Thermal Expansion and temperature changes

[stinlin]

The problem is listed. I can't figure out thermal expansion really...if there was one material, I know you can say that the displacement due to thermal expansion is equal to the displacement due to the loadings via the wall (that's why normal stresses are developed in that case). But, three different materials means, well...I don't know. I don't understand how to accommodate for the fact that as the steel expands, the bronze pushes against it so does the wall. This is the case for each material.

How do you go about determining this situation? If any of the numbers are unclear, let me know and I'd be happy to make them clearer via typing. =) Thanks...

http://img160.imageshack.us/img160/5457/picture6ky7.jpg [Broken]

 

[Q_Goest]

How much does each section 'grow' when the temperature changes?

How much would each section be compressed when put under a uniform compressive stress?

If the distance between the two walls remains constant throughout the change in temperature and change in stress levels, then isn't there a correlation between these two?

<this belongs in homework section>

 

[AlephZero]

Do the same steps as you did as for one material. Don't try to find out what happens to each rod separately.

1 If the walls were not there, find the total amount the three rods would expand.
2 Work out the equivalent stiffness of the 3 rods in series with each other.
3 Find the force applied by the walls to compress them back to the original length, from the stiffness and the displacement.

 

[stinlin]

The thing is, the equation is PL/AE for the "restoring" force...I don't know how to find things like that in this case. Normally, the L would cancel, leaving P = alpha*delta T* AE. Urgh...Confused

 

[stinlin]

Oh yeah, and this really doesn't belong in the homework section because it's not homework. Haha - I'm doing it because I need to thoroughly UNDERSTAND thermal expansion, and I picked that problem to do because it had many concept thrown into one.

 

[AlephZero]

The thing is, the equation is PL/AE for the "restoring" force...I don't know how to find things like that in this case. Normally, the L would cancel, leaving P = alpha*delta T* AE. Urgh...Confused

If you had a single rod then the force would be P = (AE/L)x where L is the length of the rod and x is the change in length (x = alpha * delta T * L for thermal expansion).

The rod is behaving the same as a spring of stiffness K = AE/L.

What you have is 3 "springs" (rods) with different stiffnesses, joined end to end. Work out the equivalent stiffness for the whole rod. If you don't remember the formula for the combining the different stiffnesss, it's easy to work it out - forget about the expansions, just consider 3 springs of stiffness k1 k2 and k3, fixed at one end and with a force F at the other end, find the total extension, and the combined stiffness = force / total extension)

Finding the extension of the three rods individually is a harder problem, and it doesn't ask you to do that. FWIW, solving the given problem would be a good first step for finding the individual extensions, not the other way round.