Structural Engineering & Statics
where the load goes
Concept

Brace — where it appears

A member added to hold another one in position rather than to carry the applied load. What it needs is stiffness and a modest strength — the force it attracts is a fraction of the force in the member it restrains — and it is one of the few elements whose requirement is written as a displacement rather than as a capacity.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

The width nobody drew. A gusset plate with a brace bolted to it over 240 mm, and the width the profession has agreed to pretend is carrying the force. Everything else on this site arrives with a cross-section; a gusset does not, because it is a piece of steel with something attached somewhere in the middle of it and there is no geometry that says how much of it is working. The answer is the Whitmore section: assume the force spreads at 30° from the first fastener and take the width it has reached at the last, b_eff = w + 2L·tan30° = 367 mm. That is 4.08 times the width anything is actually attached to, and the rule comes from a 1952 master's thesis. It has since been checked against finite element work and holds to about ten per cent, which is fortunate, because moving the assumed angle by ten degrees moves the answer by 34%. On this plate the check that governs is not the stress the rule was written for: it is the Whitmore section buckles, at 721 kN against 1564.

The width nobody drew

Every other member in this collection arrives with a cross-section. A gusset plate does not — it is a piece of steel with a brace bolted to it somewhere in the middle, and no geometry says how much of it is working. The profession's answer is a thirty-degree spread from a 1952 master's thesis, it invents three quarters of the area being checked, and the check it was written for is not the one that governs.

connections · Gusset The loop a brace has when it cannot buckle. Force against axial deformation for two braces with the same core area, cycled six times at a storey drift of 2 per cent. An ordinary brace yields at 900 kN in tension and buckles at 482 in compression — 54 per cent of it — and the buckled shape leaves a plastic hinge that does not straighten, so the compression side loses capacity every cycle and is at 12 per cent of its first value by the last. A restrained brace has a casing that carries no axial force at all and only holds the core straight, which decouples axial capacity from flexural stiffness — the coupling that makes a strut weaker than a tie — so it yields at the same force both ways and hardens instead. The energy dissipated is 2.07 times as much over the six cycles, and the casing has to satisfy one inequality: π²EI/L² above the fully hardened core force, 2.56 here, which is a buckling check on a member carrying nothing.

The brace that yields both ways

An ordinary diagonal yields in tension at its full strength and buckles in compression at half of it, and the buckle leaves a hinge that does not straighten. Stop it buckling with a sleeve that carries no load at all and the loop becomes symmetric.

dynamics · Buckling-restrained What a torsional brace buys, and the stiffness it takes. The critical moment of an 8 m beam against the stiffness of a single rotational restraint at midspan — a cross-frame or a stiffened connection to a secondary beam, resisting the twist rather than the sideways movement. It climbs from an unbraced 143 kNm to the same two-half-wave plateau of 447 a lateral brace reaches, and gets to 99% of it at 253 kNm/rad. Nothing in the calculation refers to a height, which is the difference that matters: a torsional brace cannot be put on the wrong flange because it is not attached to a flange in the sense the lateral one is.

Twice the moment, four times the brace

A lateral brace has to be on the right flange and its demand is very nearly linear in the load. A torsional brace has no flange to be wrong about, and its demand is exactly quadratic — so the restraint that is indifferent to where it is attached is the one that gets expensive fastest.

stability · Beam bracing A damper too large stops damping. The damping of a structure with 2.0% of its own, against the size of a viscous damper held by a brace — the damper's force over the brace's at the structure's own frequency, on a logarithmic scale — for braces a quarter as stiff as the structure, as stiff, and four times as stiff. With the brace at 0.25 the damping peaks at 7.9% for a damper of 0.86; with the brace at 1.00 the damping peaks at 22.7% for a damper of 0.58; with the brace at 4.00 the damping peaks at 63.9% for a damper of 0.29. Either side of the peak it falls: a small damper barely moves, and a large one barely strokes — it locks, and the brace carries the motion as a spring.

The damper is only as good as its brace

A viscous damper added to a frame is held by a brace, and the brace is a spring in series with it. That pair has a ceiling: the most damping it can add is (√(1+κ) − 1)/2, κ being the brace's stiffness over the frame's, reached at one damper size and lost on either side of it. A brace as stiff as the frame allows 20.7%; a quarter as stiff, 5.9%. A damper larger than the best one is not better — it strokes less, locks, and turns the brace into a strut that moves the resonance instead of flattening it.

dynamics · Damping Sized for one amplitude, a nonlinear damper damps only a band around it. The damping of a structure with 2.0% of its own and a brace as stiff as the structure, against the storey's amplitude over the amplitude the damper was sized for, on a logarithmic scale, for dampers whose force goes as the velocity to the power α, each sized to the brace's best at the design amplitude. All four give 22.7% there. The linear damper gives it at every amplitude. A tenth of the design amplitude, α = 0.5 gives 6.4%, α = 0.3 1.9% and α = 0.15 1.4%; a hundredth of it, 1.9%, 1.4% and 1.4% — below the structure's own, because the locked brace has stiffened it.

The damper sized for the earthquake

A fluid damper whose force goes as its velocity to the power 0.3 is chosen because it caps the force a large earthquake can push through its brace. The same exponent makes its effective size depend on how far it strokes. Sized to its brace's best at the earthquake's amplitude, it adds 20.7% of critical damping there, 5.4% at three-tenths of it, and nothing at all at a tenth — where the brace has locked, the building is stiffer, and the motion its occupants feel meets a resonance that is as sharp as an undamped frame's. Giving half the brace to a linear damper keeps 9.3% under the wind for 1.2 points at the earthquake.

dynamics · Damping The same damper behind a lever is best at a larger motion. The damping of one storey with 2% of its own damping and a brace as stiff as the storey, with an α = 0.3 damper sized to its brace's best at the design amplitude with no amplifier, against the storey's amplitude, with the same damper behind stroke amplifiers of ratio 1, 2 and 3. A ratio f makes the damper act on the storey as one f to the power 1 + α larger — 2.46 and 4.17 times — so it locks more, and its band moves up by about f to the power (1 + α)/(1 − α): its best is near 3.8 and 7.5 times the design amplitude. At a hundredth of the design amplitude all three give 1.4%; at the design amplitude 22.7%, 6.4% and 2.3%.

The lever that moves the damper and not the band

A damper whose force goes as a low power of its speed is sized for the earthquake and locks in the wind. A toggle or scissor linkage multiplies its stroke, and it seemed it might move the band of amplitudes the damper serves down towards the wind. It cannot. Behind a lever of ratio f the same damper acts on the storey as one f to the power 1 + α larger, so its band moves up, not down; re-sized to the design amplitude, it gives the storey exactly the curve it had without the lever, and what has changed is the damper — a quarter of the size, a third of the force, three times the stroke. What would widen the band is a ratio that grows with the motion, and a toggle has one — until its links lie straight and it locks.

dynamics · Damping

Named alongside it

The objects these essays reach for when they reach for this one.

DampingViscous damperEigenvalueEnergy dissipationLoss factorNatural periodNon-linear responseResonanceSeries stiffnessStiffenerWind loadBearing

All concepts