McKee Box Compression Formula (Long Form)

The long form is what McKee, Gander and Wachuta actually derived. The everyday short form is what you get after substituting a correlation into it, and the substitution is the part that can go wrong.

The general result is $BCT = 2.028 \, ECT^{0.746} (D_{MD} D_{CD})^{0.127} Z^{0.492}$, where $D_{MD}$ and $D_{CD}$ are the board's flexural stiffness per unit width in the machine and cross directions. Those are the quantities the buckling analysis genuinely calls for: a plate resists buckling in proportion to its bending stiffness, and a corrugated panel has two very different stiffnesses depending on which way the flutes run. The product raised to 0.127 is the geometric mean raised to 0.254, which is the combination that falls out of orthotropic plate theory.

Check the dimensions here too, because the same trick is at work. Bending stiffness per unit width has units of force times length — newton-metres in the ISO convention, pound-force inches in the imperial one. Force exponents: $0.746 + 0.254 = 1$. Length exponents: $-0.746 + 0.254 + 0.492 = 0$. Again exactly zero, again by construction, so 2.028 is a pure number and the equation is unit-system agnostic in the same way the short form is.

Read the exponents as a statement about what matters. $ECT$ enters at 0.746 and the stiffness pair at 0.254 between them. Double both stiffnesses and compression strength rises about 19%; double $ECT$ and it rises about 68%. That asymmetry is the reason the industry moved from bursting-strength specifications to edge-crush specifications in the 1990s, and it is the reason the short form survives at all — the term it approximates is not carrying much weight.

So when is the long form worth the extra measurements? When caliper is lying about stiffness. On ordinary uncrushed single-wall board the two track each other well, which is why the substitution works: on this site's worked example the two forms agree to within 0.03% on the same board. But board that has been crushed on the corrugator, in the printing nip or on the folder-gluer keeps most of its measured thickness while losing a great deal of its bending stiffness — the flutes are damaged but still occupy space. Caliper cannot see that and a four-point bending test can. If you are troubleshooting boxes that test below prediction, measuring $D_{MD}$ and $D_{CD}$ is the diagnostic that tells you whether the board or the converting is at fault.

One caution about running this equation backwards. Because stiffness enters at the 0.127 power, recovering a stiffness from a compression figure means raising a ratio to about the 7.9 power, and a 5% error in the input comes out as roughly 48% error in the answer. That is not a defect in the algebra — it is what a small exponent means in reverse. The forward equation barely notices stiffness, so the inverse cannot pin it down. Use the inverse brains as a plausibility check and never as a measurement.

Every honesty note from the short form applies here without change. This predicts a laboratory test, the constant was fitted to 1963 single-wall RSCs, Urbanik and Frank found large errors against modern data, and refitting to your own boxes is the normal thing to do.