The joint that softens under its own load

The panel a soft joint empties into the span gave every beam end of a two-bay, two-storey frame a rotational spring and found that a softer joint relieves its own column-web panel and loads the beam’s span, and that softening some joints and not others sends the frame’s sway to the joints that stay stiff. Every joint there kept the stiffness it started with, however much moment it carried.

A real end-plate joint does not. As the moment rises its bolts begin to stretch past their proportional range, its plate begins to form yield lines, and the column web behind it begins to shear; the joint turns further for each kilonewton-metre than it did at first, and near its resistance it turns several times as far. An interior joint under a pattern load and a push carries very different moments on its two sides, so its two halves soften at different rates. The question the linear springs left was whether that sends still more of the sway to the joints that stay stiff, so that the redistribution grows as the frame approaches its strength, or whether the frame settles into a pattern a single stiffness per joint can describe.

A joint stiff at first and soft at its resistance #

EN 1993-1-8 gives the curve a joint follows. Up to two thirds of its moment resistance $M_{j,Rd}$ the joint is linear at its initial stiffness $S_{j,ini}$. Beyond that its rotation is $\varphi = M\mu/S_{j,ini}$ with

$$\mu = \left(\frac{1.5\,M}{M_{j,Rd}}\right)^{\psi},$$

and $\psi = 2.7$ for a bolted end plate. At the resistance $\mu = 1.5^{2.7} = 2.99$, so the joint has turned three times as far as its initial stiffness alone would have taken it.

The frame is the one before: two 8 m bays, two 4 m storeys, fixed bases, 457 mm beams whose EI/L is 7.6 MN·m per radian, 254 mm columns. Every beam end is an end plate of three times the beam’s EI/L — the typical flush end plate of the earlier essay — with a resistance of 120 kN·m, about three tenths of the beam’s plastic moment: a partial-strength joint, weaker than the member it connects. The load is the earlier essay’s worst combination, 20 kN/m on the left bay only and 20 kN at each floor from the left, all multiplied together by a load factor λ.

The frame is solved at a secant stiffness for each joint: solve, read each joint’s moment and rotation, move its stiffness to the secant of the curve at that rotation, and solve again until nothing changes. At λ = 1 every joint is below two thirds of its resistance and the frame is exactly the linear one.

The loaded end stops, the unloaded one takes over #

The joint that works hardest goes first. The loaded beam’s end at the interior column carries 78 kN·m at λ = 1 — the hogging moment the pattern puts there plus the sway’s share — and passes two thirds of its resistance at λ = 1.03. From there it softens, its moment rises more slowly than the load, and at λ = 2.69 it reaches 120 kN·m and stops.

What it does not carry goes somewhere, and the figure shows where. The unloaded beam’s end across the same column, which the pattern had bent the other way and the sway had barely loaded, carries 13 kN·m at λ = 1 and 81 at λ = 3 — six times as much for three times the load. The two outer ends rise faster than in proportion too, the unloaded one most. On stiff joints the four ends at λ = 3 would carry 97, 235, 39 and 68 kN·m; on softening joints they carry 89, 120, 81 and 88. The spread between the most and least loaded ends has gone from six to one to three to two.

So the answer to the first half of the question is yes, and more strongly than the linear springs suggested. Softening is not a uniform loss of stiffness; it is a loss of stiffness exactly where the moment is largest, and the frame responds by taking moment away from there and giving it to whatever is still stiff. The joint across the column is the nearest stiff thing, and it is where most of the load goes.

Each joint at its own stiffness #

The joints started alike and end unlike. By λ = 4.2 the loaded inner end has a secant stiffness of half the beam’s EI/L — on EN 1993-1-8’s classification for a sway frame, nominally pinned — while the joint across the column from it, which carried least at the start, is at 1.56 EI/L and the outer ends somewhere between. The frame that was designed with eight identical joints is behaving, near its strength, as a frame with eight different ones, and in an order nobody chose: softest where the moment was largest.

That is the same mechanism the earlier essay found for mixed joints — soften some joints and the sway finds the others — arriving without anyone mixing them. The pattern load chose which joints would soften, and the softening then shaped how the rest of the load was shared. It is also the moment that was moved on purpose, the redistribution a continuous beam’s designer is allowed to take, happening here at the joints rather than in the members and without being asked for.

The panel the softening joint relieves #

The column-web panel at the interior joint carries the two beam ends’ moments either side of it as a pair of flange forces over the beam’s depth. Under gravity alone they oppose each other and the panel carries their difference; under the push they act in the same sense and it carries their sum. With the loaded side capped at its resistance, the sum falls even though the unloaded side is growing, because the loaded side has given up more than the unloaded side has taken: at λ = 3 the interior panel carries 386 kN where stiff joints would have given it 534. A joint that softens sheds its own panel’s shear, which is the linear result of the earlier essay holding all the way to the joint’s strength.

The exterior panel, at the outer end of the loaded beam, is barely relieved: 185 kN against 198. Its joint softens late, and in the meantime the frame’s sway has been coming its way from the softened interior. So the relief is concentrated where the softening is, and the exterior joints carry what they would have carried anyway, or more.

One secant stiffness for every joint #

EN 1993-1-8 does not ask a designer to follow each joint’s curve. It allows the frame to be analysed elastically with every joint at its initial stiffness as long as no joint’s moment exceeds two thirds of its resistance, and otherwise with every joint at $S_{j,ini}/\eta$, where $\eta = 2$ for an end plate: half the initial stiffness, for every joint, at every load.

The half secant is closer than the initial stiffness everywhere and right almost nowhere. It asks the floor’s loaded inner joint for 178 kN·m, half as much again as its resistance, so a design on it would make that joint stronger than it needs to be — the frame would carry λ = 3 with the joint at 120, having moved the rest elsewhere. It gives the unloaded inner joint 43 kN·m where that joint carries 81, and the roof’s unloaded inner joint 14 where it carries 39: those joints, designed on the half secant, would be checked for half the moment they get. And it gives the floor’s exterior panel 142 kN where the softening frame puts 185, a quarter less.

The single secant is a uniform softening, and the frame’s real softening is not uniform. Halving every joint’s stiffness changes the frame’s total flexibility about as much as the real softening does, but it leaves the joints’ relative stiffnesses as they were — and the relative stiffnesses are what decide how the load is shared. Whether the secant is conservative depends on which joint is being checked: on the safe side for the joints that soften first, on the unsafe side for the joints they hand their load to.

The drift the secant gets right, for a while #

The quantity the half secant does serve is the drift. Up to about λ = 2.5 it follows the softening frame closely, and at λ = 3 it is 8 per cent short. Drift is a global quantity, the sum of every joint’s flexibility weighted by its share of the sway, and a uniform halving gets the sum about right while getting each term wrong. That is the same distinction a pile in layered ground draws between its head deflection, which an average gets right, and its moment, which an average does not.

Past λ = 3 the secant falls behind, 23 per cent short at 4.2, because by then several joints are on their plateau and the frame is turning at them for nothing. The softening frame’s drift curves upward as it nears its strength, which a linear analysis with any fixed stiffness cannot do, and which a second-order analysis of the same frame would amplify further: a frame that drifts more under its push carries its gravity load on a larger lean.

The rotation the redistribution spends #

Redistribution at a joint is paid for in rotation, as it is at a hinge in a continuous beam. The loaded inner joint reaches its resistance at 15.6 milliradians, the rotation at which EN 1993-1-8’s curve meets its plateau. From there every increase in the load turns it further at the same moment: 18.6 mrad at λ = 3, 23.8 at 3.5, 31.5 at 4.2. The roof’s loaded inner joint follows the same path a little behind, at its resistance by λ = 3.5 and at 26.5 mrad by 4.2.

Those rotations are what the redistribution of the second figure is made of. The unloaded joint across the column can only climb from 13 to 81 kN·m if the loaded one keeps turning without dropping its moment, and the plateau that makes that possible is a property of the joint’s ductility, not of its strength. An end plate whose bolts fracture before its plate has yielded has no plateau: it reaches its resistance and lets go, and the moment it was carrying arrives on the joint across the column at once rather than gradually.

EN 1993-1-8 decides which kind of joint is in hand by its proportions. A bolted end plate has the rotation capacity a plastic analysis needs when its moment resistance is governed by the plate or the column flange yielding rather than by the bolts, and when the plate is thin enough beside the bolts — $t \le 0.36\,d\sqrt{f_{ub}/f_y}$, which for M20 bolts of grade 8.8 in an S275 plate is about 12 mm. A thicker plate, or bolts that are the weakest component, makes a joint that is stiffer and stronger and cannot turn 30 milliradians; on such a joint the curve of the first figure would end at its resistance, and so would the frame’s redistribution.

So the three joints the pattern loads hardest are the ones whose rotation capacity the frame’s strength depends on, and the frame needs them to supply twice the rotation at which they first reach their resistance before it is anywhere near its own. The joint that has to keep turning is usually discussed for its fatigue under cycles; here a single monotonic load asks for 31 milliradians of a joint that turns 3.4 at λ = 1.

Designing for the joints the frame will have #

The figures suggest three habits for a frame with partial-strength end plates.

Check the joints the load is handed to. A design on one reduced stiffness checks the loaded inner joint for more than it will carry and the joints across the column for less. The joints that matter at the frame’s strength are the receivers: the unloaded inner ends and the exterior ends, which in this frame carry from 1.3 to 2.8 times their half-secant moments at λ = 3. If they are the same end plate as the loaded joint, they have the margin; if they were made lighter because the analysis said they carried little, they are the frame’s weakest link.

Do not count on the panel’s relief. The interior panel’s shear falls as the loaded joint softens, but the relief depends on the joint reaching its plateau, which depends on its ductility. A panel sized for the softened shear is sized for a redistribution that a brittle joint would not deliver.

Read the drift from the secant, and the moments from the curve. A uniform half stiffness is a fair estimate of the frame’s sway up to about the load at which the first joint reaches its resistance, and a poor one past it. The moments are not well estimated by any single stiffness at any load past two thirds of the first joint’s resistance, because the joints by then are no longer alike.

The interior joint, by hand at λ = 3 #

At λ = 3 the loaded bay carries 60 kN/m, so its free moment is $60 \times 8^2/8 = 480$ kN·m. On stiff joints its ends carry 97 and 235 kN·m hogging, an average of 166, and its middle sags $480 - 166 = 314$. On softening joints its ends carry 89 and 120, an average of 105, and its middle sags $480 - 105 = 375$, a fifth more. The beam’s span is where the joints’ lost moment ends up first; the moment that goes round the corner has to come from somewhere, and when the corner will not take it the span does.

The interior panel’s shear is the two beam ends’ moments at the column face over the beam’s depth, less half the columns’ shears. Under the push the two ends’ moments act in the same sense, so they add. On stiff joints they are 240 and 39 kN·m at the column face, 279 in all, a flange couple of $279/0.457 = 612$ kN, from which the columns’ shears take 78: 534 kN. On softening joints they are 125 and 81, 206 in all, a couple of 451 kN, less 65: 386 kN. The loaded side has fallen by 115 kN·m and the unloaded side risen by 42, and the difference is what leaves the panel.

What the curve leaves out #

One curve for every joint. Every end is the same end plate with the same initial stiffness and resistance, and the curve is EN 1993-1-8’s, which is a design idealisation of tests rather than a measured joint. A real joint’s curve depends on its bolts, its plate thickness and the column behind it, and the plateau is a convenience: real end plates harden a little past their resistance and eventually fail by bolt fracture, which ends the plateau abruptly.

The joint is the same in both directions. An interior joint’s two halves bend in opposite senses under a pattern and a push, and an end plate with bolts above and below the beam is not symmetric: its resistance in hogging is usually not its resistance in sagging. Here both are 120 kN·m.

The load rises monotonically. Softening is drawn on a single path from rest. A frame that is pushed one way and then the other unloads each joint along its initial stiffness, keeps the rotation it had gained, and meets the reversed push with joints that are stiff again but turned; a joint that has to keep turning under a reversing load is the subject that picks that up.

First order. The frame is solved on its undeformed geometry. The drift at λ = 3 is 54 mm on a 4 m storey, 1 in 74, at which the gravity load’s lean is no longer negligible.

The pattern no stiffness was chosen for #

The linear springs had a uniform frame and a classification question: is this joint stiff enough to call rigid. The softening joints answer a different question. The frame begins uniform and ends with each joint at a stiffness set by the load it happens to carry, so the distribution of moment near the frame’s strength is a property of the load pattern as much as of the joints. The joints the pattern loads hardest soften first and pass their load across the column, the joints that receive it carry several times their proportional share, and a single reduced stiffness for every joint — what the design rule permits — keeps the frame’s flexibility roughly right while sharing its moments as though nothing had softened at all.

Still open: the joint whose resistance is the beam’s #

Every joint here is weaker than its beam, so the joints reach their resistance and the beams do not. A joint as strong as the beam — full strength, with the same initial stiffness — softens in the same way but never reaches a plateau before the beam itself begins to yield, and then the plastic hinge forms in the beam beside the joint rather than in the joint. Whether the joint’s softening on the way still redistributes enough moment to delay the first hinge in the beam, and so moves where the frame’s mechanism forms, or whether the beam’s hinge arrives first and the joint’s curve never matters, decides whether a full-strength joint’s stiffness is a serviceability question or a strength one.