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Section Properties Calculator

Moment of inertia, section modulus, and radius of gyration for standard and custom cross-sections

Structural analysis requires accurate cross-section properties. The moment of inertia (I) determines how a beam resists bending deflection. The section modulus (S) relates bending moment to maximum fiber stress. The radius of gyration (r) governs column buckling capacity. These properties depend entirely on the shape and dimensions of the cross-section.

For standard shapes like W-beams and HSS tubes, properties are tabulated in the AISC Steel Construction Manual. But for custom built-up sections, plate girders, composite sections, or unusual shapes, you need to calculate properties from geometry. This is where errors creep in, especially when the section has multiple components and the parallel axis theorem must be applied.

This calculator handles both standard shapes (rectangle, circle, I-beam, channel, angle, tube, T-section) and custom built-up sections composed of multiple rectangular elements. Enter dimensions and get Ix, Iy, Sx, Sy, rx, ry, Zx (plastic section modulus), centroid location, and cross-sectional area. All calculations follow standard engineering mechanics formulas with the parallel axis theorem for composite sections.

Pro Tip: When designing a built-up plate girder, the web contribution to moment of inertia is large but the web contribution to section modulus is small. Adding flange area is far more effective than adding web depth for increasing bending capacity once the web is sized for shear. A common mistake is to increase web thickness to gain bending strength — it barely moves the needle. Put the steel where it counts: in the flanges, as far from the neutral axis as practical.

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Section Properties Calculator
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How It Works

  1. Select Shape or Build Custom

    Choose a standard cross-section shape and enter dimensions, or switch to custom mode and define rectangular plate elements with width, height, and centroid position.

  2. Enter Dimensions

    For standard shapes, enter the overall dimensions (depth, width, flange thickness, web thickness as applicable). For custom sections, define each plate element's dimensions and position.

  3. Review Section Properties

    The calculator outputs all section properties about both the X and Y centroidal axes. The visual diagram shows the cross-section with the centroid and neutral axis marked.

Features & Capabilities

Standard Shape Library

Rectangle, circle, hollow circle, I-beam, channel, angle, T-section, and HSS tube with all dimensions parameterized.

Custom Built-Up Sections

Define custom sections from multiple rectangular plate elements. The parallel axis theorem is applied automatically to compute composite properties.

Complete Property Output

Calculates area, centroid location, Ix, Iy, Sx, Sy, rx, ry, Zx (plastic section modulus), and J (torsion constant for simple shapes).

Visual Diagram

Cross-section diagram with centroid and neutral axis location clearly marked for verification.

Parallel Axis Breakdown

Shows each component's local moment of inertia and transfer (Ad²) contribution so you can verify the composite calculation step by step.

References

  • Moment of inertia by integration for standard shapes and parallel axis theorem for built-up sections
  • Section modulus S = I / c where c is the distance from the neutral axis to the extreme fiber
  • Plastic section modulus Zx calculated from the equal-area axis for rectangular and I-shaped sections
  • Radius of gyration r = sqrt(I / A) for column buckling analysis per AISC Chapter E

Frequently Asked Questions

The parallel axis theorem (also called the transfer formula) calculates the moment of inertia of a shape about any axis parallel to its centroidal axis: I = I_centroid + A*d^2, where d is the distance between the two parallel axes. This is essential for built-up sections where each plate element has its own centroid offset from the composite section centroid.
The elastic section modulus S = I/c gives the stress at the extreme fiber when the section is in the elastic range. The plastic section modulus Z is the first moment of area about the plastic neutral axis (equal-area axis). Z is always larger than S. AISC LRFD uses Z for compact sections because the full plastic moment Mp = Fy*Z represents the true bending capacity.
AISC tabulated properties account for fillet radii at the web-flange junction, which add material and increase the moment of inertia slightly. A rectangular plate model of the same W-shape ignores the fillets and will underestimate I by 1-3%. For final design, use the AISC tabulated values. This calculator is most useful for custom built-up sections not found in the manual.
Disclaimer: This calculator provides section properties based on idealized geometry. For design purposes, verify results against AISC Steel Construction Manual tabulated values for standard shapes. Built-up section properties should be checked by a licensed structural engineer for use in final design.

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