What section properties tell you
How much a beam bends, and how much stress builds up in it, depends not only on the material but on the shape of its cross-section. Three properties describe that shape:
- Second moment of area I (also called area moment of inertia), in in⁴ or cm⁴: governs stiffness and deflection.
- Elastic section modulus S = I ÷ c, where c is the distance from the neutral axis to the outer fiber, in in³ or cm³: governs bending stress.
- Radius of gyration r = √(I ÷ A): used in column buckling.
The bending stress at the outer fiber is σ = M ÷ S, where M is the bending moment.
Formulas (bending about the strong axis)
| Shape | I | S |
|---|---|---|
| Solid rectangle b × h | b h³ ÷ 12 | b h² ÷ 6 |
| Hollow rectangle, wall t | [B H³ − (B − 2t)(H − 2t)³] ÷ 12 | 2I ÷ H |
| Solid circle, diameter d | π d⁴ ÷ 64 | π d³ ÷ 32 |
| Round tube, outside D, wall t | π [D⁴ − (D − 2t)⁴] ÷ 64 | 2I ÷ D |
| Symmetric I-beam | [b h³ − (b − t_w)(h − 2t_f)³] ÷ 12 | 2I ÷ h |
Example
A 2×10 joist has actual dimensions of 1.5 × 9.25 in. Its section modulus is 1.5 × 9.25² ÷ 6 = 21.39 in³ and its moment of inertia 98.9 in⁴. Under a bending moment of 3,000 lb·ft (36,000 lb·in), the bending stress is 36,000 ÷ 21.39 = about 1,680 psi.
Depth matters most
Because the depth is squared in S and cubed in I, doubling the depth of a rectangular beam makes it four times stronger and eight times stiffer, while doubling its width only doubles both. That is why joists are placed on edge and why I-beams put most of their material in the flanges, far from the neutral axis.
Checking a design
- Compare the bending stress with the allowable bending stress of the material: for lumber, the adjusted Fb from the NDS design values; for steel, the limits of the AISC specification.
- Steel design often uses the plastic section modulus Z, which is larger than S (1.5 times for a rectangle, about 1.1 to 1.15 for an I-beam).
- Use actual, not nominal, lumber sizes: a 2×10 is 1.5 × 9.25 in.
- Check deflection, shear and bearing as well; the beam deflection calculator covers deflection.
This calculator is a learning and estimating tool. Structural members in buildings must be designed or checked by a qualified professional under the local code.