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