1. Flexing of steel plate

An ASTM A36 STEEL PLATE of 4' by 4' has 20 tons of pressure put on it in the middle (in a rather small square). How thick has the plate to be to not flex at the edges? Another plate, same material, 4' by 2'. Same question.

You can calculate the deflections and stress with the online Deflection and Stress Calculators located here:

Beam, Plate Deflection and Stress Equations Calculators

3. You mean 20 tons of force by the way. And, simple plate deflection formulas do not take edge effects into account. I'm assuming that what you're really trying to ask, is, how thick does the plate need to be to prevent it from folding like a taco and falling through the opening. That failure mode would be very difficult to analyze mathematically, without simulation or testing.

I'd say what you're after is a code that you can reference, depending on what industry and country you're in. Your application sounds a lot like those steel plates that road crews place temporarily over pits until the underground work is done and can be re-buried and paved. Or it might be a utilities hatch in a sidewalk.

In that case, if you had to start from scratch, you'd pick an arbitrary deflection that you deem allowable and find the thickness using one of the equations that Kelly linked to. You wouldn't want to alarm the public with a plate that feels like it's going to fail when they walk on it or drive over it. To me, a deflection of 1/4 inch or 6 mm "seems" like it would be OK, but I've made a lot of wild guesses as to what the criteria are (just for the sake of discussion). Maybe it's only 1 mm. That's your null hypothesis. Then you'll find the thickness of the plate that gives you 1 mm of deflection and it might be way too thick (heavy and expensive) for your constraints, so you iterate. Maybe you find a simple plate can't work and you add some stiffening ribs.

That's how I would go about it, anyway.

4. Need more data.
How is the plate supported?
What is the shape and size of the contact load?

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