The strands that are sleeved at the ends

Four inequalities fix a prestressed section: two at transfer, when the prestress is at its largest and only the beam’s own weight is acting, and two in service, when a share of the prestress has been lost and every load is on. On a Magnel diagram they make a wedge of acceptable forces and eccentricities, and the cheapest prestress is always at the largest eccentricity the section can hold. For an unsymmetric section the wedge is cut differently, and a tee can fail at mid-span by a condition no amount of prestress cures.

Every one of those calculations was made at one section — mid-span, where the moment is largest. The four inequalities hold at every section, and the moment is not the same at every section. Towards the supports it falls, and at transfer the moment falling is the problem: the beam’s own weight, which at mid-span pulls the top fibre back into compression against the eccentric prestress, is doing nothing a metre from the end. The prestress is still there.

A post-tensioned tendon answers this by being draped: it runs low at mid-span and rises towards the centroid at the ends, and the zone it must stay inside narrows exactly as it rises. A pretensioned strand cannot be draped in the ordinary way. It is stretched between two abutments of a casting bed, the concrete is cast round it, and when the concrete has gained enough strength the strand is cut and its force passes into the concrete by bond. It runs straight from end to end, at the eccentricity chosen for mid-span.

The beam, and where it breaks #

The beam is a 300 × 700 mm rectangle spanning 12 m, carrying its own weight of 5.25 kN/m at transfer and 12 kN/m more in service. It is pretensioned with eight strands, each carrying 125 kN once it has been released and the beam has shortened, at 220 mm below the centroid — 130 mm above the soffit, room for two layers of strand and their cover. At transfer the top fibre may go into tension up to 2.5 N/mm², and the bottom may carry up to 18 N/mm² of compression; in service, with a fifth of the prestress lost, the bottom may go to 2.5 N/mm² of tension.

At mid-span the section passes with room to spare: the top fibre in a little tension, the bottom a little over half its limit. The strands were chosen there, and they are the right strands for there. Between half a metre and a metre and a half from each end the top fibre is in more tension than it may be, at up to 3.31 N/mm². A tension of that size in a fresh concrete flange at release, where the concrete has a fraction of its eventual tensile strength, is a crack across the top of a beam that has not yet been lifted out of its bed — a load that arrives before the strength does.

The figure also shows why the failure begins half a metre in rather than at the very end. A strand’s force does not arrive at the end of the beam. It is passed into the concrete along a transmission length by bond, the force that arrives along a length, so at the face of the beam the strands carry nothing and their force builds up over the first 750 mm. The top fibre’s tension grows with it, peaks just past the end of the transmission length where the full force has arrived and the self-weight moment is still small, and falls back as the moment grows towards mid-span.

What each section may carry #

The four limits, written along the beam, become a single curve: the largest prestress each section may carry at transfer.

With compression positive, the top fibre at transfer is $P/A - Pe/Z + M_0/Z$ and the bottom $P/A + Pe/Z - M_0/Z$, where $M_0$ is the self-weight moment at the section. With the strands below the kern the first falls as $P$ rises and the second rises, so each limit puts a ceiling on the force:

$$P_{\text{allow}}(x) = \min\left[\frac{M_0/Z - f_{ti}}{e/Z - 1/A},\ \frac{f_{ci} + M_0/Z}{1/A + e/Z}\right].$$

For this beam the top-fibre limit governs everywhere along the span; the bottom’s would allow more. The curve rises from 593 kN at the end of the beam, where there is no moment, to 1,507 kN at mid-span, as the self-weight moment grows like a parabola. The eight strands’ 1,000 kN sits above the curve for the first metre and a half and below it from there on.

The cure is to make the strands’ force rise along the beam as the allowed force does. A post-tensioned tendon does it by rising towards the centroid, which reduces the eccentricity and so raises the ceiling. Pretensioned strands do it by being switched on in stages: some grip the concrete from the end of the beam, and others are sleeved in a plastic tube for a set distance so that the concrete cannot grip them, and begin to carry force only where the tube stops. That is debonding, and it turns the strands’ force into a staircase that can be fitted under the curve.

The schedule #

The schedule is built one strand at a time. As many strands as the curve allows are bonded from the end; each further strand is given the shortest sleeve, in steps of a quarter of a metre, that keeps the force of all the strands bonded so far under the curve at every section.

Six strands can grip from the end: their 750 kN, ramped in over the transmission length, stays under the curve. The seventh must wait until half a metre in, and the eighth until a metre. Mirror the schedule at the other end and the beam carries the full 1,000 kN over its middle eight and a half metres, and three quarters of it close to each end.

The sleeves are short, which is the transmission length helping. A strand debonded for half a metre does not reach its full force until 1.25 m from the end, by which point the allowed force has risen by enough to take it.

How deep to put the strands #

The schedule depends on the eccentricity, and the eccentricity is the designer’s choice. Mid-span says go as low as possible: the lower the strands, the fewer are needed to keep the soffit out of tension in service.

The two halves of the figure pull opposite ways. Lowering the strands from 100 mm to 300 mm halves their number at mid-span, from twelve to six, because each is working on a longer lever. But the deeper they go the more tension they put in the top fibre at the ends, where nothing opposes them, and the more must be sleeved. Up to 200 mm no strand needs it; at 220 mm two of eight do; from 260 mm three of seven, and at 300 mm three of six — half.

The dashed line is a quarter of the strands, which is the share of debonded strands that bridge owners and several design standards commonly allow, because a beam whose ends are carried by a few strands has its anchorage, its end shear and its cracking at the ends all depending on those few. Read against it, the economic depth for this beam is 220 mm, where eight strands are needed and two are debonded — exactly a quarter. Go deeper and the mid-span saving is a strand at most while the ends need more sleeving than is allowed. The strand depth that is cheapest at mid-span is decided at the ends.

The transmission length, which does some of it unasked #

The ramp at the start of each strand is not a design choice, but it is doing design work.

With no transmission length at all — a strand reaching its full force at the face of the beam — half the strands would need sleeves. With 1.5 m, only one would. A long transmission length spreads each strand’s force over the length where the allowed force is rising fastest, so that the strand arrives at full force where the beam can take it, and it does for every strand what a sleeve does for one.

The transmission length is not chosen. It depends on the strand’s diameter and surface, the concrete’s strength at release, and whether the strands are released gradually or cut suddenly: EN 1992-1-1 gives it as a multiple of the diameter that falls as the concrete’s bond strength rises, typically 50 to 90 diameters, so 600 to 1,100 mm for a 12.5 mm strand. That makes it an uncertainty in the debonding as much as a help: a beam released early, at a low concrete strength, has a long transmission length and needs fewer sleeves, while one released later, stronger, has a shorter one and needs more. A schedule designed for the first will crack the second.

What the sleeves take away #

Debonding solves transfer by removing prestress from the ends, and the ends are where the beam’s shear is largest.

At an effective depth from the support, where the shear near a support is checked, the debonded beam has 675 kN of prestress where the bonded one would have had 867. In a member without shear reinforcement, EN 1992-1-1 counts the axial compression as adding $0.15\sigma_{cp}$ to the shear stress the concrete can carry, so the sleeves cost 27 kN of the concrete’s shear resistance there — a fifth of what the prestress contributed. And prestress does more for shear than that term: it flattens the diagonal cracks and delays their opening, and the less of it there is at the end, the more the end region behaves like a reinforced concrete beam rather than a prestressed one.

There is a sharper cost below the line the figure draws. A beam’s end is where its bottom reinforcement must be anchored past the support, and in a pretensioned beam the strands are that reinforcement. A sleeved strand anchors nothing until its sleeve ends, so the tie force a diagonal crack near the support asks of the bottom chord must be found in the strands that are bonded from the end — six here, not eight.

One strand at the top instead of two sleeves #

The commonest alternative to sleeving is to put a strand near the top of the section and bond it for the whole length. It does not remove prestress from the ends; it adds compression to the top fibre everywhere, which is where the ends were short of it.

One 125 kN strand 280 mm above the centroid puts $125{,}000/210{,}000 + 125{,}000 \times 280/24.5\times10^6 = 0.60 + 1.43 = 2.03$ N/mm² of compression into the top fibre and takes $1.43 - 0.60 = 0.83$ N/mm² of compression out of the bottom. At the worst section near the end, where the eight bottom strands left the top fibre at −3.31 N/mm², one top strand brings it to −1.28, inside the limit with room to spare; no bottom strand need be sleeved at all.

The cost is at mid-span, in service. With a fifth of the prestress lost, the soffit there sits at −1.68 N/mm² under eight bottom strands, inside its −2.5 limit by 0.82. The top strand takes 0.8 × 0.83 = 0.67 of that margin away, leaving the soffit at −2.35 — still inside, but by 0.15. So one top strand does what the two sleeves do, at the price of most of the mid-span’s service margin, and it keeps the full eight strands’ force at the ends, where the shear and the anchorage want it. On a beam with less margin at mid-span the top strand would need a ninth bottom strand to pay for it, and the comparison becomes one strand against two sleeves.

That is the general shape of the decision. Sleeves cost nothing at mid-span and take force away from the ends; a top strand costs force at mid-span and leaves the ends alone; deflecting strands in the bed costs equipment and leaves both alone. Which one a producer uses is decided as much by the bed, the hold-down hardware and the next beam on the same strands as by any one beam’s stresses.

The arithmetic, by hand #

For the beam drawn, the section has $A = 210{,}000$ mm² and $Z = 300 \times 700^2/6 = 24.5\times10^6$ mm³. At mid-span the self-weight moment is $5.25 \times 12^2/8 = 94.5$ kN·m, which on its own puts 3.86 N/mm² of compression into the top fibre. The eight strands’ 1,000 kN puts $1{,}000{,}000/210{,}000 = 4.76$ of compression into it and $1{,}000{,}000 \times 220/24.5\times10^6 = 8.98$ of tension: net, $4.76 - 8.98 + 3.86 = -0.36$ N/mm². The limit is −2.5, and the top fibre passes.

A metre from the end the self-weight moment is $5.25 \times 1 \times 11/2 = 28.9$ kN·m, worth 1.18 N/mm². With all eight strands at full force there, the top fibre is at $4.76 - 8.98 + 1.18 = -3.04$ N/mm². The allowed force at that section is $(1.18 + 2.5)/(220/24.5\times10^6 - 1/210{,}000) \div 1{,}000 = 872$ kN, so only six strands may be gripping at full force there, with less than one more strand’s worth to spare — which is why the seventh strand’s sleeve stops half a metre in, letting it ramp up as the allowance does, and the eighth’s a metre in.

A rectangle, one level of strands and a fixed transmission length #

The calculation is deliberately small, and each simplification has a direction.

One level of strands. Real pretensioned beams have strands in two or three rows, and some designs add one or two strands near the top to put the top fibre into compression at the ends — a way of cancelling the end tension without debonding at all. A top strand costs prestress at mid-span, where it works against the bottom ones, and is the alternative a designer weighs against sleeves.

Deflected strands. The other alternative is to hold some strands down at the third points of the bed and up at the ends, giving a harped profile like a post-tensioned tendon’s. It keeps the full force at the ends, which the shear and the anchorage want, and costs hold-down devices in the bed and their forces.

A linear transmission. The force in a bonded strand builds up over the transmission length more like a curve than a straight line, and the length itself scatters. A schedule built on the linear ramp is close for a schedule’s purposes, and the sleeves are set to the nearest quarter metre in any case.

No losses at transfer beyond the release. The 125 kN per strand is after the elastic shortening at release; relaxation and creep come later, and they help the transfer condition by reducing the force while making the service condition harder.

Where a sleeve ends, and the bed it was cast on #

The figures cannot show the end of a sleeve. Where a debonded strand begins to grip, its force enters the concrete over a transmission length starting at a point inside the beam, and the concrete round that point is split by the bursting stresses a bonded strand always produces at its anchorage — stresses which, at the end of the beam, the end reinforcement was detailed to carry, and inside the beam nothing was. Staggering the sleeves, as the schedule does, spreads those points out, and rules on debonding often require it for that reason.

They also cannot show the bed. A strand sleeved in a casting bed is sleeved in every beam cast on that bed in that run, because the strands run continuously through the bed from abutment to abutment; a schedule that suits a 12 m beam has to suit the 9 m beam cast after it on the same strands, or the bed has to be re-sleeved between them.

Still open: the beam that is lifted #

Transfer is not the only state in which a pretensioned beam carries its own weight differently from in service. It is lifted out of the bed by loops near its ends, stacked in a yard on bearers that may not be at its ends, and carried on a trailer whose supports overhang. Each of those puts its own-weight moment somewhere other than mid-span — over a lifting point, or hogging over a bearer set in from the end — and the top fibre at the ends, which the sleeves protected at transfer, may be in tension again when the beam is cantilevering off a bearer. Whether a debonding schedule designed for transfer still keeps the ends in their limits when the supports are a metre in from the ends, and how far in the bearers may be before it does not, is the question the yard puts to the bed.