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Technical guide · method and practical tips

Second moment of area, section modulus and radius of gyration

Section properties do not describe the material: they describe how cross-sectional area is distributed around its axes.

Reviewed

area close to the axisarea far from the axisI lowerI higherI = ∫ y² dA
Moving area away from the axis strongly increases I because every area element is weighted by the square of its distance.

Cross-sectional area A

Area is used to calculate axial stress σ = F/A, linear mass ρA and axial stiffness EA/L.

However, it says nothing about how the material is distributed.

Second moment of area I

I = ∫ y² dA

The second moment of area strongly depends on distance from the axis. The same area placed twice as far away contributes four times as much.

Always distinguish Ix and Iy: a section can be very stiff about its major axis and much weaker about its minor axis.

Section modulus W

W = I / c ; σmax = M / W

Section modulus combines I with distance c to the extreme fibre. It is directly used to estimate maximum elastic bending stress.

For the same moment M, a larger W gives a lower maximum stress.

Radius of gyration r

r = √(I/A)

Radius of gyration represents an equivalent distance describing area spread. It is used in compression-member slenderness λ = Lk/r.

The axis with the smallest r is generally the first one to check for buckling.

Axes and units: the essential check

I is expressed in length to the fourth power, W in length cubed and r in length. Confusing mm⁴ with m⁴ creates a factor of 10¹².

Always check axis name, dimension convention and displayed unit before reusing a result.

Numerical application

Worked example: 100 × 200 mm rectangle

Compare the second moments of a rectangle with width b = 100 mm and height h = 200 mm about its centroidal axes.

  1. A = b·h = 20,000 mm².
  2. Ix = b·h³/12 = 66.67 × 10⁶ mm⁴.
  3. Iy = h·b³/12 = 16.67 × 10⁶ mm⁴.
  4. The ratio Ix/Iy is 4.
  5. The section is therefore four times geometrically stiffer about x than y, all else being equal.

Result: Rotating the rectangle by 90° swaps the major and minor axes without changing area or material.

Classical analytical method taught in mechanics of materials; numerical example adapted for Formulaxis. Open exact source ↗

For checking and further study

Direct references

Each link points to the exact course, standard or publication page used, rather than a generic homepage.

  1. Mechanics & Materials IMIT OpenCourseWareUniversity course covering stress, bending, torsion and section properties.
  2. The International System of Units, 9th editionInternational Bureau of Weights and Measures (BIPM)Official reference for coherent units, symbols and conversions.
  3. NIST Special Publication 811 — Guide to the SINational Institute of Standards and TechnologyPractical rules for writing and converting physical quantities correctly.

IPE beam: a complete worked example

Major axis: 22.8 MPa and 1.81 mm. Minor axis: 156 MPa and 24.8 mm. These are model results, not an approval of the beam.