Question

When a rod of circular cross-section is fixed at one end and subjected to an axial load of 500 N, the deflection under the load is found to be 2.4 mm, what will be the amount of deflection under the same load if the dia of rod is doubled & length is reduced to half of the original length?

1. 1.2 mm
2. 0.6 mm
3. 0.3 mm
4.  0.15 mm

Answer 1

Correct Answer - Option 3 : 0.3 mm

Concept:

Deflection of circular bar under axial loading​​

\(δ = \frac{4PL}{\pi d^2E}\)

where P = load, L = length, d = diameter of cross sectional area, E = Young's modulus

Calculation:

Given:

For case-1: P₁ = 500 N, δ₁ = 2.4 mm, L₁ = L, d₁ = d, δ₁ = 2.4 mm

For case-2: P₂ = 500 N, δ₂ = 2.4 mm, L₂ = \(\frac{L}{2}\), d₂ = 2d

let E be the Young's modulus for both the cases, hence ratio of deflecction in both the case is;

\(\frac{δ_1}{δ_2} = \frac{\frac{4PL_1}{\pi d_1^2E}}{\frac{4PL_2}{\pi d_2^2E}}\)

\(\frac{δ_1}{δ_2} = \frac{\frac{L}{d^2}}{\frac{L}{2(2d)^2}}\)

\(\frac{δ_1}{δ_2} = 8\)

\(\frac{δ_2}{δ_1} = \frac{1}{8}\)

\(δ_2 = \frac{1}{8}\times 2.4\)

δ₂ = 0.3 mm

Hence deflection under the same load for second case is 0.3 mm