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STRESS ANALYSIS OF BARS OF COMPOSITE SECTIONS

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.

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.

A1 and A2 = Area of cross section of bar 1 and bar 2 respectively
E1 and E2 = Young’s modulus of elasticity for material of bar 1 and material of bar 2 respectively
P1 and P2 = Load shared by bar 1 and bar 2 respectively
σ1 and σ2 = Stress induced in bar 1 and bar 2 respectively
 
As 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 = P1 + P2
Stress induced in bar 1, σ1= P1 / A1
Stress induced in bar 1, σ2= P2 / A2
P = σ1A1 + σ2 A2

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, Ԑ1= σ1/ E1
Strain in bar 2, Ԑ2 = σ2/ E2

From above statement that strains will be same for each bar of composite bar, we will have following equation.

σ1/ E1 = σ2/ E2

Conclusion

We have secured here two very important equations as mentioned below

σ1/ E1 = σ2/ E2
P = σ1A1 + σ2 A2

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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