Prestressed Concrete (Pre-tensioned + Post-tensioned) — Engineering Reference

1. At a glance

Prestressed concrete is concrete pre-compressed by high-strength steel tendons before service loads are applied, so that combined stresses (prestress + dead + live) remain compressive — or at least below cracking — under every relevant load case. The strategy lets concrete behave like an uncracked elastic material in service, which is what allows the dramatic spans, thin slabs, and crack-free water-tight structures that ordinary reinforced concrete cannot achieve.

Two methods, distinguished by when steel is tensioned relative to concrete placement:

• Pre-tensioned: strands are tensioned against external abutments of a long stressing bed (typically 100–300 m), concrete is cast around them, allowed to gain release strength (commonly f_ci’ ≈ 24–35 MPa / 3500–5000 psi within 12–24 hr using Type III cement or steam curing), then strands are cut. Strand recoil transfers compression into the concrete by bond along the full development length. Universally factory-precast.
• Post-tensioned: concrete is cast first with embedded ducts or unbonded greased sheathed strands. After the concrete reaches stressing strength (typically f_ci’ ≈ 17–28 MPa / 2500–4000 psi), tendons are threaded (if ducts are used), jacked against the hardened concrete, and locked off with wedge anchors. Done in the field for bridges, slabs, and complex framing.

Within post-tensioned, a further split:

• Bonded — duct grouted with cementitious grout after stressing. Bond is restored; tendon failure is localized. Mandatory for bridges in most jurisdictions; common for parking, transfer girders, and complex framing.
• Unbonded — single-strand monostrand, factory-greased, HDPE-sheathed, anchored at each end only. Tendon force can re-distribute on local loss; replacement is feasible. Dominant in US suspended slabs (parking decks, podium slabs, residential PT flat plates).

Scale and applications. Prestressed concrete is the structural workhorse for:

• Highway and rail bridges (10–50 m girders pre-tensioned; 50–250 m segmental, cable-stayed pylons, and cantilevers post-tensioned)
• Parking structures (double-tees, hollow-core, PT slabs)
• Podium and floor slabs of high-rise buildings (PT flat plates, 9–15 m typical span at 200–250 mm thickness)
• Nuclear containment vessels (post-tensioned dome and cylinder, periodically re-stressed)
• LNG and water tanks (circumferential prestress for crack-free water-tightness)
• Wind turbine towers (precast spun + PT splice — Vestas EnVentus, Siemens hybrid)
• Stadium roofs (Lucas Oil Stadium Indianapolis, Singapore Sports Hub, AT&T Stadium)
• Silos, locks, dam spillway tendons (rock-anchor PT bars)
• Marine and offshore (concrete gravity-base structures — Hibernia, Troll A)

Eugène Freyssinet’s 1928 French patent on high-strength steel + high-strength concrete tendon systems is the canonical origin. Magnel’s 1948 Prestressed Concrete codified design; T. Y. Lin’s 1963 load-balancing method made PT slabs designable on the back of an envelope. The 1950s–70s saw rapid US adoption (PCI founded 1954, AASHTO girder standards 1956, monostrand PT slabs commercialized 1960s).

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2. Why it matters

The mechanical motivation is direct: concrete is ~10× stronger in compression than in tension, and even reinforced concrete cracks at modest tensile stress (≈ 3 MPa / 450 psi modulus of rupture in normal-strength mixes). Prestress eliminates the tensile-cracking limit state for the dominant gravity case, which unlocks four things ordinary RC cannot:

1. Long spans at shallow depth. A 12 m simply-supported RC beam needs h ≈ L/12–L/16 (750–1000 mm); a PT equivalent needs h ≈ L/22–L/30 (400–550 mm). For continuous PT slabs at 9 m bays, h/L ≈ 1/45 is routine. This is why every modern airport, podium deck, and parking garage uses PT.
2. Crack-free water-tight enclosures. Liquid-retaining tanks (ACI 350) and nuclear containments are designed for zero tension at service — only prestress makes this economical.
3. Camber-controlled erection. Pre-tensioned girders are designed with upward camber at release, which counteracts dead-load deflection so the bridge deck sits flat after composite action with the cast-in-place deck.
4. Material efficiency. A 1860 MPa strand carries ~6× the working stress of Gr 60 rebar at <2× the cost. The same tonnage of steel resists much more moment when stressed.

The trade-offs are real: tighter cover requirements (ACI 318 §20.6), bursting reinforcement at anchor zones, friction and time-dependent losses to budget, corrosion sensitivity of high-strength steel under stress (a Florida DOT preoccupation since the 1999 Mid-Bay Bridge failures), and the impossibility of replacing tendons in fully-grouted bonded systems.

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3. First principles

3.1 The prestress / dead-load / live-load triple state

For a simply-supported beam with axial prestress force P at constant eccentricity e below the neutral axis, the linear-elastic Bernoulli combined stress at any fibre y above the centroid is:

σ(y) = −P/A + (P·e·y)/I − (M_load·y)/I

Sign convention: compression negative, tension positive; y positive above the neutral axis. The three terms are (a) uniform precompression, (b) prestress couple putting compression at bottom, tension at top, and (c) gravity moment putting compression at top, tension at bottom. For a parabolic-profile draped tendon, e varies along the span and the second term changes sign at the supports.

Three load stages must be checked (ACI 318 §24.5):

Stage	Loads	Prestress	Concrete strength	Stress limits (ACI 318 §24.5.3 / §24.5.4)
Transfer (initial)	Self-weight + P_i (just after release/stressing)	High (P_i)	Low (f_ci’)	Top tension ≤ 0.25·√f_ci’ (MPa); bottom compression ≤ 0.60·f_ci’
Service (final)	Full DL + LL + P_e	Reduced (P_e after losses)	Full (f_c’)	Class U tension ≤ 0.62·√f_c’ (MPa); compression ≤ 0.45·f_c’ (sustained) and 0.60·f_c’ (total)
Ultimate	1.2 DL + 1.6 LL	P_e (or strain-compatibility stress f_ps)	Full (f_c’)	Flexural strength per ACI 318 §22.3

Three serviceability classes are defined (ACI 318 §24.5.2):

• Class U (“uncracked”) — extreme tension ≤ 0.62·√f_c’ (MPa) = 7.5·√f_c’ (psi); analyzed gross-section.
• Class T (“transition”) — 0.62·√f_c’ < f_t ≤ 1.0·√f_c’ (MPa); cracked-section analysis required.
• Class C (“cracked”) — f_t > 1.0·√f_c’ (MPa); cracked-section analysis with steel stress + crack-width limits.

Two-way slabs are forced Class U.

3.2 Effective prestress and losses

The jacking force P_j is reduced to the long-term effective prestress P_e by a budget of losses:

P_e = P_j − ΔP_friction − ΔP_anchorage − ΔP_elastic − ΔP_creep − ΔP_shrinkage − ΔP_relaxation

Typical total losses: 15–25 % of P_j for normal designs, expressed as a percentage drop in tendon stress:

Loss component	Mechanism	Typical magnitude (MPa)	ACI 318 / PCI ref
Friction (PT)	Curvature µ + wobble K along duct	50–150 (long PT tendons)	ACI 318 §20.3.2.6
Anchorage seating (PT)	Wedge draw-in Δ_a ≈ 4–10 mm	30–100	PTI Manual
Elastic shortening	Concrete shortens when prestress is applied	20–50 (pre-tension); 0–30 (PT, depends on sequence)	ACI 318 §20.3.2.6
Creep (long-term)	Sustained-stress concrete deformation	50–120	ACI 209 / fib MC2020
Shrinkage (long-term)	Drying shrinkage	30–80	ACI 209
Relaxation (steel)	Stress decay in tendon at constant strain	30–60 (low-relax strand)	ASTM A416
Total (typical)		150–350 MPa	(≈ 10–25 % of f_pu)

PCI Design Handbook gives the simplified lump-sum estimate of 240 MPa (35 ksi) total long-term losses for typical pre-tensioned beams in normal exposure as a fast preliminary number; refined methods (AASHTO LRFD 5.9.3.4, PCI multi-step) compute each loss separately.

3.3 Load balancing (T. Y. Lin 1963)

The cleanest design lens for PT slabs and continuous beams: choose a draped tendon profile such that its uplift exactly cancels a chosen gravity load (typically dead load + a chosen fraction of live load).

A parabolic tendon with sag a (drape) and effective force P over a span L produces upward equivalent uniform load:

w_b = 8 · P · a / L²

Choose a, P to balance w_b = w_DL + 0.25·w_LL (typical 100 % DL + 25 % LL); the slab then behaves as if subjected only to the unbalanced fraction of the live load. Combined with the elastic combined-stress check at peak unbalanced moment, this gives a one-pass design that drops out tendon force and profile simultaneously.

For continuous spans, the tendon is draped over interior supports (e_high at supports, e_low at midspans), and the equivalent load is computed segment-by-segment. Hyperstatic (“secondary”) moments arise in continuous PT structures because the tendon imposes reactions at intermediate supports — these must be carried at ultimate (ACI 318 §22.3.4).

3.4 Magnel diagram (graphical sizing)

For preliminary section sizing, plot the inverse prestress 1/P versus eccentricity e on axes with four straight-line constraints — two from transfer stress limits (top tension, bottom compression) and two from service stress limits (top compression, bottom tension). The feasible region is a quadrilateral; its vertices give the minimum P and the corresponding e. Largely supplanted by spreadsheet iteration but still taught for the geometric intuition.

3.5 Camber and long-term deflection

Initial camber at release for a simply-supported pre-tensioned girder with parabolic equivalent load w_b and self-weight w_sw:

Δ_camber = (5/384) · (w_b − w_sw) · L⁴ / (E_ci · I_g)

Long-term camber and deflection are amplified by creep + shrinkage. ACI 318 §24.2.4 gives a long-term multiplier λ_Δ = ξ / (1 + 50·ρ’), with ξ → 2.0 at 5 years; PCI Design Handbook recommends separate 2.45× multiplier on (sustained-prestress + sustained-load) deflection at midspan for normal-weight composite girders (Table 5.8.2). Always design with PCI’s multipliers, not the ACI generalized λ_Δ, for prestressed members.

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4. Materials

4.1 Prestressing strand (ASTM A416)

Seven-wire helically-wound cold-drawn high-carbon pearlitic steel; six wires laid around a slightly larger king wire; tensile strength f_pu = 1860 MPa (Grade 270) is the global default. Grade 250 (1725 MPa) is rare in new construction.

Strand	Nominal diameter	Area (mm² / in²)	f_pu (MPa / ksi)	Min breaking force (kN / kip)	Mass (kg/m / lb/ft)	Typical use
3/8”	9.5 mm	54.8 / 0.085	1860 / 270	102 / 23	0.432 / 0.290	Light precast
7/16”	11.1 mm	74.2 / 0.115	1860 / 270	138 / 31	0.582 / 0.391	Hollow-core, double-tees
1/2”	12.7 mm	98.7 / 0.153	1860 / 270	184 / 41.3	0.775 / 0.520	Most common, US
1/2” special	12.7 mm	101.9 / 0.158	1860 / 270	190 / 42.7	0.800 / 0.538	Higher-load packaging
0.6”	15.2 mm	140 / 0.217	1860 / 270	260 / 58.6	1.102 / 0.740	Bridge girders, large PT
0.6” special	15.7 mm	150 / 0.232	1860 / 270	279 / 62.8	1.180 / 0.793	Stay cables
0.7”	17.8 mm	192 / 0.297	1860 / 270	353 / 79.4	1.510 / 1.015	Long-span bridges (FDOT, TxDOT pilot)

Low-relaxation (lo-lax) strand is standard since the 1990s; stress-relieved strand persists only in old stock. Lo-lax relaxation loss after 1000 hr at 70 % f_pu ≤ 2.5 % (vs ~8 % stress-relieved).

Allowable tendon stress at jacking and after seating (ACI 318 §20.3.2.5):

Stage	Limit	Numerical (1860 MPa strand)
Maximum at jacking	0.94 f_py but ≤ 0.80 f_pu	0.80 · 1860 = 1488 MPa
Immediately after seating (pre-tension)	0.74 f_pu	1376 MPa
Immediately after seating (post-tension)	0.70 f_pu	1302 MPa
Anchor + couplers	0.70 f_pu	1302 MPa

Typical design jacking: 0.75·f_pu = 1395 MPa at the jack (just below the 0.80 limit, leaving margin for friction wobble and instrumentation tolerance).

4.2 Prestressing bar (ASTM A722)

Type I (plain) and Type II (deformed) alloy steel bars, Grade 150 (f_pu = 1035 MPa) standard, Grade 160 (1100 MPa) available. Diameters 5/8” (16 mm) through 3” (75 mm). Threaded ends for coupling. Used for:

• Short straight tendons (transfer girders, machine foundations)
• Rock and ground anchors (PTI DC35.1)
• Vertical PT in nuclear and shear walls
• Wind tower vertical PT splicing

Vendors: Williams Form Engineering (most common US), Dywidag-Systems International (DSI) Threadbar, Macalloy (UK), Stronghold (China).

4.3 Anchorage and post-tensioning systems

Multistrand and monostrand anchorages are proprietary system-engineered assemblies (wedge plate, trumpet/bursting reinforcement, grout cap or end-cap). Major suppliers:

• VSL International — flagship multistrand systems (VSL EC, VSL 6-12, 6-19, 6-37, 6-55); largest installed base globally
• Freyssinet — historical originator; C-, K-, F-range multistrand and stay cable systems
• DSI (Dywidag-Systems International) — Threadbar PT bars and multistrand DYNA systems
• Suspa-DSI — monostrand US slab anchors
• General Technologies Inc. (GTI), Continental Concrete, PSC Freyssinet (UK) — regional providers
• Tensa, BBR — European multistrand specialists
• CCL Stressing Systems — UK / Asia multistrand

Anchor types relevant to design:

• Wedge anchor — two- or three-piece tapered wedge grips strand at lock-off; nearly universal for strand systems
• Nut-and-plate — used on PT bar (Williams, Dywidag); threaded
• Dead-end anchors — strand bonded directly into concrete (hairpin, “u”-loop, or onion) at one end of pretensioned members and shorter PT runs

4.4 Ducts, grouts, and corrosion protection

Bonded systems:

• Galvanized corrugated steel duct (legacy; ID 50–100 mm for 7–19 strand)
• HDPE corrugated duct (modern; PTI Plastic Duct Specification 2012) — better corrosion isolation
• Cementitious grout per ACI 318 §20.3.2.4 and ASTM C1107: thixotropic, low bleed (≤ 0.3 %), no shrinkage, no chloride; flowable through duct (≥ 0.5 m head)

Unbonded monostrand:

• Strand factory-greased with corrosion-inhibiting grease (≥ 25 g per linear meter for typical 0.5” strand) and extruded HDPE sheath (1.0–1.5 mm wall, low-permeability, electrical isolation for cathodic protection compatibility)
• ASTM A416 + PTI Specification for Unbonded Single-Strand Tendons
• Encapsulated anchor system with watertight pocket former, end cap, and grease-injected anchor cavity — mandatory for “PT-Aggressive” exposure (PTI M50.3-12)

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5. Worked examples (SI throughout; US-customary cross-checks where useful)

Example A — Simply-supported PT beam, service stress check

Problem. Simply-supported single-span PT beam:

• Span L = 12.0 m; b = 300 mm, h = 600 mm; rectangular section
• Tendon at constant eccentricity e = 200 mm below the neutral axis
• 4 × 0.6” strand (A_p = 4 × 140 = 560 mm²), bonded
• f_pu = 1860 MPa; jacked to 0.75·f_pu = 1395 MPa → P_j = 1395 × 560 = 781 kN
• Assume long-term loss ratio 15 % → P_e = 0.85 × 781 = 664 kN
• f_c’ = 35 MPa; E_c = 4700·√35 = 27,800 MPa
• Service moment at midspan from DL + LL: M_s = 294 kN·m

Step 1 — Section properties (gross). A = 300 × 600 = 180,000 mm² = 0.18 m² I = 300 × 600³ / 12 = 5.40 × 10⁹ mm⁴ y_t = y_b = 300 mm (symmetric) S_t = S_b = I / 300 = 18.0 × 10⁶ mm³

Step 2 — Bottom-fibre stress at midspan, service. σ_b = − P_e/A − (P_e · e)/S_b + M_s/S_b = − 664,000/180,000 − (664,000 × 200)/(18.0×10⁶) + (294 × 10⁶)/(18.0×10⁶) = − 3.69 − 7.38 + 16.33 = +5.26 MPa tension

Step 3 — Compare to Class U limit (ACI 318 §24.5.2.1). Limit: 0.62·√35 = 3.67 MPa. 5.26 > 3.67 → Class U fails.

The section is Class T or C. Two responses:

• Increase prestress: add 2 strands → A_p = 840 mm², P_e ≈ 996 kN, σ_b = −5.53 − 11.07 + 16.33 = −0.27 MPa (compression — passes Class U).
• Or raise eccentricity (increase tendon drape) to lower mid-span y_e — e.g., move to e = 250 mm with current 4 strands: σ_b = −3.69 − 9.22 + 16.33 = +3.42 MPa < 3.67 MPa (passes Class U).

Step 4 — Top-fibre stress at transfer (initial, just after stressing). Use P_i = 0.93 × P_j (5 % elastic + 2 % initial relaxation losses) = 0.93 × 781 = 727 kN. Take only self-weight at transfer: w_sw = 0.30 × 0.60 × 24 = 4.32 kN/m → M_sw = 4.32 × 12²/8 = 77.8 kN·m.

σ_t = − P_i/A + (P_i · e)/S_t − M_sw/S_t = − 727,000/180,000 + (727,000 × 200)/(18.0×10⁶) − (77.8×10⁶)/(18.0×10⁶) = − 4.04 + 8.08 − 4.32 = − 0.28 MPa (compression) ✓ (well under 0.25·√f_ci’ ≈ 1.3 MPa tension limit and the 0.60·f_ci’ compression limit).

Example B — Load balancing of a continuous PT slab

Problem. Two-way PT flat plate, 9 m square panel, 250 mm thick. Design dead load (self-weight 6.0 kPa + 1.5 kPa SDL) = 7.5 kPa; design live load 2.4 kPa (office). Balance 100 % DL with parabolic tendon drape a = 130 mm (cover-adjusted top-to-bottom envelope). Required PT force per meter of band?

Step 1 — Equivalent uplift per band. For each strip 1 m wide considered as a simply-supported equivalent at the column line: w_b = 7.5 kN/m.

P · 8 · a / L² = w_b P = w_b · L² / (8 · a) = 7.5 × 9² / (8 × 0.130) = 584 kN per meter width

Step 2 — Strand count. Using 0.6” strand with effective prestress per strand (after 15 % loss) ≈ 0.85 × 0.75 × 260 = 166 kN/strand: n = 584 / 166 = 3.5 → 4 strands per meter of slab width.

Distribute 70 % of tendons in column strips, 30 % in middle strips per ACI 318 §8.7 distribution (or use banded distribution: all tendons in one direction concentrated within ¼ panel each side of column line, uniform in the orthogonal direction — the dominant US practice).

Step 3 — Net unbalanced load check. After balancing DL, the slab carries only LL = 2.4 kPa as bending. Mid-panel moment ≈ 0.05 × 2.4 × 9² ≈ 9.7 kN·m/m → very low elastic stresses → Class U service ✓ trivially.

Example C — Loss budget for a 30 m PT bridge girder

Problem. 30 m simply-supported AASHTO Type IV girder, post-tensioned with 12 × 0.6” strand in a single bonded duct draped parabolically. f_pu = 1860 MPa; jacking stress f_pj = 0.75 f_pu = 1395 MPa from one end. Friction µ = 0.20 rad⁻¹, wobble K = 0.0066/m, anchor seating Δ_a = 6 mm, jack length L_jack = 32 m. f_ci’ = 30 MPa at stressing; f_c’ = 40 MPa at 28 days. Compute long-term effective stress.

Step 1 — Friction loss at the far end. Total tendon turning angle α (from drape) ≈ 8·a/L_span in radians for a parabola ≈ 8 × 0.6/30 = 0.16 rad over 30 m.

f_x = f_pj · exp(−(K·x + µ·α)) = 1395 · exp(−(0.0066 × 30 + 0.20 × 0.16)) = 1395 · exp(−(0.198 + 0.032)) = 1395 · exp(−0.230) = 1395 · 0.795 = 1109 MPa at far end (ΔP_friction ≈ 286 MPa over the length, average ≈ 143 MPa).

Step 2 — Anchorage seating loss. Approximation (uniform friction): the seating loss propagates back from the live end over a length L_set where it equals the friction-recovery gradient.

ΔP_seat = 2 · (ΔP_friction/L_span) · L_set; with L_set = √(Δ_a · E_p · L_span / (ΔP_friction/L_span)) — use AASHTO 5.9.3.2.1 simplified table or PTI Manual chart. For our case, with Δ_a = 6 mm, E_p = 196,500 MPa, friction gradient ≈ 9.5 MPa/m → L_set ≈ √(6 × 196500 / 9.5) ≈ 352 mm — short, fully contained near the live anchor. Equivalent uniform loss across the tendon: ΔP_seat ≈ 2 × 9.5 × 0.352 / 30 = 0.22 MPa average (negligible for this geometry; matters more for shorter tendons).

Step 3 — Elastic shortening. ΔP_es = (E_p / E_ci) · f_cir, where f_cir is the concrete stress at the centroid of strands due to prestress + self-weight. With E_ci = 4700·√30 = 25,742 MPa, E_p/E_ci = 7.6. For a typical Type IV (e ≈ 580 mm, A_g ≈ 510,000 mm², I_g ≈ 1.083×10¹¹ mm⁴), f_cir ≈ 9 MPa → ΔP_es ≈ 68 MPa. PT note: for single-tendon PT, ES is zero at the last tendon stressed; for multi-tendon PT, the average ES affects the earlier tendons.

Step 4 — Time-dependent losses (AASHTO LRFD 5.9.3.4 refined).

• Shrinkage Δf_pSR ≈ 30 MPa (40 % RH)
• Creep Δf_pCR ≈ 50 MPa (sustained-load creep coefficient ~2.0)
• Relaxation Δf_pR2 ≈ 25 MPa (lo-lax)

Step 5 — Total. Average tendon stress after all losses: f_pe ≈ 1395 − 143 (avg friction) − 0.2 (seating) − 68 (ES) − 30 (SR) − 50 (CR) − 25 (R) = 1079 MPa

Total loss = 1395 − 1079 = 316 MPa = 23 % of f_pj — squarely in the 15–25 % typical range. Effective ratio P_e/P_j = 0.77.

A more refined “PCI multi-step” method would track losses at each life stage (immediately after release, at deck placement, at infinite time) and is what AASHTO 5.9.3.4 prescribes for design-of-record on bridges.

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6. Pre-tensioned practice

The factory paradigm: long stressing beds (often 200–300 m), shared by multiple successive casts.

Standard product families (PCI Design Handbook 8th ed):

• Hollow-core plank — 150–400 mm depth; spans 6–15 m; precast on long-line beds, sawed to length; floor and roof structure for hotels, apartments, schools. Spancrete, Oldcastle Infrastructure, Forterra, Florida Prestressed.
• Double-tee (DT) — 600 to 1100 mm depth; spans 12–30 m; parking structures, warehouses, office floors. NFL stadium tier slabs.
• Inverted-tee (IT) and L-beams — receiving DT or hollow-core ledges
• Solid flat slabs — wall panels, architectural cladding
• AASHTO bridge girders — depth-by-section series (Type I to Type VI), Bulb-T (FDOT, WSDOT, NU), Florida-I (FIB), and modern NU/Nebraska girders; spans up to ~50 m simply-supported

AASHTO standard girder sections:

Section	Depth (mm / in)	Span range (m)	Weight per meter	Typical strands
AASHTO Type I	711 / 28	10–14	7.0 kN/m	8–12 × 1/2”
AASHTO Type II	914 / 36	12–20	8.8 kN/m	12–22 × 1/2”
AASHTO Type III	1143 / 45	18–28	13.5 kN/m	20–32 × 1/2”
AASHTO Type IV	1372 / 54	25–35	18.0 kN/m	32–48 × 1/2” or 0.6”
AASHTO Type V	1600 / 63	30–40	25.0 kN/m	40–58 × 0.6”
AASHTO Type VI	1829 / 72	35–50	28.5 kN/m	50–66 × 0.6”
Bulb-T BT-72 (TxDOT)	1829 / 72	38–48	23.5 kN/m	60+ × 0.6”
Florida-I FIB-96	2438 / 96	50–65	26.0 kN/m	70+ × 0.6”
WSDOT WF100G	2540 / 100	55–70	27.5 kN/m	70+ × 0.6”
NU 2000 (Nebraska)	2000 / 79	40–60	22.0 kN/m	60+ × 0.6”

Strand profile shaping — Two methods to manage transfer-stress tension at girder ends:

1. Debonding (shielding) — wrap a length of plastic sleeve over selected strands at the ends so they don’t bond and contribute prestress there. Most-used in the US. ACI 318 §25.7.4 limits the fraction debonded.
2. Draped (depressed) strands — hold strands down in the middle and up at ends with hold-down devices in the stressing bed. Common in older designs and AASHTO Type V/VI.

Release timing is tied to f_ci’ (typically 24–28 MPa / 3500–4000 psi within 18 hr), often achieved with Type III cement and accelerated curing. Camber at release is the visible product success indicator.

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7. Post-tensioned practice

7.1 Bonded (grouted) multistrand

Used for bridges, large transfer girders, mat foundations, large industrial framing. Typical workflow:

1. Form concrete with corrugated steel or HDPE ducts at design profile, supported on rebar chairs at 1 m spacing
2. Cast concrete, cure to f_ci’
3. Thread strands (push or pull)
4. Stress with multistrand jack (VSL, Freyssinet, DSI) — typically two-end stressing for long tendons or one-end with sequenced restress
5. Anchor, cut surplus strand, cap anchor pocket
6. Grout duct via inlet (low point) with vents at all high points; thixotropic grout per PTI M55.1

Anchor zone (D-region) requires bursting reinforcement for the radial tensile stresses from anchor plates (ACI 318 §25.9, AASHTO LRFD 5.10.9). Local zone (just behind plate): spiral confinement supplied by anchor manufacturer per ETA. General zone: structural engineer designs with strut-and-tie (IDEA StatiCa Detail, hand STM).

7.2 Unbonded (monostrand) slabs and beams

Dominant US suspended-slab system since the 1970s. Each tendon is a single 1/2” or 0.6” strand pre-greased and HDPE-extruded at the factory; placed by tendon installers, anchored at slab edges in encapsulated pocket-formed cavities.

Typical PT slab parameters (PCI / PTI / common practice):

Slab type	Thickness (mm / in)	Span (m / ft)	Tendons per meter (0.5”)	Typical f_pe
PT flat plate, residential/office	200–230 / 8–9	8–11 / 26–36	1.3–2.0	1000–1200 MPa
PT flat plate, parking	200–230 / 8–9	8–10 / 26–33	1.5–2.0	1000–1200 MPa
PT banded slab, parking	250–275 / 10–11	9–12 / 30–40	2.0–3.0 banded	1000–1200 MPa
PT beam-and-slab, residential	250–300 / 10–12 + beams	11–14 / 36–46	2.0–3.5 in beams	1000–1200 MPa
PT one-way slab, transfer level	350–500 / 14–20	12–18 / 40–60	4–8 in band	1000–1200 MPa

Stressing sequence: in slabs, tendons are typically all stressed shortly after slab attains 17–21 MPa (3000 psi); a “pour strip” left between adjacent slab sections allows initial shortening (~3–6 mm per 30 m of slab length) before being filled. Restraint cracking of long PT slabs against stiff columns and walls is the perennial design failure mode — pour strips, delay strips, and slip joints are the mitigation.

7.3 Stressing operations and quality

• Calibrated jack + pressure gauge + elongation measurement; two independent checks (force from gauge, elongation from theoretical L·f_pj/E_p with friction correction) — agreement within 7 % is the standard PTI acceptance.
• Recording stress-elongation curves; investigate any tendon falling outside.
• Surface preparation, anchor pocket cleaning, grout-cap installation immediately after stressing.

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8. Applications

8.1 Highway and rail bridges

The single biggest market for prestress. AASHTO LRFD 9th ed governs in the US. Standard practice splits:

• Short to medium (10–50 m, ~80 % of all bridges): pre-tensioned simple-span girders (AASHTO Type II–VI, Bulb-T, FIB, NU) with composite cast-in-place deck. Multi-span continuity at piers via deck slab and link reinforcement (continuous-for-live-load), or fully continuous via splices.
• Medium-long (50–150 m): post-tensioned spliced girders (segmented pre-tensioned girders post-tensioned into continuity), or balanced-cantilever post-tensioned box girders erected segmentally.
• Long (> 150 m): cable-stayed (Penobscot Narrows ME, Cooper River SC, Sundial Bridge CA), extradosed (Pearl Harbor Memorial Bridge CT), or precast segmental cable-stayed (Sunshine Skyway FL).

Florida DOT, Texas DOT (TxDOT), Washington State DOT (WSDOT), and Nebraska DOR (NDOR) drive most innovation in girder geometry (FIB, Bulb-T, WF, NU sections). PCI Bridge Geometry Manual and PCI Zone 6 (high-strength concrete) compendium are the standard practice references.

8.2 Nuclear containment

Post-tensioned dome + cylinder containment vessel; un-grouted tendons in some designs (allowing periodic restressing every ~5 years for the design life) or grouted bonded tendons (more durable but un-inspectable). ACI 359 / ASME Section III Division 2 governs. Tendons are typically PT bar (Dywidag, Williams) in steel ducts; vertical, horizontal hoop, and dome PT all stressed in defined sequence. Periodic in-service inspection (ISI) measures residual force via lift-off testing.

8.3 LNG and water tanks

Cryogenic LNG outer-containment tanks (Cheniere Sabine Pass, Cameron LNG): post-tensioned circumferential prestress in the cylinder wall and dome to keep concrete in compression at all cryogenic shrinkage states and Boiling-Liquid-Expanding-Vapor-Explosion (BLEVE) overpressure conditions. API 620 + ACI 376 for refrigerated tanks. Water tanks (potable, treatment) per ACI 350.

8.4 Stadium roofs and long-span buildings

• Lucas Oil Stadium (Indianapolis) — PT long-span trusses and roof
• Singapore Sports Hub — 310 m dome with post-tensioned concrete compression ring
• AT&T Stadium (Arlington TX) — PT raker beams supporting cantilevered upper bowl
• Cowboys Indoor Practice Facility — PT long-span girders

8.5 Wind turbine towers

Hybrid concrete + steel towers: precast spun concrete bottom sections (Vestas EnVentus, Siemens Gamesa, Enercon), post-tensioned vertically with PT bar (Williams, Dywidag) or strand (VSL, DSI). Concrete bases up to 140–170 m hub height are economic where steel tower transport limits diameter (~4.3 m road limit). Vertical PT bars are tensioned after stacking precast rings; the same bars provide the splice continuity.

8.6 Marine and offshore

Concrete gravity-base structures (Hibernia GBS off Newfoundland; Troll A in the North Sea — 472 m total height including caissons, the largest object ever moved by humanity at the time of installation in 1995): heavily post-tensioned with multistrand systems. Crack-free water-tightness at depth + fatigue resistance under wave loading.

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9. Edge cases, failure modes, and engineering judgement

1. Tendon corrosion in bonded PT bridges. Florida DOT (Mid-Bay Bridge 1999, Niles Channel 1999, Sunshine Skyway tendon failures) and Texas DOT (Varina-Enon, San Antonio Y) all suffered grouted-tendon corrosion when bleeding and segregated grout left voids at high points exposing strand to corrosive water ingress. Industry response: PTI M55.1 thixotropic-grout spec (2003, 2012, 2019), PTI Plastic Duct Specification (2012, 2019), mandatory PT inspector certification, low-permeability HDPE ducts replacing corrugated steel.

2. Bursting and spalling at anchorage zones (D-regions). Concentrated prestress at the anchor plate generates large transverse tensile stresses ahead of the plate (bursting) and at the edges (spalling). Mandatory bursting reinforcement (ACI 318 §25.9 / AASHTO 5.10.9.6) — typically a confining spiral supplied by the anchor manufacturer (local zone) plus hairpins, ties, or stirrups closing within the general zone (designed by EOR using strut-and-tie or IDEA StatiCa Detail / CSFM).

3. Time-dependent losses — refined modelling. AASHTO LRFD 5.9.3.4 (refined) accounts for time-segment-specific creep, shrinkage, and relaxation; PCI multi-step is the building-industry analog. Bazant B4 / B4s model is the research-grade creep + shrinkage formulation used in fib MC2020. Long-term losses can grow if relative humidity is unusually low (high-altitude Mountain West) or if SCM content delays shrinkage and creep.

4. Bonded tendon replacement is effectively impossible. Once grouted, a failed bonded tendon cannot be removed or restressed. Unbonded monostrand tendons can be cut, withdrawn, and replaced individually — a major life-cycle advantage cited by US slab designers but rarely exercised in practice.

5. Cover for prestressed reinforcement (ACI 318 §20.6 / AASHTO LRFD 5.10.1). Concrete cover for prestressing steel is more stringent than mild rebar because the consequences of corrosion + section loss are worse (high-stress steel under sustained load undergoes stress-corrosion cracking — sudden, brittle, no warning). Typical: precast plant 25 mm interior, 40 mm exterior; field 38 mm interior, 50 mm exterior; cast against earth 75 mm.

6. Fire resistance. Prestressing steel loses strength rapidly above 350–450 °C — much faster than mild rebar (which holds ~70 % at 500 °C). Cover requirements for fire (typically 50 mm for 2 hr rating in beams, 30 mm in slabs per IBC Table 720.1) reflect this. Spalling-prone HSC mixes used with PT should include polypropylene microfibres for explosive-spalling protection.

7. Punching shear at PT slab columns. Less critical than for RC flat plates (the in-plane precompression suppresses cracking) but still governs design at heavy columns. ACI 318 §8.7.5 + §22.6 cover punching for PT slabs. Integrity reinforcement (bottom bars passing through the column cage and lapped) is mandatory after the lessons of Skyline Plaza (1973) and Champlain Towers South (2021).

8. Edge restraint and slab shortening. A long PT slab (50–80 m) shortens 15–30 mm under prestress + creep + shrinkage. Stiff perimeter walls or interior columns crack from restraint. Mitigation: pour strips (closure pours left for several weeks), delay strips, slip joints at first level above slab-on-grade, releasable column-to-slab connections, and limiting maximum dimension without a slip joint to ~60–80 m.

9. Hidden-tendon strikes in renovation. Drilling, coring, or saw-cutting through a PT slab can cut a live tendon. Required practice: GPR (ground-penetrating radar) survey on EVERY hole + as-built tendon drawings; in unbonded, a cut releases the entire tendon length (potential energy storage roughly equivalent to ½·k·x² for a 30 m tendon at 1000 MPa stored stress — projectile risk to workers downstream). PTI maintains incident statistics.

10. Unbonded anchorage failure can be explosive. If a corroded or improperly seated anchor releases under full prestress, the strand can fly out with significant kinetic energy. Stand clear of anchor pockets during stressing. Lock-off operations require staffed exclusion zones behind anchors.

11. Heat curing and DEF. Pre-tensioned plants commonly steam-cure; if internal concrete temperature exceeds ~70 °C during early hydration, delayed ettringite formation (DEF) can develop over years — internal expansive cracking that destroys the bond and the prestress transfer. ACI 363 + PCI MNL-116 limit curing temperature.

12. Shock losses on cutting/release in pre-tension. Sudden release of strands (oxy-fuel cut) loads the concrete dynamically. PCI MNL-116 specifies gradual de-tensioning via hydraulic jack release or flame-cut sequence to manage shock — relevant on heavily-prestressed deep girders.

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10. Tools and software

Slab and beam design

• ADAPT-Builder (Risa) — full 3-D FEM with PT modelling; ADAPT-PT (preliminary), ADAPT-Floor (FEM strip-method) — dominant US PT-slab tool
• RAM Concept (Bentley) — FEM, integrated with RAM Structural System
• ETABS + SAFE (CSI) — building FEM with PT cable-driver in SAFE
• PCA-Frame, spBeam, spSlab (StructurePoint) — older but still in service

Bridge design

• LARSA 4D — segmental and curved bridge, time-stepping creep + shrinkage
• MIDAS Civil — Korean / global; segmental, cable-stayed, curved
• CSiBridge (CSI) — AASHTO-aligned bridge design
• SOFiSTiK — German civil/bridge FEM with full PT modelling
• Bentley LEAP / RM Bridge — bridge geometry and PT analysis
• PCI Bridge Geometry Manual spreadsheets — basic span-to-depth and prestress sizing

Anchor zone and D-region

• IDEA StatiCa Detail — CSFM (Compatible Stress Field Method) for anchor zones, deep beams, corbels
• Hand strut-and-tie per ACI 318 §23 / AASHTO 5.6.3

Time-dependent and material

• B4cast — early-age thermal in mass concrete (relevant to large PT pile caps and transfer girders)
• bSim, DIANA, Atena — research-grade nonlinear FEA with full creep + shrinkage + cracking
• Life-365 — chloride-driven service life for PT bridges in marine exposure

Manufacturer software (free design aids)

• VSL, Freyssinet, DSI publish loss-calculation and friction-design spreadsheets keyed to their hardware
• PCI Designer spreadsheets for camber, deflection, prestress losses (PCI MNL-129)
• PTI publishes the Manual + design aids

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11. Cross-references

• reinforced-concrete — parent material; prestress design assumes RC mechanics + composite behaviour
• beam-theory — Euler-Bernoulli flexure that underpins the combined-stress check
• structural-analysis — frame analysis, hyperstatic moments in continuous PT
• steel-design — AISC 360 contrast for steel competition on long span
• materials-steel — prestressing steel metallurgy (pearlitic high-carbon cold-drawn)
• structural-dynamics — long-span PT floor vibration; AISC DG11 / CCIP-016 criteria
• fem-fea — modelling PT tendons as equivalent loads or as 1-D elements with prestrain
• steel-connection-design — planned; comparison and hybrid construction
• masonry-timber — planned; alternative structural systems for short-to-medium span
• soil-mechanics — planned; PT anchorages in rock and earth retention
• structural-dynamics — planned; PT detailing under ACI 318 Ch.18 + ASCE 41

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12. Citations

1. ACI Committee 318. Building Code Requirements for Structural Concrete (ACI 318-25) and Commentary, Chapters 20 (Steel Reinforcement Properties), 24 (Serviceability), 25 (Reinforcement Details), with embedded prestress provisions (American Concrete Institute, 2025).
2. ACI 423.7-14 / 423.10-16 — Unbonded Single-Strand Tendon Installation and Test Methods for Tendons.
3. Lin, T. Y. and Burns, N. H. Design of Prestressed Concrete Structures, 3rd ed. (Wiley, 1981). The canonical undergraduate / graduate text; origin of load-balancing.
4. Naaman, A. E. Prestressed Concrete Analysis and Design: Fundamentals, 3rd ed. (Techno Press 3000, 2012). Modern graduate text, deep loss modelling.
5. Nilson, A. H. Design of Prestressed Concrete, 2nd ed. (Wiley, 1987). Companion to the RC text.
6. Collins, M. P. and Mitchell, D. Prestressed Concrete Structures (Prentice Hall, 1991). Canadian / fib alignment.
7. Hewson, N. R. Prestressed Concrete Bridges: Design and Construction, 2nd ed. (ICE Publishing / Thomas Telford, 2012). The standard bridge-engineering reference.
8. Magnel, G. Prestressed Concrete, 3rd ed. (Concrete Publications, 1954; orig. 1948). Codification of design after Freyssinet’s discoveries.
9. Freyssinet, E. French Patents (1928, 1933). Origin of high-strength steel + high-strength concrete philosophy.
10. Lin, T. Y. “Load Balancing Method for Design and Analysis of Prestressed Concrete Structures.” ACI Journal, 60(6):719–742 (1963).
11. Precast/Prestressed Concrete Institute. PCI Design Handbook: Precast and Prestressed Concrete, 8th ed. (PCI, 2017). Indispensable for precast/prestressed practice.
12. Post-Tensioning Institute. PTI Manual, 7th ed.; PTI DC10.5-19 Design of Post-Tensioned Slabs Using Unbonded Tendons; PTI M55.1-19 Specification for Grouting of Post-Tensioned Structures; PTI M50.3-12 Specification for Multistrand and Grouted Post-Tensioning.
13. AASHTO. LRFD Bridge Design Specifications, 9th ed. (AASHTO, 2020, with current interims). Section 5 (Concrete Structures).
14. CEN. EN 1992-1-1:2023 — Eurocode 2: Design of concrete structures — Part 1-1: General rules. Section 5.10 (prestress).
15. CEN. EN 1992-2:2005+A1:2014 — Concrete bridges.
16. ASTM A416/A416M-18 — Standard Specification for Low-Relaxation, Seven-Wire Steel Strand for Prestressed Concrete.
17. ASTM A722/A722M-18 — Standard Specification for High-Strength Steel Bars for Prestressed Concrete.
18. ASTM A1031/A1031M-12 — Standard Specification for Steel, Concrete Reinforcement, Hot-Rolled Deformed and Plain.
19. fib (International Federation for Structural Concrete). fib Model Code for Concrete Structures 2020 (Ernst & Sohn, 2024). Sections on prestress and time-dependent behaviour.
20. fib Bulletin 75 — Polymer-Duct Systems for Internal Bonded Post-Tensioning (fib, 2014).
21. ACI 359-25 / ASME Section III Division 2 — Code for Concrete Containments (nuclear). The basis for nuclear PT design.
22. NCHRP Report 496 — Prestress Losses in Pretensioned High-Strength Concrete Bridge Girders (Tadros et al., 2003).
23. Bazant, Z. P. et al. “RILEM Draft Recommendation: TC-242-MDC multi-decade creep and shrinkage of concrete: Material model and structural analysis” (Materials and Structures, 2015) — basis for the B4 / B4s creep model.