We were
discussing basic
concept of bending stress in our previous session. We have also
discussed assumptions
made in the theory of simple bending and formula for bending stress or flexural formula for beams during our last session.

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Let us
assume that following beam PQ is horizontal and supported at its two extreme
ends i.e. at end P and at end Q, therefore we can say that we have considered
here the condition of simply supported beam.

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Original
length of the layer GH = Length of the neutral layer EF = Length of the neutral
layer E'F' = R x θ

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Let us
consider the above equation and putting the value of strain secure above, we
will have following equation as mentioned here.

Now we are
going ahead to start new topic i.e. expression for bending stress in pure
bending of beam in the strength of material with the help of this post.

Let us go
ahead step by step for easy understanding, however if there is any issue we can
discuss it in comment box which is provided below this post.

Let us
consider one structural member such as beam with rectangular cross section, we
can select any type of cross section for beam but we have considered here that
following beam has rectangular cross section.

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**Bending
stress**

Once load
W will be applied over the simply supported horizontal beam PQ as displayed
above, beam PQ will be bending in the form of a curve and we have tried to show
the condition of bending of beam PQ due to load W in the above figure.

Now let us
consider one small portion of the beam PQ, which is subjected to a simple
bending, as displayed here in following figure. Let us consider two sections AB
and CD as shown in following figure.

Now we
have following information from the above figure.

AB and CD:
Two vertical sections in a portion of the considered beam

N.A:
Neutral axis which is displayed in above figure

EF: Layer
at neutral axis

dx =
Length of the beam between sections AB and CD

Let us
consider one layer GH at a distance y below the neutral layer EF. We can see
here that length of the neutral layer and length of the layer GH will be equal
and it will be dx.

Original
length of the neutral layer EF = Original length of the layer GH = dx

Now we
will analyze here the condition of assumed portion of the beam and section of
the beam after bending action and we have displayed here in following
figure.

As we can
see here that portion of the beam will be bent in the form of a curve due to
bending action and hence we will have following information from above figure.

Section AB
and CD will be now section A'B' and C'D'

Similarly,
layer GH will be now G'H' and we can see here that length of layer GH will be
increased now and it will be now G'H'

Neutral
layer EF will be now E'F', but as we have discussed during studying of the
various assumptions made in theory of simple bending, length of the
neutral layer EF will not be changed.

Length of
neutral layer EF = E'F' = dx

A'B' and
C'D' are meeting with each other at center O as displayed in above figure

Radius of
neutral layer E'F' is R as displayed in above figure

Angle made
by A'B' and C'D' at center O is θ as displayed in above figure

Distance
of the layer G'H' from neutral layer E'F' is y as displayed in above figure

Length of
the neutral layer E'F' = R x θ

Length of
the layer G'H' = (R + y) x θ

As we have
discussed above that length of the layer GH will be increased due to bending
action of the beam and therefore we can write here the following equation to
secure the value of change in length of the layer GH due to bending action of
the beam.

Change in
length of the layer GH = Length of the layer G'H'- original length of the layer
GH

Change in
length of the layer GH = (R + y) x θ - R x θ

Change in
length of the layer GH = y x θ

Strain in
the length of the layer GH = Change in length of the layer GH/ Original length
of the layer GH

Strain in
the length of the layer GH = y x θ/ R x θ

Strain in
the length of the layer GH = y/R

As we can
see here that strain will be directionally proportional to the distance y i.e.
distance of the layer from neutral layer or neutral axis and therefore as we
will go towards bottom side layer of the beam or towards top side layer of the
beam, there will be more strain in the layer of the beam.

At neutral
axis, value of y will be zero and hence there will be no strain in the layer of
the beam at neutral axis.

Let us
recall the concept of Hook’s Law

*According to Hook’s Law, w*ithin elastic limit, stress applied over an elastic material will be directionally proportional to the strain produced due to external loading and mathematically we can write above law as mentioned here.

Stress =
E. Strain

Strain =
Stress /E

Strain =
σ/E

Where E
is the Young’s Modulus of elasticity of the material

σ/E = y/R

σ= (y/R) x
E

Therefore,
bending stress on the layer will be given by following formula as displayed
here

We can
conclude from above equation that stress acting on layer of the beam will be
directionally proportional to the distance y of the layer from the neutral
axis.

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Image
Courtesy: Google
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We will discuss another topic i.e. derivation of flexure formula or bending
equation for pure bending in the category of strength of
material in our next post.

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**Reference:**

Strength of material, By R. K. Bansal