The beam the haulier holds

The beam at the bottom of the stack put the yard’s pretensioned beam under two more of its kind and found that a stack whose bearers land a few millimetres out of line takes from its top fibre relief that its debonding schedule had already spent. The beam was standing still. It ended on the state that follows the yard: the trailer, where the beam is supported at two points chosen by the haulier — a turntable near the front, a bogie at the back — and bounced on them at road speed. Whether a road takes more relief than any stack does, and whether the haulier’s support positions belong on the schedule’s list of states, was the question.

The beam, and the schedule that has nothing to spare #

The beam is the yard’s: 12 m long, 300 × 700 mm, eight strands 220 mm below the centroid, with 5 per cent of its prestress already lost by the time it leaves the yard. Its debonding schedule was drawn for two states — the casting bed, where it is held at its ends, and the yard, on bearers a metre in from each end — and every strand that could be bonded is: five to the ends, three sleeved for 0.5, 1.25 and 1.5 m. On its yard bearers the top fibre’s worst section, 2 m from each end, is at −2.98 N/mm², two hundredths inside the yard’s tension limit of −3.0. That is what a good schedule looks like, as the strands that are sleeved at the ends found: no margin, because margin is prestress the beam’s ends could have had in service.

The top fibre is in tension at transfer because the strands sit below the centroid and bend the beam upward; the beam’s own weight, sagging it between its supports, is what relieves it. Every state that changes the beam’s moment changes that relief. The beam that leaves the bed found the bearers’ inset worth 1.29 N/mm² a metre at mid-span; the stack, 0.13 N/mm² for every beam above landing 100 mm out of line. The trailer changes the moment in two ways at once.

Where the haulier puts it #

A haulier puts the turntable where the trailer’s geometry and the road’s turning circles allow, and the bogie where the beam’s length and the axle loads put it; neither is where the yard’s bearers were, and the two are rarely symmetric. A rear support 2.5 m from the end leaves an overhang that hogs the beam by $5.25 \times 2.5^2/2 = 16.4$ kN·m over the bogie, six times the yard’s 2.6. The front overhang, at 1.5 m, hogs by 5.9.

The haulier’s choices are not arbitrary, which is why they are rarely the yard’s. A turntable has to sit far enough back from the beam’s front end for the tractor to articulate; a bogie is placed so that the beam’s weight divides between the tractor’s axles and the trailer’s in the proportions the law allows, and moving the bogie in from the rear end shifts weight off the tractor and onto the trailer’s axles, which is often where the haulier wants it. Unequal overhangs make unequal reactions: with 1.5 m at the front and 2.5 m at the rear, the bogie carries 35 kN of the beam’s 63 and the turntable 28. Nothing in that reasoning involves the beam’s prestress, and nothing in the yard’s paperwork tells the haulier it should.

The road then adds its bounce. A wheel rising into a bump lifts the beam into its supports and, for the moment, multiplies its weight; a wheel dropping into a hollow lets it lighten. A factor of ±30 per cent on the beam’s weight is a moderate value for a precast beam on a well-maintained road and a modest one on a site access track. The rising wheel takes the rear overhang’s hog to 21.3 kN·m. The falling wheel does something less obvious: it takes the sag between the supports from 31.0 kN·m to 21.7, and the sag was the beam’s relief.

Both directions of the bounce #

On its own yard bearers, with no change of support at all, a 30 per cent bounce takes the top fibre past its limit, to −3.26 N/mm², and it does so 9.8 m from the front end — between the supports, not over them. That is the falling wheel. A beam whose weight is momentarily 70 per cent of itself sags 30 per cent less, and the top fibre loses 30 per cent of the relief the self-weight gave it at every section between the bearers. The schedule had been cut to the limit at more than one section, and this one is reached first.

The bounce matters in both directions, and the direction that does the damage to a pretensioned beam is the one that lightens it. A check that applies a dynamic factor of 1.3 to the beam’s weight and nothing else is checking the wrong direction for the top fibre between the supports: the beam must also be checked at 0.7 of its weight, which is the state in which its prestress has least to work against. That is the arithmetic the weight that was dropped applied to an impact, read for a structure whose weight is on its side.

The rear overhang is most of it #

Taken one at a time, the three departures from the yard are not equal. The bounce costs 0.28 N/mm², about what two beams stacked 100 mm out of line cost the bottom of a stack. Moving the front support from 1 m to 1.5 m costs 0.61. Moving the rear support from 1 m to 2.5 m costs 1.70 — more than the three-beam stack at its worst, and the largest single loss of relief in any state these essays have checked. Together, the three take the top fibre to −4.88, sixty per cent past its limit.

The rear overhang dominates because a cantilever’s hog grows as the square of its length. Moved from 1 m to 2.5 m, the overhang’s moment at the bogie grows six and a quarter times, from 2.6 to 16.4 kN·m. And the section where that hog peaks is at full prestress: 2.5 m from the end every strand, the three sleeved ones included, is bonded and fully transferred. The schedule protected the beam’s first 1.5 m against its own end region’s lack of relief; it has nothing at 2.5 m, because in the bed and the yard nothing ever hogged the beam there.

The road against the stack #

The earlier essay’s question can now be answered directly. On the yard’s own bearers, a road’s bounce takes about as much as a careless stack: a 30 per cent bounce is the stack landing 100 mm out of line, a 50 per cent bounce is between that and the stack at 250 mm. Both are of a size a schedule with a little margin could absorb. The trailer’s support positions are not. Standing perfectly still on the haulier’s supports, the beam is already 0.70 N/mm² further past the limit than the worst stack, and the bounce adds only 0.2 on top.

So the road is not the large term; the haulier is. The bounce is a property of the road and the trailer’s suspension, and a designer can only allow for it. The support positions are a choice somebody makes, and until somebody tells the haulier what they are allowed to be, that somebody is the haulier.

Close to the ends, the bounce; further in, the overhang #

The two mechanisms trade places as the rear support moves in. With the rear support at the very end, there is no rear overhang to hog and the beam spans its whole length less the front overhang; its worst section is between the supports, where the falling wheel takes its relief, at −3.62 with the front support at 1.5 m. As the rear support moves in, the span shortens and the sag falls, but slowly; the rear overhang’s hog grows as its square. From about 1.7 m in, the hog wins, and the worst section jumps to the rear support.

A haulier who brings a support in from the end, to shorten the trailer or spread its axle loads, is therefore not trading one risk for an equal one. Up to about a metre and a half in, the beam is governed by the road and loses little; past that, it is governed by the overhang and loses a lot, and quickly.

Where a haulier may put the supports #

The map is the instruction a haulier needs, and it is narrow. Standing still, the beam can be carried anywhere the yard could carry it — about a metre from each end — because that is what the schedule was drawn for. Bouncing at 30 per cent, the front support must be within half a metre of its end and the rear within 0.74 m of its own. With the front at the yard’s metre, no rear position works at all: the falling wheel has already spent the margin between the supports, and any overhang at the rear spends what is left near the bogie.

The road’s share of the restriction can be bought back more cheaply than the haulier’s. Halving the bounce to 15 per cent — a lower speed over the bad stretches, a trailer with softer suspension, a beam bedded on rubber rather than timber — lets the rear support stand 0.94 m from its end instead of 0.74 with the front at 0.5 m. Bringing the front support to the very end, if the trailer allows it, buys a little more. None of these comes near the 2.5 m a haulier might choose unasked.

The bottom fibre, which the prestress compresses hardest, never governs: at its worst, on the trailer’s supports with the rising wheel, it carries 13.9 N/mm² against a compression limit of 18. Every number here is a top-fibre number, for the same reason as in the yard: that fibre started two hundredths from its limit.

The trailer is another state, and it belongs on the schedule’s list — with its support positions written down and its bounce stated. A schedule drawn for the yard can be carried on a trailer only with the supports close to its ends; a beam that must travel on supports further in needs a schedule drawn for them, with the sleeves extended or a strand fewer bonded near the ends, and the price paid in the end region’s prestress in service, as four inequalities and a wedge found every such state is paid for.

A schedule drawn for the road #

The alternative to restricting the haulier is to give the beam a schedule that expects the road. The schedule is built as the yard essays built theirs, strand by strand, each sleeve as short as the limits allow, with two more states to satisfy: the trailer’s supports at some distance from each end, once with the beam’s weight raised by the bounce and once with it lowered.

With the trailer’s supports a metre from each end and a bounce of 30 per cent, the schedule needs 3.75 m of sleeve on each half, against the yard’s 3.25: the same three strands, the longest sleeved to 2 m instead of 1.5. With the supports 1.5 m from each end it needs 6.50 m — the three sleeves at 1.25, 2.0 and 3.25 m, twice the yard’s — and with them 2 m from each end there is no schedule at all. The falling wheel takes so much relief between the supports, and the overhangs so much over them, that no arrangement of sleeves can carry both: the section is in the wrong place for its prestress whatever is sleeved.

So a beam that must travel on supports 2 m in needs a different beam — fewer strands, a strand moved up, a deeper section — and that decision belongs to whoever draws the beam, not to the haulier or the yard. It is a decision only somebody who knows where the trailer will hold the beam can make.

The rear overhang, by hand #

The beam weighs $25 \times 0.3 \times 0.7 = 5.25$ kN/m. A rear overhang of 2.5 m hogs it at the bogie by $5.25 \times 2.5^2/2 = 16.4$ kN·m, against the yard’s $5.25 \times 1^2/2 = 2.6$. The section’s modulus is $300 \times 700^2/6 = 24.5 \times 10^6$ mm³, so the extra hog of 13.8 kN·m takes $13.8 \times 10^6/24.5 \times 10^6 = 0.56$ N/mm² from the top fibre at the bogie relative to the yard. But the bogie’s section is at full prestress, where the strands put $-(1/A - e/Z)$ times their force into the top fibre, about −4.0 N/mm², and the yard’s sag there, which offset most of it, is replaced by a hog: the top fibre arrives at −4.68.

The falling wheel’s loss between the yard’s bearers is simpler. The self-weight sag at mid-span on bearers 1 m in is $5.25 \times 12^2/8 - 5.25 \times 1 \times 12/2 = 63.0 - 31.5 = 31.5$ kN·m, worth 1.29 N/mm² of relief at the top fibre; 30 per cent of it is 0.39 N/mm². The worst section, a little off mid-span where the schedule left least, loses 0.28 of it, which is the figure’s −2.98 to −3.26.

What the road model leaves out #

The bounce is a static factor. A beam on a trailer is a flexible body on a sprung vehicle, and the moments the road puts into it depend on the beam’s own frequencies against the trailer’s; a long, slender beam whose first mode is near the trailer’s bounce can see more than the factor, and a stiff one less. The factor here is a statement of what the road does on average to a rigid beam.

The supports are points and the beam is not twisted. A turntable and a bogie support the beam at single points, and a trailer that rolls in a corner or crosses a camber twists the beam about its own axis — a torsion the yard never applied, which hung from above and still unstable met in a lift. A beam on a trailer is also laterally unbraced over its whole length; its top flange, in tension here, is the one a sideways buckle would not need.

The concrete is the yard’s. By the time a beam travels it is older, stronger and has lost more prestress than at transfer, and both widen the margin, as the strength it had on the day and the prestress the member takes back found. A beam shipped young, straight from the bed to a site that needs it tomorrow, has neither.

Packing, straps and the driver #

They cannot show the timber packing under the beam on the trailer’s bolsters, which decides where the support really acts and is placed by the driver. They cannot show the straps that hold the beam down, whose tension adds to its weight at the supports and can crush its top corners. And they cannot show the camber — tens of millimetres on a beam this long — that the camber that lowers the hook found grows while it is stored, and which a haulier sees as a beam that does not sit flat on the trailer and packs accordingly.

Still open: the beam that is lifted off the trailer #

The trailer delivers the beam to a crane, and the crane lifts it by loops cast near its ends, swings it and sets it down on its bearings — three more states, each with its own support and its own dynamics. A lift by loops is the bed’s supports turned upside down, with the beam hanging from them; a swing adds a sideways inertia the beam’s weak axis has to carry; and setting it down on bearings that are not level puts a twist into a beam whose top fibre has no margin left. Whether the last few metres of a beam’s journey, from the trailer to its bearings, take more from its schedule than the road did — and whether the loops’ positions, fixed in the bed, can be chosen to make the lift the gentlest state rather than the harshest — is the question the trailer hands to the crane.