Shallow Box Validity Check for McKee

McKee's formula assumes the box fails by its side panels buckling. A panel has to be tall enough to buckle. Below roughly a depth of $Z/7$ it is not, and this ratio — one at exactly that limit — tells you whether you are inside the formula's world or outside it.

Why the limit exists, because this is rarely published and almost never explained. The derivation models the four side panels as thin plates loaded on edge. Under a top load they bow outward, and the box fails when they do. That is the entire reason board bending stiffness, and its stand-in caliper, appears in the formula: stiffness is what resists buckling. Now shorten the panels. A short plate cannot bow — the boundary conditions at the top and bottom edges dominate the whole height — so instead of buckling, the board simply crushes edgewise. At that point the failure is pure edge crush, the caliper term has nothing left to describe, and the equation is not merely miscalibrated. It is wrong in FORM. Refitting the constant does not repair it, because the shape of the relation is wrong: caliper should have dropped out and it has not.

That distinction matters practically. A formula that is out of calibration gives answers consistently high or low, and a factor corrects it. A formula with the wrong form gives answers that are wrong in a way that changes with the inputs, and no factor fixes that. Software that applies McKee to a shallow tray and then applies a "tray correction" is patching the second problem as though it were the first.

Who lives below the line. Retail-ready and shelf-ready cases, produce trays, pizza boxes, flat display shippers, and the low half-height cases that supermarket logistics has been standardising on for twenty years. All of them are routinely quoted McKee numbers by calculators that never checked the depth, and all of them are the boxes whose compression strength gets argued about. If you need a number for a shallow tray, the honest way to get one is to test the tray — its capacity is governed by ECT and loaded perimeter with very little contribution from caliper, and there is no widely accepted closed-form replacement.

Treat $Z/7$ as a rule of thumb rather than a switch. Different sources place the limit in slightly different spots, and the real transition is gradual: the prediction degrades as the ratio approaches one from above, because a box near the limit is already partly a crushing case. Anything between about 0.8 and 1.3 is a grey band where the answer wants testing rather than arithmetic. It is worth noticing that this is the same idea as the slenderness limit on a column — below a certain slenderness Euler's formula stops applying because the member squashes rather than buckles — and it is the same physics, one dimension apart.

One last thing worth watching for in practice: a design drifts across this line by getting WIDER, not by getting shallower. Spreading a box out at fixed depth raises $Z$ and lowers the ratio, and that change reads to everyone involved as a footprint decision rather than a structural one. Check the ratio whenever the plan dimensions change.