Stress due to radial temperature gradient in a tube

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Discussion Overview

The discussion revolves around calculating the stress in a thick-walled tube due to a radial temperature gradient. Participants explore the implications of temperature variations on stress distribution within the tube wall, considering both constrained and unconstrained scenarios.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant inquires about methods to calculate radial stress in a thick-walled tube influenced by a temperature gradient.
  • Another participant suggests that the stress depends on whether the tube can expand, providing a formula for maximum stress under constrained conditions.
  • A subsequent post clarifies that the tube is free to expand and seeks a more detailed explanation of stress as a function of radial position, emphasizing the effects of temperature gradients on hoop and radial stress.
  • A reference to Roark's Formulas for Stress & Strain is made, indicating that it contains relevant formulas for maximum stress in hollow cylinders with different temperatures on the inner and outer surfaces, and suggests extending these to find stress as a function of radial position.

Areas of Agreement / Disagreement

Participants do not reach a consensus, as there are differing views on the conditions affecting stress calculations and the level of detail required in the explanation.

Contextual Notes

Some limitations include the dependence on whether the tube is constrained or unconstrained, and the need for a clear definition of temperature distribution across the radial position.

StoneME
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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.
 
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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
 
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.
 
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).
 

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