The jacks the arch forgets

The abutment that spreads before it turns found that a fixed concrete arch’s worst enemy is its own footings. The 60 m rib, rising 6 m under 60 kN/m, shortens by 13.6 mm under its thrust, and the secondary moments that shortening causes are doubled by a footing that spreads 7 mm a side; on dense sand it spreads 8. Its last section named the obvious remedy and the reason to doubt it. An arch whose abutments are jacked outward before the crown is closed starts its life with the spread already taken. But the rib’s shortening and the ground’s spread are both time-dependent, and the essay asked whether a jack set right on the day of closure is still right a decade later, or whether the two movements “drift apart at different rates.”

They drift apart, and there are three of them, not two. The surprise is that the worst of them is the jacks themselves.

Three gaps that open after the crown is closed #

Take the arch closed at 28 days, its abutments jacked apart by exactly the gap open on that day: the 13.6 mm the thrust shortens the rib and the 8 mm the footings spread under it at once. On the day of closure the rib fits its springings with no moment at all. Then three things go on.

The rib creeps under its own thrust. EN 1992-1-1’s creep coefficient for this rib — a section 350 mm in notional size, outdoors at 70 per cent humidity, loaded at 28 days — reaches 1.82 in a century, so the 13.6 mm becomes 24.8 mm more.

The rib shrinks. EN 1992-1-1’s drying and autogenous shrinkage after 28 days total 263 millionths of its 61.6 m length: 16.2 mm, more than the whole of the thrust’s elastic shortening, and the part of the arch’s life the jacks were never asked about.

The ground consolidates. The footings’ 8 mm under the thrust was the immediate part; a clay under a sustained load goes on settling, here by as much again, 90 per cent of it in ten years.

That is 48.9 mm of gap opened after the day the jacks closed it, three and a half times the shortening they were set for. The rib does not resist all of it. It creeps as it resists, and the effective gap — the one the fixed arch acts as if held against — is 20.9 mm. Even that is half as much again as the 13.6.

The moment that does not relax, and the one that does #

The springing moment is what the gaps cost.

Not jacked, the arch starts at −179 kN·m at the springings — the 112 of the fixed rib’s shortening and the rest from its footings’ first spread — and ends at −222. Jacked to the day’s gap, it starts at nothing and ends at −173. The jacks have saved 49 kN·m at the end of the arch’s life, against the 179 they removed on the first day.

The reason is a theorem about creep that has nothing to do with arches. A load’s internal forces in a structure that does not change its supports are not altered by creep at all, because creep multiplies every deformation by the same factor and compatibility is undisturbed. That is Dischinger’s result, and it covers the thrust’s own shortening moment. Left alone, the fixed rib would carry its 112 kN·m for ever, creep or no creep.

An imposed deformation is different, as the strain that was imposed found in a member held at a fixed length: a moment forced on the rib by a displacement relaxes as the rib creeps, because the rib keeps flowing towards a shape in which it would not need the moment. The jacks are an imposed displacement. They put a correction into the arch on the day of closure that exactly cancels the load’s moment, and from that day the correction relaxes while the load’s moment does not.

Taken apart, the century’s −173 kN·m is −81 from the rib forgetting its jacks, −59 from shrinkage after closure and −32 from the ground. The largest part is not any movement of the rib or the ground. It is the correction relaxing: the arch remembers its load and forgets the jacks.

How fast it forgets depends on the creep law. EN 1992-1-1’s coefficient grows as the 0.3 power of the time under load, so it does an eighth of its century’s creep in the first day and nearly a third in the first fortnight; a displacement imposed at 28 days has fallen to 28 per cent of its first value after a century. That is faster than the age-adjusted modulus’s familiar factor of 0.8 would say, because that factor was fitted to a creep law that starts more slowly. It is checked here against a standard linear solid, whose relaxation is known exactly.

The shrinkage nobody jacked for #

Of the three movements, shrinkage is the one a closure calculation is least likely to contain, because on the day of closure there is nothing to see. EN 1992-1-1 splits it into two parts. The autogenous part, the paste consuming its own water as it hydrates, is 50 millionths in all for this concrete, and nearly all of it has happened by 28 days. The drying part is the concrete losing water to the air, and its size and pace are set by the member: in this rib’s 350 mm notional size, outdoors at 70 per cent humidity, it reaches 267 millionths, and its time constant, $0.04\sqrt{h_0^3}$, is 262 days. So the drying starts at closure and runs for years.

After closure the rib has 14 per cent of its shrinkage still to do in the first five weeks, 59 per cent within the first year, 88 per cent within five and 97 per cent within twenty. Those are the years in which the shrinkage moment in the split builds up. They are also, by the arithmetic of creep, the years in which the rib is still young enough to relax a good share of what each increment causes — which is why 16.2 mm of shrinkage costs 59 kN·m and not the 134 a fixed elastic rib would give it.

None of that is a defect of the jacking. A rib closed without jacks shrinks by the same 16.2 mm and pays the same 59 kN·m. What jacking changes is the expectation. A designer who has removed the shortening moment by jacking has a reason to think the springings are free of secondary moment, and the shrinkage arrives as the largest single movement on a structure believed to be finished with them.

The crown sags while the springings hog #

The springings are where the arch’s secondary moments are largest, but the crown carries them too, with the opposite sign. A fixed rib that has lost thrust sags in the middle and hogs at its ends, and on this arch every millimetre of effective gap puts 4.3 kN·m of sagging into the crown against 8.3 of hogging at the springings.

So the jacked arch’s crown goes from nothing to 89 kN·m of sagging over its century, and the unjacked arch’s from 92 to 114. The arch with the best single preset carries 70 kN·m of hogging at its crown on the day of closure and 70 of sagging a century later. The crown is the section that carries the arch’s live-load bending as well, so the secondary moment there is not a separate check: it shifts where in the crown’s interaction diagram the live load starts from, and it shifts it in opposite directions at the two ends of the arch’s life.

That is the practical sting of a preset. An arch jacked beyond its gap is not merely carrying a moment it did not need; it is carrying one of the wrong sign for the first months, at the crown as well as at the springings, while the concrete is youngest and its strength furthest from its design value. Loaded twice over before it is a month old made the point about floors: the load that arrives early meets the weakest concrete. A preset is a load that is made to arrive early.

A setting is one number; the movements are three clocks #

If the jacks’ correction relaxes, the obvious answer is to jack further — to set the abutments apart by more than the day’s gap, so that the correction is still big enough when the rib has crept and shrunk and the ground has settled.

Every millimetre of preset puts 8.3 kN·m of opposite moment into the springings on the day. Most of it relaxes, and only a little survives to meet the movements that come later. Because the moment on the first day rises one for one with the preset while the moment a century later falls much more slowly, the largest moment over the arch’s life is least where the two are equal and opposite: at 16.3 mm of preset, with +135 kN·m on the day and −135 a century later.

So the best single setting does not cancel anything. It trades a moment the arch carries at the start for a moment it carries at the end, and both are larger than the 112 the fixed rib would have carried had nobody touched it. A jack is one displacement at one age. The movements it is meant to cancel are three histories with three time constants, and they keep arriving after the jack has done its one thing.

Jacking again #

A second jacking — the abutments pushed apart again some years after closure, back to no moment — cancels whatever has accumulated by then, and leaves only what comes after.

The timing matters both ways. Jacked again in the first months, most of the creep and nearly all of the shrinkage and consolidation are still to come, and the moment after the second jacking grows almost as large as without it. Jacked again after ten years, the moment has already reached most of its peak before the jacks arrive. The best date is about a year: the largest moment over the life is then 119 kN·m. Combined with an 11.5 mm preset at the first closure and a second jacking at about a year and a half, it is 98.

Neither reaches zero, and nothing short of jacking for ever would. What happens after the last jacking is never cancelled, and on a creeping rib under drying, it is always something. Flat jacks left in the crown or the springings, so that they can be pumped again, are the remedy this calculation points to: the only one that treats a slow movement with a slow correction.

Ground that settles in days and ground that settles in decades #

The ground’s share depends on when it moves, as well as how far.

The ground consolidating by the same 8 mm costs 19 kN·m if it is done within days of closure and up to 37 if it takes decades. Consolidation that finishes early acts on a young rib, whose creep is at its fastest, and the rib relaxes most of what it causes. Consolidation that has hardly started by the end of the century has not yet arrived. In between, the ground’s movement lands on a rib that has done most of its creeping and can no longer relax it — and that is a sand under a quickly loaded footing at one end of the range and a soft clay at the other. A sand footing’s spread is nearly all immediate, which the jacks take; a clay’s arrives later and costs more for every millimetre.

This is the same lesson the backprop jacked tight lands on the floor below found in a building under construction: a force placed by jacking goes wherever the creep of the young concrete sends it, and the timing of the jacking against the timing of the creep decides where it ends up.

Why the arch is a worse case than a beam #

A beam on its supports does not care about any of this. A simply supported span shrinks, creeps and settles, and its moments do not change, because nothing restrains its length. The arch that gets shorter and the springings that make shortening worse showed that an arch is the structure in which length is everything: a fixed arch’s moments are proportional to the change of its length against its span, at about 8 kN·m per millimetre on this rib. Every slow movement is a length change, and every one becomes a moment at that rate.

That makes the arch the cleanest case of a general fact about concrete structures with redundants. The support that had no moment when it was cast found that a beam made continuous after it was cast picks up a support moment as it creeps, because the structure it creeps in is not the one it was loaded in. A jacked arch is the same thing done on purpose: a structure whose redundant is set by hand at one age, then left to creep in a system that remembers its load and forgets the hand.

What the calculation leaves out #

The ground is a prescribed history. Its consolidation under the thrust is taken as a known displacement, its stiffness against changes of thrust is ignored, and it neither creeps further under the larger moments it causes nor relaxes. A real footing’s spread depends on the thrust it carries, and the thrust here changes by only a few per cent, so the error is small.

The rib’s modulus does not grow. Young concrete stiffens over its first months, which slightly raises the moments an early movement causes and slightly lowers their relaxation. The creep law’s own ageing — later loading creeps less — is in the calculation; the modulus’s is not.

Temperature is left out. The summer that is worse than the last is about a bridge with no joints in a seasonal climate, and an arch is the same: its length changes every day and every year, by more than any of the movements here on the hottest and coldest days. Those are cyclic and largely elastic, and they add to these as a band either side of a drifting middle.

The creep and shrinkage are EN 1992-1-1’s means. EN 1992-1-1 states its creep coefficient and shrinkage strain as mean values with a scatter of the order of a fifth either way, and the result here depends on both twice over: once in the movements, once in the relaxation. A rib that creeps more than the mean loses more of its jacks and also relaxes more of its shrinkage; one that shrinks more pays more at every age. The split into −81, −59 and −32 is therefore a statement of proportion more than of size, and the conclusion that no single setting cancels all three does not depend on the scatter at all, since it follows from their different clocks.

The jacking is an exact displacement. Real flat jacks are pumped to a force, not a displacement, and the gap they leave is whatever the force produces in a rib and ground of uncertain stiffness. A displacement-controlled closure is the favourable case.

Still open: the arch that is never closed #

Every arch here becomes fixed on the day of closure and stays fixed. Some concrete arches are built three-hinged and left with their hinges open for months or years, so that the rib’s early creep and shrinkage and the ground’s first consolidation happen in a determinate structure that takes them without moment — and are closed, if at all, only when the fast part of every movement is over. Whether an arch closed a year after it is cast 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, is the question an arch built to wait asks of every jack above.