# Mechanics of Materials: stress due to thermal expansion

1. May 24, 2010

### steak313

1. The problem statement, all variables and given/known data

The magnesium allow tube (AM1004-T61) AB is capped with a rigid plate. The gap between E and end C of 6061-T6 alluminum alloy solid circular rod CD is d[m] when the temperature is $$T_1$$ degrees celcius.

[PLAIN]http://img27.imageshack.us/img27/8867/problemwn.jpg [Broken]

The left tube is fixed on the left end. The right shaft is fixed on the right side.

Find the maximum temperature increase from $$T_1$$ such that the normal stress at the interface between the tube and the rod is 0.

2. Relevant equations

Well for some reason no matter what I change the latex text to, something completely unrelated shows up in its place. Not sure how to fix this but I want it to say:
(change in length)= (alpha)(change in temperature)(Length).

sigma=(pressure)/(cross sectional area)

3. The attempt at a solution

Hopefully the image shows up in a timely manner. It is a pretty crude sketch so if any clarification is needed that is fine. Let me also state that I am essentially trying to learn the material as I do this so bare with me.

My First thoughts were that the tubes would elongate as the temperature increases. As the they enlongate, the distance between the two will get smaller and smaller until they are pressing against one another. My assumption is that I need to show the temperature increase such that the force from the two shafts are equal but opposite at the interface.

Before I begin to tackle the problem, would anyone be interested in commenting on my approach or my assumptions? Is this in fact what the problem is asking of me?

Last edited by a moderator: May 4, 2017
2. May 24, 2010

### PhanthomJay

The problem is asking for the max temp increase such that there is no contact force between the tube and rod (N=0). There is obviously no contact force at T1 with that gap. And there still won't be any with a temp increase when they are just on the verge of touching. There won't be any stress, either.

3. May 24, 2010

### steak313

Thank you very much for your response!

Okay so essentially increasing the temperature will elongate the shafts. I need to find the temperature when the combined length increase is d[m].

For AB $$\delta_1 = \alpha_1 (T_2 - T_1)L_AB$$

For CD $$\delta_2 = \alpha_2 (T_2 - T_1)L_CD$$

I set the sum of these two equations equal length d[m] then solved for $$T_2$$

$$d = \alpha_1 (T_2 - T_1)L_AB + \alpha_2 (T_2 - T_1)L_CD$$

after some manipulation

$$T_2 = \frac{d}{\alpha_1 L_ab + \alpha_2 L_CD} + T_1$$

Last edited: May 25, 2010
4. May 25, 2010

Looks good!