The beam at the bottom of the stack

The beam that leaves the bed took a pretensioned beam out of its casting bed and stood it on yard bearers a metre in from each end. The bearers took away the relief its own weight had given its top fibre — not only near the ends, where the overhangs hog, but uniformly between the bearers, by 1.29 N/mm² for every metre of inset — and the debonding schedule that had been drawn for the bed, with nothing to spare, had to be redrawn for the yard. The redrawn schedule has nothing to spare either. That is what a good schedule looks like: every strand that can be bonded is, and the last sleeve is as short as the limit allows.

A yard rarely stores beams one deep. Space is the yard’s scarcest resource, so beams are stacked: each on a pair of bearers — timbers or steel stools — standing on the top flange of the beam below. The bottom beam then carries its own weight on its own bearers and the weight of every beam above through two point loads on its top, wherever the bearers of the beam above happen to stand. This essay asks what those point loads do to a beam whose schedule has nothing to spare, and how carefully a yard has to place them.

The weight above goes wherever the bearers put it #

The beam is the yard essay’s: 12 m long, 300 × 700 mm, with eight strands 220 mm below its centroid, released at transfer and stored with 5 per cent of its prestress already lost to the yard’s limit of −3.0 N/mm² on the top fibre. Its own weight is 5.25 kN a metre, so each of its bearers carries 31.5 kN, and each beam stacked on it adds another 31.5 kN at each of the two points where that beam’s bearers stand.

If those points are directly over the bottom beam’s own bearers, the added load passes straight through the beam into its bearers and puts no bending into it at all. Each point load is a force on top of the beam and an equal force under it, and the beam between them is not asked to carry anything sideways. That is the ideal stack, and it costs the bottom beam nothing.

If the points land outside the bottom beam’s bearers, towards its ends, they sit on its overhangs. Each is a cantilever load: it hogs the overhang from nothing at the load to its force times its distance at the bearer. And between the bearers the hog stays — the same moment at every section, because the two point loads and the two reactions they add form a pair of couples that the beam carries uniformly from one bearer to the other. With two beams above landing 250 mm outside, that is 63 kN × 0.25 m = 15.8 kN·m of hog, everywhere between the bearers.

If the points land inside the bearers, the same arithmetic gives the same moment with the other sign: the stack sags the beam uniformly between the loads. This is the mechanism the yard essay found for the beam’s own bearers, turned round: there, moving the bearers in took a constant from the sagging moment everywhere; here, moving the load out adds a constant hog everywhere.

A schedule with no room for the beams above #

The hero figure shows what that hog does to the top fibre of a beam debonded for the yard.

Lined up, the stack leaves the top fibre exactly as the yard left it, at −2.98 N/mm², two hundredths inside the limit at two metres from the end, where the last sleeve ends. Landing 100 mm outside its bearers, the stack takes 0.26 N/mm² from every section between them, and the worst section is at −3.24. At 250 mm, −3.63.

The loss is uniform, so the debonding schedule cannot reach it where it matters most. A sleeve moves prestress away from the ends, where the top fibre had too little relief; it does nothing at mid-span, where every strand is bonded and the relief is what the self-weight sag gives. The stack’s hog lowers the top fibre at mid-span as much as at the end of the last sleeve. It lowers it everywhere, and the schedule was cut to the limit at more than one place.

Each beam above costs the top fibre 0.13 N/mm² for every 100 mm its bearers land outside the bottom beam’s. That is the beam’s reaction times the offset over the section’s modulus: 31.5 kN × 0.1 m / 24.5 × 10⁶ mm³.

Millimetres #

How far out of line can a stack land on a beam whose schedule left two hundredths to spare? The next figure answers it for stacks of up to six.

With one beam stacked on it, the bottom beam can take the upper bearers 13 mm outside its own. With two, 7 mm; with three, 4. A yard’s bearers are timbers placed by hand on a beam whose top is not marked where its bearers are underneath; the timbers are laid with a tolerance of tens of millimetres at best. A schedule built to the limit for the yard has no room for a stack that is placed by eye.

That is not a defect of the schedule but a consequence of what makes it good. The strands that are sleeved at the ends built the schedule strand by strand, each sleeve as short as it can be, and found that a schedule with margin is a schedule with prestress the beam’s ends could have had in service. Every state a schedule must satisfy has to be named before it is cut, and the yard essay named the bearers. A stack is another state, and it has to be named as well — with a tolerance in it.

Moving the bearers above inward #

There is an obvious way to place a stack safely without measuring anything: set every beam’s bearers a little further in than those of the beam below, so that the stack’s weight always lands inside the lower beam’s bearers and sags it. The bottom beam’s top fibre would then gain relief from the stack rather than lose it.

It does not work, for two reasons, and the figure shows both. The bottom beam gains almost nothing: its worst section improves from −2.98 to −2.95 N/mm² and stays there however far in the load moves. Once the stack’s sag has relieved the sections between its bearers, the worst section is over the bearer itself, where the beam’s own overhang hogs it and a load landing inside the bearers does not reach.

And the beam above pays for it. Its bearers are now further in than the yard’s metre, and it loses relief by the yard essay’s 1.29 N/mm² for every metre of its own inset. Moved 500 mm in, it is at −3.63. Every beam in the stack but the bottom one has had its own inset increased, and every one of those was cut to the yard’s metre. A stack whose bearers step inward protects the bottom beam by a few hundredths and fails every beam above it.

The only placement that costs nothing is the one that lines up: each beam’s bearers directly over those of the beam below, which is a column of timbers from the ground to the top of the stack. That is the practice good yards already follow — the stack’s bearers are in vertical lines — and the figures say why the rule is a structural one and not a tidiness.

A schedule of its own #

If a yard cannot line up its bearers to a few millimetres, the bottom beam can be given a schedule that expects them not to.

The schedule is built as the yard essay built its own, strand by strand, but with one more state to satisfy: the stack landing as far outside the bearers as the tolerance allows. A tolerance of 100 mm under two beams costs the bottom beam about a metre more sleeve on each half: the same three strands, each sleeved further. At 250 mm it needs a fourth strand sleeved and 6.25 m in all, nearly twice the yard’s. The sleeve grows faster than the tolerance, because each extra 100 mm has to be paid for at every section between the bearers and the schedule can only pay near the ends.

Each line ends where no schedule can carry the stack at all. With two beams above, that is past 600 mm; with three, past 400. Beyond it the loss of relief reaches mid-span, where every strand is bonded and no sleeve can help — the yard essay’s mid-span limit, reached here by a load the beam was not asked to carry in that essay.

The bottom beam is a different beam #

The last figure puts the two schedules side by side.

Cast from the same drawing, the two beams cannot both be right. The yard schedule fails at the bottom of a stack placed to an ordinary tolerance; the stack schedule gives every beam in service less prestress near its ends than it needed, by four strands’ worth over the last two and a half metres, where the shear check wanted it most. A yard that stacks has three honest choices: cast the beams destined for the bottom of a stack to their own schedule and mark them; line the bearers up to millimetres, with the bearer positions marked on each beam’s top flange; or stack no higher than a beam whose schedule has room for the tolerance the yard can hold.

There is a fourth choice that is not honest, and it is the common one: stack by eye on a schedule cut to the limit. The beam usually survives, because the limits have their own factors in them and the yard’s concrete is stronger than the strength it had on the day of transfer. But the beam that cracks across its top in the yard is a beam whose service life starts with a crack the designer did not draw.

The other fibre, and the order of the stack #

The bottom fibre is the one prestress loads hardest, and a hog adds to its compression, so it is worth checking that it is not the one that governs. It is not. Lined up, the bottom beam’s bottom fibre is at 12.0 N/mm² of compression; with two beams above landing 250 mm out, 12.6; the limit is 18. The top fibre, which started two hundredths from its limit, reaches it first by a wide margin, and every number in this essay is a top-fibre number for that reason.

The order in which a yard builds a stack also works for the bottom beam, and nothing here credits it. Beams are stacked as they come out of the bed, so the bottom beam is usually the oldest in the stack. It has had longer to gain strength, which widens the tension limit its top fibre is held to, and longer to lose prestress, which lowers the tension the prestress puts there. The prestress the member takes back is that second effect, and on this section each per cent of the prestress lost is worth 0.04 N/mm² at the top fibre — twice what the yard’s schedule left to spare. A yard that stacked its newest beam at the bottom would be stacking the beam with the least margin under the most weight.

The beams above are not idle either. A beam stacked on another carries its own weight on bearers at the yard’s inset for as long as it is in the stack, and its camber grows under its prestress as it would on any bearers; the stack does not change that. What the stack changes is only the bottom beam’s moment, and only by where the weight above lands. Every beam above can be checked as a beam in the yard, which the yard schedule already passes, provided its own bearers are where the yard essay put them. The stack is safe when every beam’s bearers are at the yard’s inset and in a vertical line: a rule about geometry that a yard can follow without knowing anything about prestress, which is the best kind of rule for a yard to be given.

The geometry is not as simple as a vertical line on a drawing. The bearers stand on a top flange cambered upward by tens of millimetres, a camber that the camber that lowers the hook found grows for as long as the beam is stored, so a timber placed near the end of the beam below stands lower than one placed nearer its middle. The packing that levels the stack is placed by eye, and it is one more reason the line of bearers in a real stack is seldom where the drawing put it.

The loss, by hand #

The loss per beam above follows from the reaction and the section. Each beam weighs 25 kN/m³ × 0.3 × 0.7 m² × 12 m = 63 kN, so 31.5 kN rests on each of its bearers. Landing 100 mm outside the bottom beam’s bearer, that is a hog of 3.15 kN·m at every section between them. The section’s modulus is 300 × 700²/6 = 24.5 × 10⁶ mm³, so the top fibre loses 3.15 × 10⁶/24.5 × 10⁶ = 0.13 N/mm². With the yard’s schedule leaving 0.02 N/mm² to spare at its worst section, one beam above may land about 0.017/0.129 × 100 = 13 mm out of line, the figure’s value; the margin is a little under the two hundredths it rounds to.

Where the yard’s arithmetic stops #

The bearers are points. A timber bearer is 100 to 200 mm wide and bears over its width; the load’s line of action is wherever the bearer’s contact is stiffest, which on a cambered top flange is its inner edge. The point load sits a little further in than the bearer’s centre line, and the model’s offsets are measured to the load, not to the timber.

The beams have no camber. A pretensioned beam leaves the bed with a camber of tens of millimetres, and the beam above, resting on two points of a cambered top, is supported a little higher at mid-span than its bearers suggest — which changes nothing about where the load lands, and is left out.

The stack is static. Lifting the top beam off a stack, or setting one down, puts a moment of impact into the stack’s bearers; and a stack in the wind is a tall, narrow object on four timbers, a different question from its stresses. Hung from above and still unstable is the lifting half of that.

The person laying the timbers #

They cannot show the bearers’ bearing stresses on the top flange of the beam below, which are local and governed by the timber, nor the concrete’s age in the yard, which raises its strength week by week and widens every limit drawn here. They cannot show the beams’ differing lengths in a yard that stores several products, where a short beam on a long one cannot have its bearers lined up at all. And they cannot show the person laying the timbers, who decides every number in this essay and is rarely given one.

Still open: the beam on the trailer #

A beam stacked in a yard is at least standing still. A beam on a trailer is supported at two points chosen by the haulier — usually a turntable near the front and a bogie at the back, neither at the bearers the schedule was drawn for — and bounced on them at road speed. Whether the dynamic factor of a road, applied to a beam whose supports the haulier chose and whose schedule the yard drew, takes away more relief than any stack does, and whether the haulier’s support positions should be part of the debonding schedule’s list of states, is the question the yard leaves for the road. Four inequalities and a wedge is where every one of these states first became an inequality on the same page.