Field

Dynamics

Every other field assumes the load arrives slowly and stays. When it does not, the same structure gives a different answer — twice as large for a load put down suddenly, twenty-five times for one applied at the rate the structure likes, and unbounded for one the structure's own motion creates.

20 kN applied at once and held, on a structure of 0.500 s period. Displacement against time for a single-degree-of-freedom structure of natural period 0.500 s and 2.0% damping, under 20 kN applied at once and held. The static deflection under the same peak force is 12.67 mm and the peak response is 24.56 mm — a factor of 1.94.

Twice the deflection, for the same load

A weight placed gently on a beam deflects it by one amount. The same weight let go from rest, a millimetre above the same beam, deflects it by twice as much — and the factor of two is exact, for every structure ever built.

The period nobody chose

Every structure has a natural period, it decides the answer to every question in this field, and no drawing anywhere records it. It is a consequence of a mass picked for one reason and a stiffness picked for another — and it has already been computed, by the serviceability check.

How much a harmonic force is magnified, at four damping ratios. Displacement amplitude divided by the deflection the same force would produce if it were applied slowly, against the ratio of the forcing frequency to the structure's own, at 1%, 2%, 5%, 10% of critical damping. At the natural frequency the magnification is 50, 25, 10, 5 respectively — one over twice the damping ratio, and nothing else in the problem enters it.

The only thing that stops it

Drive a structure at its own frequency and the amplitude grows without limit unless something takes energy out. What takes it out is damping, and damping is the one structural property that is never designed, never drawn, and never known until the thing is built.

Three modes of a five-storey frame. The first three mode shapes of a five-storey shear frame, from the eigenvalue problem rather than sketched. Mode 1 has a period of 0.6 s, no node and carries 88.0% of the mass; Mode 2 has a period of 0.21 s, one node and carries 8.7% of the mass; Mode 3 has a period of 0.13 s, two nodes and carries 2.4% of the mass. The nth mode crosses the axis n−1 times, which is a theorem rather than a drawing convention.

A structure has more than one period

One mass on one spring has one period. A building with eight floors has eight, each with a shape of its own, and the second one bends the building into a curve nobody drew. They are not harmonics, they do not interfere, and each behaves as though the others were not there.

How much of the mass each mode carries, over eight modes. The effective modal mass of each of the eight modes of a frame of eight storeys, as a percentage of the total. Mode 1 carries 85.6% and mode 2 9.1%; two modes are needed to reach 90% of the mass. The masses sum to exactly the total, which is a property of the eigenvectors rather than a normalisation applied afterwards.

Most of the mass moves together

A sixteen-storey frame has sixteen modes, and the first one carries eighty-four per cent of the mass. That is why a hand calculation on one mode gets the base shear right to within a tenth — and why the same calculation gets the acceleration at roof level wrong by a factor of two.

Which floor frequencies a 2 Hz pace punishes. The response factor of a floor of 12 tonnes modal mass and 3.0% damping, against its own natural frequency, under a walker at 2 steps per second. The peaks are at 2, 4, 6, 8 Hz — the harmonics of the pace — and they fall away sharply: four peaks, each smaller than the one below it, because the harmonics of walking get smaller. A floor at 2 Hz reaches R = 31 and one at 8 Hz reaches 6. Between them the floor is quiet.

The floor that is strong and unusable

A floor can satisfy every strength check, deflect less than the limit, and still be rejected by the people who work on it — because somebody walking across it at two steps a second happens to be exciting it at exactly the rate it likes to move.

What a point on a response spectrum is: three structures, three integrations, three points. Three oscillators of periods 0.3, 0.8, 1.8 s, each integrated through the whole of the same ground motion, and the peak of each one plotted against its own period on the curve at the right. The peaks are 12.54, 46.2, 108.95 mm. The complete spectrum is that done 44 times. Nothing in the curve is a property of the earthquake alone: every point on it carries a period and a damping ratio that belong to a structure.

The spectrum is not a load

A response spectrum looks like a load curve and is not one. Every point on it is the peak of a complete time integration of one particular structure, and the curve is what you get by doing that again for every structure there could be.

A ground motion, on a structure of 1.00 s period. Displacement against time for a single-degree-of-freedom structure of natural period 1.00 s and 5.0% damping, under a ground motion of 3.5 m/s² peak. The elastic peak is 74.47 mm, and the same frame given a 4th of that strength peaks at 69.87 mm and comes to rest 11.82 mm from where it started.

The earthquake asks for a displacement

A structure a quarter as strong as the elastic demand does not deflect four times as far. It deflects almost exactly as far, yields on the way, and survives — which is why no ordinary building is designed for the force an earthquake would apply if it stayed elastic.

Where the wind's energy is, and where the structure can reach it. The gust spectrum at a mean speed of 30 m/s, plotted as n·S(n) against frequency on a logarithmic axis so that equal areas are equal energy. Its total variance is 27 m²/s², which the closed form 6·K·U² gives as 27. Most of the energy is below a hundredth of a hertz — gusts lasting a minute or more. A structure at 0.2 Hz sits far out on the tail, and still takes 90% of its response from there, because the resonant part is amplified by π·f₀/(4·ζ) and the damping is 1.2%. A stiffer structure at 2 Hz takes 22%.

The wind is a spectrum

A wind load is quoted as a pressure, which suggests something steady. It is not — the energy is spread across four decades of frequency, almost all of it in gusts lasting minutes, and a tall building takes ninety per cent of its response from the sliver of that energy sitting at its own frequency.

Lock-in: the frequency the wind sheds at, and what it does to the chimney. A 1.2 m cylinder at 0.9 Hz. Vortices leave it at St·U/D, so the critical speed at which they match the structure is 6 m/s — a breeze, met many times a year rather than once in fifty. The upper panel shows the shedding frequency locking on to the structure across a band from 5.1 to 7.8 m/s; the lower shows the amplitude that results. At a Scruton number of 11.2 the peak amplitude is 180.94 mm, which is 15% of the diameter.

The wind that brings its own frequency

Every other load in this subject arrives at whatever rate it happens to arrive at. Vortex shedding arrives at a rate set by the wind speed — so for any chimney, mast or cable there is always a wind speed at which the shedding matches the structure exactly, and it is a breeze rather than a storm.

The damping a crowd leaves behind, and the number of people that uses it up. Total damping ratio against pedestrians, for a structure with 0.60% of its own and a modal mass of 120 tonnes at 0.5 Hz. Each person walking laterally puts a force in phase with the deck's velocity into it — about 300 N·s/m per person — so the crowd subtracts from the damping, and at 30.16 people what is left is nothing. Below the threshold a disturbance dies away. Above it, the structure feeds itself and the motion grows out of nothing at all — which is what makes this a different mechanism from resonance rather than a severe case of it.

The bridge that was pushed by its own sway

A crowd walking on a bridge that moves sideways adjusts its footing to stay balanced, and the adjustment pushes the bridge the way it is already going. The crowd is a damper with the sign reversed, and past a certain number of people the total damping is negative.

3% of the mass, hung on a spring, against the peak it removes. The magnification of a structure with 1.0% damping, with and without a tuned mass damper of 3.0% of its mass, tuned to 0.9709 of its frequency with 10.5% damping of its own. The bare peak is 50; with the absorber the single peak becomes two of 7.34, a reduction to 15% — a factor of 6.8. The marked points at frequency ratios 0.923 and 1.044 are the fixed points: the response there is the same whatever damping the absorber is given, which is what makes the optimum a question with an answer. The absorber's own stroke at the worse of the two peaks is 29.57 times the structure's static deflection, and that stroke is what decides whether it fits.

The mass that helps by being late

Hang three per cent of a building's mass from a spring in its roof, tune the spring so the mass arrives a quarter-cycle behind the motion, and the peak response falls by a factor of seven. Nothing was strengthened and nothing was stiffened.

What a drop height is worth, as a factor on the answer for a weight placed slowly. The peak displacement as a multiple of the static deflection, against the height a weight is dropped from divided by the deflection that weight causes when it is placed. The curve is 1 + √(1 + 2h/δ), which is conservation of energy and nothing else: the weight does work over the height it falls PLUS the distance the structure then gives, and the structure stores work only over the second. At a ratio of 40 the factor is 10.00, and at zero it is exactly 2 — the marked point, where a dropped weight becomes a placed one.

The weight that was dropped

A half-tonne load lowered onto a beam produces 5 kN. The same load dropped one metre onto the same beam produces 160 kN — and onto a beam ten times softer, 54 kN. The stiff structure is the one that suffers, which is the opposite of nearly every other rule about structures.

How much of a force a mount lets through, at three damping ratios. The force transmitted to the support divided by the force applied, against the ratio of the forcing frequency to the structure's own, at 2%, 5%, 20% of critical damping. Every curve passes through exactly 1 at a frequency ratio of root two, whatever the damping: below that ratio a mount amplifies what it was installed to isolate, and above it more damping lets more through.

The machine that shakes the building

Put a machine on springs to keep its vibration out of the floor, and below a frequency ratio of root two the springs make things worse. Every transmissibility curve ever drawn passes through exactly one at that ratio, whatever the damping — so a soft mount either works well or fails badly, with nothing in between.

The damping a wind leaves behind, and the speed that uses it up. Total damping ratio against wind speed, for a structure with 0.60% of its own and a modal mass of 25 kg per metre at 1.2 Hz. The aerodynamic damping of a section whose lift falls with angle of attack is negative and grows with speed, so at 4.14 m/s what is left is nothing. Below the threshold a disturbance dies away. Above it, the structure feeds itself and the motion grows out of nothing at all — which is what makes this a different mechanism from resonance rather than a severe case of it.

The motion that feeds itself

A steady wind contains no frequency at all, and it can destroy a bridge. The force that does it is manufactured by the structure's own movement, so there is no excitation to resonate with — there is a wind speed above which the equilibrium is unstable, and below which nothing happens.

The worst speed is not the fastest one. Peak deck acceleration against train speed, for a 20 m span at 6.25 Hz under 10 axles 18 m apart. The spikes are not a numerical artefact and they are not about how heavy the axles are: a regularly spaced train is a forcing function with a frequency v/d, and where a multiple of it lands on the bridge's own frequency each coach arrives in step with the motion the last one left. The arithmetic is v = d·f₁/k, which puts peaks at 405, 203, 135, 101 km/h — all of them operating speeds. What fails first is the acceleration rather than any stress: ballast loses its interlock at about 3.5 m/s², and strength does not appear in the equation at all. Here the limit is first passed at 376 km/h.

The train that arrives in time with itself

A single load crossing a span is a mild problem. A train is not one load — its axles are regularly spaced, so the forcing has a frequency of its own, and where a multiple of it lands on the bridge's frequency each coach arrives exactly in step with the motion the last one left behind.

The demand falls and the movement rises, by the same factor. One elastic spectrum read twice: as an acceleration on the left and as the displacement that goes with it on the right. A fixed-base building at 0.5 s sits on the plateau and is asked for 1.05 g. Put it on bearings soft enough to make its period 2.54 s and the demand falls to 0.103 g — a base shear 10.2 times smaller, bought with no strength whatever. The same shift on the right-hand plot goes the other way: displacement is S_a T²/4π², so the demand rises from 65 mm to 165. That number is the design. It is a gap all the way round the building, a moat every service has to cross, and a detail that a later contractor will fill in unless somebody says what it is for.

Made weaker on purpose

Everything else in this collection resists a load by being stiff or strong enough for it. A base-isolated building resists an earthquake by refusing to hear it — a layer of bearings under the whole structure with a lateral stiffness a twentieth of the frame's, bought with almost no strength at all.

The force falls, the drift rises, and the damping goes the wrong way. What a compliant foundation does to a 0.6 s building on a 8 × 8 m footing, against the stiffness of the ground under it. Three curves, all normalised to the fixed-base answer. The period lengthens — 1.31 times at 200 m/s — because the swaying and rocking of the foundation are flexibilities in series with the structure's own, and the rocking term carries an h², so it is the tall building that feels it. The base shear falls with the period, which is why a fixed base is usually called conservative. The displacement rises, by 51% here, and that is what breaks the cladding, the services and the gap to the building next door. And the effective damping falls rather than rises: the structure's own is divided by the cube of the lengthening — 2.2% of an original 5% — while a slender building's foundation radiates only 0.28% back, because rocking radiates almost nothing at these frequencies.

The ground is a spring

Every dynamic result in this collection has assumed a structure rising from something that does not move. Nothing does. A foundation can slide and it can rock, both are flexibilities in series with the structure's own, and the rocking one carries a square of the height — so the period lengthens, the force falls, the drift rises, and the damping goes the wrong way.

What the shape of a load in time is worth, for two load shapes. The peak displacement as a multiple of the static deflection, against the load's duration divided by the structure's natural period, for two load shapes: a load that rises linearly, then stays; a rectangular pulse, then nothing. The lines are closed forms and eight dots are the peak of a complete time integration of an oscillator of 0.300 s period under that load, agreeing with the line to within 0.19% everywhere.

The load that is over before it has moved

A blast delivers an enormous pressure for a few milliseconds. Everything else in this field asks what force a structure can carry; a load that has come and gone before the structure has travelled any distance is not asking that question, and the answer turns out to depend on the mass and the ductility with the strength barely in it.

A broad tank sloshes and a tall one does not. The liquid's division into the part that moves with the wall and the part that sloshes, against the tank's proportion. The convective masses come from the potential-flow solution and the impulsive mass is whatever is left, so the two sum to the liquid's mass exactly at every proportion rather than approximately over part of the range. A tall tank at H/R = 3 is 84% impulsive and behaves almost like a solid; a shallow one at H/R = 0.5 is 72% convective and most of its contents never notice the earthquake. This tank sits at H/R = 1.33, which is 65% impulsive.

The liquid has a period of its own

Shake a tank and its contents do not all go with it. Part of the liquid moves as though it were bolted to the wall and part sloshes at a period fixed by gravity and the radius, which the tank's stiffness has no influence over whatever. The split is decided by one proportion, and the two parts then take entirely different amounts of the earthquake.

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.

The ground is a structure, and it has a period. The transfer function of 38 m of soil at 75 m/s over bedrock at 800 m/s: how much the surface moves for a given motion in the rock, at every frequency. It is a column fixed at the bottom and free at the top, so its resonances are the odd harmonics — the peaks stand at 1, 3, 5, 7 times the first, which is a fixed-free column and nothing else. The fundamental is at 0.493 Hz, a period of 2.03 s, and it is 4H/v_s exactly. The peak amplification is 7.5 against the bound 1/(α + πξ/2) = 7.5, which agrees to 0.2% — and the α in it is the impedance ratio, 0.0554 here. That is the term that keeps the answer finite: assume rigid bedrock and α is zero, the bound becomes 13 and the model is predicting an amplification set by damping alone. What limits the surface motion is that the energy can leave downward.

The ground has a period of its own

An earthquake is measured on rock and felt on soil, and between the two is a layer that behaves exactly like a structure — a column fixed at bedrock, free at the surface, with a fundamental period of four times its depth over its shear wave velocity and a set of odd harmonics above it. The motion a building receives is the rock motion through that filter, and the filter is sharp.

$The resonance that ran out of time. The response of a 0.100 s oscillator at 2.0% damping while the driving frequency sweeps up through its own, plotted against the driving frequency rather than against time. The steady-state amplification is 1/2ζ = 25; this sweep reaches 23.6, which is 95% of it, because the time spent inside the half-power band is a limited number of build-up time constants. The whole answer depends on one group, β/ζ²ω², and over the range this figure's sibling sweeps the fraction falls from 100% to 41%. Two features fall out of the integration and neither is guessable from the steady-state picture: the peak arrives after the frequency has passed resonance, by 1.2% of it here, and the response beats afterwards at the difference between the two frequencies. A machine's instrument therefore reads its largest amplitude while it is already above its critical speed.$

The resonance that ran out of time

A resonance amplifies by one over twice the damping, which for a lightly damped structure is fifty or a hundred. That is a steady-state answer and it takes time to arrive — with a time constant containing the same small damping — so nothing that sweeps through a resonance ever collects all of it, and past a certain rate the peak reached stops depending on the damping at all.

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.

Evenly spaced modes, so one of them is always where the feet are. The first 6 modes of a 120 m stay under 3.50 MN. A taut string's frequencies are an arithmetic progression — every one of them 1.006 Hz above the last — where a beam's go as the square of the mode number and spread out. That difference is the whole of why a cable is a lively member and a beam is not: a beam has a first mode and then a gap, and a cable has a mode every 1.01 Hz for ever. The shaded band is ordinary walking, 1.6 to 2.4 Hz, and mode 2 sits inside it. Nothing about the tension can move a mode out of the band without moving another one in.

The force read off a frequency

Nothing can measure the tension in a stay cable directly — there is no gauge, no accessible end and no place to put a load cell. What there is, is a member whose frequencies are an arithmetic progression whose spacing is the square root of its own tension, so a phone taped to it for thirty seconds returns the force.

Where the demand crosses the capacity. Base shear against roof displacement for a eight-storey frame pushed with a triangular load pattern. It is elastic to 1994 kN, where the first storey reaches its shear capacity, and flattens as each of the others follows — 7 yield events in the order 3, 4, 2, 1, 5, 6, 7. The demand is a displacement rather than a force: the elastic spectrum at the building's own period of 0.93 s gives 173 mm, and the participation factor of 1.26 makes that 219 mm at roof level. Against an idealised yield of 70 mm that is a ductility demand of 3.10.

Pushed over until it will not stand

A structure in an earthquake is asked for a displacement rather than a force. A pushover answers a different question cheaply — how much base shear the frame has against how far its roof moves — and the whole art is in what happens when the two are laid over each other.

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.

The building is a filter with one very sharp tooth. What a component mounted on the roof of a eight-storey building feels, divided by what the same component would have felt on the ground. A rigid item sees the floor's peak acceleration rather than the ground's, which is already 2.29 times as much. An item whose own period matches the building's sees 6.1 times as much, at 0.94 s — a resonance inside a resonance, and the two amplifications multiply rather than add. Far above the building's period the ratio falls back toward one, because a component much softer than its support simply follows the ground. The shape of this curve is a property of the building and not of the earthquake, which is why equipment is qualified against a floor spectrum rather than a ground one.

The spectrum a floor hands on

Most of what breaks in an earthquake is not the structure. It is a transformer, a chiller, a rack, a ceiling, a pipe — and every one of them is bolted to a floor rather than to the ground. The motion it feels has already been through a filter with one very sharp tooth in it.

Where the drift went. Storey drift at the target displacement, for the same frame with and without a soft ground storey at 50 per cent of the others' stiffness and strength. The regular frame spreads 219 mm over every storey; the soft one reaches 266 mm and puts 5.58 per cent of it into the ground storey against 0.53 next to it — a concentration of 5.9 against 1.7. The roof goes 22 per cent further, and where that extra displacement lands is the whole of the difference between the two buildings.

Weaker in one place, and better on every average

Take an eight-storey frame and make its ground storey half as stiff and half as strong. Its ductility demand falls, its first mode carries more of the mass, and its period lengthens into a gentler part of the spectrum. Three global numbers all improve, and the building is the one that collapses.

$One pulse, two blocks of the same shape. Rotation as a fraction of the toppling angle, under a single 0.8 s sine pulse of 1.00 g, for two blocks of identical proportion whose sizes differ by a factor of 3. Both lift off at the same instant, because uplift depends on the shape alone. The small one reaches the toppling angle and goes over; the large one reaches 46%. The kinks are impacts: there is no dashpot anywhere in this model, and the only energy the block loses is lost when it lands on its other corner, at a velocity ratio of 0.926 per landing — 14% of the energy each time, decided by the block's shape and by nothing else.$

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.

What a damper at the anchorage can do, and the ceiling it cannot pass. Modal damping against damper size for a 200 m stay at 4500 kN, with the damper 4 m from the anchorage — 2.0 per cent of the length. Each curve is a mode, found as a complex root of the taut string with a viscous damper in it rather than from a formula. Every one of them peaks at 1.00 per cent of critical, which is x/2L exactly, and the peaks are at different damper sizes — a higher mode wants a smaller damper, because it moves faster at the same amplitude. The curves are flat near their peaks: half the optimum coefficient gives 80 per cent of the ceiling, and so does twice it.

The damper that is too near the end

A stay cable has almost no damping of its own, so it is given a damper — and the damper cannot go where the motion is, because the middle of a two-hundred-metre stay is a hundred metres above the road. What it can supply is then decided by one length, and no amount of damper changes it.

Damping that is computed rather than measured. Radiation damping of a 2.5 m block on soil with a shear wave speed of 178 m/s, against how heavy the block is made — the horizontal axis is a multiple of the 150 tonne block drawn. At that mass the three modes are at 47, 29 and 13 per cent of critical. None of this is dissipation: the energy leaves as waves travelling away into the half-space, so the quantity is a geometrical coupling and it can be computed from the size of the footing, the density of the soil and its shear modulus. Every curve falls as the block gets heavier, because a heavier block moves less for the same wave field — which is the one counter-intuitive thing here: mass buys frequency and costs damping.

The damping that is radiated

Every response in this field is quoted with a damping assumption attached, because damping is measured rather than designed and the measurement has a factor of two in it. A machine block on the ground is the exception: its damping is not dissipation at all, and it can be computed from three numbers none of which is a material property of anything that dissipates.

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.

Three quantities converging at three different rates. How much of the exact answer a truncated modal analysis of a ten-storey building reaches, against how many modes it keeps. The mass count is the rule — ninety per cent, reached at two modes. The base shear is ahead of it: one mode carries 85 per cent of the mass and 98 per cent of the base shear. The force in the top storey is behind it, at 80 per cent on one mode and 94 on two. The rule is written in the quantity that converges fastest, and it is checked against a quantity nobody plots.

The modes that were left out

Nobody runs every mode a model has, and the rule for how many is a mass count — enough to account for ninety per cent of the structure. The rule is written in the one quantity that converges fastest. On a twenty-storey frame two modes give the base shear to within one per cent and the force in the top storey to within twenty.

Two capacity curves that agree, from two answers that do not. Base shear against roof drift for the same eight-storey frame with a soft ground storey, pushed with a fixed triangular pattern and with one recomputed from the tangent stiffness at every step. At a roof displacement of 426 mm they differ by 8.7 per cent — which is the number a capacity-spectrum procedure reads, and it is the number that is nearly the same. The storey drifts underneath these two curves differ by a factor of 3.8 at the fourth floor, and the drift is what the analysis was run for.

The pattern that stopped describing the building

A pushover analysis pushes with a load pattern chosen to resemble the first mode, and by the time the structure has done anything worth analysing it no longer has that mode. Recomputing the pattern as the frame softens changes the base shear by nine per cent and the drift at the fourth floor by a factor of four.

$The same pulse on a wall that is designed to rock. Rotation as a fraction of the toppling angle under one 0.8 s sine pulse of 1.00 g, for the same 2.0 × 8.0 m wall of 400 kN three ways. Bare, it lifts at 0.250 g; with a 600 kN tendon it lifts at 0.625 g. The bare wall reaches 79 per cent of its toppling angle with one landing; with the tendon it reaches 27 per cent of its toppling angle with eight landings; with tendon and bars it reaches 12 per cent of its toppling angle with 43 landings. With its bars it comes to rest upright. The landings are where the bare wall loses energy; the bars add a loss that does not wait for a landing.$

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.

Two separate checks, and the block that is neither. Amplification of the steady displacement at a machine 2.0 m above the base of a 150 tonne block 5.0 m across and 2.0 m deep, against forcing frequency. Checked separately, the block sways at 11.2 Hz with 29 per cent of critical damping and rocks at 15.0 Hz with 13 per cent, and the rocking check's peak is 3.99. Because its mass sits above its base the two are one system, with modes at 9.9 Hz and 20.9 Hz — one below both checks and one above — and the block's own peak is 2.49 at 9.7 Hz.

The frequency below both checks

A machine block is checked for sway and for rocking as if they were two oscillators, each with its own spring, its own radiation damping and its own natural frequency. A block whose mass sits above its base is one oscillator with two modes, and neither of them is a sway or a rocking — one sits below both checks, one above both, and the lower one is where the machine resonates.

However stiff the ties, a free line stops short. The first three frequencies of stays 180, 150 and 120 m long joined by cross-ties, against the stiffness of each tie from 1.0 kN/m to 1000 MN/m, with the line free at its ends and, dashed, anchored to the deck at both ends with the same stiffness. With the line free the first frequency rises from 0.73 Hz and levels off at 0.82 Hz, however stiff the ties are made — short of the 1.01 Hz of the 120 m stay, which a free line cannot pass. Anchored, the first frequency reaches 1.09 Hz. At the 6.0 MN/m marked, the free line gives 0.82 Hz and the anchored line 1.09 Hz.

The line of ties that stops short

Cross-ties are the one intervention on a stay cable that changes its frequencies rather than damping them, and the usual account says they lift the stays out of the range that excites them. A line of ties that is not anchored to anything cannot lift the first frequency above the shortest stay's own, however stiff the ties are made. What lifts it is carrying the line to the deck.

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.

The rates a seated crowd makes worse. The stand's steady rms acceleration under 40 people jumping, against the rate they jump at, for a stand of 30 t modal mass at 6 Hz with 2 per cent damping, empty and with 15 t of spectators sitting on it. Empty, the worst rate is 3.00 Hz, where the second harmonic of the jump lands on the stand's own frequency, and the stand reaches 2.54 m/s², 25.9 per cent of gravity. Occupied, the worst anywhere from 1.5 to 3.5 Hz is 0.49 m/s² at 3.50 Hz. But from 1.64 to 2.32 Hz, shaded, the occupied stand responds more than the empty one — 1.6 times as much at 2.00 Hz.

The crowd that is also the structure

A crowd jumping to music loads a grandstand with harmonics several times those of walking, and nothing about their size is measured: they follow from a pulse that has to average one body weight. But the people who jump arrive with the people who sit, and the seated crowd is mass, stiffness and damping bolted to the stand. It lowers the worst case by a factor of five and makes the most common jumping rates worse.

The frequencies at which a moving deck makes a stay grow. Stability of the first mode of a 120 m stay at 3,500 kN inclined at 25°, whose own frequency is 1.01 Hz, when the deck at its anchorage moves at a frequency Ω against the stay's ω, plotted against the swing in tension the movement produces. Inside each shaded region a small disturbance grows. Each region's tip is at a swing of four times the damping ratio — 0.40%, 2.0% and 4.0% for damping of 0.10%, 0.50% and 1.0% — and it widens as the swing grows, to a deck between 1.970 and 2.030 times the stay's frequency at a 6.0% swing with 0.10% damping. With no damping, dashed, the region reaches down to no swing at all. A deck moving ±5, ±10 and ±20 mm vertically swings the tension by 0.70%, 1.4% and 2.8%. At ±10 mm and exactly twice the stay's frequency, marked, the stay grows with damping of 0.10% and settles with 0.50% and 1.0%.

The stay shaken along its own length

A deck that moves at a stay's anchorage pushes nothing across the stay. It stretches the stay along its own line and lets it go again, so the tension swings, and at twice the stay's frequency that swing drives the stay with no sideways force at all. Whether the swing grows is one comparison — a quarter of the tension swing against the damping ratio — and it is a comparison the capped damper wins.

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.

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.

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.

Set into the ground, the block is quieter only while its sides radiate. Displacement at the top per kilonewton of machine force for a 150 tonne block 3.2 m across and 3.2 m deep, on a logarithmic scale: on the surface, dashed, and with 1.60 m of it set into the ground, first with the side soil radiating as the side-layer model gives it and then with no radiation from the sides at all, dashed. On the surface it peaks at 191.8 µm per kN at 5.5 Hz. Embedded and radiating, it peaks at 11.0 µm per kN at 8.2 Hz. With the same embedment and silent sides the resonance moves up to 8.4 Hz and peaks at 168.7 µm per kN, because the lower mode's damping falls from 3.2 per cent to 1.6 per cent. At 5.5 Hz the three give 191.1, 8.0 and 9.5 µm per kN.

The damping that comes through the sides

A machine block set into the ground is held at its sides as well as its base, and the side soil does two things at once. It lifts the height at which the ground's resistance acts toward the block's centre of mass, which weakens the coupling between sway and rocking without ever removing it. And it radiates, which is what actually flattens the resonance — so the quiet an embedded block promises rests on the stiffness of backfill nobody measured.

The cross term has the sign of the two contributions. Four responses of the floor, each combined three ways and divided by the complete quadratic combination at ρ = 0.50. Base shear: modal 927 and 839 kN, so the root-sum-square is 0.82 of it and the absolute sum 1.15. Base torque: modal -8640 and 8640 kNm, so the root-sum-square is 1.41 of it and the absolute sum 2.00. Flexible edge: modal 60 and -6 mm, so the root-sum-square is 1.05 of it and the absolute sum 1.15. Stiff edge: modal -5 and 47 mm, so the root-sum-square is 1.05 of it and the absolute sum 1.15. Where the two modes push the same way the root-sum-square is short; where they push opposite ways it is long.

The twist the combination rule invents

Two modes close together respond together, and the square root of the sum of squares assumes they do not. The error has the sign of the two modal contributions: where they agree, as they do in base shear, the rule comes up short, and where they oppose, as they always do in torque, it comes up long — by a factor of five for a floor whose stiffness sits twenty centimetres off its mass, a torque the building does not have.

The damping that stops helping. The storey drift's white-noise root-mean-square as the bearings' damping is raised from 2 to 50 per cent, each divided by the exact value at 2 per cent. The classical analysis promises that every increment helps, and at 50 per cent predicts 0.20 of the lightly damped drift. The exact analysis flattens: 0.34 at 20 per cent and 0.29 at 50, because the damping force at the bearings is transmitted into the superstructure's own mode, which the classical analysis has decoupled from it.

The damping that belongs to no mode

Give every mode its own damping ratio and throw the rest of the damping matrix away, and an isolated building's periods and damping come out right to within a fifth of a per cent. Its storey drift at the superstructure's frequency comes out six times too small. The bearings' dashpot pushes on both modes at once, and the more of it there is the less extra damping buys: the classical analysis promises the drift keeps falling, and it stops.

Bearing damping that shakes the roof harder. The roof's white-noise root-mean-square acceleration as the bearings' damping rises from 3 per cent to 58 per cent, exact and classical, both divided by the exact value at 3 per cent, for the six-storey building. The classical analysis falls all the way, to 0.33. The exact one falls to a minimum of 0.44 at about 28 per cent and then rises, to 0.50 at 58 per cent: past the minimum, every extra per cent of damping at the bearings shakes the top floor harder, while the classical analysis says it is still helping.

The top floor the bearings shake

On a building of several storeys, the damping at the isolation bearings is tied less tightly to each higher structural mode than to the one below. It still does more harm in each, because the ground barely reaches those modes on its own. The error the usual analysis throws away gathers in the upper storeys, and past about a third of critical damping at the bearings the roof shakes harder while the analysis says it is still helping. How much damping is best depends on how tall the building is.

A mode that points wherever the asymmetry points. A square floor 24 m on a side, equally stiff in both directions, with a period of 0.80 s. Four copies are drawn on top of each other, each made stiffer by one part in 1,000,000 along a different direction: 0°, 20°, 45°, 70°. The first mode each one returns lies along 90°, 110°, 135°, 160° — at right angles to its stiffening, whatever the size of it — and the two periods differ by one part in 1,000,000. Four structures no instrument could tell apart have first modes pointing four different ways. The floor with no asymmetry at all has no first mode: every direction is one.

Two modes that are really a plane

A building equally stiff in both directions has two translational modes with one period, and they are not a pair of shapes but a whole plane of them. The pair an analysis returns is chosen by asymmetries of a millionth, so any result that depends on the pair — a square-root combination, a comparison with measured modes — inherits a choice the building never made. Damping then decides whether the difference can be seen at all.

The same mode's curvature, which can. The first mode's drift in each storey — the difference between the floors above and below it, which for a shear building plays the part a beam's curvature plays — intact, solid, and with storey 3 10 per cent less stiff, dashed, each scaled to the roof. The damaged storey's drift rises by 9.6 per cent; the storeys above it change by at most 1.8 and below it by at most 1.5. The largest change is in storey 3: the drift finds the damage and says where it is, which the frequency's fall of 0.91 per cent cannot.

What a mode shape notices that a frequency does not

Take a tenth of the stiffness out of one storey of a ten-storey building and its first frequency falls by a per cent at most, and by almost nothing if the storey is near the top. The mode shape looks unchanged, its match to the old one 0.99997. But the mode's storey drift — its curvature — rises by about a tenth in the damaged storey and hardly anywhere else, wherever that storey is. The instrument that sees the damage is the one almost nobody installs.

The modes of five masses joined by springs and held by nothing. Five equal masses joined in a line by equal springs, with nothing holding them to the ground. The first four modes, at zero, 0.62, 1.18, 1.62 times the frequency of one mass on one spring. The first is every mass moving together with no spring stretched at all: a mode at exactly zero frequency, a real solution of the eigenvalue problem, which carries all of the mass and none of the strain. Every other mode has the ends moving against each other, and carries none of the mass under a uniform acceleration.

The modes at zero frequency

A structure held by nothing — a span being launched, a segment on a crane, a pontoon — has a mode in which it moves as one body and stretches nothing, at a frequency of exactly zero. That is a real mode, not a glitch in the stiffness matrix. It carries every kilogram of the structure under a uniform acceleration and leaves the flexible modes none at all. A load that is not uniform is a different matter: pushed suddenly from one end, the structure has no static answer to give, only an acceleration with a vibration riding on it.

The vertical response on layers of different depth. The vertical amplitude of a 150 tonne block 5.0 m across on soil with a shear wave speed of 178 m/s, under a harmonic force of fixed size, as a multiple of its static deflection on a half-space, against frequency. On a half-space the mode is at 12.5 Hz and the peak is 1.10. On 10.0 m the mode is at 14.3 Hz against a cut-off of 7.4 and the peak is 1.03. On 6.0 m the mode is at 15.5 Hz against a cut-off of 12.3 and the peak is 1.74. On 4.5 m the mode is at 16.3 Hz against a cut-off of 16.4 and the peak is 5.84. The layer stiffens the footing and lifts the frequency a little; what changes the picture is the cut-off, which rises as the layer thins and, once it passes the mode, takes the radiation damping away and leaves only the soil's own 5 per cent. Radiation below the cut-off is taken as nothing and its rise above it as linear, which is the usual idealisation of the exact layered solution.

The rock that sends the waves back

A machine block on the ground is damped by the waves it launches, and a half-space lets them all escape. Put rock four and a half metres down and none of them can: below the soil layer's own lowest frequency there is no wave that travels, and a vertical mode damped at 47 per cent of critical keeps the soil's own 5. Its resonance grows fivefold, and the heavy block the textbook rule recommends is the one that loses its damping on the deepest ground.

Two blows on three foundations. The movement of a 150 tonne block struck by a hammer delivering 18 kN·s, twice, 0.75 s apart. On soil at 12.5 Hz with the 47 per cent of damping a half-space radiates, the block moves 0.86 mm and is still before the second blow. On the same soil over shallow rock, with 5 per cent, it moves 1.42 mm and rings for the whole interval. On springs at 4.0 Hz it moves 4.42 mm, and its ringing has not died when the second blow arrives. The peak force passed to the ground is 1.34, 1.32 and 0.42 MN: the heavily damped block and the lightly damped one pass almost the same force.

The blow that has no frequency

A forge hammer does not shake its foundation; it strikes it. The transmissibility curve every isolation design is drawn on has nothing to say about a blow, and the rules it teaches mislead. The force a struck block passes to the ground is least at a quarter of critical damping, not the most; five per cent and forty-seven pass almost the same; and the springs that soften every blow can bring the block down to the hammer's own rate and make a train of blows ring four times as high as one.

The same blow, three seats under the anvil. The movement of the block — the part of a 150 t hammer foundation, 30 t of it the anvil, that lies below the pad — after one blow of 18 kN·s on the anvil, over three periods of the whole foundation on soil (12.5 Hz, damping 0.47). Rigid seat: largest movement 0.86 mm at 16 ms; pad at 4.0 times: largest movement 1.14 mm at 13 ms; pad tuned to the foundation: largest movement 1.18 mm at 35 ms. On the pad at 4.0 times the block rides the anvil's ringing: a ripple at the anvil's own frequency on top of the foundation's swing, whose first crest lands near the swing's peak. On the tuned pad the block receives the blow as one slow push, peaks later and higher, and then goes on ringing long after the rigid seat has settled, because the mode in which anvil and block swing against each other is damped by the pad and hardly at all by the ground.

The pad that makes the blow worse

A forge hammer's anvil sits on a pad on its foundation block, and the pad looks like isolation: a spring between the blow and everything below it. For the block it is the opposite. Every pad an anvil can sit on makes the block move more than a rigid seat would, by half again when the pad is tuned near the foundation, and what the pad buys instead is a smaller force under the anvil. The hope that a tuned pad could act as a tuned mass works only against a train of blows, and only at a softness the anvil cannot live with.

One actuator, a tower and a floor. The peak amplification of a mode with 1 per cent damping under velocity feedback of gain 0.10, against the mode's frequency on a logarithmic scale, for control loops whose delay is 10, 25, 50 milliseconds. The dotted line is the mode without control, 50. With 10 ms: 4.6 at 0.2 Hz, 4.5 at 2 Hz and 5.0 at 8 Hz; with 25 ms: 4.6 at 0.2 Hz, 4.6 at 2 Hz and 19.0 at 8 Hz, unstable from 9.8 Hz; with 50 ms: 4.5 at 0.2 Hz, 5.3 at 2 Hz and unstable at 8 Hz, unstable from 4.9 Hz. A delay is a fixed time and a period is not, so the same loop that damps a tall building's sway is too late for a floor.

The actuator that arrives late

A tuned mass damps a structure because its force arrives a quarter of a cycle behind the motion. Replace the mass with an actuator told to push against the structure's velocity, and the same quarter-cycle is fatal: the push that was damping becomes stiffness, the stiffness becomes a source of energy, and a gain that would have cut the response tenfold makes the structure vibrate by itself. The delay is a few hundredths of a second, which is nothing to a tower and everything to a floor.

The same riser, held three ways. The peak bending stress at each floor of a water-filled DN100 riser running the full height of the eight-storey building, under the same record, held three ways. Anchored against rotation at every floor it reaches 131 N/mm², at the lower floors where the drift is largest. Guided at every floor — held in line and free to turn — the same pipe never exceeds 3 N/mm². Guided everywhere but anchored at its base, it carries 77 N/mm² at the base and 21 N/mm² one floor up, and the guided values from there on.

The pipe held at every floor

A riser runs the height of a building and is fixed at every floor, so each of its supports moves with a different floor. The floor spectrum prices the pipe's inertia, and for a pipe anchored at every floor the inertia is nearly irrelevant: the drift puts 131 N/mm² into a 100 mm riser at two thirds of a per cent, thirty times its inertia, and passes yield at 200 mm. Guide the same pipe instead of anchoring it and the drift almost vanishes — the pipe then feels only how much the drift changes from one storey to the next.

$What the actuator pushes against decides its limit. The largest feedback gain, as the damping it would add to the bare mode, at which the loop stays stable, against the delay as a fraction of the period, on a logarithmic scale, for a mode of 1.00 per cent damping carrying a tuned mass of 2.00 per cent of its modal mass at Den Hartog's tuning. Pushing against the ground, fed by the structure's velocity: 2.41 at a delay of 0.05, 0.25 at 0.2. Pushing against the tuned mass, fed by the same velocity: 0.0156 at almost no delay, rising to 0.091 at a quarter of a period. Pushing against the tuned mass, fed by the velocity across it: unlimited with no delay, 0.048 at 0.05, 0.0073 at 0.2.$

The actuator that pushes against the mass

An actuator that damps a structure by feeding back its velocity has to push against something. Against the ground, it is stable up to large gains until a delay of a quarter period removes its damping. Against a tuned mass — the arrangement meant to keep a passive damper underneath — the same feedback goes unstable at a gain of a per cent and a half with no delay at all, makes the structure worse than the passive mass at every gain below that, and drives the mass three or four times as far. What limits it is not when it acts but what it pushes against.

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