The angle made in the casting yard

The angle nobody limits found that the end rotation of a loaded beam is locked to its deflection by a coefficient with no material in it — 16/5 for a uniform load — and that the one place the angle is designed for explicitly is a bridge bearing. It listed what the bearing’s design rotation is made of: the rotation under load, the rotation of any camber that was built in, a tolerance for the bearing being set out of level, and the temperature gradient through the deck. Codes ask for the sum, and add an allowance.

A list of terms says nothing about their sizes, their signs, or when each arrives, and a bearing is not asked to permit a sum. It is asked to permit whatever the beam’s end does over its whole life, measured from whatever angle the bearing was set at. So this essay follows one beam through that life — a twelve-metre pretensioned precast beam, 300 millimetres wide and 700 deep, with a 160-millimetre slab cast on it, the same member whose slab shrinks onto it — and asks which term governs. The answer is a term that is on no load list, made before the bearing is installed, turning the ends the way no load ever turns them.

The history, from the casting yard #

A pretensioned beam is cast around strands already stretched between the ends of a bed. On the third day the strands are cut free and grip the concrete, and the beam, compressed below its centroid, bends upward off the bed. Its ends turn up. From that moment the beam carries its own weight and the prestress, and both creep, which is how a deflection arrives years late under a load that never changed.

The picture has one shape and it is worth reading before any number. Almost everything happens before the vertical line, which is the day the beam is lifted onto its bearings. Release turns each end 4.8 milliradians upward. In the next twenty-five days, with the concrete young and the prestress at its largest, creep adds 3.1 more: by the time the beam is set down it has already turned 7.9 milliradians. The slab, cast at six weeks, brings it back by 0.9, the surfacing at ten weeks by 0.1, and then the drift resumes upward for thirty years, reaching 8.1.

The shaded band is everything the load list names. Imposed load turns the ends down by 0.9, a night on which the top of the deck is colder than the bottom by 0.6, a sunny afternoon with the top warmer turns them up by 1.0. The band is 3.5 milliradians wide, from its lowest point to its highest, and it lies wholly below zero.

The same rotation, done by hand at release #

The first number can be checked with two textbook formulas, which is worth doing, because every later number is built on it. A straight tendon at eccentricity $e$ carrying a force $P$ puts a uniform hogging moment $Pe$ into the beam, and a uniform moment turns each end of a simple span through $PeL/2EI$. The beam’s own weight $w$ turns them back through $wL^3/24EI$.

At release the tendon force is 1,411 kN at 200 mm below the centroid, the concrete’s modulus at three days is about 32,000 N/mm², and the rectangle’s second moment of area is $300 \times 700^3/12 = 8.58 \times 10^9$ mm⁴. So

$$\theta_P = \frac{1.411\times10^6 \times 200 \times 12\,000}{2 \times 32\,000 \times 8.58\times10^9} = 6.2 \text{ mrad},\qquad \theta_w = \frac{5.25 \times 12\,000^3}{24 \times 32\,000 \times 8.58\times10^9} = 1.4 \text{ mrad},$$

and the difference, 4.8 milliradians upward, is what the section model gives. The bonded strands stiffen the section a little and the model includes them; the hand calculation does not, and the two agree to a tenth of a milliradian. Nothing in the rest of the essay is more exotic than this. It is the same two formulas, with a modulus that falls as the concrete creeps and a tendon force that falls as the concrete shortens under it.

Two rotations that nearly cancel, both growing #

The net rotation hides two much larger ones. Split it into the part the prestress and shrinkage make with no load on the beam, and the part the loads add, and the prestress part starts at 6.1 milliradians upward and ends, thirty years later, at 12.9. The load part starts at 1.3 downward — the beam’s own weight — and ends at 4.8, with the slab and the surfacing on it.

Both parts grow, because both are sustained and both creep. The prestress part grows by a larger factor for three reasons that point the same way. It was applied at three days, when the concrete creeps most, while the slab arrived at six weeks. Shrinkage shortens the concrete around an eccentric tendon, and a member shortened more at its bottom than its top bends upward as surely as one prestressed. And the tendon force falls as the concrete creeps — it loses twelve per cent after release here — which slows the prestress part without reversing it. So the upward rotation the beam started with is not eroded by the loads that follow. It is outpaced.

That is the first reason the list of terms misleads. The rotation of a prestressed beam is the small difference of two large numbers, each growing, each uncertain by as much as the creep coefficient is, and the net is not a load effect with a correction. It is the residue of two effects that were designed to cancel and never quite do, and it moves, as a check that depends on a date moves, with the day each effect started.

Every term, with its sign #

Laid out as a list with signs, the terms separate into two kinds. The two terms made before the bearing exists — release and the creep before erection — are 7.9 milliradians together. Every term after it, added regardless of sign, is 4.8: the slab’s 0.9, the surfacing’s 0.1, the 1.3 of creep in service, the imposed load’s 0.9, and the two temperature differences, 0.6 and 1.0. The terms made in the casting yard outweigh everything the load list is about, even when the load list is summed as pessimistically as arithmetic allows.

Two things make those terms easy to miss. The first is that they are not loads. A bearing designer tabulates actions, and the prestress’s upward turn is not an action on the bearing: it happened before the bearing was there, to a beam sitting on timber packers in a yard. The second is that they have the opposite sign to every action a bridge is designed for. Load, surfacing and a cool deck all turn the ends downward. A mental picture of a bearing rotating as the lorry crosses has the bearing starting level and tilting one way. This one starts tilted the other way and stays there.

The temperature terms use the linear differences a European code gives for a concrete beam deck — the top 15 degrees warmer than the bottom on a sunny afternoon, 8 degrees colder on a clear night — acting through the composite depth of 860 mm. They are small here because the span is short: a uniform curvature turns the ends through half the span times the curvature, so they grow only in proportion to the span, while the load terms grow as its cube.

A level bearing is turned one way for life #

With the bearing plate set level, the bearing sees the whole of the beam’s rotation from the day it is set down. Over the following thirty years that rotation stays between 9.2 and 5.7 milliradians upward. The lower figure is the beam at its most sagged — the full imposed load, on a cool night, before most of the long-term creep has arrived — and even that leaves the ends pointing up. Under every load the beam will ever carry, its ends still point up, and a level bearing is turned the same way for its whole life.

For an elastomeric bearing that matters physically — a bearing is a roller only as far as it is allowed to be one, and this is a demand it cannot shed. It accommodates rotation by compressing more on one edge than the other, and when the rotation grows past what its vertical compression under the dead load can supply, the other edge lifts off and the bearing is carrying its load on part of its area. A rotation that is always in one direction always loads the same edge — the one under the beam’s end, nearest the abutment face — and the bearing’s life is spent on half of itself. A pot bearing or a curved sliding bearing has a stated rotation capacity and is simply sized to it, but the capacity it has to be sized to is set by the casting yard, not the traffic.

The plate that takes it out #

The remedy is old and simple: a tapered plate — a sole plate or bearing plinth cut to an angle — fixed between the beam and the bearing, so that the bearing is level not when the beam’s end is level but when it is turned by the plate’s angle. The bearing then permits only the departure from that angle.

The rotation left for the bearing, as a function of the plate’s angle, is a V. Its arms rise at one milliradian per milliradian, and its floor sits at the angle that splits the beam’s range in service evenly: 7.4 milliradians upward. There the bearing is asked for half the range, 1.8, plus the allowance.

A level plate is far up the right arm: 9.2 plus the 5.0 allowance, 14.2. Cutting the plate to the rotation the beam has on the day it is set down, 7.9, brings the bearing to 7.2 — nearly all of the available improvement. The middle of the range does slightly better, 6.8, because the beam goes on turning upward after it is set and the day-of-erection plate is already a little behind it by the time the imposed load and the warm afternoons arrive.

What is left is the allowance #

Look again at the floor of the V. Once the plate has taken out the rotation made in the casting yard, what the beam’s end does over thirty years — every load, every temperature, every year of creep — spans 3.5 milliradians, and the bearing sees half of it either side of the plate’s angle. The allowance for setting-out and uncertainty is 5.0: larger than everything that was calculated.

That is not a sign that the calculation was pointless. It is the calculation’s result. The allowance exists because a bearing plinth is cast to a tolerance, a beam is placed to a tolerance, and a creep coefficient is known to perhaps a quarter either way; five milliradians is roughly the size a bridge code adds for those things combined. Before the plate, it was the second-largest term, beside a casting-yard rotation twice its size. After the plate, it is the largest, and the only way to make the bearing smaller is to know the uncertain things better — to survey the plinths, or to measure the beam’s actual camber before cutting its plate. The movement budget at a cladding joint found the same order at a different scale: the calculated deflection the smallest term, the tolerance the largest. At a bearing the order arrives only once the casting yard’s term is taken out, and that term is larger than both.

Reading the angle off a tape measure #

If the plate is to be cut to the beam, the beam’s rotation has to be known on the day, and the obvious instrument is a level and a tape: survey the camber at mid-span, and convert. The angle nobody limits supplied the conversion for a loaded beam — the end rotation times the span over the mid-span deflection is exactly 16/5 under a uniform load and 3 under a central one, with no material or section in it.

That coefficient is wrong here, and not by a little. On the day it is set down this beam has a camber of 22.0 millimetres and an end rotation of 7.9 milliradians, so $\theta L/\delta = 4.3$. The reason is the two parts again. The prestress’s part is a uniform curvature, for which the coefficient is 4 — a circular arc turns its ends through four times its rise over its chord. The loads’ part is a parabolic curvature, for which it is 16/5. The net camber is the difference of the two, and a difference of two shapes with different coefficients has a coefficient outside both: the sag takes more off the mid-span rise than off the end slope, so the ratio climbs, to 4.3 at erection and 4.7 at thirty years.

Convert a measured 22 millimetres with the loaded beam’s 16/5 and the rotation comes out at 5.9 milliradians, a quarter short — a plate cut to it leaves two milliradians on the bearing that the survey appeared to have removed. Convert it with 4, the circular arc’s coefficient, and the error is 7 per cent the same way. The instrument is right; the formula carried over from loaded beams is the wrong one for a beam whose shape was made by its prestress.

The climate’s share #

Creep and shrinkage depend on the air the beam stands in, and so does the casting-yard term. The same beam in 50 per cent humidity is turned 8.1 milliradians at erection; in 80 per cent, 7.2. Over thirty years the level-plate demand moves correspondingly, from 14.5 to 13.7. The demand with a plate at the middle of the range barely moves at all, 6.8 to 6.9, because a plate removes whatever the casting yard made, and what is left — the loads, the temperatures and the creep in service — depends on the humidity much less. It is one more way of saying the same thing: with a plate, the bearing is insensitive to the quantities that are least well known, and without one it carries every uncertainty in the creep coefficient at full size.

The day the beam is set #

The plate’s best angle depends on something no structural drawing records: the age at which the beam is lifted onto its bearings. A beam set at three days, straight from the bed, has made 4.8 milliradians of its rotation and has the steepest part of its creep still ahead. A beam that waits in the yard until forty days has made 8.1, nearly all of it.

For a level plate the day makes no difference at all, 14.2 throughout, because a level bearing sees the whole rotation whenever it started watching; the worst it will ever see was going to be reached after thirty years anyway. For a plate cut to the day’s rotation the day matters a good deal. Set at three days and matched to that day, the bearing is left with 9.4 — its plate is matched to a rotation the beam then leaves, 3.3 milliradians behind by the time the beam reaches its long-term state. Set after a fortnight and matched to the day, it is within half a milliradian of the best that any plate can do. The zero from which a bearing’s rotation is measured is a date, and the tapered plate is a decision about which date.

When the load terms would matter #

Nothing about this is special to short spans except the proportions. The load terms grow as the cube of the span and the prestress’s term, for a straight tendon at a fixed force, only in proportion to it. The same beam, with the same prestress, spanning sixteen metres instead of twelve, is set down turned 8.1 milliradians upward against 7.9, but its imposed load turns the ends back by 2.1 instead of 0.9, the slab by 2.3 instead of 0.9, and the range in service doubles, to 7.2. The casting-yard term is still the largest single one, and still the one the plate takes out; what is left for the bearing, the half-range beside the allowance, is then comparable to the allowance rather than smaller than it.

A longer beam would in practice be prestressed harder and deeper, which pushes the proportions back toward the twelve-metre case. The design of a pretensioned beam aims at a small net deflection under its permanent loads, which is exactly the aim that makes its end rotation the small difference of two large and differently-creeping parts.

The section model under the numbers #

Every sustained rotation is 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 function, its own shrinkage and its own age at loading, and the strands as a bonded steel fibre stressed before release, so that every shortening of the concrete at their level comes off their force. That calculation is linear, so the curvature at any point along the span is the curvature with no load on the section plus a share of the extra curvature the mid-span moment produces, in proportion to the moment there. The tendon is straight, so the first part is the same at every section and turns each end through half the span times itself; every load is uniform, so the second has the moment diagram’s parabolic shape and turns each end through a third of the span times its mid-span value. The short-lived terms use the composite section at the 28-day moduli, the slab transformed at their ratio.

A creep coefficient, one span, and the bearing that pushes back #

Creep. Every sustained number is carried by a creep function, and the one used is a standard code model whose own scatter is a quarter either way. Since the casting-yard term and its creep before erection are the largest terms and the prestress part is the largest of the two parts, an error in the creep coefficient moves the answer more than an error in any load would — which is part of what the allowance is there for, and why measuring the beam’s camber before cutting its plate is worth more than refining the load list.

One span. The beam is simply supported for its whole life. Most precast decks are made continuous over their piers, after which the rotation at the pier stops being free, a restraint moment grows at the joint, and the end bearings see a different history from the one drawn here.

The bearing’s own stiffness. An elastomeric pad resists rotation a little, which puts a small moment into the beam’s end and changes its rotation by an amount that is negligible here. The model assumes the support is a perfect hinge, and the bearing is the thing that makes it one.

Temperature. The differences are linear through the depth. Real ones are not, and a nonlinear profile stresses a member without bending it, which moves no bearing.

Still open: the joint over the pier #

A deck made continuous over its piers asks the same question twice. The bearings at the abutments see the end rotation of the end spans, reduced by the continuity; the joint over the pier, cast after the beams have been hogging in the yard for weeks, sees the two beams’ ends turn toward each other as their prestress goes on creeping upward, and holds them. The rotation the pier joint would have made becomes a restraint moment that sags there, while the slab’s differential shrinkage would have made one that hogs. Whether the casting-yard rotation, which this essay found to dominate a free bearing, also dominates the moment at a joint that forbids it — and whether the date the joint is cast is again the decision that settles it — is the question the continuous deck puts.