A reinforced-concrete section may fail after giving warning —through cracking and large deformation— or it may fail suddenly. The difference is not controlled only by the amount of steel. It depends on which material reaches its limiting strain first.
Does more reinforcement always mean a safer failure?
Imagine two beams with the same geometry and concrete. One has a moderate steel ratio; the other has much more reinforcement.
Build the mental model first
When a beam bends, one part of its section shortens and another elongates. Between them lies a fibre with zero strain: the neutral axis.
The section rotates
Plane sections remain plane, so strain varies linearly through the depth.
Concrete cracks
Once its low tensile strength is exceeded, concrete cracks and reinforcement carries the tension.
The position of x matters
Moving the neutral axis changes the concrete and steel strains.
The pivot diagram, step by step
A section reaches its ultimate limit state when one fibre reaches a limiting strain. The possible strain lines form families because they rotate around three fixed points: A, B and C. Select a domain and observe the section and strain line together.
Move the neutral axis
Pure or combined tension
The whole section is in tension. Concrete cracks and its tensile contribution is neglected.
- Concrete
- Cracked in tension
- Steel
- The most strained reinforcement reaches 10‰
- Physical reading
- Steel elongation governs
What each domain means
The whole section is in tension
The neutral axis lies outside the section. Concrete is cracked and ultimate response is associated with the strain of the most highly tensioned reinforcement.
Pivot A · −∞ < x < 0Highly ductile bending
A small compression zone exists, but concrete does not reach 3.5‰. Tension reinforcement reaches 10‰, so describing this whole domain as “elastic steel” is incorrect.
Pivot A · 0 < x < 0.259dSteel yields first
Concrete reaches 3.5‰ after the tension reinforcement has exceeded its design yield strain. The section can develop appreciable deformation before collapse.
Pivot B · 0.259d < x < xlimConcrete crushes first
Concrete reaches 3.5‰ while the tension steel remains below yield. Warning and redistribution capacity decrease: this is a brittle response.
Pivot B · xlim < x < dAll reinforcement is compressed
A small part of the concrete remains in tension, although every layer of reinforcement lies inside the compression zone.
Pivot B · d < x < hThe whole section is compressed
Concrete and steel are in compression. Lines rotate around C, located at 3h/7 and associated with the 2‰ pure-compression limit.
Pivot C · h < x < +∞From the diagram to the laboratory
A photograph can reveal cracks, deformation or crushing, but it cannot identify a domain on its own. We also need measured strains or at least the geometry, reinforcement, materials and applied actions. These images help connect the abstract diagram to physical mechanisms.
Worked example: where Domain 3 ends
Consider a rectangular section with d = 450 mm, B500S steel, Es = 200,000 MPa and γs = 1.15. At the boundary between Domains 3 and 4, concrete reaches 3.5‰ exactly when the reinforcement reaches its design yield strain.
fyd = 500 / 1.15 = 434.8 MPa
εyd = fyd / Es = 2.17‰
xlim / d = 3.5 / (3.5 + 2.17) = 0.617
xlim = 0.617 · 450 = 278 mm
This is why 0.63d must not be memorised as a universal constant. The limit follows from the concrete and steel strains used in the calculation.
Answer before looking back
1. Why is it called a pivot diagram?
Because the families of strain lines rotate around fixed points associated with limiting strains: A for tension reinforcement, B for the extreme concrete fibre in combined bending and compression, and C for full-section compression.
2. What separates Domains 3 and 4?
In Domain 3, tension steel has yielded when concrete reaches its ultimate strain. In Domain 4, concrete crushes before the steel yields.
3. Does adding steel always improve safety?
No. Excess reinforcement can deepen the neutral axis and produce a failure governed by concrete crushing, with lower ductility and less warning.
References and scope
- García Meseguer, A.; Morán Cabré, F.; Arroyo Portero, J. C. Jiménez Montoya: Hormigón armado, 15th edition.
- Spanish Structural Concrete Instruction EHE-08, clauses 42.1.2 and 42.1.3.
- Ruiz Martín, H. ¿Cómo aprendemos? Una aproximación científica al aprendizaje y la enseñanza. Graó.
- Sharaky, I. A. et al. “Flexural Response and Failure Analysis of Solid and Hollow Core Concrete Beams…”, Materials 14 (2021), CC BY 4.0.