Cantilever bed calculations

hi there, I wonder if I can pick some clever brains on what i guess is a simple maths/physics problem.

I am considering to build a simple cantilever bed, such that the bed appears to be floating because the legs will be recessed much like on many tables.

The question is how much can i safely recess the legs so that the bed does not tip if say 200kg weight is applied at the edge.

Attached is a sketch.
very crude sketchThe variables are:
W = the weight of the bed
a = the distance the support is recessed from the bed's edge (just consider the bed width for now)
y = the height of the bed (centre of weight height?)
X =  the maximum weight that can be applied at the edge before the bed tips
 
The constants are: the bed is king size, so 150cm x 200cm. On the sketch, b = the width of the platform support, so 2a + b = 150cm

I googled some physics homework about moments so made a simple calculation shown on the sketch. I have assumed the force due to the weight of the bed is as if applied to the centre, but not sure about this. If it is then I think Xa = Wb/2 so for example a 40kg bed (including mattress) with legs recessed 20cm would take a maximum weight of 110kg at the long edge. Do you think this is correct? I have not allowed for y, the height, though perhaps for low heights relative to width this is negligible?

Thanks for your help!
xeniumAsked:
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ozoCommented:
X*a = W*b/2,  y is irrelevant until it starts to tip, or unless the floor is tilted.
Although in practice the weight of the bed frame may not be perfectly centred, the mattress may shift, there could be other weights near the edge, and impulse weight could exceed static weight.
But you could also add ballast to increase W

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xeniumAuthor Commented:
Thanks Ozo, good to have confirmation. I will leave this open for a short while just in case anyone else has some good tips too.
Thibault St john Cholmondeley-ffeatherstonehaugh the 2ndCommented:
The sketch makes it look unstable. A table has very little load applied to it, even with diners sitting along one side with their elbows resting on it.
A bed is very likely to have a full weight of a person seated near the edge as they sit and then swing their legs around to lie on the bed.
Ballast would help as suggested above, and reducing the height to width of edge ratio.
Even with all the calculations you will probably learn more from a small scale model, perhaps just 8 to 12 inches long. You can also test your construction methods, joints, and the order of assembly by building a model first.
xeniumAuthor Commented:
THanks both for your input. The ballast idea is good. A model maybe useful for aesthetics too.

I have not touched on the construction method or materials, which obviously has to be very strong, and could be the subject of a separate question if needed. But in any case i will post details here of the final product if i complete it!
xeniumAuthor Commented:
Posting a spreadsheet for the calculations, for the record: https://docs.google.com/spreadsheets/d/1632YmRM_QnunT1i8u1cIWA4icu0PXx4cPrHLirJz1QY
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