The beam hung from its loops

The beam the haulier holds followed a pretensioned beam out of the yard: 12 m long, 300 by 700 mm, eight strands debonded near its ends so that at transfer, and on bearers 1 m in from each end in the yard, the top fibre the strands put in tension stays inside its limit. On a trailer the beam is supported where the haulier puts it, 1.5 m from one end and 2.5 m from the other, and bounced by the road; a falling wheel takes away the sag that relieves the top fibre, and the long rear overhang hogs. The top fibre reached −4.88 N/mm² against a limit of −3.0, and the rear support had to stay within three-quarters of a metre of the bearer position, or the strands needed more debonding.

That essay ended at the trailer, and the beam’s journey does not. A crane lifts it by loops cast near its ends, swings it across to the pier, and sets it down on its bearings — three more states, each with its own supports and its own accelerations. A lift by loops is the yard’s bearers turned upside down, with the beam hanging from them; a slew adds a sideways inertia that the beam’s weak axis has to carry; and bearings that are not level tilt it. Whether the last few metres take more from the schedule than the road did, and whether the loops, fixed in the bed, can be placed so that the lift is the gentlest state rather than the harshest, was the question.

Five states, one schedule #

The beam is checked as the earlier essays checked it, with one change: the top fibre is read at each of the section’s four corners rather than across its width, because a slew and a tilt bend the beam about its weak axis and put one top corner in more tension than the other. At every section and every corner the stress is the strands’ compression and their eccentricity’s moment, the self-weight moment for the state, any axial force the slings add, and the weak-axis bending, with tension negative; the limit for the stored concrete is −3.0 N/mm².

The answer to the first half of the question is in the bars. The trailer, at −4.88, is the worst state the beam passes through, and by a wide margin. Lifted by loops at the bearer positions, the beam hangs at −2.97 — the yard again, to the hundredth. Slewed it reaches −3.13, a few per cent past its limit. Set down it relaxes to −1.79, the best of the five. The last few metres of the journey take less from the schedule than the road did, except for the slew, and the slew can be designed away.

Hung from its loops, the beam is on its bearers #

A beam hanging from two loops is statically the same beam as one resting on two bearers at the same positions: the same overhangs hogging, the same span sagging between. The loops at 1 m reproduce the yard’s moment diagram, and the debonding schedule drawn for the yard holds it there. The two curves differ only in detail, because a hanging beam has two more things acting on it than a stored one.

The crane’s snatch — the moment the slings come taut and the beam leaves the trailer, and its rebound — factors the beam’s weight by ±20 per cent. That changes the overhangs’ hogging and the span’s sagging together, and since the overhangs are only a metre long, their hogging is small: the snatch moves the worst top fibre by a hundredth. The second is the slings’ inclination, which pulls the loops inwards as well as up.

The slings’ squeeze #

Two slings from one hook at an angle α from the horizontal pull each loop inwards by half the beam’s weight times cot α. The loops are cast in the top face, so between them the beam carries that pull as a compression above its centroid, which compresses the top fibre twice — once as an axial force and once as the moment of its eccentricity. With the loops 1 m in, the squeeze from slings at 60° or 30° keeps the worst section at −2.97; with vertical slings from a spreader beam it is −3.16, past the limit. Inclined slings are, for this beam, kinder than a spreader — the reverse of the usual advice, which is written for slender steel members that the squeeze would buckle.

The squeeze is limited by where it acts. It runs only between the loops, and with the loops 2 m in, the worst section is over the overhang, where the hogging of a 2 m cantilever has outgrown the strands’ relief and the squeeze has not yet arrived. There the angle makes no difference at all: −4.35 at every inclination. The slings can help a section only if it lies between them.

Where the loops should be #

The loops’ position is fixed when the beam is cast, by where they are tied into the cage before the bed is filled. That is the decision the question asked about, and it can be read off one figure.

Moving the loops in from the ends lengthens the overhangs, and the top fibre over them falls steadily: −2.40 slewed with the loops at the very ends, −3.13 at 1 m, −4.97 at 2.5 m — worse than the trailer. With the loops at the ends the beam hangs as it lay in the bed, supported at its ends, which is the one state the schedule was drawn to satisfy at the strands’ full transfer force; in the yard, with the force reduced by losses, it has margin to spare. Loops near the ends make the lift the gentlest state of the whole journey. At the bearers’ 1 m it is the yard’s state exactly, inside the limit standing still and just outside it slewed. Inward of about 0.8 m a slew of a tenth of a g is enough to cross the limit.

Lifting loops are commonly placed a short distance in from the ends — far enough for the loop’s anchorage to develop in sound concrete beyond the end zone, and often at the same positions as the yard’s bearers so that one support condition serves both. The figure gives that convention a price: each half-metre the loops move in from the end costs the slewed top fibre about 0.4 N/mm². The strands that are sleeved at the ends found that the bed state governs the debonding near the ends; loops placed there ask nothing of the schedule that the bed has not already asked.

When the snatch matters #

The crane’s snatch hardly moved the beam with its loops at the bearers, and the figure’s lighter curve shows where it does. With the loops at the very ends, the beam still hanging is at −1.42 N/mm²; with the snatch’s rebound, when the beam momentarily weighs a fifth less, it is −1.85, and the worst section has moved from near the end to 9.75 m along. A beam hung from its ends relies on its own sag to relieve its top fibre across most of its length, and a rebound takes a fifth of that relief away at the moment the slings go slack. With the loops at 1 m, −2.95 still and −2.97 with the snatch: the worst section is over a loop, where the overhang’s hogging and the span’s sagging both scale with the weight and nearly cancel in their effect. With the loops 2.5 m in, −4.68 and −4.81.

The snatch costs most where the beam leans on its sag, and least where its worst section is over a support. That makes the ends, which are the best place for the loops when the beam hangs still, a place where the lift’s dynamics have to be watched: a beam picked up abruptly from loops at its ends loses its margin faster than one picked up from loops a little in. The lift plan and the loops’ position are one decision.

A slew bends the weak axis #

A crane that slews, or a trolley that brakes, accelerates the hanging beam sideways, and the slings lean to hold it, so the beam is bent about its weak axis by its own inertia with the loops as supports. For a 300 by 700 mm rectangle the weak-axis modulus is 0.43 of the strong one, so a sideways acceleration of a tenth of a g stresses the corners as much as 23 per cent of the beam’s weight would upright. The top corner on the inside of the swing loses that much of its compression — the same corner-by-corner reading that two moments and a neutral axis that obeys neither found for any section bent about both its axes at once, where the worst point is a corner and the worst corner is decided by the signs.

With the loops at the bearers, a slew of 0.1 g takes the worst top fibre from −2.97 to −3.13, and 0.2 g to −3.37. With the loops 0.5 m in, from −2.36 to −2.76 and −3.16. The slew matters most where the beam already has little margin and least where the overhangs dominate: with the loops 1.5 m in the beam is outside its limit before it slews at all. A slew of a tenth of a g is gentle by a crane’s standards — a slew started or stopped over a few seconds — and a lift plan that limits it is the cheapest of the three remedies, after the loops’ position and before more debonding.

Set down crooked #

The last state is the beam on its bearings, and they are not quite level across its width. A slope tilts the beam, and its weight acquires a sideways component that bends the weak axis — at 1 per cent, a hundredth of the weight; at 5 per cent, a twentieth. It costs almost nothing: −1.77 N/mm² level, −1.79 at 1 per cent and −1.95 at 5. On its bearings the beam is supported at its ends, the strong-axis sag compresses its top across the middle where the sideways bending is largest, and near the ends, where the top is least compressed, the sideways bending has fallen to nothing. The beam’s harshest day was on the road, and its gentlest is the day it arrives.

A tilted beam has other concerns than its top fibre — a bearing loaded on one edge, a beam that may roll if it is slender — and the beam at the bottom of the stack found that loads landing out of line are what a pretensioned beam’s top cannot afford. Here the load out of line is the beam’s own, and the section’s sag relieves it.

Which state governs which part of the beam #

Set side by side, the five states divide the beam between them. The first three-quarters of a metre at each end is governed by the bed: there the strands’ force is still building up from the end, the self-weight moment is nearly nothing in every state, and only the bed’s transfer force at its full value — before any losses — puts the top fibre near its limit. The next metre or two is governed by whichever state hogs it most: the yard’s bearers at 1 m, the trailer’s 2.5 m overhang, loops placed inward of the bearers. The middle of the beam is never critical; every state sags it, and the sag compresses the top.

That is what makes a debonding schedule a list of states rather than a single calculation, as the beam that leaves the bed began to show when it moved the supports in from the ends. Each state is a constraint on the strands’ force at each section, and the schedule must lie inside all of them — the same picture four inequalities and a wedge drew for one section’s prestress, extended along the beam and through time. The journey’s states can be put in order: the trailer is the binding one, the bed binds at the ends, the lift and the set-down bind nowhere if the loops are where the bed or the yard held the beam. A precaster who controls where the trailer’s supports go controls the schedule.

By hand, at the loops #

With the loops 1 m in, the overhang’s hogging at a loop is $w_0 a^2/2 = 5.25 \times 1^2/2 = 2.6$ kN·m, against the strands’ eccentric force, which at 1 m from the end has developed only partly: the end strands are bonded, the sleeved ones not yet. The top fibre there is $P/A - Pe/Z + M/Z$ with $A = 210{,}000$ mm² and $Z = 24.5 \times 10^6$ mm³: the strands’ share dominates and the 2.6 kN·m adds $-0.11$ N/mm². A slew of 0.1 g bends the weak axis by a tenth of the same diagram — 0.26 kN·m at the loop and more between them — over $Z_y = 10.5 \times 10^6$ mm³.

The slings at 60° pull each loop inwards by $W/2 \times \cot 60° = 63/2 \times 0.577 = 18$ kN. Between the loops that is a compression of 18 kN at 350 mm above the centroid, adding $18{,}000/210{,}000 + 18{,}000 \times 350/24.5 \times 10^6 = 0.09 + 0.26 = 0.35$ N/mm² to the top fibre between them. That is enough to lift the sections just inside the loops, which the spreader leaves at −3.16, above the loop itself, where the overhang’s hogging peaks at −2.97 — and it is nothing at all where the worst section lies outside the loops.

What the check leaves out #

The loops are points. A real loop is a strand or bar tied into the beam’s cage and anchored over a length; its force arrives spread over that length and partly as a local tension in the concrete around it, which the loop’s own design checks separately. That anchorage is also what keeps loops from being placed right at the ends, where the figures say they are best: a loop needs a development length of sound concrete around it, as the force that arrives along a length found any bar does, and the last few hundred millimetres of a pretensioned beam are crowded with the strands’ own transmission zone. The 0.8 m inside which a slew is harmless is a window, not a point, and the loop’s anchorage has to fit inside it.

The crane’s accelerations are given, not computed. The snatch of ±20 per cent and the slew of 0.1 g are round values for a careful lift; a lift that lets the beam drop onto its slings, or a slew stopped abruptly, can exceed both, and the bounce figures of the trailer essay show how quickly a top fibre with no margin is crossed when they do.

The beam is a rectangle. Its weak axis is 0.43 as stiff as its strong one; a slender bulb-tee or I-beam’s is a fifth or a tenth, so the slew and the tilt would weigh far more for it, and a slender beam hanging from loops can also roll, which the beam that fails sideways found is a buckling question rather than a stress one.

Still open: the beam that is made continuous over its piers #

Every state here is the beam alone, from the bed to its bearings. On the bearings it is joined to its neighbours: a slab is cast across it, and over the piers a continuity joint ties its end to the next beam’s, so that the deck becomes a continuous structure — and the beam’s top fibre near its end, which every stage so far has protected, is suddenly over a support carrying a hogging moment from everything placed afterwards. Whether the debonding that kept the end’s top fibre in compression through the bed, the yard, the road and the lift leaves enough strand at the end to carry the continuity joint’s hogging, or whether the end that was made weak on purpose for the journey is the end the finished bridge needs strongest, is the question the bearings hand to the joint.