~~A thin bimetallic strip of thickness ~~*d* is straight at a temperature *T*_{0}, with length *L*_{0}. The two metals have coefficients of linear thermal expansion *α*_{1} and *α*_{2}, with *α*_{2}>*α*_{1}. If the temperature is raised to a temperature *T* not significantly greater than *T*_{0}, what is the angle *θ* through which the strip bends?

We should expect the curvature to be uniform, leaving the strip to form a circular arc of angle *θ*. If the radius of curvature for the inner half of the strip is *r*, then (with *θ* in radians), the length of that metal is *L*_{1}=*r**θ*.

We can ignore any expansion in the direction of the thickness, so the length of the other metal is

*L*_{2}=(*r*+*d*)*θ*.

Taking the difference, we see

*L*_{2}–*L*_{1}=*d**θ*,

so .

Now, the formula for thermal expansion is

(see this previous post),

so for the inner side,

,

and solving for *L*_{1},

.

Analogously,

.

Thus, the difference is

,

and so the angle *θ* is

.

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Tags: Bimetallic Strip, Friday Physics, physics

This entry was posted on December 24, 2010 at 12:04 am and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed.
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December 25, 2010 at 7:30 pm |

I think you have over simplified this problem. You have neglected the fact that, as a bimetalic strip, they are bonded on the interface and thus must expand/contract exactly the same amount on the interface. This is the same problem as that of bending a deck of cards that are free to shear versus being a deck of cards that have been glued into a solid block; they respond differently.

The problem you are looking at here was treated rather we by Stephen Timoshenko in his classic work Strength of Materials, Pt. 1 Elementary Problems, Van Nostrand, 1930. The Youngs modulii and area moments of inertia for the two segments are involved because this is a beam bending problem.

January 6, 2011 at 9:40 pm |

You’re right, and I should’ve known that. Checking, it seems that even assuming equal thicknesses and Young’s moduli, the above analysis is off by a factor of 3/2.