Generator

The first three modes of a simply supported beam

Rendered here at the parameters it defaults to, with every essay that calls it — which is the same list as the blast radius of changing it.

The first three modes of a simply supported beam. Three modes of a simply supported beam of 8 m span, drawn from the general solution with the constants fixed by the support conditions rather than assumed to be sines. Mode 1 is at 4.91 Hz with βL = 3.1416; Mode 2 is at 19.63 Hz with βL = 6.2832; Mode 3 is at 44.18 Hz with βL = 9.4248. The frequencies go as the square of βL, so the threeth mode is 9.0 times the first. The marked points are the nodes.

12 essays call beam-frequency. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

Dynamics

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.

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.

Dynamics

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.

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.

Dynamics

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.

Four guesses at one buckling mode. A pin-ended column, with four assumed shapes and the load each of them gives. The reference is a ten-term Ritz expansion solved as an eigenvalue problem, at 9.8696 EI/L² — which is π², as it must be. a half sine gives 9.870, its own sag shape gives 9.882, a mid-span sag gives 10.000, a parabola gives 12.000. Every one of them is high and none of them is low, because an assumed shape is a constraint on the column and a constraint can only stiffen it.

Stability

Guessing the shape, and getting the load anyway

A column's buckling load can be had from a shape that is wrong everywhere, because the energy criterion is stationary at the true mode. The error in the load is the square of the error in the shape, and it is always high.

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.

Dynamics

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 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.

Dynamics

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.

Every reason a building is stiffer than its model, added up. The computed natural frequency of a floor, and the same frequency after each source of stiffness that was deliberately left out is put back. Not one of them is a modelling error. Cladding and partitions are stiffness nobody is allowed to rely on for strength; a nominally pinned connection is never pinned; a slab acts with its beam whether or not shear connectors were provided; and concrete between the cracks is stiffer than a cracked section assumes. Together they multiply the stiffness by 1.83 and the frequency by 1.35, because a frequency is the square root of a stiffness, and every factor is halved on the way through. The asymmetry is the finding: leaving stiffness out makes a deflection conservative and a vibration check unconservative in the direction that matters, since a stiffer floor has a higher frequency and sits further from the footfall range. The model here reads 4.40 Hz against a 5.2 Hz criterion and fails it; the floor reads 5.95 Hz and passes. The correction that would have got it right is exactly the stiffness nobody is willing to count on.

Deflection

Stiffer than the model said

Measured natural frequencies of finished buildings come out between ten and sixty per cent above the values computed for them, consistently and in one direction only. Nothing on the list of reasons is a modelling error: every one is a real source of stiffness deliberately left out — and leaving stiffness out is conservative for deflection and unconservative for vibration.

The abutment force is the sag turned upside down. A 100 m ribbon carrying 35 kN/m at a sag of 2.0 per cent of its span. H = wL²/8f, so the horizontal force at each abutment is 21875 kN — 6.25 times the entire weight of the deck, and five times what a suspension bridge of the same span and weight at a tenth would have needed. The curve is a reciprocal and it has no flat part: halving the sag doubles the force, at any sag. What stops a designer flattening it further is not the ribbon, which is in tension and cannot buckle. It is what the ground at each end will take, and at 6.25 deck-weights that is usually rock or a very large anchor block.

Structural form

The deck that is its own cable

Every other cable structure hangs something from the cable. A stressed ribbon hangs nothing — the walking surface is the catenary, laid at a fiftieth of the span rather than a tenth, because a footbridge has to be walkable. That one decision hands the abutments six and a quarter times the entire weight of the bridge.

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.

Dynamics

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.

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.

Dynamics

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.

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.

Dynamics

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.

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%.

Dynamics

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 library, page 1 of 7 — where beam-frequency sits