STRESS ANALYSIS OF BARS OF VARYING 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 , 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

σ₁, σ₂ and σ₃= Stress induced for the section 1, section 2 and section 3 respectively

ε₁, ε₂ and ε₃= 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.