The prestress the member takes back

A prestressed member is given its load first — chosen, permanent, and pointing the wrong way — and every calculation downstream of that is proportional to the force in the tendon. The force is set by stretching the steel a known amount and locking it off.

The lock-off fixes an extension, not a force. From then on the tendon holds whatever stress that extension corresponds to, and the concrete it is anchored against spends the next fifty years getting shorter underneath it.

Why an extension is the thing that is fixed #

A tendon between two anchorages is a spring of known stiffness, $E_p A_p / L$. Stressing it to 1300 N/mm² over a 30 m length stretches it by about 200 mm, and the anchorage then holds that 200 mm.

Anything that shortens the distance between the anchorages takes some of the 200 mm back, and the force falls in proportion to what is taken. A shortening of the member by 300 microstrain over 30 m is 9 mm — four and a half per cent of the extension, and therefore four and a half per cent of the force.

The tendon has no way to tell the difference between a shortening it caused and one it did not. Creep under the prestress itself, shrinkage that would have happened with no load at all, a temperature drop, and the elastic shortening at transfer are all just the anchorages moving closer together, and all of them are paid for out of the same 200 mm.

That is the whole mechanism, and it says immediately where the loss is going to come from: from whatever makes the concrete shorter, weighted by how much of the extension each one consumes.

Creep is more than half of it #

The stacking in the hero figure is the answer to which shortening, and it is not the one a tendon supplier would give.

Creep, 107.7 N/mm², or 56 per cent of the loss. The concrete under the tendon is at about 12 N/mm² of sustained compression, its creep coefficient is 1.94, and the strain that produces — multiplied by the steel’s modulus — is the single largest term.

Shrinkage, 50.6 N/mm², or 27 per cent. This one owes nothing to the prestress: it is the concrete drying, and it would happen in an unstressed member sitting in the same air.

Relaxation of the steel, 32.5 N/mm², or 17 per cent. The only term that is about the tendon, and the smallest of the three.

That ordering is worth carrying because it locates the uncertainty. Two-thirds of the loss is a property of the concrete and its surroundings, which are the least well-known quantities in the calculation — a code creep model carries a coefficient of variation of twenty to thirty per cent — while the term that is measured on a certificate and quoted to three figures is the small one.

The time axis matters too. Half the loss has happened by 119 days and nine-tenths by eight years, so the two states usually checked — at transfer and at fifty years — bracket a period in which most of the change occurs and neither describes it. A structure first loaded in anger at a year is a structure with 131.5 N/mm² of the eventual 190.8 gone, which is neither of the design cases.

The member’s shape is in the answer #

The creep and the shrinkage both carry the notional size, $2A_c/u$ — twice the area over the perimeter that can dry — and they carry it in three separate places.

The three places are worth separating, because only one of them is the creep coefficient:

The creep coefficient’s drying term, which enters through a cube root and takes $\varphi$ from 2.18 to 1.77 across the sweep.

The shrinkage’s $k_h$ factor, which scales the whole final drying shrinkage from 1.0 at 100 mm to 0.70 above 500 — a thirty per cent reduction applied directly.

And the shrinkage’s development function, whose time constant carries $\sqrt{h_0^3}$, so a fat member has not finished shrinking at fifty years while a thin one has.

Together they give a spread of 20 per cent between the extremes at 70 per cent humidity, and 26 at 50. That is the claim this essay opened with — a thin-flanged box girder does lose more prestress than a solid slab of the same concrete at the same stress — and it is a real effect of a size a designer would not bother to chase.

The air is the larger lever #

Eight points of loss between the driest and wettest air, against four points across the whole range of member size. A tendon in a bridge deck over an estuary and the same tendon in the same section in a heated building are two different design problems, and the difference between them is larger than the difference between a box and a slab.

The humidity is also the input a designer has least control over and often gets wrong, because the value that matters is the air immediately around the member for its whole life, not the regional climate. A beam inside a heated building sits at about 50 per cent; the same beam in an unheated basement or over water sits at 80 or more; and a member that is inside a building for forty of its fifty years but stood outside for the first two of them did most of its drying in the wrong air.

One practical reading follows from the shape of that curve rather than from its ends. It is steepest in the middle of the range and flattest at the top, so the difference between 40 and 55 per cent humidity is worth seventeen newtons per square millimetre while the difference between 85 and 100 is worth thirty-five. A member’s exposure therefore has to be judged rather than looked up, and judged for the decades it will spend rather than the season it is built in.

There is a floor to it, and the floor is the interesting end. At 100 per cent humidity nothing dries at all, and the loss is still 9.7 per cent — because creep does not stop when drying does. What is left is basic creep, the part the concrete would do sealed, in the dark, for ever, and it is the largest single term even in saturated air.

Three losses that are not three losses #

The one thing in the calculation that is not a material property is the denominator, and it is the largest correction in it.

EN 1992’s expression is a fraction and the interesting half of it is below the line:

$$\Delta\sigma = \frac{\varepsilon_{cs}E_p + 0.8\,\Delta\sigma_{pr} + (E_p/E_{cm})\,\varphi\,\sigma_{cQP}}{1 + \dfrac{E_p}{E_{cm}}\dfrac{A_p}{A_c}\left(1 + \dfrac{A_c z^2}{I_c}\right)\left(1 + 0.8\varphi\right)}$$

The numerator is the three causes, computed as though each acted alone on a member whose stress never changed. The denominator is the admission that it does.

Follow one loop. The tendon loses force to creep; the concrete beneath it is therefore less compressed; less compressed concrete creeps less; so the creep loss is smaller than the calculation that assumed a constant stress. The same argument runs through the shrinkage — a shrinking member that is also unloading creeps back less — and through the relaxation, which is defined at a constant strain and is happening at a falling one, which is why the $0.8$ sits in front of $\Delta\sigma_{pr}$.

Each loss removes part of the force that drives the other two, so the three are coupled and the coupling is worth 17 per cent. That is larger than the whole relaxation term, and larger than everything the notional size does.

The $(1 + 0.8\varphi)$ in the denominator is the age-adjusted effective modulus arriving in a third setting, with the same ageing coefficient of 0.8 and the same caveat: it was fitted to a problem of this shape — an action changing gradually — and it is not a property of concrete.

The arithmetic, once, by hand #

The expression is short enough to run through completely, and doing it once shows which of its numbers are measurements and which are conventions.

Take the member in the hero figure: C40 concrete, $E_{cm} = 35{,}000$ N/mm², a tendon of 2,000 mm² of Y1860 strand at 1300 N/mm², 350 mm below the section’s centroid, in a section of 0.6 m² with a second moment of $2.2\times10^{10}$ mm⁴, at 12 N/mm² of sustained compression at the tendon’s level.

Shrinkage. The drying part comes out at 313 microstrain for a notional size of 300 mm at 70 per cent humidity, including the autogenous part that happens whatever the air does. Times the steel’s modulus of 195,000, that is 61 N/mm² of loss before the denominator.

Creep. $\varphi = 1.94$, and the concrete’s creep strain at the tendon’s level is $\varphi\sigma_{cQP}/E_{cm} = 1.94 \times 12 / 35{,}000 = 665$ microstrain. Times 195,000 again: 130 N/mm².

Relaxation. At $\mu = 1300/1860 = 0.70$, class 2 strand at 500,000 hours gives about 49 N/mm², of which the expression takes 0.8 — 39.

Those three add to 230. The denominator is $1 + (195/35)(2000/600000)(1 + 600000 \times 350^2/2.2\times10^{10})(1 + 0.8 \times 1.94)$, which is 1.21, and the answer is 191 N/mm².

Two things are worth noticing about that run-through. The creep term is a product of four numbers and every one of them is uncertain: the creep coefficient, the concrete’s modulus, the stress at the tendon, and the steel’s modulus. The relaxation term comes off a certificate. And the denominator is bigger than the relaxation term is, so the correction nobody thinks of as a loss is larger than the loss everybody names.

What a designer changes, and what a designer cannot #

Four of the inputs above are decisions and the rest are not, which sorts the problem quickly.

Transfer age is a decision, and it is worth a fifth of the creep loss. Stressing at 28 days rather than 7 takes the total from 190.8 N/mm² to 170.1. It costs a casting bed for three weeks, which is why precast yards stress at three days and accept the loss.

The tendon’s initial stress is a decision, and it does not help. Every term above is proportional either to the concrete stress — which is proportional to the tendon force — or to the tendon stress itself, so raising the jacking force raises the loss in step and buys nothing in percentage terms. The relaxation term is worse than proportional, because $\mu$ enters an exponential.

The section’s shape is a decision, and it is worth about a fifth. Thicker webs, fewer drying faces and a fatter bottom flange all raise $h_0$; so does a member that spends its life inside a building rather than over water.

The humidity is not a decision at all, and it is the largest lever of the four. That asymmetry — the biggest term being the one nobody controls — is the reason loss calculations are done with a generous margin rather than optimised, and the reason a prestressed member’s own measurements beat any of this once it exists.

Which free body produced the number #

The free body is a length of the member with the tendon cut, so that the tendon’s force appears on the cut face as an external action.

That is what makes the whole calculation a compatibility problem rather than an equilibrium one. Equilibrium is satisfied at every value of the prestress from full to zero: the tendon pulls, the concrete pushes back, and the two are equal whatever their size. Nothing in equilibrium says how big the force is, and what says it is the statement that the tendon’s extension and the concrete’s shortening have to add to the distance between the anchorages.

Three modelling steps go in, and each is a choice:

One concrete stress at the tendon. $\sigma_{cQP}$ is the compression at the tendon’s own level under the quasi-permanent combination, taken as a single number for the whole member. It varies along the length with the tendon profile and with the moment diagram, so a real calculation is done at several sections and the answer is a profile of losses rather than one number.

One notional size for the section. A box girder’s thin webs and thick bottom slab dry at different rates, and the section has one $h_0$ in this calculation. The same averaging is what the essay below this one is about, and it understates the loss in the thin parts.

And bonded, with no friction. Everything above is about the long-term losses. The immediate ones — friction along a duct, draw-in at the anchorage, elastic shortening at transfer — are a separate calculation of comparable size, and the tendon stress this one starts from is what is left after them.

What the picture cannot show #

Nothing here is a deflection or a crack. A loss of 15 per cent of the prestress is not a 15 per cent reduction in anything a reader would notice. It reduces the upward load the tendon puts back, so the long-term deflection grows by more than 15 per cent; it reduces the compression in the bottom fibre, so the moment at which the section decompresses falls; and whether that matters depends entirely on which limit state the member was designed against.

The tendon never yields and never corrodes. A calculation of loss is a calculation about a tendon that is intact for fifty years, and the durability of a duct is the failure mode that has actually taken prestressed structures down.

And the fifty years are a straight line of weather. One humidity, one temperature, one load, held from transfer to the end. A real member is heated, cooled, wetted, loaded and unloaded, and each cycle is its own superposition problem — none of which changes the total much, and all of which changes when it arrives.

The assumption that makes losses tolerable #

A structure that loses 15 per cent of its prestress and is fine is a structure that was designed for the loss, and that is easier than it sounds for one reason worth naming.

The loss is a serviceability quantity almost entirely. At the ultimate limit state a bonded prestressed member reaches its capacity with the tendon well past its initial stress — strained by the section’s own curvature rather than by whatever the anchorage left it with — so the initial prestress has nearly dropped out of the strength calculation. What the loss changes is the decompression moment, the crack width, the camber and the deflection.

That is why a fifteen per cent uncertainty on a quantity computed from a creep model with a thirty per cent scatter is not a crisis: the consequence of being wrong is a member that cracks at a lower moment than intended, in a structure whose strength is not in question. A prestressed member is unusually forgiving about its own prestress and unusually unforgiving about the duct it is in, and only one of those two gets a calculation.

Still open: the concrete that is two concretes #

These essays have now made the creep coefficient a property of the member, then of a depth within it, then of a schedule, and finally of the air. The remaining case is a member made of two concretes of different ages bonded along a plane — a precast beam six weeks dry with an in-situ slab cast on top of it while it is still wet.

Each half then has its own creep coefficient, its own shrinkage remaining and its own notional size, and they are forced to share a strain profile. The result is a stress redistribution between them that runs for years, with the young slab shrinking against a beam that has finished and the beam creeping under a load the slab has just added. It is the section-level version of the redistribution the essay below computed between members, and the same superposition integral answers it with the two halves entered separately.

Beyond that lies the non-linear regime — sustained-load strength and tertiary creep, where creep stops being proportional to stress and starts consuming the capacity rather than redistributing it.