Deriving the load point on equipment base

Isimanica

1. The problem statement, all variables and given/known data

Problem is this. I am tasked with deriving a formula or equations that I can then input in to an excel format. I am to take an equipment base that has four sides with supports at each of the four corners. I am to find the load that each one will hold base upon the Center of Gravity of the equipment upon the base plate. You will be given the Mass, the length and width of the equipment, the location of center of gravity on the base plate. Also not worried about the weight of the base plate just about the mass with respect to the center of gravity. The only unknown is the Load that each support will hold. What he wants is to be able to type in the knowns and find each supports load weight individually. I have tried for a couple of days to figure this out but I just can't seem to figure it out. It supposed to be simple but I just don't know. Any help would be welcomed.


2. Relevant equations



3. The attempt at a solution

I have tried using the Ideas behind Noncoplanar Parallel Force Systems but I think because the points line up those ideas and examples have not been of great help, only confuse me further.

PhanthomJay

If I understand your problem correctly, this is not a simple problem; for a plate supported on 4 corners, its reactions are statically indeterminate. If the cg of the equipment was located at the geometric center of a square base plate, the equipment load would distribute equally to the 4 corner supports; otherwise, the problem is difficult to solve, and I wouldn't be able to do it without a copy of Roark's 'Stress and Strain'.

Isimanica

Currently they use a formula of taking the ratio between the length or width (dependent upon the side used) of where the Center of Gravity is located at. This ratio is what is used to then find the two components that make up the corner and determine the weight that is distributed. This of course is a long process and to do by hand and can't be set up well with in an excel setting. I will try and get a scan of what the idea is tonight or tomorrow.

Isimanica

Maybe this will be more helpful

Problem

You have a wood table with length of 3 meters and width of 2 meters. Ignoring the weight of the table, a 10 Kg weight is place at the exact center of the table.
a) How much of the 10 Kg weight is each leg holding?





If the weight was moved 1.5 meter length wise from the center and 1 meter width wise from the center.
How much of the 10 Kg weight is each leg now holding.

PhanthomJay

For part (a), there is no reason to not believe (and every reason to believe) that the 10 kg weight will distribute evenly to each of the four corners (25N each, using g=10m/s^2). This is due to the symmetry of the load with respect to the supports.
For part (b), I wouldn't venture a guess. I never could afford a copy of Roark's Formulas for Stress and Strain, so I don't have one handy. As I recall in my distant memory, even Roark might have been a little weak on this particular problem, because most of its solutions involved some sort of uniform loading over continuously supported edges, either fixed or free to rotate on those edges. The point load on a 4-corner supported plate problem is a doozy. Might even be uplift on the far corners. IMHO.

Isimanica

See this is what driving me batty. I know that uniform load would distribute evenly if it is at the center of gravity. Knowing that also should the center of gravity move then the weight distribution will shift. But some where there must be a formula that gives you a way to know with some certainty what each leg or point will hold. I mean equipment manufactures must know something before just weighing each point on the equipment.
So now hearing from two people the same response of Roark's I broke down and chipped in with my dad to pick up a 7th edition of that book. I hope to find something in there that would help me out. Though I am definetely asking one of my professors at UH this question. My father and uncle spent 4 hours on a saturday trying to figure this thing out.

PhanthomJay

That will be a good investment for sure. There is a solution. Please let me know if they have the example for plate stresses and support reactions for a rectangular flat plate supported at 4 corner points and loaded by a point concentrated load (not at the cg). Note that for determining the support reactions (but not the plate stresses), a distributed load may be replaced by an equivalent concentrated point load applied at the cg of the distributed load. Thanks.