Section Modulus Calculator Formula
Understand the math behind the section modulus calculator. Each variable explained with a worked example.
Formulas Used
Elastic Section Modulus Sxx
sxx = width_b * pow(height_h, 2) / 6Plastic Section Modulus Zxx
zxx = width_b * pow(height_h, 2) / 4Shape Factor (Z/S)
shape_factor = 1.5Variables
| Variable | Description | Default |
|---|---|---|
width_b | Width (b)(mm) | 100 |
height_h | Height (h)(mm) | 200 |
How It Works
Section Modulus of a Rectangle
Section modulus links bending moment to bending stress directly.
Elastic Section Modulus
S = b h^2 / 6
Derived from S = I / c where I = b h^3 / 12 and c = h / 2.
Plastic Section Modulus
Z = b h^2 / 4
Z represents the full plastic capacity when the entire cross-section has yielded. The ratio Z/S is called the shape factor; for a rectangle it is always 1.5.
Worked Example
A 100 mm x 200 mm rectangular section.
- 01Sxx = 100 x 200^2 / 6 = 100 x 40000 / 6 = 666,667 mm^3
- 02Zxx = 100 x 200^2 / 4 = 100 x 40000 / 4 = 1,000,000 mm^3
- 03Shape factor = 1,000,000 / 666,667 = 1.5
Frequently Asked Questions
When do I use elastic vs plastic section modulus?
Use the elastic modulus S for allowable-stress (working-stress) design. Use the plastic modulus Z for limit-state (LRFD) design where plastic redistribution is permitted.
What is the shape factor for other sections?
For a solid circle the shape factor is about 1.70. For standard I-beams it ranges from 1.10 to 1.20. Higher shape factors mean more reserve capacity beyond first yield.
How do I find section modulus for standard steel shapes?
Steel manufacturers publish section property tables that list S and Z for every rolled shape. Check the AISC Manual or equivalent regional standards.
Learn More
Guide
Beam Stress Calculation Guide: From Theory to Practice
Learn how to calculate beam stress step by step. Covers bending stress, shear stress, the flexure formula, stress distributions, and practical design checks for structural beams.
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Open Section Modulus Calculator