Where the structure is allowed to move

A concrete viaduct 112 metres long gets about 45 millimetres longer between a cold night in winter and a hot afternoon in summer. Nothing can stop it: the strain is $\alpha\Delta T$ and the only choices are where the movement is allowed to happen and what resists it if it is not.

That choice is the articulation scheme, and it is a drawing rather than a calculation. It says which support is fixed, which slide, which are monolithic with their piers, and which way each sliding bearing is allowed to move. It is usually settled early, quickly, and by convention.

It decides two things nothing else does: the size of every joint a reader will eventually walk or drive over, and where the horizontal load goes.

Which free body produced the number #

The deck. Take the whole of it as one body and ask what acts horizontally on it.

Each sliding bearing delivers a friction force $\mu V$ opposing the deck’s motion, so as the deck expands the bearings to the right of the fixed point push left on it and the ones to the left push right. Those forces do not cancel unless the fixed point is in a particular place. Whatever is left over has to be taken by the fixed support, and that residue is

$$H = \left|\sum_{\text{right}}\mu V_i - \sum_{\text{left}}\mu V_i\right|$$

which is zero when the fixed point sits at the friction centroid of the sliding supports — the point that divides the weighted reactions equally.

That is the whole argument, and it produces a result that is easy to state and rarely acted on. Fixing at one end costs a pier the entire friction of the viaduct; fixing at the centroid costs it nothing. On the deck drawn that is 752 kilonewtons against zero, before any wind, braking or seismic force has been considered — a difference produced by moving one symbol on one drawing.

The reason the fixed point is so often drawn at an abutment anyway is not ignorance. An abutment is stiff, it is close to the ground, and its horizontal capacity is nearly free. The point is that this is a choice with a price, and the price is invisible unless the free body above has been drawn.

The movements go the other way #

If the force argument recommends the friction centroid, the movement argument recommends the middle — and on a symmetric viaduct those are the same place, which is why the convention works when the deck is symmetric and stops working when it is not.

Movement at a support is $\alpha\Delta T$ times its distance from the fixed point, so the largest joint is at whichever end is further away. Fixed at one abutment, one joint takes all 45 mm. Fixed at the middle, the two ends take 22 mm each.

Halving the movement halves the joint, and joints are the part of a bridge that fails. They are the only moving mechanical component in a structure otherwise designed to stand still for a hundred years, they are exposed to everything the road throws at them, and they leak — which puts chloride-laden water onto the bearing shelf and the pier head below, which is where a disproportionate share of bridge deterioration happens.

So the movement argument and the force argument are not in tension on a symmetric deck and are on an asymmetric one, and the asymmetric case is the common one: a viaduct on a curve, a deck with one long span and three short, a bridge with one abutment on rock and one on fill.

There is a third quantity that the same choice decides and that neither argument above covers: how much of the deck a single joint has to accommodate at once. A joint is a mechanical device with a rated movement, and the devices come in ranges — a sliding plate joint to about 25 mm, a single-seal joint to about 80, a modular joint beyond that and at several times the cost, with several times the maintenance. So the schemes do not differ smoothly in cost; they differ in which kind of joint is needed, and a scheme that keeps every movement under 80 mm is qualitatively cheaper than one that needs a modular joint at one end, whatever the arithmetic says about the millimetres.

That is a discreteness of the same kind the answer is continuous and the catalogue is not is about, and it has the same consequence: the sensitivity of the decision to the temperature range assumed is zero over most of its range and enormous at two or three particular values.

The scheme with no joints at all #

Take the bearings out and make every pier monolithic with the deck, and the movement problem does not disappear — it changes into a force problem.

The deck is very stiff axially compared with the piers’ flexure, so it translates as a rigid bar. It expands about a point at which the pier forces balance:

$$\sum k_i (x_i - x_0) = 0 \quad\Longrightarrow\quad x_0 = \frac{\sum k_i x_i}{\sum k_i}$$

the stiffness centroid. Each pier then carries $k_i \alpha\Delta T(x_i - x_0)$, which is the same “stiffness attracts load” result as the stiffest path takes the load, with the extra feature that the lever arm is measured from a point the stiffnesses themselves decide.

That has an unexpected consequence, and it is the finding worth carrying out of this essay.

Stiffen one pier and two things happen at once. It attracts more force, because the force is $k(x-x_0)$ and $k$ has grown. And it drags $x_0$ toward itself, because the centroid is weighted by $k$ — which shortens the lever the temperature acts on.

The second effect wins in the limit. As $k_p \to \infty$ the neutral point arrives at the pier and the force tends to

$$F_p \to \alpha\Delta T\sum_{j \ne p} k_j (x_p - x_j)$$

which contains no $k_p$ whatever. A pier’s thermal load saturates. Past a few times its neighbours’ stiffness, making it stiffer attracts no more force at all, because it has stopped being one restraint among several and become the point about which everything else expands. What it carries is then a property of its neighbours.

The saturation is easy to state in words and worth doing so, because the algebra hides how ordinary it is. A restraint that is much stiffer than everything else around it is not sharing a deformation any more — it is imposing one. What it carries is then whatever the rest of the structure would have to be pushed through to keep up with it, and that is a property of the rest of the structure. The same sentence describes a shear wall among frames, a core among columns, and a stiff foundation among soft ones: past a certain stiffness, an element stops competing and starts defining, and the force it takes stops depending on itself.

That number is available before any pier has been sized, from the geometry and the other piers alone, and it is the right number to compare a scheme against. A design in which a pier is at ten per cent of its saturation is a design in which the thermal load is not the problem; one at ninety per cent is a design in which no amount of concrete will help.

Why the whole thing is an imposed deformation #

Everything above is a consequence of one distinction that this collection keeps returning to: the movement nobody applied is not a load.

A load is a force, and a stiffer structure carrying a load has smaller stresses. A temperature change is a deformation, and a stiffer structure resisting a deformation has larger ones — force is stiffness times movement, so stiffness is the multiplier rather than the divisor.

The articulation scheme is therefore a decision about which of the two the movement will be. Bearings turn it into a movement and a small friction force. Monolithic piers turn it into a force and no movement. There is no third option and there is no arrangement in which it is neither.

Two further quantities follow the same logic and are usually forgotten in the same place. Shrinkage in a concrete deck is a one-way movement of similar size to the thermal one, and it arrives early, when the piers are young and their creep is high. And creep under the prestress shortens the deck again. All three are the same problem; only the thermal one reverses.

What else the fixed point decides #

Two horizontal loads are not thermal at all and both go wherever the articulation says.

Braking and traction are applied to the deck surface and travel to the fixed support, in the bearing schemes, entirely — and unlike the thermal force they arrive suddenly, in one direction, at a moment nobody chooses. A 900 kN braking force on a fixed abutment is a straightforward design case; the same force on a slender middle pier is a moment at its base of nine hundred kilonewton-metres per metre of height.

Wind on the deck does the same, along the bridge. Across it, the load is shared by whatever resists transversely, which is usually a different set of restraints from the longitudinal ones — so a bearing schedule has to say, for each bearing and each direction, whether it is fixed, guided or free, and the six combinations are not interchangeable.

In the integral scheme both are shared by pier stiffness instead, which is the argument for it that is not about movement: a braking force spread over four piers is a smaller demand on each than the whole of it on one. That is the stiffest path takes the load working in the designer’s favour for once — sharing a force by stiffness is a benefit, where sharing a deformation by stiffness was a penalty, and the same scheme does both at the same time under different actions.

Which is the reason the two schemes cannot be compared on one number. Bearings are better for the thermal case and worse for braking; monolithic piers are the reverse; and the seismic case is different again, because there the demand is a displacement rather than a force and the argument inverts a second time. A scheme is chosen against a set of cases that disagree, which is why it is a drawing rather than a calculation.

Where the model stops #

The deck was assumed axially rigid. For a long viaduct it is not, and the deck’s own axial flexibility joins the piers’ in series — which relieves the piers and moves the neutral point, and matters most exactly where the effect is largest.

Friction was taken as a constant coefficient. It is not: a sliding bearing’s coefficient falls as the pressure on it rises, so a lightly loaded bearing has a larger coefficient than the design value quoted for a full one. The roller that is not a roller is that argument, and it changes the friction centroid.

The abutments were treated as supports rather than as soil. An integral abutment pushes into the fill behind it, the fill’s stiffness rises with each cycle as the soil compacts — a ratcheting effect over decades — and the earth pressure it develops is a passive one rather than an at-rest one.

Nothing here is about the sequence. A deck is built at some temperature and the bearings are set at that temperature, so the movement to each side of the setting position is not symmetric unless the setting was done at the middle of the range — and it rarely is, because construction happens when it happens. A bearing set on a hot day has to accommodate almost the whole range in one direction, which is a detail that costs nothing to get right and cannot be corrected afterwards. The structure that was never complete is the general form of the same problem.

And nothing here is about the vertical. A bearing carries a vertical reaction and permits a rotation, and the rotation demand is set by the deck’s end rotation under live load — which is the angle nobody limits and is a separate reason a bearing is replaced.

The generalisation #

The habit is to ask, of any structure with more than one support, where its fixed point is — and then to check whether anybody chose it.

Every structure has one, whether or not it was drawn. A building with a stiff core at one end expands about that core. A row of columns with the same section expands about its middle. A frame with an infill panel in one bay expands about the panel. In each case the fixed point is the stiffness centroid of whatever resists, and the members furthest from it carry the largest imposed force — often members nobody considered part of the restraint system at all.

A related question is worth asking about buildings, where it is asked much less often than on bridges. A 100 metre long concrete floor plate has the same thermal movement as a 100 metre bridge deck, it is restrained by whatever columns and cores are attached to it, and the forces that follow are computed by almost nobody — because a building’s frame analysis is run for gravity and wind, and a temperature case is an optional extra. The cracks that appear at the ends of long floor plates, in the columns nearest the stiff core, are the same arithmetic arriving without having been invited. How a tall building stands still chose the core’s position for lateral load; the thermal case was decided by the same drawing and was not on it.

The second habit is to notice when a design variable has a saturation in it. The thermal force on a pier is one of a small number of quantities in this subject that stop responding to the obvious remedy, and knowing the ceiling before starting is worth more than any number of iterations toward it. A designer who knows that a pier can never be asked for more than a certain force has a bound; a designer who does not has an argument with an analysis that gets worse every time the model is refined.