Thermal Expansion of Joined Plates - Finding Tension in Copper Plate

[zorro]

Homework Statement

A copper plate of length 1 m is riveted to two steel plates of same length and same cross-section area at 0°C. The tension (in kilo Newton) generated in copper plate when heated to 20°C is 56n. Find the value of n. (Given Ys,Yc,αs,αc,A)

The Attempt at a Solution

I googled 'rivet' and found that 2 plates are joined with a rivet. We have 3 plates here. Can someone draw a diagram of the situation above?

 

[tiny-tim]

Hi Abdul!

I googled 'rivet' and found that 2 plates are joined with a rivet. We have 3 plates here. Can someone draw a diagram of the situation above?

I think the idea is that, if the copper was riveted to only one steel plate and heated, then the two plates would bend …

so they make a sandwich of it, between two steel plates, so that everything stays straight!

 

[zorro]

ok I can picture it in my mind now.
Lets solve the problem.
The copper plate is under mechanical stress.
Let the tension developed in it be F
Net expansion of the copper plate = lαcΔT - Fl/AYc
Net expansion of one of the steel plates = lasΔT + Fl/AYs

What is the next step?

 

[tiny-tim]

ok, they can't be those lengths, because they are forced to be the same length as each other …

so what are the forces involved? ​

 

[zorro]

Is the above diagram right? The plates expand along the horizontal direction. There can be a difference in their expansions which can lead to different 'l' values for each.

 

[tiny-tim]

the diagram looks ok (but what are those letters in the middle?)

There can be a difference in their expansions …

There cannot be a difference in their expansions …

they are riveted together, and so they must have the same expansion

 

[zorro]

S-Steel
C-Copper
S-Steel

They can have different expansions along horizontal direction. In simple words, there is a lateral as well as longitudinal expansion of the system. If they are riveted together, which expansion will be same?

 

[tiny-tim]

oh i see … yes, their lateral expansions won't be the same, so the lateral compressions will be different

but the question is asking about the longitudinal expansion, and the tension which that creates in the copper

 

[zorro]

I got the answer. The only confusion was in lateral/longitudinal expansion.
Thank you!