Section Modulus Calculator
The section's bending-resistance divisor, derived from first principles: I, the extreme fibre, and the S that turns moments into stresses.
Section Modulus Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Gross sections; no slender-element reductions
- Two shapes — rectangle and circle
In short: A 100 × 200 mm rectangle carries I = 66,666,666.667 mm⁴ about its strong axis and a section modulus of S = 666,666.667 mm³ — bh³/12 divided by the 100 mm to the extreme fibre, landing on bh²/6. That S is the divisor the bending-stress page consumes: the same moment over twice the modulus is half the stress, which is the whole shopping logic of beam tables.
Formula
I = bh³/12 · πd⁴/64 ··· S = I/c = bh²/6 · πd³/32
The second moment of area prices how far the section's area sits from its bending axis; the modulus divides that by the extreme-fibre distance because stress grows linearly to the surface. Depth enters cubed in I and squared in S — the arithmetic engine behind every deep beam ever chosen over a thick one.
Worked Example
- Pick the shape — rectangle or circle.
- Enter the bending depth (or diameter).
- Read I, c and the modulus S.
- Divide your moment by S — that is the bending stress.
Defaults: 100 × 200 rectangle → I 66,666,666.667 mm⁴, S 666,666.667 mm³, c 100.000000 mm. The circle check: d = 100 → I 4,908,738.521 mm⁴, S 98,174.770 mm³ — the same-area circle is the floppier section, which is why masts are not solid rods.
Strengths & Limits Of This Model
Where this engine is strong
- I, c and S from one entry
- Both axes' arithmetic shown
Where it stops
- No rolled-shape database
- No parallel-axis composites
Practical Use Cases
Section choice
compare candidates by S
Built-up members
the plate arithmetic
Teaching
why depth is cheap strength
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Section Modulus Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why is S the number tables boast about?
Because it is the one-divisor summary of bending resistance: stress equals moment over S, so capacity is yield times S. Two sections of identical area can differ in S by multiples, which is exactly the spread between paying for steel and placing it. The tables sort by S so the shopper never has to integrate anything.
Why does depth enter cubed?
Because I integrates area times distance squared, and the rectangle's distance runs to h/2 — the h³ falls out of the integration and one more h drops dividing by c. It is geometry's subsidy for spreading material vertically: every millimetre of depth works harder than the last, up to the buckling and shear limits that eventually object.
Same area, rectangle vs circle — who wins?
The rectangle with its material pushed to the extremes. A 100 × 200 rectangle (area 20,000 mm²) carries S = 666,666.667 mm³; the equal-area circle (d ≈ 159.575 mm) only about 400,000 mm³. Circles are for loads that point in every direction — shafts, poles, masts — not for gravity's single minded downward pull.
Does the modulus know about buckling?
No — and that is its fine print. S prices the fully-effective section; thin flanges and slender walls lose effectiveness by local buckling first, and deep thin beams surrender by lateral-torsional buckling before the outer fibre ever yields. The modulus is the first-order strength; classification and stability are the corrections the codes carry.
Which axis is 'strong'?
The one bending the section resists better — for a rectangle lying flat, the strong axis bends it about its depth. Enter b as the width perpendicular to bending and h as the depth parallel to the load; swap them and the modulus collapses by the aspect ratio, which is precisely how a plank on flat fails where on edge it would have shrugged.
Can I add two rectangles — a flitch or built-up?
Yes, with care: I adds about the COMMON neutral axis, so compute each part's I about the composite centroid (parallel-axis theorem) before summing, then divide by the extreme fibre. Naively adding the parts' own S values undercounts by exactly the lever arms between them — which is the entire point of building up in the first place.
Why is the circle's S πd³/32?
Same recipe as the rectangle: integrate for I — πd⁴/64 — then divide by the extreme fibre d/2, and the quarters cancel to the 32. The circle's symmetry makes every diameter a strong axis, which is its quiet superpower: no wrong way up, at the price of material spread evenly rather than where gravity's bending wants it.
How does S relate to the deflection page?
They are siblings with different parents: S comes from I divided by c and answers strength; deflection uses I directly and answers stiffness. A section can double its S with half the depth change that doubling I needs, so strength problems solve cheaper than deflection problems — the classic reason a beam can pass one page and fail the other.