Stress due to radial temperature gradient in a tube

[StoneME]

Anyone know how to use the temperature gradient in a thick-walled tube to calculate the stress seen throughout the wall (radial stress gradient)? I've been scouring the internet for a good explanation but haven't found one.

 

[edgepflow]

The stress will depend on if the tube is allowed to expand or if it can't move when it gets hot.

If it is not allowed to expand, the stress is maximum and is determined as follows:

ΔR = $\alpha$ Ro $\Delta T$

The thermal strain is now:

$\epsilon$ = ΔR / Ro = $\alpha$ $\Delta T$

And the stress is figured with Hooke's law:

σ = E $\epsilon$

 

[StoneME]

First, I should be more clear. The tube is not being constrained and is free to expand.

Second, I appreciate the response but I think I'm looking for a little more depth. What I'm looking for is a description of stress as a function of radial position given the temperature as a function of radial position. Temperature gradients will cause the hot wall (inner or outer) to expand more than the cold wall, giving rise to hoop stress as well as radial stress.

 

[edgepflow]

Roark's Formulas for Stress & Strain, Chapter 16, Section 16.6, Case 16 has formulas for max stress on the surfaces of a hollow cylinder with two different temperatures on the inner and outer surface. They look pretty easy. You may be able to extend this to determine σ(r).