The load put on backwards

Concrete carries compression at forty megapascals and tension at three, and every beam made of it therefore spends its life with half its section cracked and doing nothing but holding the reinforcement in place. That is the ordinary arrangement, it works, and it is why a cracked section has its own essay.

Prestressing is the other answer to the same problem, and it is a strange one: put the beam into compression first, hard enough that the tension the load will cause never arrives.

That figure is superposition and nothing else. Two stress distributions that each cross zero are added, and the sum does not — which is only possible because the two are shaped differently, one being a straight line from the axial force plus eccentricity and the other a straight line from bending.

The eccentricity is what makes it work #

An axial force alone would be a poor way to spend 1,500 kN. Applied at the centroid it produces 7.14 MPa of uniform compression, which offsets the load’s tension at the bottom fibre and adds to its compression at the top — helping one face and hurting the other equally.

Moving the tendon down changes that entirely. An eccentric force is an axial force plus a moment of $P \cdot e$, and that moment is a hogging one: it compresses the bottom and pulls the top. Since the load’s moment does the opposite, the two moments are in opposition at every fibre, and the beam has been given a bending moment tailored to the one it is going to receive.

$$\sigma_{\text{bottom}} = \frac{P}{A} + \frac{Pe}{Z} - \frac{M}{Z}$$

For this beam, at transfer, $P/A$ is 7.14 MPa and $Pe/Z$ is 13.47 — so the eccentricity contributes nearly twice what the force does, and its cost is nothing. The tendon is in the same place either way; it has simply been put 220 mm lower.

The limit on how low is the section, and the practical limit arrives earlier than that. Push the eccentricity to 320 mm and the transfer condition gives −8.59 MPa at the top fibre: the beam cracks along its top face while sitting in the casting yard, under no load but its own.

The empty beam is the hard case #

That is the inversion this whole subject turns on, and it catches everybody once.

Every other structure in this collection is at its worst when fully loaded. A prestressed one is at its worst when empty, because the prestress is present in full, the losses have not yet happened, and the only thing opposing it is the beam’s own weight. Transfer is a load case in which the structure is fighting itself with no help.

Two consequences follow, and both are visible on any precast yard. The prestress is applied in stages, with some strands left untensioned until the beam is in place. And the tendon is draped — pulled down to its full eccentricity at midspan and brought back up toward the centroid at the ends, where the self-weight moment that would balance it has gone to zero.

The tendon is a load, and it points up #

The stress-block reading is exact and it is hard to think with. The other reading of the same arithmetic turns the tendon into something a structural engineer already understands.

A tendon on a parabolic drape is not straight, and a force that follows a curve pushes sideways along it. The intensity of that push is the force times the curvature, and for a parabola of sag $e$ over a span $L$ the curvature is constant, so the tendon delivers a uniformly distributed upward load:

$$w_{\text{bal}} = \frac{8Pe}{L^2}$$

Load balancing is T. Y. Lin’s contribution and it changed how the subject is taught, because it removes the stress blocks from the design conversation entirely. Choose the load to be balanced — usually the permanent one — set $8Pe/L^2$ equal to it, and the beam under that load has no bending, no deflection and a uniform stress. Everything else is a variation about that state and can be checked by ordinary elastic analysis.

The reading also explains a thing about prestressed structures that is otherwise mysterious: they are often flat. A balanced beam does not sag under its permanent load because there is no net load on it, so precambering is unnecessary and long-span floors can be built without the visible droop that a reinforced concrete slab of the same span would have.

Where the tendon is allowed to be #

Combining the two conditions — no cracking at the top at transfer, none at the bottom in service — gives an inequality at every section rather than one at midspan, and the pair of them define a zone.

The shape of that zone is worth reading carefully, because it is the design. Near midspan both limits are generous, because the self-weight moment is large and it opposes the prestress. Near the supports the self-weight moment vanishes, the prestress moment does not, and the upper limit falls to something close to the kern.

The zone answers the question one section at a time and with the force already chosen. The other way of asking it takes both unknowns at once, and it is the picture the whole design used to be done on.

That last sentence is the connection this essay has been heading toward. At the end of the beam, with no applied moment at all, the requirement that the top fibre stay in compression is exactly the requirement that the force land inside the middle third:

A prestressed beam is a masonry problem at its ends and a bending problem in its middle, and the tendon profile is the curve that interpolates between the two.

The beam arrives already bent upward #

Load balancing has a corollary that is visible on any precasting bed the moment the strands are cut.

At transfer the prestress is at its full 1,500 kN and the only load on the beam is its own 5.25 kN/m. The tendon’s upward push at that moment is $8Pe/L^2 = 18.33$ kN/m, so the net load on the beam is 13.08 kN/m upward, and the beam lifts off the bed at its ends and rests on its middle.

The arithmetic is the ordinary $5wL^4/384EI$ with $EI = 274{,}400$ kNm² for this section at 32 GPa, and 12.9 mm is a serviceable number to carry around: a prestressed beam is usually built with a hog of about a five-hundredth of its span, which the applied load then partly takes out.

Three practical things follow from that, and each has caught somebody.

The hog is not constant. Creep acts on the net stress state, and at transfer that state is a hog, so the beam creeps further upward over months — the opposite direction from a reinforced beam, which creeps down. A precast floor stored in a yard for a year arrives on site with more hog than it left the bed with, and a screed poured level over it is thinner in the middle of every span.

The hog does not average out. Adjacent units cast on the same bed can differ by several millimetres because their concrete strengths at transfer differed by a day’s curing, and a floor of them has a visible ripple that no amount of levelling the supports can remove.

And the beam’s stiffness is the uncracked one, which is the whole point of prestressing it, so $EI$ is 60% higher than the cracked section a reinforced beam would settle into — which shows up as a beam that both hogs more and, once loaded, deflects less. Neither behaviour is available from a section that has cracked.

The force is not the same all along the tendon #

Every number above treats the prestress as one value applied to the whole beam. In a post-tensioned member it is not, and the variation is in the wrong direction.

A tendon is pulled through a duct, and it rubs. Two mechanisms do it: curvature friction, where the tendon presses against the outside of every bend with the same $P/R$ that produces the upward load in the first place, and wobble, the small unintended deviations of a duct that was meant to be smooth. Together,

$$P(x) = P_0\,e^{-(\mu\alpha + kx)}$$

with $\alpha$ the total angle turned between the jack and the section, $\mu$ about 0.19 for strand in a metal duct, and $k$ around 0.008 per metre.

For this beam the parabola’s end slope is $4e/L = 0.073$ rad, so a tendon jacked from one end reaches midspan having turned through 0.073 rad over 6 m: the exponent is $0.19\times0.073 + 0.008\times6 = 0.062$, and the force there is 94% of what the jack read. Six per cent, of which four fifths is wobble — on a short shallow beam the duct’s unintended wander costs more than its intended curve.

The direction is the awkward part. The force is largest at the jacking end and smallest at midspan, which is the exact reverse of where a draped tendon needs it. The remedies are all geometric: jack from both ends on a long tendon so the two profiles cross in the middle, or over-tension and release, which drags the tendon backwards through the duct and reverses the friction over the length nearest the jack.

That second operation produces a shape nobody expects. The wedges draw in a few millimetres as they lock, the tendon slides back, and friction now acts the other way over the length affected — so the force near the live anchorage falls below the value further inside, and the maximum sits some metres into the beam. A member checked at its anchorage has been checked at a point that is not the worst one, and on a short tendon the draw-in reversal can reach past midspan and undo most of what the jacking achieved.

The pretensioned beam has no force at its end at all #

The zone diagram above ended with a difficulty: near the supports the applied moment vanishes, the prestress moment does not, and the tendon has to rise nearly to the kern. A pretensioned beam resolves that in a way the elastic picture cannot express, because it has no anchorage.

A pretensioned strand is tensioned against the bed, the concrete is cast round it, and the strands are cut. There is no plate and no wedge; the force gets into the concrete by bond, building up over a transmission length of some 50 to 70 strand diameters. For 12.5 mm strand that is 600 to 900 mm.

So at the end face of the beam the prestress is exactly zero, and it reaches its full value about 750 mm in — 6% of a 12 m span. The transfer check that looked impossible at the end section is not a check on that section at all: the force is not there yet. What has to be checked instead is the station where the force has fully developed, and by then the self-weight moment has grown to $wx(L-x)/2 = 22$ kNm, which helps a little.

The same mechanism supplies the pretensioned answer to draping, and it is used constantly on products where a drape is impossible. Hollowcore units are extruded over straight strands — nothing can be bent — so the force at the ends is reduced by debonding: a plastic sleeve over selected strands for a metre or two from each end, so those strands deliver nothing there and full force further in. It is a drape achieved by switching strands off rather than by moving them, and the zone diagram is satisfied by varying $P$ along the span instead of $e$.

The two systems therefore differ in more than their construction sequence. Post-tensioning puts its whole force in at a point, and everything difficult about it — the bursting tension behind the anchorage, the friction along the duct, the draw-in at the wedge — follows from that point existing. Pretensioning has no point of application anywhere, and its difficulty is a length rather than a plate.

The force is not the force it was #

Everything above used 1,500 kN at transfer and 1,200 kN in service, and that 20% is not a factor of safety. It is a prediction of how much of the prestress will be lost, and it is made up of four separate mechanisms that are individually small.

The largest and slowest of them is creep. A beam prestressed at 28 days will, over years, shorten by two or three times its immediate elastic shortening; the tendon is bonded to it, shortens with it, and gives up force in proportion. That is why a prestressed beam’s stresses are still moving a decade after it was built, and why the service check in every figure above is a check on a state that arrives late rather than on the state the beam left the yard in.

The four are: elastic shortening at transfer, when the concrete compresses under the force it has just been given; relaxation of the steel, which holds a stress at constant strain and gives some of it back; shrinkage as the concrete dries; and creep under the sustained compression. The last two are the large ones, and they share a property that makes them awkward — they are strains imposed on the concrete, and the tendon is obliged to follow them, so the loss is proportional to the strain rather than to the force.

The arithmetic is unpleasant and the consequence is simple: a prestressed member’s condition at fifty years is different from its condition at fifty days, and both have to be checked. This site has met the same idea from the other side, where a deflection arrives three years late; here it is the force that arrives short.

What it is not #

Prestressing is often described as a way of making concrete stronger, and it is not. The compressive strength is unchanged, the tensile strength is unchanged, and the ultimate moment of a prestressed beam is very close to that of a reinforced one with the same steel in the same place.

What has been bought is behaviour at working load: an uncracked section, full stiffness, no visible cracks, and a deflection near zero. Those are serviceability properties, which is why prestressing is used for long spans, water-retaining structures and anything where cracking is unacceptable, and why it buys much less on a short heavily loaded member where the section would not have cracked much anyway.

The alternative arrangement is worth holding in mind while reading that. A reinforced section of the same size under a service moment has its neutral axis a third of the way down, everything below it cracked, its steel at a working stress and its stiffness around 60 per cent of the uncracked value. That section works. It is not failing and it is not badly designed — bending is still a pair of forces in it, with the lever arm the crack left behind. Prestressing is the choice not to be in that state, and it is a choice with a price.

One strength it does raise is the web’s, and the way it raises it is worth being exact about, because it is the one place where the compression a prestress puts in does something other than delay a crack.

The shape of that curve is the general lesson of the field arriving in a different quantity. The first newton of prestress is worth far more than the last, exactly as the first hundred millimetres of eccentricity are, and for the same reason: what is being bought is distance from a limit rather than capacity, and distance from a limit runs out.

Between the two arrangements is a continuum nobody names well. A partially prestressed member is one designed to crack under its full load and to close again when the load goes, which buys most of the stiffness and all of the appearance for a fraction of the tendon — and it is what a great many post-tensioned floors actually are, whatever the drawings call them. The choice between full and partial prestress is a choice about which load case the section is allowed to crack under, and it is made by the same reasoning as everything else in this field: decide the state the structure is to be in under the load it will actually spend its life carrying, and let the extremes be checked afterwards.

What the picture cannot show #

Every figure above is a single section. The zone diagram is the only one that admits the beam has a length, and even it treats each section independently — the real profile also has to be a curve a tendon can physically follow, with a radius the duct can take and a friction loss along it that reduces the force away from the jacking end.

The anchorage is not drawn anywhere. Delivering 1,500 kN into a 300 mm wide beam through a plate the size of a hand is a disturbed region problem with bursting tension across the load path, and it is where prestressed beams actually fail during construction.

The behaviour past cracking is absent. Everything here is elastic and uncracked, which is the state the design keeps the beam in — but the ultimate condition is a different calculation, in which the tendon yields and the section forms a plastic hinge like any other. A prestressed beam that is over-reinforced fails by crushing with no warning at all, and the warning that reinforced concrete gives by cracking has been deliberately removed.

Where the ladder goes #

The first rung is continuity. A prestressed tendon run through several spans produces secondary moments — the beam wants to lift off its middle supports, they hold it down, and the resulting moments exist with no external load at all. That is a self-stress state of exactly the kind an indeterminate structure always carries, and it is the reason continuous prestressed design is a subject of its own.

The second is the opposite arrangement: a tendon that is not bonded to the concrete, which can be tensioned later, monitored, replaced — and which cannot participate locally, so its stress is a property of the whole member rather than of a section.

The third is the general principle, which is much older than concrete. A cartwheel’s iron tyre is shrunk on to put the spokes in compression; a barrel’s hoops do the same to its staves; a gun barrel is built up from shrunk-on cylinders for exactly the reason above. Putting the stress in backwards before the load arrives is a technique rather than a material, and concrete is only where it became a system.