Structural Analysis Fundamentals / Composite beams

Strain compatibility and the transformed section

Why the stiffer material takes more load, and the trick that makes the sums easy.

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Try it

Two materials, one section — the transformed-section trick

A timber beam with a steel plate on the bottom. Strain is continuous across the bond, but stress jumps — because the two materials have different moduli. Widen the steel by the modular ratio and it becomes an ordinary all-timber section.

Bottom material

Plate thickness12mm

Modular ratio n: 20.5
Neutral axis above bottom: 92 mm
Mid-depth (for reference): 156 mm
Transformed steel width: 4100 mm (×21)
Stress just below interface (steel): 59.0 N/mm²
Stress just above interface (timber): 2.9 N/mm²
The jump: ×20.5
Peak steel stress: 67.8 N/mm²
Peak timber stress: 7.9 N/mm²

Modular ratio n = 20.5. Strain is continuous across the interface — the two materials are bonded — but stress JUMPS by a factor of 20.5 there, because the stiffer material carries n times the stress at the same strain. The neutral axis sits 92 mm above the bottom, pulled towards the stiffer bottom layer.

Things worth trying

Start with the steel plate. The middle panel is the trick: the steel is redrawn as timber, but n = 20 times wider, so the fictitious timber carries the same force at the same strain. That widened section is all one material — ordinary bending theory now applies.
Look at the stress profile on the right. Strain would be a single straight line (the two materials are bonded, so they stretch together) — but stress is TWO lines with a jump at the interface, because σ = Eε and the moduli differ.
The jump is exactly n. Just below the interface the steel carries 20× the stress of the timber just above it, at the very same height and the very same strain. That is the whole of composite action in one picture.
Find the neutral axis. It is NOT at mid-depth — it sits low, pulled down towards the stiff steel. The stiffer material attracts the neutral axis because it wants to carry more of the section's force.
Thicken the plate. The neutral axis drops further and the timber stresses fall — the steel is taking over the tension. A little steel at the bottom of a timber beam does a lot of work, which is the point of a flitch beam.
Now switch the bottom material to aluminium. n drops to 7, the transformed strip is narrower, the neutral-axis shift is gentler, and the stress jump is smaller. The modular ratio controls everything.
Switch to 'stiff timber' — n is only 3. The two materials are nearly alike, the transformed section barely differs from the real one, and the stress profile is almost a single line. When the moduli match, composite action disappears and you are back to one material.
The deep idea: bonded materials share STRAIN, not stress. Everything a composite beam does follows from that — the stress jump, the neutral-axis shift, and the transformed section that turns two materials into one.

Why this matters

Reinforced concrete, steel–concrete decks and plated timber beams combine materials with different stiffnesses. The transformed-section method relates their elastic bending stresses through compatible longitudinal strain.

By the end of this lesson you should be able to

State what is continuous at a bonded interface and what is not
Define the modular ratio
Derive the transformed-section method

What you should already know

Bending theory σ = My/I and the neutral axis (Module 9)
Hooke's law and Young's modulus (Module 7)
Centroids and the parallel-axis theorem (Module 9)

In the ideal fully bonded beam, the longitudinal strain is continuous across the interface. The longitudinal normal stress is Eε, so it changes with the material modulus. Steel with twenty times the timber modulus carries twenty times the longitudinal stress at the same non-zero strain.

This statement concerns stress parallel to the interface. The normal and shear tractions acting across the interface must still satisfy equilibrium.

Predict first

In the fully bonded elastic beam model, which longitudinal quantity is continuous across the steel–timber interface?

From first principles

The transformed-section method

We want to show: that a composite section can be replaced by an equivalent section of one material, with the second material's width multiplied by the modular ratio n.

We would like to use ordinary bending theory, but that assumes a single value of E across the section. So instead of changing the theory, we change the section. Imagine replacing the steel with an equivalent piece of timber that behaves identically. To behave identically it must carry the same force at the same strain — so it needs to be n times as stiff, where n = E_steel/E_timber. We get that by making it n times wider. Widening it, rather than deepening it, is deliberate: widening does not move the material to a different distance from the neutral axis, so the strain at that level is unchanged. Deepening it would change the geometry and break the equivalence.

Worked example

Steel-plated timber beam

Given

Timber section 200 mm wide × 300 mm deep
Steel plate 200 mm wide × 12 mm thick bonded to the underside
E_timber = 10 000 N/mm², E_steel = 200 000 N/mm²
Applied sagging moment 60 kN·m

Find

The neutral axis position and the maximum stress in each material.

Assumptions

Perfect bond between plate and timber
Both materials elastic
Transform into equivalent timber

Practice

A timber beam (E = 10 250 N/mm²) is bonded to a steel plate (E = 205 000 N/mm²). What is the modular ratio n = E_steel/E_timber?

Practice

With E_steel/E_timber = 20, the longitudinal timber stress at the perfectly bonded interface is 8.0 N/mm². What is the longitudinal steel stress at that level, in N/mm²?

Practice

With E_steel/E_timber = 20, a steel plate 100 mm wide is replaced by equivalent timber at the same depth. What transformed width is required, in mm?

Summary

Longitudinal strain is compatible at the bond; longitudinal stress changes with the modulus
n = E₂/E₁, and the transformed width is n times the real width
Widen rather than deepen, so the distance from the neutral axis is unchanged
Analyse the transformed section with ordinary bending theory
Multiply back by n to get the real stress in the transformed material