# Thermal buckling of a plate

## Summary:

Is it possible to calculate critical temperature for a plate?
Hi,

in the Timoshenko’s book "Theory of elastic stability" one can find a case of simply supported rectangular plate uniformly compressed in one direction (compressive load is applied to shorter edges). The formula for critical load is: $$N_{cr}=\frac{\pi^{2} D}{b^{2}} \left( \frac{b}{a} + \frac{a}{b} \right)^{2}$$ where: D - flexural rigidity, b - length of shorter edge, a - length of longer edge.

Here’s a scheme of this plate:

Now what if we heat up the whole plate and replace the compressive load with supports blocking translation of these edges in the normal direction? Is it possible to calculate the critical temperature knowing the thermal expansion coefficient?

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jrmichler
Mentor
If you assume:
1) Perfectly rigid supports,
and
2) Modulus of elasticity is constant over the temperature range,
then,
Yes

Calculate the critical load. Set up the equation to calculate force from temperature rise, thermal coefficient of expansion, modulus of elasticity, and area. Since you know everything except the temperature rise, rearrange to solve for temperature rise.

FEAnalyst
Thanks for reply. Is that derivation correct? $$\frac{N_{cr}}{A}=E \alpha \Delta T$$ $$\Delta T=\frac{N_{cr}}{E \alpha A}$$ where A is cross-sectional area of the plate.

Last edited:
jrmichler
Mentor
Looks right to me. Keep in mind that rigid support is a mathematical ideal, and real world support stiffness will have a large effect on actual buckling temperature.

And I missed an assumption: The supports are isothermal.

FEAnalyst
Thanks for help. I tested this equation on particular example (from FEA software’s documentation) and, to my surprise, the result is different than it should be according to reference and numerical analysis. Here’s the data for this example:
- square plate with edge length ##b=2##
- thickness: ##t=0.01##
- Young’s modulus: ##E=10^{8}##
- Poisson’s ratio: ##\nu=0.3##
- thermal expansion coefficient: ##\alpha=10^{-6}##

Units are not used in this case. My result is ##\Delta T \approx 45## while the solution from the documentation (and numerical analysis) is ##\Delta T \approx 93## (the same as critical force resulting from original Timoshenko’s equation for "regular buckling").

I wonder what may cause this difference since assumptions are the same.

jrmichler
Mentor
My preferred approach for this type of discrepancy is to start over from a different direction. I suggest carefully checking the units in all calculations, starting with the Timoshenko equation in the OP.

Does Timoshenko give the applicable range of slenderness ratios for his equation? Is the example plate thin enough for the Euler equation to apply? How does the buckling load from Timoshenko compare to Euler for the example plate? How about a narrow plate with the same dimensions and properties, except the width b is 0.2? The Euler equation should give a buckling load proportional to width (as long as width is larger than the thickness), while the Timoshenko equation does not. The difference should not be large if the width is at least several times the thickness.

When that all checks out, then calculate the strain at the buckling load, and the ##\Delta T## to get that strain for each of the above cases.

Here you can find the description of this particular case: https://abaqus-docs.mit.edu/2017/English/SIMACAEBMKRefMap/simabmk-c-buckleplate.htm

No units are used in this case since the software doesn't use them. Authors of this example claim that "the plate is rather thin (##L/t= 200##)". Here's the most interesting part:

... alternative case is where the edges parallel to the y-axis are restrained in the x-direction, and the temperature of the plate is raised. This should give the same prebuckled stress field in the plate; and, thus, critical temperature changes should be those that give the same critical stress. To investigate this case, we use a thermal expansion coefficient of ##\ 10^{−6}## (strain per unit temperature rise) so that in the prebuckled state the critical stress should occur at a temperature of 90°.

I wonder how they came to the conclusion that with this data the critical temperature should be 90°.