Circular flange bending stress problem

[n707jt]

Homework Statement

Hi guys

I've got a problem on hand which has haunted me for sometime and I thought it might be a good idea to post it here to ask for your opinions.

As you can see in the picture, I've got a circular flange. What I am intending to do is to simply find out the bending stress of the circular flange. The loading conditions are shown in the picture too. Simply put, a jack loading acts on the center bottom of the circular flange while 4 bolts that are supposed to be inserted into the smaller holes will exert a reaction force.

I've tried to use a method from Roark's (see page 502, year 2002 if you have the book) to idealised the flange into a square component (highlighted in the picture below with dimensions) but a mate told me that might not have been an accurate idea. I've included my workings below for your reference.

Homework Equations

According to Roark's

σmax = βW/t²

where,

W (force applied) = 23888.86N
β (value obtained from table in Roark's) = 0.84
t = 0.84 mm

The Attempt at a Solution

Using the formula above, I've managed to obtain a σmax = 50.1 MPa.

My mate has told me that another more accurate idea of obtaining a value is to idealised it into 2 simply supported beams (red rectangles). He said that the bending stress can be obtained by halfing the W. Then if the σmax is below the UTS of the flange material, then by similarity deduction, the circular flange will be able to withstand the full load.

Which one would have been a better and more accurate idea? Please help!

 

[PhanthomJay]

The 2 simple beam approach looks pretty good, but I would calculate the force in each bolt, then analyze a quarter of the plate as if it were a cantilever, with the single bolt force applied at the center of its hole, and the moment arm being the perpendicular distance from the bolt centerline to the tangent of the circle where the thick and thin part of the plates meet. Then it's just MC/I for the bending stress, where, in calculating I, the width of the cross section would be the length of that tangent in the quarter plate. The results will be conservative, not exact.