Structural Engineering & Statics
where the load goes
Structural form

The part that is meant to be weak

A braced frame is stiff and has nowhere to yield. A moment frame yields everywhere and is soft. Move the two diagonals a metre apart along the beam and the whole storey shear has to pass through the segment between them — which keeps most of the stiffness and puts every yielding in one member the designer chose.

Assumes How a tall building stands still, After the first yield, which is not the end and The property that appears in none of the equations.

17 min read 8 figures The load must go somewhereWhich failure arrives firstThe load that will not hold still

There are two ways to stop a steel frame from leaning, and each of them is bad at exactly what the other is good at.

A concentrically braced frame puts a diagonal from corner to corner. The storey shear becomes an axial force, the frame is very stiff, and there is nowhere in it that can yield in a useful way: the brace in compression buckles, and a buckled brace loses most of its capacity on the first cycle and does not get it back.

A moment frame has no diagonal at all. Every joint bends, hinges form at the beam ends, and the frame absorbs a great deal of energy while being about a tenth as stiff — which for a tall building means the drift limit sizes every member and the strength is never reached.

The eccentrically braced frame is neither, and it is not a compromise between them. It is a rearrangement that keeps most of the first’s stiffness and all of the second’s ductility, at the price of one design decision: choosing where the frame will yield.

Every path to the ground goes through the link. A braced bay 8 m by 4 m whose two diagonals stop 800 mm apart instead of meeting. The storey shear reaches the ground through the diagonals, and the vertical components they deliver to the beam have to pass through the segment between them: the link carries 47% of the applied shear as a shear force, at a lever arm short enough that its ends reach 0 kNm while the rest of the beam carries 0. The deflected shape drawn is the solved one, magnified — the real drift under this load is 0.008 mm. Everything outside the link is designed to stay elastic while the link is yielding, which is what makes the mechanism a choice rather than a hope.
Fig. 1 The rearrangement, in one drawing. The two diagonals stop 800 mm apart instead of meeting, so their vertical components have to pass through the beam segment between them. Every path from the storey shear to the ground goes through that segment, which is what makes it the fuse.

Which free body produced the number

Cut the beam either side of the link and take the piece between the cuts.

The two diagonals deliver their forces to the ends of that piece. Their horizontal components go into the beam as axial force; their vertical components are equal and opposite, and there is nothing else on the free body that can balance them. So the link carries the difference of the two vertical components as a shear force, along a length of 800 mm, with a moment at each end equal to half that shear times the length.

Two consequences follow immediately from the arithmetic and neither is obvious from the picture.

The shear in the link is not a fraction of the storey shear that gets smaller as the link gets shorter — it is very nearly fixed, set by the geometry of the diagonals. And the moment in the link is proportional to its length, which means a short link reaches its shear capacity first and a long one reaches its moment capacity first. That single fact organises everything else.

A short link yields in shear, a long one in bending. What a link of a given length can carry, and by which mechanism. Below 1057 mm the web reaches its shear capacity of 790 kN before the ends reach their plastic moment, and the whole length of the link yields in shear — a mechanism credited with 0.08 radians of rotation. Above 1717 mm the ends hinge first and the capacity falls as 2Mp/e, with a quarter of the rotation capacity. The two lengths are 1.6 and 2.6 times Mp/Vp, which is a property of the section and of nothing else: for this one that ratio is 661 mm. The link drawn is 800 mm, which makes it a shear link.
Fig. 2 The two mechanisms and the length at which they change places. Below 1.6Mp/Vp the web reaches 0.6 f_y over its whole area before the ends reach their plastic moment; above 2.6Mp/Vp the ends hinge first and the capacity falls as 2Mp/e. Both limits are properties of the section — for this one, Mp/Vp is 661 mm — and nothing about the frame enters them.

The stiffness that is not given up

The intuition says that pulling the diagonals apart must destroy the brace’s stiffness, because the load path now includes a piece of beam bending.

Measured, on a frame solved by the same stiffness routine as everything else here: a link a tenth of the bay long keeps 79 per cent of the concentric frame’s stiffness. At a fifth of the bay it is 55 per cent. The moment frame at the far end of the sweep is at 11.

The curve is flat where designs live and steep where they do not, and the reason is that the link is short and deep. A 400 mm-deep segment 800 mm long is not a beam in any useful sense — its span-to-depth ratio is two — and it deforms mostly in shear rather than in bending, which is a much stiffer way to deform. The frame is still a truss with a slightly soft joint in it.

The stiffness the ductility is bought with. Lateral stiffness against link length, as a fraction of the same bay braced concentrically. At a link of 800 mm — 10% of the bay — the frame keeps 79% of the concentric stiffness; at the far end of the range, where the diagonals meet the columns, it is a moment frame at 11%. The curve is steep at the left, which is the useful part of it: the first tenth of the bay costs a fifth of the stiffness and buys the whole of the yielding mechanism.
Fig. 3 Stiffness against link length, from a concentric brace at the left to a moment frame at the right. The useful part of the picture is the steepness at the very left: almost all of the loss happens in the first third of the bay, and almost all of the ductility is bought in the first tenth.

Yielding in shear, which is the unusual part

Almost every plastic mechanism in this collection is a bending one. A hinge forms, the section rotates at constant moment, and the rotation capacity comes from how much of the section can yield before something local goes wrong.

A short link does something different: the whole web yields in shear, over the whole length of the link at once. The plastic shear capacity is 0.6fytwhw0.6 f_y t_w h_w — 790 kN for the section here — and when the web reaches it the link deforms as a parallelogram, with the flanges staying parallel and the web distorting between them.

That mechanism has two properties that make it the best fuse in structural steelwork. It is stable: a yielding web that is stiffened at intervals does not lose capacity as it deforms, and its hysteresis loops are full rectangles rather than the pinched shapes a buckling brace produces. And it is capacious: a properly stiffened shear link is credited with 0.08 radians of plastic rotation, four times what a flexural link gets and an order of magnitude more than most connections.

The stiffeners are not incidental. A web deforming plastically in shear will buckle diagonally unless it is divided into short panels, so a shear link is covered in full-depth stiffeners at close centres, and the spacing of those stiffeners is what the rotation capacity is actually bought with.

Where the energy goes: one loop in force against displacement. The force the supports feel — the spring's and the damper's together — against the displacement, for one mechanism. yielding at 182.53 kN, enclosing 22.35 kJ over the record drawn. The yielding loop is a parallelogram whose area does not depend on how fast it is traced, and every circuit leaves the structure displaced from where it began.
Fig. 4 The area a yielding element encloses in one cycle, which is the energy it has removed from the structure. A shear link’s loop is close to this idealisation; a buckling brace’s is pinched to a fraction of the area, which is why the concentric frame is stiff and not ductile.

Underneath that loop is what the steel itself does under reversal. The link is asked to trace it several hundred times in a strong earthquake, at strains far past yield, and the reason mild steel is specified for links rather than a higher grade is that its loop stays open and does not harden its way into brittleness.

The rest of the frame is designed not to yield

Choosing where a structure yields is worth nothing unless the choice comes true, and making it come true is the whole of capacity design.

The link’s capacity is not a demand from the analysis — it is a property of the member, and it is bounded above rather than below. So the braces, the columns, the beam outside the link and every connection are designed for the forces present when the link is at its own capacity, with its material overstrong and strain-hardened, rather than for the forces the elastic analysis returned.

On the frame here, scaling the elastic solution until the link reaches its 790 kN puts 1,153 kN in each diagonal and 242 kNm in the column. Those are the design forces for everything that is not the link, and they have nothing to do with the loading the analysis was run for.

The failure this prevents is specific and quiet: a frame whose brace is slightly weaker than intended buckles before the link yields, and then it is a concentrically braced frame with a soft spot — which is worse than either of the two systems it was meant to improve on.

The mechanism the frame is being kept away from is the storey one: enough hinges in the columns, top and bottom, and the storey collapses on its own. Capacity design is the arithmetic that makes certain the hinges appear in the member that was chosen rather than in the member that happened to be weakest. What it is protecting against is not exotic, and the concentric arrangement shows it plainly.

The two braces balance until one of them buckles. An inverted-V brace after the compression member has gone. While both braces are elastic they carry equal and opposite forces and their vertical components cancel on the beam above, which is why the beam in a chevron bay is usually sized for gravity alone. The compression brace buckles at 445 kN and then sheds most of what it was carrying — 30% is left here — while the tension brace goes on to yield at 1065. The difference between the two vertical components is 659 kN, applied at the middle of the span with no help from either brace, and it asks the beam for 1317 kNm against the 200 kNm the gravity load asks for — 6.6 times as much. The beam drawn does not: 1517 kNm against a capacity of 731. The force is not a load case anybody applies; it is what the frame leaves behind on its way to the state it will actually be in.
Fig. 5 The same bay braced concentrically, after its compression member has gone. While both braces are elastic their vertical components cancel on the beam above, which is why a chevron beam is usually sized for gravity alone. The compression brace buckles at 445 kN and keeps 30 per cent of what it was carrying while the tension brace runs on towards 1,065 kN, so a difference of 659 kN lands at mid-span with no brace under it: the beam is asked for 1,517 kNm against a capacity of 731. That force is not a load case anybody applies. It is what the frame leaves behind on its way to the state it will actually be in, and the link exists so that this state is never reached.

The price of concentrating all the yielding in one short member is that the member’s deformation is concentrated too.

The rest of the beam is elastic and the columns are elastic, so when the storey racks through a plastic drift angle θ\theta, essentially all of that rotation appears across the link. The link’s own rotation is larger than the storey’s in the ratio of the bay to the link: L/eL/e, which here is ten.

A storey drifting 2 per cent — an ordinary demand in a severe earthquake — is asking its link for 0.2 radians if the link is a tenth of the bay. That is well past the 0.08 a shear link is credited with, which is why real designs use longer links, more of them up the height, or accept that the drift has to be limited by stiffness rather than by the link’s capacity.

The lever works both ways, and it is the reason a link is short at all: a short link is stiffer, and it also multiplies the demand on itself. Choosing the link length is choosing a point on both curves at once.

Where that demand is worst is decided by the two deflected shapes a tall building has. A braced frame racks rather than bending as a cantilever, so its drift is concentrated where the storey shear is largest; the link demand follows that profile, and the links near the base of a tall braced frame are the ones asked for the most and the ones that size the system.

The essay has offered two curves and left the choice between them open. It is not open: the two pressures meet at a single length, and the length is a property of the section rather than of the frame.

Start from the demand. The link’s plastic rotation is the storey’s, multiplied by L/eL/e, so

γ=θp Le  ≤  γcap\gamma = \theta_p\,\frac{L}{e} \;\le\; \gamma_{cap}

and γcap\gamma_{cap} is 0.08 radians while the link yields in shear, falling to 0.02 once it yields in bending — with a straight interpolation between 1.6Mp/Vp1.6M_p/V_p and 2.6Mp/Vp2.6M_p/V_p. For this section those two lengths are 1,058 mm and 1,719 mm.

Now read what the inequality permits, in plastic storey drift:

ee mechanism γcap\gamma_{cap} L/eL/e permitted θp\theta_p
800 mm shear 0.08 10.0 0.80%
1,058 shear, at the limit 0.08 7.56 1.06%
1,400 intermediate 0.045 5.71 0.79%
1,719 flexural 0.02 4.65 0.43%

The maximum is exactly at the shear–flexural boundary, and it is a maximum rather than a plateau. Below it the capacity is fixed at 0.08 and the lever is punishing; above it the lever improves and the capacity falls faster. The best link a frame can have is the longest shear link its own section allows, and that length is 1.6Mp/Vp1.6M_p/V_p — which is the number every code prints as a requirement and which falls out here as an optimum.

Stiffness does not fight it. Going from 800 mm to 1,058 moves the frame from 79 per cent of the concentric stiffness to about 71 — seven points, against a 33 per cent increase in the drift the frame can survive. The two curves the previous sections drew separately turn out to be very unevenly matched, and the rotation curve wins.

Every path to the ground goes through the link. A braced bay 8 m by 4 m whose two diagonals stop 1058 mm apart instead of meeting. The storey shear reaches the ground through the diagonals, and the vertical components they deliver to the beam have to pass through the segment between them: the link carries 47% of the applied shear as a shear force, at a lever arm short enough that its ends reach 0 kNm while the rest of the beam carries 0. The deflected shape drawn is the solved one, magnified — the real drift under this load is 0.010 mm. Everything outside the link is designed to stay elastic while the link is yielding, which is what makes the mechanism a choice rather than a hope.
Fig. 6 The optimum drawn: the same bay with its diagonals 1,058 mm apart instead of 800, which is 1.6Mp/Vp for this section. The link still takes 47 per cent of the applied shear — that share is set by the geometry of the diagonals and hardly moves with the link length — but the bay is softer, and the solved drift under the same load rises from 0.008 mm to 0.010. That is the whole of what the longer link costs, against a third more plastic drift before the mechanism runs out.

What the table also says is less comfortable. 1.06 per cent of plastic drift is not much. A severe earthquake asks a braced frame for two per cent of total drift, of which perhaps half a per cent is elastic — so a single braced bay per storey, at the best link length available, is at its rotation capacity and past it.

Three responses exist and only one of them is about the link.

Use more braced bays. Two bays per storey halve the shear each link carries and therefore halve nothing about the rotation — the lever L/eL/e is unchanged. What they do is let each link be shorter for the same total capacity, which is the wrong direction. More bays buy strength and stiffness, not rotation.

Shorten the bay. The lever is L/eL/e, so an 8 m bay asks 7.56 times the storey rotation and a 6 m bay asks 5.67 — permitting 1.41 per cent instead of 1.06. That is a 33 per cent gain from a decision made on the floor plan, and it is the reverse of the usual preference for long spans.

The stiffness the ductility is bought with. Lateral stiffness against link length, as a fraction of the same bay braced concentrically. At a link of 800 mm — 13% of the bay — the frame keeps 70% of the concentric stiffness; at the far end of the range, where the diagonals meet the columns, it is a moment frame at 12%. The curve is steep at the left, which is the useful part of it: the first tenth of the bay costs a fifth of the stiffness and buys the whole of the yielding mechanism.
Fig. 7 The same curve for a six-metre bay. An 800 mm link is 13 per cent of this bay rather than 10, so the frame keeps 70 per cent of its concentric stiffness rather than 79 — and the moment frame at the far end is at 12 per cent rather than 11. Eight points of stiffness are given away, and what is bought is a lever of 7.5 instead of 10.0. The shape of the curve is the same shape at every bay width; only where the design sits on it moves.

Or make the frame stiffer, so the drift never arrives. Which is the argument the whole system rests on and the reason the eccentric brace is worth its detailing: the link’s ductility is the reserve, and the frame’s stiffness is what keeps the reserve from being spent.

The essay has drawn one arrangement — two diagonals meeting a link at mid-span of the beam — and it is the least common of the three used.

The split-K, drawn here, puts the link between the two diagonals. It is symmetric, the link is in the middle of the beam, and the beam outside the link carries a large axial force.

The D-brace runs a single diagonal from the base of one column to a point on the beam short of the far column, making the link the segment between that point and the column. Half the members, and the link is next to a column rather than in mid-span.

The V-brace inverted puts the links at the ends of the beam, adjacent to both columns, which is the arrangement that keeps the link away from the beam’s own mid-span moment.

All three have the same free body and the same arithmetic. What differs is what else the link is carrying: a link next to a column is carrying the beam’s own gravity moment as well as the seismic shear, and a link at mid-span is carrying the beam’s largest gravity moment of all. A link is a member under shear and moment simultaneously, and the two interact.

Nothing happens, and then everything happens. The moment capacity left to a section already carrying shear, against the shear as a fraction of what the web can take. The web holds 26.1% of this section's plastic modulus and the flanges hold the rest, and only the web's share is reduced — by the factor √(1 − v²) that von Mises leaves it. So the curve is flat for most of its length: the first per cent of moment is not lost until v = 0.27, half the shear capacity costs 3.5%, and 15% is not reached until v = 0.9. The tangent at v = 1 is vertical, which is why the last tenth of the shear range costs more than the first eight.
Fig. 8 The interaction the link lives on. Shear and moment share a section, and the cost of one to the other is the web’s share of the plastic modulus times a term in the shear ratio. A shear link is being worked at V/Vp=1V/V_p = 1, where the tangent is vertical — so any moment at all is expensive, and the link is kept short partly to keep its moment small.

A system that had to be invented rather than discovered

Most of the structural forms on this site were found by building them. The arch, the truss, the portal frame and the shear wall all existed before anybody could analyse them, and the analysis arrived afterwards to explain why they worked.

The eccentrically braced frame is the other way round. It was proposed, tested and named in the nineteen-seventies, by people who had watched concentrically braced frames behave badly in earthquakes and who were looking for a way to keep the stiffness while getting a hysteresis loop that did not pinch. Nothing about it is intuitive from the outside: a frame whose diagonals deliberately miss each other looks like a mistake, and the reason for it cannot be seen without drawing the free body of the segment between them.

That origin shows in how the system is specified. The link’s length, its stiffener spacing, its material grade, its connections and the overstrength factor applied to everything around it are all prescribed rather than derived — because the system’s behaviour depends on details that a frame analysis cannot see, and every one of those details was established by cyclic testing rather than by a calculation.

It also shows in what the system is bad at. An eccentrically braced frame is a poor choice where the governing load is wind, because wind does not ask for ductility and the link is then a piece of expensive detailing that has softened the frame for nothing. The system a structure needs depends on whether the action asks for a force or a displacement, and this one is an answer to the second question.

Where the model stops

The frame is elastic and the analysis is first-order. Everything above is a stiffness calculation on a frame that has not yielded, used to identify a load path and then scaled. The behaviour the system exists for is cyclic, inelastic and dynamic, and none of the three appears in the plane-frame solve behind these figures.

The braces are pin-ended struts of infinite strength. Given a second moment a millionth of the beam’s, they carry axial force and nothing else — which is the right idealisation for a gusseted diagonal and hides the check that actually sizes the brace, which is buckling under the capacity-design force.

And the link is one member with two capacities. A real link has a slab on it, stiffeners across it, a connection at each end and an axial force from the beam; each of those changes its capacity, and the slab in particular raises its moment capacity in one direction only.

What the pictures cannot show

The deflected shape is drawn magnified by a factor printed on the figure, and the drift it is drawn from is elastic. What the system is designed for is a shape with a kink in it at the link — a parallelogram of yielded web between two elastic halves of a beam — and no elastic solve produces that shape.

Nor can any of these figures show the sequence. The braces buckle or they do not; the link yields on one cycle and again on the next in the other direction; the stiffeners hold the web flat until one of them does not. A structure designed to have a preferred failure sequence is one whose behaviour is a story, and every figure here is a still.

The assumption the figure rests on

The link’s capacity is taken as 0.6fytwhw0.6 f_y t_w h_w with fyf_y the specified yield stress. Neither factor is what will be in the building. Real mild steel is routinely 20 to 40 per cent above its specified yield, and a link that has been cycled well past yield has hardened by a further 20 or 30 per cent. Both raise the force everything else has to be designed for, which is why capacity design multiplies by an overstrength factor — and why specifying a stronger steel for a link is a way of making the frame less safe.

And there is an amplification waiting at the end of all this. A frame that has yielded is a frame whose stiffness has dropped, and a frame with a lower stiffness is closer to the load at which its own weight makes the drift run away. Stiffness is not only a serviceability matter in a structure designed to yield; it is what keeps the mechanism from becoming a collapse, and it is the reason the reserve the link provides has to be kept rather than spent.

The ladder from here

Later rungs on this anchor: the buckling-restrained brace, which puts the fuse back on the diagonal by preventing it from buckling and gets a full hysteresis loop from a concentric geometry. Link stiffener spacing, which is what the rotation capacity is actually bought with. The link-to-column connection, which is the detail that has failed in tests most often. Links up the height of a tall frame, where the demand follows the storey-shear profile and the design is a distribution rather than a member. And the same idea outside earthquake engineering — a deliberately weak element sized to protect everything else, which appears in blast design, in vehicle impact and in any structure with a defined accidental action.

Where this goes next

Two shapes that are the wrong way up for each other. Deflected shapes of a 20-storey building under a uniform wind, drawn to the same scale. The wall alone bends: its shape is flattest at the base and steepest at the top, reaching 146 mm. The frame alone shears: it is steepest at the base where the storey shear is largest, reaching 140 mm. Tied together at every floor they reach 58 mm — less than a quarter of either, and less than the 72 mm two springs in parallel would give, because each is stiff exactly where the other is not. How a tall building stands still A shear wall bends and a framed tube shears, and the two deflected shapes are the wrong way up for each other. Tie them together at every floor and the pair is stiffer than the sum of their stiffnesses — because near the base the wall holds the frame back and near the top the frame holds the wall. The collapse mechanism of a propped cantilever. A collapse mechanism, with the hinge position found by searching rather than quoted. Every position gives an upper bound on the collapse load; the lowest is 7.29, at a hinge 58.6 per cent along, which is a coefficient of 11.657 times Mp over the square of the span. After the first yield, which is not the end A steel beam whose extreme fibre has reached yield has not failed. It has started forming a hinge, and collapse waits until there are enough hinges to make a mechanism. Two materials pulled until they stop. Two stress-strain curves — mild steel, cast iron — plotted to a strain of 2.0%. One of them has a plateau, so the stress at which yielding starts is something the specimen does rather than something anyone chooses. No offset construction is drawn. The property that appears in none of the equations Ductility is in no design formula on this site. Every method on this site depends on it — and a brittle structure does not merely fail early, it makes the analysis wrong. A ground motion, on a structure of 1.00 s period. Displacement against time for a single-degree-of-freedom structure of natural period 1.00 s and 5.0% damping, under a ground motion of 3.5 m/s² peak. The elastic peak is 74.47 mm, and the same frame given a 4th of that strength peaks at 69.87 mm and comes to rest 11.82 mm from where it started. The earthquake asks for a displacement A structure a quarter as strong as the elastic demand does not deflect four times as far. It deflects almost exactly as far, yields on the way, and survives — which is why no ordinary building is designed for the force an earthquake would apply if it stayed elastic.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

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

What links here

Every essay whose body links to this one.

The objects this essay names

Each one links to every other essay that touches it.

BucklingCapacity designDuctilityEccentric braceEnergy dissipationHysteresisLateral systemLinkLoad pathPlastic hingePlastic momentRotation capacityShear yieldingStiffnessStorey drift