The flanges that lean on each other

The hinge that forms in the other flange found that an end plate bolted to a column flange is two tee stubs sharing one bolt and one contact. The component method checks each flange as its own tee stub on a rigid base and takes the weaker. But the force at the two tips is one force, pressing on both flanges at once, and the pair can collapse by a mechanism with a hinge in each flange that neither tee stub has on its own: 92 kN a bolt where the separate checks said 104 and 109.

That was strength. A bolted end-plate joint is also a rotational spring, and the component method builds its stiffness from the same parts, as springs in series. The end plate in bending is one spring and the column flange in bending another, each computed as a tee stub on a rigid base. In the joint neither flange is on a rigid base. Each tip bears on the other’s tip, and the question that essay left is whether the series of two rigid-base stiffnesses over- or understates the pair’s.

The rigid base is a plane of symmetry #

The first thing to settle is where the rigid base comes from, because it is not an arbitrary simplification. Bolt two identical tee stubs together, flange to flange, and pull their webs apart. By symmetry, whatever one flange does the other does in mirror image, so the plane between them — where the tips meet, and through which the bolt passes at its mid-length — cannot move. Each flange behaves exactly as if bolted to a rigid base through half the bolt. For two matched flanges the tee stub on a rigid base is not an approximation; it is the exact half.

So the series sum of two rigid-base stubs, each with half the bolt’s length, gives exactly the stiffness of two identical flanges back to back. An end plate and a column flange are not identical. The end plate is usually thicker than the column flange and has its bolt further from its web, because the beam flange it serves sits further from the bolt row than the column’s web does. The mirror plane is then no longer a plane of symmetry. It moves, and the error of the series sum is the measure of how far.

Two strips and one contact #

The free body is the earlier one, elastic. One bolt’s strip of each flange, 90 mm wide, is cut at its web and clamped there. The end plate is 18 mm thick with its bolt 45 mm from its web, the column flange 14 mm with its bolt 30 mm from its web; the tips are 35 mm beyond the bolt line. An M20 bolt 50 mm long joins them as a spring. A tension $T$ per bolt pulls the two webs apart, and three things hold them: each flange bends as a cantilever from its web, the bolt stretches, and at the tips the flanges press on each other with a force $Q$.

Those are three equations. The webs separate by δ; the separation at the bolt line is the bolt’s stretch, $B/k_b$, with $B = T + Q$; the separation at the tips is nothing while they touch. Each flange’s deflection towards the other comes from the bolt pulling it at the bolt line and the contact pushing it back at the tip, through the cantilever’s influence coefficients, so the equations are linear in δ, $B$ and $Q$. The tee stub is the same strip with half the bolt and its tip held at nothing.

The two shapes are the pair. Each flange is held at its web and pulled towards the other at the bolt line; at the tips they are pressed together and move together. The end plate, with the longer lever, swings its tip 3.5 thousandths of a millimetre per kilonewton relative to its web; the column flange, 0.8. In a matched pair the two would be equal, the contact would sit on the mirror plane, and that plane would not move. Here the contact moves with whichever tip gives more, and each tip bears on a tip that is moving differently from its own.

Mismatch, not thickness #

Where the end plate matches the column flange — 14 mm thick with its bolt at 30 mm — the pair is exactly its series sum, as the symmetry says. Move the end plate’s bolt further from its web and the pair falls below the series: a 14 mm end plate is 0.95 of its series sum with the bolt at 45 mm and 0.85 at 65 mm. A thicker end plate errs less at the same lever, 0.86 at 18 mm and 0.93 at 25, because a stiffer end plate’s own bending is a smaller part of the pair and its mismatch costs less.

The pair is always the softer. The series sum gives each flange a support that does not move. In the pair, the support the end plate pries against is the column flange’s tip, which yields under the prying, and the end plate’s prying is then less effective at stiffening it. In a matched pair the two tips yield equally and the effect cancels at the mirror plane; in a mismatched pair it does not.

With the lever arms at 45 and 30 mm, thickness alone moves the ratio little. Every combination drawn sits between 0.94 and 1.00, and the curves wander rather than trend, because a mismatched thickness can partly offset a mismatched lever arm. An 18 mm end plate on a 14 mm column flange, with the lever arms of this joint, is 0.97 of its series sum: 235 kN/mm a bolt against 242. For joints of ordinary proportions the series sum is good to a few per cent, and the reason is that the rigid base it assumes is a real plane in a slightly different joint.

When the lever arms part #

The error is concentrated where the two flanges’ lever arms are most unlike. With the end plate’s bolt 60 mm from its web against the column flange’s 30, an 18 mm end plate is 0.88 or 0.89 of its series sum on any of the column flanges drawn: on the 14 mm one, 146 kN/mm against 166. An end plate with its bolts well clear of the web it hangs off — an extended end plate’s outer row, or a flush plate with its bolts kept clear of a large weld — is the joint the series sum misjudges.

The pair’s own shapes, for such a joint, show why. The end plate’s tip swings 7.7 thousandths of a millimetre per kilonewton relative to its web, and it pries so hard against the column flange’s tip that it pushes that tip back past the column flange’s own web plane, 0.9 thousandths the other way. The column flange is not a support at that point. It is a second spring being bent backwards by the first, which is the furthest a pair can be from two stubs meeting on a mirror plane.

The prying they share #

The prying share shows the same thing from the side of the forces. As a stub, the end plate would draw 0.42 of the tension as contact force at its tip and the column flange 0.26. Pressed against each other they share one contact, and it carries 0.32, between the two. In a matched pair the shared contact is exactly each stub’s own, which is the symmetry again. In a mismatched pair the end plate draws less than its stub would, because its support yields, and the column flange more, because it is pried by the end plate as well as by its own bending.

For the bolt the pair is benign: 1.32 times the tension, against the end plate stub’s 1.42. The earlier essays on the force the bolt never saw applied computed bolt forces on the stub’s side of this, which is the safe side.

What reaches the joint #

The flanges and the bolts are one part of a joint’s flexibility. The column web in tension and in compression, the web panel in shear and the beam flange in compression are the rest.

The joint’s error is the pair’s diluted by the rest. Where the flanges and bolts are half the joint’s flexibility, a pair at 0.97 of its series sum leaves the joint at 0.98 of its estimate, and a pair at 0.86, with mismatched lever arms, at 0.92. Where a stiffened column makes the flanges nearly all of the joint’s flexibility, the joint inherits the pair’s error whole.

An 8% error matters only near a boundary. A joint is classified rigid in an unbraced frame if its stiffness reaches 25 times the $EI/L$ of the beam it carries, and a joint designed to that boundary with mismatched lever arms is semi-rigid in fact; the frame built with it sways a little more than the frame that leans was analysed to. Joints of ordinary proportions, with their few per cent, do not move.

Which tip gives way more #

The two tips do not move equally, and the reason is the lever arm rather than the thickness. The end plate is the thicker plate, 18 mm against 14, and on thickness alone its tip would move less. But its bolt is 45 mm from its web against the column flange’s 30, and a bolt pulling further from the web bends the strip much more at its tip: on the bolt’s pull alone, the end plate’s tip would move 2.7 times as far as the column flange’s for the same thickness. Its extra thickness takes back a factor of 2.1. The shared contact force then pushes both tips back by the same amount, and that takes back proportionally more of the column flange’s smaller movement. The end plate’s tip ends up four times as mobile as the column flange’s.

That makes the column flange the stiffer support of the two, and the end plate pries against it. In the plastic analysis the same lever arms decided which flange held the web hinge and which the bolt-line hinge of the mixed mechanism. Elastically they decide which flange’s tip the other leans on, and how far the pair is from a mirror image of itself.

Designing the mismatch out #

If the error is the mismatch, the cure is the match, and a designer has more control over it than the joint’s thicknesses suggest. The column flange’s lever arm is set by the column: the distance from its web to the bolt line, which the bolt gauge and the column’s root radius fix. The end plate’s is set by the joint’s own detailing — the bolt gauge across the plate and the weld between the beam’s flange or web and the plate — and can usually be brought close to the column’s.

The bolt gauge is shared, so the end plate’s bolts sit the same distance from the joint’s centre line as the column flange’s. What differs is where each plate’s web is. The column’s web is at the centre line; the end plate’s “web”, for a bolt row beside the beam’s web, is the beam’s web, also at the centre line. For those rows the lever arms nearly match by construction, and the series sum is close to exact. For a bolt row beside the beam’s flange, the end plate’s effective web is the flange, and the lever arm is the distance from the flange to the bolt row, set by the weld size and the clearance for a spanner. That row is where an end plate’s lever arm grows past the column flange’s, and it is the row whose stiffness the series sum overstates.

So the error is concentrated in particular bolt rows, not spread over the joint, and it is largest in the rows that carry most of the moment — the ones nearest the beam’s tension flange. A joint whose stiffness matters, near a classification boundary, is worth computing with that row’s pair rather than its stubs.

What a test of the joint sees #

A full joint test measures the pair. Its moment-rotation curve includes the shared contact, because the end plate and the column flange in the test really do bear on each other. A component calibration measures a tee stub bolted to something stiff, or two identical tee stubs bolted back to back, which is the same thing by symmetry. A joint built from calibrated components and then tested is therefore expected to come out softer than its components’ sum by the amount its flanges’ mismatch costs, before any other cause of disagreement is looked for. The direction is fixed and the size can be read from the lever arms.

Two stubs and half a bolt, by hand #

The 18 mm end plate’s strip is 90 mm wide, so $EI = 210\,000 \times 90 \times 18^3/12 = 9.18 \times 10^9$ N·mm². As a stub with half the bolt — an M20 with 245 mm² of stress area over 25 mm, 2,058 kN/mm — on a rigid base, its tip held 80 mm from its web, it carries a contact force of 0.42 of the tension and gives 420 kN/mm. The column flange’s stub, 14 mm on a strip 65 mm to its tip, gives 569 with a contact of 0.26. In series: $1/(1/420 + 1/569) = 242$ kN/mm.

In the pair, with the whole bolt of 1,029 kN/mm and the contact shared, the three equations give a contact force of 0.32 of the tension, a bolt force of 1.32, and a web separation of 4.26 thousandths of a millimetre per kilonewton: 235 kN/mm, 0.97 of the series. Move the end plate’s bolt to 60 mm from its web and the stubs give 166 kN/mm in series against the pair’s 146.

Elastic strips, no preload #

One bolt’s strip, elastic. The strip width stands in for the effective length a yield-line analysis would give each flange at that bolt row, as in how much of the plate is bending. Elastically the effective widths differ from the plastic ones; the comparison between pair and series is made with the same strips on both sides.

No preload. A preloaded bolt clamps the flanges together, and under a tension below the preload the joint is stiffer than any of this, as the bolt that was already stretched found. The pair’s stiffness is that of the joint once the clamp has been relieved.

Clamped webs. Each flange’s web is taken as rigid, which is the component method’s assumption too.

The tips touch. The shared contact needs the two tips to overlap. An end plate whose tip bears beyond the column flange’s edge bears on nothing, and then each flange is its own spring.

What a symmetry is worth #

The component method’s tee stub looks like a simplification — a flange bolted to an imaginary rigid base — and it is in fact the exact half of a symmetric joint. That is why it works as well as it does. The pair’s error is not the error of an approximation but the departure of a real joint from the symmetric one its components describe, and it can be read off the joint’s proportions before any analysis: matched flanges, nothing; matched lever arms with mismatched thicknesses, a few per cent; lever arms two to one, twelve per cent or more.

That puts the stiffness alongside the strength the earlier essay found. The pair is weaker than either flange checked alone when its thicknesses and lever arms open the window for the mixed mechanism, and it is softer than its components in series when its lever arms differ. Both come from one force acting on two plates that are not mirror images.

Still open: the pair that is preloaded #

The bolts here carry no preload, so the tips separate from the first load and the pair’s stiffness is the contact’s from the start. A preloaded bolt clamps the two flanges together over a cone of plate around it, and the joint is stiff until the external tension relieves that clamp. The decompression load of a matched pair is that of one flange on its mirror plane. For two flanges whose prying draws each other’s tips apart unequally, it is not. Whether a preloaded end plate with its bolt far from its web decompresses earlier than its stub model says — its prying lifting the column flange’s tip and opening the clamped zone from its edge — and whether the joint then drops to the pair’s stiffness or the series sum’s, is the question of what preload buys a joint whose two flanges are not mirror images.