How this instrument works
Section modulus is the geometry half of bending strength. Peak fibre stress in a loaded beam is σ = M ⁄ S — whatever moment your load imposes, divided by one figure that depends on nothing but cross-section shape and size. Formally S = I ⁄ c, second moment of area divided by distance from neutral axis out to the extreme fibre. Feed in a solid rectangle, where I = b·h³ ⁄ 12 and c = h ⁄ 2, and that ratio collapses to b·h² ⁄ 6: two multiplications standing in for an entire stress analysis.
Galileo posed this problem first, in his Discorsi of 1638, sketching a stone cantilever jutting from a wall and asking what snaps it. His answer was wrong by a factor of three, because he had the beam pivot about its bottom edge, leaving every fibre in tension. Antoine Parent argued in 1713 that tension above must balance compression below; Coulomb — better remembered for electrostatics — settled matters in his 1773 memoir submitted to the French Academy, placing a neutral axis through the centroid. Navier's lectures at the École des Ponts, published in 1826, turned σ = M·c ⁄ I into routine practice, and section tables have quoted S ever since.
Catalogue values give a feel for scale. European steel tables list elastic modulus about the strong axis in cm³: 203 × 133 UB 25 offers roughly 230 cm³, 305 × 165 UB 40 about 560. American shapes appear in cubic inches, where W12 × 26 carries 33.4 in³. Sawn 38 × 235 mm softwood joists land near 350 cm³. What b·h² ⁄ 6 assumes is strict: one solid homogeneous rectangle, bending about an axis parallel to width, stress still elastic, nothing weakened by holes or notches. It says nothing about deflection, which needs I; nothing about shear, which rules stubby spans; nothing about a slender member twisting sideways before it ever yields; and nothing about plastic reserve, since a fully yielded rectangle carries b·h² ⁄ 4, half again as much.
- Enter the dimension across the load into Beam width — millimetres, centimetres, metres or inches all work from its unit menu.
- Enter the dimension along the load into Beam depth. For a joist standing on edge, depth is its tall side, not its thin one.
- Read Section modulus (m³) for strength work: allowable bending moment is that figure multiplied by allowable bending stress.
- Read Second moment of area (m⁴) for stiffness work, since every deflection formula asks for I and never for S.
- Swap the two entries to price what laying the same piece flat would cost you.
Worked example — a 50 × 100 mm joist on edge
A softwood joist sawn 50 mm wide and 100 mm deep, standing on edge. Put 0.05 into Beam width and 0.1 into Beam depth. Section modulus (m³) returns 8.3333 × 10⁻⁵, since 0.05 × 0.1² ⁄ 6 works out as 0.0005 ⁄ 6, and Second moment of area (m⁴) returns 4.1667 × 10⁻⁶ from 0.05 × 0.1³ ⁄ 12. Written as timber tables prefer, that is 83.3 cm³ and 417 cm⁴.
Multiply by an allowable bending stress and strength falls straight out. C24 graded softwood at roughly 14 N/mm² gives 14 × 83 333 mm³, near enough 1.17 kN·m of moment. Spread as a uniform load over a 3 m simply supported span, where M = w·L² ⁄ 8, that permits about 1.04 kN per metre — plausible for a domestic floor, and rather more than most people credit a stick of wood with.
Lay that same piece flat — 100 mm across, 50 mm deep — and Section modulus (m³) drops to 4.1667 × 10⁻⁵, exactly half, while Second moment of area (m⁴) falls to 1.0417 × 10⁻⁶, exactly a quarter. Identical timber, identical weight, and well under half a joist. Depth is squared in one expression and cubed in the other, which is why nobody frames a floor with boards on their sides.
Questions
What is the difference between section modulus and second moment of area?
One sets stress, the other sets deflection. Section modulus S has units of m³ and answers strength: peak fibre stress is M ⁄ S, so allowable moment is S multiplied by allowable stress. Second moment of area I has units of m⁴ and answers stiffness, since mid-span sag under a uniform load runs as 5wL⁴ ⁄ (384EI). They connect through S = I ⁄ c, with c half of depth for a symmetric rectangle. Size a beam on S alone and it may still bounce underfoot; size it on I alone and it may still crack.
Which side of my beam counts as the depth?
Depth is measured along whichever direction your load pushes — vertically, for a floor joist under gravity — and width runs across it. Enter them the wrong way round and this sheet quietly describes a different beam. A 50 × 100 mm section standing on edge wants 0.05 in Beam width and 0.1 in Beam depth; laid flat, those two swap and section modulus halves.
Why does depth get squared while width stays linear?
Two effects stack. Material further from the neutral axis strains more, so it carries proportionally more stress, and it also pulls on a longer lever arm about that axis. Both scale with distance, which is why the defining integral ∫y² dA produces h³ inside I. Widening merely adds parallel material at unchanged distances, so b enters once. Section modulus then divides I by c = h ⁄ 2, handing one factor of h back and leaving h².
Can I use b·h² ⁄ 6 for an I-beam, a tube or a T-section?
No — that expression belongs to a solid rectangle alone. Hollow rectangles subtract their void: S = (B·H³ − b·h³) ⁄ (6H), capitals marking outside dimensions. Rolled I-shapes and channels come from published section tables rather than any formula worth memorising. Anything lacking symmetry about its bending axis — a T, an angle, a channel bent the awkward way — has two section moduli, one per face, because c differs top and bottom. Design against the smaller.
What is plastic section modulus, and should I use that instead?
Plastic modulus Z assumes an entire section has yielded rather than just its outermost fibre; for a rectangle Z = b·h² ⁄ 4, exactly 1.5 times the elastic value. Engineers call that ratio shape factor, and it drops nearer 1.12 for a rolled I-beam, which already keeps most of its material far out. Reach for elastic S when a code works in allowable stress or when yielding anywhere is unacceptable, and for Z only in plastic design of compact steel able to rotate without buckling locally. Timber and cast iron get elastic S, full stop.
Do bolt holes, notches or a service cut change the answer?
Yes, and usually by more than lost area suggests. Removing material near the neutral axis costs little, since those fibres were barely working. A notch sawn into the tension face of a joist removes precisely the fibres doing most of the job and plants a stress raiser at its root besides. Timber codes cap notch depth at some fraction of section depth and bar notches near mid-span for exactly that reason. Recompute with reduced depth, then treat what comes back as optimistic.