Section Modulus Calculator
Get the full set of section properties for common beam and structural shapes: rectangle, rectangular tube, round bar, pipe, I-beam, channel, angle and T-section. You get area, centroid, Ix and Iy, elastic section modulus at each extreme fibre, plastic modulus, radius of gyration, principal axes for angles, and the bending stress for a given moment.
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How to use the Section Modulus Calculator
- Pick the cross-section shape and the unit of your dimensions.
- Enter the dimensions shown for that shape (depth, width, wall, flange and web thickness).
- Optionally enter a bending moment and the axis to get the maximum bending stress.
- Read the section modulus and the other properties; check the drawing to confirm the shape.
- Copy the results or download the full property table as CSV.
Section property formulas
| Shape | Area A | Ix (centroidal) | Elastic Sx | Plastic Zx |
|---|---|---|---|---|
| Rectangle b × h | b h | b h³ ÷ 12 | b h² ÷ 6 | b h² ÷ 4 |
| Round bar, diameter D | π D² ÷ 4 | π D⁴ ÷ 64 | π D³ ÷ 32 | D³ ÷ 6 |
| Pipe, D outside, d inside | π (D² − d²) ÷ 4 | π (D⁴ − d⁴) ÷ 64 | π (D⁴ − d⁴) ÷ (32 D) | (D³ − d³) ÷ 6 |
| Rectangular tube B × H, inside b × h | B H − b h | (B H³ − b h³) ÷ 12 | (B H³ − b h³) ÷ (6 H) | (B H² − b h²) ÷ 4 |
| Symmetric I-beam | 2 b tf + (d − 2tf) tw | [b d³ − (b − tw)(d − 2tf)³] ÷ 12 | 2 Ix ÷ d | b tf(d − tf) + tw(d − 2tf)² ÷ 4 |
For channels, angles and tees the calculator splits the shape into rectangles, finds the centroid with ȳ = Σ Ai yi ÷ Σ Ai, and applies the parallel-axis theorem I = Σ (Ii + Ai di²). The plastic neutral axis is the line that splits the area into two equal halves, which for unsymmetric shapes is not the centroid.
Using section modulus in beam design
- Bending stress: σ = M ÷ S. With M in N·mm and S in mm³ the stress comes out directly in MPa; with M in lbf·in and S in in³ it is in psi.
- Required modulus: rearrange to Sreq = M ÷ σallow and choose the lightest section that exceeds it.
- Depth is king: S grows with the square of depth (and I with the cube), so a deeper section is far more efficient than a wider one.
- Deflection uses I, strength uses S. Check both: a beam can be strong enough but still sag too much.
- Weak-axis and torsion: open sections such as I-beams and channels have a small J and can twist or buckle sideways (lateral-torsional buckling) when not braced.
Unit conversions for section properties
| Property | Metric to inch | Inch to metric |
|---|---|---|
| Area | 1 mm² = 0.00155 in² | 1 in² = 645.16 mm² |
| Section modulus | 1 cm³ = 0.06102 in³ | 1 in³ = 16.387 cm³ |
| Moment of inertia | 1 cm⁴ = 0.02403 in⁴ | 1 in⁴ = 41.623 cm⁴ |
Frequently asked questions
What is section modulus?
The elastic section modulus S is the moment of inertia divided by the distance from the neutral axis to the extreme fibre, S = I ÷ c. Maximum bending stress is σ = M ÷ S, so for a given bending moment a larger section modulus means lower stress.
What is the section modulus of a rectangle?
About its strong axis S = b h² ÷ 6, where b is the width and h the depth. A 2×10 joist (1.5 × 9.25 in) on edge has S ≈ 21.4 in³; laid flat it is only about 3.5 in³, which is why joists are placed on edge.
What is the difference between elastic and plastic section modulus?
The elastic modulus S gives the moment at which the outer fibres first yield (M = Fy × S). The plastic modulus Z gives the moment when the whole section has yielded (Mp = Fy × Z). The ratio Z ÷ S is the shape factor: 1.5 for a rectangle and about 1.1 to 1.2 for I-beams.
Why does a channel or T-section have two section moduli?
When a shape is not symmetric about an axis, its centroid is closer to one face. The distance c to each extreme fibre differs, so S = I ÷ c is different at the top and bottom (or left and right). The smaller value governs the maximum stress.
What are principal axes and why do they matter for angles?
For an unsymmetric shape like an L-angle the product of inertia Ixy is not zero, so the strongest and weakest bending axes are rotated from x and y. The minimum principal radius of gyration is what controls the buckling of angle struts.
How accurate are the results for rolled steel sections?
The calculator uses sharp-cornered rectangles. Hot-rolled beams, channels and angles have root fillets and sometimes tapered flanges, which usually add a few percent to the properties. Use the published section tables for design of standard rolled shapes.
What is radius of gyration?
r = √(I ÷ A) is the distance from the axis at which the whole area could be concentrated to give the same moment of inertia. It is used in column buckling: the slenderness ratio is the effective length divided by r.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.