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 and types of modulus of elasticity in
our recent post.

Now we are
going further to start our discussion to understand the “Stress analysis of
bars of composite sections”, in subject of strength of material, with the help
of this post.

###
**Let us see here the stress analysis
of bars of composite sections **

First we
will understand here, what is a composite bar?

Composite bar is basically defined as a bar made by two or more than two bars of similar length but different materials and rigidly fixed with each other in such a way that it behaves as one unit and strain together against external load i.e. it behaves as single unit for compression and extension against compressive and tensile load.

Composite bar is basically defined as a bar made by two or more than two bars of similar length but different materials and rigidly fixed with each other in such a way that it behaves as one unit and strain together against external load i.e. it behaves as single unit for compression and extension against compressive and tensile load.

We can say
here from the definition of composite bar that strains will be same for each
bar of composite bar and hence stress produced in each bar of composite bar
will be dependent over the young modulus of elasticity of respective material.

So let us
see here the important points in respect of composite bar, those are very important
to keep in mind during stress analysis of bars of composite sections.

As we have
seen above that composite bar will have two or more than two bars of similar
length and these bars will be rigidly fixed with each other and therefore
change in length will be similar for each bar or we can say that strains will
be same for each bar of composite bar.

Total
external load which will be acting over the composite bar will be shared by the
each bar of composite bar and hence we can say that total external load on
composite bar will be equal to the addition of the load shared by each bar of
composite bar.

Let us see
here the following figure which represents two bar of similar length but
different materials and rigidly fixed with each other to form one composite
bar.

A

_{1}and A_{2 }= Area of cross section of bar 1 and bar 2 respectively
E

_{1}and E_{2}= Young’s modulus of elasticity for material of bar 1 and material of bar 2 respectively
P

_{1}and P_{2 }= Load shared by bar 1 and bar 2 respectively
σ

_{1}and σ_{2 }= Stress induced in bar 1 and bar 2 respectivelyAs we have already discussed that total external load which will be acting over the composite bar will be shared by the each bar of composite bar and therefore we will have following equation.

P = P

_{1}+ P_{2}
Stress
induced in bar 1, σ

_{1}= P_{1}/ A_{1}
Stress
induced in bar 1, σ

_{2}= P_{2}/ A_{2}
P = σ

_{1}A_{1}+ σ_{2}A_{2}
We have
also discussed above that composite bar will have two or more than two bars of
similar length and these bars will be rigidly fixed with each other and
therefore change in length will be similar for each bar or we can say that
strains will be same for each bar of composite bar.

Strain in
bar 1, Ԑ

_{1}= σ_{1}/ E_{1}
Strain in
bar 2, Ԑ

_{2 }= σ_{2}/ E_{2}
From above
statement that strains will be same for each bar of composite bar, we will have
following equation.

σ

_{1}/ E_{1}= σ_{2}/ E_{2}###
**Conclusion**

We have
secured here two very important equations as mentioned below

####
σ_{1}/
E_{1} = σ_{2}/ E_{2}
P = σ_{1}A_{1}
+ σ_{2} A_{2}

_{1}/ E

_{1}= σ

_{2}/ E

_{2}

_{1}A

_{1}+ σ

_{2}A

_{2}

These equations
will be required for determining the value of stresses in case of bars of
composite sections.

Do you
have any suggestions? Please write in comment box

###
**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google

We will see another important topic i.e. Thermal
stress in strength of material in our next post.

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