# Second Moment of Area Problem for Floor Plate with Cross Beams

1. May 9, 2012

### Freezer12

1. The problem statement, all variables and given/known data
So I need to know how to attack a problem for a solid metal plate.

The plate is used as a manhole cover, so it needs to withstand a certain load. The load is calculated by placing a large weight at the centre of area 25x25. I can't for the life of me figure out how to do a simple supported beam calculation on the plate so that I can calculate the amount the plate will bend under the load, and whether this is acceptable.

Basically the problem is a simple beam supported at either end with a weight in the middle.. I think. I have attached a picture of the plate.

Plate is 5mm thick, 900mm square - supported by cross beams - 20 beams 25mm high, 5mm thick, with a load in the centre of the plate 25mm square.

The load is acting down on the plate, the cross beams are supported at both ends, and the plate itself is supported on all four sides.

The second moment of inertia equations are what I'm using for this, however I'm not sure if I should be including all 20 beams in the equation, or whether I should only be using the ones under the weight itself. This is because the beams under the weight will be the ones supporting the load, whereas the other beams will not be supporting it so much. I am combining this with the second moment of the area of the plate as well.

Does anyone have any insight as to how to tackle this problem? The purpose of this is to give a very vague idea of whether the beam and plate variables I use will be able to withstand a standard load in the middle without displacing too much. I plugged the exact dimensions into ANSYS to model, however it came out as being able to withstand massive amounts of force, so I'm resorting to plugging the dimensions into a spreadsheet.

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• ###### plate.png
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2. May 10, 2012

### pongo38

There is no single 'right' answer to this. It does depend on the assumptions you make. Clearly, just taking the ribs under the point load is safe, but there can be no doubt that adjacent ribs participate to some extent. A simple model would be trapezoidal load distribution tapering from zero at edges tapering to a max at the centre. Another approach could be to calculate the flexural stiffnesses in the x and y directions. Consider a beam strip in each direction at the centre and equate the formulas for deflection in each direction. you have to partition the load qx in x direction and qy in y direction. Equating the deflection gives ratio qx/qy and combining this with q=qx + qy solves the problem of moment partition.