The joint that is not where it was drawn

Every truss figure on this site draws its members as lines through their own centroids, meeting at points. That is what makes a joint a pin: a point through which forces pass and about which nothing turns, so the members carry axial force and nothing else and the whole apparatus of resolving at joints works.

A fabricated joint is under no obligation to oblige. Bolt gauge lines are fixed by the section they are drilled in; a diagonal too shallow to reach a deep chord’s centroid cannot be pushed there; a connection detailed to keep the gusset small pulls the working lines apart deliberately. The lines then cross at different places, and the resultant of the forces at the node has a moment about the node.

That moment is real. It is in equilibrium with nothing, so it has to be carried by the members meeting there, and the question is which ones and how much.

The moment is a product, and both terms are large #

The unbalanced moment is $N e$ — the force in the offset member times the distance its line misses by. Neither term is small in a real truss.

A roof truss diagonal at 260 kN is unremarkable. Sixty millimetres of eccentricity is what a 150 mm angle bolted on a single gauge line to a 300 mm chord produces: the angle’s centroid is about 45 mm from its heel, the gauge line is at 55, and the ten millimetres of difference is multiplied by the geometry of the node into something several times that. The product is 15.6 kNm, on a joint every drawing shows as a dot.

For scale, that is roughly the moment a 3 m secondary beam carries under a person standing on it. It is not a rounding error and it does not go away.

The product form is what makes it awkward to police. An eccentricity of sixty millimetres sounds like a tolerance and is treated as one; a moment of fifteen kilonewton-metres sounds like a load case and would be treated as one. They are the same statement about the same joint, and which of the two a drawing office sees depends entirely on whether anybody multiplied. A dimension becomes a load when it is multiplied by a force, and the multiplication is nobody’s job.

Where it goes, which is not where it came from #

A moment applied at a joint divides between the members framing into it in proportion to their rotational stiffnesses: $4EI/L$ for a member held at its far end, $3EI/L$ for one that is pinned there. That is the same distribution rule the hand method uses, applied to an unbalanced moment that arrived from geometry rather than from load.

The consequence is that the moment goes to whichever member is stiffest in bending, and in a truss that is always the chord. At the node drawn, the two chord members take 48 per cent each and the diagonal that caused the whole thing takes two.

The member that produced the eccentricity is not the member that pays for it. A detailer moving a diagonal to keep a gusset small has loaded the chord, and the chord’s designer will not know unless somebody says so. That asymmetry is the practical content of the whole subject: the decision and its consequence sit in different documents.

It also means the usual instinct — make the offending member stiffer — is exactly backwards. Increasing the diagonal’s second moment increases its share of the moment, which reduces what the chord takes; increasing the chord’s, which is what a designer worried about a chord does, increases the share the chord takes. Both members get stiffer and the chord gets worse. The only moves that reduce the chord’s bending stress are reducing the eccentricity, reducing the diagonal’s force, or increasing the chord’s section modulus faster than its second moment — which, on a rolled section, means going deeper rather than heavier.

The number that decides whether it matters #

The moment on its own says nothing useful. What decides whether a noding eccentricity is a detail or a defect is the ratio of the bending stress it produces to the axial stress already there, because that is what a beam–column interaction check adds up.

At the node drawn the chord carries 139 N/mm² of axial stress and picks up 34 N/mm² of bending — 25 per cent as much again. The interaction check is roughly linear, so the member is a quarter closer to its limit than the axial analysis said, on a member that was probably sized by that analysis.

Run the offset down and the ratio comes with it, linearly. Ten per cent — which is about the level at which most people stop worrying — arrives at 28 mm. On a chord of any depth that is a detailing decision rather than a mistake: it is inside what a bolt gauge can move a working line by, and it is smaller than the tolerance on where a gusset gets welded.

Which free body produced the number #

Cut a small region around the joint and draw everything that crosses the cut: four axial forces, each along its own member’s working line.

If the four lines are concurrent, their moment about the intersection is zero and the free body is in equilibrium with forces alone — which is what “pin” means. If they are not, take moments about any point: the resultant moment is $\sum N_i e_i$, where $e_i$ is the perpendicular distance from the point to member $i$’s line. That sum does not depend on which point was chosen, because the forces themselves are in equilibrium; only the moment is left over.

So the free body is the joint, the equation is moment equilibrium, and the moment is not an approximation or a secondary effect — it is a term the concurrent drawing set to zero by construction. Restoring it is not a refinement of the truss model; it is the truss model applied to the geometry that exists.

The other eccentricity, which is worse and less discussed #

Everything above is in the plane of the truss. There is a second one at right angles to it, and it is usually larger.

A diagonal made of a single angle bolted to one face of a gusset has its centroid off the gusset plane by half its thickness plus the angle’s own centroid distance — twenty or thirty millimetres, routinely. The axial force therefore acts eccentrically about the weak axis of a member whose weak axis is very weak indeed, and that member is already using only part of itself because the load enters one leg.

The in-plane case is computed and checked; the out-of-plane case is swallowed into an effective-area factor and rarely appears as a moment at all. That is defensible for a single angle in tension and much less so for one in compression, where the eccentricity is an initial bow the member is already imperfection-sensitive to.

The gusset is not a bystander #

Between the members’ working lines and the node there is a plate, and treating the eccentricity as a point moment applied to bare members leaves it out entirely.

That is conservative for the chord, because the gusset stiffens the region where the moment is largest. It is not conservative for the gusset. The plate is carrying the whole unbalanced moment across itself, in its own plane, over a width that is whatever the load spreads to and a thickness of eight or ten millimetres. A 15.6 kNm moment across a 300 mm effective width of 10 mm plate is 104 N/mm² of bending — comparable with the direct stresses the plate was sized for, and taken on an axis the sizing calculation never mentioned.

So the honest statement is that the eccentricity is not absorbed anywhere. It is redistributed among the chord, the gusset and the bolt group by three different rules, each of which is well known, and the sum of the three is rarely assembled by anybody.

Why the codes let most of it go #

Rules for lattice structures generally permit noding eccentricities within some fraction of the chord depth to be neglected, and the permission has a real argument behind it rather than being an indulgence.

The moment is applied at a point and dies away along the chord over a length set by the chord’s own bending stiffness and the restraint at the next node. It is largest exactly at the joint — where the section is locally reinforced by the gusset, the welds and the bolts, and where the chord is not its bare self at all. So the peak bending stress computed from the bare section at the node is an overestimate, and the section that is genuinely bare is one that sees a reduced moment.

That argument has a limit and the limit is the point. It works when the joint region is genuinely stiffer and the eccentricity is small; it fails when the eccentricity is large enough that the moment is still significant a chord depth away, and it fails completely in fatigue, where the stress range at the weld toe is the whole design and the local reinforcement does not help. A noding eccentricity on a bridge truss is a fatigue problem before it is a strength one.

The tubular joint, where the geometry is the design #

The clearest case is the one where the eccentricity is chosen on purpose.

In a welded tubular truss the braces are profiled and welded straight onto the chord, and there is a minimum gap between the toes of adjacent braces that fabrication requires. Meeting that gap and keeping the working lines concurrent are usually incompatible, so the designer picks: a gap joint with a deliberate positive eccentricity, or an overlap joint with a negative one.

Both are standard, both have their own capacity formulae, and both are accompanied by a rule of the form the resulting moment may be neglected in the chord if the eccentricity lies within some band of the chord diameter. The band exists because somebody did the calculation above and decided where the ratio stops mattering.

What is worth carrying from that is the attitude rather than the numbers. The eccentricity is a design variable, chosen, recorded and checked — not an accident of fabrication discovered later. Every joint on every truss could be treated that way and almost none is.

What it does to a compression chord #

There is a stability consequence that the stress ratio understates.

A chord carrying axial compression and a moment applied at the node is a beam–column, and the moment is amplified by the axial force it shares the member with: the second-order effect multiplies the first-order bending by $1/(1 - N/N_{cr})$ over the length between restraints. A chord at half its critical load doubles the noding moment before any interaction check sees it.

Worse, the moments at successive nodes generally alternate in sign, because the diagonals alternate in direction. That produces a chord in double curvature between nodes, which is the most favourable moment distribution there is for lateral-torsional stability and the least favourable for the first-order stresses at the nodes themselves. The two effects pull in opposite directions and neither is small.

There is a third, and it is the one that turns a stress problem into a stability problem. The moment at a node deflects the chord sideways between restraints, and a compression member with a lateral deflection is a member with an imperfection — indistinguishable, as far as the buckling calculation is concerned, from the initial bow the column curve was fitted to. So a noding eccentricity does not merely add a stress to be checked in an interaction equation: it consumes part of the imperfection allowance the chord’s buckling resistance was derived with, and the two are added by nobody because they live in different clauses.

The one number worth carrying #

Everything above collapses to a ratio that can be worked out on the back of a drawing, and it is worth stating in a form that does not need the distribution calculation.

The chord takes about half the unbalanced moment. Its bending stress is therefore roughly $Ne/(2Z)$, and the axial stress it already has is $N_c/A$. So

$$\frac{\sigma_b}{\sigma_N} \approx \frac{N e A}{2 Z N_c} = \frac{e}{2}\cdot\frac{N}{N_c}\cdot\frac{A}{Z},$$

and $A/Z$ for a rolled section is about $2/(0.4d)$ — five over the depth. Putting the three together, the ratio is about $2.5\,e/d$ times the ratio of the diagonal’s force to the chord’s.

The eccentricity matters in proportion to how much of it there is measured in chord depths, multiplied by how heavily loaded the diagonal is relative to the chord. On the joint drawn, $e/d = 0.2$, the force ratio is 0.3, and the estimate gives 0.15 against the computed 0.25 — the right size, from three numbers on a drawing, which is what a rule of thumb is for.

Two ways to find it #

The eccentricity is invisible in a line-model analysis, because the line model is the concurrent drawing. So it has to be found somewhere else, and there are only two places.

In the geometry, before fabrication. Take the setting-out of the joint and compute the perpendicular distances from the intended node to each member’s centroidal axis. If they are not zero, the moment is $\sum N_i e_i$ and the arithmetic above applies. This costs a minute per joint type and it is the only method that finds the problem in time to change it.

In the built structure, by measurement. Strain gauges on opposite faces of a chord adjacent to a node separate axial from bending directly: the mean of the two is the axial stress and half their difference is the bending. It is the definitive answer and it arrives years too late to be a design tool, which is why it appears mostly in the literature on why an existing truss cracked.

Where the model stops #

The distribution assumes the members stay elastic. Once a chord yields at the node the moment redistributes to whichever member is still stiff, and the elastic shares stop describing anything.

The far-end conditions are a guess. Whether a member is $4EI/L$ or $3EI/L$ depends on what happens at its other end, which in a truss is another joint of the same uncertain kind. The shares are not sensitive to this — the chords dominate either way — but the individual numbers are.

The joint has size and the model has none. Treating the eccentricity as a point moment applied to bare members ignores the gusset entirely, which is conservative for strength at the node and says nothing at all about the gusset itself.

The moment is static and the geometry is not. Everything above computes the eccentricity from the drawing. A truss that has deflected has moved its own nodes, and a chord that has bowed has moved its own centroidal axis relative to the line the analysis drew — by an amount comparable with the fabrication eccentricity on a slender member near its capacity. The two are indistinguishable in a measurement and are attributed differently in a report.

And nothing here covers the case where the eccentricity changes under load. A bolted joint that slips takes up its clearance and moves its own working lines, so the joint that carries nothing until it slips has a different geometry before and after, and the moment computed for one is not the moment in the other.

Where the ladder goes #

Later rungs on this anchor: gap and overlap joints in tubular trusses, and the eccentricity limits that come with each. Out-of-plane eccentricity in single-angle and back-to-back members, and why it is handled as an area reduction rather than as a moment. Secondary stresses in trusses generally, of which this is one source and joint rigidity is another. The interaction of noding moments with chord buckling between nodes. Fatigue at eccentric welded joints, where the whole allowance is the stress range. Setting-out tolerance, and how much of the eccentricity in a real structure was designed and how much arrived. And the measurement problem: separating axial from bending in a member nobody can get a gauge onto.