Can thermal expansion cause buckling in a compressed plate?

In summary, the plate is heated up and the compressive load is replaced with supports that block translation of the edges in the normal direction. The critical temperature is calculated assuming that the supports are perfectly rigid and the thermal expansion coefficient is known. The critical load is found to be approximately equal to the Euler load for the same plate dimensions and properties.
  • #1
FEAnalyst
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TL;DR 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:
D38F7C10-F37B-4FCD-828A-2332A14F9796.jpeg


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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  • #2
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.
 
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  • #3
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.
 
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  • #4
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.
 
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  • #5
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.
 
  • #6
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.
 
  • #7
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°.
 
  • #8
Problem finally solved - according to Timoshenko's book, ##N_{cr}## is a force per unit length of the edge in ##\frac{N}{mm}## while I treated it as force in ##N##. So the equation should be: $$\displaystyle{ \Delta T=\frac{N_{cr}b}{E \alpha A}}$$ Now the result is correct: ##\displaystyle{ 90.381 ^{\circ} C}##. The analysis gives a value close to this one, ##\displaystyle{ 93 ^{\circ} C}## mentioned in one of my previous posts was a mistake - this value was obtained with a worse mesh.
 
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FAQ: Can thermal expansion cause buckling in a compressed plate?

What is thermal buckling of a plate?

Thermal buckling of a plate is a phenomenon that occurs when a plate or sheet of material experiences a significant change in shape or deformation due to thermal expansion or contraction. This can happen when the plate is exposed to high or low temperatures, causing it to buckle or warp.

How does thermal buckling occur?

Thermal buckling occurs when there is a temperature gradient across a plate or sheet of material. This means that one side of the plate is exposed to a higher temperature than the other side, causing uneven expansion or contraction. This can result in the plate bending or warping in order to relieve the stress caused by the temperature difference.

What factors can affect thermal buckling?

Several factors can affect thermal buckling, including the material properties of the plate, the temperature difference across the plate, and the geometry of the plate. Other factors such as the boundary conditions and the presence of external forces can also play a role in the occurrence of thermal buckling.

How can thermal buckling be prevented?

Thermal buckling can be prevented by carefully considering the design and material selection of the plate. Using materials with low thermal expansion coefficients, increasing the thickness of the plate, or incorporating stiffening elements can help to reduce the effects of thermal buckling. Additionally, controlling the temperature gradient across the plate can also prevent thermal buckling.

What are the consequences of thermal buckling?

The consequences of thermal buckling can range from minor deformations to catastrophic failure. In some cases, thermal buckling can lead to buckling-induced instability, causing the plate to collapse or fail. This can have serious implications in engineering and structural design, making it important to consider and prevent thermal buckling in these applications.

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