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Chapter 13: Problem 84

An iron bridge girder $\left(Y=2.0 \times 10^{11} \mathrm{N} / \mathrm{m}^{2}\right)$ is constrained between two rock faces whose spacing doesn't change. At \(20.0^{\circ} \mathrm{C}\) the girder is relaxed. How large a stress develops in the iron if the sun heats the girder to $40.0^{\circ} \mathrm{C} ?$

Short Answer

Expert verified
Answer: The stress developed in the iron girder is \(4.8 \times 10^{7} \mathrm{N/m^2}\).

Step by step solution

01

Calculate the thermal strain

To calculate the thermal strain, we need to find the change in temperature and multiply it by the coefficient of linear expansion (α) of the material. For iron, α = \(1.2 \times 10^{-5} \mathrm{K}^{-1}\). The change in temperature, ΔT, can be found by subtracting the initial temperature from the final one: ΔT = T_final - T_initial = \(40.0^{\circ} \mathrm{C}\) - \(20.0^{\circ} \mathrm{C}\) = \(20.0^{\circ} \mathrm{C}\). Now, we can calculate the thermal strain (ε) as: ε = α × ΔT = \(1.2 \times 10^{-5} \mathrm{K}^{-1}\) × 20.0 K = \(2.4 \times 10^{-4}\).
02

Calculate the stress using Young's modulus

Now that we have the thermal strain, we can use Young's modulus (Y) to find the stress (σ) developed in the girder. The relationship between stress, strain, and Young's modulus is given by: σ = Y × ε. We can plug in the values for Y and ε to find the stress: σ = \(2.0 \times 10^{11} \mathrm{N/m^2}\) × \(2.4 \times 10^{-4}\) = \(4.8 \times 10^{7} \mathrm{N/m^2}\). So, the stress developed in the iron girder when the temperature increases from \(20.0^{\circ} \mathrm{C}\) to \(40.0^{\circ} \mathrm{C}\) is \(4.8 \times 10^{7} \mathrm{N/m^2}\).

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Most popular questions from this chapter

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