The prestress that pushes back

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 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 $d$ over a span $L$, an upward pressure of $8Pd/L^2$;
• at a kink, a point force of $P\,\Delta\theta$ toward the inside of the bend;
• at each anchorage, a force $P$ along the tendon and a moment $P e$ about the centroid.

Apply that set to the continuous beam and solve it. The answer is the total prestress moment. Subtract the primary moment $-Pe(x)$ 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.

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 $P\Delta\theta = 654$ 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.

$$M_2(x) \;=\; \text{the moment those three reactions cause}$$

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

That is the shape of an ordinary redundancy problem and nothing more exotic: 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. It is the unit load applied where the answer is wanted, which is the classical route to any redundant; prestress enters that calculation 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 $2 \times 10^{-15}$ 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.

Areas and first moments of a moment diagram are the route to a deflection, and a straight diagram makes both of them trivial to take. So the flatness is not a curiosity to be noticed and set aside: it is what lets the redundant be found by hand on a beam of any number of spans, with a table of triangles instead of an integral.

The same sign, and more than half as large #

Over the middle support the primary moment is $-P e = -1800 \times (-0.30) = +540$ kNm and the secondary is $+306$. 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 $-580$ kNm, secondary $+150$. The secondary is eating into the benefit exactly where the benefit was wanted.

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.

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 eccentricity matters here as much as the force does, and the kern says why. The kern of a 900 mm rectangle is 150 mm deep, and the tendon over the support sits 300 mm from the centroid — 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.

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: $-M/P$ 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.

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 $-wL^2/8$ over the middle support and $+wL^2/16$ 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 $10^{-13}$ 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 balanced beam has no moments at all, secondary included #

The word parasitic is a complaint, and there is one arrangement in which the complaint is plainly wrong.

Choose the drape so that the equivalent load exactly cancels the applied one. Here that means $8Pd/L^2 = 22$ kN/m, so $d = 22 \times 144/(8 \times 1800) = 220$ mm of drape instead of the 470 the beam actually has. Now add the two loadings: the beam is carrying nothing. Not “a small residual” — nothing. Every reaction is zero, every deflection is zero, and every moment is zero, at every station, including over the middle support.

Which means the total prestress moment is exactly minus the applied-load moment, everywhere. The primary is $-Pe$, the secondary is the difference, and the secondary is doing precisely the job of making that identity hold on a beam whose middle support would otherwise have refused. The section carries $P/A = 5.0$ N/mm² of uniform compression and nothing else — no bending, no tension anywhere, and no stage at which any fibre is near a limit.

So the secondary moment is not a parasite on the mechanism; it is part of it. A determinate beam can be balanced because it is free to take the tendon’s shape. A continuous beam can be balanced because the supports push back — the pushing back is what converts a self-cancelling internal force system into a set of reactions that oppose the real load.

The beam drawn is over-balanced by a factor of 2.1, which is why it has large moments of both kinds. That is a deliberate choice rather than a mistake: full balance under permanent load leaves nothing in hand for the variable load, and the drape is usually set to balance the permanent action alone with the section sized for the variable one on top. But the limiting case is worth knowing, because it says what all the arithmetic above is arithmetic about.

The other reaction, which is horizontal #

The secondary moment comes from the supports refusing a vertical movement. There is a second refusal, in the other direction, and on a long continuous member it is the larger problem.

The prestress shortens the beam. Elastically, $\sigma/E_c = 5.0/34{,}000 = 1.47\times10^{-4}$; creep multiplies that by two or three over the years, and shrinkage adds another $3\times10^{-4}$ or so, for a total strain of about $8\times10^{-4}$. Over the 12 m from the beam’s centre to each end that is 10 mm of movement at each end, all of it inward.

If the supports let it happen, nothing follows. If they do not — a beam cast monolithically with columns, a deck on stiff piers, a slab tied into a stair core — then the restraint applies a horizontal force back into the beam and an equal one into itself.

The force is the restraint’s stiffness times the movement, so it is a property of what is holding the beam rather than of the beam. A 400 mm square column 4 m high offers about 3.4 kN per mm, so 10 mm is 34 kN — one part in fifty of the prestress, which is negligible as a loss. What it is not negligible for is the column: 34 kN at 4 m is a moment of 136 kNm, applied to a member sized for vertical load, by a movement nobody applied.

That is restrained shortening with prestress as the cause instead of temperature, and it has the same two properties: the force depends on the restraint rather than on the member, and it grows for years after the structure is finished. It is also why long post-tensioned floors are built with a temporary gap at one end, or with the stiff cores released until the stressing is done — the restraint is removed for the period during which the movement happens, and reinstated afterwards.

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.

None of which is a single number, because camber is 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 of everything that has happened to it. A continuous beam has one more term in that total, and the term 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 $P$, 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.

None of that needs a new method. The same joints can be balanced by hand, with prestress entering as one more set of fixed-end moments computed from the equivalent load rather than from the applied one; everything after that step is identical to the two-span case.

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.