The camber that lives in the member lengths

Built to the wrong shape on purpose cambered a beam: fabricated it curved upward by the amount its dead load would bend it down, so that it arrives level. It named the truss as the harder case — “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” — and left the arithmetic for later.

The arithmetic is short and its consequences are not. A beam’s camber is a curve rolled or heated into one member. A truss’s camber is a list of member lengths, each a millimetre or two away from the length the geometry draws, and the list has to be right for every member at once.

A shape made of lengths #

A truss has no curvature. Each member is straight, joined to its neighbours at pins, and the positions of the joints — the truss’s shape — are fixed entirely by how long the members are. That is why which member moved the roof could find a truss’s deflection as a sum of one term per member: under load each member changes length by its force times its length over its stiffness, and each change moves the joint being watched by an amount set by virtual work.

The same sum works in reverse. If member $i$ is fabricated longer by $\Delta_i$, bottom node $j$ moves by

$$\delta_j = \sum_i n_{ij}\,\Delta_i$$

where $n_{ij}$ is the force in member $i$ under a unit downward load at node $j$. No load is needed; the truss is simply assembled to a different shape. And because the formula is the same one that gives the dead-load deflection, a camber is exactly the reverse of a sag: build each member shorter or longer by the amount dead load will change it, and under dead load every member reaches its drawn length and the truss its drawn shape.

The truss here is a Pratt truss of eight 3 m panels, 24 m long and 2.4 m deep, its chords 4,000 mm² of steel and its webs 2,000, carrying 60 kN of dead load at each top node. Under that load its mid-span sags 48.8 mm, about a five-hundredth of its span, which is the camber the fabricator has to build in.

What a millimetre on each member does #

The influences are the unit-load forces, and they have a simple pattern. A millimetre on one of the middle top-chord members moves mid-span two and a half millimetres, because a unit load at mid-span puts a force of $L/4h$ in them — the span over four depths — and a truss ten times as long as it is deep amplifies its middle chords by two and a half. The chords further out are less effective in proportion to the unit moment at their panel, 1.88 and 1.25 on the top chord, down to 0.63 on the end bottom-chord panels. The diagonals each move it 0.80 per millimetre and most verticals 0.50.

Three verticals move it not at all. A unit load at mid-span puts no force in them, so their lengths do not enter the mid-span’s position; the member that is not worth stiffening found the same zero from the other side, a vertical that carries the whole of a panel load and contributes nothing to the deflection being watched. The camber at mid-span does not care how long those verticals are — though the camber elsewhere does, since each node has its own influences.

Every member, the chords, or one strain #

Three ways of building the camber can be compared directly. The exact one builds every member to its own dead-load strain: the top chord longer by the amount its compression will shorten it, the bottom chord shorter by its tension’s stretch, each diagonal and vertical adjusted by its own. Under dead load the truss then comes to its drawn shape exactly, at every node. It is also the most tedious to fabricate, since every member has its own length.

The obvious simplification is the chords only, on the reasoning that the chords carry the bending. They do, and they are not the whole deflection. In this truss the top chord is 46 per cent of the mid-span sag, the bottom chord 28 per cent and the webs 26 per cent, so a camber built into the chords alone takes out 36.2 mm of the 48.8 and leaves 12.6 mm of sag. A beam’s shear deflection is small beside its bending deflection; a truss’s web deflection is not, because a truss’s webs are slender members at low stress rather than a solid plate.

The simplest to set out in a shop is one strain on the whole top chord: every top-chord member cut long by the same 1.45 mm per metre — 4.3 mm on each 3 m member — scaled so that mid-span comes out right. The truss then bends into a nearly circular arc, since a uniform strain on one chord is a uniform curvature, while its dead-load sag has the shape of a moment diagram. At the quarter point the arc gives 32.5 mm against a sag of 34.2, and near the supports it is a few millimetres light. It takes out the webs’ share at mid-span only because it was scaled to, by building extra length into the chord that compensates for the webs.

The top chord or the bottom #

If one chord is to carry the camber, it matters which. Lengthening the top chord and shortening the bottom chord both lift mid-span, and per unit of strain they are not equally effective. The top chord’s six members have influences summing to 11.3, and each is 3 m long, so a strain $\varepsilon$ on them lifts mid-span $33{,}750\varepsilon$ mm. The bottom chord’s eight members sum to 8.8 — its end panels have moment centres at the first top nodes, where the unit moment is small — so the same strain on the bottom chord lifts it $26{,}300\varepsilon$ mm. A camber built into the top chord needs a fifth less strain than the same camber built into the bottom, which matters when the strain is carried by packing plates in the splices rather than by cutting, and each millimetre of pack is a separate item on the drawing.

The choice is also a choice about which chord is shimmed and which is short, and a top chord in compression closes onto its packs while a bottom chord in tension pulls away from them. A pack in a compression splice stays where it was put; one in a tension splice depends on the bolts.

The camber is set out as joint positions #

In a fabrication shop a truss is rarely built from a cutting list. It is assembled in a jig — a set of stops on the shop floor fixing where each joint goes — and the members are cut and fitted between the stops. The camber is then specified as the jig’s geometry: the bottom-chord joints set out on the cambered profile, the top-chord joints above them. Every member’s length follows from the joint positions, webs included.

That changes what the three schemes above mean in practice. A jig set to the dead-load profile at every joint builds the exact camber, every member at its own dead-load strain, without anybody computing a single strain; the arithmetic of this essay is done by the geometry of the jig. A jig set out with only the bottom-chord joints raised, and the top-chord joints left at their drawn heights, builds something else again — a truss whose verticals are all the wrong lengths — and a jig with the camber only at mid-span and straight lines either side builds a camber whose shape is two straight lines. The profile given to the jig is the whole of the specification, and it has to be the dead-load deflection at every panel point, which is exactly what one deflection, without solving everything is not.

A millimetre on every member #

The influences also set how sensitive the camber is to how accurately the members are cut. Every member carries a fabrication error — a sawn length, a drilled hole position, a bolt slip in a splice — and each error arrives at mid-span multiplied by its influence.

Random errors add as the root of the sum of their squares, each weighted by its influence. For this truss that is 6.4 times the per-member error, so a millimetre of scatter on every member is 6.4 mm of scatter at mid-span, an eighth of the camber. Nearly half of a batch of such trusses would miss the specified camber by more than a tenth, and the ones that miss are not the ones with a badly cut member: they are the ones where the small errors on the middle chords happened to lean the same way.

A systematic error is worse. If every top-chord member is cut a millimetre long — a tape that reads short, a jig set out once and used for the whole run — the six top-chord influences add directly, and mid-span rises by their sum, 11.3 mm, nearly a quarter of the camber. The camber specification that allows a few millimetres at mid-span is therefore a specification on the members’ lengths of a fraction of a millimetre, which is tighter than the usual tolerance on a member’s length, and much tighter than the play in a bolted splice with clearance holes.

The splices that slip #

A bolted splice with clearance holes is a length error waiting for its first load. Holes are drilled two millimetres larger than the bolts, and until the bolts bear on the sides of their holes the splice can slide. In a top-chord splice under compression the slip closes the joint and shortens the chord; in a bottom-chord splice under tension it opens and lengthens it. Both move mid-span down. A single splice at mid-span in each chord, each slipping its full two millimetres, costs $2.50 \times 2 + 1.88 \times 2 = 8.8$ mm of camber — 18 per cent of it — the first time the roof is loaded, and never comes back.

That is why cambered trusses with bolted chord splices use preloaded bolts designed not to slip, or fitted bolts in close-tolerance holes, and why the camber of a truss with ordinary bolts in its splices is best measured after the first loading rather than before. A truss’s camber is a property of its joints as much as of its members, and stiffness is not strength in exactly this sense: a splice that is perfectly strong can still give away a fifth of the camber the day it is loaded.

Deeper trusses keep more in their webs #

The two shortfalls move with the truss’s depth, in opposite directions from what one might guess. A deep truss has short, lightly stressed chords and long webs, so its webs carry nearly half of its sag: the deeper the truss, the less a chords-only camber achieves. A shallow truss has influences that grow as its span over depth, so each millimetre of length error is amplified more — but its camber, which grows as the span squared over the depth, grows faster, and the tolerance’s share of the camber falls. The shallow truss needs a larger camber and forgives its members’ errors more; the deep one needs a smaller camber and has to have its webs built into it.

The sun on the top chord #

Temperature is a length change with no load behind it, and the same arithmetic prices it. A uniform change of temperature lengthens every member in proportion and moves nothing out of shape — the truss grows, and stays similar. A difference between the chords is a strain on one of them, and it lifts mid-span by 0.40 mm per degree. Ten degrees between a top chord in the sun and a bottom chord in the shade of the deck is 4 mm, a twelfth of the camber. That matters twice: it is the error in a camber measured in a fabrication yard at midday, and it is a movement the finished roof will make every sunny afternoon, which the movement nobody applied treats as a load case of its own.

The camber by hand #

The influences need no computer. A unit load at mid-span of a simply supported span puts a moment of $x/2$ at a distance $x$ from the nearer support, and the chord force in the panel whose moment centre is there is that moment over the depth. At the middle, $M = L/4 = 6$ m and the chord force is $6/2.4 = 2.50$; one panel out, $4.5/2.4 = 1.88$; two panels out, $3/2.4 = 1.25$. A diagonal’s force is the panel shear, a half, over the sine of its angle: $0.5 \times 3.84/2.4 = 0.80$.

The uniform camber strain follows. A strain $\varepsilon$ on every top-chord member lengthens each by $3{,}000\varepsilon$ mm, and mid-span rises by $\sum n_i L_i\varepsilon = 3{,}000\varepsilon \times (1.25 + 1.88 + 2.50 + 2.50 + 1.88 + 1.25) = 33{,}750\varepsilon$. To lift it 48.8 mm needs $\varepsilon = 0.00145$, or 4.3 mm on each 3 m member. And the systematic error of a millimetre on each of those six members is their influences summed, 11.3 mm.

Where the method stops #

The joints are pins and the truss is statically determinate. In a determinate truss, a member built to the wrong length changes the shape and nothing else: there is no other member to resist it, so no force is locked in. A redundant truss — a counter-braced panel, a continuous truss over three supports — cannot take a misfit so quietly. Its self-stress resists the misfit and puts forces in the members before any load arrives, which built to the wrong length found in proportion to stiffness. A camber built into a redundant truss is a prestress as well as a shape, and it has to be designed as one.

The members are straight and their lengths are all there is. Real trusses have gusset plates, eccentric connections and bolted splices with clearance holes; a bolt that slips into its clearance lengthens the member by up to two millimetres at the first heavy load, and the slip of many joints together is a systematic length error the arithmetic above prices at eleven millimetres per millimetre on the top chord.

The dead load is known. A camber built for the calculated dead load is right only if the dead load is: a heavier roof finish, an added plant platform or a thicker slab than drawn and the camber is taken out and more, while a lighter one leaves the truss humped. The camber for a truss, like a beam’s, is a bet on the load, placed with the members’ lengths.

What the pictures cannot show #

That a truss is fabricated and erected, not built whole. A long truss is usually made in segments and spliced on site, and each splice is a joint whose gap, bolt slip and fit-up decide a length change on the chord right where the influences are largest. The camber is decided as much by the splice details as by the shop’s cutting list, and the drawing that is right except for a rotation is a reminder that the shape of a truss follows from its members’ lengths only after the support conditions are fixed — a truss cambered on the ground and lifted onto bearings that are not level acquires a rotation its camber did not include.

Still open: cambering a truss that cannot take a misfit #

Every result here is for a determinate truss, where a length change is free. A continuous truss over an interior support, or a counter-braced one, resists any camber that is not compatible with its own self-stress states. Whether a camber can be chosen for a redundant truss that takes out its dead-load sag while locking in no force — which requires the camber’s length changes to be orthogonal to every self-stress state — or whether some locked-in force is the unavoidable price of a level redundant truss, is the question that joins this essay’s cutting list to the self-stress of the truss whose forces follow its sections.