The arch that waits with its hinges open

The jacks the arch forgets took a 60 m fixed concrete rib, rising 6 m under 60 kN/m, closed it at 28 days with its abutments jacked apart to take up the day’s gap, and followed what happened next. Three movements went on opening the gap after closure — the rib’s creep under its own thrust, its shrinkage, and the ground’s consolidation under the thrust — and the rib, creeping itself, let the jacks’ correction relax while the load’s own shortening moment stayed. A hundred years on, the springing carried −173 kN·m, half as much again as the fixed rib’s whole shortening moment of −112. The best single preset left ±135, and a second jacking did best at about a year and still left 119.

Every arch in that essay became fixed on the day of closure. Some concrete arches are not. They are struck as three-hinged arches, with hinges at both springings and the crown, and left that way for months or years while the fast part of every movement happens, and only then are the hinges concreted shut. That essay ended on whether such an arch carries less over its life than one jacked at closure and again at a year — and whether the hinges’ own rotation in the meantime puts a shape into the rib that the closed arch then has to live with.

Three hinges cost nothing to move #

A three-hinged arch is statically determinate: the arch that is only its three hinges found its thrust by one moment equation about the crown, with no stiffness in it anywhere. That has a consequence for every movement the fixed arch had to restrain. If the rib shortens, the two halves rotate about their springings and the crown comes down; if the abutments spread, the same. The thrust adjusts by the ratio of the old rise to the new, which is a fraction of a per cent, and no moment appears, because nothing resists the change of length.

So the three movements are taken for free while the hinges are open. The question is only how much of each happens before the hinges are closed, and what is left after.

The two solid and dashed curves always add to the same 48.9 mm — the whole of the creep, shrinkage and consolidation that open the gap after striking — and the date only decides who takes it. At three months the open arch has absorbed 18.4 mm and leaves 30.6. At a year it has absorbed 29.9 and leaves 19.1. At ten years it has absorbed 45.9 and leaves only 3.0. Concrete’s creep and shrinkage both run fastest early, so the first year takes three-fifths of the century’s movement; the consolidation is the slow one, and most of what is left after a year is ground.

The third curve is the cost. The crown drops 48 mm in three months, 78 mm in a year and 120 mm in ten — two and a half times the gap, because a shallow arch turns a small change in the length of its halves into a large change in its rise. That drop is where the open arch’s price is paid, and it comes back below.

What the closed arch restrains #

Closing the hinges makes the arch fixed, and from that day it behaves exactly as the arch of the earlier essay did, with two differences: it starts with no gap to correct, and its concrete is older. The movements left after closure are restrained by a rib that is now a year old, whose creep is slower than a 28-day rib’s, so the restraint it develops relaxes less. Both effects go into the same step-by-step creep integral, now begun on the closure date.

The dotted curve is the earlier essay’s: fixed and jacked at striking, rising steadily to −173. The dashed one jacks it back to nothing at a year; within a few months it is at −50, and it ends at −117. The solid curve is the arch left open for a year. It starts at nothing on its closure date, like the rejacked one, and ends at −84 — a third less than the rejacked arch, and half the arch fixed at striking. Left open for three years it ends at −48.

The rejacked arch and the arch closed at a year both start their second life at no moment, with the same movements still to come. The difference between them is the memory of the first year. The fixed arch spent that year resisting the gap, and its resistance is a creep history: the restraint it built up and the jacks’ first correction both go on relaxing after the second jacking, and that relaxation is a moment at the springing that the open arch never acquires. A jack can set the moment on a date. It cannot reach back and remove the history that the moment was the sum of, because the creep that follows depends on how the stress got there and not only on what it is.

Closing beats jacking at every date #

The comparison holds for any date, and the next figure draws it as one.

At three months, closing gives −124 kN·m and jacking again gives −140. At a year, −84 and −117. At ten years, −15 and −76: the arch closed at ten years carries almost nothing, while the arch jacked at ten years still carries two-thirds of what the arch jacked at a year ends with, because a decade of early restraint is still relaxing. The solid line falls faster than the dashed one at every date, and the gap between them widens as the date moves out, which is the memory getting longer.

This is the same result the arm connected late found for an outrigger, put in the form of a choice of structural system rather than a choice of connection date: a structure that does not exist yet cannot be loaded by a movement. And it is the converse of the arm that is reset by hand, whose re-shimming, however often repeated, ended a little above the arm connected late, for the same reason — each reset starts from a history.

The shape the open hinges leave #

The open arch’s advantage is paid for in geometry. While the hinges are open each half turns about its springing as the rib shortens and the ground spreads, and the crown drops. The three hinges are still on the funicular of the dead load — a determinate arch’s thrust line always passes through its hinges — but the rib between them is no longer on it: each half is the original parabola, shortened and turned, while the funicular through the three displaced hinges is a slightly flatter parabola.

The mismatch is a moment, zero at the hinges and largest near the quarter points: −50 kN·m after three months open, −82 after a year, −114 after four. It is not a restraint moment — the arch is still determinate when it appears — but a load moment, the thrust acting at an eccentricity the rib has acquired. When the hinges are closed it is locked in, and creep does not relax it, because a load moment in a homogeneous indeterminate structure does not change as the structure creeps. The open arch trades a moment at its springings that grows after closure for a moment at its quarter points that grows before it.

Its size has a short derivation. Turning a half by a small angle θ about its springing drops the crown by $\delta = \theta L/2$, and the rotated half sits below the new funicular by $\delta \cdot (2x/L)(1 - 2x/L)$, which is largest at the quarter point, $\delta/4$. The thrust times that eccentricity is $H\delta/4$: with $H = 4{,}500$ kN and $\delta = 78$ mm, about 88 kN·m, against the 82 computed on the deformed geometry, where the rib’s shortening along its own length takes a few kN·m off.

Without a camber, wait about a year #

Both moments belong to one rib, and the figure that decides a closure date has to add them along its length.

With nothing done about the drop, the largest moment along the rib falls from 124 kN·m for closure at three months to 90 for closure at a year, and then rises again — to 126 at ten years, the locked-in shape having overtaken the restraint it saved. The best closure is about a year, where the springing moment still to come and the quarter-point moment already made are about equal, and it is half the arch fixed at striking.

A camber changes the answer completely. If the falsework sets the rib high by the drop the open period will produce, so that the shortened and turned halves arrive on the funicular at closure, the shape term vanishes and the largest moment falls with every year waited — to 15 kN·m at ten years. That is the same idea as a falsework set high for the elastic drop at striking, which is always done and is excluded from everything here. But a camber for the open period’s drop is a prediction of a creep, a shrinkage and a consolidation, and those are known to about a third. Cambered for 30 per cent less than happens, the arch closed at ten years ends at 37, the unpredicted part of the drop locked in as shape. Waiting longer is then only worth what the prediction is worth.

The ground decides how much waiting buys #

The earlier essay found that the ground’s consolidation had a clock of its own, and that clock decides how much a wait can absorb.

On ground that consolidates in days, closing at a year leaves −60 kN·m: the year has absorbed the whole of the ground’s movement and most of the rib’s. On ground still consolidating after a century it leaves −93, because most of the consolidation is still to come when the hinges close, and a year-old rib relaxes less of it than a young one would. The benefit over the rejacked arch holds across the whole range — 36 kN·m on fast ground, 30 on slow — but its share of the fixed arch’s moment does not: on fast ground the open year takes away more than half, on slow ground less than half. A wait absorbs only the movements that finish inside it.

The design consequence is that a long open period is wasted on soft clay and well spent on rock. On clay the ground will move after any closure date a programme can afford, and the arch’s protection against it is an abutment that spreads before it turns, or a tie, as the thrust that never reaches the ground found — an arch whose thrust is taken internally does not care what its ground does.

A preset at closure, on an older rib #

The arches above are closed with no jack at all. The earlier essay’s best single preset, applied at striking, pushed the abutments apart beyond the day’s gap so that the springing started sagging and ended hogging by the same amount, ±135 kN·m. The same idea is available at closure: the hinges can be locked with the abutments jacked a few millimetres apart, so that the closed arch starts with a sagging moment the movements still to come will carry through zero.

Computed for the open arch, the best preset at a year’s closure is 7 mm, and it leaves the springing at +58 kN·m on the closure day and −57 a century later — ±58 over the whole life, against the ±135 of the best preset at striking and the 98 the earlier essay reached with a preset and a second jacking together. At three months the best preset is 10.5 mm and leaves ±91; at three years, 3.5 mm and ±32. The preset shrinks with the date because less is still to come, and it works better on an older rib for the same reason the unjacked closure did: a preset is an imposed gap, and a year-old rib relaxes its preset less, so less of the jack’s work is lost before the movements arrive to cancel it.

That is the clearest statement of the method’s logic. A jack is a correction to a history; the earlier the jack, the more history follows it and the more of the correction creep takes away. The open hinges postpone the moment at which any correction is needed until most of the history is over, and then a small one, on a rib that holds it, does the rest. The quarter points still carry the open year’s shape, which a preset at the springings does nothing for, so the camber remains the other half of the method.

By hand, for closure at a year #

In the first year after striking, the rib creeps under its thrust by $\delta_{el}\,\varphi(t, t_0)$: the day’s elastic gap of 13.6 mm times a creep coefficient of 1.30 at a year for 28-day loading, 17.7 mm. It shrinks by 9.6 mm over its 61.6 m length, and the ground consolidates 2.6 mm. Together, 29.9 mm of gap absorbed.

The crown drop that buys: the 29.9 mm of horizontal shortening turns each half by $\theta \approx g/(2f) = 0.0299/12 = 2.5$ mrad, which drops the crown by $gL/4f = 0.0299 \times 60/24 = 75$ mm; the rib’s shortening along its own height takes the crown down another $\varepsilon f = 0.00045 \times 6 = 3$ mm. The crown hinge has turned through 5 mrad in all, which a concrete hinge of the Freyssinet type takes without difficulty.

What is left after the year is 19.1 mm of gap — 0.9 years’ worth of creep from a one-year-old rib, the shrinkage still to come, and three-quarters of the consolidation. Fixed and unrelaxed, 19.1 mm would put $19.1 \times 8.2 = 157$ kN·m into the springing (the fixed rib takes 8.2 kN·m per millimetre of gap). The year-old rib relaxes it to −84, a factor of 0.54. The 28-day rib of the earlier essay relaxed the same kind of gap by more, but it had all 48.9 mm to relax.

Why a hinge is worth more than a jack #

The two methods do the same thing to the same rib on the same date: they set the springing moment to nothing. The jack does it by adding a correction to a stressed structure, and a correction is a new increment of restraint that relaxes on its own clock while the old increments go on relaxing on theirs. The hinge does it by never having had the stress. In a creeping material those are different states with the same moment, which is the whole difficulty that the strain that was imposed found in relaxation: the stress is the sum of a history, and only a history that was never written is free of it.

That makes the three-hinged erection a structural version of a familiar rule in prestressed concrete — that the losses after transfer depend on the age at which the load arrived — applied to the structural system. The joint poured twice over the pier found the same arithmetic in a bridge deck made continuous in two stages: the continuity made later restrained only what came after it, and the first stage’s own history was its own account.

Where the calculation stops #

The hinges are perfect. A real concrete hinge — a Mesnager or Freyssinet throat, a narrowed section with crossed bars — has a small rotational stiffness, and it carries a little moment while open; a steel pin has friction. Either puts a fraction of the restraint into the open period, which a few per cent of a fixed rib’s rotational stiffness would make a few per cent of the fixed arch’s moment.

The drop is treated as a rigid turn of shortened halves. The rib’s own creep curvature under the locked-in shape, after closure, is ignored, as is the change in the thrust’s second-order effect as the rise falls by 78 mm out of 6,000. Both are small at this rise; on a flatter arch, the arch that gets shorter showed how quickly the rise ratio amplifies everything.

The movements are the earlier essay’s, unchanged. EN 1992-1-1’s creep and shrinkage for a 350 mm notional size at 70 per cent humidity, a Terzaghi consolidation curve, and a rib loaded only by its own funicular dead load. Temperature is left out of both arches here, and it is where the permanent three-hinged arch below would differ most: a fixed arch restrains a season’s swing, and an open one turns at its hinges.

Still open: the hinge that is never closed #

Every arch here is closed in the end, and the comparison is between dates. Some arches never are: a three-hinged concrete arch can be left determinate for its whole life, carrying its shortening, its shrinkage, its ground and its temperature without moment, and paying instead in a crown that drops for decades and in a hinge that must carry the thrust, the shear and a century of rotation. Whether the permanent three-hinged arch, cambered for its long-term drop and exposed to every season’s temperature swing at its hinges, is lighter over its life than the fixed arch closed at a year — whose quarter points carry the year’s shape and whose springings carry what came after — or whether the hinges’ own durability is what decides between them, is the question an arch that waits forever puts to the one that waits a year.