The splice that finds its own axis

A flange splice in a beam is a cut through the whole section, bridged by plates bolted to both halves — the remedy for the lap joint that is crooked by construction, applied to every plate of a member at once: an outer cover on each flange, a pair of inner covers either side of the web beneath it, and a pair of web plates. The splice that bends as one section took a rolled beam, doubly symmetric, and found that across the gap the plates are the only section there is, so they share the beam’s moment by plane sections about their own neutral axis — not by area flange by flange, and not by the lever arms that decided how two unequal covers share a plate’s load. In that beam the plates’ axis and the beam’s were the same line, by symmetry.

A great many girders are not symmetric. A crane girder carries its rail on a heavier top flange; a plate girder in a building may have a wider compression flange to stabilise it; a beam that will later act compositely with a slab may be given a lighter top flange than its bottom one. In each the girder’s neutral axis is off its mid-depth, and a splice whose plates are sized flange by flange — each flange’s covers to that flange’s force — has a centroid somewhere else again. The question here is what the mismatch does. It sounds as though it ought to make an axial force across the gap, which the girder does not have. It does not; but it does something else.

A girder whose top flange is the heavier #

Keep the rolled beam’s web and bottom flange — a web 573 mm deep and 11.9 mm thick, a bottom flange 229 mm by 19.6 — and give it a heavier top flange, 300 mm by 30 mm, as a crane girder’s would be for its rail. The top flange has twice the bottom one’s area. The girder’s centroid moves up to 374 mm above its underside, against the 312 mm that would be mid-depth, and at 700 kN·m — the moment the symmetric beam’s splice carried — its top flange works at 133 N/mm² and its bottom one at 201: the lighter flange, further from the axis, is the more highly stressed.

Size the splice the way it is usually sized. Each flange’s force is its stress at its own centre times its area: 1,126 kN in the top flange, 877 kN in the bottom. Each flange gets covers whose area carries that force at a design stress of 250 N/mm², half the area in the outer cover and half in the inner pair, so the top covers are 7.5 and 8.4 mm thick and the bottom ones 7.7 and 8.9. The web plates are the same pair as before, 480 mm deep and 8 mm thick, centred on the web.

The plates sized flange by flange put their centroid at 326 mm, 48 mm below the girder’s. The reason is the web plates. They are centred on the web, at 306 mm, and they are a large part of the plates’ area — two 480 mm strips — but nothing in the flange-by-flange design moves them toward the girder’s axis. Across the gap the plates are the only section, and plane sections place the stresses in them about their own centroid. The figure shows the result: two straight lines of stress through one cut, one about each axis, crossing about 234 mm above the underside.

So at the top of the section the plates are less stressed than the girder’s own profile, scaled, would suggest, and at the bottom more. The bottom outer cover works at 269 N/mm² — over its design stress of 250 — while the top outer works at 244. That is half of what the offset does, and it is the smaller half.

The force the web carried #

A girder whose neutral axis is not at the web’s mid-depth has a web that is not symmetrically stressed. In this one the web’s centre is 68 mm below the axis, so more of the web is in compression than in tension, and the web carries a net compressive force.

The flange forces are not equal — 1,126 kN of tension in the top flange, 877 kN of compression in the bottom — and the girder is nonetheless in equilibrium, carrying no net force, because the web makes up the difference with 249 kN of compression of its own. The flange-by-flange design hands each flange’s force to its covers, and hands the web plates the web’s moment about its own centre. It hands the web’s 249 kN to nobody. On paper, the design’s forces at the gap do not balance.

Across the gap they cannot fail to balance. The plates are bolted to two halves that each carry 700 kN·m and no net force, so a free body cut through the gap carries exactly that. What plane sections do with the 249 kN that the design did not place is visible in the figure: the top covers carry less than the top flange did, 1,031 kN; the bottom covers carry more than the bottom flange did, 904; and the web plates, given no force by the design, carry 127 kN of compression. The web’s force is still there. It has gone partly to the web plates and partly to the bottom covers, both of which were sized without it.

The load path is the whole of it: the girder’s web carried a force, the splice cuts the web, and the plates crossing the cut carry that force among them in proportion to how stiffly they resist it, which plane sections decide and the design does not.

The axial force that is not there #

The obvious worry about two axes 48 mm apart is axial force. If the girder bends about one axis and the splice about another, the splice seems to need a net force to make the two agree — or, if nothing supplies it, the girder seems to have to change length at the splice. The first is impossible; the second is real and very small.

There can be no net force in the plates, because the free body says so: the cut carries what the girder carries. What happens instead is that over the gap the plates bend about their own axis, so a fibre at the girder’s axis height is not at the plates’ neutral axis: it is 48 mm above it, in tension, and over the gap it lengthens. The girder’s axis gets a little longer across the splice. A girder free to change length simply does. A girder held at both ends — between stiff abutments, say, or in a frame with rigid columns — resists that lengthening, and the resistance is an axial force.

It is negligible, because the gap is short. The curvature of the plates at the gap is small, the offset is 48 mm, and the lengthening is curvature times offset times the gap’s length: for a 10 mm gap, under two thousandths of a millimetre. Held against it by a girder 12 m long, that makes 0.66 kN — a ten-thousandth of the girder’s squash load. Even a splice with a metre of plates spanning between the girder’s ends alone, which no splice has, would make 66 kN, under one per cent.

So the question whether an unsymmetric splice can be balanced for moment and axial force at once has a short answer: there is no axial force to balance. A misfit locked into a structure is proportional to the stiffness resisting it, and the only part of this girder whose length disagrees with the rest is a few millimetres of gap. The offset between the axes is not a problem of force. It is a problem of where in the splice the stresses fall.

Which plate the offset overloads #

Three designs can be compared at the gap: the flange-by-flange design as it stands; the same design checked by plane sections across the gap and each flange’s covers thickened until none works above 250 N/mm²; and a design that forces the axes together — the bottom covers chosen so that the plates’ centroid falls on the girder’s, then the top covers scaled until the worst cover works at 250.

The flange design’s bottom outer cover is 8 per cent over, at 269 N/mm², and its web plates’ foot is 30 per cent over what the design gave it, at 212 against 163. Checking at the gap fixes the covers by thickening the bottom ones — from 7.7 and 8.9 mm to 8.6 and 10.0 — which moves the plates’ centroid further down, to 56 mm below the girder’s, and does not fix the web plates, whose foot falls only to 195. A check at the gap has to check the web plates too, and the honest design would deepen or thicken them as well.

Forcing the axes together does something less useful than it sounds. To pull the plates’ centroid up to the girder’s, the bottom covers must be lightened or the top ones made heavier; the bottom covers are already as light as their own stress allows, so the top covers grow — to 12.7 and 14.2 mm — and then work at 168 and 140 N/mm², two-thirds of their design stress. The web plates’ foot is still at 204. The girder and the splice now bend about the same axis, and nothing in particular is better for it, because nothing was wrong with the axes being different.

The reason the balanced design leaves the web plates overstressed is the same as the reason the others do: the web plates carry the moment the plates’ section gives them, and at their foot they are 260 mm or more from whichever axis the section bends about. The flange-by-flange design gave them only the web’s moment about the web’s own centre, 100 kN·m, which at their edge is 163 N/mm². What they actually carry depends on the whole section.

The further from symmetric, the more it matters to the web plates #

Vary the top flange’s thickness, from 15 mm — where its area very nearly equals the bottom flange’s — to 45 mm, three times it, sizing the splice flange by flange each time.

The covers hardly care. In a lap of two plates the thicker one takes the bend, because the plates’ own stiffness decides it; here the covers are thin plates far from either axis, and a shift of the axis by a few tens of millimetres changes their distances from it by a few per cent. The worst cover is 6 per cent over its design stress when the flanges are equal and 8 per cent over at three times — and the 6 per cent at equal flanges is the symmetric beam’s own result, that across the gap the outer cover works harder than the inner ones because it is further from the axis. The asymmetry adds two points to it.

The web plates care a great deal. At equal flanges they are slightly under their design stress, as the symmetric beam’s were. As the top flange grows, the web’s net force grows with it, the plates’ axis falls further below the girder’s, and the web plates’ foot climbs: 1.30 times its design stress at the girder drawn, 1.56 at three times the area. The web plates are where the offset lands, because they were sized for a moment about the web’s centre and are carrying a share of the section’s moment, and of the web’s force, about an axis well below it.

A web splice is often the part of a girder splice designed last and quickly: two plates the depth the bolt rows allow, checked for the shear and a share of the moment. In a symmetric girder that habit is safe, and the earlier essay found the web plates carrying less than the web did. In an unsymmetric one it is the part that the asymmetry overloads.

What each design spends #

Put the four designs side by side by the steel in their covers — two outer covers and two pairs of inner covers — the web plates being the same in all of them.

The flange-by-flange design is the lightest and the only one with a cover over its design stress. Checking it at the gap costs 5 per cent more cover steel and brings every cover to 250 N/mm² or below. Forcing the axes together costs 38 per cent more than the flange design and buys nothing the check does not. Copying the symmetric beam’s covers onto both flanges — what a detailer might do with a standard splice — costs nearly as much as balancing and leaves the covers working at under 200 N/mm², which is safe and wasteful: the heavy top flange’s covers are thin for their flange and the bottom flange’s thick for theirs, and the result happens to be fine because the symmetric beam’s covers were ten per cent heavier than its flanges to begin with.

The bolts follow the covers. Each flange’s bolts are in double shear through the outer cover on one plane and the inner covers on the other, so the bottom flange’s bolts carry the bottom covers’ 904 kN rather than the bottom flange’s 877 — three per cent more than the flange-by-flange design gave them. That is small, but it is in the direction a designer would not have guessed, toward the lighter flange.

The cut, the plates and what is assumed #

Everything here is elastic, with no slip in the bolts: the halves turn as rigid bodies and the plates across the gap bend as one section. Slip in clearance holes would let the plates carry less than plane sections say until the bolts bear, and the order in which they bear is a question about the bolt group, not the plates. A pretensioned friction-grip splice is closer to the model drawn.

The design stress of 250 N/mm² is an arbitrary common level, chosen so the designs can be compared; a real splice is sized at its plates’ yield stress with a partial factor, and the comparisons scale with it. The girder’s flanges are compact and the moment well below first yield; where the section yields the plastic neutral axis is a different line again, and a splice designed for the girder’s plastic moment would be proportioned to that axis, by area, not by stiffness.

And the web plates are kept at the rolled beam’s 480 mm, centred on the web. Moving them up toward the girder’s axis would relieve their foot, but between the girder’s axis and the top flange’s inner covers there are only 210 mm, so plates 480 mm deep cannot be centred on the axis, and shorter ones carry less of the moment and need more of the covers. A slender web’s own steel is a separate question from its splice’s.

Not a matter of force #

The worry that an unsymmetric girder’s splice must carry an axial force, or be balanced against one, turns out to be a worry about a force that equilibrium forbids. The splice finds its own axis, bends about it, and the girder on either side bends about its own; the two disagree in length only over the gap, which is too short for the disagreement to matter even in a girder held at both ends.

What the asymmetry does is real, and it is a matter of load path. The girder’s web carries a net force because its centre is not on the neutral axis, and when the web is cut that force has to cross the gap in some plate. Flange-by-flange design assigns it to none. Plane sections assign it to the web plates and the lighter flange’s covers, both of which were sized without it — the web plates by 30 per cent. The cure is not to balance the axes but to check the plates where they are the only section, which is at the gap, and to give the web plates the share of the section’s moment they actually take.

Still open: the splice whose plates slip #

Every splice here has rigid bolts, so the plates are one section from the first newton. A splice in clearance holes carries nothing until its bolts slip into bearing, and they do not all slip at once: the plates that plane sections load most heavily — here the bottom covers and the web plates’ foot — reach their bolts’ slip load first, slip, and shed their share to the plates still gripping. Whether a splice that slips redistributes toward the plates the design meant to carry the load, or away from them — which is a question of where a group of bolts can be said to have an axis at all; whether the web plates, slipping last because their bolts are lightly loaded at mid-depth, end up carrying more than plane sections said; and whether a girder that has slipped bends about yet a third axis, is what the bolts’ own movement would ask of a splice that was elastic here.