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Section Properties Calculator: Area, Moments of Inertia, Torsion Constant
A, Iy, Iz, J and section moduli of rectangles, round bars, tubes, box and I-sections.
Try an example: I-section like an IPE 200 (100 × 200, tf 8.5, tw 5.6) · Timber joist 100 × 200 mm, C24 · Steel tube 88.9 × 4 mm · Aluminium box section 100 × 50 × 4 mm · Solid round bar 30 mm, stainless
How it works
Enter the shape and dimensions of a beam cross-section — solid rectangle, solid circle, circular tube, rectangular hollow section or doubly symmetric I-section — and get the area, the second moments of area about both axes, the St. Venant torsion constant, the elastic section moduli, the torsion modulus, the radii of gyration and, with a typical steel, stainless, aluminium or timber material, the mass per metre and the bending and torsional stiffness. A drawing shows the section with its local axes, and a ready line feeds the 3-D frame model of Proofline’s truss and frame solver. Part of the Proofline math notebook.
- Local axes as in Eurocode 3: y across the width b, z along the depth h, so Iy is the strong axis of an upright rectangle or I-section. A, Iy and Iz from the exact formulas of each shape (I-sections and box sections without root or corner radii).
- Torsion: circles and tubes J = πd⁴/32 (difference for the tube), τ = T·r/J; solid rectangles by Roark’s formulas, J = a·c³·[1/3 − 0.21·(c/a)·(1 − c⁴/(12a⁴))] and τmax = 3T/(a·c²)·[1 + 0.6095r + 0.8865r² − 1.8023r³ + 0.9100r⁴]; box sections by Bredt’s thin-wall formula J = 4Am²t/p; I-sections as thin open rectangles J = (2·b·tf³ + (h − 2tf)·tw³)/3, warping ignored.
- Section moduli W = I/(distance to the extreme fibre), torsion modulus Wt = T/τmax, radii of gyration i = √(I/A); with a material the mass per metre ρ·A and the stiffnesses E·I and G·J.
- Verified against closed forms (b·h³/12, πd⁴/64), the square-bar torsion constant 0.1406·a⁴ and τmax = T/(0.208·a³), Bredt’s formula for a thin square tube and steel-table values of IPE 200 (Iz 142 cm⁴; A and Iy within 6 % because of the fillets).
Nominal geometry without fillets, corner radii or tolerances; open sections ignore warping torsion. Material values are typical, not those of a certified grade. Use the manufacturer’s section tables for design.
Sources
- W. C. Young, R. G. Budynas, A. M. Sadegh: Roark’s Formulas for Stress and Strain, 8th ed. (McGraw-Hill, 2012), table 10.7 (torsion of solid and thin-walled sections) (McGraw-Hill)
- EN 1993-1-1 (steel, E = 210 GPa, G = 81 GPa), EN 1999-1-1 (aluminium), EN 338 (timber strength class C24): typical material values (CEN)
- W. McGuire, R. H. Gallagher, R. D. Ziemian: Matrix Structural Analysis, 2nd ed. (Wiley, 2000), chapters 2–4 and 7 (Wiley / Cornell eCommons)
Part of Proofline
-
Proofline Math notebook
Solve, graph, test and visualise in one notebook — with data, findings and a report.
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- Plate Stress Analysis with Holes (2-D FEA) – Stress map of a plate with holes under tension: von Mises, peaks and the concentration factor.
Calculate area, moment of inertia and torsion constant of a beam section
Get A, Iy, Iz, J, section moduli and mass per metre of rectangles, round bars, tubes, box and I-sections, ready for a 3-D frame model.
What it is for
Before you can check a beam, a shelf bracket or a frame you need the numbers of its cross-section: the area for axial stress, the second moments of area for bending stiffness, the torsion constant for twisting and the section moduli for bending stress. Steel tables list them for standard profiles, but not for the box tube or timber batten you actually have.
This calculator covers five shapes: a solid rectangle, a solid round bar, a circular tube, a rectangular hollow section and a doubly symmetric I-section. Pick a typical material (S235 or S355 steel, stainless, aluminium 6061/6082 or C24 timber) and it adds the mass per metre and the stiffnesses EI and GJ.
The local axes follow Eurocode practice: y runs across the width b, z along the depth h, so Iy is the strong axis of an upright rectangle or I-section.
Worked example
An I-section of 100 × 200 mm with 8.5 mm flanges and a 5.6 mm web — the outline of an IPE 200 — gives A = 27.25 cm², Iy = 1,845.6 cm⁴ and Iz = 141.93 cm⁴, J = 5.165 cm⁴, Wy = 184.6 cm³ and Wz = 28.39 cm³. In steel it weighs 21.39 kg/m and has a bending stiffness EIy of 3,876 kN·m². The steel table gives A = 28.5 cm² and Iy = 1,943 cm⁴ for a rolled IPE 200: the difference is the root fillets this calculator leaves out. Iz matches the table's 142 cm⁴.
A 100 × 200 mm C24 timber joist gives Iy = 6,666.7 cm⁴, Wy = 666.7 cm³ and J = 4,577.6 cm⁴ at 8.4 kg/m. Notice how much larger J is for a solid section than for the open I-section, even though the I-section is stiffer in bending per kilogram.
How to read the result
A governs axial stress N/A. Iy and Iz set the bending stiffness about each axis; deflections scale with 1/I. Wy and Wz turn a bending moment into the extreme-fibre stress M/W. J is the torsional stiffness constant, and Wt turns a torque into the largest shear stress, τ = T/Wt. The radius of gyration i = √(I/A) enters the slenderness KL/i of a column.
The drawing shows the section to scale with its centroid and axes. The code block holds a section … shape=… line you can paste into the 3-D frame model of the truss and frame solver.
Tips and common mistakes
- Enter dimensions in millimetres; results come in cm² and cm⁴, the units of steel tables.
- For open sections like I-beams, J is small and warping torsion, which this calculator does not include, carries much of a torque. Avoid twisting open sections.
- Thin walls buckle locally before they yield. The calculator warns when a tube wall is thinner than 1/50 of the section.
- The material values are typical, not certified grades. Timber uses its characteristic bending strength as the reference stress.
- A box section is far better in torsion than an I-section of the same weight.
FAQ
Why do my values differ from the steel table?
Rolled sections have root fillets and tubes have rounded corners that add a few per cent of area and stiffness. Use the manufacturer's table for design.
How accurate is J for a rectangle?
It uses Roark's formula, accurate to about one per cent for any aspect ratio; a square bar gives 0.1406·a⁴.
Which axis is y?
The horizontal axis across the width b. Bending about y means the beam deflects up or down along z.
Can I enter my own values instead?
Yes: in the 3-D frame model write section S1 A=… Iy=… Iz=… J=… Wy=… Wz=… with values from any table.
Guide last updated:
For scripts and AI agents
- JSON API:
POST https://prooflinesectionproperties.usefulweb.app/api/v1/calculate(same calculation as this page) - Interface description: OpenAPI 3.1 · AI guide: llms.txt
- MCP server (Streamable HTTP):
https://prooflinesectionproperties.usefulweb.app/mcp· tool nameproofline_section_properties
Example request
curl -s -X POST https://prooflinesectionproperties.usefulweb.app/api/v1/calculate \
-H 'content-type: application/json' \
-d '{"shape":"i","b":100,"h":200,"tf":8.5,"tw":5.6,"material":"steel","name":"IPE200"}'