We were
discussing the concept of stress and strain and also we have discussed
the different types of stress and also different types of strain as well as concept of Poisson ratio in our previous posts. We have also
seen the concept of Hook’s Law , types of modulus of
elasticity in
our recent post.

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As we have already discussed above that when we will go to determine the total change in length of the bar of varying sections, we will have to add change in length of each section of the bar.
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Now we are
going further to start our discussion to understand “Stress analysis of bars of
varying sections”, in subject of strength of material, with the help of this
post.

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**Let us see here the stress analysis of bars of varying sections **

Let us see
the following figure, where we can see a bar of having different length and
different cross-sectional area and bar is subjected with an axial load P. As we
can see here that length and cross-sectional areas of each section of bar is
different and therefore stress induced, strain and change in length too will be
different for each section of bar.

Young’s
modulus of elasticity of each section might be same or different depending on
the material of the each section of bar.

Axial load
for each section will be same i.e. P. When we will go to determine the total
change in length of the bar of varying sections, we will have to add change in
length of each section of bar.

P

_{ }= Bar is subjected here with an axial Load
A

_{1}, A_{2}and A_{3 }= Area of cross section of section 1, section 2 and section 3 respectively
L

_{1}, L_{2}and L_{3 }= Length of section 1, section 2 and section 3 respectively
Ïƒ

_{ 1}, Ïƒ_{ 2}and Ïƒ_{ 3 }= Stress induced for the section 1, section 2 and section 3 respectively
Îµ

_{1}, Îµ_{ 2}and Îµ_{ 3 }= Strain developed for the section 1, section 2 and section 3 respectively
E= Young’s
Modulus of the bar

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*Let us see here stress and strain
produced for the section 1*

*Let us see here stress and strain produced for the section 1*

Stress, Ïƒ

_{1}= P / A_{1}
Strain, Îµ

_{1}= Ïƒ_{1}/E
Strain, Îµ

_{1}= P / A_{1}E####
*Similarly stress and strain produced
for the section 2*

*Similarly stress and strain produced for the section 2*

Stress, Ïƒ

_{2}= P / A_{2}
Strain, Îµ

_{2}= Ïƒ_{2}/E
Strain, Îµ

_{2}= P / A_{2}E####
*Similarly stress and strain produced
for the section 3*

*Similarly stress and strain produced for the section 3*

Stress, Ïƒ

_{3}= P / A_{3}
Strain, Îµ

_{3}= Ïƒ_{3}/E
Strain, Îµ

_{3}= P / A_{3}E
Now we
will determine change in length for each section with the help of definition of
strain

Î”L

_{1}= Îµ_{1}L_{1}
Î”L

_{2}= Îµ_{2}L_{2}
Î”L

_{3}= Îµ_{3}L_{3}###
**Total change in length of the bar **

As we have already discussed above that when we will go to determine the total change in length of the bar of varying sections, we will have to add change in length of each section of the bar.

Î”L= Î”L

_{1}+ Î”L_{2}+Î”L_{3}
Let us
consider that Young’s modulus of elasticity of each section is different, in
this situation we will have following equation to determine the total change in
length of the bar.

Do you
have any suggestions? Please write in comment box

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**Reference:**

Strength
of material, By R. K. Bansal

Image
Courtesy: Google

We will
see another important topic i.e. stress analysis of bars of composite sections in
our next post.

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