The support that had no moment when it was cast

A structure built in pieces carries whatever was present at the moment each piece became structural, and a section that changed while it was being loaded has no single section modulus to divide by. Both of those are elastic arguments: a stress arrives, it is locked into a state, and the state is the answer.

Concrete does not leave a state alone. Everything locked in at the moment a connection is made goes on creeping afterwards, and the connection is there to stop it.

Why a connection acquires a moment nobody applied #

Take the joint apart and watch what it is preventing.

Two simple spans under a sustained load rotate at their ends. The rotation has an elastic part, which arrives when the load does, and a creep part, which goes on growing for years. If the two ends are separate, the rotation is free and no force is needed.

Stitch them at 60 days. The elastic rotation and the creep of the first month have already happened, and the joint takes no notice of them — a connection cannot undo a movement made before it existed. What it can do is prevent the remaining rotation, and the remaining rotation is everything the creep has left to deliver.

So from 60 days onward the compatibility condition is that the relative rotation at the joint does not change. The load’s creep goes on asking for one; a hogging moment at the joint is what refuses it. The moment therefore grows from nothing, and it grows on exactly the schedule the creep does.

That is the mechanism, and it is the force method — release a redundant, compute the movement at the release, and choose the force that closes it — with the movement supplied by time rather than by load.

The arithmetic is a relaxation problem wearing a different hat #

The equation is short enough to write down, and writing it down is what makes the numbers in the figures checkable.

Let $\theta_{el}$ be the relative rotation per unit of the applied load on the released structure, and let $\varphi(t, \tau)$ be the creep coefficient of concrete first loaded at $\tau$. From the connection at $t_c$ onward:

$$\theta_{el}\,\big[\varphi(t,t_0) - \varphi(t_c,t_0)\big] \;=\; \sum_j \Delta X_j \, f_{el} \, \big[1 + \varphi(t,\tau_j)\big]$$

The left-hand side is the rotation the load’s continuing creep wants to produce. The right-hand side is the rotation the redundant produces, with each increment of $X$ carrying its own creep from the day it was applied — which is why it is a sum over the history rather than a product.

The two creep coefficients in that equation are different and the difference is the whole subject. On the left is $\Delta\varphi = \varphi(t,t_0) - \varphi(t_c,t_0)$, which is 1.20 for the beams above: the creep the load still has coming. On the right is $\varphi(t,t_c)$, which is 2.06: the creep available to the redundant, larger because the redundant is applied to concrete that is young relative to the rest of its own history.

Dividing through, the redundant cannot reach the monolithic value, because the numerator is smaller than the denominator’s creep. Solved step by step, the beams above reach 46 per cent.

It arrives over twenty years and then stops #

Three routes are drawn there and two of them agree.

The superposition integral is the definition. It is the equation above, discretised, forward-substituted, and it needs nothing but the creep function. Its answer moves by three parts in ten thousand between 160 steps and 2,560, so it is the number the other two are approximating.

The age-adjusted effective modulus replaces the whole history with one stiffness: $X = \Delta M \cdot \Delta\varphi / (1 + \chi\,\varphi(t,t_c))$, with $\chi$ an ageing coefficient quoted as 0.8. It gives 46 per cent, which is the integral’s answer to within a point.

Dischinger’s method of 1937 gives $1 - e^{-\Delta\varphi}$, which is 70 per cent. That is not a small disagreement, and the assumption that produces it is visible in the equation above: Dischinger’s rate-of-creep formulation takes $\varphi(t,\tau)$ to depend only on $t - \tau$ in a way that makes the two creep coefficients interchangeable. They are not interchangeable — 1.20 against 2.06 — and the error is in the direction that matters, because it over-predicts the moment a designer is checking a support for.

A designer using the older method therefore designs the stitch for half as much moment again as it will ever see, which is conservative for the support and unconservative for the midspan, because the two always add to the simple-span moment. Being wrong about a redistribution is being wrong twice in opposite directions, which is why an over-prediction is not automatically safe.

The connection never catches up #

The curve in that figure flattens well below the line at 1, and the gap is not an artefact of the twenty-year cut-off. Run it to a thousand years and it stays there.

The reason is the sentence three sections up. The elastic rotation happened before the joint existed. Of everything the load asks the spans to do, the elastic part — a unit of rotation, against the creep’s eventual 2.4 — was delivered when there was nothing to resist it, and no later mechanism can go back for it. The redundant is competing for the remainder only.

That puts a ceiling on the whole phenomenon which is worth having in advance:

$$\frac{X_\infty}{X_{mono}} \;\approx\; \frac{\Delta\varphi}{1 + \chi\,\varphi(t,t_c)}$$

With $\Delta\varphi$ at its largest — continuity made the instant the load arrives — the ratio is about $\varphi/(1+\chi\varphi)$, which for $\varphi = 2.4$ and $\chi = 0.8$ is 0.82. The stepwise integral gives 76 per cent for that case.

Three-quarters is the ceiling, not the answer. Every real programme is below it, and the distance below it is a date.

What the date is worth #

The curve is steep at the left-hand end and that is the practical content. A month of delay between casting the beams and stitching them is worth thirty points of support moment, and nothing on any drawing records which month it was.

Two consequences follow for a designer rather than for an analyst.

The moment is bounded and the bound is useful. Whatever the programme, the support moment lies between nothing and 76 per cent of the monolithic value, and the midspan between the monolithic value and the simple-span one. Designing the support for the upper bound and the midspan for the simple span covers every programme at once, at a cost of a little steel in two places. That is a lower-bound argument — any distribution in equilibrium with the load, with capacity everywhere — and it is why this phenomenon does not collapse structures.

What it does do is crack them. A support detailed with nominal top steel because the analysis said the moment there was zero will crack over the support at about year three, and the crack will be where the wearing surface is. That is a durability defect produced by a calculation that was correct on the day it was done.

The coefficient that is not a material property #

The age-adjusted shortcut tracked the integral to within a point at every age above, which is a strong result for a one-line formula and is worth being suspicious of.

The relaxation of a restrained strain is the same superposition integral with a different right-hand side, and the essay on restrained shrinkage already records that its implied χ is about 1.33 rather than 0.8. Here the same integral on the same creep function implies 0.79 to 0.90.

One material, two questions, and a factor of 1.5 between the coefficients that make the shortcut work. So χ is not a property of concrete. It is a number fitted to a class of problem, and the 0.8 that every text quotes was fitted to this one — a growing action — which is why it works here and does not work there.

The distinction has a physical reading. χ measures how much of the stress history was applied early enough to have creeped fully. A redundant that grows smoothly from nothing over twenty years has most of its increments applied late, so most of them have creeped little, so the effective stiffness is nearer the elastic one and χ is below 1. A strain imposed all at once and held has its whole stress history opposing a creep that started immediately, and the shortcut needs a χ above 1 to reproduce it.

Quoting χ as a material constant is the error, not quoting 0.8. Used on the problem it was fitted to, 0.8 is excellent; used on the other one it is out by forty per cent in the unsafe direction for cracking.

A change of restraint is a load case with a date on it #

The stitched pair of beams is the textbook case, and the arithmetic is not about stitches. It applies wherever a structure’s restraints change after a load has been sitting on it, which is more often than the word “staged” suggests.

A temporary prop removed. A slab propped back to the floor below carries very little until the props come out. Take them out at six months and the load transfers to a slab that has already crept under the prop reactions; the moments that arrive are not the ones an analysis of the unpropped slab gives, and the difference is this calculation with the sign reversed.

A bearing replaced. A simply supported deck jacked up and set down on new bearings has had its restraint removed and restored, and whatever creep happened in between is locked in.

A settlement that occurs slowly. A support that moves produces moments proportional to the structure’s stiffness, and if it moves over years rather than in an afternoon the structure’s stiffness for that purpose is its effective one. The moments come out at roughly $1/(1+\chi\varphi)$ of the elastic value — a third of them — which is why differential settlement is so much less damaging when it is slow, and it is the same shortcut with a held displacement instead of a growing force.

And an arch or a frame decentred. Falsework struck at 28 days hands the structure its own weight in one act, and everything afterwards is a redundant arriving on a schedule.

What links them is worth stating as the rule, because it is more portable than any of the cases: an elastic analysis assumes the load and the structure arrived together, and where they did not, the answer depends on a creep function rather than only on a stiffness. The question to ask of any staged structure is not what its moments are but what its moments were when, and the answer is a pair of numbers with dates attached.

Why the older method survived for thirty years #

Dischinger published the rate-of-creep method in 1937 and it was the standard tool for a generation, although its central assumption — that the creep of concrete depends only on how long the load has been on it, and not on how old the concrete was when it arrived — had been known to be wrong since the first test programmes.

The reason it survived is in its form. The rate-of-creep assumption turns the superposition integral into a differential equation, with $\varphi$ as the independent variable, and a differential equation could be solved on paper in 1937 while an integral equation could not. Every closed form in the older literature — the $1 - e^{-\varphi}$ above, and its relatives for every arrangement of restraint — is that substitution paying for itself.

Trost’s ageing coefficient of 1967 is the repair, and what makes it a good one is that it keeps the one-line form. It does not solve the integral; it replaces the whole stress history with a single effective modulus $E/(1 + \chi\varphi)$ chosen so that the one-line answer matches the integral’s. Bažant then showed that χ could be computed from the creep function rather than tabulated, which is what the sweep above is doing numerically.

So the three routes in the figures are not three levels of approximation. They are a differential equation that is exactly solvable and physically wrong, an integral equation that is physically right and was not solvable by hand, and a one-line formula calibrated to reproduce the second. The middle one is the definition and the last one is what anybody actually uses — which is why knowing what χ was fitted to matters more than knowing its value.

Which free body produced the number #

The free body is one span, cut at the stitch, with the hogging moment $X$ drawn on the cut face.

What it does not contain is any statement about the concrete. The equilibrium of that free body fixes the relationship between $X$, the load and the reactions, and it is satisfied for every value of $X$ from zero to the monolithic one. Equilibrium cannot choose, which is the ordinary condition of a redundant structure — and what usually chooses is stiffness, evaluated once.

Here the choosing is done by a compatibility condition evaluated over time, and the quantity it is written in is not a stiffness but a creep function. So this is an indeterminate structure whose redundant depends on a material property that does not appear in any elastic analysis at all, and whose value changes for twenty years after the last person left the site.

Two modelling steps go into the numbers and both are stated rather than derived:

One creep function for both beams and the stitch. The in-situ concrete of the stitch is weeks younger than the precast it joins and creeps differently, and its notional size is different too. The calculation above treats the whole as one material, which is the ordinary simplification and is worth a check on a structure where the stitch is a significant length.

Uncracked, linear creep. The support region is exactly where the hogging moment is arriving, so it is exactly where cracking would reduce the stiffness that the compatibility condition is written in. A cracked support redistributes less, because the joint is less able to refuse the rotation.

What the picture cannot show #

Nothing in these figures is a deflection. The redistribution takes moment away from midspan, which reduces the long-term sagging deflection — by less than the moment ratio suggests, because the deflection accumulated before the stitch is already there and is not given back. A structure that ends with 46 per cent of the monolithic support moment does not end with the monolithic deflection.

The load never changes. Every figure here has one load applied at 28 days and left. A real pair of beams gets a deck slab at 60 days, a surfacing at 120 and a live load thereafter, and each of those is a separate problem with its own $t_0$ and its own share of the structure — the ones arriving after the stitch act on the continuous structure directly and do not redistribute at all. Superposing them is legitimate and it is four calculations rather than one.

And the ceiling assumes the stitch is rigid from the day it is made. An in-situ stitch is a few days old when it is asked to start resisting, its own modulus is climbing, and its own shrinkage is pulling the two beams together. That last is an imposed deformation at the joint of the opposite sign, and nothing here contains it.

Still open: the tendon that is consumed by the member’s own drying #

The redistribution above is driven by creep and resisted by creep, and its answer is a ratio of two creep coefficients in which the concrete’s strength barely appears. Prestress loss is the same arithmetic with the answer read as a force rather than as a moment: a tendon is stretched, it is locked off against the concrete, and the concrete then shortens underneath it by exactly the drying creep and shrinkage of the member it is in. The tendon’s extension is consumed, and the force falls.

What makes it the next essay rather than a footnote is that it inherits the size effect of the creep that belongs to the member directly: a thin-flanged box girder has a small notional size, dries fast, creeps more, and loses more of its prestress than a solid slab of the same concrete at the same stress. And the steel relaxes at the same time, into a concrete that is shortening, so the two losses are not additive — each reduces the driving force for the other.