The pier that moves in jumps

A bridge deck grows by a millimetre or two an hour on a warm morning, and at every pier it carries a sliding bearing whose job is to let it. The bearing presses 3000 kN onto a PTFE surface, and the drawing shows a roller, meaning that the horizontal force in the pier is whatever the friction happens to be — a number the bearing’s own coefficient decides, and a small one.

What the drawing cannot show is that the deck does not slide over the pier at a millimetre or two an hour. It does not slide at all for hours at a time, and then it slides six millimetres in a sixth of a second.

The mechanism was named, in words, in the essay on why friction has two coefficients: a contact whose static coefficient is larger than its kinetic one, pushed slowly, holds and releases instead of drifting, so the force in whatever pushes it oscillates between two values. The two values are what this essay is for, because neither of them is the one a designer would guess, and the size of the swing turns out to have nothing to do with the thing designers usually adjust.

Where the force goes when the bearing lets go #

Take the pier head as what it is: a mass on a spring. The spring is the pier’s lateral stiffness, 20 kN/mm here. The mass is the part of the pier and its cap that moves with its top, about 50 tonnes. And the thing dragging it is the deck, which is so much heavier and stiffer than the pier head that it can be treated as a surface moving at a fixed speed.

While the bearing grips, the pier head goes wherever the deck goes. The pier bends, and its force rises in proportion to the distance the deck has travelled: 20 kN for every millimetre. That continues until the force reaches what the static coefficient allows, 0.05 times 3000, or 150 kN, after 7.5 mm of travel. At 1.7 mm an hour that is four and a half hours.

Then the bearing breaks away, and the force that resists the pier springing back is no longer 150 kN. It is the kinetic friction, 90 kN. The spring is pushing with 150 against a friction of 90, so the pier head accelerates back toward the level at which the two would balance — a spring force of 90 kN, 4.5 mm from its unloaded position.

It does not stop there, for the reason no mass on a spring stops at its equilibrium: it arrives with speed. It passes 90 kN and carries on to the far side, and with no damping the far side is exactly as far below 90 as the start was above it. The pier head comes to rest relative to the deck at 30 kN, 1.5 mm from its unloaded position, and at that instant the deck and the pier head are moving together again. The static coefficient takes over and the bearing grips.

So the swing is

$$\Delta F = 2\,(\mu_s - \mu_k)\,N$$

— the drop between the coefficients, twice. The first half is the drop itself. The second half is the overshoot, which is a dynamic amplification of that drop by exactly the factor of two a suddenly applied load always carries. The slip is a load step of 60 kN applied to an undamped oscillator in a fraction of its period, and a load that arrives suddenly and stays takes a structure to twice the displacement the same load would give it if it were eased on.

The energy closes the account. The spring stored ½·20·7.5² kN·mm at the start of the slip and ½·20·1.5² at the end, a difference of 540 J. The pier head slid 6 mm over a surface resisting it with 90 kN, and 90 × 6 is 540 J. Everything the spring gave up went into the sliding surface as heat, which is why the model needs no damping to stop: the friction is doing the stopping, and what it cannot do is stop the pier head anywhere but at the far end of a swing.

The stiffness is not in the swing #

Now change the pier. The obvious move for a designer worried about a bearing that grips and gives is to stiffen the pier, on the reasoning that a stiffer support resists the movement better. Double its stiffness twice over.

The swing is identical, and the formula already said it would be, because $k$ does not appear in it. Both of the numbers the pier force moves between are set by friction and by the dynamics of a mass on a spring, and the stiffness of the spring cancels out of the ratio in the same way the mass does. The stiffness decides how far the pier head has to travel to build up 150 kN and how far it swings back, and both of those shrink in proportion as the pier gets stiffer.

A pier four times as stiff therefore does not reduce the load case. It delivers the same load case four times as often, in quarter-length jumps, and the total travel of the deck over the bearing is unchanged because that was set by the temperature all along.

That chart reads as a trade rather than an improvement, and the two ends of it are different engineering problems.

The soft pier jumps rarely and far. A 24 mm jump on a pier of 5 kN/mm is a deck lurching over its bearing twice a day, with an impact at the end of each lurch wherever something else stops it — an expansion joint, a guide, the gap somebody sized for the movement without allowing for its arriving all at once.

The stiff pier jumps often and imperceptibly. A third of a millimetre at a time is nothing anybody hears, and more than a hundred of them a day is a fatigue spectrum: a force range of 120 kN on the bearing’s fixings, the pier cap and its reinforcement, tens of thousands of times a year, each cycle of it a load that never came near failing anything. It is also a wear spectrum for the sliding surface, which travels the same total distance in more starts, and a start is when a PTFE sheet wears.

Neither end is safe by default, and the chart says which question to ask of which pier. What neither end offers is a smaller swing.

Damping takes the overshoot and leaves the drop #

If the stiffness cannot touch the swing, the next candidate is damping, and it does something — exactly half of what would be wanted.

With a damping ratio $\zeta$ in the pier head, the swing back after breakaway loses a fraction of its amplitude on the way, and the overshoot below the kinetic level shrinks by the factor $e^{-\pi\zeta/\sqrt{1-\zeta^2}}$ that any damped oscillator loses in half a cycle. The drop itself is not an oscillation and has nothing to damp. The swing becomes

$$\Delta F = (\mu_s - \mu_k)\,N\left(1 + e^{-\pi\zeta/\sqrt{1-\zeta^2}}\right)$$

which starts at twice the drop and falls toward once the drop, and never below it.

The shaded strip on that chart is where real piers sit, and it is at the steep end of the curve only in the sense that the curve starts steeply. A concrete pier with two to five per cent of critical damping takes back somewhere between a twentieth and a sixth of its overshoot. Damping in an ordinary structure is small and mostly unknown, and a mechanism that needs a damping ratio of 0.5 to take a quarter off a load case is not a mechanism anyone can design with.

The same distinction explains why a friction device used deliberately — a damper whose slip load is chosen — has the spike at the start of each traverse that its essay names as a demand on the members behind it. The cap is the kinetic force. The breakaway is the static one. And the member behind the cap sees the difference twice, once on the way up and once as the overshoot on the way back, whatever the device is bolted to.

Only a smaller drop makes a smaller swing #

What is left, once the stiffness and the damping have been ruled out, is the drop itself. The swing is proportional to $\mu_s - \mu_k$, and that is a property of the surface.

A quarter of the drop gives a quarter of the swing, as the formula says, and it gives four times as many slips, which the formula also says: the jump is the swing divided by the stiffness, and a smaller swing is a shorter jump. The load case shrinks and becomes more frequent at once, and the fatigue arithmetic has to be done again with the new range and the new count — but a range that falls by four and a count that rises by four is a large net gain, because fatigue damage goes roughly as the cube of the range.

This is why sliding bearings are specified by what their surfaces do from rest and not only by their coefficient. Lubricated PTFE with dimples to hold the grease exists precisely so that the static and kinetic values nearly coincide, and the value that matters to the pier is the one no single coefficient on a drawing records: the gap between them.

The gap is also not a constant. A contact that has sat still has had time to grow its real area of contact, so its static coefficient rises with the time it has been at rest. A slow drive gives a contact a long time at rest between slips. So the conditions under which thermal movement arrives — slowly, over hours, with long pauses in the night — are the conditions that make the breakaway largest.

A spring, a mass and a surface that is faster #

The free body the whole argument lives on is the pier head, cut at the sliding surface and at the pier. Crossing the cut at the surface is the friction force. Crossing it at the pier is the pier’s own lateral force. The deck does not appear as a force at all; it appears as a speed.

That is the distinctive thing about the problem, and the reason it cannot be settled by a statics calculation of the ordinary kind. While the bearing grips, the load on the pier is displacement-controlled: the deck imposes a movement and the pier’s force follows from its stiffness. The moment the bearing slips, the load on the pier head becomes force-controlled — the friction, a fixed number — and the pier head’s response to a change from one kind of control to the other is dynamic. The breakaway force comes from friction and statics. The force it is left with comes from friction and dynamics. Both are needed and neither is enough.

It is the same structure as a roof that jumps when its load passes a peak on a path that falls: a system driven slowly along a path, with a capacity that drops the instant it is exceeded, releases its stored energy at once, and ends up somewhere the static path could not have predicted. The bearing does it hundreds of times a year and more quietly.

The same drop, under a bow and along a fault #

The mechanism is not structural in origin, and the places it turns up are the best evidence of how general it is.

A violin string under a bow is a spring dragged by a surface. Rosin gives the contact a static coefficient well above its kinetic one, and the string grips the bow, travels with it, lets go, flies back and grips again — hundreds of times a second rather than a few times a day, with the string’s own period deciding the timing the way the pier head’s does here. The note is the stick-slip.

A geological fault is a surface dragged by a spring as well: the elastic rock either side of it, loaded by a plate motion of a few centimetres a year. Brace and Byerlee proposed in 1966 that shallow earthquakes are stick-slip on that surface, and the idea has survived because it explains the thing that most needs explaining, which is why the rock does not simply creep. The part of their argument that carries over to the pier is the part this essay has not yet used. On a fault the friction does not drop instantly: it falls away over a small slip distance. That gradual weakening has a stiffness of its own, and whether the surrounding rock is stiffer or softer than it decides whether the fault slips in events or creeps steadily.

A step from the static to the kinetic coefficient has no such distance in it, so in the model used here every pier stick-slips, however stiff. A real sliding surface is somewhere between the two. It is what makes the stiffness chart above an upper bound on the number of events for a very stiff pier, and possibly on whether a very stiff pier has events at all.

The evening, when the deck turns round #

Everything so far has been one direction of travel, and a deck does not expand for ever. A movement nobody applied reverses every evening, and the reversal is where the daily force range on the pier is actually set.

At the end of the morning the bearing is gripping with the pier carrying 30 kN in the direction the deck was moving. When the deck starts to contract, the bearing does not slip back at once, because nothing about the contact has changed: it is still gripping, the pier head is still carried with the deck, and the pier’s force now falls by 20 kN for every millimetre of contraction. It passes through zero, keeps going, and only when it reaches 150 kN in the other direction does the static coefficient give way.

That is a change of 180 kN before the first evening slip, not 120 — 9 mm of contraction held entirely by the pier, about five hours at the same rate. After it the events resume with exactly the swing of the morning, mirrored: breakaway at 150 kN one way, grip again at 30 kN the same way, in 6 mm jumps.

So the pier’s day has two scales in it. The slips deliver a 120 kN range several times over. The turn of the day delivers a 300 kN range once — from 150 kN one way to 150 kN the other — set by the static coefficient twice and by nothing else. For fatigue the once-a-day range is the larger contribution: three hundred and sixty-five cycles a year of 300 kN do more damage than two thousand of 120, since damage goes roughly as the cube of the range. The seasonal movement adds one more, larger loop on top, which is the same accumulation of small daily movements into a yearly load that an abutment with no bearing at all meets as a ratcheting earth pressure.

What the oscillator leaves out #

The friction drops in a step. Real interfaces lose their static grip over a small distance and a small time, and a very stiff support can follow that loss without ever storing enough energy to jump. The swing is right for a soft or ordinary pier and an overestimate for one stiff enough that a fraction of a millimetre of pre-sliding matters.

The coefficients are fixed. PTFE’s coefficient rises steeply in the cold and falls with contact pressure, and its static value grows with the time at rest. All three change the drop, which is the only quantity the swing depends on.

The pier head is one mass on one spring. A real pier has modes, a real deck has flexibility, and a deck that is not infinitely stiff adds a second spring in series with the pier. The effective stiffness is lower than the pier’s own, and the jump is correspondingly longer.

The load on the bearing is constant. Traffic arriving and leaving changes $N$, and with it both the breakaway force and the kinetic level, in the middle of a slow thermal drive. That changes more than the size of the swing.

The drive is steady. The deck is taken to move at one speed, and a real thermal movement accelerates in the morning sun and stalls under cloud. Since the swing does not depend on the speed and the time between slips does, an uneven drive changes when the events happen and not how large they are — while the static coefficient’s growth with time at rest means a long stall makes the next breakaway a little larger.

Still open: which force the pier is left with when it stops #

Every slip above ends with the bearing gripping at a force that is not the kinetic one and not the static one: 30 kN in the base case, 39 with some damping, 75 with a better surface. The pier holds that force until the deck moves again, and nothing in the present state of the bridge — its temperature, its traffic, its geometry — says what it is.

It says instead how the pier got there. A bearing that stops slipping is left at whichever of its admissible forces the last event delivered it to, and if the load changes while it is gripping, or the deck turns round, the next event starts from that force rather than from zero. That is the general form of a fact the ordinary friction problem already contained: the friction at a contact is a range of possible values, and which value a structure is actually carrying is decided by the order in which things happened to it.