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Section Properties — Rectangle, Circle, Pipe, I-Beam

Cross-section geometric properties: area A, moment of inertia I, section modulus S, radius of gyration r, plastic section modulus Z. Strong-axis (x-x, bending) and weak-axis (y-y) values where applicable.

in
in
in (I-beam only)
in (I-beam only)
in (pipe only)
—in²
—in⁴
—in³
—in
—in³
—in⁴
—in

Defaults: 6 × 12 in rectangle. For I-beam, the "Height" field is total depth d; flange width bf, flange thickness tf, web thickness tw. Plastic modulus Z is reported for symmetric sections.

Rectangle (b wide × h deep):
$$ A = bh, \quad I_x = \frac{bh^3}{12}, \quad I_y = \frac{hb^3}{12}, \quad Z_x = \frac{bh^2}{4} $$
Circle (diameter D):
$$ A = \frac{\pi D^2}{4}, \quad I = \frac{\pi D^4}{64}, \quad Z = \frac{D^3}{6} $$
Hollow circular (Do, Di):
$$ I = \frac{\pi (D_o^4 - D_i^4)}{64} $$
I-beam (parallel-axis theorem, two flanges + web):
$$ I_x = \frac{b_f \, d^3}{12} - \frac{(b_f - t_w)(d - 2t_f)^3}{12} $$
I moment of inertia · S = I / c section modulus, c = distance to extreme fiber · r = √(I/A) radius of gyration · Z plastic section modulus (= sum of |y| dA over half the section).

S vs Z — elastic vs plastic

Section modulus S = I/c is the elastic section modulus. The maximum elastic moment a section can carry is My = S σy — the moment at which the extreme fiber just reaches yield. Beyond this, plastic strain spreads inward.

Plastic section modulus Z is the area moment of one half-section about the plastic neutral axis. The maximum moment for a fully-plastic section is Mp = Z σy. Always Z ≥ S (always); the ratio Z/S is the shape factor.

AISC LRFD design uses Z (plastic) for compact sections; ASD historically used S (elastic) but now also uses Z with a safety factor.

Why two axes

For non-square sections, bending strength depends on which axis is the bending axis. The "strong axis" (x-x) is the one with greatest I — usually the dimension perpendicular to the long side of the cross-section. The "weak axis" (y-y) is perpendicular to that.

An I-beam loaded weak-axis bends about y-y and carries 1/15 to 1/30 the load of strong-axis loading. Always orient I-beams so loads bend about strong axis (web vertical for downward load).

Composite sections — go beyond this calculator

For built-up sections, transformed sections, or asymmetric shapes (channels, angles, tees), use the parallel axis theorem. Each component contributes (Iown + A × d²) to the total I, where d is the distance from the component's centroid to the section's neutral axis. The neutral axis is at ΣAȳ / ΣA.

Composite steel-concrete sections require modular ratio n = Esteel/Econcrete to convert concrete area into "equivalent steel" before computing I. AISC and ACI handle this differently; check both codes for composite floor design.

Where to find AISC W-shape properties

The AISC Steel Construction Manual has tabulated A, Ix, Iy, Sx, Sy, rx, ry, Zx, Zy, J (torsion), Cw (warping) for every standard W, M, S, HP, C, MC, L, WT, MT shape. Free CD-ROM with the manual; some online sources tabulate as well. This calculator is for non-standard shapes and quick-checks.

Reference: AISC Steel Construction Manual, 16th ed. (2023). Hibbeler, R.C. (2014). Mechanics of Materials, 9th ed., Pearson, Appendix A.

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