Effect of thermal expansion on restraints

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Discussion Overview

The discussion revolves around calculating the force reactions within restraints when a beam undergoes thermal expansion. Participants explore the implications of fixed supports, the application of superposition, and the relevant equations for determining displacement and stress in the context of thermal effects on materials. The scope includes theoretical and mathematical reasoning related to structural mechanics.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant inquires about calculating force reactions in fixed restraints when a beam expands thermally, suggesting that deflection could be related to thermal expansion using the formula Delta=PL/AE.
  • Another participant hints at using the principle of superposition to analyze the problem, suggesting a two-step loading approach.
  • A participant questions whether the initial formula is applicable and seeks clarification on the relationship between thermal expansion and displacement.
  • There is a discussion about using the thermal expansion formula Δx = l α ΔT to determine the change in length due to temperature change, with a suggestion that the PL/AE equation might still be relevant.
  • Participants discuss the implications of different materials with varying Young's moduli on the method of calculation, raising questions about how the approach would change if different sections of the beam are made of different materials.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of certain formulas and methods for calculating forces and displacements due to thermal expansion. There is no consensus on the best approach, and multiple competing views remain regarding the equations and methods to be used.

Contextual Notes

Participants have not fully resolved the relationship between thermal expansion and the equations discussed, and there are assumptions regarding material properties and constraints that remain unexamined.

Ry122
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When a beam undergoes thermal expansion and the restrains that support the beam are fixed so that no movement is allowed, how do you calculate the force reactions within the restraints? The restraints are two walls, and before expansion occurs no horizontal reaction forces are present.

Since the amount by which the beam is deflected would be equal to the amount by which the beam thermally expands could it be determined with the deflection formula Delta=PL/AE?
 
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Hint: think superposition

Assume that there is intially only one wall, and then a force that pushes the beam back "into place".
 
ok but do I use the formula i mentioned above to determine the Force that does that?
 
Yes, assume that the loading happens in two steps. The principle of superposition states that
The net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually.

In layman's terms, it means that you can take the effect of each load component and add them up to get the sum. Since your bar is fully constrained, the sum effect is no displacement.

So, use your formula to find the displacement that a certain delta T would cause. Then, using stress/strain equations, determine the force and stress required to "push" the beam back into place. That will be your second loading.
 
Delta=PL/AE doesn't have anything to do with delta T though.
Do you mean DeltaTx(Alpha)x(L)=Displacement?
So I don't use Delta=PL/AE at all?

For the stress strain equation would i just use
y=mx
where m = young's modulus
x=percentage strain
and y=stress?

How would the method for doing this change if the first 1/3 of the Beam was made of a material with E=100,000 and the next 2/3 were made of a material with E=200,000?
 
Last edited:
I didn't check your equations, I was simply saying that the thermal expansion causes a change in length.
<br /> \Delta x = l \alpha \Delta T
The constraints will then exert a force back on the beam with the magnitude that negates expansion. I'm not going to do it for you, but your PL/AE equation may come in handy.
 
Would the method for doing this change if the first 1/3 was made of brass of a different cross sectional area to the 2/3 of the beam which is made of steel?
 
The method is going to stay the same, but the implementation would be a little trickier.
 

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