## 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**

**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**

**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