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Section Properties Calculator

Inertia governs deflection, section modulus governs strength — they are not interchangeable.

Also known as section properties · moment of inertia · radius of gyration

Quick answer

With the default inputs: Section shape Rectangle b×h · Width b 200 mm · Depth h 400 mm etc.Second moment I 106667 cm⁴

Use your own dimensions ↓

Inputs

Result

Second moment I
106667cm⁴
Section typeRectangle
Area A800.0cm²
Second moment I106667cm⁴
Section modulus W5333cm³
Plastic modulus Wpl8000cm³
Radius of gyration r115.5mm
Mass per metre628.0kg/m
Shape factor1.50
How this is worked out

Second moment and section modulus are two numbers doing two jobs, and swapping them gives a wrong answer. Inertia, in cm⁴, governs stiffness — deflection, distortion, vibration. Section modulus, W = I/(h/2) in cm³, governs strength — how much moment the section takes. So “is this stiff enough” asks about I and “will this fail” asks about W. For a rectangle I goes with depth cubed and W with depth squared: 20% more depth adds 73% to I but only 44% to W, so deflection improves faster than bending. That is the point of an I-section — material moved away from the neutral axis gives two to three times the inertia of a rectangle of the same area, at the cost of a thin web prone to local buckling and poor minor-axis stiffness, often a tenth of the major axis. An I-beam therefore always needs a lateral stability check and a rectangle does not. The plastic modulus runs 12 to 15% above the elastic one for an I-section and 50% for a rectangle; that ratio is the shape factor, the plastic reserve.

Next up: Steel Beam Span Calculator · Fillet Weld Strength Calculator

Formula

RectangleI = bh³/12, W = bh²/6
Circular hollowI = π(D⁴−d⁴)/64, d = D−2t
I-sectionI = [bf·h³ − (bf−tw)(h−2tf)³]/12
Section modulusW = I/(h/2)
Radius of gyrationr = √(I/A)
Mass per metrem = A·7850 (A in m²)
A 200×400 rectangle: area 800.0 cm², 106667 cm⁴ of inertia, 5333 cm³ of section modulus, radius of gyration 115.5 mm, 628.0 kg/m and a shape factor of 1.50.

Basis

FAQ

What is the difference between inertia and section modulus?
Inertia is the integral of y² over the area and governs stiffness; section modulus is inertia divided by the extreme fibre distance and governs strength. Deflection uses I, bending capacity uses W. For a 200×400 rectangle, I is 1.067×10⁹ mm⁴ and W is 5.33×10⁶ mm³ — 200 times apart because the units differ, mm⁴ against mm³, so neither can be substituted into the other formula.
Why is an I-section so efficient in bending?
Because the material sits far from the neutral axis. Capacity scales with the integral of y squared, so the same area at 200 mm out contributes sixteen times as much as at 50 mm. Thin the web and widen the flanges and inertia can go up two to three times for the same area. The penalties are local buckling in a thin web, needing stiffeners, torsion in thin flanges, and a weak minor axis — often a tenth of the major — so lateral restraint is essential.
Why is a hollow section the same in every direction?
Because a circle is symmetric about any axis through its centre. That is valuable in columns, where buckling can go any way, so a tube needs no preferred orientation. It makes spiral-welded tube and steel pipe columns economical under biaxial moment or uncertain direction. The section modulus is I/(D/2), slightly below a rectangle of the same area, because the material sits on a ring rather than at the extremes.
What is the shape factor?
It is the plastic modulus over the elastic one: 1.5 for a rectangle, about 1.12 to 1.15 for an I-section and 1.27 for a tube. It measures the reserve after first yield — a rectangle carries 50% more moment than its elastic limit, an I-section only 12%, so once the flanges yield an I-beam loses capacity quickly while a rectangle has a long plastic plateau. That matters in seismic design, where members expected to form hinges benefit from a higher shape factor because rotation capacity follows it.

What to calculate next

These jobs are done back to back on site — follow the steps to work through the whole sequence.

Steel: span to connectionStep 2 of 4

  1. 1Steel Beam Span Calculator
  2. 2Section Properties Calculator
  3. 3Column Buckling Calculator
  4. 4Fillet Weld Strength Calculator

Cite this page

Use this format in reports, method statements or coursework. The link is permanent.

Building Calculator. Section Properties Calculator. 2026-10-25. https://jianzhujisuan.com/en/section-modulus

Results are estimating aids only and are not a design basis. Construction and final accounts must follow the drawings, the applicable standards and an approved method statement. Anything affecting structural safety, shoring or falsework must be checked by a suitably qualified engineer.