Section Modulus Calculator
Section modulus, moment of inertia and the bending moment a shape can carry — for rectangles, rounds, pipes, tubes and I-beams. Pick a section, get the number that actually sizes the beam.
The Number That Sizes a Beam
Section modulus is the single geometric property that tells you how much bending a cross-section can take. Bending stress is simply the moment divided by it:
So for a given allowable stress, the moment a section can carry is M = S × σ. Double the section modulus and you double the bending capacity. That is why beam tables are sorted by S, not by weight or depth alone.
Elastic S vs Plastic Z
The elastic section modulus S applies while the whole section stays below yield — the basis of allowable-stress design. The plastic modulus Z applies when the entire section has yielded, used in limit-state and steel plastic design. Their ratio, the shape factor, is about 1.5 for a rectangle and 1.1 to 1.2 for a typical I-beam. The calculator reports both.
Shapes It Covers
| Section | Elastic modulus S |
|---|---|
| Rectangle | b h² / 6 |
| Solid round | π d³ / 32 |
| Round tube | π (D⁴ – d⁴) / (32 D) |
| Rectangular tube | (B H³ – b h³) / (6 H) |
| I-beam | I / c, web and flanges |
Frequently Asked Questions
Section modulus or moment of inertia?
Moment of inertia I governs deflection (stiffness); section modulus S governs bending stress (strength). S = I / c ties them together. Size for both.
Why is my I-beam value below the steel table?
This calculator assumes square corners and ignores the fillets at the web-flange junction, which add a little material. Rolled-section tables include them, so use published values for final design of standard shapes.
Which axis is this?
The strong (major) axis, bending about the horizontal centroidal axis with the depth vertical — the usual orientation for a loaded beam.
Related calculators
- Moment of Inertia Calculator — the I behind S, and the basis for deflection.
- Mechanical Stress Calculator — bending stress from moment and section.
- Beam Deflection Calculator — the stiffness side of beam sizing.
- Beam Load Calculator — loads and moments on the beam.
- Factor of Safety Calculator — margin against the limit.