Section Modulus Calculator
Calculate the elastic section modulus S = I/c and the moment of inertia for common cross-sections — rectangle, solid round, or hollow round tube. Enter the dimensions in inches, millimeters, or centimeters to size a beam against bending stress.
Example: with Cross-section shape Rectangle (b × h) · Width b (rectangle) 2 · Height / depth h (rectangle) 6 · Outer diameter (round) 4 · Inner diameter (hollow) 3 → Section modulus S: 12 in³.
- Moment of inertia I36 in⁴
- What it meansBending stress = moment / S; a larger S carries the same moment at lower stress.
Computed by the calculator below using its default values. Change any input to see your own numbers.
Section modulus S = I/c bundles a beam's shape into one number for bending: stress equals bending moment divided by S. Push material away from the neutral axis and S climbs fast.
How shape becomes strength
When a beam bends, one face stretches and the other compresses, and the stress is highest at the surface farthest from the neutral axis. Section modulus captures exactly that geometry: S = I/c, the moment of inertia divided by the distance c to the extreme fiber. Bending stress is then simply the applied moment divided by S, so for a target stress a bigger S lets a beam carry a bigger load.
Because the moment of inertia depends on the cube (or fourth power) of depth, orientation dominates. A 2×6 laid flat has a section modulus of only 3 in³, but stood on edge it jumps to 12 in³ — four times stiffer in bending from the same board. That is why joists are installed tall and why I-beams pile their material into the top and bottom flanges.
How it’s calculated
Rectangle: I = b·h³/12, c = h/2, so S = b·h²/6. Solid round: I = π·D⁴/64, S = π·D³/32. Hollow round: I = π·(D⁴ − d⁴)/64, S = I/(D/2), with D outer and d inner diameter. Units follow the input length: results in in³/in⁴, mm³/mm⁴, or cm³/cm⁴.
Gives the elastic section modulus about the horizontal (bending) axis for symmetric shapes only. It is a geometry calculator, not a code check — have a licensed engineer size any load-bearing member for actual loads, material, and stability.
Section modulus of common shapes
| Shape | Dimensions | Section modulus S |
|---|---|---|
| Rectangle (flat) | 6 × 2 in | 3.00 in³ |
| Rectangle (on edge) | 2 × 6 in | 12.0 in³ |
| Solid round | 4 in dia | 6.28 in³ |
| Hollow round | 4 in / 3 in | 4.30 in³ |
| Rectangle | 2 × 8 in | 21.3 in³ |
Computed with S = b·h²/6 and S = π·D³/32; rounded.
Common mistakes
- Swapping b and h on a rectangle — depth h is squared, so getting them backwards changes S dramatically.
- Using the section modulus where the moment of inertia is needed; deflection uses I, bending stress uses S.
- Applying symmetric-shape formulas to a tee or channel, whose neutral axis is off-center.
- Treating this geometry number as a safe load — allowable stress, span, and material still govern the design.
Frequently asked questions
What is the section modulus formula?
S = I/c, the moment of inertia divided by the distance from the neutral axis to the outermost fiber. For a rectangle it reduces to S = b·h²/6; for a solid round, S = π·D³/32.
What is the difference between S and I?
Moment of inertia I measures resistance to bending and drives deflection. Section modulus S = I/c relates bending moment to peak stress, so it is what you use to check strength.
Why is a beam stronger on edge than flat?
Section modulus depends on the square of the depth, so standing a 2×6 on edge instead of flat multiplies S about fourfold. Depth in the bending direction is what buys strength.
Can I size a real beam with this?
Use it to compare shapes and get S and I, but not to certify a design. Allowable stress, span, load type, and buckling matter, so have a licensed engineer size load-bearing members.