I for one would really like to know who is correct here. It seems like the fail point would be from the screws fastened into the stud, so wouldn't the structural cals need to be for the screws?

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Chris's approach did not take into account the length of shelf, so his approach would yield the same for a 1 mile long beam with hundreds of screws compared to the 36 inch beam in question each supporting only 200lbs

Keeping things simple, here are the calculations and I will round numbers for the purpose of simplicity

Lets say the cleat was 2" high, the shelf is 10" deep. We have three sets of screws, each set to its individual stud.

The cleat has one screw 1/4" from the top and one screw 1/4" from the bottom.

0 = the distributed load x the distance from the wall - the force of the top screw x 1 1/2" - the force of the bottom screw x 1/2"

0 = 200/3sets of screws x 5" distance from the wall - F x 1 1/2 - F x 1/2

it follows 333 = 2 1/2F

F = 133

So it tells us roughly that each cabinet screw needs to hold 133 lbs if we put 200lbs on the shelf

KCMA rates a maximum weight for an overhead cabinet as 600 lbs and some of those are held by no more than 4 cabinet screws. The bending moment would be much less than the shelf example due the height of the cabinet, but we still have the shear stress. I have seen some cabinet screw data saying they are rated for 75lbs and some data saying their holding force is 100lbs per inch of spruce penetration, so who knows. I have also seen some tests done on pocket screw joints failing at around 700lbs, who knows

All I know Chris thought it was funny that a shelf held 200lbs without issues, and a few people including the customer witnessed the test. He seems to be a bit of a bear, and he knows EVERYTHING, but that is OK

The key with floating shelves is flexing. As long as they are stiff enough, the customers love them.

Over and out.