Two steel rods and an aluminium rod of equal length $l_{0}$ and equal cross-section are joined rigidly at their ends as shown in the figure below. All the rods are in a state of zero tension at $0^{\circ} C$ .
Find the length of the system when the temperature is raised to $θ$ . Coefficient of linear expansion of aluminium and steel are $α_{a}$ and $α_{s}$ respectively. Young's modulus of aluminium is $Y_{a}$ and of steel is $Y_{s}$ .

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Solution

Initial length of system (two steal rods + aluminimum rod )
at $0^{0} C = l_{0}$
Coefficient of lines expansion of steal $l$ at all $α_{a}$ and $α_{s}$
$△ T = θ - O^{0} C = θ$

If $l$ is final length at $θ$
strain on system $= \frac{l - l_{0}}{l_{0}}$

$Y = \frac{S t r e s s}{s t r a i n}$

$∴ s t r a i n = \frac{T o t a l s t r e s s o n s y s t e m}{T o t a l Y o u n g^{'} s m o d e o f s y s t e m}$

Total stress $= Y_{s} α_{s} θ + Y_{s} α_{s} θ + Y_{A l} α_{A L} θ$

$Y = 2 Y_{s} + Y_{A l}$

$S t r a i n = \frac{2 Y_{s} α_{s} θ + Y_{A l} α_{A l} θ}{2 Y_{s} + Y_{A l}} - \frac{l - l_{0}}{l}$

$l = l_{0} [\frac{1 + 2 Y_{s} α_{s} θ + Y_{A l} α_{A l} θ}{2 Y_{s} + Y_{A l}}]$

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Two steel rods and an aluminium rod of equal length $l_{0}$ and equal cross section are joined rigidly at their ends as shown in the figure below. All the rods are in a state of zero tension at $0^{\circ} C$ . Find the length of the system when the temperature is raised to $θ$ . Coefficient of linear expansion of aluminium and steel are $α_{a}$ and $α_{s}$ respectively. Young's modulus of aluminium is $Y_{a}$ and of steel is $Y_{s}$ .