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Section Modulus Calculator

Enter the beam width and height to find its section modulus and moment of inertia.

Result

333,333.3 mm³
33,333,333.3 mm⁴
333.333 cm³

How it is calculated

For a solid rectangle bending about its horizontal centroidal axis, with width b (horizontal) and height h (vertical, the direction of the load), both in millimetres: the elastic section modulus is S = b × h² ÷ 6 and the second moment of area (moment of inertia) is I = b × h³ ÷ 12. The tool also shows S in cm³, which is S ÷ 1000. S is in mm³ and I in mm⁴, shown to one decimal. Both dimensions must be above zero. Only solid rectangles are covered; I-sections, hollow tubes and circles use different formulas. Stress in the outer fibre is then σ = M ÷ S, with M in N·mm giving σ in MPa.

Worked example

A timber joist 50 mm wide and 200 mm deep, standing on its edge. S = 50 × 200² ÷ 6 = 50 × 40,000 ÷ 6 = 2,000,000 ÷ 6 = 333,333.3 mm³. I = 50 × 200³ ÷ 12 = 50 × 8,000,000 ÷ 12 = 33,333,333.3 mm⁴. In cm³ the section modulus is 333.333. If the same joist lay flat (b = 200, h = 50), S would be only 83,333.3 mm³, a quarter as much, because h counts squared.

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Common mistakes

  • Swapping b and h. The height h, measured in the direction of the load, is squared, so a 50 × 200 beam on edge has four times the section modulus of the same beam laid flat.
  • Entering dimensions in centimetres or inches. The outputs assume mm, so 20 cm must be typed as 200; with cm input the S in mm³ would be wrong by a factor of 1,000.
  • Comparing the mm³ and cm³ results as if they were the same quantity. The cm³ line is the same S divided by 1,000.
  • Using this for a non-rectangular or built-up section. A hollow box, an I-beam or a notched joist needs its own S; the b·h²/6 formula applies only to a solid rectangle.
  • Confusing S with I. Deflection depends on I (and the material stiffness), while bending stress depends on S; neither alone checks a beam.

About this calculator

When a beam bends, the bending stress it experiences at the outer fiber follows σ = M/S, where M is the applied bending moment and S is the cross-section's elastic section modulus — a purely geometric property that tells you how efficiently a shape resists bending regardless of material. For a solid rectangular cross-section of width b and height h, S = b·h²/6, derived from the moment of inertia I = b·h³/12 divided by the distance c = h/2 from the neutral axis to the extreme fiber.

Structural engineers use section modulus, alongside allowable stress values from codes like AISC (steel) or Eurocode, to size joists, beams and lintels so bending stress under design loads stays safely below the material's yield strength — it's one of the first checks in any beam-sizing calculation, whether for a timber floor joist or a steel I-beam.

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Comments (1)

  • Editorial team

    Once you have S, bending stress is the moment divided by S. For the 50 by 200 mm joist at 333,333 mm cubed, a moment of 5 kN-m, which is 5,000,000 N-mm, gives 15 MPa, to be compared with the allowable stress for your timber grade and code. Deflection needs I, not S. Are you checking a joist, a lintel or a machine part, and which shapes would you like added?

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