A state no load could reach

Two earlier arguments treat a redundant structure’s free direction as a liability. Statics cannot fix its amplitude; a tolerance fixes it instead, and so does a temperature change, and nobody chose either.

Nobody has to leave it to them. The free direction is a pattern of internal forces that satisfies every equilibrium equation identically, so it can be put there deliberately and it stays — and where it is put is somewhere no external load could take the structure. That is prestressing, and two of the most consequential pieces of ordinary engineering are it.

Moving a stress state rather than raising a strength #

Concrete is about forty megapascals strong in compression and about three in tension, and the ratio is the whole of the design problem for anything that bends. A simply supported beam under load has tension in its bottom half by construction — that is what bending is — so a concrete beam cracks along its underside at a small fraction of the load its compressive strength would allow.

The ordinary answer is reinforcement: put steel bars in the tension zone and let the concrete crack, with the steel carrying the tension across the cracks. It works, it is what most concrete is, and it accepts that the beam is cracked in service. That acceptance is a real design choice rather than a compromise — resolving a load into the two directions that suit the problem is what a reinforced section does, sending compression through the concrete and tension through the steel — and it leaves the concrete doing nothing at all across the bottom third of the beam. Cracks admit water, water corrodes the steel, and the section is stiffer before it cracks than after.

Prestressing does something categorically different. It does not help the concrete carry tension; it arranges for there to be none. A tendon threaded below the centroid and pulled tight applies two things at once — a compression across the whole section, and a moment that bends the beam the opposite way to the load — and the sum of those with the load’s own bending can be made to leave every fibre in compression.

Nothing about the material has changed. The concrete in the hero figure’s second case and its fourth case is identical concrete; what differs is where on its stress–strain range it is being asked to work. That is a move worth naming, because it recurs: rather than strengthening a component, shift the state it operates in so that the weak direction is never visited.

Glass does the same thing in a single piece of material. Cool the surface of a hot sheet quickly and the surface solidifies first; when the interior then cools and tries to contract, the surface is already rigid and is put into compression, with balancing tension in the middle. That is toughened glass, and the surface compression has to be overcome before any crack can open — which is why it is several times stronger in bending, and why, when it does fail, the stored tension in its middle tears it into dice rather than shards. Glass fails from surface flaws under tension, so a surface that is never in tension has no flaws that can open — the same reasoning as a crack needing a barrier before it can grow, with the barrier supplied rather than found. It is the same self-equilibrating state as the three bars heated together, created on purpose and frozen in.

The condition nobody expects, which is the empty beam #

The hero figure has four cases and the one that decides the design is the one with nothing on it.

A tendon below the centroid pushes the beam’s bottom fibre harder than its top, which bends the beam upwards. In service that is exactly what is wanted, because the load bends it down. Before the load arrives there is nothing to oppose it, and the top fibre goes into tension — 1.7 megapascals in the section drawn, which is over half of what the concrete will stand.

What saves the beam is its own weight, and only just: the self-weight moment brings the top back to a third of a megapascal of compression. A beam of the same section on a longer span would be in more danger at transfer than in service, and a beam supported along its whole length while the tendons are released would have nothing to help it at all.

This is not a curiosity. It is the reason the prestress is applied to a beam supported at its ends rather than on the casting bed, the reason precast members are handled at specified lifting points, and the reason the check that limits how far down the tendon may go is a check made on an unloaded beam. A structure whose worst condition is the one where nothing is happening to it is a structure whose stress state is not a function of its loads — which is the whole content of the self-loading argument, met again from the constructive side.

Where the tendon is allowed to be #

That makes the design problem an unfamiliar shape. There is not one condition to satisfy but four, they apply at every section along the span, and two of them concern a beam that is doing nothing.

Each of the four conditions is linear in the eccentricity, so each is a bound on it: two upper bounds from the transfer condition and two lower bounds from the service condition. The tendon must lie between the largest lower bound and the smallest upper bound, at every section, and the figure draws that band.

Two things about it are worth having.

The first is that the cable’s curve is not a design decision. It is forced. The lower bound rises towards midspan because the service moment does, so a straight tendon at a constant depth cannot satisfy the service condition at midspan and the transfer condition at the support simultaneously. Every prestressed beam’s cable is draped, and it is draped in the shape of the bending moment diagram because that is the only shape the inequalities leave.

The second is that the band can close. At eight hundred kilonewtons rather than fifteen hundred the two curves cross and no position works at all — the figure checks that, because a band drawn without ever being empty is a constraint that constrains nothing. That crossing is the minimum prestress the section needs, and finding it is the first step of a design rather than an afterthought.

The force that will not stay where it was put #

There is a complication in that figure hidden in the words “after losses”, and it is large enough to change the answer.

A prestressing tendon does not keep the force it was tensioned to. The steel relaxes under sustained stress, losing a few per cent. The concrete creeps — it goes on deforming under a constant load for years — which shortens the member and lets the tendon retract. The concrete shrinks as it dries, with the same effect. Together those take twenty to twenty-five per cent of the initial force away over the life of the structure, and there is no way to avoid them.

So the design has to be safe at two quite different values of a quantity that drifts between them over decades: the full force on the day of stressing, when the beam is empty, and the reduced force in fifty years, when it is fully loaded. Those are the two ends of the band above, and the band is narrow because it is the intersection of conditions evaluated at both.

It also explains a piece of practice that looks like superstition. Prestressing is done with a small number of very highly stressed wires — steel at twelve hundred megapascals rather than the two hundred and fifty a structural section works at — because the losses are an absolute shortening, of order half a millimetre per metre, and the force lost is that shortening times the tendon’s stiffness. A tendon stressed to a small strain would lose all of its force to the same shortening; one stressed to six parts per thousand loses a fifth of it. The steel has to be strong so that the loss is a small fraction of what it holds, which is a reason for a material choice that has nothing to do with strength in the ordinary sense.

The same trick in a joint, for a different reason #

A bolted joint is the other place this is done everywhere, and what it buys there is not about avoiding tension at all.

A preloaded bolt and the flanges it clamps are two springs sharing a displacement. When an external load tries to pull the joint apart, the bolt stretches a little more and the members relax a little, and because the members are much stiffer than the bolt, almost all of the load is taken by the members giving up compression rather than by the bolt gaining tension.

The consequence is about fatigue. A bolt in a fluctuating load fails by fatigue, and fatigue depends on the range of the stress far more strongly than on its mean — a bolt cycling between 30 and 34 kilonewtons will outlast one cycling between 0 and 20 by a very large factor, even though its mean force is higher and its peak force is higher. Preloading trades a higher mean, which costs little, for a fifth of the range, which is worth almost everything.

And then it stops. At the separation load the clamping compression reaches zero, the members are no longer in contact, and from there the bolt carries the whole external load: the two curves meet and the benefit ends abruptly. That is why a preload is a specified number with a tightening procedure attached rather than a matter of how hard somebody pulls, and why joints that come loose fail so much faster than the same joint never tightened — it has been designed for a stress range it no longer has.

The energy a preload puts in #

There is a quantity none of the figures draws and it decides how prestressing is done in practice.

Straining a material stores energy, and the density stored is $\sigma^2/2E$. Structural steel at its working stress of two hundred and fifty megapascals stores about 160 kilojoules per cubic metre. Prestressing steel at twelve hundred stores 3.6 megajoules per cubic metre — twenty-three times as much, because the stress is squared and the modulus is the same.

Put numbers on the beam above. Fifteen hundred kilonewtons at twelve hundred megapascals needs a tendon area of 1,250 square millimetres, which over an eight-metre span is a hundredth of a cubic metre of steel holding about thirty-six kilojoules. That is the kinetic energy of a small car at thirty kilometres an hour, stored in a bundle of wire the thickness of a wrist, and it is released in milliseconds if an anchorage fails.

So the energy is why a stressing operation is conducted with nobody standing behind the jack, why anchorages are the most heavily proof-tested component in the assembly, and why a tendon that corrodes and snaps inside a bridge does more than lose its share of the force. The same arithmetic explains why a snapped wire rope is dangerous and a snapped chain of the same strength is much less so: the rope is elastic over its length and has been storing the energy, and the energy released is the area under a force–extension curve rather than anything about the breaking load.

It is also the reason preloading is not free in a joint. The energy stored in a tightened bolt has to be put in by turning it against friction, which is why a large bolt needs a torque multiplier and why the tightening itself is most of the labour of erecting a bolted structure.

What cannot be prestressed #

The rule that decides where any of this is available is worth stating precisely, because the usual shorthand is wrong in an informative way.

A structure with exactly as many unknowns as equilibrium equations has a unique set of reactions determined by the loads. No self-equilibrating state exists, so nothing can be preloaded into it, and nothing has to be: it also develops no thermal stresses and no residual stresses.

But a simply supported prestressed beam is determinate — two supports, a determinate pair of reactions — and it is plainly prestressed. The resolution is that structural determinacy and internal determinacy are different counts. A cross-section always has infinitely many stress distributions giving the same resultant force and moment, so a body is internally redundant even when its reactions are not, and prestressing exploits that internal redundancy. The tendon and the concrete are two members sharing one displacement, exactly as the bolt and the flanges are.

The distinction earns its keep the moment a beam runs over more than two supports. Prestressing a continuous beam produces reactions — the beam tries to lift off its middle support and cannot, so the support pushes back, and that reaction produces bending moments throughout the structure that nothing in the section calculation predicts. They are called secondary moments, they are comparable in size to the ones the designer intended, and getting their sign wrong is one of the classic ways to design a prestressed structure badly. The reactions they produce are also what a load cell under a silo reads and cannot separate from the weight — a support force that is not a response to any load. The prestress excites the structure’s own free directions as well as the section’s, which is the self-loading phenomenon arriving as a consequence of this one’s technique.

Two more of the same thing #

The pattern is general enough to be worth collecting, because none of these is usually filed with the others.

A bicycle wheel is a rim held by spokes that are only useful in tension. Tensioned to a few hundred newtons each during building, the spokes at the bottom of the wheel lose tension as the load comes on and none goes slack, so the rim is supported by a reduction in pull from below rather than by a push. Build the wheel with slack spokes and it collapses at the first bump; the preload is what keeps every spoke inside its useful range. A slack spoke is a member that has dropped out of the count, and a structure that loses members as it deflects is one whose force network is not reproducible in the way a granular pack’s is not.

A tennis racket and a drum are the same, with a membrane rather than spokes. A flywheel spun to high speed is prestressed by winding a carbon-fibre hoop on under tension, because the hoop stress at speed would otherwise exceed what the rim can take. And an autofrettaged gun barrel is pressurised past yield once, deliberately, so that the bore is left in compression by the elastic outer material — the same idea as toughened glass, performed with plasticity rather than with heat.

In each of them a body has been put into a state that has nothing to do with what it will be asked to carry, and in each the benefit is that the changes the loads then produce happen somewhere useful.

Creep, cracking, and the load cases not drawn #

Everything here is elastic and the concrete is treated as though it were. Concrete creeps, and the creep is the reason for the losses above; it is also nonlinear in the stress and depends on the age at loading, the humidity and the section’s surface-to-volume ratio. Losses computed from a code’s creep coefficient are estimates with a spread of tens of per cent, and the tendon band’s lower curve moves with them.

The section is uncracked throughout. The whole design above is arranged so that it stays that way under service load, and the calculation is invalid the moment it does not — an overloaded prestressed beam cracks, its stiffness drops, and its behaviour from there is governed by the tendon acting as reinforcement rather than as prestress. The ultimate strength of such a beam is computed by an entirely different method.

The bolt calculation takes the load as applied at the joint face. Where in the joint the external load is introduced changes the load factor substantially, and for a load applied part-way along the clamped members the bolt can take a much smaller fraction than the stiffness ratio suggests. The figure’s number is the worst case and the honest one for design.

The members are treated as rigid except where they are not. The bolt figure gives the flanges a stiffness and the beam figure gives the concrete one, and both take everything else as unyielding — which is the idealisation that has to be given up the moment a count runs short, and which is being given up here selectively rather than consistently.

And the tendon zone is drawn for one loading. A real beam has several load cases — full load, no load, load on half the span, a construction case — and the permissible band is the intersection over all of them, which is narrower than any one and can be empty when each separately is not.

Four lines that belong to four different decades #

The stress-block figure draws four straight lines through a section and each of them is a state the beam occupies at a different time: one before the tendons are released, one at release, one when it is first loaded, one in fifty years. Nothing in a plot of stress against depth carries a time, and the design problem is precisely that the four have to be satisfied by one tendon that was placed once.

The tendon-zone figure draws a band and cannot show what a point inside it means. A tendon is not a mathematical line at a depth; it is a duct fifty millimetres across containing several strands, with a tolerance on its placement of perhaps ten millimetres, inside a beam whose own dimensions carry tolerances. A band seventy-five millimetres wide at midspan is not generous, and part of the reason designs use more prestress than the minimum is to open it.

And the bolt figure draws two forces and hides the displacement that produces them, which is a few tens of micrometres. Everything about the joint — the load factor, the separation point, whether it stays tight — turns on a movement far too small to see, and on surface finish and embedded dirt that relax a fraction of it in the first hours after tightening.

Still open: what a preload is when nobody measured it #

Every number in this essay assumes the preload is known. In a prestressed beam it is: the jack has a pressure gauge, the tendon’s extension is measured, and the two are checked against each other.

In a bolted joint it usually is not. Tightening is done by torque, and the relation between torque and preload runs through the friction under the head and in the threads — which accounts for about ninety per cent of the torque applied and varies by a factor of two with the surface condition, the lubricant and whether the bolt has been used before. The scatter in the preload achieved by a torque wrench is routinely plus or minus thirty per cent, which is why critical joints are tightened by measuring the bolt’s extension, by turning through a specified angle past snug, or with bolts designed to shear off a spline at a known tension.

That scatter interacts badly with everything above. A preload thirty per cent low moves the separation load down by the same fraction; a preload thirty per cent high eats into the bolt’s remaining strength. And the quantity in question is a self-equilibrating force in a redundant assembly, which means it cannot be measured afterwards without unloading the joint — the same difficulty a redundant structure’s locked-in state has, in the one place where somebody deliberately put the state there.

The habit worth carrying away is the one this whole essay turns on. A structure with a free direction has a state that is chosen rather than determined, and somebody chooses it. Left alone, the tolerances and the temperature choose; taken in hand, it becomes the cheapest strengthening move available, and it works by moving where the material operates rather than by changing what the material is.