The pier that bends the wrong way

The support that had no moment when it was cast stitched two beams together over a middle support a month after they had started carrying their own weight, and watched a hogging moment grow at the stitch for twenty years: the beams went on creeping under their weight, the stitch would not let their ends turn, and a moment nobody applied arrived toward the one a monolithic beam would have had. The slab that shrinks onto a finished beam followed a precast pretensioned beam with an in-situ slab and found that the slab’s shrinkage, not its creep, is what takes the beam’s precompression. Both beams were left free at their ends.

Most precast decks are not. The slab is cast continuous over the pier together with a joint — a diaphragm of in-situ concrete between the ends of the beams — so that the deck carries its later loads as a continuous beam. From that day the joint forbids the two beams’ ends from turning relative to each other. And a pretensioned beam’s ends do go on turning, for years — the same creep that makes a deflection arrive three years late — in both directions at once: upward as the prestress creeps, downward as the dead load creeps and as the slab shrinks onto it. Which of those wins decides the sign of the moment at the pier, and the joint was reinforced for one sign only.

What the joint forbids #

The two beams are the member of the earlier essays: 12 metres long, 300 mm wide and 700 deep, pretensioned with strands 200 mm below the centroid and released at three days, carrying a 160 mm slab 1.2 m wide. The slab and the joint over the pier are cast together; everything before that happens to two simply supported beams.

The first thing to know is what the joint will have to cancel. Take away the joint and follow one span’s pier end from the day the slab is cast. Its rotation changes by the part the prestress and shrinkage make — a uniform curvature along the span, turning the end through half the span times the curvature — and by the part the dead load’s creep makes, a curvature shaped like the moment diagram, turning it through a third of the span times its mid-span value.

For beams joined at 28 days the net drift is 1.8 milliradians upward over thirty years. The prestress has been on the concrete since it was three days old, it is still creeping fast when the slab arrives, and it keeps lifting the beam’s middle and turning its ends up faster than the slab’s weight, the beam’s own weight and the slab’s shrinkage can bring them down. For beams joined at 90 days most of that upward creep has already happened; the drift is 0.3 upward. For beams a year old it has reversed: what is left after the joint is the slow downward creep of the dead load and the slab’s shrinkage, and the drift is 1.3 milliradians downward.

The moment the joint grows #

A joint that forbids a rotation grows whatever moment it takes to cancel it — the support that moved, turned from a displacement into a rotation — and a moment that forbids an upward turn of the beams’ ends is a sagging moment: tension at the bottom of the joint. Joined at 28 days, the moment passes +128 kN·m within four months, +175 by the end of the first year and reaches +214 by thirty years. The underside of the joint is plain diaphragm concrete, poured with the slab, whose mean tensile strength cracks it at 159 kN·m. It crosses that about seven months after the joint is cast.

The pier’s design was for the opposite. A continuous beam under load hogs over its supports — tension at the top, where the slab’s reinforcement is — and the joint is detailed for that: bars in the slab over the pier, and very little across the bottom of the joint, where the two beams’ ends face each other across a few hundred millimetres of in-situ concrete. A sagging moment of this size opens a crack across the bottom of the joint that nothing crosses, and the load the continuity was built for is then carried across the pier by a joint hinged at its underside.

Beams joined at 90 days end at +59 kN·m — sagging, but a third of what cracks the joint. Beams joined at a year end at −164, hogging, which is the direction the design expected, although its size is a load nobody drew.

Three causes, two of them holding the third back #

The calculation is linear, so the moment separates exactly into its causes. For beams joined at 28 days the prestress’s continuing creep contributes +473 kN·m — more than twice the result, and more than the dead-load moment the beam carries at mid-span. Against it, the creep of the beams’ and slab’s own weight contributes −153: the same redistribution the stitched beams of the earlier essay found, moving the pier toward the hogging moment a monolithic beam would have. And the slab’s shrinkage relative to the beam contributes −105: a slab that shortens more than the beam it sits on bends the composite section into a sag, and a sagging curvature at every section of both spans turns the pier ends downward, which the joint resists with a hogging moment.

So the joint is bent the wrong way by the largest of three causes and held back by the other two. That is why the sign can change. None of the three is large compared to the others by an order of magnitude, all three depend on the dates, and the answer is the small difference between them.

The dates decide the sign #

Sweep the day the joint is cast and the answer moves from +320 kN·m for a joint cast a week after release to −164 for one cast a year after. It exceeds the cracking moment of the joint for any joint cast before about 45 days, and it changes sign at about 130.

The three parts move in different directions as the beams age, and two of them move together. The prestress’s share falls, from +555 kN·m to +204, because a beam joined late has already made most of its upward creep. The differential shrinkage’s share grows, from −57 to −285, because a young beam shrinks alongside the slab while an old one has finished shrinking, so that the slab’s shrinkage is then all difference. And the dead load’s share eases, from −178 to −83, because the dead load too has done most of its creeping before a late joint exists. The first two both push the total toward hogging as the beams age. That is what makes the date so decisive: waiting helps twice.

That is also where one bridge-design rule comes from. American practice allows the restraint moments at such a joint to be neglected altogether when the beams are at least ninety days old when the joint is made, and asks for a bottom connection across the joint otherwise. On this member ninety days is on the safe side of the crack and short of the sign change — the moment is small, not zero — which is roughly what a rule that neglects it is betting on. The bet is about the date, and nothing on the drawing records the date.

Three ways out, and what each costs #

The figure also says what can be done about it, and each remedy is a trade.

Wait. Every week the beams stand in the yard before they are joined takes a share of their upward creep out of the joint’s future. It costs storage, programme and a crane that comes back later, and it is the only remedy that works on all three causes at once, since the older beam is also the one against which the slab’s shrinkage is most fully differential.

Connect the bottom of the joint. Bars or extended strands across the underside let the joint carry the sagging moment instead of cracking under it. It costs detailing at the most congested place in the deck, where two beam ends, their bearings and the diaphragm meet, and it accepts the moment rather than removing it — the spans still lose the precompression.

Prestress less eccentrically. The prestress’s share is proportional to the moment the tendon makes about the section, and a tendon nearer the centroid cambers the beam less. It costs the beam’s capacity under its own design loads, which is what the eccentricity was for, and it is decided long before anybody knows when the joint will be cast.

None of them is free, and the cheapest — waiting — is the one decided by someone other than the designer.

The joint’s arithmetic, by hand #

The orders of magnitude can be checked without the section model. A joint that forbids a rotation $\theta$ at the ends of two spans grows, in a structure that did not creep, the moment that turns each end back by $\theta$. A moment $M$ at the pier varies linearly along each span from nothing at the abutment, and turns the pier end through $ML/3EI$. So the elastic answer would be

$$M = \frac{3EI\,\theta}{L}.$$

With the composite section’s second moment of area, $26 \times 10^9$ mm⁴, a 28-day modulus of 37 kN/mm², a span of 12 m and the 1.8 milliradians the joint forbids, that is 440 kN·m. The computed moment is 214 — about half — because the restraint moment is not applied at once. It grows as the rotation it cancels grows, and the concrete creeps under the moment itself while it grows, giving back part of every increment as it arrives. A moment that builds up over years under a creep coefficient of one and a half keeps roughly half of its elastic value, which is the same relaxation the stitched beams showed from the other direction.

The cracking moment is equally short. The bottom of the joint is 540 mm below the composite section’s centroid; its concrete’s mean tensile strength is 2.9 N/mm². With the section’s second moment of area, that is a sagging moment of 159 kN·m at which the underside of the joint cracks.

The spans pay for it too #

The restraint moment is not confined to the joint. A moment at the pier of a continuous beam varies linearly along each span, from its full value over the pier to nothing at the abutment, so half of it acts at mid-span. For beams joined at 28 days that is 107 kN·m of sagging added at the middle of each span, on top of the beam’s own weight and the slab’s, and it arrives slowly, over the same thirty years, as the prestress creeps.

At mid-span the beam’s soffit is where the prestress has put its precompression, and the composite section’s bottom-fibre modulus is about $48 \times 10^6$ mm³ in beam concrete. The added moment therefore takes about 2.2 N/mm² off the precompression at the soffit — a loss of prestress as real as the fifteen per cent the tendon gives back to creep and shrinkage, and one that no loss calculation includes, because it is not a loss of force in the tendon. It is a moment the continuity added. The earlier essay found a moment nobody chose moving from one section of a continuous beam to another; this one is not redistributed from anywhere. It is made by the joint, out of the beams’ camber.

So a joint cast early is a double error with one cause. The pier receives a sagging moment it was not reinforced for, and the spans receive a sagging moment that eats into the precompression they were designed with. A late joint reverses both: its hogging restraint moment adds to the precompression at mid-span and to the hogging the pier was designed for.

What the traffic does to a joint that sags #

The permanent restraint moment is only the base on which the live load acts. A lorry on one span puts a hogging moment into the pier, as a continuous beam should, and a joint already sagging by 214 kN·m under permanent load swings toward zero with every lorry and back again as it leaves. A joint that has cracked across its underside opens and closes that crack with the traffic, and a joint on the edge of cracking is cracked by the first heavy vehicle. Either way the joint’s underside sees a stress range on every crossing, which is the quantity that decides fatigue, in the one place of the deck that was detailed as if it would never be in tension at the bottom.

This is why the rules that do address the positive moment ask for a bottom connection sized to more than the cracking moment rather than to the computed restraint: once the joint cracks, the computed moment is no longer what it carries, and what matters is that something crosses the crack.

Wet air is the worse case #

It would be natural to expect a dry climate to be worse, since dry air makes concrete creep more. It is not. Drier air makes the prestress creep more, pushing the joint toward sagging, and makes the slab shrink more, pulling it back, and over the ordinary range of climates the two nearly cancel: for a joint cast at 28 days the thirty-year moment stays between +212 and +225 kN·m from 40 to 70 per cent humidity.

In wet air the balance goes. The slab hardly shrinks, one of the two causes holding the joint back falls away, and the sagging moment rises to +260 kN·m at 80 per cent and +328 at 90. For a joint cast at 90 days, comfortably below the cracking moment in ordinary air at +54, wet air takes it to +224, over it. A bridge over water, in a humid climate, is where a joint cast at the age the rule accepts can still crack at its underside — because the rule is about the prestress’s creep, and the slab’s shrinkage was part of what made the prestress’s creep tolerable.

The two runs under every number #

Every number comes from one cross-section followed step by step in time, fibre by fibre — the beam’s concrete and the slab’s, each with its own creep, shrinkage and age at loading, the strands as a bonded steel fibre — which the slab essay built. Because the section is linear, the curvature anywhere along a span is the curvature with no load on it plus a share of the extra curvature the mid-span moment produces, so the free pier-end rotation is a sum of two terms. The joint’s own moment needs one more thing: the curvature a moment added on a given day produces at every later day. That is found by running the section twice on the same time grid, once with a moment added on that day and once without, and taking the difference, so that the steps of the time integration cancel out of it. The moment at the joint is then built up step by step so that at every step the joint’s total rotation since it was cast is zero. The three parts come from switching off the prestress, and making the slab shrink by the beam’s law instead of its own, and taking differences.

One joint, two spans, and a hinge that is not there yet #

The joint is rigid. A real diaphragm is concrete between the beam ends, cast against them; it has a finite stiffness, it cracks, and once cracked at the bottom it behaves as a hinge for further sagging. The moment computed here is the moment an uncracked joint would carry; a cracked one carries less and rotates instead, which is the damage.

Two equal spans. A deck of several spans has joints whose moments interact, and an end span differs from an interior one.

Only the permanent loads. The surfacing and the traffic add their own hogging moments at the pier, as a continuous beam should. A sagging restraint moment reduces the hogging those loads produce and can remove it entirely under a light traffic load, which is the other half of why it matters: the joint cycles through zero with every lorry.

The creep and shrinkage laws. Every number carries a code model’s scatter, a quarter either way on creep, and the sign change at 130 days would move by weeks with it.

Still open: the joint that is cast before the deck #

Here the joint and the slab are cast together, on one day. Often they are not: the diaphragms are cast first, to make the beams continuous for the weight of the wet slab, and the slab follows days or weeks later. The joint then carries the slab’s own weight as a continuous beam — a hogging moment at the pier from the first day — and the prestress’s continuing camber starts from a joint already hogging, while the slab’s shrinkage begins after the joint rather than with it. Whether casting the joint first buys enough hogging at the start to keep the pier out of sagging for thirty years, or only delays the day it crosses zero, is a question about three dates rather than two.