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(OP)
I uncovered something that I may have overlooked in past design calculations.  The question comes up when analyzing the stresses on a tapered padeye (with a long base to hole dimension) or on a tapered beam.  The tapered member is loaded perpendicular to the longitudinal axis of the member.  For the padeye, what I have done in the past is calculated the radius needed around the hole of the padeye (based on allowable tearout stress) and calculated the section height needed at the base of the padeye (based on allowable bending stress).  Then I simply would make the taper line from the base to the tangent of the radius around the hole.  My assumption was that the max bending stress was always going to be the largest at the base of the padeye.  I have now found that not to be the case - that there are sometimes bending stresses that are higher at some intermediate point between the base and the hole.  The same problem holds true for tapered beams.

Can anyone let me know have they have handled that issue in calculations?  Is there a formula that will give the taper angle (as a function of the height of the section) that is needed in order to assure that the max stress is at the base of the padeye/beam?  I saw a formula in appendix F of the ASD spec that possibly could be used.  How would I use that formula to determine "dL" the depth at the larger end of the member?

Thanks in advance for any help with this problem.

Tony Billeaud
Mechanical Engineer

Replies continue below

### RE: Tapered beam/long padeye design

Look up Roarks, I think it has a section on tapered beams.

### RE: Tapered beam/long padeye design

Tony,

I've also always approached it the same way you described. If your finding is correct, then it looks like I will have to change my approach also.

For the padeye (or the beam, for that matter, but starting with the padeye), could you derive an expression for the cross section of maximum bending stress using calculus? Sketch a generic padeye, apply a force P at the pin hole, and define all your padeye dimensions with variables. From this, try to obtain a general formula for bending stress at any cross section of the padeye. Then, take the derivative of the bending stress with respect to the variable that defines "distance from pin hole to base of padeye". Set this derivative equal to zero, and solve for "distance from pin hole to base of padeye". The expression you obtain will be a max/min, and in this case should be the location of maximum bending stress.

I always find myself less concerned about padeye sizing than about whether the structure can handle the introduction of forces from the padeye. As an example, consider a 1-inch thick by 15 inch-deep-at-the-base padeye plate attached to the edge of the flange of a W16x40, with the 15-inch dimension running parallel to the longitudinal axis of the W16x40. Now say you have forces applied at the pin hole in all 3 orthogonal directions. How in the world do you check that flange? And even if there are stiffeners plates in the web of the W16x40 directly beneath the padeye, still, how do you check a situation like this? Tony, do you (or anyone else out there) know of any text or published paper that deals with stuff like this?

### RE: Tapered beam/long padeye design

tnteng:  If you want the maximum bending stress to be located at the tapered cantilever fixed end, then you would need to have a taper angle (relative to a horizontal line) on each edge (top and bottom edge) less than or equal to alpha = atan(0.5*ho/L), where ho = beam depth at cantilever tip (or lug width at hole centerline), and L = beam length.

### RE: Tapered beam/long padeye design

As I see it "alpha = atan(0.5*ho/L)" as posted by vonlueke is another way of stating that for the maximum bending stress to be located at the fixed end, then the depth at that end must not be greater than twice the depth at the cantilever tip.
Alternatively, the depth at the cantilever tip (d) should not be less than half the depth calculated as required for (and used at) the fixed end (D), otherwise the allowable bending stress will be exceeded.
I plotted length vs. required depth for a couple of examples and the above relationship (d > 0.5D) does seem to be correct.

### RE: Tapered beam/long padeye design

You can also divide the beam into several slices, compute S and Mallow, then plot and compare to Mact.

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