Structural Engineering & Statics
where the load goes
Concept

Non-linear response — where it appears

A structure's response once its force is no longer proportional to its displacement, through yielding, uplift, contact or a gap closing. Superposition stops working, so two loads do not add, and the history has to be followed step by step rather than read off a spectrum.

Named by 9 essays across one field — each of them below, with the objects they name alongside it.

A restoring moment that gets smaller the further it leans. Restoring moment against rotation for a block 0.90 m wide and 4.20 m tall, both normalised — the moment by its value at first uplift, the rotation by the angle α = 12.1° at which the block topples. It is mgR·sin(α − θ), which is a weight times a geometry with no material property in it at all, and it has two features an elastic system does not. There is a jump at the origin: the moment is whatever the ground demands until uplift and then it is mgR sinα, so the law is discontinuous where a spring's is steepest. And the slope is negative — the further it leans the less it pushes back — so the equilibrium at θ = 0 is stable only because of that jump, and there is no stiffness to divide into a mass. The straight line is what a spring of the same first-uplift strength would have done.

The block that is safer for being bigger

A block resting on the ground lifts off at an acceleration that depends only on its shape and not at all on its size, and then falls over at one that depends strongly on its size. Two objects of identical proportion begin rocking at the same instant and only the smaller one topples — which is why the slender water towers stood in Chile and the squat tanks beside them did not.

dynamics · Rocking 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.

dynamics · Rocking One collision, four contact laws, one impulse. Contact force against time for one collision between 500 t and 300 t closing at 1.94 m/s, under four contact laws sharing one contact constant of 2.8×10^9 N/m^1.5. The elastic linear spring peaks at 19.7 MN and lasts 58 ms; the Hertz law peaks at 22.0 MN over 61 ms. Set to a restitution of 0.65, the spring and dashpot peaks at 16.8 MN and ends pulling at 3.5 MN, a tension two faces in contact cannot carry, while the damped Hertz law peaks at 20.1 MN and returns a restitution of 0.77 instead. The areas under the curves are the impulses: 600 kN·s for the spring and dashpot, which is what momentum requires at 0.65, 646 for the damped Hertz law, and 727 for both elastic laws.

The force that belongs to the model

When two buildings meet, momentum decides what each of them feels, and no contact law can change it. What the contact law decides is the force, and the force it reports is the contact stiffness somebody assumed, raised to a power. Four laws and a hundredfold change of stiffness move the buildings by a few per cent and the force by a factor of eight.

dynamics · Pounding A filler sets the force, until it runs out of thickness. Contact force against the movement since first touch for one collision between 500 t and 300 t on a 40 mm filler crushing at 4 MPa over 5 m², 60 per cent of it crushable, at closing speeds of 1.00, 1.94 and 2.85 m/s. The filler loads elastically, then crushes at 20.0 MN for as long as it has thickness to give. At 1.00 m/s it crushes 4 mm and the force never passes 20.0 MN; at 1.94 m/s it crushes 17 mm and the force never passes 20.0 MN; at 2.85 m/s it crushes all 24 mm it can and the faces meet through it, peaking at 23.7 MN. Dashed, the bare contact at 2.85 m/s peaks at 35.0 MN.

The force a filler can promise

A crushable filler in the gap between two buildings replaces a contact stiffness nobody knows with a crush strength somebody chose, and the force of a collision becomes that strength — for as long as the filler has thickness left to crush. Energy decides how much thickness that has to be, and in a gap sized for buildings that were never meant to meet it is not much.

dynamics · Pounding 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.

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.

dynamics · Rocking What the soft layer costs the strong one. For a 40 mm filler whose outer 50 per cent is the soft layer, the least crush force its inner layer must have for the filler to stop a collision at 2.85 m/s — 761 kJ — without running out, against the soft layer's crush stress. The uniform filler that just does it crushes at 31.7 MN; every softer outer layer needs a stronger inner one, and at 1.5 MPa outside the inner layer needs 56.0 MN. The bare concrete's contact at that speed is 35.0 MN (dashed): only an outer layer of at least 5.7 MPa keeps the inner one below it.

The filler that cannot be gentle twice

A crushable filler in the gap between two buildings caps a collision at its crush force, but a filler soft enough to make the everyday contacts gentle runs out on the hardest one. Grade it — soft at the face, strong behind — and the soft layer takes the small contacts while the strong layer stops the large. The energy says otherwise. Every kilojoule the soft layer does not absorb, the strong layer must, in less depth: in a 40 mm filler, an outer half at 1.5 MPa needs an inner half at 56 MN, far above the 35 MN of the bare concrete it was meant to improve on.

dynamics · Pounding A fuse caps the force the strand ever sees. The force in the tendon line of a 2.0 × 8.0 m rocking wall of 400 kN, prestressed to 600 kN, with bars of 400 kN, against the opening of the base under the tendon, loaded to 79 mm and back: the strand alone, 20 kN/mm yielding at 1,370 kN, and the strand with a fuse yielding at 1,000 kN with a stiffness of 500 kN/mm in series, together 19.2 kN/mm and capped at 1,000 kN. The strand alone yields at 39 mm and is slack at upright after any opening past 69 mm; the fused line yields at 21 mm and is slack past 52 mm. In the fused line the strand carries the line's force and never more than 1,000 kN, so it stays elastic and all the plastic stretch, 58 mm at this opening, is in the fuse.

The fuse that protects the strand and not the wall

A rocking wall's tendon loses its prestress when it yields, so give it a fuse: a short link in series that yields first and is replaced afterwards. The strand then never passes 1,000 kN and stays elastic. But the fuse caps the restoring moment lower, so under the same 1.5 g pulse the wall rocks to 110 mrad instead of 89, and the line loses all its prestress just as the strand did. A new fuse gives the prestress back; the bars' stretch stays, and the next pulse takes the wall to 134 mrad.

dynamics · Rocking 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.

Energy dissipationImpulseRestitutionDampingOverturningRockingDuctility demandHysteresisNegative stiffnessPoundingPrestressSeries stiffness

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