Section Modulus Calculator

What Is Section Modulus?

Section modulus $S$ is a geometric property of a cross section that relates bending stress to bending moment. In elastic beam theory, maximum bending stress $\sigma = M/S$, where $M$ is the bending moment.

Common Formulas

Rectangular section (width $b$, depth $h$):

$$S = \frac{bh^2}{6}$$

Circular solid section (diameter $d$):

$$S = \frac{\pi d^3}{32}$$

I-beam (approximate, symmetric):

$$S = \frac{I}{c}, \quad I = \frac{b_f h^3 - (b_f - t_w)(h - 2t_f)^3}{12}$$

$c = h/2$ is the distance to the outer fiber. Example: a 200 mm × 300 mm rectangle gives $S = 200 \times 300^2 / 6 = 3{,}000{,}000$ mm³.

Design Use

Compare section modulus to required $M/\sigma_{\text{allow}}$ when selecting beams. A larger $S$ means the section can carry more moment before reaching the allowable stress.

Frequently Asked Questions

What units does section modulus use?

This calculator outputs mm³ (cubic millimeters). In SI, m³ is also common: divide mm³ by $10^9$ to convert.

How is section modulus related to moment of inertia?

$S = I/c$, where $I$ is the second moment of area and $c$ is the distance from the neutral axis to the outermost fiber.

Is the I-beam formula exact?

It assumes a symmetric I-shape with uniform web and flanges. Real rolled sections have fillets and taper; use manufacturer tables for final design.

Which dimension is height for a rectangle?

Height $h$ is the dimension perpendicular to the neutral axis in bending (the depth resisting flexure), not necessarily the vertical direction on a drawing.

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What Is Section Modulus?

Common Formulas

Design Use

Frequently Asked Questions

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