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Structural Analysis Fundamentals / module 12

Composite beams

Two materials working together: strain compatibility, the modular ratio and the transformed-section method.

What you will understand

  • Explain why strain is continuous across a bonded interface but stress is not
  • Derive the transformed-section method from compatibility and equilibrium
  • Use the modular ratio to convert one material into an equivalent area of another
  • Find the neutral axis and stresses in a steel-reinforced timber beam
  • Analyse a cracked reinforced-concrete section elastically

Lessons

  1. Strain compatibility and the transformed section

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

  2. Reinforced concrete sections

    Cracked-section analysis: where the neutral axis goes when the concrete cannot pull.

  3. Steel and concrete composite beams

    Making a slab and a beam act as one member, and what it is worth.

Module summary
  • At a bonded interface the two materials must stretch by the same amount, so strain is continuous
  • Stress is not continuous: σ = Eε, so the stiffer material carries proportionally more
  • The modular ratio n = E₁/E₂ converts one material into an equivalent width of the other
  • Transform, find the centroid of the transformed section, then use ordinary bending theory
  • Recover the real stress by multiplying the transformed stress by n for the transformed material
  • In reinforced concrete the concrete below the neutral axis is assumed cracked and carries no tension
Teaching basis and review status

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

Read the lesson register and outstanding review notes

Check your understanding

Check yourself

In a transformed section, which quantity is multiplied by the modular ratio?

Practice

A cracked RC section has Icr = 2.0 × 10⁹ mm⁴, x = 180 mm, d = 420 mm and n = 15. Under a moment of 120 kN·m, what is the steel stress?

Check yourself

In this fully bonded elastic bending model, which longitudinal quantity is continuous across the material interface?