Plate bending and effective section

[tubelight]

A simple plate bending problem. A 1/4" plate (say 8" x 4&quot is bolted by one bolt and has about 400# load at the edge.
What will be the effective area to calculate the section modulus? I believe I can not use the full anle length. And what is the reference for this? I have in past taken effective width as 4 x t. Thanks.

 

[dlew]

tubelight,

The book "Formulas for Stress and Strain", by Roark, discusses the subject under the section "Beams of Relative Great Width". The effective width depends on the type of supports (free or fixed), the manner of loading (concentrated or uniform, and the ratio of width to span.

For a cantilever plate with a concentrated load on the edges, as is the case of hoist wheels hung from the bottom flange of monorail beams, I have used an effective width of 2 times the distance from the edge of the flange to the web of the beam.

For plates spanning between two continuous supports, I believe the effective width is close to 16 times the plate thickness.

 

[vonlueke]

tubelight: I didn't check, but I suspect your exact problem might not be in Roark. If I understand your problem, you have one bolt midway along the long side of your rectangular plate. And I assume your concentrated, transverse load is applied at the midpoint of the opposite long side such that it will cause the load application point to cross and deflect below the interface plane your plate is bolted to, rather than away from that interface. The direction of the applied load in this problem might make a difference because, when applied in the direction I'm assuming, the short plate edges lift off of the bolted interface as the load application point deflects downward, whereas these edges would tend to press into the interface (in contact compression) if the load were reversed (applied upward). For the downward load assumed, a more complex surface contact plays a role, because you don't automatically know the length of the contact zone before the plate lifts off of the bolted interface.

I also assume your bolt has a nominal diameter of about 5 mm and a hardened, thick washer with OD of about 11 mm. I assume overlap of your plate with the bolted interface is 20 mm, which makes your cantilever length about 80 mm. I also assume your plate is bolted to a relatively very stiff interface.

For this specific problem (which should be assumed nonscalable to other thicknesses, plate aspect ratios, and load values unless proven otherwise), and under all the above assumptions, the effective width of your cantilever beam w.r.t. plate deflection appears to be about 11.0*t, whereas the effective width w.r.t. plate stress appears to be about 9.3*t.

Why the effective width is different w.r.t. deflection versus stress I can only attribute to the unusual boundary conditions (BCs), stress concentrations, etc.; i.e., the BCs are not the same as a plain cantilever beam. And note that the above answer is different from dlew's 16*t, because you have different BCs (as he qualified).

Given your applied load and an effective width 9.3*t, your plate appears highly overstressed (assuming steel A36). It appears you would need to increase material tensile yield strength to something like 600 MPa (87 ksi), or increase your A36 plate thickness to, say, 8.5 mm. Also check the bolt stress. The above assumes the bolt holds, but I didn't check it. Another possible option, though I didn't check it, might be to add another bolt, to reduce a detrimental stress concentration you're getting near your one bolt (caused by having only one bolt).