Structural Engineering & Statics
where the load goes
Concept

Transfer — where it appears

The moment at which a prestressing force is released onto the concrete, when the force is at its largest and the only load is the member's own weight. It is a governing case in its own right, and its critical fibre is the opposite one from the service case.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Two triangles that cross zero, and a block that does not. Stress across a 300 × 700 mm section at each stage, compression positive. The prestress alone gives -6.33 MPa at the top and 20.61 at the bottom; at transfer, with only self-weight on it, the top is at -2.47 MPa and in service the section runs from 7.61 to 3.82 MPa — compression everywhere. The same beam with no prestress reaches -12.67 MPa at the bottom fibre, which is 4.2 times what the concrete can hold.

The load put on backwards

Every other structure in this collection waits for its load and then resists it. A prestressed one is given a load first — chosen, permanent, and pointing the wrong way — so that when the real one arrives the two nearly cancel and the material never has to do the thing it is bad at.

internal-forces · Prestress Four inequalities, and the wedge between them. The Magnel diagram: every limit on a prestressed section, plotted as a bound on 1/P against the eccentricity. Two of the four come from transfer, when the force is largest and the only moment is the beam's own weight, and two from service, when 20% of the force has been lost and the moment is 640 kNm. Each is linear in 1/P, which is the substitution that makes the problem a picture rather than a search. The shaded region is every force-and-eccentricity pair the section will accept: it is a wedge opening to the right, so the cheapest prestress is always at the largest eccentricity the cover allows — 1029 kN at e = 400 mm here. The section's kern is 241 mm, and every useful answer is outside it.

Four inequalities and a wedge

A prestressed section has to satisfy two stress limits when the force is largest and the load smallest, and two more when the force has relaxed and the load has arrived. Each is linear in one over the force — which turns a search for a prestress into a region on a page, and turns an impossible section into an empty one.

sections · Prestress limits Each fibre has to earn its own prestress. For each section on a 12 m span carrying its own weight and 20 kN/m more: bars, the top and bottom moduli it has; ticks, the moduli its two pairs of limits need — the top fibre holding the service compression and the transfer tension, the bottom holding the service tension and the transfer compression, both from the same prestress. The symmetric I: top 28.9 against 17.4 needed, bottom 28.9 against 23.5 (× 10⁶ mm³) — a prestress exists; the tee: top 48.5 against 17.8 needed, bottom 22.7 against 24.0 (× 10⁶ mm³) — no prestress exists; the bulb-tee: top 46.0 against 17.7 needed, bottom 37.4 against 23.8 (× 10⁶ mm³) — a prestress exists.

The flange the prestress cannot use

The four stress limits on a prestressed section pair up by fibre: the bottom fibre has to hold the service tension and the transfer compression from the same force, and the top fibre the service compression and the transfer tension. Each pair is possible only if that fibre's own section modulus is large enough, whatever the eccentricity. On a symmetric section the two pairs are close to balanced. On a tee they are not: its wide flange multiplies the top fibre's modulus and hardly touches the bottom's, so a tee with half as much concrete again as a symmetric I cannot be prestressed for a load the I carries.

sections · Prestress limits The force each section may carry rises from the end, and the strands must climb under it. The prestress each section of a pretensioned beam 300 × 700 mm spanning 12 m, with eight strands of 125 kN at 220 mm below its centroid may carry at transfer without its top fibre passing −2.50 N/mm² or its bottom 18 (thick), against the distance from the end: 739 kN at 0.5 m, 1,101 at 2 m, 1,507 at mid-span, where the self-weight moment is largest. All eight strands bonded from the end give 1,000 kN once their transmission length is past (dashed), which the beam can carry only near mid-span. Debonded — six bonded from the end, one debonded for 0.50 m, one debonded for 1.00 m — their force (solid) climbs in steps that stay under the line everywhere.

The strands that are sleeved at the ends

A post-tensioned tendon can be draped up towards the centroid near the supports, where the self-weight moment that made its eccentricity safe has gone. A pretensioned strand cannot: it runs straight between the abutments of the casting bed. Choose eight strands at 220 mm below the centroid of a 12 m beam and mid-span passes every limit at transfer, while a metre in from each end the top fibre cracks. The cure is to stop some strands gripping the concrete near the ends, and how many is a trade: every strand saved at mid-span by lowering the strands is a strand to sleeve at the ends.

sections · Prestress limits

Named alongside it

The objects these essays reach for when they reach for this one.

KernPrestressPrestress lossesBending stressCrackingMagnel diagramSection modulusAsymmetric sectionBondCable zoneCoverCreep

All concepts