A rod of steel is 20 m long at a temperature of 20°C. What will be the temperature stress produced when the temperature of the rod is raised to 70°C? The rod is permitted to expand by 6 mm. [Coefficient of linear expansion = 12 × 10^-6 per °C, Modulus of elasticity. = 2 × 10⁵ N/mm²]

Explanation:

The formula for thermal expansion is given by:

ΔL_thermal = L_initial × α × ΔT

Where:

• ΔL_thermal = Thermal expansion (in meters)
• L_initial = Initial length of the rod = 20 m = 20,000 mm
• α = Coefficient of linear expansion = 12 × 10^-6 per °C
• ΔT = Temperature change = 70°C - 20°C = 50°C

Substituting the values, we get:

ΔL_thermal = 20,000 mm × 12 × 10^-6 per °C × 50°C

ΔL_thermal = 20,000 × 0.000012 × 50

ΔL_thermal = 12 mm

Step 2: Determine the actual expansion allowed.

According to the problem, the rod is permitted to expand by 6 mm.

Step 3: Compute the difference between the unconstrained expansion and the actual allowed expansion.

The difference in expansion, ΔL_difference, is given by:

ΔL_difference = ΔL_thermal - ΔL_allowed

Substituting the values, we get:

ΔL_difference = 12 mm - 6 mm

ΔL_difference = 6 mm

Step 4: Use the modulus of elasticity to find the stress produced due to this difference in expansion.

The formula for stress, σ, is given by:

σ = (E × ΔL_difference) / L_initial

Where:

• σ = Stress (in N/mm²)
• E = Modulus of elasticity = 2 × 10⁵ N/mm²
• ΔL_difference = Difference in expansion = 6 mm
• L_initial = Initial length of the rod = 20,000 mm

Substituting the values, we get:

σ = (2 × 10⁵ N/mm² × 6 mm) / 20,000

σ = (1,200,000 N/mm²) / 20,000

σ = 60 N/mm²

Therefore, the temperature stress produced in the steel rod is 60 N/mm².