A hinge one way and a spring the other

The column on the other side of the joint followed a partial-strength joint through a braced frame, where its moments were a nuisance to the column and nothing else. It ended by taking the bracing out. Semi-continuous frames are proposed for low-rise buildings mainly on the ground that the same joints which cut the beam’s midspan moment can also stand the frame up against the wind, saving a bay of bracing. That asks one joint for two things at once: gravity has already pushed it some way up its curve, or all the way to its resistance, and the wind then arrives to ask for what it has left.

The concern is simple to state. A joint at its resistance has stopped resisting, so a joint gravity has yielded is a hinge, and a frame on pinned feet whose beams are hinged at the column is a mechanism. If the design moment is around a joint’s resistance, as it is meant to be in a partial-strength design, then the frame’s sway stiffness is going to disappear at about the working load. Two questions followed from it: whether a yielded joint still contributes anything to a sway frame’s stability, and whether the critical load then depends on the order in which the joints reached their resistance. The answer to the first is that the concern has confused two directions of a moment, and the answer to the second is that it does, but not in the way expected.

A storey that the joints have to hold up #

The frame is one storey with two 8 m bays of the beam used through the earlier essays, $I = 2.9 \times 10^{-4}\ \text{m}^4$, on three 4 m columns with pinned feet. With pinned feet the joints are all there is: take away their stiffness and the columns are free to rotate as a mechanism. Each column carries 300 kN from the floors above, and the beams 20 kN/m, for a total of 1,220 kN in the storey. Every joint has an initial stiffness of three times the beam’s $EI/L$, and is elastic up to its resistance and perfectly plastic after that, with no difference between one sense of moment and the other until a later section.

The load path is followed from one joint’s arrival at its limit to the next. Within a step, a joint at its limit is free if the load pushes it further and elastic if the load brings it back. The storey’s second-order behaviour comes in through a geometric term in the sway stiffness, $-\sum P/h$, 305 kN/m with the whole load on.

With joints that never yield, the gravity load puts 50 kN·m on the outer joints and 67 kN·m on the inner, about half the fixed-end moment of 107 kN·m. The storey has a critical factor of 8.27, and a sway stiffness, with geometry included, of 2,216 kN/m. Those are the numbers a designer starts from: under the conventional limit of 10 for ignoring second-order effects, though not by much.

Where gravity leaves the joints, and where the wind takes them #

Give the joints a resistance of 60 kN·m. The gravity load carries the inner two to their limit and leaves the outer two at 52. A rightward push then does the same thing to every beam: it adds hogging at the beam’s left end and removes it at the right end. The two ends of a beam that sways go opposite ways, and that fact decides everything in this essay.

Follow the four lines in the figure. The first bay’s left end, at 52, rises to its limit almost at once. The second bay’s left end is at its limit already and stays there. Those two have nothing further to give, and they are the hinges the concern was about. The other two are the right-hand ends, and the first of those is also at its limit, yet nothing about that limit restrains it: sway is taking moment off it. It falls straight down from 60 through zero to −60, at a steady rate, with the joint’s full elastic stiffness at work over a range of 120 kN·m. Gravity pushed that joint to its resistance, and sway reads its resistance as the far end of a range it has not begun to use.

This is the point about unloading that after the first yield made of a steel section: the plastic flow is not a loss of the elastic stiffness but a movement of where the elastic range sits. A section yielded in one direction unloads at the original modulus. Here it is the joint, and the moment of unloading is delivered by a different load from the one that yielded it.

The factor, read in two ways #

The critical factor makes the difference plain. Reading each joint at its limit as a hinge, as any analysis program would if told to remove stiffness from a yielded joint, the factor runs in steps. Above about 67 kN·m nothing has yielded and it is 8.27. Between 50 and 67 the inner joints have yielded and it is 3.71. Below 50 all four have yielded and the reading is zero, since a frame on pinned feet with hinges at every joint is a mechanism. That reading says a partial-strength sway frame stops existing at the very resistance a design would want.

Now let each joint respond the way sway actually finds it. With a resistance of 60 kN·m only one of the two inner joints is a hinge for sway, and the factor is 6.48, not 3.71. Below 50 kN·m, where every joint has yielded, the two right-hand ends are elastic, and the factor is 3.71, not nothing. It stays at 3.71 for every resistance below 50, however small. That is, the factor is 45 per cent of the value with no yielding at all, and it never goes lower, because half of the joints always remain.

So the frame’s critical factor does depend on gravity’s history, and it has a floor that does not depend on how much gravity there is, once it is enough to yield everything. A partial-strength frame with all its joints spent has lost half its stiffness against sway. The reading that says it has lost all of it is reporting on a direction no sway will go in. And since these are the same joints read in the opposite sense when the wind reverses, the frame’s factor for a leftward push is the same number from the other two joints.

What the wind then finds #

The stiffness only tells part of the story, because the range of those unloading joints is finite. Under gravity then wind, the storey sways at a tangent stiffness of about 830 kN/m once its first hinges have formed at 5 kN, where a frame with no yielding has 2,216. At a storey shear of 10 kN its drift is 9 mm against 4.5 mm for joints too strong to yield, so a joint chosen for gravity has doubled the service drift by the time the wind is a fifth of the way to the peak.

The frame then goes on to a peak of 44 kN. A rigid-plastic mechanism would give a simple answer: four joints at their resistance, each worth 60 kN·m over a 4 m column, a shear of 60 kN. The peak is lower by the storey’s own second-order moment at the drift it has reached, and that is a sum a designer can do with a pencil. $\sum P/h = 305$ kN/m, the drift at the peak is 52 mm, so the geometric term takes off 15.9 kN and the peak is 60 − 15.9 = 44.1 kN. The load that makes itself worse reached this from the column’s side: the vertical load’s moment grows with the sway it causes, and here it comes straight off what the joints can deliver. The same sum does the 120 kN·m joint, 120 less 0.305 × 81 = 95.3 kN, against 95.1 computed.

The one number needed to do that sum is the drift at which the last joint arrives, and it does not follow from anything elastic. It comes from how far the joints have to travel: with a resistance of $M$ the unloading joints traverse $2M$ at a stiffness of the order of $S$, along with what the beams and columns add, so the drift at the peak grows almost in proportion to the resistance.

Along that path the factor falls in steps, as each joint arrives. It stays at 3.71 for most of the frame’s useful range, and does not fall below it until the two unloading joints reach the sagging limit, just short of the peak. Then the last one arrives and it is gone at once. This is a stiffness that has nothing gradual about it, and a designer watching it would have no warning from the factor that the peak was near: from 5 kN to 43 kN the reading does not move at all.

The factor, by hand #

Both numbers can be had with a pencil, which is worth doing because a figure this unexpected deserves a second route. A column on a pinned foot, restrained at its top by a rotational spring $k$, has a sway stiffness of $(3EI_c/h^3) \times k/(k + 3EI_c/h)$. Here $3EI_c/h^3 = 2{,}727$ kN/m and $3EI_c/h = 43{,}630$ kN·m.

With every joint elastic, a beam swaying antisymmetrically offers each end $6EI_b/L = 45{,}680$ kN·m, in series with its joint’s 22,840: 15,230 per arm. An outer column has one arm and stiffness 705 kN/m; the inner column has two and 1,120. The storey has 2,530 kN/m, and over 305 that is a factor of 8.3, against 8.27 computed.

Now free the left end of each beam, the two that sway loads further. Each beam is then pinned at one end, offering its other end $3EI_b/L = 22{,}840$, in series with its joint: 11,420. The left column has nothing to restrain it and contributes nothing. The middle column has one arm, as does the right column, and each contributes 566 kN/m. The storey has 1,132 kN/m, and over 305 that is 3.71, exactly. There is no hinge pattern in which all four are free, since no sway makes that the case; the mechanism in the first reading is a state the frame cannot reach by swaying.

Whether the order matters #

The uniqueness theorem says the plastic collapse load of a frame does not depend on the path to it. That theorem has no $P$–Δ in it. Here the three orders of loading give three peaks, because the peak is the mechanism’s shear less the geometric term at the drift the path has reached, and the path decides the drift.

For strong joints the difference is small. At a resistance of 60 kN·m the three peaks are 44.0, 44.7 and 46.0, and at 120 they are 95.1, 97.3 and 99.8, a spread of 5 per cent, because the joints take similar drifts to arrive by any path. The order in which they get there hardly matters, and the orders do not rank as expected: applying gravity last is the best, since the storey has not been carrying its full vertical load while it was swaying.

For weak joints the difference is a factor of two, in the reverse order. At 15 kN·m, gravity then wind gives a peak of 11.0 kN at a drift of 13 mm. Applied together, or with the wind first, the peak is 5.5 or 5.7 kN at 35 mm. A joint that begins the sway at its hogging limit has the whole of its range $2M$ before it, on the elastic side, and arrives at the opposite limit quickly; so that frame reaches its mechanism at a small drift and loses little to the geometric term. A frame loaded the other way drifts further before its joints have all arrived. The peak of the weak frame, then, depends on whether the gravity load got there first, and a real building has its gravity load in place first almost always: the dead load always, and the live load usually. It is one of the few cases where the ordinary order is the favourable one.

A factor that is true and tells very little #

The worked comparison of the first section’s two numbers is worth doing at the lowest resistance. At 15 kN·m the critical factor after gravity is 3.71. At 50 kN·m it is also 3.71. A classification by critical factor treats the two as the same frame. At 50 the peak shear is 37 kN; at 15 it is 11 kN and arrives at a drift of 13 mm. A designer who took the 3.71 as permission to amplify a first-order analysis by $1/(1 - 1/3.71) = 1.37$ would be using a stiffness that the frame has only for those 13 mm, at the end of which the factor is zero.

The critical factor reports how stiff the joints are. How far that stiffness lasts is a question about their resistance, and no elastic analysis can answer it. So a semi-continuous sway frame has two checks that do not substitute for each other: the factor for second-order effects while elastic, and a peak shear against the wind, from the mechanism less its geometric term. The joint that has to keep turning has the same message about rotation capacity: a joint’s numbers are not interchangeable.

The argument for the semi-continuous frame survives this, with its terms changed. Suppose the storey has a wind shear of 25 kN. A joint of 60 kN·m carries it with a peak of 44, at a drift of about 27 mm, a hundred-and-fiftieth of the 4 m storey. That is not stiff, and a drift limit would have something to say about it, but it stands. What the same joint does for gravity is the reason it is there, as the moment that was moved on purpose showed: it moves moment out of the span. A bay of bracing would give a far higher stiffness. Whether that is worth a bay, and the door it blocks, is what the frame’s advocates are really proposing, and neither the claim that yielded joints are useless nor the claim that a factor of 3.71 shows them adequate settles it. One column is enough to find the sway read a fixed-base portal’s sway from a column’s two end moments; with a pinned foot only the moment at the joint is left, and that is the one this essay has been following.

The bottom bolts #

Everything so far has given the joint the same resistance in both directions, and a real one rarely has it. An end-plate joint is laid out for hogging, with most of its bolts in tension at the top; under sagging, the few bolts at the bottom have to do the work, and the compression moves to the top flange. In a joint set out only for gravity, the sagging resistance is commonly a fraction of the hogging. Neither pinned nor rigid noted that the curve a joint is classified by is one curve, measured in one sense.

Sway stiffness does not care about this, because it depends only on which joints are free, and the critical factor stays at 6.48 for any sagging resistance. The range and the peak do. With a sagging resistance a third of the hogging, the unloading joints reach their limit having travelled 80 rather than 120 kN·m, the mechanism’s shear is $(2 \times 60 + 2 \times 20)/4 = 40$ kN rather than 60, and the peak falls from 44.0 to 29.4 kN. A third of the peak has gone to a limit no gravity check ever reads, since gravity loads every joint in hogging. A sway frame’s wind capacity is set in large part by bolts the gravity design has no reason to provide, and that is a detail the two uses of a joint disagree about.

What the one storey is not saying #

The rest of the joint. Every joint here has a flat plateau and no rotation limit. A yielded end plate has a finite rotation capacity, and at a peak where the first joints have been plastic through 52 mm of drift, the rotation they have used is about 17 milliradians, in addition to whatever gravity took. Reversing cycles of wind are different again: the hysteresis of a real bolted joint pinches, and a joint that has opened a gap under one sense of moment does not unload at its initial stiffness all the way down. The figure of 3.71 is therefore an upper bound on a joint that has already yielded and is not new.

One storey on pinned feet. Feet with any fixity would hold the frame up on their own, and the case with pins is deliberately the case that puts the whole of the lateral stiffness on the joints. A frame of several storeys would add column continuity, which shares a joint’s work between the storey above and the one below, and a mechanism that has to choose its storey, as the frame that leans sets out for the plain portal.

Elastic members. The beams are assumed to have spare strength at midspan, and the columns to have spare strength at their tops. Partial strength often assumes this, and it is checked elsewhere. A storey with a 60 kN·m joint has a beam whose midspan moment under gravity is 160 − 56 = 104 kN·m, comfortably short of a 405 kN·m plastic moment, so it holds here.

Static wind. A gust is a load with a period, and a frame with a low critical factor has a long period as a result. What it does in resonance is not on these pictures.

Still open: the joints the wind has cycled #

Each path here goes once in one direction. A frame in a real wind goes to a drift one way and returns, and each joint that went to its sagging limit in the first excursion comes back with a new centre for its range. Under repeated sway the joints would settle into a stable loop if the excursions stay inside twice their range; the structure that settles down described exactly that bound for a section. But these are bolted joints, and each cycle takes out some of the plate’s rotation capacity and loosens some of its preload. Whether a partial-strength sway frame shakes down under its wind climate, or whether the slow ratchet from the gravity load’s permanent hogging walks the frame’s lean in one direction a little further in every storm, is what a wind that blows more than once would ask of a joint that was only ever pushed once here.