Structural Engineering & Statics
where the load goes
Internal forces

The load put on backwards

Every other structure in this collection waits for its load and then resists it. A prestressed one is given a load first — chosen, permanent, and pointing the wrong way — so that when the real one arrives the two nearly cancel and the material never has to do the thing it is bad at.

Assumes Bending is a pair of forces, pushing and pulling, What a cut reveals, and why it was there all along and The middle third.

17 min read 8 figures The load must go somewhereWhich failure arrives first

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.

Two triangles that cross zero, and a block that does notStress across a 300 × 700 mm section at each stage, compression positive. The prestress alone gives -6.33 MPa at the top and 20.61 at the bottom; at transfer, with only self-weight on it, the top is at -2.47 MPa and in service the section runs from 7.61 to 3.82 MPa — compression everywhere. The same beam with no prestress reaches -12.67 MPa at the bottom fibre, which is 4.2 times what the concrete can hold.prestress alone-6.3320.61tensionat transfer-2.4716.76tensionin service7.613.82with no prestress12.67-12.67tension
Fig. 1 Stress across a 300 × 700 mm section at each stage, compression positive. Prestress alone gives −6.33 MPa at the top and 20.61 at the bottom. Self-weight brings the top back to −2.47 and the bottom to 16.76 — that is transfer. In service, with losses and the full load, the section runs from 7.61 to 3.82 MPa: compression everywhere. The same beam without prestress reaches −12.67 MPa at the bottom fibre, which is four times what the concrete can hold.

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⋅eP \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.

σbottom=PA+PeZ−MZ\sigma_{\text{bottom}} = \frac{P}{A} + \frac{Pe}{Z} - \frac{M}{Z}

For this beam, at transfer, P/AP/A is 7.14 MPa and Pe/ZPe/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 triangles that cross zero, and a block that does notStress across a 300 × 700 mm section at each stage, compression positive. The prestress alone gives -12.45 MPa at the top and 26.73 at the bottom; at transfer, with only self-weight on it, the top is at -8.59 MPa and in service the section runs from 2.71 to 8.71 MPa — compression everywhere. The same beam with no prestress reaches -12.67 MPa at the bottom fibre, which is 4.2 times what the concrete can hold.prestress alone-12.4526.73tensionat transfer-8.5922.88tensionin service2.718.71with no prestress12.67-12.67tension
Fig. 2 The same beam with the tendon 100 mm lower. In service it is comfortable — 2.71 MPa at the top and 8.71 at the bottom, further from cracking than before. At transfer the top fibre is at −8.59 MPa, which cracks it. The change that improved every service number made the beam unbuildable, and no calculation of the loaded beam would ever say so.

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 ee over a span LL the curvature is constant, so the tendon delivers a uniformly distributed upward load:

wbal=8PeL2w_{\text{bal}} = \frac{8Pe}{L^2}

The tendon is a load, pointing the other wayA 12 m beam with a parabolic tendon dropping 220 mm to midspan, stressed to 1200 kN after losses. Its curvature pushes the beam up along its whole length with an intensity of 8Pe/L² = 14.67 kN/m, against an applied 17.25 kN/m — so 2.58 kN/m is left to bend anything, and the beam carries 46.5 kNm where an unstressed one carries 311 kNm. What the section then feels is 5.71 MPa of uniform compression and very little else.applied 17.25 kN/mtendon pushes back 14.67 kN/mnet 2.58 kN/m — the beam hardly bends at allmidspan moment 46.5 kNm, against 311 kNm unstressed
Fig. 3 The tendon read as a load. A drape of 220 mm over 12 m at 1,200 kN after losses pushes up with 14.67 kN/m along the whole span, against an applied 17.25 kN/m of self-weight and imposed load. What is left to bend the beam is 2.58 kN/m, giving a midspan moment of 46.4 kNm where an unstressed beam carries 310.5. The section feels 5.71 MPa of uniform compression and almost nothing else.

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/L28Pe/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 zone the tendon has to stay insideThe eccentricities that keep the top fibre out of tension at transfer and the bottom fibre out of tension in service, along a 12 m beam. The two limits cross the section at different rates, and the parabolic profile drawn between them is the tendon: 220 mm at midspan, where the zone is 148 mm deep, and on the centroid at the ends, where a tendon left low would crack the top of a beam carrying nothing but itself.024681012-300-200-1000100200300distance along the span (m)eccentricity below the centroid (mm)above this line the top cracks at transferbelow this line the bottom cracks in service
Fig. 4 The eccentricities that satisfy both conditions along the span. The upper limit comes from transfer and the lower from service, and they close in on each other toward the supports: at midspan the zone is 148 mm deep and at the ends it is bounded above at 166 mm, which is why a tendon that stayed at its full drape would crack the beam near its ends. The parabola drawn between them is the profile.

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.

Which 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:

The middle third, computedThe kern of a 300 × 700 mm rectangle, computed by asking, for every direction, how far the resultant can move before the far face would be pulled. It reaches ±116.7 mm vertically and ±50.0 mm horizontally, which are h/6 and b/6 exactly, and the region between is a rhombus rather than the ellipse the two numbers suggest — a resultant on a diagonal has less room than either axis allows.rectangle±117 of 700 mm33.3% of the depth
Fig. 5 The kern of the same 300 × 700 mm section: ±116.7 mm vertically, which is h/6, and ±50.0 mm horizontally. A prestressing force landing inside this rhombus puts no part of the section into tension on its own; the whole draped-tendon arrangement exists because at midspan the applied moment lets the force sit far outside it, and at the ends it does not.

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/L2=18.338Pe/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 deflected shape is the moment, integrated twiceA loaded beam with its deflected shape above and its bending-moment diagram below. The shape was obtained by integrating the moment twice and fitting the constants to the supports; the vertical scale is exaggerated enormously.the largest movement, at x = 6.00momentdrawn at roughly three hundred times the real deflection —a beam at its serviceability limit moves about a three-hundredth of its span
Fig. 6 The deflected shape of a 12 m beam under 13.08 kN/m, drawn hugely exaggerated as every deflection figure on this site is — the real movement is 12.9 mm on a 12 m span, which is a five-hundredth of it and thinner than the line the beam is drawn with. For the prestressed beam this shape is upside down: the net load is upward and the beam hogs by that amount before it has been used for anything.

The arithmetic is the ordinary 5wL4/384EI5wL^4/384EI with EI=274,400EI = 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 EIEI 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 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 deflection that arrives years lateThe multiplier on a concrete member's deflection under a sustained load, against time. The elastic deflection arrives on the day the load does and is the 1.0 at the left. After a year it has been multiplied by 3.00, after five years by 3.29, and it approaches 3.38. Nothing has been added to the load and nothing about the strength has changed: this is a serviceability failure arriving on a structure that passed every strength check on the day it was built.1 d10 d100 d2.7 yr27 yr0123time under loaddeflection ÷ the deflection on day one1 year: ×3.005 years: ×3.29the deflection the calculation gives
Fig. 7 Concrete’s creep coefficient against time. A beam prestressed at 28 days will, over years, shorten by two or three times its immediate elastic shortening — and the tendon, which is bonded to it, shortens with it and loses force in proportion. This is the largest of the four losses and the slowest, and it is the reason a prestressed beam’s stresses are still moving a decade after it was built.

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 neutral axis is wherever the first moment vanishesA 300 by 500 section with 1200 mm² of steel at a depth of 450, carrying 150 kNm after the tension side has cracked. The neutral axis is no longer at mid-depth: it has risen to 137.0 mm from the top, which is where the first moment of the compression zone plus the transformed steel vanishes. The compression is 18.0 N/mm² at the top fibre and the steel carries 309 N/mm²; the resulting couple is 371 kN on a lever arm of 404 mm, which multiplies back to the 150 kNm applied. The uncracked section would have had 3422×10⁶ mm⁴ against the cracked 1139×10⁶ — a loss of 67% of the stiffness.x = 1371200 mm² of steel, n = 7.5b = 30018.0 N/mm²371 kN in the steelz = 404C = T = 371 kN · C·z = 150.0 kNm = the applied momentcracked I 1139×10⁶ mm⁴ against uncracked 3422×10⁶ — 67% of the stiffness gone
Fig. 8 The alternative arrangement, for comparison: a reinforced section under 150 kNm with its neutral axis 137 mm from the top and everything below it cracked. The section works, the steel is at 309 MPa, and the beam is 60% of its uncracked stiffness. Prestressing is the choice not to be in this state.

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.

Where this goes next

neutral axiscompressiontensionI = 29.97 × 10⁶Z = 299.7 × 10³peak stress 200.2σ = M y ÷ I, at every height Bending is a pair of forces, pushing and pulling A bending moment is not a mysterious twisting. It is a push near the top of a section and a pull near the bottom, separated by a lever arm — a couple, made out of stress. rectangle±100 of 600 mm33.3% of the depth The middle third A material that cannot be pulled imposes a condition on where the load may land, and the condition is a region rather than a point. For a rectangle it is the famous middle third; for every other section it is a shape nobody quotes, and one ordinary section's is nearly twice as generous as the rule allows. x = 1371200 mm² of steel, n = 7.5b = 30018.0 N/mm²371 kN in the steelz = 404C = T = 371 kN · C·z = 150.0 kNm = the applied momentcracked I 1139×10⁶ mm⁴ against uncracked 3422×10⁶ — 67% of the stiffness gone When half the section has given up Bending theory puts the neutral axis through the centroid. That is a consequence, not a rule — and when the tension side cracks, the same reasoning moves the axis somewhere else entirely. 1 d10 d100 d2.7 yr27 yr0246810time since the strain was imposedstress, N/mm²E × strain: 9.6tensile strength: 3.5what is left: 2.32the one-line shortcut at χ = 0.8: 3.31 The strain that was imposed, and the stress that leaked away Multiply a restrained shrinkage strain by the modulus and the answer is three times the tensile strength — which predicts that every restrained concrete member ever cast has cracked. Most have not, and the reason is that the material creeps while it is being stressed. 1 d10 d100 d2.7 yr27 yr0123time under loaddeflection ÷ the deflection on day one1 year: ×3.005 years: ×3.29the deflection the calculation gives The deflection that arrives three years late A concrete beam that passes every check on the day it is built goes on deflecting for a decade, and ends up three times where it started. Nothing about the load changed, and nothing about the strength was ever in question.

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Bending stressCrackingKernLoad balancingPrestressPrestress lossesSuperpositionTransfer