The column on the other side of the joint

A partial-strength joint is a connection between a beam and a column that is deliberately weaker than the beam. Under a growing load it carries moment elastically until it reaches its own resistance, then rotates at that moment while the beam goes on loading its span, and the beam’s collapse load is set by the joint’s strength and not its stiffness. The design is attractive for exactly the reason that makes it possible: the joint is cheap — an end plate rather than a fully welded and stiffened connection — and the beam still gets a useful share of continuity from it, chosen rather than classified.

Every one of those arguments put the joint between the beam and something that did not move. The far side of the joint was a support. The calculation of the beam needed nothing more, and the essay on rotation demand ended on the question it had not asked: a joint that delivers its resistance to the beam delivers it to the column too, and a column designed for axial load now has a moment at every floor.

That question has two halves. At an interior column under a symmetric load, the beams on the two sides balance each other and the column turns hardly at all — the support really is still, and the joint’s resistance is a cap on what the column can be given. At the edge of a frame there is no beam on the other side. The column is the only thing resisting the beam’s end moment, it rotates under it, and that rotation is part of the joint’s.

A frame, joint by joint #

The frame is a braced six-storey line, 4 m storeys, pinned at the base. Its edge column is a 254 × 254 UC 89 for the lower three storeys and a 203 × 203 UC 60 above a splice. Every floor carries the 9 m beam of the partial-strength essays — flexural rigidity $61{,}700\ \text{kN·m}^2$, plastic moment 522 kN·m — on joints of stiffness 41,130 kN·m per radian and resistance 261 kN·m: half the beam’s strength, at the stiffness that balances the beam’s end and span moments. The beam’s far end meets an interior column that does not turn, through an identical joint.

The frame is solved as what it is — a set of rotations: the edge column’s node at each floor, and each beam’s two ends — with each joint an elastic–perfectly-plastic spring. The load is raised until the next joint reaches 261 kN·m; that joint is then held at its resistance, and the load is raised again.

The six interior joints yield first, all between 44 and 50 kN/m. They are on a column that does not turn, so each sees the whole of its spring stiffness and reaches the resistance at the load the beam calculation predicted. The edge joints are slower. Each is a spring of 41,130 kN·m/rad in series with the column’s own resistance to rotating — the column is simply one more spring in the chain a joint is made of — and the series spring is much softer: for the top floor, where a single 203 UC 60 length below the joint does all the resisting, it is about a third as stiff as the joint alone.

So in service, an edge joint is a smaller moment than it was designed to be. At 60 kN/m the top floor’s edge joint carries 171 kN·m, two thirds of its resistance. That would be reassuring if it were the end of the story, but the moment the interior joints yield the beam’s extra load has nowhere to go but the span and the edge. From 50 kN/m on, the edge joints’ moments climb faster, and the edge joints on the three lower floors — whose columns are the stiff 254 UC 89 — reach 261 kN·m by about 60.

What the column receives #

What an edge joint delivers is what the column carries, and the comparison that matters is with what the column would have carried otherwise.

Two comparisons, and they point opposite ways.

Against a rigid frame, the partial-strength joints have changed almost nothing at the edge. The rigid joints deliver 282 to 156 kN·m, the partial-strength ones 259 to 171 — within six per cent on average, with the partial joints delivering more at the top floor. The joints differ enormously at the interior, where a rigid joint takes 466 kN·m to the partial joint’s 261. At the edge, both are limited by the same thing, the column’s flexibility, and a joint’s strength cannot cap a moment that its stiffness never lets it reach. The fuse is fitted where the moment would have been small anyway, and is not at the edge where it would matter. Where the partial-strength joint does relieve the frame is at the interior, and its effect there is on the beam: the span carries 347 kN·m against the rigid frame’s 233.

Against the simple-construction rule, the comparison runs the other way. The rule gives the edge column the moment the beam left behind — the beam’s end reaction acting 100 mm from the column face — and nothing more: 270 kN times about a quarter of a metre, 62 kN·m below the splice and 55 above it. A designer who chooses partial-strength joints for the beam, and then designs the columns as if the frame were simple, has given the column a quarter of what it receives.

The moment up the column #

The moments arrive at every floor, and they arrive the same way round.

Each beam pulls the top of the joint towards the span. A floor’s moment therefore rotates the column node in the same sense at every level, and each column length, between two such floors, is bent by one moment at its top and an equal-sense one at its bottom: double curvature, with a point of contraflexure part-way up the storey. The first storey is the exception: its base is pinned and carries nothing, so it bends in single curvature from zero to 97 kN·m.

The pattern is familiar from a frame read as a stack of joints, and its consequence for a column is the next figure. The largest moment is not at the bottom, where the axial load is greatest, but at the top of the top storey, where the column has only one length to share the floor’s moment with and the full 171 kN·m arrives at its upper end.

The interior column, where the fuse works #

The contrast with the interior is worth drawing out, because it is where the partial-strength joint earns its reputation.

At an interior column the two beams’ end moments arrive from opposite sides and, under the same load on both, cancel. The column is given nothing but the difference, and that difference is what the joints’ resistance caps. With rigid joints, a loaded span beside an unloaded one sends the unloaded span’s whole fixed-end moment’s worth of imbalance into the column — the full elastic moment of one side less the small moment of the other. With partial-strength joints the loaded side’s contribution can never exceed 261 kN·m, so the imbalance is bounded by the joint’s resistance less whatever the unloaded side carries under its own dead load. That is a cap the column designer can rely on, and it is the sense in which the joint really is a fuse.

The edge has no other side. There is no imbalance to cap, only a moment, and the moment is set by how stiff the column is against the joint. The fuse is installed in both places and does its work in one.

The buckling check and the section check #

A column in a braced frame is checked twice: as a member, against buckling over a length its ends decide under its axial force and the moments along it, and as a section, at its most heavily loaded point, against yielding — the second being the interaction of axial force and moment at a single cross-section.

The buckling check hardly notices. Moments three times larger change it by a few hundredths in the lower storeys and by 0.17 at the top, and the reason is the double curvature. A member bent in an S has its largest moments at its ends, where it cannot buckle, and the moment that matters for buckling is an equivalent uniform one: $C_m = 0.6 + 0.4\psi$, with $\psi$ the ratio of the end moments, which for opposite-sense ends near equal in size is close to $-1$. The factor is floored at 0.4, and every storey above the first is on the floor. The double curvature that a column’s edge position forces on it is the reason the buckling check can absorb the joints’ moments — it discounts them by sixty per cent.

The section check is not discounted. It is made at the end, where the moment is, and the moment arrives there in full. In the top storey the section check rises from 0.24 under the rule to 0.74 at the design load — three times — while the buckling check rises from 0.25 to 0.42. The check that governed the column under the simple rule, and that a designer would normally look at first, is the one that has stopped governing. The column’s critical point has moved from the middle of a storey to the top of it, and from the bottom of the building to the top.

A bigger column attracts a bigger moment #

The obvious repair is a heavier column in the top storey, and here the series spring turns against the designer.

A stiffer column is a stiffer spring behind the joint, and the series spring stiffens with it, so the joint delivers more. The lightest column drawn, a 203 UC 46, attracts 148 kN·m at the design load; the 203 UC 60 in the frame, 171; a 254 UC 89, 232. At the beams’ collapse load the same pattern holds until the joint reaches its resistance — the heavier columns all attract exactly 261 kN·m there, because the joint will not deliver more.

The comparison with what each column can carry is the design answer. The 203 UC 46 attracts more at collapse than its section can take, 195 against 167, so it would yield at the floor before the joint did. The 203 UC 60 carries what it attracts — 225 against 230 — by a margin of two per cent. Every heavier column is comfortable, and it is comfortable because the joint has capped the moment at 261 kN·m. The joint protects the column only once the column is stiff enough to make the joint yield. Below that, a column chosen to carry its moment is chasing a moment that grows as it is chosen.

The two per cent is the part worth worrying about, and it is fragile in a way the design resistance does not show.

A joint stronger than it was drawn #

A joint’s resistance in a design is a calculated minimum. The joint as built is stronger: the end plate’s steel has a yield strength above its nominal grade, the bolts are a size up because that is what was on the shelf, a plate is thicker than specified. None of that is an error, and in the beam’s calculation all of it is safe — a stronger joint raises the beam’s collapse load.

On the column’s side of the joint it is not safe. Ten per cent of extra joint strength takes the top storey’s section past its limit at the beams’ collapse load; thirty per cent takes the storey above the splice past its own, where the lighter section meets an axial force accumulated from three floors; forty per cent takes the bottom storey’s buckling check past one. A joint built stronger than designed is a column built weaker, in the only sense that matters for the collapse mechanism the beams’ strength was computed from: that mechanism needs a hinge at the joint, and if the column yields first, the hinge forms in the column instead.

That is the principle seismic design writes down as capacity design — the member that is meant to stay elastic is designed for the overstrength of the member that is meant to yield — and it is not usually applied to a braced frame under gravity load, because a braced frame is not usually designed around a hinge at a joint. A semi-continuous frame with partial-strength joints is.

The top floor, by hand — and what the hand misses #

The top storey’s numbers can nearly be reached without the frame, and the place where they cannot is instructive.

The joint at the top floor sits on one 203 UC 60 length, 4 m long. If the column’s far end at the floor below turns in the same sense and by the same amount, the length resists with $6EI/h$: $6 \times 210 \times 10^6 \times 6.125\times10^{-5}/4 = 19{,}300$ kN·m/rad. In series with the joint’s 41,130 that is $1/(1/41{,}130 + 1/19{,}300) = 13{,}100$ kN·m/rad — a third of the joint alone. The beam’s fixed-end moment at 60 kN/m is $60 \times 81/12 = 405$ kN·m. By 60 kN/m the interior joint at the beam’s far end has yielded and holds 261 kN·m, so the slope-deflection equations give the near end

$$M_{\text{near}} = \frac{405 + 261}{2}\cdot\frac{13{,}100}{13{,}100 + 3EI_b/L} = 333 \times \frac{13{,}100}{13{,}100 + 20{,}600} = 130 \text{ kN·m}.$$

The frame says 171. The difference is the floor below. Its own edge joint delivers 246 kN·m into the node beneath the top storey and turns that node further than the top node turns, so the top length is being rotated from below as well as loaded from above, and it resists the top joint more stiffly than $6EI/h$ allows. A one-floor sub-frame — the hand method a designer would reach for — underestimates the top joint’s moment by a quarter, because the floor below is part of the top floor’s joint as well.

At collapse the top storey’s column carries its axial force of 342 kN, an eighth of its squash load, so its plastic moment of 233 kN·m is reduced only to 230. The joint’s 261 kN·m is more than that, so a top-storey column that the joint actually loads to its resistance yields first. The frame has the joint at 225 when the beams collapse, because the column keeps relieving it; with ten per cent more joint strength, the beams’ collapse load rises to 79.9 kN/m and the delivered moment with it.

An edge column, a braced frame, a design that stops at the section #

The calculation rests on a set of choices.

The frame is braced. No storey sways, so the joints’ moments are the only moments and the column’s buckling length is its storey height. In an unbraced frame the sway adds its own moments at the same joints, the edge column’s double curvature becomes the sway frame’s, and the joints’ rotation demand accumulates up the height in a way this frame does not ask.

The interior column does not turn. Every beam’s far end is on a node held still, which is the symmetric interior case. Pattern loading would put a loaded span beside an unloaded one at some interior columns, and the unbalanced moment there — capped on the loaded side at the joint’s resistance and elastic on the unloaded side — is the interior column’s own version of this argument.

The joints are elastic–perfectly-plastic. A real end-plate joint is neither pinned nor rigid at any load, softens before its resistance and hardens a little after it; the softening delays the moment’s arrival at the column and the hardening is part of the overstrength.

The column checks are code checks, EN 1993-1-1’s interaction for a class 1 or 2 section with buckling curves b and c, a floor-to-floor buckling length, and lateral–torsional buckling between floors excluded. A column checked by another rule would give other numbers; the shift of the governing check from the member to the section at the floor does not depend on which rule is used.

The panel zone, and the splice above it #

The figures cannot show the panel zone. A beam that delivers 261 kN·m into a column flange puts a shear into the column web between the flanges, and a column web sized for axial load may need a doubler plate to carry it. That is a third check on the column side of the joint, local rather than member or section, and it is the one a fabricator meets first.

They also cannot show the column splice, which in this frame sits at the top of the third storey — exactly where the storey-4 section check is high — and which in simple construction is often detailed to carry axial load and little else. A splice placed at a column’s contraflexure point carries almost no moment; this column’s contraflexure moves with the load and the joints’ state, and nothing guarantees that the splice is where the moment is not.

Still open: the sway frame that the joints are holding up #

The frame here is braced, so the joints’ moments are a nuisance to the column and nothing else. If the bracing is removed and the semi-continuous joints are asked to provide the frame’s lateral stiffness as well — which is the main reason semi-continuous frames are proposed for low-rise buildings — the same joints carry gravity moments and sway moments together, the edge joints yield earlier on the windward side and later on the leeward, and every joint that yields softens the frame against sway exactly when the sway is largest. Whether a partial-strength joint that has yielded under gravity load still contributes anything to a sway frame’s stability, and whether the sway frame’s critical load then depends on the order in which its joints reached their resistance, is the question a braced frame never has to ask.