Question

A steel rod 15 m long is at a temperature of 15°C. The values of α = 12 × 10^-6m and E = 200 GN/m² are adopted. When the temperature is raised to 65°C, what is the free expansion of the length; and if this expansion of the rod is fully prevented, what is the temperature stress produced?
1. 5 mm and 120 MN/m²
2. 9 mm and 120 MN/m²
3. 5 mm and 150 MN/m²
4. 9 mm and 150 MN/m²

Answer 1

Correct Answer - Option 2 : 9 mm and 120 MN/m²

Concept:

Ordinary materials expand when heated and contract when cooled, hence, an increase in temperature produces a positive thermal strain.

Change in Length, δL = α L ΔT

Strain = δL /L = αLΔT/L = αΔT

Stress = strain × E = α.ΔT.E

Where α = coefficient of linear expansion for the material, L = original Length, ΔT = temp. change

Calculation:

Initial Temperature, T₀ = 15°C

Final Temperature, T_F = 65°C

L = 15 m, α = 12 × 10^-6 m, E = 200 GN/m²

Free expansion in length, ΔL = L ∝ ΔT

= 15 × 12 × 10^-6 × (65 - 15)

ΔL = 9 × 10^-3 m

Now thermal strain, \({\varepsilon _{th}} = \frac{{{\rm{\Delta }}L}}{L} = \frac{{9 × {{10}^{ - 3}}}}{{15}}\)

ε_th = 6 × 10^-4

Thermal stress induced when free expansion is prevented.   

σ_th = (ε_th)_prevented × E

σ_th = (6 × 10^-4) × (200 × 10³) MN/m²

∴ σ_th = 120 MN/m²