Section Modulus Calculator
Find the elastic section modulus S, plastic section modulus Zp, and moment of inertia I for rectangular and solid circular cross-sections — the key geometric properties for beam bending design.
Cross-section shape
mm
mm
Bending stress σ = M / S, where M is the applied moment
- 1
Moment of inertia I = b·h³ ÷ 12
100 × 200³ ÷ 12 = 66,666,666.67 mm⁴ - 2
Distance to extreme fibre c = h ÷ 2
200 ÷ 2 = 100 mm - 3
Elastic section modulus S = I ÷ c
66,666,666.67 ÷ 100 = 666,666.67
How does this calculator work?
The elastic section modulus S = I/c determines bending capacity: σ = M/S. For a rectangle S = bh²/6; for a solid circle S = πd³/32. Enter width and height (rectangle) or diameter (circle) in mm to get S, Zp, moment of inertia, and cross-section area.
Formula
How this is calculated
The elastic section modulus S = I / c links the applied bending moment M to the maximum bending stress σ via σ = M / S. Here I is the second moment of area (moment of inertia) about the neutral axis, and c is the perpendicular distance from the neutral axis to the extreme fibre (the point furthest from the axis). A larger S means the cross-section can resist the same moment with less stress, which is why deep I-beams are more efficient than solid rectangles of equal area.
For a solid rectangle (width b, height h bending about the horizontal axis): I = bh³/12 and c = h/2, giving S = bh²/6. The plastic section modulus Zp = bh²/4 applies at full plastic (ductile) yielding, where the entire cross-section has yielded. For a solid circle of diameter d: I = πd⁴/64, c = d/2, S = πd³/32, and Zp = d³/6.
This calculator assumes bending about the strong (major) axis and a symmetric, homogeneous cross-section. For hollow sections, composite sections, or I-beams, the formulas differ and a full section-properties tool is needed. All dimensions are in millimetres and results are in mm³ or mm⁴; multiply by the appropriate power of 10³ to convert to cm units.
Frequently asked questions
The elastic section modulus S is used to find the maximum bending stress in a beam: σ = M / S, where M is the applied bending moment. Designers compare this stress to the material's yield strength to ensure the beam stays in the elastic range.
The elastic section modulus S applies when stresses are within the elastic range (no permanent deformation). The plastic section modulus Zp applies at full plasticity, when the entire cross-section has yielded. The ratio Zp/S is called the shape factor; for a rectangle it is 1.5, meaning a rectangular beam can carry 50% more moment before full collapse than at first yield.
No — this calculator covers only solid rectangular and solid circular cross-sections. I-beams, hollow rectangular sections, and hollow circular (pipe) sections require separate formulas. For those, use a dedicated section-properties calculator that accepts flange and web dimensions separately.
Also known as
TG we-Calculate Editorial Team. (2026). Section Modulus Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/section-modulus-calculator
TG we-Calculate Editorial Team. "Section Modulus Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/section-modulus-calculator.
TG we-Calculate Editorial Team, "Section Modulus Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/section-modulus-calculator
@misc{wecalculate_section_modulus_calculator, title = {Section Modulus Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/section-modulus-calculator}}, year = {2026}, note = {TG we-Calculate} }
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