The brace that yields both ways

A diagonal brace in a frame is asked to do the same thing in both directions. The frame sways one way and the brace is a tie; it sways back and the same brace is a strut. Spending an afternoon as each is the ordinary condition of a brace, and a brace that behaved identically in both would be the simplest energy-dissipating device there is.

It does not, and the reason is one line of the buckling calculation. A tie’s capacity is $A f_y$, decided by the area. A strut’s capacity is $\chi A f_y$, decided by the area and the second moment and the length — and those are properties of the same piece of steel, so they cannot be chosen independently.

Axial capacity and flexural stiffness are coupled, and the coupling is what makes a strut weaker than a tie. Everything on this page follows from breaking it.

What an ordinary brace does over six cycles #

The brace drawn has a 3,000 mm² core at 300 N/mm², so it yields in tension at 900 kN. At a slenderness of 1.1 the Perry–Robertson reduction gives $\chi = 0.535$, so it buckles in compression at 482 kN — 54 per cent of its tension capacity.

That asymmetry alone would be tolerable. What is not tolerable is what happens next.

A buckled brace has a plastic hinge at midspan. When the frame reverses and pulls the brace back into tension the hinge straightens — but not completely, because straightening a plastic hinge requires reverse yielding and the tension force is applied along a member that is now bent. The residual bow that is left is an imperfection for the next compression excursion, and a larger imperfection buckles lower.

So the compression capacity falls cycle by cycle. On the model used here it loses 35 per cent of what remains each time, and after six cycles it is at 12 per cent of its first-cycle value. The loop is pinched on one side and the pinch deepens.

What the sleeve changes #

A buckling-restrained brace separates the two functions that an ordinary brace performs with one piece of steel.

A core — a flat steel plate or a cruciform, sized for the axial force — carries all of the load. A casing around it, usually a steel tube filled with mortar, carries none of it: the core is wrapped in a de-bonding layer, a thin compressible material that lets it slide and lets it expand laterally as it yields.

The casing’s only job is to stop the core deflecting sideways. Since the core cannot buckle, its compression capacity is its yield force, and it yields at the same load in both directions.

The de-bonding layer is the whole invention. Without it the casing is a composite member sharing the load, which reintroduces the coupling. With it, the core has an axial capacity decided by its area and a flexural restraint decided by something else entirely — and that is the decoupling the ordinary brace cannot achieve.

The casing requirement, which is one inequality #

The casing carries no axial force, so it has no strength requirement in the usual sense. What it has is a stiffness requirement, and it is a buckling check:

$$P_E = \frac{\pi^2 E I_{casing}}{L^2} \;>\; \omega\,\beta\,N_y,$$

where $\omega$ is the strain-hardening factor and $\beta$ the compression overstrength — because what the casing must hold straight is not the core at yield but the core fully hardened, at the largest force it will ever develop.

For the brace drawn, $P_E$ is 3,485 kN against a demand of 1,361, a ratio of 2.56. Design practice asks for something above about 1.5.

Two things about that inequality are unusual enough to be worth naming.

It contains no strength term at all. The casing’s yield stress does not appear; a casing of mild steel and one of high-strength steel with the same second moment are equally good.

And it is the only requirement of its kind in ordinary structural practice: a member whose entire specification is that it must not buckle under a load it does not carry. The force is in the core and the check is on the casing, which is a separation most engineers find uncomfortable the first time they meet it.

Which free body produced the number #

Cut across the brace assembly at midspan and separate the core from its casing.

The core carries the whole axial force, $N$. It is also bowed slightly — no member is perfectly straight — so the axial force acting on a curved member produces a transverse pressure of $N\kappa$ per unit length, the same deviation force a curved tendon puts on the concrete around it.

That pressure has to go somewhere and it goes into the casing, through the de-bonding layer, as a normal contact force. The casing, carrying no axial load, resists it purely by bending — which is why its second moment appears and its strength does not.

The core is a column and the casing is its foundation, and the arrangement is a member on a continuous elastic restraint with the restraint stiff enough that the buckling wavelength never develops.

The one force the casing does carry is that contact pressure, which is concentrated wherever the core’s bow is largest and which is what limits how large the bow may become. A core that has yielded a great deal has shortened, thickened, and bowed into higher modes inside its casing, and the contact pressures rise accordingly.

The energy, which is the point #

A brace in an earthquake is a damper. What matters is the area of its hysteresis loop, integrated over however many cycles the ground motion supplies.

For the six cycles drawn at a storey drift of two per cent, the restrained brace dissipates 2.07 times as much energy as the ordinary one. That single number understates the difference, because the ratio is not constant: it is 1.46 on the first cycle and 2.69 on the last.

The gap widens with every cycle, which is the correct direction. A long-duration earthquake is exactly the case where an ordinary braced frame has least left, and it is exactly the case where the restrained one is worth most.

The ductility the core is asked for #

The core’s deformation demand is worth computing, because it is the quantity a qualification test is written around and it is much larger than most structural strains.

At a storey drift of two per cent on a 3.5 m storey with the brace at 40 degrees, the axial deformation is $0.02 \times 3500 \times \cos 40° = 53.6$ mm. The core’s yield deformation, over its 5 m length, is $N_y L/EA = 7.14$ mm.

So the core is being asked for a ductility of 7.5, once, and for several times that cumulatively over a full record. That is a very large demand for a steel member and it is achievable only because the core is restrained: an unrestrained member asked for the same thing buckles on the first compression half-cycle and never reaches it.

The consequence is that qualification testing for these devices is written in terms of cumulative inelastic deformation rather than peak force — typically a requirement to accumulate two hundred times the yield deformation without failure. That is a fatigue criterion in disguise, applied at strains far into the plastic range, and it is why low-cycle fatigue of the core is the type’s governing limit state rather than any strength.

Two factors the connections have to know about #

The symmetric loop is bought with two overstrengths, and both of them are demands on everything the brace connects to.

Strain hardening, $\omega$. The core yields at $A f_y$ and then hardens; at the strains a design-level earthquake imposes it may be delivering 1.3 to 1.5 times that.

Compression overstrength, $\beta$. The core in compression is slightly stronger than in tension, typically by 5 to 15 per cent, because it is confined by the casing and because it thickens as it shortens while a tensile core necks.

Together they say that the connections, the beams, the columns and the foundations must be designed for $\omega\beta N_y$ rather than $N_y$ — around 1.5 times the nominal yield force on the brace drawn.

That is capacity design applied to a device whose whole purpose is to be the weak element. The brace is chosen to yield; everything else must not; and the force that decides “must not” is the brace’s real strength rather than its specified one.

Where the loop’s shape comes from #

An ordinary brace’s loop is pinched and a restrained one’s is a parallelogram, and it is worth saying precisely which feature of each produces which shape, because the words “pinched” and “full” are used loosely.

The tension side is the same in both: elastic to yield, then a plateau. Nothing about buckling affects a member being pulled.

The compression side is where they differ. The ordinary brace reaches its buckling load, and then its force falls as it shortens further, because the midspan deflection grows and the moment at the hinge with it. That descending branch is the pinch: the loop is not merely lower on that side, it slopes the wrong way.

The reversal is the second difference. An ordinary brace that has buckled has to be pulled straight before it can take tension, and straightening it takes deformation at almost no force. That is the flat portion at the left of the loop, and it is where most of the missing area is.

So “pinched” is two separate defects — a descending compression branch and a slack reversal — and the restrained brace removes both with one mechanism. The area lost is not mainly the difference between 900 and 482 kN; it is the descending branch and the slack, which is why the energy ratio is 2.07 rather than the 1.35 the capacities alone would suggest.

What it costs #

The type is not free and the costs are worth stating alongside the mechanics.

It is a proprietary component. The core, the de-bonding layer and the casing are a system, qualified by testing rather than by calculation, and the qualification is specific to the manufacturer and often to the size.

It cannot be inspected after an earthquake. The yielded core is inside a mortar-filled tube. A brace that has done its job looks exactly like one that has not, and there is no non-destructive way to tell — which turns a post-earthquake decision about replacement into a judgement about how much the building moved.

The frame it goes into gets softer. A restrained brace has the same axial stiffness as an ordinary one of the same core area, but the core area needed is smaller — because it is fully effective in compression — so a frame braced this way is often less stiff than the equivalent conventionally braced one. Drift under wind, which is a serviceability check with nothing seismic about it, can end up governing the core size instead of the earthquake.

It has no useful reserve past its plateau. The loop is flat, which is what makes it a good damper and means the frame relies on a device with almost no strain-hardening margin. That is the trade a designer is making: a predictable, symmetric, non-degrading element in exchange for one that has nothing extra to give.

The residual displacement problem #

There is a consequence of the flat loop that the energy argument hides, and it is the type’s main practical weakness.

A structure whose lateral resistance comes from elastic–perfectly-plastic devices has no restoring force once they have yielded. It stops wherever the ground motion left it, and a frame braced entirely with restrained braces can end an earthquake permanently out of plumb by a substantial fraction of its peak drift.

That is a serviceability failure and often an economic write-off: a building that is undamaged in every structural sense and two hundred millimetres off vertical is not usable and is very hard to correct.

The remedies are all about restoring some stiffness after yield — a parallel elastic system such as a rocking frame or a set of post-tensioned tendons, or a dual system in which a moment frame stays elastic. Each of them re-introduces something for the flat loop to be added to, and each is a live area of design rather than a settled one.

It is worth noticing that the problem is created by the same feature that makes the device good. A loop with a rising branch after yield would restore the frame toward plumb and would dissipate less energy per cycle, because the area of a parallelogram is larger than the area of the same excursion with a slope on it. Energy dissipation and self-centring are in direct conflict, and every system that offers both does so by having two elements rather than one — which is, again, the decoupling this whole page is about.

The same trick elsewhere #

Decoupling two functions that one member had been performing together is a move that appears in several places, and it is worth setting them beside each other because the pattern is more useful than any one instance.

A laced or battened column does the opposite of a restrained brace: it separates the material into two chords to get a large second moment from a small area, accepting a shear flexibility in return. The coupling being manipulated is the same one; the direction is reversed.

A tension-only bracing system removes the compression requirement altogether by providing two diagonals and letting one go slack. That solves the asymmetry by declining to use half of it, which costs a member.

And an eccentric brace moves the yielding out of the brace entirely, into a short link in the beam, so the brace stays elastic and never has to be symmetric.

All four are answers to one question — what to do about a diagonal that is worth less in compression than in tension — and they differ in where they put the yielding. The restrained brace is the only one that puts it in the diagonal and makes the diagonal symmetric; the others avoid the problem rather than solving it, and each is cheaper.

Where the model stops #

The degradation law is a fitted trend. Real braces degrade at rates depending on their slenderness, their section shape and the drift amplitude, and the 35 per cent per cycle used here is representative rather than derived.

The loop is idealised as a parallelogram. Real loops have rounded corners, a Bauschinger effect on reversal and a gradual rather than sudden transition.

The casing requirement is a global buckling check. Real casings also have to resist the local contact pressures where the core bows in higher modes, which is a plate check the inequality above does not contain.

The core is treated as a bar with a stress–strain curve. A real core has transition regions at each end, where it widens into the connection, and those regions must stay elastic while the yielding length yields — which is a proportioning problem the arithmetic here does not contain.

And the energy comparison is at one amplitude. At small drifts the ordinary brace does not buckle at all and the two are identical; the whole advantage appears only past the point where the ordinary one has failed.

Where the ladder goes #

Later rungs on this anchor: the de-bonding layer, and what it has to allow in three directions at once. Core shapes and the transition regions between the yielding length and the connections. Cumulative ductility demand as a qualification criterion. Residual displacement and the self-centring systems that address it. Dual systems combining restrained braces with moment frames. Replaceability after an event, and the details that make it possible. Low-yield steels, which extend the plateau further. And the general principle this is the clearest example of: a structural function is easier to design when its two requirements are carried by two different members.