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.

Now we are going ahead to start new topic i.e.
moment of resistance of a 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.

###
**Moment
of resistance**

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 as displayed in following figure.

As we have discussed that when a beam will be
subjected with a pure bending, as displayed in above figure, 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

Recall here the concept of bending stress, which is
mentioned below, and use the value of bending stress (σ) in above equation and
we will have equation for bending moment of the layer about the neutral axis.

σ = (E/R) 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

dM = (E/R) x y

^{2 }dA
Where M will be termed as moment of resistance

We will
discuss another topic i.e. Section modulus in the category of strength of
material in our next post.

###
**Reference:**

Strength
of material, By R. K. Bansal

Image
Courtesy: Google

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