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.

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.

### 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
A1, A2 and A3 = Area of cross section of section 1, section 2 and section 3 respectively
L1, L2 and L3 = Length of section 1, section 2 and section 3 respectively
σ 1, σ 2 and σ 3 = Stress induced for the section 1, section 2 and section 3 respectively
ε1, ε 2 and ε 3 = Strain developed for the section 1, section 2 and section 3 respectively
E= Young’s Modulus of the bar

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

Stress, σ1= P / A1
Strain, ε1 = σ1/E
Strain, ε1 = P / A1E

#### Similarly stress and strain produced for the section 2

Stress, σ2= P / A2
Strain, ε2 = σ2/E
Strain, ε2 = P / A2E

#### Similarly stress and strain produced for the section 3

Stress, σ3= P / A3
Strain, ε3 = σ3/E
Strain, ε3 = P / A3E
Now we will determine change in length for each section with the help of definition of strain
ΔL1= ε1L1
ΔL2= ε2L2
ΔL3= ε3L3

### 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.
ΔL= ΔL1+ ΔL2+ΔL3
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

### Reference:

Strength of material, By R. K. Bansal