Question

A steel beam is used in the roadbed of a bridge. The beam is mounted between two concretesupports when the temperature is 23 [oC], with no room provided for thermal expansion. Whatcompressional stress must the concrete supports apply to each end of the beam, if they are to keep the beam from expanding when the temperature rises to 42 [oC]

Answer 1

13.86 MPa

Step-by-step explanation:

We need to find the change in length ΔL of the steel beam and thus, the strain in the beam, ε.

ΔL = L₀αΔθ

Strain, ε = ΔL/L₀ = αΔθ where α = coefficient of linear expansion of steel = 11 × 10⁻⁶ °C and Δθ = temperature change of steel beam = θ₂ - θ₁ where θ₁ = initial temperature of steel beam = 23 °C and θ₂ = final temperature of steel beam = 42 °C

Substituting the values of the variables into the equation, we have

ε = αΔθ

ε = α(θ₂ - θ₁)

ε = 11 × 10⁻⁶ /°C(42 °C - 23 °C )

ε = 11 × 10⁻⁶ /°C(42 °C)

ε = 462 × 10⁻⁶

ε = 4.62 × 10⁻⁴

Now, the Young's modulus of concrete is Y = 30 GPa = 30 × 10⁹ N/m²

Y = σ/ε where σ = compressional stress from concrete and ε = strain on steel

So, σ = Yε

σ = 30 × 10⁹ N/m² × 4.62 × 10⁻⁴

σ = 138.6 × 10⁵ N/m²

σ = 1.386 × 10² × 10⁵ N/m²

σ = 1.386 × 10⁷ N/m²

σ = 13.86 × 10⁶ N/m²

σ = 13.86 MPa