Chapter 11 of 11

Pre-stressed Concrete

In the TGPSC Manager (Civil) syllabus under RCC Design · 2 parts

Prestressed Concrete — Materials, Systems & Losses · Prestressed Members in Flexure

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📑 Contents (17 sections)

Part 1 of 2

Prestressed Concrete — Materials, Systems & Losses

Last reviewed 16 Sept 2026 · 6 min read

Principle

Prestressing applies a permanent compressive force to concrete before service loads act, so that the tension later produced by loads is cancelled or reduced. Concrete, strong in compression and weak in tension, then remains largely uncracked.

Three ways of looking at prestress:

Stress concept — prestress and loads are combined elastic stresses ( $P / A \pm P e / Z \pm M / Z$ ).
Strength (internal couple) concept — like RCC, steel tension and concrete compression form a couple, but the steel is high-strength and pre-tensioned.
Load balancing concept — a curved or draped tendon exerts upward forces that balance part of the gravity load (for a parabolic tendon of sag $e$ : $w_{b} = 8 P e / L^{2}$ ).

Advantages over RCC

Section remains uncracked (or cracks are controlled) → better durability, watertightness.
Smaller, lighter members for the same span → longer spans (bridges, roofs, sleepers, poles).
High-strength steel and concrete used efficiently.
Lower deflections (camber from prestress offsets load deflection).
Better shear resistance (inclined tendons and compression).
Higher fatigue resistance.

Disadvantages: needs high-quality materials, specialised equipment and skilled labour; higher unit cost for short spans; losses must be accounted for.

Pre-tensioning and post-tensioning

Aspect	Pre-tensioning	Post-tensioning
Sequence	Tendons stretched between abutments → concrete cast → released after hardening	Concrete cast with ducts → tendons threaded and stressed after hardening → anchored
Transfer of force	Bond along the transmission length at the ends	End anchorages (bearing)
Typical use	Factory-made precast units: sleepers, poles, piles, hollow-core slabs	Cast-in-situ and large members: bridge girders, long-span beams, slabs, containment structures
Tendon profile	Usually straight (or deflected)	Curved profiles easy
Losses	Larger (elastic shortening full)	Smaller elastic shortening; friction and anchorage slip additional
Grouting	Not needed	Ducts grouted for bond and corrosion protection (bonded), or unbonded greased tendons

Systems

Pre-tensioning: Hoyer's long-line system — several units cast in a line between two abutments with continuous wires.
Post-tensioning:
- Freyssinet — conical wedge anchorage for wire groups; flat jacks.
- Magnel–Blaton — wires anchored in pairs by flat metal wedges in sandwich plates.
- Gifford–Udall — each wire anchored by split conical wedges in a barrel.
- Lee–McCall — high-tensile bars with threaded ends and nuts.
- Modern multi-strand systems (VSL, CCL, etc.) use wedge anchorages for 7-wire strands.

Materials

Code ProvisionIS 1343 — materials (key points)

Concrete: minimum grade M40 for pre-tensioned and M30 for post-tensioned members (higher grades are common).
Steel: high-tensile wires, 7-wire strands and bars (ultimate tensile strength roughly 1400–1900 N/mm²), low-relaxation strands preferred.
Maximum initial prestress in the tendon is limited to a fraction of the ultimate tensile strength (about 80% during stressing).

Why high-strength materials: total losses are typically 150–300 N/mm². With mild steel stressed to, say, 150 N/mm², all the prestress would be lost; high-tensile steel stressed to 1000–1400 N/mm² retains most of it. High-strength concrete has lower creep and shrinkage, higher bearing strength at anchorages and allows smaller sections.

Losses of prestress

The initial force applied by the jack reduces with time. Effective prestress = initial prestress − losses.

Loss	Pre-tensioned	Post-tensioned
Elastic shortening of concrete	Yes (full)	Only if tendons stressed one after another (average ≈ half)
Shrinkage of concrete	Yes	Yes (smaller — part has occurred before stressing)
Creep of concrete	Yes	Yes
Relaxation of steel	Yes	Yes
Friction (curvature and wobble)	—	Yes
Anchorage slip	— (small at abutments)	Yes

FormulaLoss calculations

Elastic shortening (pre-tensioned): $Δ f = m f_{c}$ , where $m = E_{s} / E_{c}$ and $f_{c}$ is the concrete stress at the tendon level due to prestress (and self-weight): $f_{c} = \frac{P}{A} + \frac{P e ^{2}}{I}$ (– self-weight moment term). Post-tensioned with $n$ tendons stressed in sequence: average loss ≈ half of $m f_{c}$ ; zero if all stressed together.

Shrinkage (IS 1343): strain $ε_{s h}$ = $3 \times 1 0^{- 4}$ for pre-tensioned; $\frac{2 \times 1 0 ^{- 4}}{lo g _{10} ( t + 2 )}$ for post-tensioned ( $t$ = age of concrete at transfer in days). Loss $= ε_{s h} E_{s}$ .

Creep: loss $= ϕ \frac{f _{c}}{E _{c}} E_{s} = ϕ m f_{c}$ ; creep coefficient $ϕ$ = 2.2 (7 days), 1.6 (28 days), 1.1 (1 year) at the age of loading.

Relaxation of steel (IS 1343, 1000-hour values in the absence of test data): initial stress 0.5 $f_{p}$ → 0; 0.6 $f_{p}$ → 35 N/mm²; 0.7 $f_{p}$ → 70 N/mm²; 0.8 $f_{p}$ → 90 N/mm² ( $f_{p}$ = characteristic strength).

Friction (post-tensioned): $P_{x} = P_{0} e^{- (μα + k x)}$ — $μ$ = coefficient of friction between tendon and duct, $α$ = cumulative angle change (rad), $k$ = wobble coefficient per metre, $x$ = distance from the jack.

Anchorage slip (post-tensioned): $Δ f = \frac{Δ E _{s}}{L}$ , $Δ$ = slip (typically a few mm), $L$ = tendon length — more significant for short tendons.

Typical total losses: roughly 15–25% for pre-tensioned and 10–20% for post-tensioned members (depends strongly on materials, age at transfer and tendon profile). Preliminary design often assumes about 20% and 15% respectively, then checks.

Part 2 of 2

Prestressed Members in Flexure

Last reviewed 16 Sept 2026 · 6 min read

Load stages

A prestressed member is checked at two critical stages:

Stage	Prestress	Loads	Critical fibre
Transfer (release of prestress)	Highest (before long-term losses), $P_{i}$	Self-weight only	Tension at the top, compression at the bottom
Service	Lowest (after losses), $P_{e} = η P_{i}$	Self-weight + superimposed dead + live	Compression at the top, tension at the bottom

Permissible stresses (compression and tension) at each stage depend on the member class (IS 1343 Type 1 — no tension; Type 2 — tension without visible cracking; Type 3 — limited crack width).

Stresses in a section

With compression positive, for a beam with prestress $P$ at eccentricity $e$ below the centroid, moment $M$ (sagging), section moduli $Z_{t}$ and $Z_{b}$ :

FormulaExtreme-fibre stresses

f_{t o p} = \frac{P}{A} - \frac{P e}{Z _{t}} + \frac{M}{Z _{t}}

f_{b o tt o m} = \frac{P}{A} + \frac{P e}{Z _{b}} - \frac{M}{Z _{b}}

Each term: uniform compression from $P / A$ ; bending from eccentric prestress (hogging, which compresses the bottom); bending from loads (sagging, which compresses the top).

Kern points and the pressure line

Kern (core) distances — the limits within which a compressive force produces no tension anywhere in the section:

k_{t} = \frac{Z _{b}}{A} (above centroid), k_{b} = \frac{Z _{t}}{A} (below centroid)

For a rectangle of depth $h$ : $k_{t} = k_{b} = h /6$ (middle third).

Pressure line (line of thrust): the location of the resultant compressive force in the concrete. Under external moment $M$ , it shifts upward from the tendon by

a = \frac{M}{P}

If the pressure line stays within the kern, the section has no tension.
At transfer (small $M$ ), a large eccentricity keeps the pressure line near the bottom kern; under service load it moves up towards the top kern — prestressed beams use the whole kern range efficiently.

Cable profiles

Simply supported beams under UDL: parabolic profile with maximum eccentricity at mid-span and zero (at centroid) at the supports, matching the bending moment diagram.
Point loads: draped (harped) straight segments.
Continuous beams: prestress induces secondary reactions and moments. A concordant cable produces no secondary reactions; any profile obtained from the bending moment diagram of the continuous beam under some loading is concordant. Linear transformation — moving the cable at interior supports without changing its shape within spans — does not alter the pressure line (Guyon's theorem).

Magnel's graphical method

For a chosen section, write the four stress limits — top and bottom fibres, at transfer and at service — as inequalities. For example, the top-fibre tension limit $f_{tt}$ at transfer:

\frac{P _{i}}{A} - \frac{P _{i} e}{Z _{t}} + \frac{M _{min}}{Z _{t}} \geq - f_{tt} \Rightarrow \frac{1}{P _{i}} \geq \frac{e - k _{b}}{M _{min} + f _{tt} Z _{t}}

Written this way, each of the four conditions is a straight line in the $e$ – $(1/ P)$ plane. The region satisfying all four gives feasible combinations; the design usually takes the maximum practical eccentricity (limited by cover) and the corresponding minimum prestress. This is Magnel's diagram.

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