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 flexure formula for beams during our last session.

Now
we are going ahead to start new topic i.e. derivation of flexure formula or bending equation for
pure bending 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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First of all we will find here the expression for bending stress in a layer of the beam subjected to pure bending and aftre that we will understand the concept of moment of resistance and once we will have these two information, we can easily secure the bending equation or flexure formula for beams.

So let us first find out the expression for bending stress acting on a layer of the beam subjected to pure bending.

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So let us first find out the expression for bending stress acting on a layer of the beam subjected to pure bending.

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**Bending stress**

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.

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.

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.

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

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

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

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

σ/E = y/R

σ= (y/R) x E

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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Moment of resistance

As we have discussed that when a beam
will be subjected with a pure bending, layers above the neutral axis will be subjected
with compressive stresses and layers below the neutral axis will be subjected
with tensile stresses.

Therefore, there will be force acting on
the layers of the beams due to these stresses and hence there will be moment of
these forces about the neutral axis too.

Total moment of these forces about the
neutral axis for a section will be termed as moment of resistance of that
section.

As we have already assumed that we are
working here with a beam having rectangular cross-section and let us consider the
cross-section of the beam as displayed here in following figure.

Let us assume one strip of thickness dy and
area dA at a distance y from the neutral axis as displayed in above figure.

Let us determine the force acting on the
layer due to bending stress and we will have following equation

dF = σ x dA

Let us determine the moment of this
layer about the neutral axis, dM as mentioned here

dM = dF x y

dM = σ x dA x y

dM = (E/R) x y x dA x y

dM = (E/R) x y

^{2 }dA
Total moment of the forces on the
section of the beam around the neutral axis, also termed as moment of
resistance, could be secured by integrating the above equation and we will have

Let us consider the above equation of
moment of resistance and equation that we have secured for bending stress in
case of bending action; we will have following equation which is termed as
bending equation or flexural formula of bending equation.

We will
discuss another topic in the category of strength of material in our next post.

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**Reference:**

Strength
of material, By R. K. Bansal

Image
Courtesy: Google