Structural Engineering & Statics
where the load goes
Internal forces

The prestress that pushes back

On a simply supported beam a tendon is an internal matter and changes no reaction. Put the same beam on three supports and the tendon lifts it off the middle one, the support refuses, and the force it takes to hold the beam down is a reaction produced with no load applied at all.

Assumes The load put on backwards, One support too many, and what it costs to know and The moment over the support, and what it buys.

20 min read 12 figures One support too manyThe load must go somewhereDrawing as calculation

A tendon stressed inside a simply supported beam is an entirely internal transaction. It shortens the concrete, it bends it, it lifts it — and every reaction stays exactly where it was, because the beam is free to take whatever shape the tendon asks for and the supports have no opinion about it.

Add a third support and the beam is no longer free. The tendon still tries to lift it; the middle support will not let it; and the force required to hold it down is a reaction produced by prestress with no external load anywhere. That reaction bends the beam, and the moment it causes has a name — secondary, or parasitic, the second word chosen by people who wished it were not there.

It is not small and it does not helpfully cancel anything. On the beam below it is 57% of the primary moment over the middle support, with the same sign.

The tendon is a load, pointing the other wayA 12 m beam with a parabolic tendon dropping 320 mm to midspan, stressed to 1440 kN after losses. Its curvature pushes the beam up along its whole length with an intensity of 8Pe/L² = 25.60 kN/m, against an applied 21.00 kN/m — so -4.60 kN/m is left to bend anything, and the beam carries -82.8 kNm where an unstressed one carries 378 kNm. What the section then feels is 4.00 MPa of uniform compression and very little else.applied 21.00 kN/mtendon pushes back 25.60 kN/mnet -4.60 kN/m — the beam hardly bends at allmidspan moment -82.8 kNm, against 378 kNm unstressed
Fig. 1 The tendon as a load, which is the reading that makes all of this arithmetic rather than mystery. A draped tendon presses upward along its whole length with an intensity 8Pd/L28Pd/L^2, where dd is the sag measured from the chord joining its ends. On a determinate beam that is the end of the story.

The equivalent load, applied to the beam that cannot move

Everything here comes from one substitution and no others. Replace the tendon by the forces it exerts on the concrete:

  • along a parabolic drape of sag dd over a span LL, an upward pressure of 8Pd/L28Pd/L^2;
  • at a kink, a point force of P ΔθP\,\Delta\theta toward the inside of the bend;
  • at each anchorage, a force PP along the tendon and a moment PeP e about the centroid.

Apply that set to the continuous beam and solve it. The answer is the total prestress moment. Subtract the primary moment −Pe(x)-Pe(x) that each section sees from its own tendon, and what is left is the secondary one.

For the beam drawn — two 12 m spans, 1,800 kN of prestress, the tendon 300 mm above the centroid over the middle support and 320 mm below it at each midspan — the drape measured from the chord is 470 mm, so the equivalent load is 47 kN/m upward in both spans. That is a good deal more than the 22 kN/m the beam is actually carrying, which is the point of prestressing it.

A reaction with no load, and the moment it bends the beam withThe prestress moments in a 2-span beam. The primary moment is −P·e, the tendon acting on its own section, and it reaches 540 kNm over the middle support. The secondary moment is what is left when the primary is taken off the total, and it is 306 kNm — 57% of the primary, with the same sign, so it does not cancel anything. It comes from the middle support refusing to let the beam lift: 51.0 kN pressing down there and 25.5 kN lifting at each end, a reaction set that sums to -2e-13 because nothing external was applied. Its diagram is straight between supports to 2.0e-13% of its own peak, which it has to be: reactions are point forces and a point force puts no curvature in a span.05101520-600-400-200200400distance along the beam (m)bending moment (kNm)primarysecondarythe loadeverythingpushed down by 51.0 kN with nothing applied at all
Fig. 2 The four diagrams. Primary is −Pe-Pe, the tendon acting on its own section. Secondary is what is left when the primary is taken off the total. The applied load’s diagram is the ordinary continuous-beam one, and the fourth curve is everything at once — which is the only one the concrete has any knowledge of.

Which free body produced the number

The reactions are the whole mechanism, so take the beam itself as the free body with the tendon replaced by its equivalent load.

Vertically, the equivalent load sums to nothing: 47 kN/m upward over 24 m is 1,128 kN up, the kink over the middle support presses down with PΔθ=654P\Delta\theta = 654 kN, and the two anchorages pull down with 237 kN each. That has to be so — the tendon is inside the beam, and a body cannot apply a net force to itself.

The supports, however, do not know that. Solving the continuous beam under that self-cancelling load set gives reactions of 25.5 kN upward at each end and 51 kN downward at the middle, which also sum to zero and which are not individually zero. The middle support is being pushed down by 51 kN with nothing on the beam at all.

M2(x)  =  the moment those three reactions causeM_2(x) \;=\; \text{the moment those three reactions cause}

and over the middle support that is 25.5×12=30625.5 \times 12 = 306 kNm.

One support too manyThe same uniformly loaded beam with three sets of restraints, and the bending moment in each. Adding restraint moves moment from mid-span to the supports and lowers the peak — but only the first case can be solved by statics.simply supportedstatics alonesag 396.0propped at one endneeds stiffnesssag 222.8hog 396.0built in at both endsneeds stiffnesssag 132.0hog 264.0the load never changes; only what is holding the endsthe built-in case peaks at two-thirds of the simple span's moment
Fig. 3 The same structure as an ordinary redundancy problem: a beam that would deflect one way, a support that says it may not, and a force that appears to enforce the disagreement. Prestress differs only in what is trying to make the beam move.

There is a second route to the same reaction, and it is the one that makes the secondary moment feel less like an accident. Release the middle support, let the tendon lift the beam as far as it likes, measure how far — and then ask what downward force at that point would put it back. That force is the redundant, and 51 kN is the answer either way.

The deflection at x = 6, by virtual workThree diagrams: the moment from the real load, the moment from a unit load placed where the answer is wanted, and their product. The area under the third, divided by EI, is the deflection — 12690.03 here. No standard case was consulted, so the method works for any load pattern at all.real Mpeak 846.0a unit load, here and nowhere elseunit mM × marea ÷ EI = 12690.03the unit load is the only place the question 'deflection where?' is asked
Fig. 4 The unit load applied where the answer is wanted, which is the classical route to any redundant. Prestress enters it as a load like any other, and the fact that the load happens to be self-cancelling makes no difference at all to the compatibility equation.

The property that says the arithmetic is right

The secondary moment diagram is straight between supports, everywhere, always.

It has to be. It is caused by reactions, reactions are point forces, and a point force puts no curvature into a span it does not act in. Nothing in the computation imposes that — the total moment is a continuous-beam solve under a distributed load and the primary is a parabola, so their difference is a difference of two curved things that comes out flat. Measured on the beam drawn, the departure from a straight line is 2×10−152 \times 10^{-15} of the peak, which is the arithmetic saying so rather than the writer.

It is also the most useful practical fact in the subject, because it means the secondary moment is fully described by its value at each support. Two numbers for a two-span beam; three for a three-span; and linear interpolation between.

The area is the rotation, and its first moment is the movementA 12 m cantilever under a tip load of 10, with the M/EI diagram beneath it. The shaded area is 6336.0, which by the first theorem is the change of slope along the whole member. Its centroid is at 3.000 m, and the first moment about the tip is 57024 — which by the second theorem is the deviation from the tangent, and for a cantilever that tangent is horizontal, so it is the deflection itself. Integrating the curvature twice instead gives 57024.centroid at 3.00 mM/EIarea = 6336.0 · first moment = 57024by double integration: 57024
Fig. 5 Areas and first moments of a moment diagram are the route to a deflection, and a straight diagram makes both of them trivial. The secondary moment’s flatness is not a curiosity — it is what lets the redundant be found by hand for a beam of any number of spans.

The same sign, and more than half as large

Over the middle support the primary moment is −Pe=−1800×(−0.30)=+540-P e = -1800 \times (-0.30) = +540 kNm and the secondary is +306+306. Both hogging; both adding.

The reason is worth having in words rather than in signs. The tendon is high over the support, so it is pushing the beam up there — but the beam cannot go up, so the support pushes back down, and a downward force at midspan of a two-span beam causes hogging over that support. The primary and the secondary are both consequences of the same drape, and there is no mechanism by which they would have opposed each other.

At midspan the two do disagree: primary −580-580 kNm, secondary +150+150. The secondary is eating into the benefit exactly where the benefit was wanted.

over the support at midspan
primary, −Pe-Pe +540 −580
secondary +306 +150
prestress total +846 −430
applied load, 22 kN/m −396 +201
everything +450 −229

The bottom row is the only one the concrete has any knowledge of. A calculation that used the primary alone would have found +144 over the support instead of +450 — a third of the real number, on the section that governs.

2 continuous spans against 2 simple onesThe bending moment in a continuous beam, solved by the stiffness method, drawn over the moment in the same spans made simply supported. The peak sagging moment falls from 396.0 to 222.8, and a hogging moment of 396.0 appears over the supports where there was none.moment222.8 sagging396.0 hogging396.0 if the spans were simplereactions 99.0 330.0 99.0 — the inner supports carry far more than a sharethe continuous case needed stiffness; the comparison did not
Fig. 6 The applied load’s own diagram, for comparison. A continuous beam hogs over its middle support and sags in its spans; the prestress diagram above does the opposite in the spans and the same over the support, and the design question is entirely about what is left when they are added.

What the section actually sees

None of the moments above is a stress. The section carries an axial force of 1,800 kN as well, spread over 360,000 mm² of concrete — a uniform 5.0 N/mm² of compression before any bending is added — and the design question is whether the total leaves the extreme fibres inside their limits at every stage.

Over the middle support the governing combination is the +450 kNm of the bottom row, on a section modulus of 54 × 10⁶ mm³, which is 8.3 N/mm² of bending against 5.0 of uniform compression. The top fibre goes to 13.3 in compression and the bottom to −3.3 in tension, which is about where an uncracked design stops being one.

Had the secondary moment been left out, the same calculation would have found +144 kNm, 2.7 N/mm² of bending, and a bottom fibre comfortably in compression. The omitted term is the difference between a section that cracks and one that does not.

The middle third, computedThe kern of a 400 × 900 mm rectangle, computed by asking, for every direction, how far the resultant can move before the far face would be pulled. It reaches ±150.0 mm vertically and ±66.7 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±150 of 900 mm33.3% of the depth
Fig. 7 The kern, and the reason the eccentricity matters as much as the force. The tendon here sits 300 mm from the centroid on a 900 mm section whose kern is 150 mm, so it is two kern distances out — well past the point where the prestress alone would leave the far fibre in tension, and entirely dependent on the applied load arriving to correct it.
Bending is a push and a pullA section carrying a bending moment, with the stress at every height computed as the moment times the distance from the neutral axis divided by the second moment of area. It is compression above and tension below, and zero exactly at the neutral axis.neutral axiscompressiontensionI = 17.33 × 10⁶Z = 173.3 × 10³peak stress 346.2σ = M y ÷ I, at every height
Fig. 8 The bending half of the sum, drawn on its own. Every stress on this page is this triangle added to a rectangle of uniform compression, and prestressed design is the business of choosing the rectangle so that the triangle never quite reaches the bottom of the page.

The pressure line, and the profile that has no secondary moment

There is a second way to read all of this that makes one particular fact obvious.

Divide the total prestress moment by the force: −M/P-M/P is a distance, and it is the line along which the prestress force is actually acting — the pressure line. On a determinate beam it is the tendon itself. On this beam it sits 470 mm above the centroid over the middle support while the tendon sits at 300, and the 170 mm between them is the secondary moment divided by the force.

Where the force is, and where it actsThe tendon's own line down the beam, and the line the prestress force actually acts on — M/P taken from the total prestress moment. On a simply supported beam the two are the same curve, which is why nothing on a simple beam ever needs this figure. On this two-span beam they differ by exactly the secondary moment divided by the force, so the pressure line sits 0.170 m away from the tendon over the middle support. A profile for which the two coincide everywhere is called concordant, and it produces no secondary moment at all — a statement about the shape of the tendon reached from an analysis that never mentions its shape.05101520-0.50.5distance along the beam (m)height above the centroid (m)the tendonwhere it actsequal ⇒ concordant
Fig. 9 The tendon, and where its force acts. On a simply supported beam these are the same curve, which is why nothing on a simple beam ever needs this figure. The gap between them here is the secondary moment, in units of length.

Which raises an obvious question: is there a profile for which the two coincide? There is, and it is called concordant. A tendon laid on a shape proportional to any bending-moment diagram the continuous beam could have under some loading produces no secondary moment at all.

For two equal spans under a uniform load that diagram is −wL2/8-wL^2/8 over the middle support and +wL2/16+wL^2/16 at each midspan — a ratio of exactly minus two. Lay the tendon 300 mm above the centroid over the support and 150 mm below it at midspan, and the secondary moment comes out at 10−1310^{-13} of the primary. Move the support ordinate the other way, keeping everything else, and it comes back at a third of the primary.

That is a statement about the shape of a curve, reached from an analysis that never mentions its shape. The solver is told a profile and a force and it returns moments; concordance is a property nobody put in.

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: 320 mm at midspan, where the zone is 330 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.024681012-400-2000200400distance 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. 10 The zone the tendon has to stay inside for the stresses to work at both stages. Concordance is one more curve to fit in that zone, and the reason concordant profiles are not simply always used is that the concordant one is very often outside it.

The deflection, which does not have a secondary anything

The beam’s movement under prestress is worth a separate look, because it is the one quantity where the two-span case is simpler than the simple-span case rather than harder.

On a simply supported beam the tendon’s equivalent load lifts the beam and the camber is whatever that load produces. On the continuous beam, the middle support is holding it down — so the upward movement in each span is reduced by the deflection the 51 kN would have caused on its own, and the beam over the support does not move at all, because the support is there.

The result is that a continuous prestressed beam cambers less than a simple one of the same span under the same drape, which sounds like a loss and is not: the load balancing is doing its job in both, and the difference has gone into the reactions rather than into movement.

Cambered against the wet loadA 12 m composite beam whose flexural rigidity rises from 94 to 260 kN·m² when the slab sets, so the first two loads are carried by the bare steel and the rest by the composite section. Fabricated with 25.3 mm of camber, it moves through -20.7, 0.0, 7.5, 12.7 mm as the four stages arrive — 0.0 mm on the day the slab is poured, and 12.7 mm at the end, which is one part in 947 of the span. The largest curvature it ever has is 25.3 mm of hog, and it has that with nothing on it. Every shape is drawn at the same exaggeration and the drawing is a diagram of a proportion: the vertical scale is 119484 times the horizontal.levelas fabricated: 25.3 mm of camber-20.70.07.512.7self-weight of the steel: 4.6 mm on EI = 94wet concrete: 20.7 mm on EI = 94finishes and services: 7.5 mm on EI = 260imposed load: 5.2 mm on EI = 260
Fig. 11 Camber as a sequence rather than a number. Prestress puts the beam up, the load brings it down, creep takes it further over years, and the shape at any moment is a running total. A continuous beam has one more term in that total and it is a reaction rather than a load.

Where the model stops

Everything is elastic and uncracked. That is the honest range for a prestressed member in service and it is not the range at collapse: once hinges form, the secondary moment participates in redistribution like any other moment field in equilibrium with a set of reactions, and how much of it survives to the ultimate limit state is a question this model cannot answer.

The prestress force is one number. It is not: it varies along the tendon with friction, drops at transfer with elastic shortening and anchorage draw-in, and falls further over years with creep, shrinkage and relaxation. Every moment above scales with PP, so the secondary moment falls with the losses in exactly the proportion the primary does — which is one of the few things in this subject that is simpler than it looks.

And the profile is a parabola in each span. Real tendons are parabolas joined by short reversed parabolas over the supports, because a kink is a point force and a point force on a support region is not a thing anybody wants. The reversed curve spreads that force over a metre or so and changes the drape slightly; the arithmetic is the same arithmetic with one more segment.

What the pictures cannot show

The moment diagrams are drawn at one instant. A prestressed beam has at least three that matter — transfer, service, and years later — and the secondary moment is present at all of them, scaled by whatever force is left.

Nor can they show the beam’s own construction. A two-span beam is often built as two simply supported spans, stressed, and only then made continuous, in which case the secondary moment for the tendons stressed before continuity is zero, and for those stressed after it is not. The structure has two histories in it and the diagram has one.

Three spans, and why the middle one is where the trouble is

Everything above generalises without a new idea, and the arithmetic is worth stating once because the answer changes.

A three-span beam has two interior supports, so the secondary moment diagram has two ordinates and a straight line between them. Under a tendon draped the obvious way — low in the spans, high over both supports — the equivalent load is upward in all three spans, both interior supports are held down, and the secondary moment is hogging over both.

What differs is the outer spans. Their ends are anchorages rather than continuity, so a tendon that is concordant for the interior of the beam is generally not concordant near the ends, and the secondary moment does not go to zero until the end support itself. On a long viaduct the diagram is a sawtooth of straight segments whose peaks sit over the piers, and the pier that carries the largest one is not always the one carrying the most load.

The answer arrives in instalmentsThe hogging moment at support 1 of a three-span beam, cycle by cycle. It starts at the fixed-end moment of 264.0 kNm — the value with every joint clamped — and settles at 316.8 kNm against an exact 316.8. The error falls by about a factor of four per cycle: 69.30, 17.32, 9.67, 2.92 kNm after one, two, three and four. Two cycles is an engineering answer and nobody had to invert anything.01234560100200300400cycles of distributionmoment at the support (kNm)exact: 316.8all joints clamped
Fig. 12 The same joints balanced by hand. Prestress is one more set of fixed-end moments in that table — computed from the equivalent load rather than from the applied one — and everything after that step is identical.

The ladder from here

Later rungs on this anchor: linear transformation, which is the theorem that moving a tendon’s support ordinates without changing its drape leaves the total moment unchanged — the primary and secondary swap magnitudes exactly. The concordant profile as a construction rather than a coincidence, drawn from the beam’s own influence lines. Secondary moments at collapse and the redistribution allowance. Secondary shear, which is the derivative of the diagram above and is often the term that decides the links near a support. The same effect in a frame, where prestressing a beam pushes its columns sideways. And the case where the secondary moment is deliberately used: a tendon profile chosen so that its parasitic moment relieves a support the load overloads.

Where this goes next

prestress alone-6.3320.61tensionat transfer-2.4716.76tensionin service7.613.82with no prestress12.67-12.67tension 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. simply supportedstatics alonesag 32.0propped at one endneeds stiffnesssag 18.0hog 32.0built in at both endsneeds stiffnesssag 10.7hog 21.3the load never changes; only what is holding the endsthe built-in case peaks at two-thirds of the simple span's moment One support too many, and what it costs to know Add a redundant restraint and the load has two routes to the ground. Equilibrium cannot say how it splits, and the answer turns out to depend on stiffness — which is a different kind of question. moment19.6 sagging24.5 hogging30.6 if the spans were simplereactions 14.0 38.5 38.5 14.0 — the inner supports carry far more than a sharethe continuous case needed stiffness; the comparison did not The moment over the support, and what it buys Run a beam over its supports instead of stopping at each one, and the mid-span moment falls by a third while a new moment appears where there was none. Nothing was added but continuity. this support 0.01 lowmoment19.6 sagging24.5 hogging30.6 if the spans were simplereactions 14.0 38.5 38.5 14.0 — the inner supports carry far more than a sharethe continuous case needed stiffness; the comparison did not The support that moved A redundant structure knows things statics cannot see. Settle one support by ten millimetres and a complete set of bending moments appears — in equilibrium with no load at all, and larger for a stiffer beam.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

Concordant profileContinuityEccentricityEquivalent loadIndeterminacyLoad balancingPressure linePrestressReactionSecondary momentSelf equilibratingSuperpositionTendon profile