What Temperature Must an Iron Band Be Heated to Fit a Barrel?

[e(ho0n3]

Homework Statement

A barrel of diameter D at 20 degrees Celsius is to be enclosed by an iron band. The circular band has an inside diameter of d at 20 degrees Celsius. It is w wide and t thick. (a) To what temperature must the band be heated so that it will fit over the barrel? (b) What will be the tension in the band when it cools to 20 degrees Celsius?

Homework Equations

For (a) I will need the equations to calculate the volume of the band and the change in volume of the band when heated. For (b), I will need the equation for determining thermal stress.

The Attempt at a Solution

For (a), is it safe to assume that the wideness of the band remains the same while the thickness of the band expands linearly? If so, then I can calculate the the desired volume after heating and use the volume expansion formula to determine the change in temp.

For (b), the equation for thermal stress has area as one of its variables. The area is some cross-section of the material. In the case of the band, will the cross-section used be t x w?

 

[Doc Al]

For (a), is it safe to assume that the wideness of the band remains the same while the thickness of the band expands linearly? If so, then I can calculate the the desired volume after heating and use the volume expansion formula to determine the change in temp.

All linear dimensions expand by the same factor. No need to consider volume.

For (b), the equation for thermal stress has area as one of its variables. The area is some cross-section of the material. In the case of the band, will the cross-section used be t x w?

That seems reasonable.

 

[e(ho0n3]

All linear dimensions expand by the same factor. No need to consider volume.

Does that mean that the radius of the band expands linearly? I'm playing it safe by dealing with the volume because I don't know the answer to this question.

 

[Doc Al]

Does that mean that the radius of the band expands linearly?

Yes it does.

 

[e(ho0n3]

Why is that? Suppose the band is a thin ring that is not closed. Does the radius still expand or does the gap in the ring just get smaller when the ring is heated? I would suspect the latter.

 

[Doc Al]

All linear dimensions expand the same way (assuming the material is isotropic). The radius will expand, as will the gap.