section modulus calculator - Aaron Graves, PhDude Replica

Section Modulus Calculator (Elastic)

Enter dimensions in mm. The calculator returns section modulus Z (mm³), second moment of area I (mm⁴), and optional bending stress if moment is provided.

Section Type

Width, b (mm)

Depth/Height, h (mm)

Outer Width, B (mm)

Outer Depth, H (mm)

Inner Width, b (mm)

Inner Depth, h (mm)

Diameter, d (mm)

Outer Diameter, D (mm)

Inner Diameter, d (mm)

Optional Bending Moment, M (kN·m)

What Is Section Modulus?

Section modulus is a geometric property that tells you how efficiently a cross-section resists bending. In beam design, higher section modulus usually means lower bending stress for the same applied moment. It is one of the most practical “quick-check” values in structural and mechanical engineering.

The elastic bending stress relationship is: σ = M / Z, where σ is bending stress, M is bending moment, and Z is elastic section modulus. If Z increases while M stays the same, bending stress drops.

Why Engineers Use a Section Modulus Calculator

To size beams and members quickly during preliminary design.
To compare candidate shapes (solid vs hollow, wide vs deep, etc.).
To estimate bending stress from a known moment.
To check whether a section change improves strength efficiently.

Formulas Used in This Calculator

1) Solid Rectangle

Second moment of area: I = b h³ / 12
Distance to extreme fiber: c = h / 2
Section modulus: Z = I / c = b h² / 6

2) Hollow Rectangle (RHS)

I = (B H³ - b h³) / 12
c = H / 2
Z = I / c = (B H³ - b h³) / (6H)

Inner dimensions must be smaller than outer dimensions.

3) Solid Circle

I = π d⁴ / 64
c = d / 2
Z = I / c = π d³ / 32

4) Hollow Circle (Tube)

I = π(D⁴ - d⁴) / 64
c = D / 2
Z = I / c = π(D⁴ - d⁴) / (32D)

How to Use This Tool Correctly

Select a section type.
Enter dimensions in millimeters.
Optionally enter bending moment in kN·m.
Click Calculate to get Z, I, and estimated stress.

The stress value is returned in MPa because the calculator converts moment to N·mm and divides by Z in mm³, giving N/mm² (which equals MPa).

Practical Design Notes

Section modulus depends strongly on depth. Increasing depth is often the fastest way to increase bending capacity.
Hollow sections can provide excellent strength-to-weight efficiency.
Always confirm axis direction. This calculator assumes major-axis bending based on entered depth/diameter.
For final design, include code checks: deflection, local buckling, lateral-torsional buckling, fatigue, and safety factors.

Elastic vs Plastic Section Modulus

This page calculates elastic section modulus, which is used in elastic stress analysis and serviceability-oriented calculations. Plastic section modulus is a different property used for plastic moment capacity in ultimate limit state design and depends on full-yield stress distribution.

Example Quick Check

Suppose a rectangular beam has b = 100 mm and h = 200 mm: Z = b h² / 6 = 100 × 200² / 6 = 666,666.7 mm³. If M = 40 kN·m, then stress σ = M / Z = 40,000,000 / 666,666.7 = 60 MPa.

Final Thoughts

A reliable section modulus calculator is a great first-pass engineering tool. Use it to compare options, estimate stress quickly, and build intuition about how geometry affects strength. Then pair those insights with full structural checks for safe, code-compliant design.