The roller that is not a roller

A roller symbol on a structural drawing means a support that carries a vertical force and no horizontal one. It is one of the first abstractions anybody learns, it makes a beam determinate, and it is the reason half the free bodies in this collection have three unknowns instead of four.

The component that gets built is a stainless steel plate sliding on a sheet of PTFE. It is not a roller, it carries a horizontal force, and the force is a few per cent of whatever it is holding up.

Which free body produced the number #

The deck, cut at the bearing.

Sliding contact delivers a force $\mu V$ opposing the relative motion, and the free body of the deck has one of them at every sliding support. The force that is whatever it needs to be is the general character of friction: it is a reaction below the sliding threshold and a known force at it, and a bearing that is moving is at it.

That distinction is the first practical point. A bearing that is not moving delivers whatever horizontal force the structure asks of it up to $\mu V$, and one that is moving delivers exactly $\mu V$. So the same component is a restraint and a load, on different days, and both cases have to be checked.

The coefficient is a function #

PTFE’s coefficient against polished stainless steel is not a material constant. It falls as the contact pressure rises, and the standard fit is

$$\mu = \frac{1.2}{10 + \sigma}$$

with $\sigma$ the contact pressure in newtons per square millimetre. At the 30 N/mm² a bearing is usually designed for, that is 0.030. At 5 N/mm² it is 0.080. At 45 it is 0.022.

The mechanism is that PTFE is a polymer with a very low shear strength that transfers a thin film onto the mating surface, and the friction is that film’s shear resistance times the real contact area. Real contact area grows less than proportionally with load, so the friction force grows less than proportionally and the coefficient falls.

The consequence for design is that the force is strongly non-linear in the load. A bearing at a fifth of its design load has two and a half times the coefficient and half the force, not a fifth of it.

Which end of the range governs what #

Two readings follow, they point at opposite ends of the load axis, and only one of them is usually taken.

The largest force is at full load: the bearing drawn is at 30 N/mm², so $\mu = 0.030$ and the force is 108 kN. That is the number a pier is designed for and it is the number in the calculation.

The largest nuisance is at light load. The same bearing carrying a fifth of that has a coefficient of 0.080 and delivers 58 kN — against a wind force on the same span of 140 kN. So the “free” bearing is contributing forty per cent of the horizontal load it was installed to release the structure from, and doing it in a case nobody looked at because the vertical load was small.

The second reading matters most on structures whose bearing loads vary a great deal: a light roof with large uplift, a bridge with a heavy live load fraction, a lifting structure, anything where a bearing can be nearly unloaded.

And cold makes it worse again. Below about −5 °C an unlubricated PTFE bearing’s coefficient roughly doubles, which is a factor applied on top of everything above — and it applies exactly when the thermal movement is largest, because the coldest day is one end of the temperature range.

Why it is not a small effect #

It is tempting to file a few per cent of the vertical load as a rounding error, and there are three reasons not to.

It is compared with a horizontal load, not a vertical one. A bridge’s vertical reactions are ten or twenty times its horizontal ones, so three per cent of the vertical is a third of the horizontal. The comparison that decides whether a number is small is with the quantity it is added to.

It accumulates along a deck. Where the structure is allowed to move shows that the fixed support takes whatever does not cancel across it, and on a viaduct fixed at one abutment that is the friction of every sliding bearing on the structure — 750 kN on an ordinary four-span deck. That is a force of the same order as the braking load, arriving from a component that appears on the drawings as an open circle.

And it is applied at the worst time. Friction opposes motion, and the motion is largest when the temperature change is largest, which is also when the deck is delivering its largest thermal force to everything else. The two arrive together and they are not independent.

The other half of the same component #

Everything above is about the sliding surface. A bearing has a second job, and it has a second friction to go with it.

A bearing has to permit rotation as well as translation, because the deck end rotates under live load — the angle nobody limits is that rotation, and it is a few milliradians on an ordinary span. The rotation is provided by a curved sliding surface, an elastomeric pad, or a pot of confined rubber, and each of them resists it.

A pot bearing’s rotational resistance is the friction of its own seal against the pot wall, and it produces a moment on the pier that the model has released. It is the same species of error as neither pinned nor rigid in a steel frame: a connection idealised at one end of its range and built somewhere inside it. On a large bearing that moment can be tens of kilonewton-metres, applied at the top of a pier that was designed as pinned there.

So a bearing drawn as a roller has, in practice, a horizontal force it was not supposed to have and a moment it was not supposed to have, both of them small, both of them at the top of a member whose design is sensitive to exactly those two things. The roller symbol suppresses two degrees of freedom’s worth of reality, not one.

What it does to the bearing itself #

A bearing is a maintenance item and friction is why.

The PTFE wears. It is a sacrificial layer a few millimetres thick, it is consumed by sliding, and the total sliding distance over a bridge’s life is the daily thermal cycle times the number of days plus the live-load cycles — kilometres, on a long span. When the PTFE is gone the sliding surface is steel on steel and the coefficient is not 0.03, it is 0.3 — which is the force that is whatever it needs to be arriving with a ceiling ten times higher than the one the pier was designed against.

That is a factor of ten arriving slowly, and it arrives on a component nobody can see, inside a structure that was designed on the small number. A bearing that has seized or worn through is delivering ten times the horizontal force its pier was designed for, and the pier finds out first.

The failure is common enough to be one of the standard bridge deterioration mechanisms, and it is usually compounded: a leaking movement joint above puts chloride-bearing water onto the bearing shelf, the bearing corrodes, the sliding surface roughens, the friction rises, the movement is resisted rather than accommodated, and the joint above is loaded further and leaks more.

The number that should be on the drawing #

There is a practical suggestion that follows from all of this and is worth stating, because the alternative is what happens now.

A bearing schedule records a vertical capacity, a movement capacity and a rotation capacity. It does not usually record a friction force, and the pier below it was designed for one — computed by somebody, from a coefficient chosen by somebody, at a load case chosen by somebody, and then not written down.

That is a chain with no record in it. If the bearing is later replaced with a different type, or a different manufacturer’s, or a lubricated one, nothing on the drawings says what the substitution has to match. The bearing is specified by what it must permit and not by what it must not exceed, and the second is the structural requirement.

The same argument applies to every component whose unwanted action matters. A movement joint’s stiffness, a cladding bracket’s restraint, a services penetration’s fixity: each is specified by what it should do and each also does something the structure was designed around. The connection is not a point makes the case that a joint has properties beyond the force it transfers, and a bearing is the clearest instance — its whole purpose is to not transfer something, and how well it fails to is a number.

The one place the friction is welcome #

There is a case where all of this is a benefit rather than a cost, and it is worth including because it is the reason bearing friction has been studied so carefully.

Under an earthquake, a sliding bearing is a friction damper. The horizontal force it can transmit is capped at $\mu V$ whatever the ground does, so it limits the force the substructure can be asked for; and the sliding dissipates energy, because the force opposes the motion through every millimetre of it. That is exactly what a base isolation system does, and the simplest isolators are sliding bearings with a curved surface — the curvature supplies a restoring force and the friction supplies the damping.

Made weaker on purpose is the design philosophy; a friction pendulum bearing is that philosophy built from the component this essay has spent its length complaining about. The same property — a horizontal force proportional to the vertical load, with a coefficient nobody controls precisely — is a nuisance under temperature and a design tool under an earthquake, and nothing about the component changed between the two.

Which is the honest summary of the whole subject. Friction at a bearing is neither a defect nor a feature; it is a property, and whether it helps depends entirely on which load case is being asked about.

Where the model stops #

The fit is one standard’s. Different codes and different manufacturers give different expressions, and the spread between them at low pressure is large — which matters, because low pressure is where the interesting case is.

Lubrication was ignored. Dimpled PTFE holding silicone grease has a much lower coefficient, and a much lower one still on its first movements before the grease has been displaced. It also has a different behaviour after twenty years.

Nothing here is about the movement’s speed. PTFE’s coefficient rises with sliding velocity by a factor of two or three between a thermal movement, which takes hours, and a seismic one, which takes tenths of a second — so a bearing that is nearly free under temperature is much less free under an earthquake, which is either a problem or a damping mechanism depending on the design.

The vertical load was assumed known. It is not, quite: a bearing’s reaction includes a live-load fraction that varies through the day, and the friction force varies with it — so the horizontal force from friction is a function of another load case rather than a load case of its own. Combining it correctly means combining it with the vertical load that produced it, which no combination rule expresses.

And the bearing was assumed to be sliding at all. A guided bearing restrained transversely has friction in the guides as well, and a bearing whose movement is smaller than the elastic deformation of its own components does not slide — it deflects, and delivers a force proportional to displacement rather than to load.

Why the idealisation survives anyway #

None of this is an argument against drawing rollers. The idealisation is enormously valuable and it is used because it works, and it is worth saying why it works despite everything above.

A friction force of three per cent of a vertical reaction changes the bending moments in the deck by almost nothing, because the deck is stiff in its own plane and the force is applied along it. What it changes is the horizontal design of the substructure — the pier, the abutment, the foundation — and those are designed by a different calculation, from a different set of load cases, by somebody who has to be told the number.

So the abstraction is correct for the analysis it was introduced into and incorrect for the design it feeds into afterwards, which is a pattern worth recognising: an idealisation is safe inside the calculation it was made for and dangerous when its output crosses a discipline boundary. The roller is fine in the deck analysis. It is a missing load case in the pier design.

The same is true of the other releases. A pinned base is a fine idealisation for a frame’s moments and a bad one for the base plate’s design. A rigid diaphragm is fine for distributing storey shears and bad for the slab’s own reinforcement. In each case the model’s simplification is a real force somewhere else, and the somewhere else is usually a different drawing.

The generalisation #

The habit is to ask, of every idealisation on a drawing, what component will be built to satisfy it and how nearly it does.

A pin is a bolt in a hole and has a moment capacity — the joint that is not a pin is that case in a truss. A roller is a bearing and has a friction force. A fixed base is a base plate on bolts and has a rotational flexibility. A rigid diaphragm is a slab and has a shear stiffness. Each idealisation is an approximation whose error runs in a known direction, and each is written on the drawing in a symbol that gives no hint of the component.

The second reading is more specific and is worth carrying into any structure with movement in it. A release is never free. Every component that permits a movement charges for it, in one of three currencies: a friction force, a stiffness that is not zero, or a maintenance obligation. A bearing charges in all three. The design question is not whether to pay but which of the three to pay in — which is why an integral bridge, which has no bearings at all, is not a bridge that has avoided the cost but one that has chosen to pay it in pier bending instead.