The fuse that protects the strand and not the wall

The tendon that forgets its prestress found the limit on a post-tensioned rocking wall. Its tendon pulls the wall home, it stretches as the wall rocks, and once it is stretched past its yield it keeps its yield force less its stiffness times the stretch, whatever it was prestressed to. Every guarantee a design wants is then a guarantee about rotation: past a certain lean the tendon has nothing left, and more prestress only reaches that lean sooner.

It ended on the move that might lift the limit. Put a short link in the tendon line that yields at a force below the strand’s and can be replaced after an earthquake — a fuse. The fuse takes the plastic stretch, the strand stays elastic, and a new fuse after the earthquake gives the wall its prestress back. The essay said what the fuse would cost as well: it caps the rising moment at its own lower force, and until it is replaced it leaves the same slack a yielded strand does.

This essay puts a fuse in the same wall and measures both halves. The fuse does exactly what it is for, and what it is for is the strand. The wall it was meant to protect rocks further, loses its prestress just as completely, and is not restored by a new fuse.

The line, with and without a fuse #

The wall is the one before: 2 m wide, 8 m tall, 400 kN, rocking on its toes, held by a central unbonded tendon prestressed to 600 kN that stiffens by 20 kN for every millimetre the base opens beneath it and yields at 1,370 kN. Two yielding bars, 400 kN each, cross the base 0.6 m either side of the centre and take energy out on every cycle.

The fuse yields at 1,000 kN, about three quarters of the strand’s yield. In mild steel of 355 N/mm² that is a bar of 2,800 mm², and with a yielding length of 1.2 m — a short link in the line, at the anchorage — it has a stiffness of 500 kN/mm, twenty-five times the strand’s.

Two springs in series carry one force and add their stretches, so the line is as stiff as $k_e = k_{pt}k_f/(k_{pt} + k_f)$: 19.2 kN/mm, barely softer than the strand alone, because the fuse is short and stiff. What changes is the ceiling. The line’s force can never pass the fuse’s 1,000 kN, so the strand, which carries the same force, never gets within 370 kN of its own yield. The strand is protected absolutely, at every opening, for as long as the fuse is in the line. That is a force capped on purpose, exactly as a friction damper caps the force in the member behind it.

The rest of the figure is the price, and it is all visible in the line’s own terms. The fused line yields at 21 mm of opening, where the strand alone yielded at 39. It unloads along its elastic slope from a lower peak, so it reaches zero force sooner on the way back: slack at upright after any opening past 52 mm, against 69 for the strand. The fuse yields earlier, holds less, and goes slack sooner. Each of those is the strand’s own behaviour, translated down to a lower yield force.

The fused wall stops stiffening sooner #

What the wall feels is the moment, and the moment is the weight’s, falling as the wall leans, plus the tendon line’s force times its lever arm, plus the bars. The strand alone keeps the line’s force rising until 40 mrad and the wall’s moment peaks there at 2,506 kN·m. The fused line stops rising at 23 mrad, and the wall’s moment peaks at 2,164 kN·m, 14 per cent lower. Past those points both walls’ moments fall slowly as the weight’s lever arm shortens, and from 40 mrad on the fused wall resists the same push with 370 kN·m less.

That deficit is the fuse’s yield force subtracted from the strand’s, times the half-width: $(1{,}370 - 1{,}000) \times 1 = 370$ kN·m. It is a constant offset from the rotation where the strand would have yielded onward, and in a rocking wall a constant offset in the restoring moment is a large thing. The wall is held up by a moment that falls as it leans plus whatever the tendon adds, and the tendon’s share is most of what keeps the backbone from sloping downward.

The fused wall rocks further in the same earthquake #

Under the 1.5 g pulse the earlier essay used, the strand-alone wall rocks to 89 mrad. Its strand yields at 39 mrad and keeps stretching; by the time the wall comes back, the strand is slack, with none of its 600 kN prestress left. That was the earlier finding.

The fused wall rocks to 110 mrad, 24 per cent further. Its strand is never above 1,000 kN — the fuse saw to that — and the fuse has taken 89 mm of plastic stretch. And when the wall comes back, the tendon line is slack, with none of its 600 kN left. The strand is undamaged and the line is as empty as if it were not.

That is the first half of the earlier essay’s question answered. A fuse cannot protect the strand without spending the re-centring. It protects the strand by yielding at a force below the strand’s; yielding at a lower force means its plastic stretch starts sooner and the wall rocks further; and the prestress a line keeps after an excursion is its yield force less its stiffness times the stretch, which for a short, stiff fuse is nothing well before 110 mrad. The fuse moves the yield from a part that cannot be replaced to one that can. It does not make the yield smaller. It makes it larger.

The pulse is the one wall and one record the earlier essays used, so the 24 per cent is not a general number. The direction is: any fuse yielding below the strand lowers the restoring moment by a constant offset past its own yield, and a lower restoring moment under the same ground motion is a larger rotation.

More energy taken out, and further still #

A fuse is a yielding element, and yielding elements are usually welcomed in a rocking wall as damping: the bars are there for exactly that, because the only damping a bare block has is the landing. So it is worth asking whether the fuse’s yield at least took more energy out of the pulse than the strand’s did. It did.

The strand alone yielded at 38.5 mm of opening and the wall went on to 89 mrad, so the strand took about 50 mm of plastic stretch at 1,370 kN: some 69 kJ. The fused line yielded at 21 mm and the wall went on to 110 mrad, so the fuse took 89 mm at 1,000 kN: 89 kJ, nearly a third more. The fused wall dissipated more energy in its tendon line and rocked further anyway.

That is not a paradox once the two numbers are put in the right order. The energy a pulse puts into a rocking wall is not fixed in advance; it depends on how far the wall moves while the ground is pushing it, and a wall with a lower restoring moment moves further and takes in more. The fuse’s extra dissipation is a consequence of the extra rotation, not a remedy for it. A rocking wall’s peak rotation is governed by what pulls it back, and only secondarily by what takes energy out on the way — the same reason a block of the same shape is safer for being bigger: size changes how hard it is pulled back against the ground’s push, and that is what decides the excursion.

The same arithmetic says what the fuse is good for as a damper: nothing, compared with the bars it sits beside. The bars yield on every cycle in both directions and are designed to; the fuse yields once, in one direction, and then is slack. Its plastic work is a one-off charge, paid in the first excursion and not repeated.

Replacing the fuse restores the tendon, not the wall #

The second half of the question is what a new fuse buys. After the earthquake the yielded fuse is taken out, a new one is fitted, and the line is pulled to 600 kN again. Is the wall then new?

Unrepaired, both walls meet the second pulse badly. The strand-alone wall, with a slack tendon and stretched bars, goes from 89 mrad to 174, seven tenths of the way to toppling. The fused wall, also slack, goes from 110 to 161 — a little less far, and no different in kind.

With a new fuse the fused wall reaches 134 mrad in the second pulse. The prestress is back, and the wall does far better than either unrepaired one — but it does 22 per cent worse than it did the first time. The bars are still stretched. They yielded in the first pulse and kept their plastic set, and near upright that set pushes the wall away from upright rather than toward it; the wall then starts the second pulse with dissipators that resist less and a restoring moment that has to fight them. Only when the bars are renewed as well does the second pulse reach 110 mrad, the same as the first, which is what a new wall would do.

So a fuse restores the tendon exactly and the wall partly. The wall’s state after an earthquake is in everything that yielded, and a fuse is one of them. That is not a defect of fuses; it is a statement about what “replaceable” has to cover. A wall designed to be returned to service after an earthquake is a wall whose every yielding part can be taken out, and the bars that dissipate the energy are the first of them. A fuse makes the tendon one more replaceable part, which is valuable — a strand anchored inside a wall is not something anybody replaces — but it is one part among several.

A weaker fuse costs more rotation; a stronger one has no margin #

The fuse’s yield force is the design choice, and the figure shows what it trades. A weak fuse, at 650 kN — just above the prestress — protects the strand by a wide margin and lets the wall reach 154 mrad in the first pulse and 169 in the second even with a new fuse. A strong one, at 1,310 kN, is nearly the strand: 92 mrad, then 112 with a new fuse — almost as good as the strand alone in the first pulse, and far better than the unrepaired strand’s 174 in the second.

The figure suggests making the fuse as strong as possible, and the reason not to is outside it. The fuse is modelled as elastic-perfectly-plastic, yielding at exactly its nominal force. A real fuse of mild steel yields above its nominal strength by the scatter in its material, and then strain-hardens as it stretches: after 89 mm of plastic stretch in a 1.2 m link — seven per cent strain — its force has risen well above its yield. The strand is protected only if the fuse’s largest force stays below the strand’s yield. A fuse at 1,310 kN leaves a margin of 5 per cent, which the fuse’s own hardening consumes before the earthquake is over, and the strand yields after all. A fuse at 1,000 kN leaves 37 per cent, which is about what capacity design asks of a fuse and the member it protects. The same rule protects the member behind any yielding part — a brace that yields in both directions is designed so that its overstrength, not its yield, is what the frame around it must carry.

So the fuse’s force is pinned from both sides. It must be low enough that its overstrength stays below the strand’s yield, and every kilonewton it is lowered costs rotation in the first earthquake. There is no fuse that protects the strand and leaves the wall’s first-earthquake response as it was.

Only a very long fuse keeps the prestress #

The fuse’s stiffness is the other choice, and in a bar of fixed area it is set by its length: the longer the yielding part, the softer the link.

The earlier essay found that a longer tendon buys rotation: a softer line stretches less force per millimetre of opening, so it yields later and loses its prestress more slowly. A long fuse does the same, and the dashed line shows it — the rotation past which the line is slack at upright climbs from 52 mrad for a short fuse toward the toppling angle as the fuse lengthens.

The lengths it takes are the finding. A fuse as long as the wall is tall, 8 m, still leaves the line slack after this pulse. A fuse 100 m long keeps 384 kN and still falls short of the 400 kN the wall needs to re-centre against its bars, and it lets the wall rock to 135 mrad because it is so soft. A fuse is short by definition, and a short fuse is stiff, so the slack is set by the fuse’s yield force divided by a stiffness about equal to the strand’s: $1{,}000/(19.2 \times 1{,}000) = 52$ mrad, and every pulse that rocks the wall further than that empties the line.

The numbers for the fused line, by hand #

The fuse: $A_f = 1{,}000{,}000/355 = 2{,}820$ mm², over a yielding length of 1.2 m, $k_f = 210 \times 2{,}820/1{,}200 = 493$ kN/mm, taken as 500. In series with the strand’s 20 kN/mm, $k_e = 20 \times 500/520 = 19.2$ kN/mm.

The line yields when $600 + 19.2\,e = 1{,}000$, at $e = 21$ mm of opening — 21 mrad of rotation, since the tendon sits 1 m from the toe. It is slack at upright after an opening $e$ if $1{,}000 - 19.2\,e \le 0$ on the way back, that is past $1{,}000/19.2 = 52$ mm.

The restoring moment past the fuse’s yield at 50 mrad is the weight’s $400 \times 4.12 \times \sin(0.245 - 0.050) = 320$ kN·m, plus the line’s $1{,}000 \times 1 = 1{,}000$, plus the bars at yield, $400 \times 1.6 + 400 \times 0.4 = 800$: 2,120 kN·m. The strand alone at the same rotation has 1,370 in place of the 1,000, which is the 370 kN·m the backbone shows between them.

Two springs in a line and a pulse that starts from upright #

The fuse and the strand are springs in series, elastic-perfectly-plastic, carrying one force and unable to push; the fuse’s hardening is left out, which is why the yield-force figure needs the overstrength argument beside it. The strand’s yield is its 0.1 per cent proof force, as before.

The bars, the landing and the integration are the earlier wall’s, unchanged: elastic-perfectly-plastic bars carried through their history, a Housner restitution at each landing, and the same single sine pulse.

Each pulse starts from upright. A wall whose bars leave it leaning after the first pulse is taken to have been pushed back before the second, as the earlier essay’s sequence assumed. A wall left leaning that met the next earthquake from its lean would do worse than the figures show.

Replacing the fuse is instantaneous and exact: the old link out, a new one in, the line pulled to 600 kN again. In practice the line has to be de-tensioned to take the fuse out, which with a slack line after an earthquake is already done; re-tensioning is the part that needs access and a jack, and is the order in which things were done once more deciding a force.

What a strand is worth #

Every figure counts rotation, and by that measure the fuse loses in every first earthquake. Its value is in a currency the figures do not have: a strand that has yielded inside a wall has to be assessed, and probably replaced, and replacing an unbonded tendon threaded through eight metres of wall is a major repair, while a fuse at an accessible anchorage is an afternoon’s work. Whether 24 per cent more rotation in one earthquake is a fair price for that is a judgement about a wall’s life, not its response — and it is the same judgement as deciding what yields first, made with replacement rather than collapse in mind.

Still open: whether the bars can be fuses too #

The wall is restored only when everything that yielded is renewed, and the bars are the parts that yield most. A wall designed to be returned to service needs bars that can be replaced as easily as the fuse — external dissipators bolted across the base joint rather than bars grouted into it — and then a whole replaceable set: fuse and dissipators together. Whether a wall whose every yielding part is outside it, and whose tendon is protected by a fuse, then re-centres after each earthquake once its parts are changed, or whether the concrete at the rocking toe, which crushes a little on every landing, is the part that finally decides what the next earthquake finds, is the question that turns a replaceable fuse into a repairable wall.