Composite Beam Transformed Section Calculator
This calculator finds the transformed moment of inertia (I_tr) for a steel-concrete composite beam under full composite action. The concrete slab is transformed to an equivalent steel area by dividing the effective slab width by the modular ratio n = Es/Ec. The centroid of the composite transformed section is found by statics, then the parallel-axis theorem gives the total moment of inertia about that centroid. The transformed I is used for elastic deflection calculations and stress checks at service load levels. Full composite action assumes all required shear connectors (studs) are provided between the steel and concrete.
Transformed section method
Ac_tr = (beff × tc) / n [transformed concrete area]
y_conc = ds + tc/2 [centroid of slab from bottom of steel]
y_steel = ds/2 [centroid of steel from its base]
y_bar = (As × y_steel + Ac_tr × y_conc) / (As + Ac_tr)
I_tr = Is + As(y_bar - y_steel)² + (beff×tc³/12)/n + Ac_tr(y_conc - y_bar)²
This gives the total elastic transformed moment of inertia for full composite action, suitable for deflection and elastic stress calculations.
Composite beam design notes
- Full composite action requires sufficient shear studs: AISC 360 Section I8.2 specifies minimum and maximum stud spacing.
- For partial composite action (50-100% of full shear), use the lower bound I formula from AISC 360 Commentary Section I3.2.
- Short-term deflection (under construction, before composite): use the steel section I alone.
- Long-term deflection (creep under sustained loads): use a modified n (typically 2n or 3n) to account for concrete creep.
Frequently asked questions
What is a composite beam?
A composite beam consists of a steel beam connected to a concrete slab above it by shear studs. When connected, the concrete and steel act together, significantly increasing the effective moment of inertia and flexural stiffness compared to the steel beam alone.
What is the transformed section method?
The transformed section method converts the composite section into an equivalent all-steel section by dividing the effective concrete width by the modular ratio n = Es/Ec. The resulting transformed section has a single elastic modulus (steel) and can be analysed with standard section property formulas.
What is the effective width of the concrete slab?
Per AISC 360 Section I3.1a, the effective slab width on each side of the beam centreline is the lesser of: one-eighth of the beam span, half the distance to the adjacent beam, or the distance to the slab edge. Add both sides for the total effective width beff.
What modular ratio n should I use?
n = Es / Ec, where Es = 29,000 ksi and Ec = 33 * wc^1.5 * sqrt(fc') (psi). For normal-weight concrete (wc = 145 pcf) with fc' = 4,000 psi, Ec approximately 3,600 ksi and n = 29,000/3,600 approximately 8. AISC typically uses n = 8 or 9 for normal-weight concrete.
Is the lower bound moment of inertia the same as the transformed I?
No. AISC 360 Commentary to Section I3.2 defines a lower bound moment of inertia Ilb that accounts for partial composite action and is used for deflection calculations. The fully transformed I (used in this calculator) applies only to full composite action.
Official sources
- American Institute of Steel Construction: AISC 360-22 Chapter I Composite Members.
- American Concrete Institute: ACI 318-19 Chapter 26 Construction Documents and Inspection.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.