The splice that bends as one section

The thinner cover carries more spliced a 20 mm plate with a 15 mm cover on one face and an 8 mm cover on the other, cut the joint across the gap where the two halves of the main plate meet, and found that only the two covers cross that cut. With the load held on the main plate’s centre line by grips far away, two forces must balance about that line, and they share by lever arm, not by area. The thin cover sits nearer the line, so it carries more than half the load on less than half the steel: 471 kN of 900, against the 313 its area would give it, and its outer face works at 313 N/mm² where the area rule says 196.

That essay ended by naming the place where the same joint is most often built. A beam’s flange splice is two cover plates on the two faces of a flange: a wide one on the outside, and on the inside, where the web is in the way, a pair of narrower plates either side of it. They are often of different thickness, for the reasons that essay gave. The web, meanwhile, has its own splice plates, and every one of these plates is bolted to the same two halves of the beam. This essay asks what happens to the lever rule when the plate is a beam’s flange.

A beam spliced where its moment is 700 kN·m #

The beam is a rolled I-section 612 mm deep, with flanges 229 mm wide and 19.6 mm thick and a web 11.9 mm thick, spliced at a section where it carries 700 kN·m — about 60 per cent of the moment at which its flanges first yield. Each flange has a full-width outer cover 15 mm thick, 3,435 mm², and two inner covers 95 mm wide and 8 mm thick either side of the web, 1,520 mm² together; the three plates hold 4,955 mm² against the flange’s 4,488, ten per cent more than the flange they replace. Two web plates, 480 mm deep and 8 mm thick, splice the web. Eight bolts of 20 mm diameter pass through each flange on each side of the gap, every one in double shear, through the outer cover on one plane and an inner cover on the other.

The usual hand design of this splice divides the beam’s moment by stiffness between the flanges and the web, as the uncut beam divides it: the flanges hold 81 per cent of its second moment of area, so they take 566 kN·m, which on a lever arm of $h - t_f = 593$ mm is a flange force of 955 kN. The web plates carry the other 19 per cent. The flange force is then shared between the flange’s covers by area — the outer cover 662 kN and the inner pair 293, both at 193 N/mm².

The plate splice’s lever rule shares the same 955 kN differently. The outer cover’s centre is 17.3 mm from the flange’s centre line and the inner covers’ 13.8 mm, so the inner pair carries $955 \times 17.3 / 31.1 = 531$ kN and the outer 424. On 1,520 mm² the inner covers are then at 350 N/mm², a whisker under yield; on 3,435 mm² the outer is at 123.

Two different cuts #

The difference between the two joints is in what crosses the cut and what holds the pieces either side of it.

In the plate splice, the free body to the left of the gap has the load on its far end and the two covers on the cut, and nothing else. The load’s line is fixed: the grips hold the main plate on its centre line, a long way off, and the free body that answers the question has two unknown forces and two equations to find them with. The covers share by lever arm because there is no other way for them to share and still balance.

In the beam the same cut passes through six plates on each side of the neutral axis: the outer cover and the two inner covers on the top flange, the same on the bottom, and the two web plates. Two equations still apply — the forces along the beam add to nothing and their moments add to the beam’s 700 kN·m — but there are many more unknowns than two. The splice is one support too many several times over, and statics cannot say how the plates share.

What can say it is the way the beam’s halves move. Each half is a rigid body in its own plane, a deep stiff thing compared with the few millimetres of plate that bridge the gap, and every splice plate is bolted to both halves at once. If the bolts do not slip, the halves’ relative rotation is imposed on every plate across the gap together: the outer cover, the inner covers and the web plates stretch or shorten in proportion to their distance from one line. That is plane sections, applied not to the beam but to the plates. Across the gap, the splice plates are one section, and the beam’s moment bends them as one.

The lever rule needed a line on which the load was held. A beam’s flange has no such line. Its halves can turn, and when they turn they take the flange with them, so the flange force has no fixed point of application for the covers to balance about — only the beam’s moment, which the whole section of plates carries together.

The beam relieves the inner covers #

The hero figure is the stress through that section. It is a straight line from nothing at the neutral axis to 211 N/mm² at the outer cover’s outer face, and every plate sits on it at its own height. The outer cover’s centre, 313.6 mm from the axis, is at 206 N/mm²; the inner covers’ centre, 282.5 mm out, is at 186. The ratio is the ratio of the distances, 1.11, whatever the covers’ areas or thicknesses.

The three readings disagree in a pattern worth reading slowly. The area rule is the nearest to the section’s answer, 193 N/mm² for both covers against 206 and 186, and its error is that it is level where the truth slopes: it gives the outer cover 7 per cent too little and the inner covers 4 per cent too much. The lever rule is wrong by much more and in the opposite direction. It puts the inner covers at 350, nearly double what they carry, and the outer cover at 123, two-fifths below. It is not merely inaccurate in a beam; it ranks the plates backwards. The thin plate it calls the critical one is the second-most relaxed plate in the splice, and the thick one it calls lightly worked is the hardest-worked.

The reason the lever rule gets the direction wrong is that its arm is measured from the wrong line. In the plate splice the arms were distances from the main plate’s centre line, because that is where the grips held the load, and the thin cover was nearer the line. In a beam the line that matters is the neutral axis, a few hundred millimetres away, and from there the inner covers are nearer the axis than the outer cover is — so they are strained less, not more. Both covers still sit on either side of the flange; what changed is which side of them the fulcrum is on.

The flange’s own gradient decides nothing #

The question that essay closed on suspected a different mechanism. A flange in a bent beam does not carry a force on its centre line: its stress is larger at its outer face than its inner, because it is further from the neutral axis there, and the flange force’s resultant therefore sits a little outside the centre line. If the covers balanced about that resultant, the lever rule would merely be measured from a shifted line.

The shift can be computed exactly. A flange of thickness $t_f$ whose centre is $\bar y$ from the axis, with stress proportional to height, has its resultant at $\bar y + t_f^2/12\bar y$. For this flange, $19.6^2/(12 \times 296.3) = 0.108$ mm. The covers’ arms about the centre line are 17.3 and 13.8 mm, so moving the line by a tenth of a millimetre changes the lever rule’s answer by well under one per cent. The force on the flange is very nearly on its centre line, and that is not why the lever rule fails. It fails because the flange force is not a given quantity applied at a given point at all; it is a consequence of how the plates share a moment, and the plates share it as a section.

Thinning the inner covers #

The plate splice’s most striking result was what happened as its second cover thinned: the thin cover moved nearer the load line and had to carry more, on less steel, so its stress rose steeply. The next figure does the same to the beam’s inner covers.

By the lever rule the inner covers’ stress climbs as they thin, past yield below about 8 mm and to 482 N/mm² at 6 mm, while the outer cover’s falls. As one section nothing so dramatic happens. Thinner inner covers make the whole section of plates less stiff, so every plate’s stress rises together — the outer cover from 156 N/mm² with 20 mm inner covers to 218 with 6 mm — and the inner covers stay a fixed fraction below the outer, at the ratio of their distances from the axis, which changes only through the few millimetres the inner covers’ centre moves as they thin.

So a thin inner cover is not the danger in a beam that a thin second cover is in a plate splice. The plate that runs out first is the outer cover, at its outer face, and it runs out because the inner covers are too thin to give the section the stiffness that would have relieved it, not because the inner covers are overloaded. The cure in the plate splice was to balance the covers’ first moments about the main plate’s centre line. With inner covers 95 mm wide, that balance needs them 17.1 mm thick, and the lever rule then gives both covers 143 N/mm². As one section the same splice puts the outer cover at 166 and the inner at 147, still 13 per cent apart. No choice of covers makes a beam’s flange splice even, because the covers can never be at the same height.

How deep the beam is #

The lever rule has no depth in it: it reads only the flange’s and the covers’ thicknesses. The section’s answer depends on the depth at every step.

On a deep beam the outer and inner covers are nearly the same distance from the axis, because the flange thickness is a small fraction of the depth, and they share almost evenly: at 1,200 mm the ratio is 1.05. On a shallow one the flange and its covers occupy a larger fraction of the half-depth, and the outer cover is relatively much further out: at 300 mm the ratio is 1.25, and the outer cover’s extreme fibre is worked far harder than any average says. The lever rule’s 0.35 belongs to no beam at all. It is what the same plates would do as a plate splice held on the flange’s centre line, which a flange in a beam never is.

The dotted line is the comparison a designer is most likely to make, the outer cover’s peak stress against the area rule’s uniform one, with the web plates kept in the same proportion to the web at every depth. It hardly moves: 1.09 at 300 mm, 1.09 at 600, 1.12 at 1,200. Two errors in the area rule trade places along it. On a shallow beam the covers sit at very different heights and the area rule’s level stress misses the gradient across them; on a deep beam the gradient is gentle, but the web carries a third of the uncut beam’s stiffness and the web plates only about a quarter of the splice’s, so the covers take moment the area rule gave to the web. Either way the outer cover’s face works about a tenth harder than the hand design says, which is a margin that a splice designed to the hand rule’s full resistance does not have.

The web plates take less than the web did #

The section reading moves moment between the flange covers and the web plates as well as between the two covers.

The hand design hands the web plates the web’s share of the moment, 19 per cent, because that is what the web carried before it was cut. But the web plates are not the web. Two 8 mm plates 480 mm deep have a second moment of area of $1.47 \times 10^8$ mm⁴, against the web’s $1.87 \times 10^8$, while the flange covers, ten per cent heavier than the flange and further from the axis, are stiffer than the flange they replace. As one section the web plates take 14 per cent of the moment and the covers 86. The flange force through the covers is 990 kN, not 955, and the web plates’ edge works at 158 N/mm² instead of 218.

The shift is on the safe side for the web plates and the unsafe side for the covers, and it is small here because the splice was proportioned close to the beam. A splice with short web plates — the shallow ones that a moment connection’s bolt rows sometimes leave room for — moves almost all the moment into the covers: at 160 mm deep the web plates carry about one per cent. The hand design would then be sizing the covers for 81 per cent of a moment of which they carry 99. Splices are often checked for exactly this by giving the flange covers the whole moment, which is the section’s answer in its limiting case, and the reason that habit is sound is the reason the lever rule is not: the plates share by stiffness, and stiffness is set by distance from the axis.

The bolts’ outer plane runs out first #

Every flange bolt here carries the outer cover’s share on one shear plane and an inner cover’s on the other, so each reading of the moment puts a different load on each plane.

As one section the outer cover’s 707 kN shared among eight bolts puts 88 kN on each outer plane, 94 per cent of the plane’s resistance, and the inner covers’ 282 kN puts 35 kN on each inner plane. The area rule is close — 83 and 37 — because it gives the outer cover most of the flange force through its larger area, and its error on the outer plane is 6 per cent. The lever rule puts 53 kN on the outer plane and 66 on the inner. A designer who checked the bolts by it would find the inner plane governing with 30 per cent to spare, while the outer plane was in fact 6 per cent from its resistance.

The bolt check also exposes the assumption on which all of this stands. Plane sections across the gap require the plates to move with the beam’s halves, which requires the bolts not to slip. Bolts in clearance holes do slip, by up to the hole’s clearance, before they bear — unlike a preloaded joint, whose bolts, as the joint that carries nothing until it slips found, carry no shear at all until friction gives way — and a plate whose bolts slip first sheds load to the others until they slip too. The bolts that do not share followed what a long joint’s flexibility does to the share each bolt takes along its length; across the section, slip redistributes between plates, and the plate most heavily loaded — the outer cover — is the one whose bolts slip soonest. Preloaded bolts in a slip-resistant splice satisfy the assumption until slip, and that is the class of splice where the section’s reading matters most.

The section of plates, by hand #

The section’s second moment of area is the sum of each plate’s own and its area times its distance squared, for six plates and the web plates. Each outer cover contributes $3{,}435 \times 313.6^2 = 3.38 \times 10^8$ mm⁴ and each pair of inner covers $1{,}520 \times 282.5^2 = 1.21 \times 10^8$; both flanges together, $9.18 \times 10^8$; the web plates, $2 \times 8 \times 480^3/12 = 1.47 \times 10^8$; the plates’ own second moments add a little more. In all, $I = 1.066 \times 10^9$ mm⁴, nine per cent more than the beam’s own $9.75 \times 10^8$.

The outer cover’s centre is then at $700 \times 10^6 \times 313.6 / 1.066 \times 10^9 = 206$ N/mm², and the inner covers’ at $700 \times 10^6 \times 282.5 / 1.066 \times 10^9 = 186$. The two cover forces are $206 \times 3{,}435 = 707$ kN and $186 \times 1{,}520 = 282$ kN, and the splice first yields at its outer face when the moment reaches $355 \times 1.066 \times 10^9 / 321.1 = 1{,}178$ kN·m, against the beam’s own first yield at 1,131. A splice with ten per cent more flange steel than the beam, worked as a section, is stronger than the beam at first yield, by four per cent. Worked by the lever rule, its inner covers would have been declared yielded at about 710 kN·m, and the splice judged weaker than the beam by more than a third.

Plane sections, no slip and an elastic gap #

The section’s reading is itself a model with edges.

The bolts are taken as rigid. The halves’ rotation is passed to each plate exactly, which is true for a slip-resistant splice below slip and for a fitted-bolt splice; a splice in normal clearance holes is softer and its plates share by stiffness modified by slip, which moves load away from the stiffest plates first.

The gap is short and the plates are elastic. Over a few millimetres of gap the plates cannot shear-lag — a wide outer cover fed through its bolts is a flange that is not all there only over the length it takes the stress to spread — or buckle; a wide gap, or a compression flange whose cover plate is slender between bolt rows, would need its own check — the thicker plate takes the bend found how far an eccentric plate can turn under its own load once it has room to.

The plates’ own bending is in the section, as their own second moments of area, and for plates this thin it is a hundredth of a per cent of the whole — the reverse of a single lap, where the joint that is crooked by construction found the plate’s own bending to be three parts in four of its peak stress, because nothing else was there to carry the moment. And the moment is taken as constant across the splice, although a splice near a support sees a moment gradient and a shear that the web plates carry by their bolt group’s own eccentricity, which is a different calculation.

Still open: the splice that is not symmetric #

Everything here has been a doubly symmetric beam, where the neutral axis of the plates sits at mid-depth and the top and bottom splices are mirror images. A plate girder with an unequal top flange, a composite beam whose slab moves the axis up, or a crane girder with a heavier top flange for its rail, has a neutral axis that is not at mid-depth — and a splice whose plates are sized flange by flange may put the splice section’s axis somewhere else again. Then a splice under pure moment also carries an axial force across the gap unless its plates are proportioned to keep the two axes together, and the bolt group that has no neutral axis is the question of where, in a group of discrete fasteners, that axis can even be said to lie. Whether an unsymmetric splice can be balanced for moment and axial force at once, or whether it always trades one against the other, is the question the symmetric section leaves.