The beam built towards its own middle

The beam stressed one bay at a time built a continuous prestressed beam from one end, casting each bay against the last and stressing its length of tendon onto the beam that existed then. A tendon stressed on a free-standing bay does its work as camber, not as stress, because a beam on two supports cannot resist a curvature; only supports that are interior when a bay is stressed collect its secondary moment. The finished beam kept about half the secondary moment of a beam stressed at once, and more of it at the end where it was finished: 164, 160 and 219 kN·m over the three supports of four spans, against 350, 234 and 350.

That beam was built from one end, and the lopsided diagram recorded it. A long viaduct is more often built from both abutments towards a closure bay in the middle, or outwards from a central pier in both directions. The question the earlier essay left was whether a beam built towards a central closure is symmetric, whether its closure bay — the one stage stressed onto a beam with supports on both sides of it — carries the largest secondary moment, and whether the order of construction can be chosen to put the secondary moment where the applied load needs it.

Five spans, four orders #

The beam is the earlier one, lengthened to five equal spans of 16 m: a tendon carrying 2,400 kN, 300 mm above the centroid over every interior support and 320 mm below it at every mid-span, on a parabola in each span. Stressed all at once, its secondary moments over the four interior supports are 344, 258, 258 and 344 kN·m, sagging — the supports holding down a tendon that tries to lift the beam off them.

Built in stages, each new bay is cast against the bays beside it, its length of tendon coupled on, and stressed. The bay’s curvature then acts on whatever continuous piece it has just joined — and nothing else, because the rest of the beam is not yet connected to it. Built from one end, that piece is always the beam so far. Built from both ends, there are two pieces, one growing from each abutment, until the closure bay joins them. Built from the middle outwards, there is one piece growing in both directions.

The beam built from one end has the shape the earlier essay found, its largest moment at the support where it was finished. The beam built from both ends is symmetric, as a symmetric order on a symmetric beam must be, and nearly flat: 161 at the first interior supports, 172 at the two beside the closure. The beam built from the middle outwards is symmetric too, but the other way up: 215 at the first interior supports and 120 in the middle. Every staged beam keeps between a third and two thirds of the at-once moment, and where each keeps the most is decided by the order.

Two half-beams, and then one #

The first stage casts a bay at each abutment, and each sits on two supports and cambers freely: no secondary moment anywhere. The second stage adds a bay to each, and each half is now the two-span beam of the earlier essay, with exactly its 204 kN·m over its middle support — the first interior support of the finished beam, from each end. The two halves are separate structures; neither knows the other exists.

The closure bay is cast between them and stressed, and its tendon does something no other tendon in the beam does: it acts on the whole beam at once.

The closure’s curvature tries to lift the beam off both of its supports at once, and both refuse: 172 kN·m at each. One support further out, the moment changes sign and is about a quarter of the size — the carry-over of the three-moment equation, $-43/172 = -0.25$ — so the closure takes 43 kN·m back from the first interior supports, which keep 161 of their 204.

That answers the earlier essay’s questions directly. The beam is symmetric, and the closure’s supports carry the most — 172 against 161 — but only by seven per cent. A support made interior by the last stage keeps everything that stage put there, because nothing comes later to take any of it back; that is why the beam built from one end keeps 219 over its last support. The closure’s supports are both made interior by the last stage, but the closure’s tendon puts less into each of them than a new end bay puts into one, because the closure’s curvature is resisted at both ends at once and shared between them.

Why the closure puts less into each support #

It is worth seeing why a closure bay, stressed last onto the whole beam, puts only 172 kN·m into each of its supports when an end bay added last puts 219 into its one. Both are tendons stressed onto a beam that resists them. The difference is how the resistance is shared.

An end bay joins the beam at one support; its far end sits on the abutment, which cannot resist rotation. Its curvature is held at one place only, and the whole of the resistance arrives there. The closure bay joins the beam at both ends; each end is held by a long continuous piece that resists rotation. Its curvature is held at two places, and each takes part of it. The closure provokes more secondary moment in total — 344 kN·m across its two supports, against 219 — but less at each of them, because a span held at both ends refuses the same curvature at two supports instead of one. That is why the closure’s supports stand only a little above their neighbours in the finished beam, and why the support a closure cannot help is the one at the other end of a half-beam’s last bay.

The order that relieves the ends #

The supports that matter most in a continuous beam are not in the middle. Under a uniform load the first interior supports carry the largest hogging moment of any — $0.105wL^2$ against $0.079wL^2$ further in, for five equal spans — because the end span has a pinned end that cannot help. The secondary moment, being sagging over the supports, relieves that hogging. A designer would want the most relief at the first interior supports, and the beam built from either end gives them the least.

The middle-out order does the opposite. The middle bay is cast first and cambers freely. The bays either side of it are added together, making a three-span piece, and the two supports they make interior collect their stage’s moment. The end bays are added last, and the first interior supports — made interior by the final stage, with nothing after it to take any back — keep 215 kN·m, while the middle supports, having had the end stage’s carry-over subtracted, fall to 120.

Built from the middle out, the first interior supports keep about 63 per cent of the at-once moment, against 47 per cent built from either end, and the advantage does not fade with the beam’s length. It is the same carry-over argument in reverse. A support’s moment is set mainly by the stage that made it interior, less a quarter of whatever the next stage on its side puts into the next support along, less a sixteenth of the stage after that. Built from the ends, the first interior supports are made interior early and lose a carry-over to every stage that follows on their side; built from the middle out, they are made interior last and lose nothing.

The middle supports pay for it, but where they pay depends on how the middle is started. With an odd number of spans the middle span is cast alone, cambers freely, and its supports receive nothing until the bays either side are added; they end at 0.47, no worse than the ends-first beam gives them. With an even number the two middle spans are cast together as a two-span piece, their shared support collects its moment at once, and every support keeps at least 0.64. A beam of six or eight spans built from its middle pair outwards keeps more secondary moment at every support than the same beam built from either end.

What the relief is worth #

Under 60 kN/m on every span the load alone hogs the first interior supports by 1,617 kN·m. Built from either end, the secondary relief there is 161 and the worst support carries 1,456. Built from the middle out, the relief is 215 and the worst support carries 1,402 — 54 kN·m less, at the section that governs. The middle supports, relieved less, go from 1,041 to 1,093, still well below the first supports. The middle-out order has moved relief from where the beam had spare capacity to where it had none.

The answer depends on the spans, and it can reverse. Viaducts are often given end spans about four fifths of the interior ones precisely to even out the support moments, and with end spans of 12.8 m beside three of 16 m the load’s hogging is nearly level: 1,256 kN·m over the first interior supports, 1,285 over the middle ones. Built from the middle out, the first supports keep 192 and the middle ones 125; built from both ends, 179 and 173. Now the middle supports govern, and the middle-out order leaves them carrying 1,160 against 1,112 — the worse of the two by 48 kN·m. The order that helps is the one that makes the governing supports interior last, and which supports govern is decided by the spans before it is decided by the programme.

Fifty-four kilonewton-metres is about three per cent of the governing moment, which is not a large number by itself. It is not free, either, and it is a number a designer can choose by programme rather than by concrete. And it is largest exactly when the beam is weakest: on the day the last bay is stressed, before the concrete of the newest bays has gained its strength, with the tendon at its highest force and before the losses the member takes back.

Creep takes back most of the choice #

The staged moments are what the beam has when it is finished, and the concrete then creeps under the tendon’s continuing curvature, now resisted by every support the finished beam has. The support that had no moment when it was cast found the one-line rule for this: the moment moves from its staged value towards the at-once value by the fraction $\varphi/(1 + \chi\varphi)$, with an ageing coefficient $\chi$ of about 0.8. At a creep coefficient of 2 that fraction is 0.77, whatever the order.

Applied to two orders that start 54 kN·m apart over the first support, the rule leaves them $54 \times 0.23 = 12$ kN·m apart, both near the at-once 344. The order of construction decides the secondary moments on the first day, and creep spends the following decade erasing most of that decision. A designer who chose the middle-out order for the relief it gives over the first supports gets all of it at transfer and a quarter of it in service; one who did not gets the at-once moment, nearly, either way.

At collapse the order matters still less. A secondary moment is made by reactions that sum to nothing, so a beam whose support sections can rotate will redistribute it as it forms hinges, and reach the same collapse load whatever secondary moment it started with — the moment that was moved on purpose is the same freedom used deliberately.

That makes the choice of order a transfer decision, not a service one. It matters where the transfer check governs: a beam whose support sections are near their limit when the last bay is stressed, with young concrete and full prestress, can be relieved there by choosing which bays to cast last. Where the long-term check governs, the order hardly matters, and the at-once analysis is close to the truth.

The closure, by hand #

The closure’s numbers can be checked with the three-moment equation. Before the closure the beam is two separate two-span halves, each with $M = 204$ kN·m over its middle support, the earlier essay’s two-span value. The closure’s tendon on the complete five-span beam provokes equal moments $M$ at its own two supports and a carry-over at each first interior support. With equal spans and the outer ends pinned, the carry-over from a support to the next across an unloaded span is a factor of about $-0.25$ here, and the figure gives $M = 172$ and $-43$: the first interior supports end at $204 - 43 = 161$.

The middle-out beam runs the same arithmetic the other way round. The three-span piece of the second stage puts 163 kN·m on each of the two middle supports; the end bays’ stage then puts 215 on each first interior support and carries 43 of it, the other way, onto each middle support, which end at $163 - 43 = 120$. The same carry-over of 43 appears in both orders, taken from different supports.

Joints at the supports and one creep law #

The joints between bays are at the supports. In practice they sit about a fifth of a span past them, where the dead-load moment is small, and each pour carries a short cantilever into the next bay; part of each next bay’s tendon is then stressed with the earlier stage. That moves the numbers without moving the argument, because each length of tendon still acts only on the piece that exists when it is stressed.

The self-weight is left out, so the moments are the tendon’s alone. A beam’s self-weight is also carried by the structure that exists when each bay’s falsework is struck, and its moments are staged in exactly the same way; the structure that was never complete followed that half of the problem for composite beams. The two staged fields add, and a designer choosing an order for its prestress relief must count what it does to the self-weight moments too, which favour the same supports for the same reason.

Every bay creeps alike. A beam built over months has bays of different ages, and the bays cast first have crept furthest by the time the last is stressed. In the middle-out order the oldest bays are in the middle, where the beam has relief to spare; in the ends-first order they are at the ends, where it has none. Real creep therefore favours the middle-out order slightly beyond what a single coefficient shows.

What the order is worth #

The earlier essays on secondary prestress established that the secondary moment is a property of the finished beam, then that it depends on how the tendon is described, then that it depends on when the tendon was stressed. This one adds that the order of stressing is a free variable with a definite effect: it decides which supports keep the secondary moment, by deciding which supports are made interior last.

The rule that falls out is short. A support keeps most of what the stage that made it interior put there, less the carry-overs of every later stage on its side. Built from one end, that favours the far end; from both ends, the closure, slightly; from the middle out, the first interior supports, where a continuous beam’s load needs the most relief. It is the kind of decision the prestress that pushes back would call a secondary effect, and it is one of the few in prestressed design that costs nothing but a page of the construction programme.

Still open: the beam that is jacked at its closure #

Every stage here is stressed and left. A closure can also be made actively: before the closure bay is cast, the two half-beams can be jacked apart, or one end lifted, so that the closure locks in a chosen moment rather than the one the tendons happen to produce. Bridge builders do this to correct for the very imbalance the staged analysis predicts. Whether a closure jacked by a calculated amount can give the finished beam its at-once secondary moments on the first day — and whether creep then carries it past them, the way the jacks an arch forgets are partly taken back — is the question of whether the order’s deficit can be bought back with a jack instead of waited out with creep.