Topic -B) Bars of varying sections

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 the bar is different and therefore stress-induced, strain and change in length too will be different for each section of the bar.

Axial load for each section will be the 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 the change in length of each section of the 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

e1, e 2 and e 3 = Strain developed for the section 1, section 2 and section 3 respectively

E= Young’s Modulus of the bar

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

 

Section 1	Section 2	Section 3
Stress, σ1= P / A1	σ2= P / A2	σ3= P / A3
Strain, e1 = σ1/E Strain, e1 = P / A1E Strain e1= δL1 /L1	Strain, e2 = σ2/E Strain, e2 = P / A2E Strain , e2= δL2 /L2	Strain, e3 = σ3/E Strain, e3 = P / A3E Strain,e3= δL3/L3
Change in Dimension(δL1 ) δL1  =[(P1  L1 / A1  E)]	Change in Dimension(δL2 δL2  =[(P2 L2 / A2  E)]	Change in Dimension(δL1 ) δL3  =[(P3 L3/ A3  E)]

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 the change in length of each section of the bar. (ΔL=δL)

ΔL= ΔL1+ ΔL2+ΔL3   

Let us consider that Young’s modulus of elasticity of each section is different, in this situation we will have the following equation to determine the total change in length of the bar.