Mechanics of Materials: stress due to thermal expansion

[steak313]

Homework Statement

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

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.

Homework 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)

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?

 

[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.

 

[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

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

 

[PhanthomJay]

Looks good!

 

[DeeNos]

Your approach seems to be on the right track. When materials are subjected to thermal expansion, they can experience stress due to the change in dimensions. In this case, the magnesium alloy tube and the aluminum alloy rod will have different coefficients of thermal expansion, causing them to expand at different rates. This will result in a gap between the two components when the temperature is increased.

To find the maximum temperature increase, you will need to consider the stress at the interface between the tube and the rod. As you mentioned, the force from the two shafts should be equal but opposite at the interface. This means that the stress at the interface should be 0, indicating that the forces are balanced.

To solve this problem, you will need to use the equation for thermal expansion: change in length = alpha * change in temperature * length. You will also need to use the equation for stress: sigma = force / cross-sectional area. By setting the stress at the interface to 0 and solving for the change in temperature, you can find the maximum temperature increase.

It's great that you are trying to learn the material as you go. Just make sure to double check your equations and units to ensure that your final answer is correct. Good luck!