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.

18 min read 9 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 linkA 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.storey shearthe link — 800 mm0.69 kN in each diagonal per kN of shearlink shear 0.47 kN · storey drift 0.008 mm · deflection magnified 5435×
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 bendingWhat 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.05001000150020000200400600800link length (mm)link capacity (kN)shear yieldinghinges at the ends1.6 Mp/Vp2.6 Mp/Vp0.08 radians of rotation on the left, 0.02 on the right
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 withLateral 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.0.00.20.40.60.81.000.20.40.60.81link length ÷ baystiffness ÷ the concentric brace's79% at e/L = 0.10a moment framea concentric bracethe link carries 47% of the storey shear at every length drawn
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 displacementThe 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.-40-202040-200-100100200displacement (mm)restoring force (kN)yielding at 182.53 kN — 22.35 kJ
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.
Yielding one way makes it easier to yield the otherMild steel taken to a strain of 0.60% and then pushed back the other way. The stress falls by 550 N/mm² before it yields again, against a yield stress of 275 — the elastic range is twice the yield stress and not once it, which is the Bauschinger effect and is a consequence of the yield surface sliding rather than growing.-0.6%-0.4%-0.2%0.2%0.4%0.6%-300-200-100100200300strainstress, N/mm²550 N/mm² of elastic range — twice the yield stress
Fig. 5 What the steel itself does under reversal. The link is being asked to do this 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 this 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 collapse mechanism of a propped cantileverA 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 43.71, at a hinge 58.6 per cent along, which is a coefficient of 11.657 times Mp over the square of the span.sagging hinge at 4.69hinge at the fixed endlowest upper bound: 43.71every hinge position gives an upper bound on the collapse loadassumed position of the sagging hingecoefficient 11.66 Mp ÷ L²
Fig. 6 The mechanism the frame is being kept away from. A mechanism forms when there are enough hinges, and a storey whose columns hinge top and bottom is a mechanism that collapses the storey. 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.

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.

Two shapes that are the wrong way up for each otherDeflected 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.wall 146 mmframe 140 mmtogether 58 mm20 storeys at 3.5 m · 40 kN per floor
Fig. 7 The two deflected shapes a tall building has. A braced frame racks like the lower shape, with its drift concentrated where the shear is largest; the link demand follows that profile, so 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 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 happensThe 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.00.20.40.60.810%20%40%60%80%100%shear, as a fraction of the web's capacitymoment capacity leftthe sectionthe web alone√(1 − v²)96.5%85.3%web 26.1% of the plastic modulus · M_pl 744 kNm · V_pl 959 kN
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.

The load that makes itself worseThe amplification of a deflection against the ratio of applied load to buckling load. A structure at half its buckling load deflects twice as far as first-order analysis predicts, and the curve runs away well before the load is reached.00.20.40.60.80246810applied load ÷ buckling load1.3×1.7×2.5×5.0×10.0×first-order analysis says the answer is always 1×one over one minus the ratio
Fig. 9 The 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.

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

wall 146 mmframe 140 mmtogether 58 mm20 storeys at 3.5 m · 40 kN per floor 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. sagging hinge at 4.69hinge at the fixed endlowest upper bound: 7.29every hinge position gives an upper bound on the collapse loadassumed position of the sagging hingecoefficient 11.66 Mp ÷ L² 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. 00.5%1%2%2%050100150200250300350strainstress, N/mm²mild steelcast iron 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. 02468101214161820-60-40-20204060time (s)displacement (mm)a ground motion of 3.5 m/s² peakelastic, and the same frame at a 4th of its strengthelastic: peak 57.71 mmyielding: peak 65.46 mmleft 14.13 mm off plumb 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.

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.

BucklingCapacity designDuctilityEccentric braceEnergy dissipationHysteresisLateral systemLinkLoad pathPlastic hingePlastic momentRotation capacityShear yieldingStiffnessStorey drift