Two planes of braces and one circumference

The joint whose lines may not meet found that a welded tubular K joint’s noding eccentricity is not a matter of care in setting out. It follows from three dimensions chosen for other reasons — the brace’s diameter, its angle and the gap between the toes — and for a third of ordinary proportions the one gap at which the working lines would meet on the chord’s axis is a gap no fabrication rule allows. Every joint in that essay was planar: two braces and a chord in one plane.

Its last section named the joint that carries most of the world’s tubular towers. A tower’s leg, a triangular girder’s chord and a space truss’s node all receive braces from two planes at once. Each plane’s K has its own gap and its own eccentricity, at right angles or at sixty degrees to the other’s, and the chord receives moments about two axes at the same joint. The essay asked 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.

Both answers turn out to depend on something the planar drawing does not show at all: the two planes’ braces are welded to the same tube, and there is only so much tube.

Where a brace meets a chord #

A brace welded to a chord meets it along a curve, the intersection of two cylinders — a saddle-shaped loop that is longest along the chord at the crown and reaches furthest round it at the sides. That curve is where the weld goes, and it is where one brace’s weld has to stay clear of another’s.

The curve can be found exactly. Each line drawn along the brace’s surface, parallel to its axis, meets the chord’s surface at one point, which is the root of a quadratic; going round the brace gives the whole loop. Unrolling the chord — cutting it along a line and laying it flat — turns the loops into closed shapes on a plane, where the distance between two of them can be measured.

The joint drawn is the earlier essay’s: 114.3 × 6.3 mm braces at 45 degrees on a 168.3 × 8 mm chord, each K with a 12.6 mm gap between its own toes — the smallest in which both toes can be welded, $t_1 + t_2$. That K is legal on its own. Its working lines miss the chord’s axis by 3 mm, well inside the band in which the planar rules allow its moment to be neglected.

Put a second, identical K on the same section of the chord in a plane 90 degrees round, as a square tower’s leg has. On the unrolled surface the four footprints sit in a square: two across the chord’s axis, two above them. Within each K the toes are 12.6 mm apart, as designed. Between the two planes, a brace of one comes within 6.5 mm of a brace of the other. Two fillet welds back to back need 12.6. The joint, legal in both planes, cannot be welded.

Every brace takes the same arc of the chord #

The reason is plain once the chord is seen in section.

A brace whose axis lies in its plane reaches round the chord exactly as far as its own radius reaches across that plane: the footprint’s furthest point from the plane is where the brace’s side, $d_1/2$ from its axis, meets the chord, and that is an angle of $\arcsin(d_1/d_0)$ either side of the plane, whatever the brace’s angle and whatever its gap. For a 114.3 mm brace on a 168.3 mm chord, $d_1/d_0 = 0.68$ and the footprint spans 85.6 degrees. Two planes 90 degrees apart, each needing 85.6, leave 4.4 degrees between them: 6.5 mm of surface.

That is the result that matters. Moving a K’s toes apart or together slides its two footprints along the chord; it does not move them round it. Nothing a designer can choose within a plane changes how much of the chord’s circumference the plane’s braces take. The room between planes is fixed by the brace’s diameter, the chord’s, and the angle between the planes — the three things the planar checks never compare.

Move the planes to 60 degrees, as a triangular girder’s chord has them, and there is no room at all. Each plane still needs 85.6 degrees and the two are only 60 apart. The footprints overlap, and the braces of one plane would have to be cut to fit the braces of the other.

How slender the braces have to be #

Against the brace’s diameter the room between the planes falls almost linearly, since it is the arc left after two arcs of $2\arcsin(d_1/d_0)$. At 90 degrees it passes below the welds’ 12.6 mm at a diameter ratio of about 0.66. At 60 degrees it passes below at about 0.43, and the footprints touch at 0.5.

Those are not extreme proportions. A brace two-thirds the chord’s diameter is not unusual in a tower leg, where the bracing is sized for buckling and wants a stocky tube, and the joint’s strength rises with the diameter ratio, which pushes designers towards the larger braces. On a triangular girder 0.43 is slender: half the chord’s diameter is common, and at half the planes’ footprints meet.

How often both planes can meet at one point #

The earlier essay’s question — how often a joint legal in both planes can be concurrent in either — now has two parts. Each plane has to have a gap at which its working lines meet on the chord’s axis that the welding and overlap rules allow, which is the planar question. And with both planes concurrent at the same section, their braces have to leave room to weld between them, which is not.

For a square tower’s leg, of 77 proportions from slender braces to stocky ones and from 30 to 60 degrees, 46 can have both planes concurrent at one section. Ten more are concurrent in each plane but have no room between the planes; ten have room but no legal concurrent gap in either plane, the earlier essay’s forbidden band; eleven have neither. The planar question removes about a quarter of the proportions, at the steeper angles and the stockiest braces. The multiplanar one removes the stocky braces at every angle.

For a triangular girder’s chord the multiplanar constraint is the dominant one. Only 21 of the 77 proportions can be concurrent at one section, and 35 that pass every planar check fail on the room between the planes. On a chord shared by planes 60 degrees apart, a brace more than about four-tenths of the chord’s diameter cannot meet the other plane’s braces at the same section however its own K is set out.

Moving one plane’s node along the chord #

The remedy that keeps each plane concurrent is to separate the two nodes along the chord: each K’s working lines meet the chord’s axis, at two different points.

For the square tower’s joint the room between planes grows as the second K slides along the chord, until its leading brace has passed between the first K’s braces. With the nodes 48 to 84 mm apart the braces can be welded. Further, the second plane’s brace comes up against the first plane’s other brace, and the room shrinks again to 6.5 mm at about 130 mm; it opens for good only past 180 mm, when the two Ks have passed each other. For the triangular girder there is no window short of 288 mm: the braces’ footprints overlap round the chord, so they have to be separated along it by more than a whole K.

A stagger is not free. Each plane is concurrent, but the space truss’s node is now two points 48 mm or more apart, and every member of the third dimension — a horizontal diaphragm brace, a redundant, the leg’s own splice — has to choose one of them. The joint’s statics moves from an eccentricity within a plane to an eccentricity along the chord, which six equations, and the drawing shows three would recognise: equilibrium in three dimensions has moments about axes the planar drawings never draw.

The other remedy is to let the planes’ braces overlap and weld one plane’s brace onto the other’s, as a planar overlap joint welds one brace onto its neighbour. That is a detail with its own rules and its own load path, through a brace wall loaded across its thickness — the direction a plate was never tested in is the general warning about that — and it is outside this essay.

Two eccentricities, one chord #

The second half of the question is about the band. The planar rules allow a K joint’s moment to be neglected in its own check when the eccentricity lies between $-0.55d_0$ and $0.25d_0$, and the joint whose lines may not meet already found that this licence belongs to the joint, not to the chord. A multiplanar joint makes the chord’s position worse.

Each plane’s eccentricity bends the chord in that plane, about the axis normal to it. Two moments about two different axes add as vectors, and for equal moments at an angle $\varphi$ the resultant is $\sqrt{2 + 2\cos\varphi}$ times either. For a square tower that is $\sqrt{2} = 1.41$, for a triangular girder $\sqrt{3} = 1.73$. Round the chord, the peak bending stress falls between the two planes, at a fibre neither plane’s check looks at.

So two K joints each at the gap edge of the band — 42 mm on the 168.3 mm chord — give the chord 41 or 73 per cent more bending than the band was written for. That is not an argument for a narrower band; the band is a statement about the joint’s own resistance, and the joint’s resistance is checked plane by plane with a reduction factor for the second plane. It is an argument for doing the chord’s member check with both moments, at the fibre where they add.

Only for planes 120 degrees apart is the factor exactly one, the moments’ axes being far enough apart that they add to no more than either. Neither a square tower’s leg nor a triangular one’s has planes that far apart: both towers’ faces meet at their legs at 90 and 60 degrees, the two cases drawn.

A compressed chord with moments from two sides #

The chord does not carry its node moments alone. A tower leg is a compression member, and the chord is a continuous beam between its nodes: a moment put into it at one node is shared between the leg lengths above and below, and travels some way along it before it dies out. In a planar truss the node moments alternate in sign from one node to the next, because the K joints’ eccentricities do, and the chord’s bending in that plane is a ripple along its length.

With two planes the ripples are in two directions, and nothing makes them stay in phase. If the two planes’ panels are the same length and their nodes are at the same sections, the two eccentricities alternate together and the chord bends in a fixed direction between the planes, at the $\sqrt{2}$ or $\sqrt{3}$ found above. If the stagger that made the joints weldable moves one plane’s nodes along the leg, the two ripples are offset, and the chord’s bending wanders round its circumference from one node to the next. Either way the fibre that carries the peak is between the planes, where no single-plane check looks.

A compressed member carrying a moment carries more of it than the moment it was given. The load that makes itself worse is the general statement: the axial force acting through the deflection the moment causes adds to the moment, by a factor that grows as the force approaches the member’s buckling load. A tower leg is sized for buckling, so it carries its compression at a good fraction of that load, and the amplification multiplies whatever the node moments are. Under-counting those moments by $\sqrt{2}$ at the node becomes a larger under-count along the leg.

The node a tower is made of #

None of this would matter much if multiplanar nodes were rare. They are the commonest node there is in tubular structures. A lattice tower is two or more planes of bracing on every leg at every panel point; a guyed mast’s shaft, held by something that goes soft, is a triangular lattice whose legs receive two faces of bracing at each node; a triangular girder’s chords do the same along a whole roof. Every level is a longer span described how a structure’s hierarchy repeats one arrangement at larger and larger scales; a tower repeats the multiplanar node hundreds of times up a single leg, and every repetition has the same circumference to share.

That is why a designer’s first decision on such a structure is not a gap. It is the ratio of brace to chord. A ratio under about 0.4 lets a triangular lattice’s nodes be points; between 0.4 and 0.66 a square lattice’s can be and a triangular one’s cannot; above 0.66 neither can, and every node is a stagger or an overlap. The choice is usually made by buckling — a stockier brace buckles later — and the node’s geometry follows it without anyone choosing.

What the planar checks do not compare #

None of this is visible in a planar check, because a planar check is made in one plane. The rules for multiplanar joints in EN 1993-1-8 apply a reduction factor to each plane’s resistance and otherwise treat them as two planar joints. They do not set the two planes’ braces beside each other on the chord, and nothing in the planar dimensions — the gap, the eccentricity, the band — measures the circumference the planes share.

That makes the brace’s diameter ratio the first thing to check on a multiplanar node, before any gap is chosen: $2\arcsin(d_1/d_0)$ of circumference per plane, plus the welds, against the angle between the planes. A designer who checks that first knows whether the node can be one point. One who checks it last finds out at the fabricator’s, where a node that cannot be one point is resolved by whatever stagger or overlap fits, and the analysis model is no longer the structure.

There is a pattern in this collection worth naming. The joint that is not where it was drawn began with the working lines that a truss diagram draws meeting at points; the essay before this one found that in a planar tubular joint they often cannot; this one finds that in a multiplanar joint the planes often cannot even share a section. Each step is the drawing assuming a point where the steel needs room, and each is invisible until the members are given their real diameters.

What the calculation leaves out #

The welds are a fixed gap. Two fillet welds back to back are taken to need $t_1 + t_2$ between the footprints, the planar rule applied across the planes. Between planes the welds meet at an angle and the access for a welder’s torch is round the chord rather than along it, which is usually harder; a fabricator may want more.

The footprints are of the braces’ outside surfaces on a perfect chord. Ovality in the chord, the brace’s end profile cut to fit, and the weld’s own size all change the room by a millimetre or two, and on a margin of 6.5 against 12.6 a millimetre or two does not change the answer; on one closer to the limit it might.

Only identical Ks are drawn. Planes with different braces, different angles or different gaps move their footprints differently. Changing one plane’s gap alone does slide its braces against the other plane’s, but on this joint not far enough: a second plane with a 100 mm gap, and an eccentricity of 47 mm, outside the band, still leaves only 11.8 mm.

The chord moments are of equal eccentricities. Real planes carry different brace forces and so different moments, and the factor is then between one and the figures’ values, peaking where the two planes’ moments are equal.

Still open: the leg that is also a splice #

Tower legs are spliced, usually a short distance above a node, with flanged or sleeved joints whose own size is fixed by the leg’s axial force. A splice in the stretch of chord that a staggered pair of Ks would use takes the room the stagger needed, and a splice placed for erection — where a crane can reach, where a section can be carried — is placed for reasons that have nothing to do with the bracing. Whether a multiplanar node and a splice can share the length of a leg panel, how much of the stagger window a flange takes, and whether the answer pushes the designer to the overlapped detail the planar rules were written to avoid, is the question the leg’s full length puts to the node.