Moment of Inertia of a T-Section
Iₓ, Iₖ, Centroid & Unequal Section Moduli • Tee Beams
Σ Live equations — formulas + values for the current shape
💡 What-if coach — design insights from current values
Moment of Inertia of a T-Section — Formula and Worked Example
A T-section is symmetric about the vertical axis only, so the centroid sits well above mid-height, close to the flange. The consequence is the one that catches people out: the top and bottom fibres are different distances from the neutral axis, so the section has two different section moduli.
Formulas
| Quantity | Expression |
|---|---|
| Area | A = Btₜ + t₷(H − tₜ) |
| Centroid from base | yᶜ = Σ(Aᵢyᵢ) / ΣAᵢ |
| Second moment about x | Iₓ = Σ[Iᶜₓ + Aᵢ(yᵢ − yᶜ)²] |
| Second moment about y | Iₖ = tₜB³/12 + (H − tₜ)t₷³/12 |
| Section modulus, top | Sₓ,ₜₒₚ = Iₓ / (H − yᶜ) |
| Section modulus, bottom | Sₓ,ₛₒₜ = Iₓ / yᶜ |
Because yᶜ ≠ H/2, the bottom fibre is further from the neutral axis than the top, so Sₓ,ₛₒₜ is the smaller value and governs whenever the bottom is in tension. Always check which fibre your load actually stresses — using the larger modulus is unconservative.
Worked Example — A 100 × 120 mm tee with a 12 mm flange and 10 mm web
| Quantity | Value |
|---|---|
| B (flange width) | 100 mm |
| H (overall depth) | 120 mm |
| t₷ (web) | 10 mm |
| tₜ (flange) | 12 mm |
| A | 2,280 mm² |
| yᶜ from base | 85.6 mm |
| Iₓ | 3.110 × 10⁶ mm⁴ |
| Iₖ | 1.009 × 10⁶ mm⁴ |
| Sₓ top fibre | 90.4 × 10³ mm³ |
| Sₓ bottom fibre (governs) | 36.3 × 10³ mm³ |
Values computed by the calculator above. Change any dimension and every property updates live.
Where is the centroid of a T-section?
It lies on the vertical axis of symmetry but well above mid-height, pulled up by the flange. Compute it as yᶜ = Σ(Aᵢyᵢ)/ΣAᵢ using the flange and web as two rectangles. For a 100 × 120 mm tee with a 12 mm flange and 10 mm web the centroid is about 85.6 mm from the base.
Why does a T-section have two section moduli?
Section modulus is S = I/c, and c is the distance to the fibre being checked. Since the centroid is not at mid-height, the top and bottom fibres have different c values and therefore different S. The smaller S corresponds to the fibre furthest from the neutral axis, and that is the one that reaches allowable stress first.
Which section modulus should I use for design?
The one for the fibre in the critical stress state, usually the smaller value. For a simply supported tee with the flange on top, the bottom of the web is in tension and furthest from the neutral axis, so S at the bottom governs.
Other Cross-Sections
The same calculator handles all eight standard sections:
- I-Beam
- Channel
- Angle
- T-Section (this page)
- Rectangle
- Circle
- Hollow Circle
- Hollow Rect
For the full section-property reference and the parallel-axis theorem, see the Moment of Inertia Calculator hub.