We were discussing basic
concept of bending stress and derivation for beam bending equation 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. Section
modulus of beams 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. So let us come to the main subject i.e. Section
modulus.

Strength of any section indicates the moment of
resistance offered by the section. Let us recall the concept of moment of
resistance which is defined as the total moment of the forces, due to bending
stresses, about the neutral axis of the section.

Where,

Z: Section modulus of the section

M: Moment of resistance

σ = Bending stress

####
*Now
we will define here the section modulus of the beam*

*Now we will define here the section modulus of the beam*

Section modulus of a beam will be defined as the ratio
of area moment of inertia of the beam about the neutral axis or centroidal axis
of the beam, subjected to bending, to the distance of the outermost layer of
the beam from its neutral axis or centroidal axis.

If we consider the moment of resistance offered by a
section for a given value of bending stress, we can easily say that moment of
resistance will be directionally proportional to the section modulus of the
section.

Therefore, strength of any section will be dependent
over the section modulus and we can say that if a beam has higher value for section
modulus then beam will have more capacity to bear the bending moment for a
given value of bending stress or beam will be stronger and hence section
modulus of the beam will indicate the strength of the section.

####
**Unit
of section modulus**

Unit of section modulus = Unit of area moment of
inertia / Unit of distance

Unit of section modulus = m

^{3}####
**Section
modulus of a beam having rectangular cross-section**

Let us consider a beam, as displayed in following
figure, with rectangular cross-section. Let us consider that width of the rectangular
cross-section is B and depth or height of the rectangular cross-section is D.

I

_{XX}= Area moment of inertia of the rectangular section about the XX axis
Recall the derivation and concept of area moment of inertia of the rectangular section and we will have

I

_{XX}= BD^{3}/12
y = distance of the outermost layer of the beam from
its neutral axis or centroidal axis

y = D/2

Section modulus of the rectangular section about XX
axis could be secured as mentioned here

**Z**_{XX}= BD^{2}/6
Similarly, we can find out here the value of section
modulus of the rectangular section about YY axis

Z

_{YY}= DB^{2}/6####
**Section
modulus of a beam having circular cross-section**

Let us consider a beam, as displayed in following
figure, with circular cross-section. Let us consider that diameter of the
circular cross-section is D.

I

_{XX}= Area moment of inertia of the circular cross-section about the XX axis
I

_{YY}= Area moment of inertia of the circular cross-section about the YY axis
Recall the derivation and concept of area moment of inertia of the circular cross-section and we will have

I

_{XX}= I_{YY}= ПD^{4}/64
y = distance of the outermost layer of the section
from its neutral axis or centroidal axis

y = D/2

Section modulus of the circular cross-section about
XX axis and YY axis could be secured as mentioned here

*Z*_{XX}= Z_{YY}= ПD^{3}/32
Similarly
we can secure the value of section modulus for various cross-sections as we
have discussed above.

We will
discuss another topic i.e. 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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