so there's no there's no
torsion occurring so in both of these
cases if those other cases existed we
would have to check the full stress
state as it is this will suffice for the
stress state that's occurring on the
inside edge of that break and on the
outside edge of the strip the break
looking at these two and there would not
be any other stresses in any of the
other directions no other normal
stresses nor any other shear stresses so
these will define our stress state at
those locations