Reinforced Concrete Design / Prestressed concrete
The idea, and the two stages
One sentence of principle, three terms of algebra, and two load cases that pull in opposite directions.
Lesson review: expert verification remains open
This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.
Corrections checked in this pass
- Corrected the top- and bottom-fibre minimum section moduli and the associated stress-limit derivation.
Checks needed for project use
- A concrete-design reviewer must check the stated first-generation clauses, National Annex choices, detailing rules and factor sets. Numerical agreement alone does not establish code compliance.
- Check actual losses, load history, transfer/end zones, tendon development, ultimate resistance, shear and deflection.
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 notesWhy this matters
Reinforced concrete accepts that the concrete will crack and puts steel across the crack to carry the tension. Prestressing introduces compression to reduce tensile stress and control cracking. Some members are designed to remain uncracked under specified service conditions; others permit controlled tension or cracking. Both serviceability and ultimate resistance must be checked at the relevant stages.
By the end of this lesson you should be able to
- State the principle in one sentence
- Write the three stress terms and identify what each does
- Explain why the prestress is eccentric
- Say why transfer and service are opposite cases
What you should already know
- Bending stress and the section modulus (Structural Analysis Fundamentals, Module 9)
- Combined bending and axial stress (Structural Analysis Fundamentals, Module 9)
- Creep and the effective modulus (Module 6)
The principle
Apply a compression to the concrete before the load arrives, so that the tension the load would have caused never appears.
That is the whole idea. A beam that never goes into tension never cracks, and a beam that never cracks keeps its full uncracked stiffness, has no crack widths to check, and needs no cover for crack control — only for durability and fire.
The consequences are large. Prestressed members span perhaps 50% further than reinforced ones of the same depth, they deflect far less, and they can be made in sections that would be absurd in reinforced concrete because they are working entirely in compression.
What it calculates: The stress at the extreme fibres of a prestressed section.
- P
- Prestress force (N)
- e
- Eccentricity of the tendon below the centroid (mm)
- A
- Cross-sectional area (mm²)
- Zt
- Section modulus to the top fibre (mm³)
- M
- Applied sagging moment (N·mm)
This assumes
- Compression NEGATIVE, tension positive
- The section is uncracked and elastic — which is the point of prestressing it
- Plane sections remain plane
In plain terms: Three terms, and it is worth naming what each does. −P/A is uniform compression, the same at every fibre. Pe/Z is the hogging moment the eccentric prestress applies: it relieves the top and compresses the bottom. M/Z is the applied sagging moment, doing exactly the reverse. Design is choosing P and e so the sum stays inside the limits.
Two stages, and they are opposites
A prestressed member has at least two load cases that both matter, and the trap is that they are the reverse of each other.
At transfer, when the prestress is released into the concrete:
- The prestress is usually high, but immediate losses appropriate to the stressing and transfer sequence must already be included.
- The load is at its smallest — usually self-weight alone, since the member is still in the casting yard.
- So the member hogs, and the bottom fibre is most compressed while the top may go into tension.
In service, years later:
- The prestress has fallen by 20 to 25%.
- The load is full.
- So the member sags, and the top fibre is most compressed while the bottom may go into tension.
The section must satisfy both. Designing at the initial force alone is unconservative for service; designing at the final force alone is unconservative for transfer.
And the concrete is younger and weaker at transfer, so the limits are tighter exactly when the prestress is largest.
Try it
Two stages, one section
Compare signed fibre stresses at one section under the two stated load cases. The effect of changing prestress depends on eccentricity and the fibre considered; end zones and tendon development are separate checks.
- Section modulus Z
- 37.3 × 10⁶ mm³
- Transfer top
- 2.28 N/mm²
- Transfer bottom
- -18.00 N/mm²
- Transfer limits
- -19.2 to +3.02
- Service top
- -11.04 N/mm²
- Service bottom
- -1.21 N/mm²
- Service limits
- -20.3 to +0.00
- Governing
- transfer bottom / service top
The displayed section stresses meet the adopted limits. End-region, tendon and ultimate checks remain separate. Both fibres are inside the transfer limits, with transfer bottom closest. Both fibres are inside the service limits, with service top closest.
Things worth trying
- Raise P₀ and inspect both fibres at both stages. The direction of each stress change depends on eccentricity relative to the kern.
- Watch the service TOP fibre as you raise P₀. Beyond the kern distance Z/A — 133 mm for this section — the Pe/Z relief outweighs the P/A compression, so more prestress relieves the top rather than compressing it. Set e below 133 mm and the direction reverses.
- Set e to zero. The prestress becomes uniform compression only, and the section needs far more force for the same benefit. Eccentricity is what makes prestress efficient.
- Push e to its maximum. Transfer usually fails first, at the bottom fibre, because that is where full prestress and least load coincide.
- Lower α towards 0.6. That is a member with severe losses, and it fails in service while transfer is untouched — losses only affect the later stage.
- Set both moments equal. The two diagrams differ only by α, which isolates what the losses alone are doing.
Worked example
Checking a prestressed beam at both stages
Given
- Beam 350 mm wide × 800 mm deep, spanning 14.0 m
- C45/55 concrete, with fck(t) = 32 N/mm² at transfer
- Initial prestress P₀ = 2200 kN at an eccentricity of 250 mm
- Losses assumed at 22%, so α = 0.78
- Imposed load 18 kN/m in service; self-weight only at transfer
Find
Check the elastic fibre stresses and necessary minimum section moduli at transfer and service. This is not a complete prestressed-member design.
Assumptions
- Tension is positive and compression negative; uncracked elastic section at the two stated stages
- For this illustration adopt transfer limits −0.6fck(t) to +fctm(t), and a service range −0.45fck to zero; these chosen limits do not define an EC2 “Class 1” route
- Use constant force at each stage and a parabolic tendon with zero eccentricity at both supports and 250 mm at midspan; all transverse loads are uniform
- Tendon development, transfer-zone stresses, reinforcement for transfer tension, shear, ultimate resistance and long-term deflection need separate checks
Predict first
A prestressed beam is found to be overstressed in compression at the bottom fibre at transfer. Which change helps?
Practice
A 350 × 800 mm rectangular section carries a prestress of 2200 kN at zero eccentricity. What is the stress at both fibres, in N/mm²? Give the magnitude of the compression.
Practice
For that section (Z = 37.33 × 10⁶ mm³), what stress does an eccentricity of 250 mm add at the bottom fibre, in N/mm²? Give the magnitude.
Check yourself
Why is the transfer stage often more critical than the service stage?
Summary
- Compress the concrete before the load, so the tension never appears
- Three terms: −P/A, ∓Pe/Z, ±M/Z
- Eccentricity is what makes prestress efficient, not proximity to the tension face
- Transfer: maximum P, minimum load, youngest concrete — the bottom fibre governs
- Service: reduced P, full load — the top fibre governs
- Both stages must be satisfied, and transfer is often the tighter
- Draping and debonding exist to relieve the transfer condition near the ends