The core idea
A beam's resistance to bending is its second moment of area, I (mm⁴) — how the cross-section's area is distributed about the bending axis. For a rectangle: I = b·h³ / 12, where b is the width and h is the depth measured in the direction of bending. The cube is the whole story: depth earns a cubic vote, width only a linear one. Stand a 45×195 mm joist on edge and I ≈ 27.8 million mm⁴; lay it flat and I ≈ 1.48 million mm⁴ — same board, ~19× stiffer standing up, because the outer fibres (which strain most) have been moved far from the neutral axis, where they leverage more area to resist.
Deflection follows: δ = F·L³ / (3·E·I) for an end-loaded cantilever — force F (N), span L (mm), stiffness E (MPa), second moment I (mm⁴). Two levers dominate: δ shrinks with I and grows with L³ — double the span and deflection multiplies by 8.
Real-world example
House floor joists: every carpenter stands them on edge — not for strength folklore, but for the ~19× gain above, which is what keeps floors from bouncing. The same geometry scales down to shelf brackets (mount the vertical plate tall, not wide) and up to steel: I-beams and H-beams concentrate their flanges far from the neutral axis, buying even more I per kilogram of steel than a solid rectangle. One principle, three industries.
Common pitfall
Choosing more area instead of better-placed area. Doubling a board's width doubles I; standing the original on edge multiplies it by ~19. Beginners also forget the L³ term — a shelf bracket that barely sags at 400 mm can droop embarrassingly at 800 mm with the same load, and no extra thickness will quietly absorb an 8× penalty.