The chord is a continuous beam

A truss analysis returns one number per member. Assemble the joint equations and eliminate, and every member comes back with an axial force and nothing else, because that is all a pin-jointed bar can carry.

The steel does not know that. A top chord running through eight panels is one length of section, continuous over the points where the web members meet it, and anything that lands on it between those points makes it bend.

Where the second load path comes from #

The truss above was solved with the whole 600 kN applied at the nodes. Real load does not arrive that way unless something puts it there.

If the roof deck spans between purlins and the purlins sit exactly over the panel points, the idealisation is true and the chord carries axial force alone. If the deck sits directly on the chord — a metal deck, a concrete slab, a walkway — the load is distributed along it, and the chord is a beam spanning 3 m between its own nodes.

The two demands are computed by two different calculations and neither knows about the other. The truss solver reports −605 kN. A beam analysis of the same piece of steel reports 22.5 kNm. The section carries both.

That is not a refinement. For a 200 × 200 × 10 mm hollow section the axial force uses 23 per cent of the squash load and the moment uses 12 per cent of the plastic moment — so the bending nobody reported is a third of the utilisation, and it arrived from a load path the drawing shows and the analysis does not.

How much of the section each demand takes #

Putting the two numbers on one section makes the size of the effect concrete, and it is worth doing because the instinct is to treat chord bending as a correction.

Take a 200 × 200 × 10 mm square hollow section for the top chord: an area of 7,400 mm², a plastic modulus of 5.3 × 10⁵ mm³, and a yield strength of 355 N/mm². Its squash load is 2,630 kN and its plastic moment 188 kNm.

The truss reports 605 kN, which is 0.23 of the squash load. The beam analysis reports 22.5 kNm, which is 0.12 of the plastic moment. Neither alone is alarming. Added on a linear interaction they reach 0.35, and the bending contributes a third of that — from a load path the analysis did not model, at a member the analysis said was carrying axial force only.

The proportion is not a constant. Bending scales with the deck load and the square of the panel length; axial force scales with the total load and the inverse of the depth. So a deep truss with short panels carrying a light deck has almost no chord bending, and a shallow truss with long panels carrying a slab has a chord in which bending is the larger of the two. The idealisation is excellent in one corner of the design space and misleading in another, which is a much more useful statement than calling it approximate.

There is also a stability consequence hiding in the pair. A member carrying axial compression and bending buckles at a lower axial force than one carrying compression alone, because the moment gives the imperfection somewhere to grow from — which is what an interaction surface exists to describe. The chord is therefore not merely more heavily stressed than the truss solution says; it is also less stable than a member at 605 kN would be.

Continuity redistributes and does not reduce #

The chord is continuous over its nodes rather than simply supported at them, and continuity is worth having: the peak moment falls from 28.1 kNm to 22.5. It is worth being exact about what that buys, because it is easy to read as a capacity.

The free moment $wL^2/8$ belongs to the free body of one panel and to nothing else. Whatever the end moments are, the sagging ordinate at mid-panel is the free moment minus their average, so raising one lowers the other by exactly as much. Continuity moves the peak; it does not remove any of it.

That has a practical reading for a chord. A chord that is continuous carries 22.5 kNm at a node and 18.0 in a panel; one that is spliced at every node carries 28.1 in every panel and nothing at the splices. Which is better depends entirely on where the axial force is largest — and in this truss the largest axial force and the largest hogging moment are at the same place, which is the worst possible arrangement and is also the usual one.

Which free body produced the number #

Two free bodies, taken at different scales of the same structure, and the reason the second is invisible to the first is worth stating precisely.

The truss solver’s free body is a joint — a disc cut out with every member severed just outside it. That cut is what forces the answer to be axial: it is taken at a point, the forces on it are concurrent, and a concurrent force system has no moment equation. Nothing about it is wrong, and nothing about it can see a load between two joints.

The chord’s free body is a length of member, cut with two vertical sections a panel apart. Crossing those cuts are an axial force, a shear and a moment, and the moment is what the panel-length beam analysis returns. The load standing on the chord between the two cuts is inside this free body and outside every joint free body in the structure.

So the two calculations are not competing answers to one question. They are answers to two questions about the same steel, and the pin-jointed idealisation is not an approximation here but a change of subject — a fact about which free body was drawn rather than about how accurately anything was computed.

The load arrangement matters, and there are eight of them #

If the load on the chord is partly permanent and partly variable, the worst moment is not produced by the worst total.

Loading alternate panels maximises the sagging moment in the loaded ones; loading adjacent panels maximises the hogging over the node between them. The chord therefore has a design moment that no single state of the structure produces, which is the ordinary property of an envelope arriving in a place nobody expects it — inside a member of a truss.

It has a second consequence that matters more. Pattern loading on the chord does not change the truss’s axial forces much, because those depend on the total delivered to each node. So the two demands vary independently: the arrangement that maximises the moment is not the one that maximises the force, and combining the worst of each is conservative by an amount nobody usually quantifies.

Purlin spacing is the design variable #

Because the moment goes as the square of the span between supports, the cheapest way to remove it is to support the chord more often.

A factor of four for one extra line of purlins is a better return than anything available inside the truss, and it explains a detail that otherwise looks arbitrary: purlin spacing on a trussed roof is usually a divisor of the panel length rather than a round number.

It also explains the opposite case. Where the deck sits directly on the chord and cannot be supported more often — a walkway, a slab, a crane rail — the chord’s bending is fixed by the panel length, and the panel length is then being chosen by a bending calculation rather than by the buckling argument that usually decides it. Two quite different considerations reach for the same number, and they do not agree.

What to do with the moment once it is there #

A chord with 22.5 kNm in it is a beam-column, and there are three ways of dealing with that.

Carry it. Check the section for combined axial force and bending on an interaction curve, size for the pair, and accept the section that results. This is what is usually done and it is correct.

Redistribute it. The chord is a redundant beam, so the elastic distribution is not the only one in equilibrium.

Redistribution is legitimate because the lower-bound theorem says so, and the currency it is paid for in is rotation capacity. A chord already carrying 23 per cent of its squash load has less rotation capacity than a beam does, because the axial force pushes the section toward a class in which the compression flange buckles before the hinge turns.

Remove it. Splice the chord at the nodes so that it really is a series of pin-ended bars. This is what a bolted lattice does and it is the reason a lattice tower’s members are single angles bolted through one leg: the idealisation is not assumed, it is built.

The bottom chord has the same problem and a different answer #

Everything above is about the compression chord because that is where a deck usually sits. The tension chord picks up bending too, and what it does with it is not symmetric.

A bottom chord carries a ceiling, services, a walkway or a suspended floor, and each of those delivers load between the nodes exactly as the deck does above. The moment is computed the same way; the interaction is not. A section in tension and bending has no stability problem at all, so the two demands add on a plain interaction with no buckling term and no second-order amplification — and a tension chord at 567 kN and 22.5 kNm is in genuinely less trouble than a compression chord at the same pair.

That asymmetry is the reason a hung load is easier to accommodate than a supported one, and it is the same asymmetry that runs through the whole subject: a load hung from below has to be lifted back into the structure and, once it has been, the member carrying it is not at risk of the failure that governs everything above it.

The one thing the tension chord does worse is fatigue. Bending stress adds to axial stress at the extreme fibre, the range is what matters, and the surface where the two add is a welded connection — so a chord that is comfortable at the ultimate limit state can be the detail that decides the life.

The moment that is not this one #

There is a second source of chord bending that is often confused with the first, and separating them is the point of the last figure.

Secondary bending comes from the joints not being pins. The truss deflects, the angles between members change, and a welded joint forces the member ends to rotate with the joint rather than freely — so a moment appears with no load applied between nodes at all.

The two moments differ in every way that matters:

Cause. One is a load on the member; the other is a rotation imposed on its ends.

Where it is largest. Chord bending is largest where the load is; secondary bending is largest where the truss’s curvature is, which is mid-span, and in the stubbiest members.

What relaxes it. Secondary bending is a displacement effect, so a hinge anywhere in the member releases it and it does not reappear. Chord bending is a load effect and does not go away; a hinge in the chord makes it larger.

Whether it is checked. Codes routinely permit secondary bending to be ignored for strength when the members are slender enough. No code permits chord bending to be ignored, and the distinction is exactly the one above.

What a truss analysis is claiming #

Stepping back from the arithmetic, the useful thing here is a statement about what a model asserts and what it merely does not mention.

The pin-jointed solution asserts that the forces it reports are in equilibrium with the loads it was given, at the joints it was given them at. That claim is exact — the residual at every joint in the truss above is a few parts in 10¹⁵ — and it remains exact when the chord is bending, because bending in a member does not disturb the equilibrium of the nodes at its ends.

What the solution does not assert is that those are the only forces in the members. A model that is exactly right about what it computes can be silent about a demand of comparable size, and the silence looks identical to an absence. That is not a shortcoming of trusses; it is the ordinary condition of every idealisation on this site, and choosing which free body to draw is the act that decides what a calculation can see.

The practical form of the rule is a habit rather than a check: for every member, ask what is physically attached to it between its ends. If the answer is nothing, the truss solution is the whole answer. If the answer is a deck, a purlin off the node, a services run or a person, there is a second calculation owed and no gate anywhere will ask for it.

Where the model stops #

The chord is treated as a beam of uniform section. A real chord changes section along its length, and the moment redistributes toward the stiff parts when it does.

The panel points are treated as rigid supports. They are not: a node deflects as the truss deflects, and the differential settlement between adjacent nodes changes the chord’s support moments. On a stiff truss that is small; on a shallow one it is not.

Axial force and bending are computed separately and combined afterwards. With 605 kN of compression in a 3 m member there is a second-order amplification of the moment, which this arithmetic does not include and which is worth about a few per cent here and much more in a slender chord.

The load is taken as vertical and in the plane of the truss. A chord carrying a deck also carries wind on the cladding, out of plane, and out of plane it spans between bracing points rather than between nodes — a longer span and a weaker axis.

The deck is assumed to deliver a uniform line load. A profiled metal deck delivers its load at its ribs, a slab delivers it continuously, and a walkway delivers it wherever somebody stands. The first is a series of point loads at 150 or 300 mm centres, which for a 3 m panel is indistinguishable from a uniform load; the last is not, and a single heavy point load at mid-panel produces half again the moment that the same total spread out does.

And nothing here is a torsion calculation. A load applied to one flange of an open chord section, off the shear centre, twists it as well as bending it, and that demand is invisible to every figure on this page.

There is one more thing the model cannot be asked. Nothing in either calculation says where along the chord the section may change, because both assume it does not. A chord spliced at a third point rather than at a node has a discontinuity in stiffness inside a panel, and the moment diagram there is neither of the two drawn above — which is a reminder that the panel is a unit of the analysis and not necessarily a unit of the steelwork.

The ladder from here #

Later rungs on this anchor: counters and tension-only diagonals, where a member is present under one load case and absent under the other. The counting rule in three dimensions, where a satisfied count can still hide a mechanism. The chord’s out-of-plane length, which belongs to the bracing system rather than to the truss. Eccentricity at the node, where the members’ centrelines do not meet at a point and the offset produces a moment that is primary rather than secondary. Fatigue at the weld toe, where the moment described here and the one before it add and the sum is what cracks. And the transition to the Vierendeel, in which the diagonals are removed and chord bending stops being a correction and becomes the entire structural mechanism.

The distinction on this page is old and was drawn the hard way. Nineteenth-century lattice bridges were analysed as pin-jointed frames and built with riveted gusset plates, and the cracks that appeared at the joints over the following decades were attributed for a long time to poor workmanship. They were the two moments described here, doing what they do, in members that had been checked for neither.