Force in a bar due to thermal expansion or compression
If a metal bar
is heated it will expand, but if we prevent it from expanding then large
forces will be set up within the bar. The force will be that which would
be needed to compress the bar to its original length (L) from the
expanded position.
Consider the bar shown in Figure
1.
Let the linear expansivity of the material be α and its Young
modulus be E.
For a rise in temperature of θo the bar will expand by an amount e, given
by e = Lαθ but the extension produces a force
in the wire of EeA/L
Therefore Lαθ = FL/eA and so:
Force (F) in bar due to thermal expansion or compression:
F = EAαθ
Example problems
A steel bar with a cross-sectional area of 2 cm2 is heated, raising its temperature by 120oC and prevented from expanding. Calculate the resulting force in the bar.
Young modulus for steel = 2.1x1011 Pa, linear expansivity = 0.000012 m-1
Force (F) = EAαθ = 2.1x1011x2x10-4x0.000012x120 = 6.05x104 N
Student investigation
You may have seen the classic experiment where a cast iron peg is broken by the large forces set up in a steel bar due to thermal expansion or com¬pression (Figure 2).
Look up the breaking stress for an iron peg such as one used in the experiment (diameter 4 mm) and use it to estimate the mean temperature of the steel bar.
The strength of rivets is explained by this experiment.
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