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Queensferry

Module 4 · Lesson 4.1

How a section behaves, from zero load to failure

Four distinct states, and the one the design calculation actually describes.

Why this matters

The design equation for a reinforced-concrete section describes one instant: the moment of failure. It says nothing about the three states the section passes through on the way, and a designer who only knows the equation cannot tell whether a beam will crack, deflect, or fail with any warning. Walk the whole path first, and the equation becomes obvious.

By the end of this lesson you should be able to

  • Name the four states a section passes through
  • Say what governs the transition between each
  • Explain what 'under-reinforced' means and why it is wanted
  • Distinguish ductile from brittle flexural failure

What you should already know

  • Bond, cracking and composite action (Module 1)
  • Strain compatibility (Module 1)
  • Bending stress and the neutral axis (Structural Analysis Fundamentals, Module 9)

Load a reinforced beam slowly and watch four distinct states go past.

1. Uncracked. The whole section acts, tension concrete included. The neutral axis is near mid-depth, the section is stiff, and the steel is barely stressed — it has the same strain as the concrete beside it, and that strain is tiny. Behaviour is elastic and the gross section governs.

2. Cracked, elastic. The extreme tensile fibre reaches the concrete's tensile strength and cracks. Below the neutral axis the concrete stops carrying tension; the steel picks it all up. The neutral axis rises, stiffness drops sharply, but both materials are still elastic and stresses are still proportional to strains. This is the state a beam spends its service life in, and it is what serviceability calculations describe.

3. Steel yields. As load rises the steel strain reaches its yield value — about 0.00217 for the design strength of ordinary reinforcement. The bar force now stops growing. Further load makes the section rotate rather than take more tension, cracks widen visibly, and deflection increases rapidly.

4. Concrete crushes. With the tension force capped, equilibrium forces the concrete compression to stay the same, so the compression zone must shrink and its stress must rise. Eventually the concrete reaches its crushing strain — around 0.0035 — and the section fails.

The design calculation describes state 4 only. That is why it uses design strengths, why it ignores tension concrete, and why it says nothing about deflection or crack width — those belong to state 2 and are checked separately.

Predict first

Two beams are identical except that one has twice the tension steel. Which fails at a higher load, and which fails more safely?

Cross-section of a rectangular reinforced concrete beam 300 mm wide and 550 mm deep, with a link inside the cover, three 25 mm bars near the bottom, a dashed neutral axis about 157 mm from the top, and the compression stress block hatched above it.neutral axis3H25b = 300h = 550d
The section at failure: compression block above the neutral axis, three bars carrying the whole tension below it. Overall depth h, effective depth d to the bar centres — the distinction that most errors in concrete design come down to.

Worked example

Following one section through all four states

Given

  • Rectangular section 300 mm wide × 550 mm deep, C30/37 concrete
  • 3 no. 25 mm bars, area 1473 mm², at an effective depth of 490 mm
  • Reinforcement fyk = 500 N/mm², so fyd = 435 N/mm² and the yield strain is 0.00217

Find

The moment at each transition, to see the size of each stage.

    Practice

    A section has 1473 mm² of tension steel with fyd = 434.8 N/mm². What tension force can the steel deliver once it has yielded, in kN?

    Practice

    For equilibrium the concrete must supply that same 640 kN. With η fcd = 17.0 N/mm² and a width of 300 mm, what depth of stress block is needed, in mm?

    Practice

    With x = 157 mm and d = 490 mm, and an ultimate concrete strain of 0.0035, what is the strain in the tension steel at failure?

    Summary

    • Four states: uncracked, cracked-elastic, steel yielding, concrete crushing
    • The design calculation describes only the last of them
    • Service life is spent in the cracked-elastic state, which serviceability checks describe
    • Under-reinforced means the steel yields first, giving ductile, announced failure
    • Over-reinforced means the concrete crushes first: stronger, but sudden
    • x/d is a ductility control, not a strength control
    Progress is kept in this browser only.

    This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint