A shelf that sags.

For a simply supported beam with a load at its centre, beam deflection theory gives δ = PL³/(48EI), where I = bh³/12 is the cross-section's second moment of area — substituting that in leaves δ = PL³/(4Ebh³), which is the form shown above so the h³ is visible directly rather than hidden inside I. The board is a standard "2×6" stud, actual dimensions 38×140 mm, in spruce–pine–fir framing lumber (E = 9.65 GPa, the No. 2 grade design value). Turning it on edge swaps which dimension is h and which is b, making it about 13.6× stiffer — not because the wood changed, but because the equation's cubed term did. The gauge's limit line is L/360, a deflection commonly used in building codes as a serviceability limit for many spans. Only elastic bending is modelled: at the sliders' extreme corner (longest span, heaviest load, flat) the formula predicts several centimetres of sag, but a real board that thin would almost certainly crack or fail outright long before bending that far — the equation doesn't know that, so treat results far past the limit line as "this shape is a bad idea," not a literal prediction.