Free moment of inertia calculator

Elastic, warping and torsion (J, C_w), DSM, and full plastic analysis for any cross section - standard shapes, composite built-up sections, or arbitrary polygons.

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How to use the free moment of inertia calculator

1
Choose a section type
Pick from standard shape templates (I-section, T-section, C/Z-section, angle, rectangle, circle, hollow section) or select a standard steel section from the built-in library.
2
Define the geometry
Enter dimensions for each shape element. Use the corrosion offset option to automatically reduce section dimensions for weathered or corroded steel.
3
Select analysis depth
Choose from elastic, warping and torsion constants, DSM buckling factors (cold-formed steel), or full plastic analysis depending on what your design check requires.
4
Read the full property set
Results appear instantly: area (A), centroid, I_x/I_y, elastic moduli (S_x/S_y), plastic moduli (Z_x/Z_y), radii of gyration (r_x/r_y), torsion constant (J), warping constant (C_w/I_w), and shear areas.

About this calculator

Four analysis modes - elastic, warping and torsion constants (J, C_w), Direct Strength Method (DSM), and full plastic - covering every cross-section property a structural engineer needs for design. Standard shape templates cover I-sections, T-sections, angles, Cees, Zeds, hollow sections, and more - or select directly from a steel section library. Outputs include area, centroid, I_x/I_y, elastic and plastic section moduli, radii of gyration, torsion constant (J), warping constant (C_w/I_w), and shear areas.

How to use the free moment of inertia calculator

This calculator analyses any arbitrary cross section and returns the full set of geometric properties needed for structural design. Build sections from standard shape templates, combine multiple shapes into composite or built-up sections, or enter a fully custom polygon outline.

Section types and input modes

Standard shape templates cover the most common section types: I-sections, T-sections, C-sections (channels), Z-sections, angles, rectangles, hollow rectangles, circles, and hollow circles. Each template requires only the key dimensions - depth, flange width, thicknesses - and fills in the rest automatically.

For non-standard sections, you can combine multiple shapes using the built-up section mode - for example, adding a cover plate to a standard I-section or combining two angles back-to-back. You can also select standard steel sections directly from the built-in database and use them as a starting point.

Analysis modes

The calculator offers four analysis depths. Elastic analysis returns area, centroid, I_x/I_y, elastic section moduli (S_x/S_y), and radii of gyration. Warping and torsion analysis adds J and C_w/I_w. DSM mode adds Direct Strength Method buckling factors for cold-formed steel sections. Full plastic analysis additionally computes plastic section moduli (Z_x/Z_y) and shear areas.

Using properties in design

Copy section properties directly into the free beam analysis calculator to compute deflection and bending moment for any loading. For a standard section not in the library, use the free section properties library. Sign up for a free Calcs.com trial to link your custom sections directly to steel, timber, and concrete design calculators.

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Frequently asked questions

What is moment of inertia (second moment of area)?

The moment of inertia (also called the second moment of area) measures how a cross section resists bending about a given axis. A higher I value means the section is stiffer in bending. It appears in the beam bending formula M = EI × d²y/dx² and is the fundamental input for deflection and stress calculations.

What cross section shapes does this calculator support?

The calculator supports I-sections, T-sections, C-sections (channels), Z-sections, angles (L-sections), rectangles, hollow rectangles (HSS rectangular), circles, and hollow circles (HSS round). Multiple shapes can be combined into composite or built-up sections. You can also select standard AISC steel sections directly from the built-in library, or enter a fully arbitrary polygon outline.

What is the difference between elastic and plastic section modulus?

The elastic section modulus (S = I/c) assumes the section remains elastic and is used for bending stress checks at first yield. The plastic section modulus (Z) accounts for full yielding across the entire cross section and is used for plastic moment capacity checks. This calculator computes both.

Can I calculate properties for a built-up or composite section?

Yes. You can combine multiple shape elements - for example, adding a cover plate to a standard W-section or combining two angles back-to-back for a double-angle member. The calculator applies the parallel axis theorem automatically and returns properties for the combined section as a single result.

What does a Calcs.com trial add over the free calculator?

The free calculator gives you the full property set instantly. A Calcs.com trial connects your custom section directly to steel, timber, and concrete design calculators - so properties flow into member checks automatically - and lets you save sections, reuse them across projects, and export a signed PDF report.

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