Question

A tension member of square cross-section of side 10 mm and Young’s modulus E is to be replaced by another member of square cross-section of same length but Young’s modulus E/2. The side of the new square cross-section, required to maintain the same elongation under the same load, is nearly
1. 14 mm
2. 17 mm
3. 8 mm
4. 5 mm

Answer 1

Correct Answer - Option 1 : 14 mm

Calculation:

Given:

L₁ = L₂ = L, P₁ = P₂ = P, ∆₁ = ∆₂

Initially side of square = 10 mm

A₁= 100 mm²

Let the new side be ‘a’

Deflection of the beam is given by:

\(\Delta = \frac{{{\rm{PL}}}}{{{\rm{AE}}}}\)

∵ ∆₁ = ∆₂

∴ \(\frac{{{\rm{PL}}}}{{{{\rm{A}}_1}{{\rm{E}}_1}}} = {\rm{}}\frac{{{\rm{PL}}}}{{{{\rm{A}}_2}{{\rm{E}}_2}}}\)

\(\frac{{{{\rm{A}}_2}}}{{{{\rm{A}}_1}}} = {\rm{\;}}\frac{{{{\rm{E}}_1}}}{{{{\rm{E}}_2}}} = {\rm{}}\frac{{\rm{E}}}{{{\rm{E}}/2}} = 2\)

∴ \({{\rm{A}}_2} = 2{\rm{}} \times {{\rm{A}}_1}\)

a² = 200

a = \(\sqrt {200}\) = 14.14 mm ≃ 14 mm