## Subway Hand Rail

Consider a 10 m long hand rail down a subway station is rigidly attached at both ends. The ambient temperature is 35°C during the summer, and drops to -15°C during the winter. What axial stress does the hand rail experience due to temperature fluctuations if the material's Young's Modulus = 200 GPa, and Coefficient of Thermal Expansion = 12 x 10^-6/°C?

Hint
$$\delta =\alpha \Delta TL$$$where $$\delta$$ is the deformation caused by change in temp, $$\alpha$$ is the coefficient of thermal expansion, $$L$$ is the member length, and $$\Delta T$$ is the temperature change Hint 2 Engineering strain is the change in length divided by the original length: $$\varepsilon =\frac{\delta}{L}$$$
Thermal deformations:
$$\delta =\alpha \Delta TL$$$where $$\delta$$ is the deformation caused by change in temp, $$\alpha$$ is the coefficient of thermal expansion, $$L$$ is the member length, and $$\Delta T$$ is the temperature change $$\delta =(12\cdot 10^{-6}\cdot ^{\circ}C^{-1}) (35^{\circ}C-(-15^{\circ}C))(10m)=0.006\:m$$$
Since engineering strain is the change in length divided by the original length:
$$\varepsilon =\frac{\delta}{L}=\frac{0.006m}{10m}=0.0006$$$Finally, to find the axial stress due to temperature deformation: $$\sigma=E \times \varepsilon$$$
where $$E$$ is the Young's Modulus and $$\varepsilon$$ is the engineering strain. Thus,
$$\sigma= (200\cdot 10^9Pa)(0.6\cdot 10^{-3})=120\:MPa$$\$
120 MPa