The summer that is worse than the last

Where a structure is allowed to move is a design decision, and for most of the twentieth century the answer on bridges was: at a joint, over the abutment, with a bearing underneath it.

The maintenance record of that decision is not good. An expansion joint is a hole in the deck. Water gets through it — water carrying de-icing salt — and lands on the bearing shelf, the bearings, and the top of the abutment: the three parts of a bridge that are hardest to reach and most expensive to replace. A very large fraction of what has been spent on bridge maintenance since the nineteen-sixties has been spent within two metres of an expansion joint.

The integral bridge removes it. The deck is built monolithic with the abutments, there is nothing to leak through, and there are no bearings to replace.

The movement does not go away.

The movement, and where it goes #

A deck of length $L$ subject to a temperature range $\Delta T$ changes length by $\alpha \Delta T L$, half of it at each end. For a 60 m concrete deck over a 25 °C range that is 9 mm at each abutment.

Nine millimetres is not much. Expressed as a fraction of the abutment’s height it is 0.15 per cent, which is well below the strain at which a granular backfill mobilises full passive resistance — a few per cent — so the pressure after one summer is nowhere near passive.

The at-rest coefficient for a dense granular fill at 38 degrees of friction is $K_0 = 1 - \sin\phi = 0.384$. The passive coefficient is $K_p = 4.20$. After one cycle the wall is somewhere just above the first.

That would be the end of it if the soil were elastic. It is not.

Ratcheting, which is why there is a page #

Granular soil under repeated shear strain densifies. Grains work into the gaps between other grains, the void ratio falls, and the packing becomes tighter. That is a one-way process: the grains do not work back out when the strain reverses.

The consequence for the abutment is that each cycle leaves the backfill a little stiffer and a little denser than the last. Achieving the same 9 mm of movement next summer requires a higher pressure, and the summer after that a higher one again.

The pressure therefore climbs, cycle by cycle, toward a limiting state. The calibrated end point is

$$K^* = K_0 + \left(\frac{d}{0.05H}\right)^{0.6} K_p,$$

with $d$ the movement at the top of the wall — an empirical form, fitted to model and full-scale tests, and this page says so. For the abutment drawn it gives $K^* = 0.897$: 2.33 times the at-rest force, reached to within three per cent after about a hundred and twenty cycles.

A hundred and twenty cycles is a hundred and twenty years. The governing load case on an integral abutment is one that does not exist when the bridge opens.

Why the length limit is a length #

The force grows with the deck’s length by two routes at once, which is why integral bridges have a maximum length rather than a maximum stress.

The movement is proportional to $L$, directly. The pressure goes as roughly $d^{0.6}$, so as $L^{0.6}$. And the force integrates that pressure over the wall height, which does not change. So the force per unit width of abutment goes as $L^{0.6}$ — sub-linear, but relentless.

The numbers for the abutment drawn: 201 N/mm of wall at a 10 m deck, 256 at 28 m, 332 at 65 m, 418 at 120 m. Nothing bounds it below passive, and passive is a long way above where a 60 m deck ends up.

So the limit is not the soil running out. The limit is what the abutment, the piles under it and the deck’s own axial capacity can take, and it is expressed as a length because the length is the variable that decides all three. Typical limits are 60 m for concrete and 60 to 100 m for steel, and they are limits on the thermal movement dressed as limits on the span.

Which free body produced the number #

Take the abutment wall as the free body and cut it at its base.

Acting on the back face is the earth pressure, distributed roughly triangularly with depth, resultant $\tfrac12 K \gamma H^2$ per unit width acting at $H/3$ above the base. Acting at the top is the deck, delivering an axial force and a moment. At the base is whatever the foundation supplies.

Two things about that free body are worth naming.

The pressure is a displacement-controlled action, not a load. It exists because the deck pushed the wall into the soil, and its magnitude is whatever the soil requires to accommodate that push. If the wall were free to move away it would fall to the active value; if it were rigid it would rise toward passive. It is the reaction to an imposed movement, which is exactly the character of every restrained thermal action.

And the deck’s own axial force is the mirror of it. The abutments push back on the deck, so a jointless deck is in compression in summer and tension in winter — 7,200 kN if it were fully restrained, and rather less because the abutments yield. That is a force nobody applied, in a member designed for bending.

What it does to the deck #

The deck’s axial force is the half of this subject that gets least attention and is the one that decides the deck’s own design.

A concrete deck in axial compression is not troubled by it. A steel deck, or a composite one, is: the axial force is applied to a member that is also in bending, and the interaction is a beam–column check on a bridge girder that nobody thinks of as a column.

Worse, the force reverses. Winter puts the deck in tension, which for a composite deck means tension in the concrete slab over the abutment — a location that is already hogging, already cracked, and already the worst place on the bridge for durability.

A jointless bridge trades a durability problem at the joint for a durability problem at the deck ends, and whether that is a good trade depends on whether the second is easier to detail than the first. Usually it is, because a crack in a slab that is properly reinforced is a serviceability matter and a corroded bearing is a replacement.

Two temperature ranges, and the deck has both #

The 25 °C used above is the annual range, and it is the one the ratchet cares about. There is a second, smaller range that acts far more often, and separating them matters because they load different things.

The annual cycle is large and slow. It produces the full 9 mm at each abutment, it strains the backfill through its whole range, and it is what the densification argument is built on. It happens once a year.

The daily cycle is a few degrees on the deck’s surface and rather less through its depth. It produces perhaps a millimetre of movement, and it happens every day for forty-four thousand days. Whether many small strains densify a fill as effectively as a few large ones is a question about soil behaviour that the design rules do not answer — they count annual cycles and ignore the rest.

There is also a gradient through the deck’s depth, which is a different action altogether: the top warms faster than the bottom, the deck tries to curl, and the abutments prevent it. That produces a moment rather than a length change, and in an integral bridge it is restrained at both ends by a connection that was designed for something else.

Three thermal actions, one structure, and only one of them appears in the ratcheting calculation.

The abutment has to be able to move #

The design response is not to make the abutment strong enough for $K^*$. It is to make it flexible enough that $K^*$ never develops.

An abutment on a single row of slender piles, oriented with their weak axis in the direction of movement, can accommodate the deck’s expansion by bending rather than by pushing. The pile’s characteristic length decides how much force that costs, and a slender pile in soft ground costs very little.

That inverts the usual instinct. A stiff abutment attracts the pressure this page computes; a flexible one sheds it. The stiffest possible abutment is the worst possible abutment, and integral bridges are therefore built on arrangements that a bridge engineer’s training says are inadequate — one row of small piles, no raking piles, deliberately weak in the direction of movement.

Raking piles are specifically excluded, because a raked pile is stiff axially in the direction the deck wants to move and will attract the whole force.

Articulation is a decision, and this is one answer to it #

Setting it beside the alternatives makes the trade explicit. A fully jointed bridge puts a joint over every pier and accumulates nothing; it has many joints, each leaking. A conventional arrangement fixes one support and slides the rest, accumulating all the movement at one joint; it has one large joint, leaking harder. An integral bridge has none and accumulates the movement in the soil.

Each is a decision about where a length change is allowed to happen, and each moves the difficulty rather than removing it — into a joint, into a bearing, or into a backfill. What differs is which of the three the owner would rather maintain, and that is a question about access as much as about mechanics.

The details that make it work #

Three arrangements appear on nearly every integral bridge and each addresses one part of the mechanism.

A run-on slab. The pavement immediately behind the abutment settles, because the soil there is being cyclically strained and densified — the same mechanism that produces the pressure produces a void. A reinforced slab spanning from the abutment out over the settled region carries the traffic across it, and it is the single most important detail on the bridge for the road user.

Compressible fill or a drainage layer. Placing a compressible material between the abutment and the fill absorbs the movement without straining the soil, which suppresses the ratchet at source. It works, and its long-term performance is the question: the material has to stay compressible for a hundred and twenty years.

Selected backfill. A well-graded granular fill compacted to a known density has a predictable $K^*$; a mixed fill does not. This is one of the few structural details where the specification of a soil is a structural specification.

The joint was not free either #

It is worth being precise about what is being compared, because the argument for integral construction is often made as though the alternative had no structural cost.

A jointed bridge on bearings has its own set of actions that the integral one does not. The bearings supply a friction force at every movement, which is a horizontal load on the abutment of the same character as the earth pressure — displacement-driven, present twice a day, and difficult to bound because the friction coefficient of a bearing changes over its life. A roller that is not a roller is a real and well-documented failure mode, and a seized bearing loads the abutment with the deck’s full restrained thermal force.

So the choice is not between a load and no load. It is between a load that arrives through soil and grows predictably, and a load that arrives through a mechanical device and grows unpredictably as the device deteriorates. The integral bridge’s advantage is not that its actions are smaller but that they are more knowable, which is a different and better argument than the one usually offered.

What it is worth #

The comparison is a whole-life one and it is not close.

An expansion joint costs relatively little to install and is replaced perhaps three times in a bridge’s life, each time with lane closures. The bearings under it are replaced at least once, which requires jacking the deck. The abutment and bearing shelf are repaired for chloride damage. Against that, an integral bridge costs a slightly heavier abutment, a run-on slab, and a deck designed for an axial force.

The integral bridge wins by a wide margin on any life-cycle assessment, which is why highway authorities in several countries now require it below the length limit rather than merely permitting it.

The honest qualification is that it wins on a forecast. The joint’s costs are measured, from decades of records. The integral bridge’s costs are predicted, from a mechanism whose empirical constants were fitted to tests much shorter than the structure’s life. The first bridges built to these rules are not yet forty years old.

The measurement, and what forty years of it says #

Integral bridges are among the most heavily instrumented structures there are, precisely because the mechanism was uncertain when they were first required.

What has been measured is broadly consistent with the model and differs in two places worth knowing.

The pressures are real and they do climb. Instrumented abutments in several countries show earth pressures rising over the first ten to twenty annual cycles and then flattening, which is the shape the saturating expression above produces. The measured end states are generally somewhat below the design values, which is what a calibrated rule ought to do.

The settlement behind the abutment is worse than expected. The void that forms under the run-on slab is the most consistently reported problem on integral bridges, and it appears earlier and more severely than the pressure argument alone predicts — because the fill is not merely densifying, it is also being carried away by water entering the gap that opens each winter.

That second finding is worth carrying because it is the shape of most in-service surprises. The mechanism was right and a second one was operating alongside it, and the second was not in any calculation because it is a hydraulic problem rather than a structural one.

Where the model stops #

The ratcheting expression is empirical. Its form comes from a physical argument and its coefficients from model tests and a limited number of instrumented structures. Extrapolating it to a hundred and twenty cycles is extrapolating well past the data.

The saturating approach to $K^*$ is a chosen shape. The end state is calibrated; how quickly it is approached is much less well established, and it matters for assessment of an existing bridge.

The daily cycle is ignored. The analysis counts annual cycles. Daily cycles are smaller and there are four hundred times as many of them, and how much they contribute to densification is not settled.

And the soil is treated as one material with one density. A real backfill is layered, variably compacted, and drains differently at different depths; the ratchet acts on the top of it hardest, which is where the pressure has the least lever arm.

Where the ladder goes #

Later rungs on this anchor: the run-on slab and the settlement trough behind an abutment. Compressible inclusions and their long-term behaviour. Semi-integral arrangements, where the deck is jointless and the abutment is not. Integral bridges at skew, where the deck’s movement is at an angle to the abutment and the ratchet acts unevenly along it. Pile flexibility as a design variable, and the pile orientation that follows. Instrumented bridges and what forty years of measurement has shown. Assessment of existing integral structures, where the current $K$ is unknown. And the general question: what to do about a load case that is created by the structure’s own life rather than by anything applied to it.