Structural Engineering & Statics
where the load goes
Dynamics

The tendon that forgets its prestress

A rocking wall comes home because a tendon pulls it, and the tendon must not yield — yet the rotations that test the wall are exactly the ones that stretch it. Past its yield, the force a tendon keeps is its yield force less its stiffness times the stretch, whatever it was prestressed to, so the guarantee a design needs is a limit on rotation, and more prestress only reaches that limit sooner.

Assumes The block that is safer for being bigger, The load put on backwards and The only thing that stops it.

16 min read 7 figures The load that will not hold stillThe statics of things that do not moveWhich failure arrives first

A wall that is allowed to lift is kept coming home by a tendon. Anchored at the top of the wall and in its foundation, unbonded so that it stretches over its whole length, it pulls the wall down onto its base with a force no gravity supplies, and as the base opens it stretches and pulls harder. That rising moment is what turns a block that is easier to push over the further it leans into a wall that resists more the further it goes. Every figure in that essay kept the tendon elastic, and it ended on the question the assumption hides: how much of the tendon’s prestress survives the rocking it is there to undo.

The question has a sharper edge than it looked. The wall there was 2 m wide, 8 m tall and weighed 400 kN, with a tendon prestressed to 600 kN that stiffened by 20 kN for every millimetre the base opened. A tendon 8 m long with that stiffness is 820 mm² of strand, and 820 mm² of strand with a 0.1 per cent proof stress of 1,670 MPa yields at 1,370 kN. The wall reaches that at a rotation of 38.5 mrad. Under the 1.5 g pulse that essay drew, the elastic tendon let it rotate to 147 mrad, where the same tendon would have been carrying 3,540 kN.

After a yield, the prestress left does not remember the prestress put in. The prestress left in the tendon of a 2.0 × 8.0 m rocking wall of 400 kN once it has rocked to each rotation and come back upright, for a tendon yielding at 1,370 kN and stiffening by 20 kN a millimetre as the base opens, prestressed to 300 kN, 600 kN and 900 kN. Each keeps all its prestress until it yields — at 53.5 mrad, 38.5 mrad and 23.5 mrad — and then loses 20 kN for every further milliradian, so past 53.5 mrad the three lines are one: the yield force less the stiffness times the stretch. Every one of them is slack at upright past 68.5 mrad. With bars of 400 kN the wall re-centres only while 400 kN remains, dashed, and it keeps that only up to 48.5 mrad, whatever it was prestressed to.
Fig. 1 The prestress left in the wall’s tendon once the wall has rocked to each rotation and come back upright, for the strand yielding at 1,370 kN and prestressed to 300, 600 or 900 kN. Each keeps all of its prestress until it yields — at 53.5, 38.5 and 23.5 mrad — and then loses 20 kN for every further milliradian, so past 53.5 mrad the three lines are one. All three are slack at upright past 68.5 mrad. With bars of 400 kN the wall needs 400 kN to re-centre, dashed, and keeps it only to 48.5 mrad whatever it was prestressed to.

A yield puts every tendon on the same line

A tendon stretched past its yield flows plastically and comes back longer than it went out. When the wall returns upright the base closes, the tendon’s elastic stretch is recovered, and what remains is a tendon that is too long for the wall by exactly its plastic stretch. It carries its prestress less the stiffness times that stretch.

The arithmetic of that stretch is where the surprise is. The tendon yields once the wall’s rotation has added the difference between its yield force and its prestress; everything beyond that is plastic. So the plastic stretch is the half width times the rotation, less the yield force over the stiffness, plus the prestress over the stiffness — and when it is multiplied by the stiffness and taken off the prestress, the prestress cancels. What is left at upright is the yield force less the stiffness times the half width times the largest rotation. Nothing in that expression was chosen at the jack.

The figure is that cancellation drawn. The tendon prestressed to 900 kN yields first, at 23.5 mrad, because it had least far to go. The one at 300 kN yields last, at 53.5. Past 53.5 mrad all three are carrying the same force, falling by 20 kN a milliradian, and past 68.5 mrad all three are slack: the tendon hangs loose in its duct when the wall stands upright, and pulls only once the base has opened far enough to take up the slack again. The prestress is a setting, and an earthquake past yield overwrites it with a number that belongs to the strand.

The loss also runs one way only. The bars across the base yield out and yield back on every cycle, alternating in a way that never settles. A tendon cannot be pushed, so it never yields back: it yields again only when a later rotation stretches it past its longest stretch so far. Its prestress ratchets down with the largest rotation the wall has ever reached, a step for each new record, and nothing the wall does afterwards gives any of it back.

That changes what re-centring means for a wall that has been tested. The ratio that decides whether a wall comes home is the weight and the prestress over twice the bars’ yield force, and with bars of 400 kN the wall needs 400 kN of prestress to keep that ratio at one. It keeps 400 kN up to 48.5 mrad of rotation. That limit is the same for all three tendons, because it is a statement about the force a yielded tendon keeps, and that force does not remember where it started.

The moment stops rising where the tendon yields

The rising moment was the whole case for the tendon, and it is worth seeing how much of it survives the yield.

A tendon turns a moment that falls into one that rises. Restoring moment against rotation for a wall 2.0 m wide and 8.0 m tall weighing 400 kN, three ways. As a bare block it lifts at 400 kN·m and the moment falls as it leans — 238 kN·m at 100.0 mrad. With a central tendon prestressed to 600 kN it lifts at 1000 kN·m, and because the tendon stretches as the base opens the moment rises instead, until the tendon yields at 1,370 kN at 38.5 mrad, with 1708 kN·m; past that the tendon's force is fixed and the moment falls with the weight's, to 1608 kN·m at 100.0 mrad. With yielding bars of 150 kN added at 0.6 m either side of the centre it reaches 1908 kN·m, against 3138 kN·m if the tendon never yielded, dashed. The first line is inherited from the wall's weight and shape; the others are chosen.
Fig. 2 Restoring moment against rotation to 100 mrad for the wall three ways. Bare, the moment falls from 400 kN·m. With the tendon it rises from 1,000 kN·m until the tendon yields at 38.5 mrad with 1,708 kN·m, then falls with the weight’s, to 1,608 kN·m. With bars of 150 kN it reaches 1,908 kN·m at 100 mrad, against 3,138 kN·m if the tendon never yielded, dashed.

Up to 38.5 mrad the tendon does exactly what the earlier essay said: the moment climbs, from 1,000 to 1,708 kN·m. At the yield the tendon’s force stops growing, and from there its contribution is a constant 1,370 kN times the half width. A constant moment has no slope. What is left with a slope is the weight, and the weight’s slope is negative, so past its yield the tendoned wall is a bare block again with a fixed extra moment attached — easier to push further the further it has gone, which is the property the tendon was installed to remove.

The dashed line is the wall the elastic model drew, and at 100 mrad it has 3,138 kN·m where the real wall has 1,908. Every rotation that model reported beyond 38.5 mrad was the wall resisting with moment it did not have.

What the yield costs round a cycle

The loop the bars draw, and the moment left when the wall closes. Moment against rotation for the wall carried through one full cycle to ±60.0 mrad and back. The moment jumps from +1000 kN·m to −1000 kN·m as the wall passes upright, because the pivot changes corner, and between those jumps the bars yield on the way out and yield back on the way in, which is what gives the loop its width. The loop encloses 91.7 kN·m, exactly the energy the bars dissipate, 70.5 kN·m, and the 21.2 kN·m spent stretching the tendon past its yield on the first excursion, which it never gets back — an equivalent damping ratio of 12.3 per cent. When the wall first closes after its excursion, 270 kN·m is still pushing it upright, so it comes back to upright with nothing holding it.
Fig. 3 Moment against rotation through one full cycle to ±60 mrad and back, with the tendon yielding on the first excursion. The loop encloses 91.7 kN·m: 70.5 kN·m dissipated by the bars and 21.2 kN·m spent stretching the tendon past its yield, which it never gets back — an equivalent damping ratio of 12.3 per cent. When the wall first closes, 270 kN·m is still pushing it upright.

Carried once round a cycle to 60 mrad, the wall draws a loop with a larger area than the bars alone account for. On the way out the tendon stretches 21.5 mm past its yield, and its yield force does 29.5 kN·m of plastic work doing it. It comes back to upright carrying 170 kN instead of 600, so it also gives up the elastic energy its prestress was holding, 8.3 kN·m of it, and the difference, 21.2 kN·m, is the extra area in the loop. On the excursion to the other side it does no more plastic work, because it would have to stretch past its new length to yield again.

That area reads as damping — the equivalent ratio rises from the bars’ own to 12.3 per cent — and it is the most expensive damping the wall has. It was paid for with the re-centring. When the wall closes after the excursion, the moment still pushing it home is 270 kN·m. With the tendon intact it would have been 700 kN·m: its 400 kN of weight and 600 kN of prestress over the half width, less the bars’ 300 kN·m. The yield took 430 kN of prestress out of that margin, and what the loop records as energy dissipated is a tendon made permanently longer than its wall.

The pulse the elastic tendon rode out

The same wall under the pulse that essay drew makes the difference concrete.

The same pulse on a wall that is designed to rock. Rotation as a fraction of the toppling angle under one 1.0 s sine pulse of 1.50 g, for the same 2.0 × 8.0 m wall of 400 kN three ways. Bare, it lifts at 0.250 g; with a 600 kN tendon it lifts at 0.625 g. The bare wall overturns; with the tendon the wall overturns; with tendon and bars it reaches 90 per cent of its toppling angle with seventeen landings. With its bars it comes to rest upright. Its tendon, which yields at 1,370 kN, is left with 0 kN of its 600. The landings are where the bare wall loses energy; the bars add a loss that does not wait for a landing.
Fig. 4 The wall under one 1.0 s sine pulse of 1.5 g with its tendon yielding at 1,370 kN, three ways. Bare, it overturns. With the tendon it overturns too. With tendon and bars it reaches 90 per cent of its toppling angle, landing seventeen times, and comes to rest upright — with its tendon left carrying nothing.

With an elastic tendon this pulse took the wall with bars to 60 per cent of its toppling angle and the wall with the tendon alone to 77 per cent, and both came home. With the strand given its yield, the wall with the tendon alone overturns: once the tendon has yielded early in the pulse it adds a fixed moment and no more, and the wall is a bare block that lifts later. The wall with bars reaches 90 per cent of its toppling angle, 219 mrad, and does come to rest upright.

It comes to rest upright on its weight. Its tendon was taken far past the 68.5 mrad at which it goes slack, so when the wall stands again the tendon carries nothing, and the only thing re-centring it is 400 kN of weight against bars of 150 kN each. The figure’s last frame shows a wall that survived. What it does not show is that the wall standing there is no longer a post-tensioned wall. It lifts off at a quarter of gravity rather than 0.625 g, it has no rising moment at all, and it will meet the next earthquake as the block that is safer for being bigger, with a pair of yielding bars attached.

The strand survives the stretch in the model because nothing in the model can break it. At 219 mrad the tendon has been stretched 219 mm over 8 m on top of the strain its prestress put into it, a total strain near 3.1 per cent, and strand is not made to stretch much further than a few per cent before it breaks. Whether this one would be intact is a question the elastic–plastic tendon cannot answer, and it is not a margin anybody would choose.

More prestress yields sooner and buys no rotation

The cancellation says what a designer can and cannot do about this, and it is cleaner drawn against the one quantity it removes.

More prestress yields sooner and buys no rotation. Three rotations of a 2.0 × 8.0 m rocking wall of 400 kN against the prestress its tendon is given, for a tendon yielding at 1,370 kN and stiffening by 20 kN a millimetre. The rotation at which the tendon first yields falls as the prestress rises, from 68.5 mrad with none to nothing at the yield force itself; at 600 kN it is 38.5 mrad. The rotation past which the tendon is left slack at upright is 68.5 mrad at every prestress. With bars of 400 kN, the rotation past which the wall no longer re-centres is 48.5 mrad at every prestress from the 400 kN it needs up to the yield force, and below 400 kN it never re-centres at all.
Fig. 5 Three rotations of the wall against the prestress put into its tendon, for strand yielding at 1,370 kN. The rotation at which the tendon first yields falls from 68.5 mrad with no prestress to nothing at the yield force; at 600 kN it is 38.5 mrad. The rotation past which the tendon is slack at upright is 68.5 mrad at every prestress. With bars of 400 kN, the rotation past which the wall no longer re-centres is 48.5 mrad at every prestress from the 400 kN it needs to the yield force.

One line falls and two are flat. The falling line is the one a stress check sees. A design that keeps the tendon below its yield at the rotation it is designed for is a design whose first-yield rotation lies beyond that rotation, and that rotation does depend on the prestress: less prestress leaves more room. The flat lines are what an earthquake larger than the design one sees, and on them the prestress has no influence whatever. Prestress decides when the tendon yields. It has no say in what a yield leaves.

So the two questions a rocking wall has to answer need two different checks. Will the tendon stay elastic in the design earthquake is a check on stress, made at the design rotation, and prestress is one of its levers. Will the wall still re-centre after an earthquake that exceeds it is a check on rotation, and its only levers are the strand’s yield force, the tendon’s stiffness and the bars: the limit is the yield force less the prestress the bars demand, over the stiffness times the half width. More strand raises it. Weaker bars raise it. More prestress does not appear.

Prestress still has work to do; it simply has a narrower brief. It sets the ground acceleration at which the wall first lifts, and a wall must stay closed under the wind and the smaller earthquakes it will meet often, so there is a least prestress below which the wall rocks when it should not. Above that, every extra kilonewton brings the first yield closer and buys nothing in an earthquake that exceeds the design one. And since what an earthquake asks of a structure is a displacement, the check that governs survival, a limit on rotation, is the check stated in the quantity the earthquake was specified in.

That is the same shape as a strand that relaxes under a stretch it is held at: the force in a tendon is a consequence of the length it is given, and a tendon given more length than it can hold elastically keeps the force its material allows rather than the force it was set to.

A longer tendon buys rotation and gives up stiffness

The tendon’s stiffness is the strand’s axial stiffness over its unbonded length, and length is the lever a design has most control over.

A longer tendon buys rotation and gives up stiffness. Rotations of a 2.0 × 8.0 m rocking wall of 400 kN against the unbonded length of its tendon, for 820 mm² of strand with a modulus of 195 GPa yielding at 1,370 kN, prestressed to 600 kN. Its stiffness is the strand's axial stiffness over its length, 40.0 kN a millimetre at 4 m and 10.0 at 16 m, so every rotation that ends in a yield grows in proportion to the length: the tendon goes slack at upright past 34.3 mrad at 4 m and 137.1 mrad at 16 m. With bars of 400 kN it stops re-centring past 24.3 mrad and 97.1 mrad. The price is the moment the tendon adds as the wall leans: at 20 mrad, 800 kN·m at 4 m and 200 kN·m at 16 m. The wall's own tendon is 8.0 m long, marked.
Fig. 6 Rotations of the wall against the unbonded length of its tendon, for 820 mm² of strand prestressed to 600 kN. The stiffness falls from 40 kN a millimetre at 4 m to 10 at 16 m, so every rotation that ends in a yield grows in proportion to the length: slack at upright past 34.3 mrad at 4 m and 137 mrad at 16 m, and with bars of 400 kN no longer re-centring past 24.3 and 97 mrad. At 20 mrad the tendon adds 800 kN·m at 4 m and 200 kN·m at 16 m. The wall’s own tendon is 8 m long.

Every rotation limit on the figure is proportional to the tendon’s length. A tendon twice as long stretches half as far in strain for the same opening of the base, so it yields at twice the rotation, goes slack at twice the rotation, and stops re-centring the wall at twice the rotation. The 8 m tendon of the wall goes slack past 68.5 mrad; a 16 m tendon, anchored 8 m below the base, would hold on to 137.

The price is the moment the earlier essay was built on. The tendon adds moment in proportion to its stiffness, so the longer tendon adds half as much for the same rotation — 400 kN·m at 20 mrad from the wall’s own tendon, 200 from one twice as long. The stiffness that makes a tendon push back hard is the stiffness that makes it yield early, and a design chooses where between the two to sit. A wall whose tendon never yields in any earthquake it can survive is a wall with a soft tendon and a restoring moment that barely rises; a wall whose moment rises steeply is a wall whose tendon will be spent by the first rotation large enough to test it.

The same pulse again

The last question is the one the earlier essay’s final section asked: what the wall is after the earthquake that tested it, when the next one arrives.

The same pulse again, on a wall its tendon has already given up. Rotation as a fraction of the toppling angle for a 2.0 × 8.0 m rocking wall of 400 kN with a tendon prestressed to 600 kN and bars of 400 kN, under two identical 1.0 s pulses of 1.50 g in a row, each starting from the state the one before left. With a tendon yielding at 1,370 kN: the first pulse takes it to 36 per cent, leaves 0 kN of prestress, and it rests 2.5 mrad off upright; the second pulse takes it to 71 per cent, leaves 0 kN of prestress, and it rests 93.7 mrad off upright. With a tendon that never yields, dashed: the first reaches 32 per cent and it rests upright; the second reaches 37 per cent and it rests upright.
Fig. 7 A wall with a 600 kN tendon and bars of 400 kN — a re-centring ratio of 1.25 — under two identical 1.0 s pulses of 1.5 g in a row, each starting from the state the last one left. With the tendon yielding at 1,370 kN the first pulse takes it to 36 per cent of its toppling angle, leaves no prestress, and it rests 2.5 mrad off upright; the second takes it to 71 per cent and leaves it resting 94 mrad off upright. With a tendon that never yields, dashed, it reaches 32 and 37 per cent and rests upright both times.

This wall was designed to re-centre with margin: 1,000 kN of weight and prestress against 800 kN from its bars, a ratio of 1.25. With an elastic tendon it does exactly that twice, reaching a third of its toppling angle each time and coming to rest upright.

With the strand’s yield, the first pulse looks almost the same — 36 per cent against 32 — and it is not the same wall afterwards. The rotation stretched the tendon past its slack rotation, so it carries nothing at upright, and the wall’s ratio is its 400 kN of weight against 800 kN of bars: 0.5. Its bars, left in compression by the excursion, push it off upright, and it settles 2.5 mrad away. That is a lean nobody would see, and it is the whole of the warning.

Nothing else about the wall would give the loss away. The tendon is inside a duct, unbonded, and slack; the wall stands; the bars look as they did. The one direct measurement is to jack the tendon at its anchorage until it lifts off its seating and read the force, and a wall that has been through an earthquake large enough to test it is a wall whose tendon force is the first thing worth reading, because it is the one quantity the earthquake may have changed without leaving a mark. A block that tips or slides shows what happened to it; a tendon that has been spent shows nothing until the next earthquake.

The second pulse meets a different structure. It lifts off at a quarter of gravity instead of 0.625 g. It has no rising moment, only the weight’s falling one and a slack tendon that engages again only once the base has opened further than it did the first time. Its bars resist its return more strongly than anything pulls it home. The same pulse takes it to 71 per cent of its toppling angle and leaves it resting 94 mrad off upright, a lean of more than five degrees. The pulse is the same; the wall is not, and the difference between them is written entirely in the tendon’s plastic stretch — a record of the order things happened in, as every residual state after an earthquake is, carried into the next one.

What the elastic–plastic tendon leaves out

Strand hardens. Real strand does not stop at its proof stress; its force keeps rising, slowly, towards its tensile strength, so a yielded tendon keeps a little more than the flat line says, and the lines on the first figure lean rather than meet exactly.

Strand breaks. Nothing here limits the tendon’s elongation, and the largest rotation drawn brings its strain to the order at which strand fractures. A fractured tendon is a slack tendon that never engages again.

The anchorages are perfect. Wedge anchorages seat and slip under cyclic load, which loses prestress without any yield at all, and a tendon partly bonded near its anchorage concentrates its stretch into a shorter length and yields at a smaller rotation than its full length implies.

The prestress does not keep leaking after the earthquake. A tendon left at high stress relaxes, and concrete under prestress creeps; both take more force out of a tendon between one earthquake and the next.

The toe does not crush and the bars do not harden. Both shorten the margin the ratio is computed on, as the earlier essay noted, and both do so more in the second earthquake than in the first.

And no figure here shows the anchorage. The whole of this essay is a statement about the length of a piece of steel, and the length that matters is decided by details at its two ends — where the strand is gripped, and where the duct lets it stretch freely — that a rocking-wall model reduces to one stiffness.

The assumption the figures rest on is a tendon that is unbonded over its whole length and elastic–perfectly plastic, so that the stretch the base imposes is shared evenly along it and a yield leaves a clean plastic length behind.

Still open: whether a tendon can be given a fuse

Every limit on the figures above is a limit on rotation, and the one design move that could lift them all is to stop the tendon’s stretch from growing with the rotation. A yielding element in series with the tendon — a short link that yields at a force below the strand’s, and is replaced after an earthquake — would take the plastic stretch itself and leave the strand elastic, in the way a device capped on purpose protects everything behind it. But a fuse that yields below the strand’s force also caps the rising moment at that lower force, and a fuse that has yielded leaves the same slack a yielded strand does until it is replaced. Whether a fuse can be proportioned so that it protects the strand without spending the re-centring it was meant to preserve, and whether replacing it after every earthquake restores the wall exactly, is a question about the connection between the tendon and the wall — and it decides whether re-centring is a property of the wall or a property of its last earthquake.

Where this goes next

The same pulse on a wall that is designed to rock. Rotation as a fraction of the toppling angle under one 0.8 s sine pulse of 1.00 g, for the same 2.0 × 8.0 m wall of 400 kN three ways. Bare, it lifts at 0.250 g; with a 600 kN tendon it lifts at 0.625 g. The bare wall reaches 79 per cent of its toppling angle with one landing; with the tendon it reaches 27 per cent of its toppling angle with eight landings; with tendon and bars it reaches 12 per cent of its toppling angle with 43 landings. With its bars it comes to rest upright. The landings are where the bare wall loses energy; the bars add a loss that does not wait for a landing. A wall that is allowed to lift A block that rocks inherits everything that decides whether it survives — a restoring moment set by its weight and shape that falls as it leans, and a loss of energy set by its proportions at each landing. Put a tendon through a wall and yielding bars across its base and both become design quantities, and the ratio between them decides whether the wall comes home. One pulse, two blocks of the same shape. Rotation as a fraction of the toppling angle, under a single 0.8 s sine pulse of 1.00 g, for two blocks of identical proportion whose sizes differ by a factor of 3. Both lift off at the same instant, because uplift depends on the shape alone. The small one reaches the toppling angle and goes over; the large one reaches 46%. The kinks are impacts: there is no dashpot anywhere in this model, and the only energy the block loses is lost when it lands on its other corner, at a velocity ratio of 0.926 per landing — 14% of the energy each time, decided by the block's shape and by nothing else. The only damping is the landing A rocking block has no dashpot in it. The only energy it ever loses is lost at the instant it lands on its other corner, and how much is a property of the block's proportions — 14 per cent for a slender one and 38 for a stocky one. That single number decides whether it settles or goes over, and a real base does not deliver the value the theory computes. The stress that leaks away. A restrained shrinkage strain of 300 microstrain in concrete of modulus 32000 N/mm². Ignoring creep it produces 9.60 N/mm², which is above the tensile strength of 3.5 and predicts that every restrained concrete member ever cast has cracked. Counting creep by the superposition integral leaves 2.32 N/mm² after 27 years, and the one-line age-adjusted shortcut at the usually quoted ageing coefficient of 0.8 leaves 3.31. The two disagree — this creep function implies an ageing coefficient of 1.32, not 0.8 — and both are below the tensile strength, so the conclusion turns on counting creep at all rather than on how it is counted. The strain that was imposed, and the stress that leaked away Multiply a restrained shrinkage strain by the modulus and the answer is three times the tensile strength — which predicts that every restrained concrete member ever cast has cracked. Most have not, and the reason is that the material creeps while it is being stressed. Same deck, same load, and two pier forces. A deck bearing on a pier of 20 kN/mm with μ = 0.03, taken to the same final state two ways: the deck moves 4 mm over the pier, and the bearing's load rises from 2000 to 4000 kN. Moved first, while the bearing carries 2000 kN, the pier force reaches the limit of 60 kN and the bearing slides for the rest of the movement; the load arriving afterwards raises the limit and changes nothing, and the pier is left carrying 60 kN. Loaded first, the limit is 120 kN before the deck moves, the bearing grips throughout, and the pier carries 80 kN. Both states are at the same displacement under the same load, and both satisfy equilibrium and the friction bound; the order is the only difference, and it appears in neither. The order the loads arrived in Statics allows a contact with friction a whole range of forces and has no way to choose between them. A real structure does choose, and what it chooses by is the order in which things happened to it — so the force in a pier under a sliding bearing is a record of its history, not a function of its loads.

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.

Ductility demandEnergy dissipationHysteresisLocked in stressNegative stiffnessNon-linear responseOverturningPrestressRelaxationRocking