Inputs
Each part supplies its area A, its position y on a common reference axis, its own second moment of area I about its own centroidal axis, and its elastic modulus E; a reference elastic modulus is also chosen.
Statics & strength · Composite sections
Transformed composite section
Use the classical transformed-section method to calculate the modulus-weighted neutral axis, the transformed second moment of area and the flexural rigidity EI of a composite section built from several parts with different elastic moduli.
This covers a rigidly bonded composite section in the linear-elastic range, with parts positioned along a common reference axis. Interface slip and shear-connector behaviour are not included.
EIn=E/E_ref
02
Transformed section
Enter the section parts and calculate.
Inputs and method
Transformed composite section
Combine multiple parts of different elastic moduli into a modulus-weighted neutral axis, a transformed second moment of area and a flexural rigidity EI.
Calculation
Each part is scaled by the transform factor n=E/E_ref. The modulus-weighted neutral axis follows from ȳ=Σ(n·A·y)/Σ(n·A), the transformed second moment of area from I_tr=Σ(n·I_own+n·A·(y−ȳ)²), and the flexural rigidity from EI=E_ref·I_tr.
Example
Two parts of A=1,000 mm² each with negligible own second moment, at y=0 mm (E=100 MPa) and y=10 mm (E=200 MPa), reference E_ref=100 MPa: n=1 and 2, giving ȳ=6.667 mm, I_tr≈66,667 mm⁴ and EI≈6.667·10⁶ N·mm².
Sources and limits: Classical transformed-section method for composite beams (e.g. Hibbeler, Mechanics of Materials); no normative dependency, parts are assumed rigidly bonded.
Transformed composite section in detail
Combine multiple parts of different elastic moduli into a modulus-weighted neutral axis, a transformed second moment of area and a flexural rigidity EI.
What is a composite section and the transformed-section method?
A composite section consists of several parts of different materials, rigidly bonded together. The transformed-section method conceptually replaces this section with a homogeneous section made of the reference material, by scaling each part by the ratio of its elastic modulus to the reference modulus.
Formula and variables
ȳ = Σ(nᵢ·Aᵢ·yᵢ) / Σ(nᵢ·Aᵢ)
nᵢ = Eᵢ/E_refI_tr = Σ(nᵢ·I_own,ᵢ + nᵢ·Aᵢ·(yᵢ−ȳ)²)EI = E_ref·I_tr
| Symbol / input | Meaning |
|---|---|
| Aᵢ, yᵢ | Area and position (on the common reference axis) of part i. |
| I_own,ᵢ | Second moment of area of part i about its own centroidal axis, parallel to the composite section's neutral axis. |
| Eᵢ, E_ref | Elastic modulus of part i and the chosen reference elastic modulus. |
| nᵢ | Transform factor of part i, nᵢ=Eᵢ/E_ref. |
| ȳ, I_tr, EI | Modulus-weighted neutral axis, transformed second moment of area and flexural rigidity of the overall section. |
Choose the inputs correctly
Each part supplies its area A, its position y on a common reference axis, its own second moment of area I about its own centroidal axis, and its elastic modulus E; a reference elastic modulus is also chosen.
How to use the calculator
Define, for each part, its area A, its position y on a common reference axis, its own second moment of area I (about its own centroidal axis, not the eventual neutral axis) and its elastic modulus E. Then choose a reference elastic modulus -- usually the modulus of one of the materials involved.
Worked example
Two parts of A=1,000 mm² each with negligible own second moment, at y=0 mm (E=100 MPa) and y=10 mm (E=200 MPa), reference E_ref=100 MPa: n=1 and 2, giving ȳ=6.667 mm, I_tr≈66,667 mm⁴ and EI≈6.667·10⁶ N·mm².
How should the transformed neutral axis be interpreted?
Under the transformed-section method, the neutral axis ȳ no longer sits at the geometric centroid but at the modulus-weighted centroid -- a stiffer material 'pulls' the neutral axis toward itself. The flexural rigidity EI is the quantity that actually matters for the composite section's bending deflection, regardless of which material was chosen as the reference.
Area in mm², position in mm, own second moment of area in mm⁴, elastic modulus in MPa; results in mm, mm⁴ and N·mm².
Typical applications
Analyzing composite cross-sections made of several materials with different elastic moduli, e.g. steel-reinforced concrete, sandwich sections, or bonded/bolted multi-layer beams, where the overall flexural rigidity EI is needed.
Assumptions, limits and common mistakes
Classical transformed-section method for composite beams (e.g. Hibbeler, Mechanics of Materials); no normative dependency, parts are assumed rigidly bonded.
Common mistake: For the 'own second moment of area', don't accidentally enter the second moment about the eventual common neutral axis -- that is added by the calculator itself via the parallel-axis term. Choosing the reference material only changes which material EI is expressed in, not the physical result; but a wrong reference modulus leads to misinterpreted transformed areas.
Frequently asked questions
What does the transform factor n mean?
n=E/E_ref indicates how a part 'behaves' relative to the reference material: a part with n=2 contributes to flexural rigidity as much as twice its area of the reference material would.
Why is a reference material needed?
Because second moments of area from different materials cannot simply be added -- only after scaling by n is the section conceptually homogeneous, so the usual beam-theory formulas apply.
Does the result change if I choose a different reference material?
The neutral axis and the flexural rigidity EI stay physically the same; only the transformed second moment of area I_tr by itself changes numerically, since it is expressed relative to the chosen reference modulus. EI=E_ref·I_tr is the actual meaningful, reference-independent quantity.
What is the parallel-axis term in the formula?
n·A·(y−ȳ)² is the parallel-axis (Steiner) term: it accounts for a part sitting away from the common neutral axis. It is added on top of the part's own transformed second moment n·I_own.
What does this model assume?
A rigid bond with no slip between the parts (no relative displacement at the interfaces), and linear-elastic behaviour of all materials.