# Deflection of thin plate with 1 free edge and deflection > thickness

I have an thin uniform rectangular steel plate supported on 3 sides and free on one, and supporting a uniform load.

I'm trying to determine what thickness of plate I need to keep deflection below a given value, but the deflection may not be "small" compared to the plate thickness, so standard formulae for maximum deflection may not be accurate, and I'm stuck on how to check my answer, or how to calculate it more accurately.

Data:

The plate is a flat rectangular plate (stainless 316 or mild, undecided) anything from 1.5 -15mm thick with an unsupported area at its edge of 1250x500mm total 0.625m^2. It is simply supported along the 500-1250-500 edges and free on the other 1250 edge. The unsupported area carries a static load of 4800 N/m^2 (approx 360kg spread uniformly over the unsupported area) plus its self weight.

The 3 supported edges are unrestrained simple supports that can slide or rotate; they don't resist any movement except in the Z-direction (a bit like it's resting across the 3 edges of a "u" shaped pit).

My question is the thickness of steel plate I need, to ensure maximum deflection (at the middle of the free edge?) stays under possible values of 3mm / 5mm / 10mm (the most likely permitted deflections or at least a good selection to choose between)

The problem is that I am guessing solutions could be thicknesses around 1.5 - 5mm, which means that the deflection might not be "small" compared to its thickness and the usual simple calculation may not be very accurate or trustworthy. But I'm not sure....

Thanks - and any hints how I can work this out myself appreciated but not essential :)

You are correct that this problem falls within the category of 'large-deflection' problems since the deflection is larger then about $\frac{t}{2}$. To correctly analyse a plate under large-deflections would require solving the Föppl–von Kármán equations, which is generally not possible except for some very specific cases. Numerical solutions to some cases have been published (for example in this book), unfortunately your case does not appear to be one of them. Using non-linear finite element analysis may be the only option.

However, if the edges were restrained from sliding membrane action will occur (some of the load will be carried by direct tension in the plate and not just bending) and the stiffness of the plate would be higher than that predicted by small deflection theory [Roark (2002) pg. 448]. Therefore, if a small displacement analysis predicts the displacement will be less than your requirements, a large-displacement analysis would result in a smaller deflection and therefore also meet your requirements. Note in this case stresses for a given load will also be less than predicted by small-displacement theory.

Reference:

Roark, R. J., Young, W. C., & Budynas, R. G. (2002). Roark's formulas for stress and strain. New York: McGraw-Hill.

• This provides the info needed to move on and solve the practical issue,thanks. Jun 7 '16 at 22:15
• It is my understanding that sliding supports are not required to generate membrane action. An internal compression ring forms within the plate which forms an equilibrium state with the tensile membrane forces. A non-linear FEM with roller supports confirms that membrane forces develop even without horizontal restraints. Nov 21 '17 at 0:25

The small deflection approximation is not based on deflection vs thickness, it is based on deflection over span. So for your maximum case of 10mm deflection on a 1250mm span, you are only looking at a 0.8% deflection.

Another way to look at is from a statics point of view. Where the Deflection is one side of a triangle and half the span in the hypotenuse. This would give you an angle of 0.5 degrees. Then you can look at how much force is being transferred into the part horizontally instead of vertically with $F_{_{x}}=F*sin\theta$. Therefore, with your maximum deflection case you are only going to get about $0.008*F$ transferring into the plate in the non-normal direction.

I think the normal equations are fine for you what you are trying to do.

• Apart from the relevant span being the lesser of the two (500??) I think, that does seem to make sense. You're sure its deflection vs span not deflection vs thickness though? May 8 '16 at 18:22
• Unfortunately the sources seem to confirm it is the ration of deflection to thickness not span that makes the classic simple solutions unreliable: files.engineering.com/… May 8 '16 at 19:40
• @Stilez, interesting I know for beam theory that small deflection theory is based on length, not thickness and assumed it was the same for plates. I'll have to check my Roark's at work tomorrow. You are also correct that I should've used the 500 instead of the 1250. May 8 '16 at 21:44
• Any update on this ? :) May 11 '16 at 13:29

If you use Mathematica or similar program you can create finite difference solution for bending of a plate. I have done similar calculation based on Timoshenko's work and rather accurate if you use reasonable number of elements. You can vary thickness and dimension and test your formulas, also you can upgrade code to include large deflection.

I have tested my code with FEM but is too large to paste it here. It is based on this article: Sertic