curved plate under gravity

huangyhg

curved plate under gravity
if an initially sphereically curved rectangular plate is placed on a solid flat table, is it simple to determine to what level gravity would be able to push the plate flat?
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if it started as a rectangular plate that was then curved, presumably it touches the table at 4 points. these points are then free to slide. it sounds like a very complex problem and i wouldn't like to tackle it by hand alone. for a check on a fe model you might consider it as two arches going across the plate diagonals and take some notional effective width of plate in order to calculate the stiffness of the arch and hence it's deflection under the load of the plate. in effect you will have two arches but with varying stiffness from very little at the support points to quite a lot at the apex.
carl bauer
further info: the plate is the other way up, with the corners sticking in the air. i'm not worried about friction.
if you need a first order approximation i suggest to use the uniformly distributed load that, for a plate supported at the corners only, would give the center deflection in your plate. for a square plate this is given by
f=0.025pa4/b
where the symbols are those commonly used in plate analysis.
if you need to be sure that the plate is really flat i would suggest a test, guessing that you'll need 5 to 10 times the figure calculated as above.
prex
depending upon the plate stiffness, gravity will be unable to "flatten" the initially spherical plate.
the deflections for such setup with the error assumed in the modelization can ge gained through modeling with curved shell elements, that will have aberrations at the central support (contact stress and fem derived).
then, initial residual stresses may have a say.
and the deflections will be quadrant symmetrical, but the plate very likely won't get flat nor all its points at level zero at the same time. a symmetrical contour will.