When we examine bars with varying shapes, material characteristics, and dimensions, which are subjected to axial forces, we aim to comprehend the resulting stress and strain values. This analysis is rooted in Hooke’s law, a fundamental equation expressing the relationship between stress (f) and strain (e):
f=e⋅E (Eq.1)
Where ‘E’ stands for the modulus of elasticity. In this exploration, we will delve into the analysis of strain or deformation in bars with varying sections.
Consider a bar with distinct lengths, cross-sectional areas, and diameters, exposed to an axial load ‘P,’ as depicted.
In this scenario, the stress and deformation in each section of the bar differ, despite the uniform axial load. The derivation is grounded in the principle that the total change in length (dL) is the sum of changes in length in Section 1, Section 2, and Section 3:
dL=dL1+dL2+dL3 (Eq.2)
In Section 1, stress (f1) is calculated as the load (P) divided by the area of Section-1 (A1):
f1=PA1
The corresponding strain (e1) is determined by the change in length (dL1) divided by the original length (L1):
e1=dL1L1=f1E=PA1E (Eq.3)
Therefore, the change in length (dL1) is given by:
dL1=PL1A1E
Similar calculations are performed for Section 2, where stress (f2) and strain (e2) are expressed in terms of the load, area, and modulus of elasticity:
dL2=PL2A2E (Eq.4)
Section 3 undergoes a comparable analysis, with stress (f3) and strain (e3) derived as:
dL3=PL3A3E (Eq.5)
The total change in length (dL) across all sections is the sum of dL1, dL2, and dL3:
dL=PL1/(A1E)+PL2/(A2E)+PL3/(A3E) (Eq.6)
This equation is applicable when the bar sections share the same modulus of elasticity. If ‘E’ varies across sections, a different relationship emerges.