Structural Engineering & Statics
where the load goes
Concept

Stay cable — where it appears

A straight cable running from a pylon to a deck, carrying part of the deck's weight in tension. It has almost no damping and a frequency every fraction of a hertz, so wind, rain and deck motion all find a mode, and dampers, cross-ties and sheath treatments are how its vibration is controlled.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

dynamics · Cable dynamics 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.

dynamics · Cable dynamics 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.

dynamics · Cable dynamics A sine's requirement grows with the deck, and a random deck's with its square. The damping a 120 m stay at 3,500 kN, 1.01 Hz needs to hold still, against the rms of the deck's movement at the anchorage, with the deck tuned to twice the stay. A sine of that rms: √2·σ·(dε/dx)/4, a straight line. Against it, a random deck with 1.0% damping of its own: the swing's variance over sixteen times the deck's damping ratio, a parabola for a typical history and, dotted, twice that for the mean square, with dots from integrating the stay and the deck together over five 2,000-second histories each. With the stay's own 0.10% the sine grows above an rms of 2 mm and the random deck above 9.1 mm — 6.4 mm for its mean square. At 7.5 mm the integration asks for 0.06% against an average of 0.07%, at 15 mm the integration asks for 0.25% against an average of 0.27%, at 23 mm the integration asks for 0.49% against an average of 0.61%, past the range the average claims and at 30 mm the integration asks for 0.71% against an average of 1.1%, past the range the average claims.

The deck that forgets its own rhythm

A deck at twice a stay's frequency grows the stay whenever a quarter of the tension swing beats the damping. But a deck pushed by gusts is not a sine: its amplitude comes and goes and its phase drifts, and it holds a rhythm for seconds where the stay needs minutes to answer one. Against that deck the requirement is not a quarter of the swing but its square over sixteen times the deck's own damping — a sixth of the sine's at service, and a number set by how much the deck remembers rather than by how far it moves.

dynamics · Cable dynamics Double the span and the deck is nearly three times nearer buckling. The factor on its own dead load at which a stayed deck buckles, against the main span, on logarithmic axes, for a deck of 0.50 × 10¹² N·m² under 150 kN/m with stays every 15 m from towers 0.40 of the half-span tall. At 300 m it is 40.7; at 600 m, 14.7; at 1,000 m, 6.9; at 2,000 m, 2.3. Beyond 600 m the line falls with a slope of 1.53; it crosses ten, below which a design has to follow the deck's deflections into its forces, at 805 m.

The deck the stays squeeze

Every stay of a cable-stayed bridge pulls the deck towards its tower, and the deck carries all those pulls as a compression that is largest at the tower — 95 MN in a 1,000 m span under 150 kN/m. The stays hold it up and hold it straight, as springs under a strut. The compression grows with the square of the span and the springs soften with it, so the factor on its own weight at which the deck would buckle falls as the span to the power one and a half: 41 at 300 m, 6.9 at 1,000 m, 2.3 at 2,000 m. A deck four times stiffer only doubles it, and the long stays' sag, which softens them for traffic, hardly enters.

structures · Stiffening girder A scatter of a few per cent in the headways and the rhythm is gone. The damping a 120 m stay at 3,500 kN, 1.01 Hz, over a deck mode at 2.01 Hz damped at 1.0 per cent needs to hold a typical history still, under identical vehicles, each giving the deck the impulse with which a random stream of them every 2.0 s on average gives it an rms of 7.1 mm, arriving every 1.987 s on average — 4 deck periods — against the scatter of their headways (coefficient of variation, logarithmic), from integrating the deck and the stay together over four 1,500 s histories at each point. Exactly regular, 0.99 per cent; scattered by 1 per cent, 0.86 per cent; by 2 per cent, 0.58 per cent; by 5 per cent, 0.13 per cent; at random, 0.05 per cent. The requirement halves at a scatter of 2.2 per cent.

The convoy that keeps the deck's time

A deck shaken by traffic is shaken by vehicles, and vehicles arrive with a rhythm of their own. The same lorries that, arriving at random, move a stay's deck by 7 mm rms and ask the stay for six hundredths of a per cent of damping, move it by 20 mm and ask for a whole per cent if they arrive every four deck periods exactly. But the rhythm is fragile in a precise way: scatter the headways by two per cent and half the danger has gone, and the timing that has to be kept is a tenth of the deck's period whatever the headway and however damped the deck. Ordinary traffic is counted by a stay, not timed; a convoy driven by a machine might not be.

dynamics · Cable dynamics

Named alongside it

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

Cable dynamicsDampingResonanceParametric excitationTaut stringViscous damperEigenvalueFatigueMode shapeNatural frequencyVortex-sheddingBuckling

All concepts