Structural design / section analysis

Section properties

Get area, second moment of area, section modulus and radius of gyration for rectangular, circular, hollow rectangular and hollow circular sections — and see how much inertia a hollow section buys for the same outside size.

ShapeSolid rectanglevs solid rect100.0% IExtreme fibre c150.0 mm

Second moment I

3.375e+8 mm⁴

Resistance to bending

Section modulus S

2.250e+6 mm³

I divided by the extreme fibre distance

Cross-section area

4.500e+4 mm²

353.25 kg/m in steel

Radius of gyration

86.60 mm

Governs buckling resistance

Shape comparison

All four sections at the same outside dimensions, so the material efficiency is directly comparable.

Solid rectangle

Shape comparison
Shape	Area (mm²)	I (mm⁴)	S (mm³)	r (mm)
Solid rectangle	4.500e+4	3.375e+8	2.250e+6	86.60
Solid circle	7.069e+4	3.976e+8	2.651e+6	75.00
Hollow rectangle	8600	9.969e+7	6.646e+5	107.66
Hollow circle	9111	9.589e+7	6.393e+5	102.59

Hollow sections carry almost the same inertia with far less material — that is why tubes beat solid bars in bending.

Inertia by shape

Second moment of area at the same bounding box.

Section modulus vs depth

For a fixed width, S grows with the square of depth.

Canonical equations

The exact relationships implemented by this engine.

I = ∫y²dA · about the centroidal axis

S = I/c · c is the extreme fibre distance

r = √(I/A) · used in column buckling

σ = M/S · the design check this feeds

Show model notation and computationOpen the symbolic form, the variable definitions, and the step-by-step derivation when you need the detail.

A clear path from inputs to results.

Bending resistance depends on how far the material sits from the neutral axis, not on how much material there is. That is why a hollow section carries almost the same moment as a solid one at a fraction of the weight. This engine computes the four elastic properties that structural checks actually use.

Step-by-step methodWorked exampleAssumptions explained

How to use

Choose the section shape.

Enter the width and depth, or the diameter for circular sections.

Set the wall thickness for hollow shapes.

Read the area, second moment, section modulus and radius of gyration.

Compare against the solid rectangle in the table — the difference is the material efficiency.

Use the modulus for bending stress and the radius of gyration for buckling.

The formula

I_{\text{rect}} = \frac{bh^3}{12}, \quad I_{\text{circ}} = \frac{\pi d^4}{64}, \quad S = \frac{I}{c}, \quad r = \sqrt{\frac{I}{A}}, \quad \sigma = \frac{M}{S}

Reference: Gere & Goodno, "Mechanics of Materials", appendix tables; EN 1993-1-1 for steel section properties.

Worked example

150 × 300 mm solid rectangle (default inputs)

Inputs

Width 150 mm.
Depth 300 mm.
Solid rectangle.

Calculation path

1
Area: 150 × 300 = 45,000 mm².
2
Second moment: 150 × 300³ / 12 = 337.5×10⁶ mm⁴.
3
Section modulus: I / 150 = 2.25×10⁶ mm³.
4
Radius of gyration: √(337.5×10⁶ / 45,000) = 86.6 mm.
5
Mass in steel: 45,000 mm² × 7,850 kg/m³ = 353 kg/m.

Result

45,000 mm² · 337.5×10⁶ mm⁴ · 2.25×10⁶ mm³ · r = 86.6 mm · 353 kg/m in steel.

Interpreting the result

Section modulus is what converts a bending moment into a stress: σ = M/S. Doubling the depth quadruples S but only doubles the area, which is why deep beams are efficient and why a 150×300 section is far stronger in bending than a 300×150 one laid flat. The radius of gyration matters separately — it governs column buckling, where the same section can behave very differently.

Assumptions & limitations

Elastic properties about the strong (major) axis. The weak axis has a different second moment.

The section is homogeneous and remains elastic; plastic modulus and shape factor are separate properties.

No local buckling of thin walls is considered. Slender hollow sections fail locally before the full modulus is reached.

Corners of hollow sections are assumed sharp; real rolled sections have fillets that add a little area and inertia.

Torsion constant and warping constant are not derived — they need a separate calculation.

Frequently asked questions

What is the difference between I and S?

I is the second moment of area, a geometric property independent of the extreme fibre. S = I/c is the section modulus, which includes the distance to the extreme fibre and is what converts moment to stress.

Which axis should I use?

The axis about which bending occurs. For a beam spanning horizontally with vertical loads, that is the strong axis. For a column, the weaker axis usually governs buckling.

What about plastic design?

Plastic modulus Z is roughly 1.1–1.2 times S for common shapes. Plastic design allows the full section to yield, giving a higher moment capacity, but requires compact sections and adequate bracing.

Why are hollow sections so efficient?

Because material near the neutral axis contributes almost nothing to bending resistance. Moving it outward increases I quadratically with distance while barely changing the area.

How does this feed a bending check?

Compute the maximum moment, divide by S to get stress, and compare against the allowable or yield strength with a safety factor. This engine gives you the S.

Can I use this for timber or concrete?

The geometry is the same, but the allowable stress and failure mode differ. Concrete is rarely designed on elastic section modulus because it cracks in tension.