BENDING OF BEAM
When a beam is loaded under pure moment
M, it can be shown that the beam will bend
in a circular arc.
If we assume that plane cross-sections
will remain plane after bending, then to form
the circular arc, the top layers of the beam
have to shorten in length (compressive
strain) and the bottom layers have to elongate
in length (tensile strain) to produce the
M curvature. The compression amount will
gradually diminish as we go down from the
Compressed top layer, eventually changing from
layer compression to tension, which will then
gradually increase as we reach the bottom
layer.
Elongated
Thus, in this type of loading, the top layer
M layer
will have maximum compressive strain, the
bottom layer will have maximum tensile
Un-strained
layer strain and there will be a middle layer where
the length of the layer will remain unchanged
and hence no normal strain. This layer is
known as Neutral Layer, and in 2D
representation, it is known as Neutral Axis
(NA).
NA= Neutral Axis Because the beam is made of elastic
Compression material, compressive and tensile strains will
N A
also give rise to compressive and tensile
stresses (stress and strain is proportional –
Hook’s Law), respectively.
Unchanged More the applied moment load more is
Elongation the curvature, which will produce more
strains and thus more stresses.
Two Dimensional View Our objective is to estimate the stress
from bending.
, We can determine the bending strain
and stress from the geometry of
bending.
Let us take a small cross section of
width dx, at a distance x from the
left edge of the beam. After the
beam is bent, let the section dx,
subtends an angle dφ at the center of
curvature with a radius of curvature
r at NA. Then,
rdφ=dx
dφ 1
or, dx = r
Let us consider an arbitrary layer at
a distance v from the NA. If we
draw a line BC parallel to AO 1, then
the angle CBD = dφ. The elongation
of this layer = CD= v.dφ. The
original length of this layer was dx.
Hence the strain ε = vdφ = v …………..(1)
dx r
E
Within elastic limit stress σ = E ε = v ……(2)
r
where E = elastic constant.
Since r is fixed for a loading condition, and E is also a constant,
then the stress will be proportional to the distance v of any layer
from the neutral axis. If we know the radius of curvature due to
bending, we can find the bending stress and bending strain using
these formulas.
When a beam is loaded under pure moment
M, it can be shown that the beam will bend
in a circular arc.
If we assume that plane cross-sections
will remain plane after bending, then to form
the circular arc, the top layers of the beam
have to shorten in length (compressive
strain) and the bottom layers have to elongate
in length (tensile strain) to produce the
M curvature. The compression amount will
gradually diminish as we go down from the
Compressed top layer, eventually changing from
layer compression to tension, which will then
gradually increase as we reach the bottom
layer.
Elongated
Thus, in this type of loading, the top layer
M layer
will have maximum compressive strain, the
bottom layer will have maximum tensile
Un-strained
layer strain and there will be a middle layer where
the length of the layer will remain unchanged
and hence no normal strain. This layer is
known as Neutral Layer, and in 2D
representation, it is known as Neutral Axis
(NA).
NA= Neutral Axis Because the beam is made of elastic
Compression material, compressive and tensile strains will
N A
also give rise to compressive and tensile
stresses (stress and strain is proportional –
Hook’s Law), respectively.
Unchanged More the applied moment load more is
Elongation the curvature, which will produce more
strains and thus more stresses.
Two Dimensional View Our objective is to estimate the stress
from bending.
, We can determine the bending strain
and stress from the geometry of
bending.
Let us take a small cross section of
width dx, at a distance x from the
left edge of the beam. After the
beam is bent, let the section dx,
subtends an angle dφ at the center of
curvature with a radius of curvature
r at NA. Then,
rdφ=dx
dφ 1
or, dx = r
Let us consider an arbitrary layer at
a distance v from the NA. If we
draw a line BC parallel to AO 1, then
the angle CBD = dφ. The elongation
of this layer = CD= v.dφ. The
original length of this layer was dx.
Hence the strain ε = vdφ = v …………..(1)
dx r
E
Within elastic limit stress σ = E ε = v ……(2)
r
where E = elastic constant.
Since r is fixed for a loading condition, and E is also a constant,
then the stress will be proportional to the distance v of any layer
from the neutral axis. If we know the radius of curvature due to
bending, we can find the bending stress and bending strain using
these formulas.
