Solution to Problem 268 Thermal Stress

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Problem 268
The rigid bar ABC in Fig. P-268 is pinned at B and attached to the two vertical rods. Initially, the bar is horizontal and the vertical rods are stress-free. Determine the stress in the aluminum rod if the temperature of the steel rod is decreased by 40°C. Neglect the weight of bar ABC.

Solution 268

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Contraction of steel rod, assuming complete freedom:
$\delta_{T(st)} = \alpha \, L \, \Delta T$

$\delta_{T(st)} = (11.7 \times 10^{-6})(900)(40)$

$\delta_{T(st)} = 0.4212 \, \text{mm}$

The steel rod cannot freely contract because of the resistance of aluminum rod. The movement of A (referred to as δ_A), therefore, is less than 0.4212 mm. In terms of aluminum, this movement is (by ratio and proportion):

$\dfrac{\delta_A}{0.6} = \delta_{al}{1.2}$