Structural Engineering & Statics
where the load goes
Internal forces

The joint stronger than its beam

A full-strength joint never reaches its own resistance: the beam beside it hinges first. But an end plate softens from two thirds of its resistance, and a joint only a little stronger than its beam is already on the soft part of its curve before the beam yields. The softening sheds moment from the beam's end into its span, so it delays the first hinge — by 8 per cent for a joint at 1.2 times the beam's strength, 18 for one at exactly the beam's, and not at all from 1.5 on. It never moves the collapse. And too much shedding hands the first hinge to the span, so the first hinge comes latest at the stiffness where the end and the span yield together.

Assumes The moment that goes round the corner, Neither pinned nor rigid, which is every real connection and After the first yield, which is not the end.

15 min read 6 figures The load must go somewhereWhich failure arrives firstOne support too many

A rigid knee carries the beam’s moment round the corner into the column; a real one does it through bolts and a plate that turn as they carry it. The joint that softens under its own load put bolted end plates at every beam end of a two-bay frame and let them follow EN 1993-1-8’s curve: straight up to two thirds of the joint’s resistance, then softening, until at its resistance the joint is a third as stiff as it began. Every joint there was weaker than its beam, so the joints were the frame’s fuses. The most loaded one softened first and reached its plateau, and the frame sent its share across the column to the joint on the other side.

That essay ended on the other kind of joint: one at least as strong as the beam it joins. A full-strength joint never reaches its plateau, because the beam beside it reaches its plastic moment first and hinges there. The question it left is whether the joint’s curve still matters. The joint softens on the way to the beam’s strength, and softening sheds moment. If it sheds enough, the beam’s first hinge comes later, and the joint’s curve is a question of strength. If the beam yields before the joint has done anything, the curve is only about stiffness, and a designer who chose a full-strength joint can forget it.

A frame whose beams are its fuses

The frame is the same two bays of 8 m and two storeys of 4 m, loaded by λ times 20 kN/m on its left bay and 20 kN at each floor from the left. Its beams have a plastic moment of 405 kN·m. Its columns are heavier than before, with a plastic moment of 538 kN·m, so that the beams are what yields — the order a designer wants, and the one that makes a beam’s own mechanism the frame’s.

That mechanism needs no analysis. The loaded beam fails when it has hinged at both ends and in its span, and a beam with the plastic moment at its two ends and its middle carries 16Mp/L2=10116M_p/L^2 = 101 kN/m: λ = 5.06. No full-strength joint can change that number, because no full-strength joint can be the place a hinge forms. What a joint can change is the order in which the three hinges arrive, and the load at which the first of them does.

The first hinge matters. It is the load at which the frame stops being elastic, where a plastic analysis begins counting rotation and an elastic one has already stopped. It sets how far a frame designed elastically is from its limit, and how much rotation the beam’s hinges must find between the first and the last.

How much of the curve lies below the beam’s strength

A full-strength joint softens before its beam yields only if it is barely full strength. The moment against the rotation of a beam-to-column end plate of 25 times the beam's EI/L initial stiffness, on EN 1993-1-8's curve, up to the beam's plastic moment of 405 kN·m, where the beam hinges beside the joint and the joint turns no further. At 1.00 times the beam's moment the joint softens from 270 kN·m and reaches the beam's plastic moment at 6.4 mrad; at 1.20 times the beam's moment the joint softens from 324 kN·m and reaches the beam's plastic moment at 3.9 mrad; at 1.50 times the beam's moment the joint softens from 405 kN·m and reaches the beam's plastic moment at 2.1 mrad; dashed, the initial stiffness, 2.1 mrad at the plastic moment.
Fig. 1 The moment against the rotation of a beam-to-column end plate of 25 times the beam’s EI/L initial stiffness, on EN 1993-1-8’s curve, up to the beam’s plastic moment of 405 kN·m, where the beam hinges beside the joint. At 1.00 times the beam’s moment the joint softens from 270 kN·m and reaches the beam’s plastic moment at 6.4 mrad; at 1.20, from 324 kN·m, at 3.9 mrad; at 1.50, from 405 kN·m, at 2.1 mrad, which is where the initial stiffness reaches it.

EN 1993-1-8’s curve is straight up to two thirds of the joint’s resistance and softens above it. For a full-strength joint the part that matters is the part below the beam’s plastic moment, since the beam hinges there and the joint goes no further. A joint exactly as strong as its beam softens from 270 kN·m, two thirds of 405, and by the time the beam yields it has turned 6.4 mrad, three times what its initial stiffness would give. A joint 1.2 times as strong softens from 324 kN·m and has turned 3.9 mrad. A joint 1.5 times as strong begins to soften at exactly 405 kN·m, the moment at which its beam hinges, and so never softens at all.

So “full strength” is not one kind of joint. Every full-strength joint is protected by its beam, but a joint barely stronger than its beam is soft for its last third, and one half as strong again is straight to the end.

The softening holds the inner end back

The softening joint holds the inner end back and the span takes it. The loaded beam's moments at its inner end, its outer end and in its span, against λ, for the two-bay frame of 8 m bays and 4 m storeys, its beams of plastic moment 405 kN·m, under λ times 20 kN/m on the left bay and 20 kN at each floor, with end plates of 25 EI/L initial stiffness and 1.20 times the beam's plastic moment: softening as EN 1993-1-8 has it (solid) and kept linear (dashed), each drawn until the span reaches the plastic moment. The first hinge forms at the inner end at λ = 3.78 with softening and at the inner end at λ = 3.51 without; the span reaches the plastic moment at λ = 4.78 with softening and 4.78 without, since by then the only joint past the straight part of its curve is the one beside the hinge. The beam collapses at λ = 5.06 either way, when it has hinged at both ends and in the span.
Fig. 2 The loaded beam’s moments at its inner end, its outer end and in its span against λ, with end plates of 25 EI/L and 1.20 times the beam’s plastic moment: softening (solid) and kept linear (dashed). The first hinge forms at the inner end at λ = 3.78 with softening and 3.51 without; the span reaches the plastic moment at λ = 4.78 in both; the beam collapses at 5.06.

The loaded beam’s inner end carries the most moment: the gravity load on that bay and the push from the left both hog it. With joints of 25 EI/L kept linear it reaches 405 kN·m at λ = 3.51. Let the joint soften and the inner end’s moment bends away from its straight line as it passes 324 kN·m, the joint’s two thirds, and reaches the plastic moment at λ = 3.78 instead. The moment it did not take has gone into the span, whose line rises a little more steeply, and into the outer end — the same emptying of a joint into its span that the panel a soft joint empties into the span found for joints soft from the start, arriving here only near the beam’s strength.

After the inner end has hinged, the two analyses are the same frame. The hinge carries 405 kN·m whatever the joint does, and the other joints are still on the straight parts of their curves, so the span reaches the plastic moment at λ = 4.78 in both. The softening buys the frame a later first hinge and nothing after it.

That 3.51 is itself later than rigid joints give. A rigid frame puts its first hinge at λ = 3.25; joints of 25 EI/L, the stiffness EN 1993-1-8 calls rigid in an unbraced frame, are still flexible enough to shed 8% of it. A rigid classification is a statement that the joint’s flexibility may be ignored in working, as neither pinned nor rigid put it, and not a statement that the frame yields where a rigid frame would.

Why the collapse cannot see the joint

That the collapse load does not move is not a property of this frame. It is the plastic collapse theorem, which after the first yield used for a single beam: a structure that can form enough hinges to become a mechanism collapses at the load that mechanism needs, and the elastic history on the way — every stiffness, every residual moment, every softened joint — decides only the order in which the hinges arrive. Nothing elastic appears in 16Mp/L216M_p/L^2.

What the theorem asks in return is that every hinge can rotate for as long as the others take to form. A first hinge at λ = 3.25 has to keep turning at its plastic moment until λ = 5.06; a first hinge at λ = 4.52 has to keep turning only from there. So the softening that delays the first hinge does not raise the collapse load, but it shortens the stretch over which the first hinge must turn while carrying its full moment — the redistribution the moment that was moved on purpose did by design, done here by the joint’s curve without anyone choosing it. A frame whose first hinge comes at 89% of its collapse needs much less ductility from that hinge than one whose first hinge comes at 64%.

This is the sense in which the full-strength joint’s curve is a strength question after all. It cannot change what the frame carries, but it changes what the frame’s hinges must survive to carry it.

A joint half as strong again does not soften

The softening delays the first hinge only in a joint barely stronger than its beam. The load factor at which the first plastic hinge forms in the two-bay frame of 8 m bays and 4 m storeys, its beams of plastic moment 405 kN·m, under λ times 20 kN/m on the left bay and 20 kN at each floor, against the strength of its end plates over the beam's plastic moment, the end plates' initial stiffness 25 times the beam's EI/L. At 1.0 the first hinge forms at λ = 4.15, at the inner end; at 1.2, 3.78; at 1.5 and beyond, 3.51, the value for the same joint kept linear, because a joint that strong is still on the straight part of its curve when the beam hinges. Rigid joints give 3.25; the beam collapses at 5.06 whatever the joint.
Fig. 3 The load factor at which the first plastic hinge forms against the strength of the end plates over the beam’s plastic moment, the end plates’ initial stiffness 25 times the beam’s EI/L. At 1.0, λ = 4.15, at the inner end; at 1.2, 3.78; from 1.5, 3.51, the value for the same joint kept linear. Rigid joints give 3.25; the beam collapses at 5.06 whatever the joint.

The weaker the full-strength joint, the more of its curve lies below the beam’s strength and the more it sheds. A joint exactly as strong as its beam delays the first hinge to λ = 4.15, 18% later than the same joint kept linear. At 1.2 times the beam’s strength, 3.78, 8% later. From 1.5 times the beam’s strength on, the softening starts at the plastic moment and buys nothing.

That last number is worth knowing for a reason that has nothing to do with this frame. Seismic design asks a beam-to-column joint to be stronger than its beam by a margin for the steel being stronger than specified and for strain hardening, which comes to about 1.4 times the beam’s plastic moment. A joint designed to that rule is nearly straight up to the beam’s hinge: in this frame it delays the first hinge from 3.51 to 3.60. The rule exists to make the beam yield and the joint stay out of it, and its consequence is that the joint’s curve barely matters before the beam yields.

Too much shedding hands the first hinge to the span

A softening joint behaves, below the beam’s strength, like a joint of lower stiffness. So the question of how much softening helps becomes a question of how much flexibility helps, and flexibility does not help without limit.

The first hinge comes latest where the beam's end and its span yield together. The load factor at which the first plastic hinge forms in the two-bay frame of 8 m bays and 4 m storeys, its beams of plastic moment 405 kN·m, under λ times 20 kN/m on the left bay and 20 kN at each floor, against the initial stiffness of its full-strength end plates (1.20 times the beam's plastic moment), softening as EN 1993-1-8 has it and kept linear. With softening, 3.39 at 100 EI/L, 3.78 at 25 and 3.91 at 3; kept linear, 3.32, 3.51 and 3.91. Stiff joints hinge first at the inner end and flexible ones in the span; the first hinge comes latest between, 4.52 at 10.0 EI/L with softening and 4.31 at 5.0 kept linear. Rigid joints give 3.25, and the beam collapses at 5.06 for every joint.
Fig. 4 The load factor at which the first plastic hinge forms against the initial stiffness of full-strength end plates at 1.20 times the beam’s plastic moment, softening and kept linear. With softening, 3.39 at 100 EI/L, 3.78 at 25 and 3.91 at 3; kept linear, 3.32, 3.51 and 3.91. Stiff joints hinge first at the inner end and flexible ones in the span; the first hinge comes latest between, 4.52 at 10 EI/L with softening and 4.31 at 5 kept linear. Rigid joints give 3.25, and the beam collapses at 5.06 for every joint.

As the joint’s initial stiffness falls from 100 EI/L, the first hinge comes later, because the beam’s end takes less of the load and the span more. That works until the span is carrying as much as the end. Below about 10 EI/L the span reaches the plastic moment first, and from there a still more flexible joint makes things worse: it pushes more moment into a span that is already the first to yield. At 3 EI/L the first hinge is in the span at λ = 3.91.

The first hinge comes latest at the stiffness where the beam’s end and its span yield together. For a joint 1.2 times as strong as its beam and softening, that is about 10 EI/L, where the first hinge arrives at λ = 4.52, 89% of the collapse load. Kept linear, the same balance is reached at about 5 EI/L and λ = 4.31. The softening moves the balance to a stiffer joint, because a joint that softens near the beam’s strength is stiffer in working and softer exactly when the beam’s end is approaching its limit.

That balance is the inelastic relative of the joint that was chosen, whose stiffness made a single beam’s end and span moments equal so that the beam could be sized for the smaller of two numbers. There the balance was elastic and set by stiffness alone. Here it is set by the stiffness and the curve together, in a frame being pushed sideways, and its reward is not a smaller beam but a first hinge that arrives near the collapse rather than at two thirds of it.

At working load the curve is invisible

At working load the joint's curve is invisible. The roof's drift against λ for the two-bay frame of 8 m bays and 4 m storeys, its beams of plastic moment 405 kN·m, under λ times 20 kN/m on the left bay and 20 kN at each floor, with rigid joints and with three full-strength end plates (1.20 times the beam's plastic moment), each until the span reaches the plastic moment. At λ = 1 the drift is 5.4 mm with rigid joints; 5.8 mm at 25 EI/L, kept linear; 5.8 mm at 25 EI/L, softening; 6.6 mm at 8 EI/L, softening: the softening joint and the same joint kept linear agree, because no joint has passed two thirds of its resistance. They part only as the frame approaches its first hinge.
Fig. 5 The roof’s drift against λ, with rigid joints and with three full-strength end plates at 1.20 times the beam’s plastic moment. At λ = 1 the drift is 5.4 mm with rigid joints; 5.8 mm at 25 EI/L kept linear; 5.8 mm at 25 EI/L softening; 6.6 mm at 8 EI/L softening. The softening and linear joints part only as the frame approaches its first hinge.

At working load no full-strength joint in this frame has come anywhere near two thirds of its resistance. That is the opposite of the situation the redistribution nobody chose described, where a connection assumed to be a pin carried moment in service that no analysis had given it; here the joint behaves in service exactly as its initial stiffness says, and its surprises are saved for the approach to the hinge. The most loaded end carries about 115 kN·m at λ = 1, and a joint 1.2 times as strong as the beam does not soften until 324. So the drift a serviceability check reads is set by the joint’s initial stiffness alone — 5.4 mm with rigid joints, 5.8 with joints of 25 EI/L, 6.6 with joints of 8 — and a softening joint and the same joint kept linear give the same number to the tenth of a millimetre. They part only above λ = 2.8, where the inner end passes 324 kN·m.

That divides the joint’s two properties cleanly between the two checks that use them. Its initial stiffness is a working-load property, and its curve is a property of the approach to the first hinge. A full-strength joint chosen for its drift has had one of its two numbers chosen; the other decides when the beam starts to yield.

The rotation the joint spends first

The rotation the joint spends before its beam yields. The rotation of a full-strength end plate of 25 times the beam's EI/L initial stiffness at the moment the beam beside it reaches its plastic moment of 405 kN·m, against the joint's resistance over the beam's plastic moment. At 1.0, 6.4 mrad, 3.0 times the 2.1 the initial stiffness gives; at 1.2, 3.9 mrad; from 1.5 on, 2.1. That rotation is the joint's, and the beam's plastic rotation comes after it.
Fig. 6 The rotation of a full-strength end plate of 25 times the beam’s EI/L initial stiffness at the moment the beam beside it reaches its plastic moment of 405 kN·m, against the joint’s resistance over the beam’s plastic moment. At 1.0, 6.4 mrad, 3.0 times the 2.1 the initial stiffness gives; at 1.2, 3.9 mrad; from 1.5 on, 2.1.

The delay is not free. A joint that sheds moment does so by turning, and a joint exactly as strong as its beam has turned 6.4 mrad by the time the beam beside it hinges — three times what its initial stiffness would give, and all of it inside the bolts and the plate before the beam has done anything plastic. A joint 1.2 times as strong turns 3.9 mrad, and from 1.5 times the beam’s strength the joint turns only its elastic 2.1.

What comes after is the beam’s. Once the beam has hinged beside the joint, the joint’s moment stays at the beam’s plastic moment and the joint turns no further; the plastic rotation the frame needs between its first hinge and its collapse is supplied by the beam, which is what a full-strength joint was chosen to arrange. That is the difference from the joint that has to keep turning, a partial-strength joint that hinges first and must go on rotating at its full moment until the span catches up, and whose ductility therefore decides whether the collapse load is reached at all. A full-strength joint asks its own ductility for only the softening below the beam’s strength — a few milliradians, which a well-detailed end plate has — and hands the rest to a rolled section.

The first hinge, by hand

With rigid joints at λ = 1 the loaded beam’s inner end carries 124 kN·m, so the first hinge comes at 405/124=3.25405/124 = 3.25. With linear joints of 25 EI/L it carries 115, and the hinge comes at 405/115=3.51405/115 = 3.51.

A joint 1.2 times as strong as the beam has a resistance of 486 kN·m and softens from 23×486=324\tfrac{2}{3} \times 486 = 324. At the beam’s 405 its stiffness modification is (1.5×405/486)2.7=1.252.7=1.83(1.5 \times 405/486)^{2.7} = 1.25^{2.7} = 1.83, so its secant stiffness there is 25/1.83=13.725/1.83 = 13.7 EI/L. Every joint in the frame kept linear at that secant would put the first hinge at λ = 3.72, a little earlier than the softening joints’ 3.78. The two are not the same frame: the uniform secant makes every end that soft, while the softening joints are that soft only at the one end approaching the beam’s strength, and stay at their initial stiffness everywhere else. A single reduced stiffness is a fair stand-in for a full-strength joint’s curve, as it was not for the partial-strength joints that softened under their own load, because only one joint in the frame is on the soft part of its curve when it matters.

The collapse needs the plastic moment at both ends and in the span: wL2/8=2MpwL^2/8 = 2M_p, so w=16×405/82=101w = 16 \times 405/8^2 = 101 kN/m, λ = 5.06.

Where the strong-column frame stops

The columns stay elastic. They are chosen heavier than the beams, and at the beam’s collapse the largest column moment is about 450 kN·m against a plastic moment of 538. A frame with weaker columns would hinge them first, and its mechanism would be a sway of the storey rather than the loaded beam’s.

The span has no hinge of its own. Every curve stops where the span reaches the plastic moment, which is the last state the model describes; the collapse is the beam’s mechanism, which needs no analysis. Between the span’s hinge and the outer end’s, which this model cannot follow, the frame is stiffer than a mechanism and softer than an elastic frame.

The joint’s curve is the standard’s. EN 1993-1-8’s curve is a design idealisation of tests: the exponent 2.7 is a bolted end plate’s, and a welded joint, whose curve is stiffer and straighter, would shed less. Strain hardening in the beam would raise its moment past 405 kN·m and drive a barely full-strength joint further up its curve, towards its own resistance — the case a seismic overstrength rule exists to prevent.

Every joint is the same. A designer can choose different joints at different ends. A softer joint at the inner end alone would delay the first hinge where it forms without sending moment into the outer end, which is where this leaves off.

Still open: the joint chosen end by end

Every joint here is the same, so the balance between end and span was reached by softening every joint at once. The loaded beam’s ends are not equal: the inner end is hogged by the gravity load and the push together, and the outer end by the gravity load against the push. A frame whose inner joints were made softer than its outer ones — or whose inner joints were made barely full strength while its outer ones were made strong — would shed moment only from the end that yields first, and could bring the inner end, the outer end and the span to the plastic moment almost together. Whether a frame tuned that way reaches its first hinge at nearly its collapse load, and whether a push from the other side, which reverses which end is the inner one, undoes the tuning, is the question the moment that was moved on purpose puts to a joint chosen to move it.

Where this goes next

The loaded end stops; the unloaded one takes over. The moments at the four floor beam ends of the frame of two 8 m bays and two 4 m storeys, its beam ends joined to the columns by end plates of 3.0 EI/L initial stiffness and 120 kN·m resistance, under λ times 20 kN/m on the left bay and 20 kN at each floor, against λ: on the softening joints (solid) and on joints that keep their initial stiffness (dashed). At λ = 1 the loaded beam's inner end carries 78 kN·m and the unloaded beam's inner end, across the same column, 13. The loaded end reaches its resistance at λ = 2.69; at λ = 3 the four ends carry 89, 120, 81, 88 kN·m, against 97, 235, 39, 68 on stiff joints. The unloaded inner end has gone from 13 to 81, 6.2 times as much for three times the load. The joint that softens under its own load A bolted end plate is stiff until its moment passes two thirds of its resistance and then softens until, at its resistance, it is a third as stiff. Put such joints at every beam end of a frame under a pattern load and a push, and the most loaded joint softens first and hands its share across the column to the joint on the other side, which carries six times as much for three times the load. The redistribution grows as the frame approaches its strength. Every joint ends at its own stiffness, so the single secant stiffness an elastic analysis is allowed asks the loaded joint for half as much again as it can carry, and gives the exterior panel three quarters of what it gets. Where the span moment and the support moment cross. Span moment and support moment against the joint's stiffness, for a 6 m beam under 30 kN/m. They move in opposite directions because they add to a constant — the simple-span 135 kN·m is fixed by statics and the joint only decides how it is split. They cross at 68 kN·m, where the joint is delivering 75 per cent of the fixed-end moment, and neither the crossing nor the fraction depends on the beam, the span or the load: it is the point where f·wL²/12 equals wL²/8 − f·wL²/12, which is f = 0.75 for every beam there has ever been. The joint that was chosen A joint's stiffness decides how a beam's moment divides between its span and its supports, and the two add to a constant. So there is a stiffness at which they are equal, the beam is sized by the smaller of two numbers rather than the larger of one, and the design moment is half what a simple connection leaves behind. The joint yields first and the span takes the rest. The end and span moments of a beam of span 9 m and flexural rigidity 61,700 kN·m², with a plastic moment of 522 kN·m, on joints of rotational stiffness 41,133 kN·m/rad, 6.00 EI/L, and resistance 261 kN·m, as a uniform load rises to collapse. While elastic the joints carry 0.75 of the fixed-end moment. The joints reach their resistance first, at 51.6 kN/m, and from there every further increment of load goes to the span, which reaches 522 kN·m at 77.3 kN/m. The collapse load is 8(Mp + Mj)/L² = 77.3 kN/m, between the 51.6 at which the beam would collapse on pins and the 103.1 it would reach on rigid full-strength ends, and it contains the joint's strength and not its stiffness. The joint that has to keep turning A beam on partial-strength joints collapses at 8(Mp + Mj)/L² however stiff the joints are. Stiffness decides only which yields first, and a joint that yields first has to go on rotating at full moment until the span catches up. A weak joint on a stiff connection has the most turning to do — more than a rigid full-strength one.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

End plateJoint classificationJoint stiffnessMechanismMoment rotationPattern loadingPlastic hingeRedistributionRotation capacitySway