The joint whose lines may not meet

The joint that is not where it was drawn treated a noding eccentricity as a fact of fabrication: the working lines of the members at a truss joint miss one another by some distance, the resultant has a moment about the node, and the question was where that moment goes. It ended on the welded tubular truss as the case where the eccentricity is chosen rather than suffered, and it drew the two standard choices — a gap joint with the lines meeting below the chord’s axis and an overlap joint with them meeting above — as the same node with the sign reversed, about which the chord’s arithmetic is indifferent.

That is true of the chord and it is not the whole of it. In a tubular joint the eccentricity is not a variable a designer sets and then checks. It is a consequence of three dimensions set for other reasons, and following it through shows something the concurrent drawing hides completely: for a large share of ordinary proportions, the joint whose working lines meet on the chord’s axis is forbidden by the fabrication rules, and every buildable joint misses by a margin nobody picked.

Three dimensions, and no fourth #

A K joint in a welded tubular truss is two braces landing on a chord side by side. Each brace is cut to the saddle shape of the chord’s surface and welded all round, with a weld whose strength depends on which way it is loaded. Where the two braces arrive, their toes — the edges nearest each other — are either separated by a gap along the chord’s crown, or one brace is laid partly over the other in an overlap.

Once the chord diameter $d_0$, the brace diameter $d_1$, the brace angle $\theta$ and the gap $g$ are fixed, the geometry is fixed. The brace centrelines run down through the chord and meet at a point a distance $e$ from its axis, positive when the point is below the axis on the side away from the braces:

$$e = \left(\frac{d_1}{2\sin\theta_1} + \frac{d_2}{2\sin\theta_2} + g\right)\frac{\sin\theta_1\sin\theta_2}{\sin(\theta_1+\theta_2)} - \frac{d_0}{2}$$

with an overlap entered as a negative gap. There is no fourth dimension to adjust. The eccentricity is not set out; it arrives.

The joint drawn at the top of this page is a 168.3 × 8 mm chord with two 114.3 × 6.3 mm braces at 45 degrees and a gap of 12.6 mm. Its working lines meet 3.0 mm below the chord’s axis — an eccentricity of 0.02 chord diameters, which is as close to concurrent as a drawing office would ever ask for. That is not the result of anybody’s care. It is what that gap produces with those tubes at that angle.

Why a band of gaps is forbidden #

The eccentricity is a straight line in the gap. The line itself is continuous; what is not continuous is the set of gaps anyone is allowed to build.

A gap joint needs room to weld. Both braces are welded all the way round their saddle, including along their toes, and the two toe welds need space for a welding torch and for the weld metal itself. The rule is that the gap must be at least the sum of the two brace walls, $g \ge t_1 + t_2$ — here 12.6 mm. Below that, the toe welds run into each other and neither can be made properly.

An overlap joint needs enough overlap to work as one. When one brace sits on the other, part of the force passes directly from brace to brace through the weld where they lie together, bypassing the chord wall. That is the whole reason overlaps are strong, and it only happens if the overlap is large enough to carry the shear: the rule is at least a quarter of the overlapping brace’s contact length $d_1/\sin\theta$, which here is an overlap of 40.4 mm.

Between those two limits nothing may be built. Between a gap of 12.6 mm and an overlap of 40.4 mm lies a band 53 mm wide, and the line of eccentricity crosses zero right in the middle of it, at a gap of 6.7 mm. The one joint whose lines meet is a joint in which the toe welds cannot be made and the overlap cannot carry anything.

So the designer has two choices, and neither of them is concurrent. The smallest legal gap puts the lines 3.0 mm below the axis. The smallest legal overlap puts them 23.5 mm above it. The truss was analysed with them meeting exactly on it.

The overlap, and why it misses the other way #

Built as an overlap, the joint moves its node to the other side of the chord’s axis. A 50 mm overlap puts the working lines 28.3 mm above it. The sign has changed because the brace centrelines now land on the chord closer together than they would need to for their lines to reach the axis; the geometry is the same formula, fed a negative gap.

The earlier essay made the point that the chord’s bending does not care about the sign — a moment of the same size bends the chord the same amount the other way. What it could not show, because it took the eccentricity as given, is that the joint cares a great deal. Overlaps and gaps are different joints with different strengths, and the choice between them is a choice about the joint first and the eccentricity second.

The chord face is strongest where the gap is smallest #

The commonest failure of a tubular K joint is not in the braces or the welds. It is the chord’s own wall, pushed in by the compression brace and pulled out by the tension brace, folding locally between them. The resistance to that is a plastic mechanism in the chord wall, and the design formula carries a gap factor that captures how much of the wall between the braces is available to fold. A short gap between stiff toes leaves a short, stiff strip of chord wall; the two braces work against each other through it, and the mechanism needs more force.

So the joint wants the gap small. At 60 mm it resists 464 kN per brace; at the smallest gap that can be welded, 564 kN, a fifth more; and at the smallest legal overlap 627 kN, where part of the force has stopped going through the chord wall at all. The curve continues smoothly through the forbidden band, because the formula knows nothing about welding torches.

Put that beside the eccentricity and the picture is consistent in a way worth noticing. On the gap side, the strongest legal joint is also the one whose lines come closest to meeting: both the chord face and the concurrency want the gap closed, and the one thing that stops both is the welder’s minimum. A designer who takes the smallest gap the rule allows has made the best joint on both counts, and the eccentricity left over is the price of being able to weld it.

That is a real design rule and it is not the usual one. Gaps in tubular trusses are often drawn generous — 30, 40, 60 mm — because a generous gap is easy to fabricate and easy to inspect, and nothing in the frame analysis objects. Here that generosity costs a fifth of the chord-face resistance and moves the working lines 12 to 27 mm off the axis.

A steeper brace leaves no room at all #

Change the brace angle from 45 to 60 degrees and the same braces on the same chord behave very differently. The line of eccentricity is steeper — each millimetre of gap now moves the node 0.87 mm — and it has shifted upward, because steeper braces land further apart for the same gap.

The smallest weldable gap, 12.6 mm, now puts the working lines 41 mm below the axis, 0.24 chord diameters. The band inside which the joint rules let the moment be neglected ends at 0.25. The designer of this joint has one and a half millimetres of gap to play with before the eccentricity leaves the band, and every generous gap a fabricator might prefer is outside it.

The concurrent joint has not disappeared at 60 degrees; it has moved into the overlap region. The lines would meet at an overlap of 34.8 mm, which is 26 per cent of the contact length and therefore legal, just. So at 45 degrees these braces can meet only in the forbidden band, and at 60 degrees they can meet only by overlapping. Neither of them allows the joint every tubular truss drawing shows: a gap joint with the lines through the axis.

Where a concurrent joint can be built at all #

The two cases generalise into a map. Over brace angles from 30 to 70 degrees and brace diameters from three to nine tenths of the chord’s, the gap at which the working lines meet falls in one of three places. For small braces at shallow angles it is a positive gap wide enough to weld: those joints can be built concurrent, as gap joints. For large braces at steep angles it is an overlap of more than a quarter: those can be built concurrent, but only as overlaps. And between the two regions, over a third of the map, it falls in the forbidden band.

The band runs diagonally, which is the geometry’s way of saying that brace size and brace angle trade against each other: a larger brace needs a shallower angle to leave a weldable gap at concurrency, and a steeper brace needs a smaller one. The joint drawn at the top, with braces at 0.68 of the chord’s diameter and 45 degrees, sits just inside that band — neither small enough nor steep enough, and so neither concurrent option exists for it.

A truss designer choosing member sizes by force and member angles by the panel geometry lands somewhere on this map without having looked at it. A third of the time, the concurrent joint the analysis assumed does not exist, and some eccentricity — whichever the nearest legal gap or overlap produces — has been chosen by default.

What the band lets the joint ignore, and the chord cannot #

The joint rules deal with all this by permitting an eccentricity band: the moment may be neglected in the joint’s own resistance check if $e$ lies between $-0.55\,d_0$ and $+0.25\,d_0$. The usual explanation is that the joint resistance formulae were fitted to tests whose specimens had eccentricities spanning roughly that range, so their effect is already inside the formula’s calibration. It is a statement about the joint, and about what the tests covered.

It is not a statement about the chord. The two chord members on either side of the joint still carry the unbalanced moment, divided by their stiffnesses exactly as moment distribution divides any moment at a joint, and a compression chord carrying a moment is a beam–column whose bow adds to the one it was never straight without. The rules say so: inside the band, the moment may be dropped for the joint and for tension members, and must be kept for the compression chord.

The size of what is being carried is not small at the edges. With braces at 250 kN each, their components along the chord change its force by 354 kN across the joint, and that change acts at the eccentricity. At the gap side’s limit of $0.25\,d_0$, 42 mm, the chord’s bending reaches 24 per cent of its axial stress. At the overlap side’s limit of $-0.55\,d_0$, 93 mm, it reaches 53 per cent. Both are eccentricities the joint check is entitled to ignore, and on a compression chord both are moments that second-order effects will then amplify before any interaction check sees them.

At the joints actually drawn here the numbers are modest: 2 per cent at the smallest gap, 14 at the smallest overlap. That is the practical case for taking the smallest gap — it is the strongest joint and it nearly removes the chord’s bending — and the practical danger of a generous gap on a steep brace, which can put the chord at the band’s edge while every joint check passes.

Why the band is lopsided #

The band is more than twice as wide on the overlap side as on the gap side: $0.55\,d_0$ against $0.25$. Nothing in the chord explains that; the chord’s bending is symmetric in $e$, as the figure shows. The asymmetry is the joint’s.

In a gap joint all of each brace’s force enters the chord wall, which is the thing the gap factor is about, and a moment in the chord adds to the stresses in exactly the wall that is folding. In an overlap part of the force goes from brace to brace without touching the chord wall at all, and the joint’s governing mechanisms are in the braces and the weld between them rather than in the chord face. A moment in the chord matters less to a joint that is not relying on the chord face, and so the tested range could run further in that direction before its effect became visible. That reading is consistent with the rules; it is an inference about why they are the shape they are, and the figures here do not test it.

The concurrent drawing, again #

The earlier essay made the point that a truss analysed as pin-jointed is drawn with every member meeting at a point, and that the drawing sets the unbalanced moment to zero by construction. That is before the joints’ own rigidity adds its bending. In a steel truss with gussets, the eccentricity is then something to find: a setting-out decision somewhere in the detailing that moved a working line.

In a welded tubular truss it is not something to find, because it was never a decision. It is determined by $d_1$, $\theta$ and $g$, and $g$ was chosen for the welder. The eccentricity is a derived quantity that nobody computes, because every input to it was decided by somebody else for a different reason: the brace size by the member force, the angle by the panel geometry, and the gap by the fabricator’s preference or the smallest the rule allows. The one person who could compute it — the truss analyst — works from a drawing on which it is zero.

The practical remedy costs one line of arithmetic per joint type, the formula at the top of this page, and it answers three questions at once: whether the concurrent joint can be built, which legal joint comes nearest, and whether that joint’s eccentricity is inside the band. A spreadsheet column would do it. Almost no truss design carries one.

The joint’s geometry, and the resistance formula behind it #

The eccentricity is geometry alone: the two brace centrelines, drawn from where each meets the chord’s crown at the stated gap, carried down until they intersect. The drawing at the top computes that intersection from the picture’s own coordinates and checks it against the formula, and the two agree to the precision of the arithmetic.

The chord-face resistance is Eurocode 3’s for a circular hollow section K joint: $N_{1} = k_g k_p f_{y0} t_0^2 (1.8 + 10.2\,d_1/d_0)/\sin\theta_1$, with the gap factor $k_g = \gamma^{0.2}\left(1 + 0.024\,\gamma^{1.2}/(1 + e^{0.5g/t_0 - 1.33})\right)$ and $\gamma = d_0/2t_0 = 10.5$. The chord-stress factor $k_p$ is taken as one, which is right for a tension chord and generous for a compression chord, and the partial factor as one, so the numbers are characteristic resistances. For an overlap the same formula is fed a negative gap, which is how the rules state it; an overlap joint also has checks of its own on the overlapping brace that are not drawn.

The chord’s bending takes the free body of the joint region: two brace forces of 250 kN at 45 degrees and the chord on either side. Their components along the chord differ by 354 kN, that difference acts through the node at the eccentricity, and the moment it makes is shared equally by the two chord members, which have equal stiffness.

The weld, the saddle and the chord’s own force #

The weld. A gap of 12.6 mm is the rule’s minimum, not a guarantee that a real welder with a real torch can reach the toe. Fabricators often want more, and the figures price what more costs.

Out-of-plane geometry. A real K joint is a saddle, and the braces’ toes meet the chord at points that are not in the plane of the drawing. The gap is conventionally measured along the crown, and the formula is exact there; the weld and the mechanism are three-dimensional.

The chord’s force. The chord-face resistance falls when the chord is heavily compressed, through $k_p$, and the figure has held it at one. On a heavily loaded compression chord the curve is lower everywhere and the ranking between gaps unchanged.

Fatigue. At a welded tubular joint the governing check for a bridge or an offshore structure is the stress range at the weld toe, and there a small added stress is most of the allowance. The band in which a moment may be neglected for static strength has no counterpart in fatigue.

The fabrication limits are rules, not physics #

That the rules’ two fabrication limits — a gap of at least $t_1 + t_2$ and an overlap of at least a quarter — are hard. They are rules, not physics, and a fabricator with a narrow torch and a good procedure can make a joint inside the band; some codes permit smaller overlaps with extra checks. What the map shows is where the concurrent joint falls against the rules as written, which is what a designer has to satisfy; it does not show that a joint inside the band would fail.

Still open: the eccentricity that is not in the plane of the truss #

Every tubular joint here is planar — two braces and a chord in one plane, the eccentricity measured in it. A space truss, a triangular girder or a multiplanar tower leg has braces arriving from two planes at once, and their working lines have to meet the chord’s axis in both. The gap in one plane and the gap in the other are set by different braces, the eccentricities they produce are at right angles, and the chord receives moments about two axes at the same joint. Whether the permitted band has any meaning for a joint with two eccentricities — and how often a multiplanar joint that is legal in both planes can be concurrent in either — is a question the planar formula cannot answer, and it is the joint that carries most of the world’s tubular towers.