How much an unevenly heated glass plate bows?

Question

In relation to this question about 3D printing, I wonder how much a 200 mm square glass plate heated only in a central 15 cm square region would bow upwards or downwards.

According to the thermal expansion of glass, the linear expansion for 15 cm and 100°C deltaT would be 0.05 mm.

Using the formulas provided in https://planetcalc.com/1421/ I input 200 mm as chord length and 200.05 as arc length. The result is 2 mm height which seems a lot to me. Using 150 mm and 150.05 mm I get 1.7 mm which is still a lot.

Answer 1

You are right that your numbers are over-estimates. They would be "correct" for a plate with no resistance to bending, which was only supported along two edges and heated across its full width between the unsupported edges.

This is a hard problem to produce a simple "formula", because in the real situation the bending will be reduced by the stiffness of the glass, and because the deformed shape is not part of a circle but a bulge in the center of a square. In real life there will be some transition between the temperature of the heated part and the surrounding "frame" as well, and not a sudden temperature change.

The best way to get a theoretical answer would be to model this with finite element analysis. Alternatively, learn from other people's practical experience, which is quicker and cheaper than learning from your own practical experience!

FWIW the simpler case of a circular plate was apparently studied as part of a PhD back in 1960, with this report as a spin-off: http://naca.central.cranfield.ac.uk/reports/arc/rm/3245.pdf