Two
rods of different materials bur having the same lengths
L and
cross-sectional area A are arranged end-to-end between fixed rigid
supports. The temperature is T and
there is no initial stress. The rods are heated, so that their
temperature increases by
where
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Solution: Click For PDF Version As the rods gets heated they undergo linear thermal expansion, but as there ends are clamped with rigid supports stress develops at the joint of the two rods. Let S be the stress at the interface when the temperature has increased by .
The change in length of each rod is the combined effect of thermal
expansion and elastic change due to stress.
As shown in
the diagram (b) the length of the rod 1 after heating becomes
Because of the
stress length of rod 1 will decrease by
We have two
linear equations and two unknowns,
and
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and
are the coefficients of linear expansion and
,
are Young’s moduli of the materials. We may ignore changes in
the cross-sectional areas. (b) We
have to find the stress at the interface after heating.
and that of the rod 2 becomes
.
and that of the rod 2 by
.
And because of change in temperature the length of the rod 1 will
increase by
and the length of the rod 2 will increase by
.
The resultant changes in lengths are
and S. By solving the
algebraic equations, we find