Rectangular Section Modulus

S = bh²/6 and I = bh³/12, with optional bending stress from a moment.

InputS = b·h²/6 I = b·h³/12 σ = M/S

Width b (mm)or enter in — converts to mmDepth h (bending direction) (mm)or enter in — converts to mmBending moment (optional) (kN·m) — optionalor enter lbf·ft — converts to kN·m

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The engineering

Depth squared in S is the whole story of beam design: orient a 2×8 flat and it carries a quarter of what it does on edge — same wood, same weight, one rotation. The h³ in I does the same for deflection, only harsher. This is why joists stand tall, why I-beams push material away from the neutral axis, and why 'lay it flat to be safe' is exactly backwards.

The bending-stress row divides the moment by S and hands you the number to compare against allowable stress — the two-second check before anything gets welded to anything. Remember h is measured *in the bending direction*; swapping b and h is the classic units-of-embarrassment error this card cannot protect you from.

Where this math comes from

Galileo opened beam theory in 1638 with the famous cantilever problem — and got the stress distribution wrong, pivoting the section about its bottom edge. Antoine Parent placed the neutral axis correctly in 1713, but the definitive linear-elastic bending theory is Navier's, from 1826: stress proportional to distance from the neutral axis, capacity summarized by one geometric number.

Thomas Tredgold's 1820 'Elementary Principles of Carpentry' brought section arithmetic to working builders — the first English design handbook where a carpenter could size a beam by rule rather than lore. The rolled I-beam and the steel handbooks that followed are Navier's σ = M/S sold by the tonne.

1638Galileo GalileiThe cantilever problem posed — beam strength becomes science.
1713Antoine ParentNeutral axis located correctly.
1820Thomas Tredgold'Principles of Carpentry' — section design for practitioners.
1826Claude-Louis NavierLinear bending theory: σ = M·c/I.

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