Structural Engineering & Statics
where the load goes
Series

Pounding — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Buildings that sway alike need almost no gap between them. The separation two adjacent buildings need, against the ratio of their periods. The obvious answer is the sum of what each can do — 340 mm — and it is wrong, because the two peaks do not occur at the same instant. The right combination is the one modal responses use, √(u₁² + u₂² − 2ρu₁u₂), with ρ the cross-correlation coefficient of the two responses. ρ depends on the period ratio and behaves the opposite way to intuition: at a ratio of one the two buildings sway together, ρ = 1, and the gap collapses to the difference of the two, 100 mm. At the 0.57 drawn ρ is 0.029 and the gap is 248 mm — 27 per cent less than the sum, and 99 per cent of the square root of the sum of squares.

    The gap between two buildings

    Two towers side by side in an earthquake need a gap. The obvious answer is the sum of what each can move, and it is wrong — because the two peaks do not happen at the same instant. What decides the answer is the ratio of the two periods, and buildings that sway alike need almost no gap at all.

    part 1 · dynamics
  2. Where two buildings touch, and what is there when they do. Closure between an 8-storey building 24 m tall and a 4-storey one 15 m tall, drawn against height, with the 50 mm gap between them. The two swaying out of phase close on each other more the higher up they are, so they first touch at 7.6 m and are in contact above it. The dots are the taller building's floors: its storeys are 3.00 m and the other's are 3.75, so two of them arrive part way up a column rather than at a slab. A blow at mid-height of a column asks it for a shear of half the impact and an end moment of Fh/8, neither of which is a demand any part of the design contains.

    The floor that arrives at a column

    The gap between two buildings is computed from their roof displacements, which is where each of them moves most. It is not where they touch, and it is not what is there when they do — a slab edge meeting a column part way up its height is a different event from two slabs meeting, and it is the one that appears in the photographs.

    part 2 · dynamics
  3. 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.

    part 3 · dynamics
  4. A link that keeps two buildings apart has made them one. The largest closing movement between a 500 t building with a 0.8 s period and a 300 t building with a 1.2 s period, 50 mm apart, under one 1.0 s sine pulse of 0.50 g, and each building's largest displacement, against the size of a viscous damper joining them across the gap, from 0.01 MN·s/m to 541.3 MN·s/m. With no link they close by 515 mm, and the stiffer building moves 220 mm and the softer 419 mm. The link that takes the most energy out, 0.64 MN·s/m, still lets them close by 214 mm. The least that keeps the 50 mm gap is 4.8 MN·s/m, where the stiffer building moves 269 mm and the softer 280 mm. At the largest link the two move together, 281 mm and 281 mm.

    The damper that ends up as a joint

    A damper across the gap between two buildings acts on exactly the motion the gap is sized for, and it can be sized for two different things. The size that takes the most energy out of the pair still lets the buildings collide. The size that keeps them apart has nearly stopped moving: it has joined them into one building, and the stiffer of the two pays for it in drift.

    part 4 · dynamics
  5. 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.

    part 5 · dynamics
  6. 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.

    part 6 · dynamics
  7. A filler that knows the speed meets the collision hardest at its start. The contact force against the depth crushed in the hardest collision, 500 t and 300 t closing at 2.85 m/s with 761 kJ, on a 40 mm filler between 500 t and 300 t, crushable over 24 mm: uniform (dashed), with 50 per cent of its strength at that speed depending on the speed, and with all of it. Each is as strong as it must be to stop the collision in 24 mm. Uniform: 30.5 MN at first contact; 50 per cent on speed: 38.4 MN at first contact; all on speed: 63.4 MN at first contact. The crushing speed is highest at the first instant and falls to nothing, so a filler whose strength follows it spends its force at the start; the same area under a falling line needs a higher peak than under a level one, and a purely viscous filler's is twice the uniform's.

    The filler that knows how fast it is hit

    A filler graded through its depth meets the small collisions softly and the hard one with a strong layer that has to be very strong, because every kilojoule the soft layer misses must be absorbed in less depth. A filler whose strength rises with the speed it is crushed at ranks collisions by speed instead, with no layers at all, and treats the middling collisions better than grading does. But it meets every collision hardest at its first instant, when the crushing speed is greatest, and a force that falls as the collision proceeds needs a higher peak to absorb the same energy in the same depth. A purely viscous filler that stops the hardest collision in 24 mm hits it with exactly twice the force of a uniform one.

    part 7 · dynamics
  8. Where each filler's depth goes. The crushable depth left in each filler between buildings of 500 t and 300 t, periods 0.8 and 1.2 s, 50 mm apart, through a record of six one-second sine cycles peaking at 0.35 g, for 40 mm fillers crushable over 24 mm, each just strong enough to stop a 2.85 m/s collision from new. Uniform: 5 mm when its hardest contact arrives at 2.76 m/s, 0 mm at the end. Graded: 12 mm when its hardest contact arrives at 2.76 m/s, 6 mm at the end. 50% on speed: 0 mm when its hardest contact arrives at 2.88 m/s, 0 mm at the end. The graded filler's soft layer goes in the first middling contacts and its strong layer is barely reached; the other two lose depth to every middling contact and have little or none left when the largest arrives.

    The filler that is used up before the worst

    A crushable filler between two buildings is sized for the hardest collision it might meet, meeting it new. An earthquake does not oblige: the hardest contact arrives after a dozen middling ones, and each has taken some of the depth. Through a record of six cycles, a filler half of whose strength follows its crushing speed is dense before the worst contact arrives, and a uniform one has five millimetres left. A graded filler still has twelve. Yet the graded filler meets the worst contact at 56 MN and the spent rate-sensitive one at 39. Keeping depth is not the same as being gentle, and the order of the contacts barely matters.

    part 8 · dynamics

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