Structural Engineering & Statics
where the load goes
Deflection

The camber that lowers the hook

A camber is built into a girder so that it ends up level in the finished structure, and nothing about it matters until then — except on the day it is lifted. A cambered girder hanging from its ends is a shallow arch, its centre of gravity two-thirds of the camber above the line through its lifting points, and it rolls about that line as though its hook were that much lower. On a long precast girder already close to its lifting limit, a hundred millimetres of camber takes the factor of safety against roll from 2.3 to 1.4. The camber grows while the girder waits in the yard, so the same girder becomes harder to lift every week it is stored. Pick it a fifth of its length in from each end and the camber drops out of the problem altogether.

Assumes Built to the wrong shape on purpose, Hung from above and still unstable and The deflection that arrives three years late.

16 min read 6 figures The statics of things that do not moveWhich failure arrives firstScale changes everything

A cambered beam is built to the wrong shape on purpose: curved upward in fabrication by about the amount its permanent load will bend it down, so that in the finished structure it is level. Nothing in its analysis changes, no stress anywhere is altered, and almost every mistake made with it is a bookkeeping mistake about which loads were meant to take it out. The camber is a shape, and in service it is meant to vanish.

Before it vanishes it has to be lifted. A precast girder is cast on a bed, stored in a yard, lifted onto a lorry, lifted again onto its bearings, and every lift hangs it from two points, usually close to its ends, from a crane hook above. A girder hung from above can still roll over: tilted, the sideways component of its own weight bows it laterally and moves its centre of gravity outward, and past a certain length that happens faster than the tilt brings back a restoring couple. That essay treated the girder as straight. A precast girder never is.

An arch on two slings

The girder drawn is the one that essay found near its limit: 40 m long, 11 kN/m, a lateral stiffness of 2.55×105 kN⋅m22.55 \times 10^5\ \text{kN·m}^2, its roll axis — the line through its lifting points — 0.9 m above its centre of gravity when straight, picked 0.8 m in from each end.

A cambered girder is an arch on its slings. A 40 m precast girder of 11 kN/m, lateral stiffness 2.55 × 10⁵ kN·m², hung 0.90 m below its roll axis, picked 2 per cent of its length in from each end, with 100 mm of camber at mid-length, drawn in elevation with the camber exaggerated. The girder's centre of gravity is two-thirds of the camber above the line through its ends, 67 mm; its lifting points, on the arc 2 per cent in from each end, are 8 mm above it. It rolls about the line through the picks (dashed), so the camber has lifted its centre of gravity 59 mm towards the roll axis: the girder hangs as though its hook were that much lower.
Fig. 1 The 40 m girder with 100 mm of camber, drawn in elevation with the camber exaggerated. Its centre of gravity is two-thirds of the camber above the line through its ends, 67 mm; its lifting points, on the arc 2 per cent in from each end, are 8 mm above that line. It rolls about the line through the picks (dashed), and its centre of gravity has been lifted 59 mm towards it.

With a camber of 100 mm at mid-length, the girder is a shallow arch. For a parabolic camber the arch’s centre of gravity sits two-thirds of the rise above the chord through its ends — 67 mm — because there is more girder near the crown than the chord suggests. The lifting points sit on the arc too, a short distance in from the ends, and at 2 per cent in they are only 8 mm above the chord.

The girder rolls about the line through its picks, and its stability against rolling depends on how far its centre of gravity hangs below that line. The camber has lifted the centre of gravity by 67 mm and the picks by 8, so it now hangs 59 mm less far below the roll axis than a straight girder would. In general, with the picks a fraction α\alpha of the length in from each end and a camber cc,

Δ=c(23−4α(1−α))\Delta = c\left(\frac{2}{3} - 4\alpha(1-\alpha)\right)

is how far the camber has lifted the centre of gravity towards the roll axis. A cambered girder rolls as though its hook were Δ lower.

The two-thirds is the same kind of number as the one the sweep carries in the lateral direction. A girder bowed sideways by a sweep has its centre of gravity offset from the chord by 2/π2/\pi of the sweep for a half-sine bow, and the lifting check counts that offset as the eccentricity that starts the roll. A camber is a bow in the other plane, and its centre of gravity is offset from the chord by two-thirds of it for a parabola. The sweep is a tolerance — something the casting bed was meant not to produce, a few tens of millimetres at most. The camber is deliberate, often three or four times larger, and it is in the plane the lifting check measures its most important height in. Nobody would omit the sweep from a lifting check, and the camber is routinely left out of it, because it lives on a different drawing.

The margin it comes out of

Fifty-nine millimetres against a 900 mm hook height sounds like six or seven per cent. It is not, because the hook height is not what the girder has to spare.

Every millimetre of camber comes off the margin. The factor of safety against roll of a 40 m precast girder of 11 kN/m, lateral stiffness 2.55 × 10⁵ kN·m², hung 0.90 m below its roll axis, picked 2 per cent of its length in from each end, by Mast's method, against the camber at mid-length. Straight it is 2.31, with 153 mm between the roll axis and the height at which the girder's own sideways bow would roll it over. Every metre of camber lifts the centre of gravity 0.59 m towards the roll axis, so 50 mm of camber leaves 1.86, 100 mm leaves 1.42, and the factor reaches one at about 150 mm.
Fig. 2 The factor of safety against roll of the 40 m girder, by Mast’s method, against its camber at mid-length. Straight it is 2.31, with 153 mm between the roll axis and the height at which the girder’s own sideways bow would roll it over. With 50 mm of camber it is 1.86, with 100 mm 1.42, and it reaches one at about 150 mm.

Mast’s analysis of a hanging girder compares the height of the roll axis above the centre of gravity, yry_r, with zˉ\bar z, the sideways deflection of the centre of gravity when the girder’s whole weight is applied laterally. If yry_r exceeds zˉ\bar z the girder settles at a tilt ei/(yr−zˉ)e_i/(y_r - \bar z), where eie_i is the centre of gravity’s initial offset from its sweep, and the factor of safety is how far that tilt is from the one at which it would be lost. The margin is yr−zˉy_r - \bar z, not yry_r.

For the 40 m girder zˉ\bar z is 747 mm, because it goes as the fourth power of the length, so the margin is 153 mm. The camber’s 59 mm comes out of that 153, not out of 900. The factor of safety falls in proportion, from 2.31 to 1.42 for 100 mm of camber, and it reaches one — the girder rolls over — at about 150 mm. A girder that was checked straight with a comfortable factor of two can be lifted cambered with a factor of one and a half, and nobody altered the hook.

The camber grows while it waits

The camber of a pretensioned girder is not fixed at casting. The prestress bends it upward, its own weight bends it down, and both bendings creep. The prestress wins, and its upward bow grows with the concrete’s creep coefficient.

The longer it waits in the yard, the harder it is to lift. The factor of safety against roll of a 40 m precast girder of 11 kN/m, lateral stiffness 2.55 × 10⁵ kN·m², hung 0.90 m below its roll axis, picked 2 per cent of its length in from each end, against the days since it was cast, as its camber grows by creep from 60 mm at release: 92 mm at a week, 113 mm at four weeks, 133 mm at three months, 160 mm at a year. The factor falls from 1.77 at release to 1.31 at four weeks and 1.12 at three months, and below one at about 240 days: the same girder, the same slings and the same hook, lifted later.
Fig. 3 The factor of safety against roll of the same girder against the days since it was cast, as its camber grows by creep from 60 mm at release: 92 mm at a week, 113 mm at four weeks, 133 mm at three months, 160 mm at a year. The factor falls from 1.77 at release to 1.31 at four weeks and 1.12 at three months, and passes below one between six and eight months.

Taking the camber to grow as c0(1+φ)c_0(1 + \varphi), with φ\varphi Eurocode 2’s creep coefficient for a girder loaded at two days in 70 per cent humidity, the 60 mm the girder had at release becomes 92 mm in a week, 113 mm in four weeks and 133 mm in three months. The same growth turns its bearings: the casting yard’s upward rotation is the largest term in a pretensioned girder’s bearing design, and it is the same creep at work.

On the lift it means that the factor of safety is a function of the date. Lifted at release the girder has 1.77. Lifted at four weeks, 1.31. Lifted after three months in the yard, waiting for a site that was not ready, 1.12 — the same girder, the same slings and the same hook. A lifting check is a check on a particular day, and the girder that waited longest is the one that has drifted closest to rolling.

Storage is not the only thing that changes the camber. A girder stored on supports set in from its ends has a smaller self-weight moment than on the bed, and so a larger camber; a girder in strong sun curves upward on its warm top face. Both add to the camber the lift sees, and neither appears in a calculation that uses the camber on the drawing.

The longest girder hardly moves

The obvious question is whether camber changes how long a girder can be before it cannot be lifted at all. It hardly does, and the reason is the fourth power.

Camber hardly moves the longest liftable girder, and eats the margin just short of it. The factor of safety against roll of a precast girder of the same section and slings as the length changes, straight and with a camber of L/400 and L/300. The length past which it cannot be lifted at all is 41.9 m straight, 41.2 m at L/400 and 40.9 m at L/300 — barely moved. What moves is the margin before it: at 40 m the factor is 2.31 straight, 1.42 at L/400 and 1.12 at L/300, because the girder's own sideways bow grows as the fourth power of its length and has used up nearly all of the hook's height by then, so the camber's few centimetres are a large share of what is left.
Fig. 4 The factor of safety against roll as the girder’s length changes, straight and with cambers of L/400 and L/300. The length past which it cannot be lifted at all is 41.9 m straight, 41.2 m at L/400 and 40.9 m at L/300. At 40 m the factor is 2.31 straight, 1.42 at L/400 and 1.12 at L/300.

The longest liftable girder moves only from 41.9 m straight to 40.9 m with a camber of L/300. The girder’s sideways bow zˉ\bar z goes as the fourth power of its length and the camber only as the first, so close to the limit zˉ\bar z is changing so fast that a few centimetres of camber shifts the limit by a metre. A table of maximum lengths for a section barely notices camber.

What it does notice is the margin just short of the limit. At 40 m, the factor of safety falls from 2.31 to 1.42 at L/400 and 1.12 at L/300. That is where real girders are: a precast section is usually chosen so that its longest span is near what it can carry, and what it can carry is often near what it can be lifted. The camber is a small number in the length and a large one in the margin, because the margin is small.

A fifth of the way in

The camber’s lever depends on where the girder is picked, and at one position it vanishes.

Pick it a fifth of the way in and the camber drops out. How far a girder's camber lifts its centre of gravity towards the roll axis, c(2/3 − 4α(1 − α)), against how far in from each end it is picked as a fraction α of its length, for cambers of 50, 100 and 150 mm. Picked at its ends the whole two-thirds of the camber counts: 67 mm for 100 mm. At 0.21 of the length the picks are on the arc at exactly the height of its centre of gravity and the camber does nothing; further in, the centre of gravity hangs below the picks and the camber helps.
Fig. 5 How far the camber lifts the girder’s centre of gravity towards the roll axis, against how far in from each end it is picked, for cambers of 50, 100 and 150 mm. Picked at its ends the whole two-thirds counts: 67 mm for 100 mm. At 0.21 of the length the picks are on the arc at exactly the height of its centre of gravity and the camber does nothing; further in it helps.

Picks at the very ends sit on the chord, and the camber lifts the centre of gravity by its full two-thirds. As the picks move inward they climb the arc, and the line through them rises towards the centre of gravity. At α=(1−1/3)/2\alpha = (1 - 1/\sqrt{3})/2 = 0.211 of the length the picks are exactly at the centre of gravity’s height, and the camber has no effect on the roll at all. Further in, the picks are higher than the centre of gravity and the camber helps — the girder hangs from its arch rather than from its ends.

That position is close to another one. Moving the picks inward shortens the span between them, so the girder’s sideways bow under its own weight, zˉ\bar z, falls steeply — a tenth of the length in from each end cuts it by more than half — and it is near its smallest when the overhangs and the span between the picks balance, at about a fifth of the length, as for any beam whose supports are free to move.

The picks that remove the camber also remove most of the bow. The factor of safety against roll of a 40 m precast girder of 11 kN/m, lateral stiffness 2.55 × 10⁵ kN·m², hung 0.90 m below its roll axis, against how far in from each end it is picked, straight (dashed) and with 100 mm of camber (solid). Picked 2 per cent in, 2.31 straight and 1.42 cambered; at a tenth, 9.70 and 9.23; the cambered girder is best at about 0.23, at 13.32. Moving the picks in shortens the span between them, so the girder's sideways bow under its own weight falls steeply; near a fifth of the length that bow is close to its smallest and the camber has dropped out, so the two curves meet.
Fig. 6 The factor of safety against roll against how far in from each end the girder is picked, straight (dashed) and with 100 mm of camber (solid). Picked 2 per cent in, 2.31 and 1.42; at a tenth, 9.70 and 9.23; the cambered girder is best at about 0.23 of the length, at 13.32.

So the two curves converge. Picked near its ends, the cambered girder has a much smaller factor than the straight one; picked a tenth of the way in, 9.23 against 9.70; at a fifth, they are the same. The picks that remove the camber’s effect are the same picks that remove most of the bow, and a girder picked there has a factor of safety above thirteen whatever its camber.

That is not a free remedy. Picks well inside the ends make the overhangs cantilevers, and a pretensioned girder’s ends already carry the prestress’s hogging with little self-weight to balance it; the extra hogging from an overhang can crack its top. The first essay on this problem found the overhangs’ own stability to be the other price. Picks somewhere between the ends and a fifth of the length are what lift plans settle on, and the camber is one more reason to push them inward.

The other days it is carried

The lift is one of several days on which a precast girder is held by something other than its bearings, and on every one of them the camber works the same way.

On the lorry the girder sits on two bolsters near its ends, which tilt with the trailer on the camber of the road and in the bends. It is now supported from below rather than hung from above, and its stability depends on how high its centre of gravity is above the roll centre of the bolsters: the higher, the worse. The camber raises the centre of gravity above the line through the bolsters by the same c(2/3−4α(1−α))c(2/3 - 4\alpha(1-\alpha)), and here every millimetre of it counts against the girder directly, with no hook to subtract it from. A girder that has waited three months in the yard rides on the lorry with twice the camber it had at release.

Set on its bearings at the site, before any deck or bracing ties it to its neighbours, the girder is seated at its ends on elastomeric pads whose rotational stiffness resists its roll. The camber lifts its centre of gravity above the pads by two-thirds of itself, and that height multiplies the overturning moment of any tilt. This is the state in which nothing has been drawn: a member that the finished bridge holds in every direction, held for a week by two rubber pads and whatever temporary bracing was remembered. Its camber is at its largest, because the creep has run for the whole of its storage and the deck that will take it out has not arrived.

In all three states the camber is a vertical offset of the centre of gravity from the supports, and in all three it is growing until the deck is cast. The stability checks for each day use the girder’s section and length, which are on the drawing, and its camber, which is not a fixed number on any drawing.

What a lift plan can say

The findings reduce to three instructions.

Use the camber on the lifting day. A lift plan prepared when the girder was cast and used three months later is a plan for a different girder. The camber can be predicted from the age at lifting with Eurocode 2’s creep coefficient, which depends on how thick the member is and how dry its air, or measured with a level on the bed the day before. Measured is better, because the creep coefficient is uncertain by a quarter either way and the camber also carries every thermal and storage effect.

Move the picks in. Picks a tenth of the length in remove two-thirds of the bow and half the camber’s lever; picks at a fifth remove all of the camber’s lever and most of the bow. The price is the overhangs’ hogging and their own stability, which is a strength check and usually passes for the first and needs a tie for the second.

Do not treat the factor of safety as a property of the section. For a girder near its lifting limit, the factor depends on the camber, the date, the sweep on the day and where the picks are, and each of them moves it more than the hook height does. A section table that quotes a maximum liftable length is quoting it for one camber, usually none.

The girder, by hand

The straight girder’s margin is the essay’s starting point: yry_r = 0.9 m and zˉ\bar z = 0.747 m, so yr−zˉy_r - \bar z = 0.153 m. The sweep tolerance is L/960 = 42 mm, and the centre of gravity of a half-sine bow sits 2/π2/\pi of the way out, eie_i = 26.5 mm. Mast’s factor of safety against a limiting tilt of 0.4 radians is

FS=θmax(yr−zˉ)ei=0.4×0.1530.0265=2.31.FS = \frac{\theta_{max}(y_r - \bar z)}{e_i} = \frac{0.4 \times 0.153}{0.0265} = 2.31.

With 100 mm of camber, picked at α\alpha = 0.02, the camber lifts the centre of gravity by 0.1×(0.667−4×0.02×0.98)=0.0590.1 \times (0.667 - 4 \times 0.02 \times 0.98) = 0.059 m, so the margin is 0.094 m and the factor 0.4×0.094/0.0265=1.420.4 \times 0.094 / 0.0265 = 1.42.

The camber that would take the factor to one leaves a margin of ei/θmaxe_i/\theta_{max} = 66 mm, so it uses up 87 mm of the 153, and is 0.087/0.5880.087 / 0.588 = 148 mm.

A parabola, and a creep law

The calculation rests on choices that limit it.

The camber is a parabola. A pretensioned girder with straight strands has a nearly circular camber, and one with harped strands a flatter one; the centre of gravity’s height above the chord stays close to two-thirds of the rise for the shapes that occur, a little more for a camber with a flat middle, which changes the numbers slightly and the finding not at all.

The camber when hanging is the camber on the bed. A girder on its bed is supported near its ends, and hanging from picks near its ends it is supported the same way, so its self-weight bending is the same and so is its camber. Picks well inside the ends change the self-weight bending and the camber with it, an effect left out here.

The creep law is Eurocode 2’s, applied to the net camber. The prestress’s upward bending and the self-weight’s downward one creep at the same rate from the same age, so their difference does too; but the prestress also loses force by creep and shrinkage, which slows the growth. The figures are on the side of more camber, by perhaps a tenth.

The roll axis is the line through the picks. With inclined slings meeting at a hook above, the roll axis is higher — the sling angle sets it — and a spreader beam keeps it at the picks. The camber’s effect on the centre of gravity’s height is the same either way; only the height it is measured against changes.

Still open: the girder that is lifted on its side

Some long girders are rotated before they are lifted — turned so that the camber is horizontal and a sling can pass under them, or tilted on the lorry to clear a bridge. A girder lifted on its side has its camber in the plane in which it bends easily, so the camber becomes a sweep: its centre of gravity is offset sideways by two-thirds of the camber rather than by two-thirds of the sweep tolerance, which for this girder is two and a half times as much. Whether that offset, combined with the far smaller lateral deflection a girder has about its strong axis, makes a girder lifted on its side more stable or less — and how far it must be turned before the camber stops lowering the hook and starts pushing it sideways — is a question about the angle between a girder’s two bows.

Where this goes next

Four camber rules, and what each leaves on the finished beam. The same 12 m composite beam, cambered against four different things, followed through its own load history. Positive is a sag and negative a hog, and the point at the left of each line is the shape it was fabricated to. Cambering against the wet concrete leaves 12.7 mm of sag at the end and a flat beam on the day the slab is poured; cambering against the total load leaves the beam dead flat when fully loaded and hogged 37.9 mm — one part in 316 of the span — before anything is on it at all. Built to the wrong shape on purpose A cambered beam is fabricated curved upward so that load bends it down to something like straight. Nothing in the analysis changes, no stress anywhere is altered, and almost every mistake made with it is a bookkeeping mistake about which loads count. A fourth power, and then a cliff. The factor of safety against rolling, against beam length, for one section hung from a roll axis 0.9 m above its centre of gravity. Nothing about the section changes along this axis. z̄ goes as the fourth power of the length — 0.236 m at 30 m becomes 0.747 m at 40 — and the factor of safety is proportional to (y_r − z̄), so it does not decline gently: it falls away and then stops existing. The working factor of 1.5 is lost at about 41 m, and past 42 m there is no hook height at all at which this beam hangs stably. Which is why long girders are lifted with the picks moved inboard, or with the beam braced, or not in one piece. Hung from above and still unstable A rigid body hanging from a point above its centre of gravity is a pendulum and cannot fall over. A beam is not rigid, and tilting it puts a component of its own weight sideways — which bows it, which moves its centre of gravity further out. Past a length there is no hook height at which it hangs stably at all, and the length arrives as a fourth power. The rotation a bearing sees is made before it arrives. The end rotation of a 12 m pretensioned beam, 300 mm wide and 700 mm deep, with a 160 mm in-situ slab 1.2 m wide, in 60 per cent humidity, from the prestress's release at three days to thirty years, with sagging positive. Release turns each end 4.8 milliradians upward; creep takes it to 7.9 by day 28, when the beam is set on its bearings. The slab brings it back by 0.9, the surfacing by 0.1, and thirty years of creep take it to 8.1 upward. The band after erection is the imposed load (0.9 down) and a night with the top cooler (0.6 down) above the line, and a sunny day with the top warmer (1.0 up) below it. Every value in the band is below zero: under every load it will carry, the beam's ends still point up, and a level bearing is turned the same way for its whole life. The angle made in the casting yard A bridge bearing is designed for the rotation of the beam it carries, and the rotation is listed as a sum of load, temperature, creep and a tolerance. Followed through the life of a pretensioned beam, the largest term is none of those. It is the upward turn the prestress gives the beam's ends in the casting yard, before any bearing exists, and under every load the beam will ever carry its ends still point up.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

CamberCentre of gravityConstruction sequenceCreepLift stabilityPrecastRoll stability