Composite Section Calculator

Parallel Axis Theorem · T-Beam · Reinforced Concrete · Steel-Concrete Composite

Composite Section Calculator

Reference axis (e.g. bottom edge = 0)
Verbund ← y₂ (flange centroid))
← y₁ (web centroid)

y_i = distance of sub-section centroid from reference axis (bottom).
Empty rows are ignored.
Part 1
A_i [cm²]
I_i [cm⁴]
y_i [cm]
Part 2
A_i [cm²]
I_i [cm⁴]
y_i [cm]
Part 3
A_i [cm²]
I_i [cm⁴]
y_i [cm]
Part 4
A_i [cm²]
I_i [cm⁴]
y_i [cm]

Formulas & Theory

1. Centroid location (first moment of area):
ȳ = Σ(A_i · y_i) / Σ(A_i)
y_i = distance of sub-section centroid from reference axis
Sum over all parts i = 1…n
2. Steiner distance:
d_i = y_i − ȳ
Distance between the sub-section centroid
and the overall centroid (positive = above centroid)
3. Steiner term (parallel axis theorem):
I_S,i = A_i · d_i²
Contribution due to shift of sub-section axis
to the overall centroidal axis. Always positive!
4. Total second moment of area:
I_total = Σ(I_i + A_i · d_i²)
I_i = own second moment of area of part i
(Rectangle: I = bh³/12 | Circle: I = πd⁴/64)
5. Section moduli:
W_bot = I_total / ȳ (bottom fibre)
W_top = I_total / (H − ȳ) (top fibre)
H = total height of cross-section
6. Modular ratio (composite sections):
n = E₁ / E₂
Convert material 2 into material 1 equivalent:
A₂,eq = A₂ / n  |  I₂,eq = I₂ / n
Steel/Concrete: E_s = 210 000 N/mm², E_c ≈ 30 000 N/mm² → n ≈ 7
Own Second Moments – Standard Shapes
ShapeI [cm⁴]
Rectangle b×hbh³/12
Circle ∅dπd⁴/64
Hollow circle D/dπ(D⁴−d⁴)/64
Triangle b×h (centroid h/3 from base)bh³/36


Composite Sections – Fundamentals & Applications

What is the Parallel Axis Theorem (Steiner's Theorem)?

The parallel axis theorem is the key tool for calculating the second moment of area of composite cross-sections. It quantifies how the second moment of area of a sub-section increases when its own centroidal axis does not coincide with the overall centroidal axis.

Steiner's Theorem:
I_total = Σ (I_i + A_i · d_i²)
d_i = distance between the sub-section centroidal axis and the overall centroidal axis

The Steiner term I_S,i = A_i · d_i² is always positive and can exceed the own second moment of area I_i by orders of magnitude when d_i is large. This is why flanges located far from the centroid – as in I-sections and T-beams – deliver exceptional bending stiffness from minimal material.

Typical Composite Sections
  • T-beams (steel): welded web + flange plate combinations
  • Flanged beams (RC): web + effective compression slab
  • Composite beams: steel section + concrete slab (shear studs)
  • Timber-concrete: CLT deck + concrete topping
  • Sandwich sections: two stiff face sheets + core
Why does a T-beam have a high I?

In a T-beam the flange sits far from the centroid. Its Steiner term A_F · d_F² dwarfs its own I_F. The same principle explains the efficiency of IPE and HEB sections: both flanges are positioned at maximum distance from the centroid, delivering high stiffness with minimum steel weight.

Worked Example: Welded T-Beam

Web 12×300 mm + Flange 200×20 mm
  1. Sub-sections and positions:
    Web: A₁ = 36 cm², y₁ = 15 cm  |  Flange: A₂ = 40 cm², y₂ = 31 cm
  2. Centroid:
    ȳ = (36·15 + 40·31)/(36+40) = 1780/76 = 23.4 cm
  3. Own second moments:
    I₁ = 1.2·30³/12 = 2700 cm⁴  |  I₂ = 20·2³/12 = 13.3 cm⁴
  4. Steiner distances:
    d₁ = 15−23.4 = −8.4 cm  |  d₂ = 31−23.4 = +7.6 cm
  5. Steiner terms:
    36·8.4² = 2541 cm⁴  |  40·7.6² = 2310 cm⁴
  6. Total:
    I_total = (2700+2541)+(13.3+2310) = 7564 cm⁴
  7. Section moduli:
    W_bot = 7564/23.4 = 323 cm³  |  W_top = 7564/8.6 = 880 cm³

Steel-Concrete Composite Beam – Transformed Section

In a composite beam the steel section and concrete slab are connected by shear studs, forcing both to act as one section. Because the materials have different stiffnesses, a transformed section is used: the concrete area is converted to an equivalent steel area by dividing by the modular ratio n = E_s/E_c.

A_c,eq = b_eff · h_c / n  |  I_c,eq = b_eff · h_c³ / (12·n)
Typical result: IPE 270 + 1200 mm wide slab 120 mm → I_total ≈ 3.9 × I_s

Frequently Asked Questions

Steiner terms (A_i · d_i²) quantify the increase in second moment of area due to the offset of a sub-section's centroidal axis from the overall centroid. They grow with the square of the offset distance d_i – even a moderate offset produces a large contribution. In T-beams the Steiner term of the flange often exceeds the own I of all parts combined by a factor of 10–100.

Using the first moment of area: ȳ = Σ(A_i · y_i) / Σ(A_i), where y_i is the distance from any convenient reference axis. The result is independent of the choice of reference. Important: I_total is only minimum for the centroidal axis. Always reference I to the overall centroid, not to the reference axis.

State I (uncracked): the full concrete section participates. I_I is valid for loads below the cracking moment M_cr.
State II (cracked): concrete in tension is cracked and ignored. Only the concrete compression zone (depth x) and reinforcement are effective. I_II << I_I. Deflection calculations use an effective mean value (EC2 Method 2: I_eff).

For typical proportions the ratio I_total / I_s lies between 3 and 5 for full shear connection. An IPE 270 with 1200 mm wide, 120 mm thick slab gives I_total ≈ 22 500 cm⁴ versus I_s = 5 790 cm⁴ – a factor of approximately 3.9. The bending stiffness EI_comp / EI_s is identical (both referenced to steel modulus).

Due to shear lag, the compressive stress in a wide concrete slab is not uniform. Only an effective width b_eff is assumed to act: b_eff = b₀ + Σ b_eff,i (EC4 clause 5.4.1.2). For a simply supported beam: b_eff ≤ L_e/4 + web width, where L_e is the equivalent span. For floor beams b_eff ≈ beam spacing.

Summary

Centroid

ȳ = Σ(A_i·y_i) / Σ(A_i)

Steiner Term

I_S,i = A_i · d_i²
d_i = y_i − ȳ

Total Second Moment

I_total = Σ(I_i + A_i·d_i²)

Practical Applications
  • Welded plate girders: optimise geometry for minimum weight at target stiffness
  • RC flanged beams: transformed section (State I) for cracking moment and deflection
  • Composite floor beams: profiled sheeting + concrete – full and partial interaction per EC4
  • Timber-concrete composites: glulam beam with concrete topping
  • Sandwich sections: masts, facade elements, lightweight structures

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