Section Properties for Common Shapes
Second moment of area, polar moment, radius of gyration and section modulus for rectangles, circles, tubes, I-sections and thin-walled sections.
Definition
The second moment of area (I, also called area moment of inertia) describes the resistance of a cross-section to bending. The polar moment of area (J) describes resistance to torsion. The section modulus (Z = I/c) relates bending moment to maximum bending stress.
Common Section Properties
| Section | I (bending) | Z | J (torsion) | r (gyration) |
|---|---|---|---|---|
| Rectangle (b × h) | bh³/12 | bh²/6 | bh³·(1−0.63·h/b)/3 (approx) | h/√12 |
| Hollow rectangle (b×h, t) | (bh³ − bᵢhᵢ³)/12 | (bh³ − bᵢhᵢ³)/(6h) | — | — |
| Circle (diameter d) | πd⁴/64 | πd³/32 | πd⁴/32 | d/4 |
| Hollow circle (d, t thin) | πd³t/8 | πd²t/4 | πd³t/4 | d/√8 (approx) |
| I-section (flange bf×tf, web tw×hw, total h) | (bf·h³ − (bf−tw)·hw³)/12 | I/(h/2) | — (use FEA or torsion charts) | — |
| Thin-walled tube (d, t) | π(d⁴ − dᵢ⁴)/64 | πd³/32·(1−(dᵢ/d)⁴) | π(d⁴ − dᵢ⁴)/32 | — |
Useful Relationships
Section modulus: Z = I / c Radius of gyration: r = √(I / A) Bending stress: σ = M·c / I = M / Z where c = distance from neutral axis to extreme fibre Parallel axis theorem: I_x = I_c + A · d² where I_c = centroidal second moment, d = distance to parallel axis
Variables
| Symbol | Definition | Units |
|---|---|---|
| I | Second moment of area (bending) | mm⁴ |
| J | Polar moment of area (torsion) | mm⁴ |
| Z | Section modulus | mm³ |
| r | Radius of gyration | mm |
| A | Cross-sectional area | mm² |
| b, h | Width and height of section | mm |
| d | Diameter of circular section | mm |
| t | Wall thickness | mm |
| c | Distance from neutral axis to extreme fibre | mm |
Notes and Limitations
- I-section torsion constant J is not the polar moment — use torsion charts or FEA for open thin-walled sections
- For asymmetric sections, the principal axes are not necessarily horizontal and vertical
- For composite sections, use the transformed section method with modular ratio
- All formulae assume the neutral axis passes through the centroid
Related Knowledge
See the Structural Analysis Knowledge category for bending theory and the Loads Development category for section selection.