Structural Engineering & Statics
where the load goes
Concept

Dynamic amplification — where it appears

The ratio of the response to a time-varying load to the response the same load would produce if applied slowly. It is two for a suddenly applied constant load, less than one for a load slower than the structure, and very large near resonance.

Named by 13 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

dynamics · Dynamic amplification 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.

dynamics · Impact factor 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.

dynamics · Moving load resonance 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.

dynamics · Blast The force nobody applied, and the speed it wins at. Lateral force per unit weight for a vehicle on a 400 m curve, against speed. The rising curve is what the free body demands — v²/gR, which is the body's own acceleration written on the other side of the equation — and the flat line is what 6.0° of cant supplies from the weight. They cross at 73 km/h, which is the speed the curve was set out for; below it the deficiency has the other sign and the rail is pushed the other way. The upper line is overturning, at b/2h = 0.399 — and there is no mass in that number, so a loaded vehicle and an empty one go over at the same 160 km/h and only the height of the load decides. At the 108 km/h drawn the deficiency is 0.124 of the weight, which is 49 kN on this 40 tonne vehicle.

The force that is really an acceleration

Every other load in this collection is applied by something. This one is applied by nothing at all — it is the body's own acceleration, written on the other side of the equation so that statics can be used on a problem statics has no business with. The move is legitimate, it is a hundred and eighty years old, and it is exactly half done more often than it is done.

equilibrium · Centrifugal load 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.

dynamics · Transient resonance The pier grips, gives, and grips again. A sliding bearing carrying 3000 kN on a pier head of 20 kN/mm, dragged by a deck expanding at 1.7 mm an hour, with a static coefficient of 0.05 and a kinetic one of 0.03. The force in the pier climbs while the bearing grips, reaches 150 kN, and falls in a fraction of a second to 30 kN: the pier springs back under only the kinetic friction, overshoots the 90 kN that friction would hold it at, and grips again. The swing is 120 kN — 2.00 times the 60 kN between the two coefficients — and the pier head jumps 6.00 mm each time, three times in 12 hours.

The pier that moves in jumps

A sliding bearing whose static friction is larger than its kinetic friction does not release a slow thermal movement as a drift. It grips, gives and grips again, and each time the force in the pier swings by twice the difference between the two coefficients — whatever the pier is made of.

equilibrium · Friction 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.

dynamics · Mode shapes 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.

dynamics · Vibration isolation 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.

dynamics · Vibration isolation 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.

dynamics · Tuned mass damper Three members after a diagonal goes. The forces in three members of the counter-braced truss when the diagonal 9–2, carrying 242 kN, is removed instantaneously, with 2 per cent damping, over one and a half of the damaged truss's first periods (0.57 s); dashed, the static force each settles to. The counter-diagonal 1–10 goes from −112 kN to −354 kN and peaks at −491 kN, 1.57 times its change. The bottom chord 3–4 ends exactly where it started, 750 kN, and on the way peaks at 1,115 kN. The bottom chord 2–3 settles lower, at 556 kN, after a first swing the other way, to 905 kN.

The factor of two belongs to one mode

A member that fails suddenly hands its force to the structure around it all at once, and the convention is to double the static answer: a load applied suddenly to a spring overshoots to twice its static deflection. A truss is not one spring. Take a diagonal out of a counter-braced truss in an instant and some members swing to three times their change of force, one swings the wrong way first, and a bottom chord whose force does not change at all passes through half as much again as it carries — because every mode overshoots by two, at its own time, and a member is a sum of modes.

structures · Robustness The cable pays the factor the plateau saved. The force a beam of two 8.0 m spans over the column, 600 mm deep, plastic moment 800 kN·m, its far ends held against rotation and against spreading (plastic collapse load 400 kN) reaches after a sudden loss divided by the load it carries — the dynamic factor — against that load as a share of the plastic collapse load. Below the plateau the beam is a linear spring and the factor is 2. On the bending plateau it falls to 1.18 at 0.90 times the collapse load, because a flat resistance does its work at full strength from the start. Past it the beam becomes a cable, whose resistance is again straight, and the factor climbs back: 1.73 at 1.5 times, 1.88 at 2, 1.97 at 3.

The cable that pays the factor back

A frame that loses a column carries the floor across the gap first by bending and then, as the beam sags, as a cable. The hope is that a path which stiffens as it deflects needs less than the factor of two a suddenly loaded spring does. On the bending plateau it does — 1.18 at nine tenths of the collapse load. But past one depth of sag the cable is a straight line again, and the factor climbs straight back toward two: 1.73 at one and a half times the collapse load. And the connections are asked for a tenth of a radian on the way.

structures · Robustness

Named alongside it

The objects these essays reach for when they reach for this one.

DampingImpact factorImpulseNatural periodMode shapeDamping ratioFree bodyModal massNatural frequencyResonanceRobustnessVibration isolation

All concepts