Section Properties Calculator: Moment of Inertia and Section Modulus

How to use this calculator

There are two ways in. For a standard shape, pick it from the list and fill the dimensions: b is the overall width or the flange width, h is the overall depth, the flange or wall thickness goes in the third box, and the web thickness in the fourth. Press build and the section is drawn to scale with its centroidal axes and every property computed. For anything the presets do not cover, build the section from parts: add rectangles and circles at their centre coordinates, tick the hole box for openings, and the tool assembles the composite for you. Bolt holes, notched beams, castellated webs, timber built up sections and odd precast shapes all fall out of the parts builder.

Everything is in consistent millimetre units, so areas come out in square millimetres, second moments in millimetres to the fourth, and section moduli in cubic millimetres. Work in metres or inches if you prefer, and the outputs simply follow the same powers of your chosen unit.

What each property is for

The area carries axial load and resists shear. The centroid is where the neutral axis sits in pure bending, and locating it correctly is the whole game for unsymmetrical shapes: a tee section has far more material near the top, so its centroid rides high and the bottom fibre is much further from it than the top fibre.

The second moment of area, Ix, is the stiffness term. It appears in every deflection formula, and because depth enters as a cube, it rewards putting material far from the neutral axis. Doubling the depth of a rectangle multiplies Ix by eight. That single fact is why beams are deep and thin rather than square, and why an I section beats a solid rectangle of the same weight.

The section modulus is what you use for stress: bending stress equals moment divided by section modulus. The tool reports Sx for the top and bottom fibres separately, and for any unsymmetrical section those two numbers differ. Always check the smaller one, because that is the fibre that reaches yield first.

The radius of gyration is the buckling term. Slenderness is the effective length divided by r, and columns buckle about the axis with the smaller radius, so it is ry that usually governs and not rx.

The plastic modulus and shape factor

Alongside the elastic properties the tool finds the plastic neutral axis, the line that divides the section into two equal areas, and computes the plastic modulus Zx about it. Where the elastic modulus tells you when the outer fibre first yields, the plastic modulus tells you the moment at which the whole section has yielded and a plastic hinge forms. Their ratio is the shape factor: roughly 1.5 for a rectangle, near 1.7 for a solid circle, and typically 1.12 to 1.18 for a rolled I section. A high shape factor means a lot of reserve strength between first yield and collapse, which is exactly why plastic design is worth doing for rectangular and solid shapes and much less rewarding for I sections.

The mistake to avoid

The parallel axis theorem is where most hand calculations go wrong. The second moment of a part about the section centroid is its own second moment plus its area times the square of the distance from the part centroid to the section centroid. People remember the first term and forget the second, which for a flange sitting far from the neutral axis is by far the larger of the two. In the worked I section, each flange contributes about 133 thousand from its own shape and 78 million from the transfer term. The tool does this for every part automatically, which is the main reason a composite builder beats a formula sheet.

A worked check you can repeat

Load the default I section: flanges 200 wide by 20 thick, a web 10 thick, overall depth 300. Area comes out at 10,600 square millimetres, the centroid sits at 150 by symmetry, Ix is 171.7 million and Iy is 26.7 million, giving rx of 127.3 and ry of 50.2 millimetres. The elastic section modulus is 1.145 million and the plastic modulus 1.289 million, a shape factor of 1.126. Every one of those numbers can be checked by hand in a few minutes, which is the point: use the tool for speed, not for faith.

Related Calculators and Reference

• Beam Deflection Calculator: feed Ix straight into deflection and moment.
• Euler Buckling Calculator: use the smaller radius of gyration for slenderness.
• Rebar Weight and Spacing Calculator: for the reinforcement side of a concrete section.
• S355 Structural Steel: the strength that turns section modulus into moment capacity.
• Steel Weight Calculator: the weight of plate, bar, pipe and tube in steel and other metals.
• AISC: published section property tables for rolled shapes.