Built to the wrong shape on purpose

Nothing on a construction site is straight, and one of the crooked things is crooked deliberately. A cambered beam leaves the fabricator’s shop bent upward by a chosen amount, is bolted into a frame that is nominally level, and is then loaded until it is nearly straight. It is one of the few operations in structural engineering that alters no force, no stress and no capacity, and it is also one of the few that is got wrong routinely — because everything about it is bookkeeping, and the books have four columns nobody agrees on.

The first thing to be clear about is what has and has not been done to the steel.

A change of shape is not a change of stress #

Bend a beam elastically with a load and it acquires curvature and stress together, in the fixed ratio $M = EI\kappa$. Fabricate a beam already curved — by rolling it, by heating one flange, by cutting the web to a curve before welding — and it acquires curvature with no stress at all, because there is no moment anywhere on it. The shape is its unstressed shape, and every subsequent calculation is made from that shape as the datum.

So a cambered beam and a straight one of the same section carry the same load with the same moments and the same fibre stresses. There is nothing to compute; the generator behind these figures returns exactly zero for the difference, which is worth returning as a number rather than asserting in prose, because it is the property every misunderstanding of camber denies.

What changes is where the beam is. That is a serviceability matter and nothing else, and serviceability is usually what governs — so a member’s whole reason for being the size it is may be a limit that camber addresses without touching the strength calculation at all.

The beam that is two beams #

The complication that makes camber interesting is that a composite floor beam has two different flexural rigidities during its own construction. Before the slab sets, the steel section is on its own: here $EI = 94{,}000$ kN·m². Afterwards the slab acts with it, and $EI = 260{,}000$ — 2.77 times as stiff, from concrete that was already sitting on the beam as a dead weight before it did any structural work at all.

That factor of 2.77 is the difference between two pieces bending separately and the same two bonded into one section. The shear connectors welded through the decking are the restraint on slip that makes the second case true, and until the concrete has set there is nothing for them to restrain — so the connectors are present, installed and inert through exactly the two stages that produce most of the movement.

So the four stages of the beam’s life divide into two by stiffness, not by kind:

The third row carries a long-term multiplier: it is sustained, the concrete creeps under it, and three years later the deflection is three times what it was on the day. That is the whole of the 7.48 mm — 2.49 mm elastic and 4.99 mm arriving over the following decade.

Most of that creep arrives in the first year and it is still accruing at ten thousand days, which is why a camber decision made about elastic deflections is a decision about the wrong number: the shape the beam is cambered to is fixed in the shop, and the shape it is being cambered against is still changing a decade later.

Which free body produced the number #

Each row of the table is a separate analysis of a separate beam. Cut the member at mid-span at the moment the wet concrete is placed: the free body carries 8.8 kN/m over 12 m, the section on the cut is the bare steel one, and $5wL^4/384EI$ with $EI = 94{,}000$ gives 25.28 mm of cumulative sag. Cut the same member after the finishes are on and the cut face is a different section — a composite one, with the neutral axis up in the slab — and the same formula with $EI = 260{,}000$ gives 2.49 mm for the 2.4 kN/m.

The two cuts are through the same beam at the same station and they reveal different sections, which is the fact all camber bookkeeping turns on. Superposition still holds, because each increment is elastic; it just cannot be applied to the total load, because there is no single section the total load was ever carried by.

Four rules and what each leaves behind #

Against nothing. The beam finishes 37.9 mm down, which is one part in 317 of the span and outside every deflection limit worth having. Under dead load alone it is at 32.8 mm, so the floor is visibly dished before anybody moves in.

Against the wet concrete, 25.3 mm. This is the common rule and the figure shows why: the beam reads exactly zero at the end of the second stage, so the slab is poured to a level top surface and the concrete thickness is uniform. It finishes at 12.7 mm, one part in 945.

Against the dead load, 32.8 mm. Flat under everything permanent, 5.2 mm of sag under the full load, and 7.5 mm of hog on the day of the pour — which means the slab must be poured thicker in the middle, and the extra concrete is a load nobody counted.

Against the total load, 37.9 mm. Dead flat when fully loaded and hogged at every other moment of its life. Its worst hog is 37.9 mm, with nothing on it.

The rule between the two is the one most commonly specified in practice, and it splits the difference in both directions at once.

That last line is the finding, and it generalises: a cambered beam is at its most curved when it is empty. The largest departure from straight that any of these members ever shows is the one it was built with, and it shows it on the day it is erected — when the cladding is being set out against it, when the fit-out contractor is measuring from it, and when nobody is thinking about deflections at all.

The rule that is chosen by what is being measured from #

The choice between the four is not a structural question and there is no structurally correct answer. It is a question about which stage somebody is going to lay a straightedge on, and the limit that straightedge is checked against is written about the finished member rather than about any moment in its construction.

Three practical constraints narrow it, and none of them is in the analysis.

Concrete is placed to a level, not to a thickness. If the beam is hogged when the slab is poured, the slab is thin over the beam and thick between them, and the extra concrete in the middle of the bay is real load. Cambering against the wet concrete makes that term vanish, which is the reason the rule is common rather than an argument that it is right.

Fabrication tolerance is a substantial fraction of the number. A camber is a length quoted to a millimetre on a member whose own second moment of area is known to a few per cent, and The camber of 25 mm on this 12 m member is 0.2% of the length, and the tolerance on a rolled and cambered beam is commonly ±5 mm or more. A rule that distinguishes 32.8 mm from 37.9 mm is distinguishing quantities inside the noise.

And some of that noise is a camber nobody asked for.

So the shape a beam arrives in is the sum of the camber that was specified and the one the welding sequence supplied, and only the first of the two is on the drawing.

Camber cannot be removed. It is fabricated in. A beam cambered for a load case that then changes is a beam with a permanent hog, and a structure is rarely finished in the sequence it was designed in.

A member is cambered and a floor is a grid #

Every calculation above treats the beam as though its two ends stay where they were put. On a real floor they do not, because a secondary beam lands on a primary and the primary is deflecting too.

The level of a point at the middle of a secondary beam is its own mid-span sag plus the movement of whatever its ends are sitting on. Camber the secondary against its own loads and it is straight relative to its own supports — and those supports have gone down. A 9 m secondary that sags 20 mm, landing at the mid-span of the 12 m primary above that sags 37.9 mm, is 58 mm low at the worst point of the bay. Cambering the secondary perfectly recovers 20 mm of that and leaves 38.

So a camber specified member by member does not produce a level floor, and the correction is not a refinement — it is most of the number. What a secondary beam has to be cambered for is its own deflection plus the deflection of its supports, which makes camber a property of a position on the floor plate rather than of a member. Two identical secondaries, one landing at the middle of a primary and one landing near a column, need different cambers.

Two consequences follow, and both are the reason large floor plates are set out as a surface rather than as a schedule.

The camber diagram is two-dimensional. The right object is a contour map of how much the finished floor would drop at every point, and the camber of each member is read off it. That is what a bridge deck’s camber diagram is, and buildings with long spans have quietly started to be set out the same way.

And the slab thickness follows it. Concrete is placed to a level, so a slab poured over a grid that is sagging non-uniformly is thickest where the sag is deepest — which is the middle of the bay, which is where the load does most. That extra concrete is a real load, applied exactly where it is least welcome, and it is ponding without the water: a deflection that recruits its own extra weight. The loop gain is far too small to be unstable and the load is not negligible, and on a large lightly loaded floor plate the unplanned concrete can be a measurable fraction of the design dead load.

The one case where the sign matters structurally #

Almost everywhere, camber is cosmetic. On a flat roof it is not, because there the deflected shape decides the load.

Water collects in the sag, the extra weight increases the sag, and the loop closes with a gain that depends on how stiff the bay is relative to how fast it fills. The essay on ponding sets that out; what belongs here is the consequence for camber. The amplification is $1/(1-\lambda)$ acting on the initial sag, so a bay built with a hog starts with a negative initial sag, and the same multiplier that would have magnified a defect now magnifies a margin.

That is the only argument on this page where camber changes a force, and it changes it by changing the load rather than by changing the member.

Camber, precamber, and the thing that is not camber #

Three operations get the same word and only one of them belongs here.

Camber is a fabricated change of unstressed shape. No stress, no force, no capacity.

Prestressing is a fabricated change of stress, usually accompanied by a change of shape. Putting the stress in backwards before the load arrives is a real structural operation with real stresses, and a prestressed member’s hog is a consequence of it rather than the point of it.

Jacking, packing and propping change the load path during construction, and therefore change the forces. Propping a composite beam while the slab sets moves the wet-concrete row of the table from the bare-steel column to the composite one and reduces the total deflection by about 13 mm here — which camber cannot do, because camber does not move a load. The distinction matters most where a member is handing its load to something beneath it, since a transfer structure’s settlement is inherited by everything above and a camber is the cheapest way to give some of it back.

The bookkeeping in its general form is a member whose flexural rigidity is not constant, which shows up as a step in the $M/EI$ diagram. A composite beam’s step is in time rather than along the length, and every camber calculation on this page is the same sum of areas evaluated stage by stage.

The number that decides how much it matters #

Camber is worth arguing about in proportion to how large the deflections are, and deflections are decided by span before they are decided by anything else.

Deflection goes as the fourth power of the span at a constant section and a constant load per metre, so doubling a span multiplies the movement by sixteen and a rule that is a nicety at 6 m is a fabrication instruction at 12 m and a governing constraint at 18 m.

The 12 m beam above moves 37.9 mm uncambered. The same section and loading at 6 m would move 2.4 mm, at which point every one of the four rules produces a beam nobody could distinguish by eye and the whole question is not worth the fabricator’s time. At 18 m it moves 192 mm, which is a member that cannot be built straight in any useful sense.

So camber is a long-span practice, and the threshold is a length rather than a ratio. The common trigger — camber beyond about 10 m of span, or beyond about 20 mm of dead-load deflection — is the fourth-power law meeting a tolerance, and it moves whenever either does.

Every profile drawn on this page is one curve translated vertically by a constant. That is the whole geometric content of camber, and it is the reason the four rules never cross: they are not four different shapes, they are one shape read from four different datums.

Where the model stops #

Every deflection above is elastic and small. The camber itself is 0.2% of the span, so the change of geometry it produces alters nothing about the analysis — but a deep-cambered member, a curved girder or an arch is a different object, and its curvature is part of its structural behaviour rather than an offset to it.

The creep multiplier is a single number standing in for a process. The 2.0 used here belongs to a particular concrete, a particular age at loading and a particular humidity, and the honest range is roughly 1.5 to 3.0. The rows of the table move with it, and so does every camber rule that includes the third one.

Nothing here accounts for the slab’s own restraint. A composite beam’s long-term deflection is complicated by shrinkage of the slab, which curves the member downward with no load applied at all — an imposed deformation rather than a load, and one of the same order as the creep term on a lightly loaded floor. A restrained member develops force rather than movement, and a composite slab is restrained by the beam it is cast onto.

Fabrication is not exact and neither is erection. A cambered beam that is bolted into a frame whose columns are not plumb acquires a shape nobody computed. The four rules differ by a few millimetres and the site differs from the drawing by a few millimetres, which is a fact about the usefulness of the fourth decimal place rather than about the method.

And camber does not survive a change of section. A member that is later strengthened, or that cracks, has a different stiffness and therefore a different deflection under the same remaining load, while its fabricated shape is fixed.

What the pictures cannot show #

Every profile on this page is drawn at an exaggeration of about a thousand. The 12 m beam moves 37.9 mm at its worst; drawn to scale on the page it would be a straight line with a bump one twentieth of a millimetre high, and every distinction the figure is about would be invisible. The honest reading of the shapes is the ratio in the caption — one part in 945, one part in 317 — and not the curvature.

The stage curves are drawn as parabolas of the computed amplitude, which is what a uniform load gives to within about a per cent. The real profiles differ slightly from each other in shape as well as in size, because a point load and a spread load bend a beam differently, and no drawing at this exaggeration can show the difference.

And the fabricated shape is drawn as a smooth curve. A cambered beam is often not smooth: heat-cambering works in discrete passes, and press-cambering in discrete bites, so the real unstressed shape is a chain of straight segments with kinks between them. The kinks are within tolerance and they are not within the drawing.

The ladder from here #

Later rungs on this anchor: camber in composite bridge girders, where the deck pour sequence makes the load history genuinely two-dimensional and the camber diagram is a surface. Precamber of trusses, where the shape has to be built into member lengths rather than into a curve, and where a millimetre on one chord moves the whole profile. The interaction of camber with lateral-torsional stability during erection, since a cambered beam hanging from two points is not a straight member. Camber of concrete members by prestress, which is the same shape obtained by an operation that does change stresses. Setting-out and tolerance, and the survey question of what “level” means on a 60 m floor plate. And the long-term case where the third and fourth stages are separated by decades, so that the beam a reader sees is at a point on a creep curve nobody chose.

The word itself is old and comes from the same root as chamber — a vault, an arched space — by way of the French cambré. Which is the right etymology for the operation: it is the arch made small enough to be invisible, and used for its shape rather than for its structural action.