The jack that creep carries past the mark

The beam built towards its own middle took a five-span prestressed beam of 16 m spans, its tendon draped from 0.30 m above the centroid at the supports to 0.32 m below it at mid-span, and built it in different orders. Stressed all at once, the tendon’s curvature on the whole continuous beam puts 344 kN·m of secondary moment at the first interior supports and 258 at the middle ones. Built from both ends towards a closure bay, each half is stressed while it is a shorter beam, and the finished beam keeps 161 and 172. Creep then moves a structure whose statical system changed during construction towards the state it would have had built at once, and with a creep coefficient of 2 it gave back most of the difference — but only most.

That essay ended on a practice bridge builders use to correct for exactly this. Before the closure bay is cast, the two half-beams can be lifted or lowered on jacks at their supports, so that the closure locks in a chosen moment rather than the one the tendons happen to produce. The question was whether a jacked closure can give the finished beam its at-once moments on the first day — and whether creep then carries it past them, the way the jacks the arch forgets were partly taken back.

What the order took #

The deficit — what building in stages took from the at-once moments — is 183 kN·m at the outer interior supports and 86 at the inner ones. Both are sagging: the staged beam carries less sagging secondary moment everywhere, and most less where the tendon’s curvature was first imposed on short pieces.

Creep closes 77 per cent of that deficit, and the number is exact rather than empirical. By the age-adjusted effective modulus, a structure whose statical system changes after loading moves from its staged moments towards its at-once ones as $\rho = \varphi/(1 + \chi\varphi)$; with φ = 2 and the ageing coefficient χ = 0.8, ρ = 2/2.6 = 0.769. The first interior support ends at 302, still 42 short of 344.

What the at-once moments are for #

The at-once moments are not sacred; they are simply the ones a designer computes. A continuous prestressed beam is designed with its secondary moments found from the tendon’s curvature on the finished structure — the calculation the prestress that pushes back set out — and the support sections are proportioned so that the load’s hogging, less the secondary relief, stays inside the section’s limits. A beam built in stages has less relief at its supports than that calculation credited it with, so its support sections carry more net hogging than they were designed for, and its spans carry correspondingly less sag.

The relief also arrives as shear. The relief that belongs to the pressure line found that a secondary moment’s diagram is straight between supports, so it brings a constant secondary shear into each span equal to the difference of its end moments over the span; and the beam stressed one bay at a time found that staging changed that shear most in the end spans, where the abutment’s moment is zero and the first interior support’s is what staging cut. A jacked closure therefore restores two things at once — the support section’s net moment and the end span’s shear — and the overshoot carries both past their designed values together.

A jacked moment is an imposed deformation #

Lowering a support of the finished continuous beam is a settlement, and a settlement imposes moments by the three-moment equation. For a beam of n spans there are n − 1 interior supports and n − 1 interior support moments, so the equation is square: a jack at every interior support can impose any set of support moments, including exactly the deficit. The beam can be made to start its life at the at-once moments.

But the moment a jack imposes is a moment from an imposed deformation, and a concrete beam relaxes those. Under the same age-adjusted modulus, a moment imposed at closure decays to $J/(1 + \chi\varphi)$ — at φ = 2, to 38 per cent of itself. The staged moments, meanwhile, go on creeping towards the at-once ones whatever has been jacked. So the moment at any support is

$$M(\varphi) = M_{st} + d\,\frac{\varphi}{1+\chi\varphi} + \frac{J}{1+\chi\varphi},$$

with d the deficit and J the jacked moment. The two creep terms act on different things. One is the system change still working itself out; the other is the jacked correction relaxing. They would cancel exactly if χ were 1, and it never is.

Jacked by the whole deficit, the first support starts at 344 kN·m, exactly the at-once moment, and rises to 372 by the time creep has reached 2. Creep carries it past the mark by 28 kN·m, a seventh of the jack. The overshoot is $d\,\varphi(1 - \chi)/(1 + \chi\varphi)$, which at φ = 2 and χ = 0.8 is 0.154 d: the jacked correction relaxes by $1 - 1/(1+\chi\varphi) = 0.615$ of itself while the system change returns 0.769 of the deficit, and the difference between those two fractions is what the beam ends with beyond its target.

Set for the end instead — jacked by $1 + \chi\varphi - \varphi = 0.60$ of the deficit — the support arrives at 344 exactly when creep reaches 2, and starts at 271, 73 short. Between those two settings lies a balanced one.

The jack that splits the difference #

The day’s distance falls linearly as the jack grows, from −183 not jacked to nothing at the full deficit; the end’s falls more slowly, because only 38 per cent of each unit jacked survives creep, and crosses zero at 0.60. The larger of the two is least where they are equal and opposite, at a share of

$$\frac{2 - \rho}{1 + 1/(1 + \chi\varphi)} = \frac{1.231}{1.385} = 0.89,$$

leaving the support 21 kN·m short on the day and 21 over at the end — a ninth of the deficit either way, against −183 and −42 for the beam not jacked at all. No setting of a jack applied once puts the beam at its at-once moments for its whole life, because the jack’s moment and the order’s deficit age at different rates. A jack applied once is a single number set against two clocks.

This is the same structure as the arch’s in the earlier essay, where jacking the springings apart at closure to take up the day’s gap left the springing moment 173 kN·m at a century, because the jacks’ correction relaxed while the load’s own shortening did not. In the arch the overshoot was the whole of the relaxed correction; in the staged beam most of it is paid back by the system change still in progress, and what is left is the difference between two creep fractions.

Why the two clocks run at different rates #

The ageing coefficient is the whole of the overshoot, so it is worth seeing what it measures. When a stress in concrete changes gradually rather than at once — as the restraint stress in a structure whose system has changed builds up while it creeps — the concrete that carries each increment has already aged by the time it receives it, and creeps less under it than it would under a stress applied on the first day. χ is the average discount: 1 for a stress applied all at once, about 0.8 for one that develops over the creep’s own course, as the strain that was imposed found when it compared the age-adjusted shortcut with the full superposition integral.

The jacked moment is a stress that arrives all at once at closure and then decays, and the system change is one that develops slowly over the same years. Both pass through the same ageing concrete, and the age-adjusted modulus gives the first a relaxation of $\chi\varphi/(1+\chi\varphi)$ of itself, 0.615, and the second a recovery of $\varphi/(1+\chi\varphi)$ of the deficit, 0.769. The two fractions differ by exactly the factor χ: the jack loses a little less of itself than the system change delivers, and the difference is left in the beam as the overshoot. If concrete did not age, χ would be 1, both would be the same fraction of the same thing, and a jack set for the day would be right for ever. The overshoot is what concrete’s ageing costs a correction applied early.

How well the creep is known #

The balanced share depends on φ and χ, and a creep coefficient is a prediction. EN 1992-1-1’s coefficient for a given concrete, humidity and member size is a mean, and real creep scatters about it by a fifth to a third.

The jacked settings are far less sensitive to the creep than the unjacked beam is. Not jacked, the first support ends 66 kN·m short if creep reaches only 1.3, and 27 short if it reaches 2.7 — the whole deficit is being returned by creep, so the result depends on how much creep there is. Jacked for the end, it ends within 13 either way; balanced, within 25. Most of the deficit was put back by the jacks, and the jacks’ share does not depend on creep at all; only the correction’s relaxation and the remaining system change do.

That is the argument for jacking that does not depend on getting the creep right. A beam that relies on creep to recover its order’s deficit carries an uncertainty proportional to the deficit; a beam that has jacked most of the deficit back carries an uncertainty proportional to what creep still has to do.

Where the jacks go, and how far #

Every interior support goes down, the inner pair further than the outer: 54 mm at the first interior supports and 78 at the middle ones, with the beam’s bending stiffness taken as $1{,}000\ \mathrm{MN\,m^2}$. The displacements are large because the deficit is a sag at every support at once, and lowering all four interior supports together is nearly a rigid movement of the beam — the moments come only from the difference between the supports’ displacements and the abutments’. Only the supports’ levels relative to one another matter, so the same correction can be made with the abutments raised and the interior supports lowered by less, but not with the abutments alone.

That has a practical consequence. Seventy-eight millimetres on a 16 m span is a gradient of 1 in 200 at the abutments, which a bearing has to accept and a deck profile has to absorb, and the beam is then left permanently out of its cast level by that amount at its supports — the vertical profile is part of the correction, and the road surface has to be regulated to it.

One pair of jacks is not enough #

The deficit has a shape — 183, 86, 86, 183 — and a set of jacks reproduces it only if there are as many jacks as there are interior supports. With fewer, the best that can be done is a compromise, and the compromise is worse than it looks.

Lowering only the outer interior supports — where the deficit is largest, and where a designer would be tempted to put the only pair of jacks — puts 93 kN·m there against the 163 wanted, and hogs the inner supports by 51. They wanted 76 more sag and got 51 less: further from their at-once moments than if nothing had been jacked. Jacking the inner pair does worse at both. Lifting both abutments together, the cheapest scheme of all, gets the outer supports nearly right, 142 against 163, and still hogs the inner ones by 28.

The reason is in the three-moment equation. A single support’s settlement puts a sagging moment at that support and hogging moments at its neighbours, decaying along the beam — the same pattern the support that moved found for an unplanned settlement. The deficit sags every support at once, and no combination of fewer settlements than supports can sag every support: each one jacked pays for its own sag with its neighbours’.

No order lets fewer jacks do #

The construction order sets the deficit, so it is natural to ask whether some order leaves a deficit that fewer jacks can correct. Built from the middle outwards, the earlier essay’s other symmetric order, the beam keeps 215 kN·m at its outer interior supports and 121 at its inner ones; the deficit is 129 and 138, flatter than the closure order’s 183 and 86. Built from one end, it is 184, 83, 98 and 125, lopsided because the beam is.

In every case the deficit is a sag at every interior support, because every order imposes the tendon’s curvature on a beam with fewer supports than the finished one, and fewer supports always means less of the sagging reaction. So in every case a pair of jacks or a lift at both abutments hogs the supports it does not reach. The middle-out beam’s flatter deficit is, if anything, harder to correct: its balanced jacks lower the interior supports by 56 and 87 mm, a little more than the closure order’s 54 and 78, because a uniform sag along the beam is closer to the rigid movement that produces no moment at all. The order chooses how much the jacks must put back; it never chooses how many jacks there must be.

The jacked closure in two lines #

At the first interior support the deficit is $d = 344 - 161 = 183$ kN·m. With φ = 2 and χ = 0.8, $\rho = 2/2.6 = 0.769$ and $1/(1 + \chi\varphi) = 0.385$.

Not jacked, the support ends at $161 + 0.769 \times 183 = 302$. Jacked by the whole deficit, it ends at $161 + 0.769 \times 183 + 0.385 \times 183 = 161 + 141 + 70 = 372$, which is $344 + 0.154 \times 183 = 344 + 28$. Jacked by 0.60 of it: $161 + 141 + 0.385 \times 110 = 344$, and on the day $161 + 110 = 271$. Jacked by 0.889: on the day $161 + 163 = 324$, and at the end $161 + 141 + 0.385 \times 163 = 365$; 344 − 324 = 20 and 365 − 344 = 21.

For the jacks’ displacements: a single interior support of a long continuous beam of equal spans lowered by δ takes a sagging moment of roughly $3EI\delta/L^2$ if its neighbours stay put — $3 \times 10^6 \times 0.001/256 = 11.7$ kN·m per millimetre. Lowered together, the four supports’ moments mostly cancel each other’s hogging, which is why each needs several times the millimetres that one alone would.

What one ageing coefficient and no self-weight leave out #

The creep law is the age-adjusted modulus with a single χ. A step-by-step integral, as the arch essays used, gives a slightly different fraction for an imposed deformation applied at closure than for a system change, because the two load histories are different shapes in time; χ = 0.8 is a good average for both and exact for neither. The structure of the result — two fractions that do not cancel — does not depend on the value.

Self-weight is left out. The beam’s own weight was cast in stages too, and it has its own staged and at-once moments, its own deficit, and its own creep. A real closure jack is set for the sum, and the shape of the self-weight deficit is not the tendon’s.

The jacks are released into a hardened closure. The beam is lowered while the closure is wet and held there until it has hardened, and then the jacks are removed with the supports at their new levels; the moment is applied at the closure’s age, not at the age of the half-beams, and the half-beams’ older concrete relaxes it less than the young closure would. The joint poured twice over the pier found the same difference of ages inside one continuity joint; here it would make the balanced share a little lower.

Still open: the beam that is jacked twice #

Every jack here is set once, at closure. But a jack is also a measurement: a jack holding a support at a known level reads the reaction there, and a reaction is the support moments in another form. A beam whose closure jacks are left in place for its first year could be read as it creeps, and reset — a second correction, made when much of the creep has happened and the remaining uncertainty is smaller. The arch that waits with its hinges open found that an arch closed late carries less than one jacked again on the same date, because the jacked arch remembers its first year. Whether a beam whose jacks are reset at a year, from the reactions they have read, can be brought within a few kN·m of its at-once moments for the rest of its life — or whether the first year’s jacked correction, still relaxing, carries the second past its mark the way creep carried the first — is the question a jack that measures puts to a jack that only pushes.