T Section Properties Calculator
Enter the flange width, depth, stem thickness and flange thickness of a T section and get its area, centroid, second moments, section moduli at both fibres, radii of gyration, torsion constant and shear centre, drawn to scale.
Section dimensions
Enter the dimensions in the selected unit — the drawing and the results update as you type. The drawing uses centroidal coordinates: the bold gridlines are the x and y axes through the centroid C.
Section
- Section with its dimensions
- Centroid C, centroidal axes x and y
- Principal axes 1 and 2, angle θp
- Shear centre S
T sections come from splitting an I beam along the web, from hot-rolled tee bars, and from welded stiffeners with a strip of plate acting as flange. The distinctive feature is a centroid that sits close to the flange, so the extreme fibre at the stem tip is far from the neutral axis and carries a much higher bending stress than the flange — the calculator reports separate section moduli Sx,top and Sx,bot for that reason.
The centroid is found by taking moments of the two rectangles about the stem tip, then Ix follows from the parallel-axis theorem. The section is symmetric about the vertical axis, so Ixy = 0. Its shear centre lies at the junction of the flange and stem mid-lines, above the centroid: a transverse load applied at the centroid would twist a thin T.
How the t section properties are calculated
- Areas
- Af = b·tf, Aw = (h − tf)·tw, A = Af + Aw
- Centroid above the stem tip
- ȳ = [Af·(h − tf/2) + Aw·(h − tf)/2] / A
- Second moment about the centroidal x axis
- Ix = b·tf³/12 + Af·(h − tf/2 − ȳ)² + tw·(h − tf)³/12 + Aw·((h − tf)/2 − ȳ)²
- Second moment about the centroidal y axis
- Iy = [tf·b³ + (h − tf)·tw³] / 12
- Elastic section moduli about x
- Sx,top = Ix / (h − ȳ), Sx,bot = Ix / ȳ
- Torsion constant (thin-walled)
- J ≈ [b·tf³ + (h − tf)·tw³] / 3
Principal moments and axes follow from Ix, Iy and Ixy with I1,2 = (Ix + Iy)/2 ± √[((Ix − Iy)/2)² + Ixy²] and tan 2θp = −2·Ixy/(Ix − Iy); the radii of gyration are r = √(I/A) and the elastic section moduli S = I/c for each extreme fibre. Definitions of every property are in the glossary on the section properties overview.
Assumptions. Flange on top, stem vertical and centred, sharp corners and no fillet. Torsion constant and shear centre use thin-walled theory.
Other sections
Frequently asked questions
Which section modulus do I use for a T beam?
It depends on the sign of the moment. With the flange on top and a sagging moment, the stem tip is in tension and the flange in compression; the stem tip is farther from the neutral axis, so S_x,bot is smaller and its stress higher. For a hogging moment the roles swap. Check both fibres when the material has different tensile and compressive strengths, as cast iron does.
Where is the shear centre of a T section and why does it matter?
At the intersection of the flange and stem mid-lines, i.e. inside the flange, not at the centroid. A transverse load must act through the shear centre to bend the beam without twisting it. The calculator shows it as the point S on the drawing and reports its offset from the centroid.
How do I get the properties of a T cut from an I beam?
Enter the beam’s flange width and flange thickness, and half its depth as h. A T cut from a 200 × 100 × 6 × 10 I section is therefore b = 100, h = 100, t_w = 6, t_f = 10 — this is the “half I” often used for tee stubs and for the local buckling check of the compression zone.
References & further reading
- Second moment of area — Wikipedia — definition, parallel-axis theorem and the sign convention for the product of inertia.
- Shear centre — Wikipedia — the point a transverse load must pass through to bend a beam without twisting it.
- Section modulus — Wikipedia — elastic and plastic section modulus, with a table of formulas by shape.
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