The beam that leaves the bed

The strands that are sleeved at the ends dealt with the moment a pretensioned beam is born. Its straight strands are cut free of the casting bed’s abutments and squeeze the concrete, and the top fibre, pulled into tension by the strands’ eccentricity, is relieved only by the beam’s own weight bending it the other way. At mid-span that relief is largest. Near the ends it falls to nothing, and the top fibre cracks unless some strands are sleeved there so that they do not grip. For a 12 m beam 300 × 700 mm with eight 125 kN strands at 220 mm below its centroid, two strands sleeved for half a metre and a metre did it.

Its last section named the next state the beam passes through. Transfer is not the only time it 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, carried on a trailer whose supports overhang. The essay asked whether a schedule designed for transfer still keeps the ends inside their limits when the supports are a metre in from the ends, and how far in the bearers may be before it does not.

The relief that is taken away everywhere #

The beam’s self-weight is 5.25 kN a metre. In the bed it is held at its ends and its moment is the familiar parabola, 94.5 kN·m at mid-span. Lifted by loops or stood on bearers set in by $a$ from each end, its moment changes in two ways.

Over the overhangs the beam hogs: a cantilever $a$ long under its own weight, $w a^2/2$ at the bearer, which for a metre’s inset is only 2.6 kN·m. That is the change everyone expects, and it is small.

The other change is not small and is easy to miss. Between the bearers, each reaction acts $a$ closer to mid-span than the bed’s did, and the moment there is the bed’s moment less a constant $w a L/2$ — 31.5 kN·m for each metre of inset, the same at every section between the bearers. The moment is not altered near the ends and kept at mid-span; it is lowered uniformly along the whole length.

For the top fibre that is a uniform loss of relief: $waL/2Z$, 1.29 N/mm² for every metre the bearers are moved in, at every section between them. The schedule that kept the top fibre inside its limit was built section by section against the relief each section had in the bed. In the yard every section has 1.29 less per metre.

A schedule with nothing to spare #

The bed’s schedule was built strand by strand, sleeving each just far enough that the force bonded so far stays under what each section may carry. A schedule built that way leaves almost no margin by construction: in the bed its worst top fibre is −2.43 N/mm² against a limit of −2.5. On bearers half a metre in, the same section is at −3.08. A metre in, −3.71, at 1.25 m from the end, where the strand sleeved for a metre has just begun to grip and the overhang’s hogging and the uniform loss of relief add.

In transfer’s own conditions — the prestress at transfer, the young concrete’s tension limit — the bearers may come in 51 mm. That is not a yard practice anyone could follow.

Why a good schedule has no margin #

The 0.07 N/mm² the bed’s schedule leaves is not carelessness. It is what a good schedule looks like. Debonding is designed by the same arithmetic the earlier essay used: bond as many strands from the end as the end can take, then sleeve each further strand just far enough that the force bonded so far stays under what every section may carry. Each strand is sleeved by the least length that works, in steps of a quarter-metre, because every extra quarter-metre of sleeve is prestress lost to the end of the beam in service. A schedule built that way meets its limit somewhere by construction; if it did not, one of its sleeves could have been shorter.

So the transfer check is not a check with a margin that the yard can spend. It is the design condition, met as closely as the steps allow. Anything that removes relief from the top fibre after transfer — a bearer moved in, a lift with a jolt, a beam stacked on a point load — is spending relief that was never there to spare. Four inequalities and a wedge put the point generally: a prestressed section is designed to sit on the edge of the region its four limits allow, because that is where it uses least prestress, and any state not among the four inequalities it was designed to can move it off.

The remedy is not a bigger margin everywhere; it is putting the yard’s state among the inequalities. The schedule then sits on the edge of a smaller region, the one where both the bed and the yard are satisfied, and it costs exactly the extra sleeving the next section counts.

The yard’s own conditions #

The yard is not transfer, and two things move in the beam’s favour by the time it is stored. The strands have lost some of their force — to the concrete’s early creep and shrinkage and to their own relaxation, which the prestress the member takes back worked through — and less prestress means less tension at the top. And the concrete is older and stronger; the strength it had on the day traced how quickly a young concrete gains, and its tensile strength rises with it, so a limit of −3.0 N/mm² in the yard is reasonable where −2.5 was at transfer.

With 5 per cent of the prestress gone and that limit, the bearers may come in 0.57 m. That is an inset a careful yard can work to. A metre — a figure a yard might well choose, keeping the bearers clear of the end blocks and the bearing pads — is still too far: at a metre the top fibre is past −3.0 near 1.25 m from the end.

So the answer to the earlier essay’s question, for this beam, is no: a schedule built for transfer does not keep the ends inside their limits with the bearers a metre in, and the yard has to be told so. It is a figure that appears on no drawing unless somebody computes it, and the person placing the bearers will choose where they go by where the beam’s end blocks and lifting points are, not by where its top fibre is.

What a metre’s inset asks of the schedule #

If the yard needs bearers a metre in, the schedule can be designed for it. The beam has to pass the bed first, so the schedule must satisfy both states at once: the bed, with the beam held at its ends and the full transfer prestress, and the yard, with the bearers in and the yard’s conditions.

The bed alone needs two strands sleeved, for 0.50 and 1.00 m. The bed and a metre’s inset together need three, for 0.50, 1.25 and 1.50 m: a third strand, and the second strand’s sleeve lengthened by a quarter of a metre. That is the price of a metre in the yard, and it is paid in service. The crack that never reached forty-five degrees found that the prestress near a support is what keeps a pretensioned beam’s web uncracked in shear; every metre of sleeve is prestress the end does not have when the beam is in its bridge, carrying its traffic, with its shear largest exactly there.

The count rises in steps as the bearers move in. In the yard’s conditions, two strands serve up to 0.7 m and three from there to 2.2 m; with transfer’s conditions applied to the yard too, the count rises sooner, to three at 0.2 m and four at 0.6 m. A common owner’s rule allows no more than a quarter of the strands debonded — two of these eight — so even the yard’s line passes the rule at 0.7 m, and transfer’s at 0.2.

Each line stops. Past 1.6 m in transfer’s conditions, or 2.2 m in the yard’s, no schedule fits at all.

The section no debonding can reach #

The reason the lines stop is the uniform loss of relief. Mid-span needs every strand — that is what the strands are for — and its top fibre was at −0.36 N/mm² in the bed, comfortably inside the limit. Every metre of inset takes 1.29 N/mm² from it as from every other section between the bearers. At 1.66 m of inset it reaches transfer’s limit; at 2.22 m, the yard’s. Debonding works only near the ends, where strands can be spared. At mid-span none can, so past those insets the beam cannot be stood on its bearers however its ends are detailed.

That is the general shape of the answer. The schedule protects the ends against a moment the ends do not have; it cannot protect the middle against a moment the middle loses. Four inequalities and a wedge drew the four stress limits as a wedge of permissible prestress and eccentricity for each section, and moving the bearers in shrinks every section’s wedge from the top at once.

The bottom fibre, which the yard also costs #

The top fibre is not the only one the bearers move. The bottom fibre at transfer is in compression, the strands’ eccentric force squeezing it and the self-weight sagging relieving it; the limit there is the young concrete’s compressive strength, 18 N/mm² for this beam. In the bed the worst bottom fibre is 11.8 N/mm², 1.75 m from the end. The uniform loss of relief that costs the top fibre 1.29 N/mm² a metre costs the bottom fibre the same in the other direction, and on bearers a metre in it reaches 13.1, two metres in 14.2.

For this beam that is never the limit — the top fibre fails long before the bottom gets near 18 — because the strands are well below the centroid and the section is deep. On a shallower beam, or one prestressed harder to carry a heavier service load, the bottom fibre’s compression at transfer is closer to its limit, and moving the bearers in takes it there from below while the top fibre is being taken there from above. The section’s wedge of permissible states shrinks from both sides at once.

Where the bearers would do no harm #

The bed itself shows where a pretensioned beam is happiest to be held: at its ends, where the bed held it and where the schedule was designed. Bearers right under the end blocks would leave every number as it was at transfer, with the losses and the stronger concrete as a margin. The reasons yards do not do this are practical. The end blocks carry the bearing pads that will sit on the bridge’s bearings, the beam’s camber lifts its middle and makes an end bearer rock, and a beam stacked on bearers at its very ends has nowhere to put a lifting strop.

So the useful number for a yard is not “at the ends” but how far in is allowed. For this beam, with the bed’s schedule, it is 0.57 m once losses and the yard’s limit are counted, a figure that ought to be on the beam’s drawing next to its lifting points. Loaded twice over before it is a month old found that the loads a slab meets during construction are the ones nobody drew; the bearer’s inset under a pretensioned beam is the same kind of number, decided in the yard and invisible from the design office.

The day it is lifted #

The yard’s conditions are a kindness the beam earns with time. The strands lose force over weeks and months, and the concrete’s tension limit grows over days. A beam lifted from the bed the morning after its strands are cut has neither: it is in transfer’s conditions, and its loops may be set in by 51 mm. A precast works that strips its beds daily lifts every beam on the day after casting, and that is the state the dashed lines describe.

That makes the lifting loops the critical position, not the bearers. A loop is set where the beam can be picked up safely and kept stable, which hung from above and still unstable found pushes the loops in from the ends to keep a slender beam from rolling. The further in a loop goes for stability, the more relief the top fibre loses, on the youngest concrete the beam will ever have.

Lifting, and the trailer #

A lifting loop is a bearer that moves. Lifting loops are set in from the ends for the same reasons bearers are, and the statics are the same, with one addition: a beam swung off the bed is accelerated, and its self-weight moment is multiplied by whatever dynamic factor the lift imposes. A factor of 1.2 on the self-weight multiplies every self-weight moment by the same fifth. Over the overhangs that is more hogging on a top fibre already short of relief; between the loops it is more sagging, which helps the top fibre where the beam sags a good deal and hardly at all near the loops, where it sags little. The sections just inside the loops — the ones that set the limit — gain least and lose most, so a loop’s inset is worth less than the same inset on a bearer.

A trailer’s supports are the extreme case. A beam carried on a bolster at one end and a dolly some way in from the other has an overhang at one end of perhaps a fifth of its length, with the road’s bumps adding their own dynamic factor. The camber that lowers the hook dealt with the stability of a long girder hanging from its loops; the stresses in a short pretensioned beam on its transport are a different check with the same lesson, that the states between casting and service are each a load case of their own.

What the calculation takes for granted #

The section is a rectangle. Real pretensioned beams are I-sections or inverted tees, with a wide top flange whose section modulus at the top differs from the bottom’s. The uniform loss of relief is the same moment over a different modulus; the shape of the answer does not change.

The losses are a single share. Five per cent of the prestress is taken off uniformly between transfer and storage. Losses grow with time, so a beam stored for months before it is used is better off than one lifted in the first days — and a beam lifted the day after transfer is in transfer’s conditions, the dashed lines.

The strands’ force ramps up linearly over the transmission length, as in the earlier essay. A sleeve’s end is where the strand starts to grip, and the force then builds over 750 mm. The worst top fibre on bearers sits inside one of those ramps, so the ramp’s shape and length move its position and its value slightly.

The bearers are rigid and level. A beam on bearers at different heights is on a twisted support, and a beam stacked on another beam is carried at whatever points the lower beam’s camber puts under it.

Still open: the beam stacked on another #

Yards stack beams to save space, each beam on bearers resting on the beam below. The lower beam then carries its own weight on its bearers and the upper beam’s weight through two point loads on its top flange, wherever the upper beam’s bearers happen to be — usually set in from the ends, often not directly over the lower beam’s own bearers. A point load set in from the bearers below adds sagging between them and relieves the top fibre there, but it adds its own hogging over the overhangs if it lands outside them, on a top fibre already at its limit. Whether a stack of identical beams can be built at all with bearers a metre in, and whether the lowest beam in a stack of three needs a schedule of its own, is the question the yard’s space puts to the yard’s bearers.