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Question

Two steel rods and an aluminium rod of equal length l0 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 C. Find the length of the system when the temperature is raised to θ. Coefficient of linear expansion of aluminium and steel are αa and αs respectively. Young's modulus of aluminium is Ya and of steel is Ys.

SteelAluminiumSteel


Solution

Let the initial length of the system at 0 C =l0

When temperature changes by θ

Final length of the a tool at system temperature

So, strain of the system =(ll0)l0

SteelAluminiumSteel

But the total strain of the system

=total stress of systemtotal Young's modulus of system

Now, total stress due to two steel rod + stress due to aluminium

=γsαsθ+γsαsθ+γAl αAlθ

=2γs αs θ+γAl αAlθ

Now Young's modulus system

=γs+γs+γAl=2γs+γAl

Strain of system =2γsαsθ+γAlαAlθ2γs+γAl

  lll0=2γsαsθ+γAlαAlθ2γs+γAl

    l=l0[1+2γsαsθ+2γAlαAlθ2γs+γAl]

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