The tendon that can be moved

A prestressed continuous beam pushes back against its own supports, and the moments that follow are called secondary — a word that has done a great deal of damage, because it suggests something small enough to correct for afterwards.

Half as large as the primary and in the same direction. It is not a correction, and this page is about the fact that its size is a choice.

Move the tendon and hold the drape #

Lift the tendon over the middle support from 450 mm to 150 mm, and lower it at mid-span so that the drape — the sag measured from the chord joining the support ordinates — is unchanged at 675 mm.

Two profiles, two completely different splits, one identical total. The primary lost 960 kNm and the secondary gained exactly 960. Nothing about what the beam experiences has changed.

That is the theorem of linear transformation: moving a tendon’s ordinates at the supports of a continuous beam, by any amounts, without changing the drape in any span, leaves the total prestress moment unchanged everywhere.

Which free body produced the number #

The theorem looks like a coincidence and is a consequence of one free body.

Take a span of the beam and replace the tendon by the loads it applies — a curved tendon is a load pointing the other way. A parabolic drape $d$ over a span $L$ at force $P$ delivers an upward intensity $8Pd/L^2$. At each end, where the tendon leaves the span, it delivers a force $P$ along its own line and an eccentric anchorage moment.

The upward load depends on the drape and on nothing else. Moving both ends of the chord vertically does not change the drape, so the equivalent load on the span is untouched — and the beam’s total moment is the response to a load set that has not moved.

What does change is the anchorage moment at the ends, $Pe$, and that is the primary. Since the total is fixed and the primary has changed, the secondary changes by the opposite amount. The theorem is that arithmetic, made once.

Why “secondary” is the wrong word #

Before going further it is worth saying what the two moments are and are not, because the vocabulary actively misleads.

The primary moment is $-Pe$: the tendon pulling on a section at an eccentricity, computed as though the beam were free to do whatever it liked. On a simply supported beam it is the whole answer, because the beam is free to lift.

The secondary moment is what the supports add by refusing. A prestressed beam with a drape wants to arch upward; an interior support holds it down; the force required to hold it down is a reaction with no applied load anywhere, and the moment diagram of that reaction set is the secondary moment.

Neither word describes size. In the second figure the secondary is three and a half times the primary, which is a fair description of most real continuous tendons: the interior support ordinate is usually modest and the drape is usually large, so the primary is small and almost all of the prestress moment is the reaction set. Calling that secondary is a statement about the order in which it was calculated and about nothing else.

The better name, and the one some texts use, is the parasitic moment — which is worse, because it suggests something unwanted. It is neither unwanted nor optional; it is the price of continuity, exactly as a support that moves puts a moment into a redundant beam, and its sign is often helpful.

The pressure line is the thing that is real #

There is a single curve that stays fixed under linear transformation, and it is the one worth drawing.

The pressure line is $M_{\text{total}}/P$: the position at which the prestress force would have to act to produce the moment the beam actually has. It is 0.675 m from the centroid at the support in both cases, because the total moment is 2,160 kNm in both and the force is 3,200 kN in both.

That gives the cleanest statement of the whole subject. A tendon can be moved; the pressure line cannot. The stresses in the concrete depend on the total moment, so they depend on the pressure line, so they are unchanged by any linear transformation — and the tendon’s position at the supports is free for the designer to choose on other grounds entirely.

Concordance, which is not a virtue #

A profile whose tendon and pressure line coincide everywhere has no secondary moment at all, and is called concordant.

It is a real and useful construction: since the pressure line is fixed, and every profile with the same drape shares it, the concordant profile is simply the pressure line itself used as a tendon. Draw the pressure line for any convenient profile, put the tendon on it, and the secondary moments vanish.

What it is not is better. The total moment is identical, the concrete stresses are identical, the required force is identical. The only thing that changes is where the bookkeeping puts 720 kNm, and a designer who spends effort finding the concordant profile has bought a tidier calculation and nothing else.

It is also, usually, unbuildable. A concordant profile has whatever ordinates the pressure line has, and the pressure line does not care about cover, duct diameter, or the reinforcement it has to pass between. The practical profile is the one that fits, and the secondary moment it produces is then computed rather than avoided.

There is a second reading of the pressure line that is more useful than the first. Because the concrete stresses depend on $M_{\text{total}}$ and on $P$, and the pressure line is their ratio, the pressure line is the eccentricity at which a simply supported beam would need its tendon to produce the same stresses. So a continuous beam can be checked section by section exactly as a simple beam’s four inequalities are checked, with the pressure line’s ordinate used in place of the tendon’s.

That is the reason the construction survives. It converts a redundant problem into a determinate one for the purposes of the section check, at the cost of one analysis to find the total moment — and everything about tendon zones, Magnel diagrams and permissible stresses then applies without modification.

What the choice is actually made on #

If the total is fixed, what does a designer gain by moving the tendon?

Room. Over an interior support of a bridge deck there is a diaphragm, column reinforcement, bearings and often several tendons from adjacent spans. Lifting the tendon 300 mm may be the difference between a profile that can be built and one that cannot, and the theorem says it costs nothing.

Cover and duct curvature. A tendon has a minimum radius, and a profile with an abrupt lift at a support cannot achieve it. Linear transformation gives a family of profiles to choose the buildable one from.

Anchorage geometry. The tendon’s slope at the end anchorage sets where the stressing jack goes and what the anchorage zone has to carry.

None of those is a structural quantity, which is the point. The theorem hands a genuinely free variable to the person detailing the reinforcement, and free variables in this subject are rare.

The same theorem, read as a self-stress state #

Everything above has a shorter statement in the language the rest of this collection uses for redundant structures.

A prestressed continuous beam carries a self-stress state: a set of internal forces in equilibrium with no external load. Every indeterminate structure can carry one, a settlement produces one, a lack of fit produces one, and prestress produces one deliberately.

The primary and secondary moments are two ways of splitting that state, not two states. The split is a bookkeeping choice — “what would the tendon do if the beam were determinate” against “what did the supports add” — and linear transformation is the observation that a family of tendon geometries produce the same self-stress state with different splits.

That framing explains something the primary–secondary language obscures. The reason the total is what matters is that the concrete does not know which part of the moment came from where. A section carrying 2,160 kNm of prestress moment carries 2,160 kNm, and asking how much of it is parasitic is a question about the calculation rather than about the beam.

Where it stops being free #

Two boundaries matter and neither is small.

The drape must be held. Everything above depends on it. Changing the ordinate at a support without compensating at mid-span changes the drape, changes the equivalent load, and changes the total. In the first two figures the drape was held at 675 mm deliberately; had it not been, the totals would have differed by the ratio of the drapes.

The tendon must stay inside the section. The transformation is unlimited mathematically and bounded physically. Lifting the support ordinate needs the mid-span ordinate to drop by half as much per span, and both run into the concrete face. On this 1,400 mm section the family runs out at about 600 mm either side of the centroid.

There is a third boundary that is easy to miss: the theorem is about a beam with unyielding supports. If the supports settle, the secondary moment is not what this calculation says — the beam is carrying an imposed displacement as well as a prestress, and the two self-stress fields add.

Three spans, and the pattern #

The effect grows with the number of redundancies, and the split moves with the profile in each span.

The secondary is always a straight line between supports, and the figure asserts that it is — flat to within 5 × 10⁻¹³ per cent of its own peak. That is not a numerical curiosity; it is the identifying property. The secondary moment is the moment diagram of a set of point reactions, and point forces put no curvature into a span.

That gives a check anybody can make on any output: if the “secondary” moment diagram a program prints is curved inside a span, it is not the secondary moment.

What it adds to #

The prestress moments are not the whole story, and the two have to be put together on the right terms.

At the ultimate limit state the total prestress moment — primary plus secondary — is added to the applied moment with its own sign. The secondary is hogging here and so is the applied moment over the support, so they add rather than cancel, and the support section is being designed for more than the applied analysis alone reports.

That is the practical reason the secondary matters. It is not a correction to be neglected; it is a moment of the same order as the applied one, in the same direction, at the section that governs. Codes vary in whether it may be redistributed and by how much, which is a question about whether a self-equilibrating field can be shed the way an applied one can — and it is genuinely harder, because shedding it changes the reactions.

A check anybody can make #

Three properties of a correctly computed secondary moment are worth having, because between them they catch most mistakes.

It is straight between supports. Point reactions produce no curvature. A curved secondary diagram inside a span means the primary has been subtracted wrongly, or the equivalent load has been applied to the secondary as well as to the total.

Its reactions sum to zero. The secondary field is self-equilibrating: nothing external was applied, so the reactions it implies must add to nothing. In the two-span case they are 102.9 kN down at the middle and 51.4 kN up at each end, summing to zero to within a part in 10¹³.

It vanishes on a determinate beam. A single simply supported span has no redundancy, so it has no secondary moment at all, whatever the tendon does. If a program reports one, the model has a restraint in it nobody intended — which is the commonest way a spurious secondary moment appears, from an axial restraint at both ends of a beam that was supposed to slide.

Those three take a minute and are worth more than re-reading the analysis, because each of them fails in a different way for a different reason.

Where the model stops #

The tendon is parabolic in each span, with no reverse curvature. A real profile has a reverse curve over each support, and its equivalent load has a downward patch there rather than a point force. The total is close and the local moments near the support are not.

The force is uniform along the tendon. It is not: friction and wobble mean the force at mid-span is lower than at the jack, so the equivalent load is not quite uniform and the primary is not quite $-Pe$.

Losses are applied as a single factor. Creep, shrinkage and relaxation all change $P$ with time, and the secondary moment scales with $P$ while the applied moment does not — so the ratio between them drifts for decades.

The section is uncracked and prismatic. A haunched member has a different stiffness distribution, so the same equivalent load gives a different total moment and the transformation theorem still holds but the numbers do not.

The beam is prismatic and uncracked along its whole length. A partially prestressed member cracks in service, its stiffness falls where it cracks, and the secondary moment — which is a compatibility quantity — moves with it.

And nothing here is an ultimate-limit-state calculation. The secondary moment is computed elastically; at collapse the beam has hinges in it and the redundancy that produced the secondary moment may no longer exist.

What to carry away #

The drape is the structural variable and the support ordinate is not. The equivalent load, and therefore the total moment, depends on the sag measured from the chord joining the support ordinates. Move the chord and nothing happens; change the sag and everything does.

The pressure line is the only curve worth drawing. It is fixed for a family of profiles, it is what the concrete stresses depend on, and it converts a redundant beam into a determinate section check.

And the secondary moment is usually the larger half. Half again as large as the primary in the first profile here and three and a half times as large in the second, hogging over a support where the applied moment is also hogging, and added to it. A design that treats it as a correction has left out the larger of two terms.

Two other essays deal with the same distinction between a force a structure was given and a force it developed because it is continuous. A moment that was moved on purpose is redistribution used as a design tool, and a strain that was imposed is the general case of an action that exists only because the structure is restrained.

The ladder from here #

Later rungs on this anchor: the concordant profile constructed from the beam’s own influence lines rather than found by trial. Secondary shear, which is the derivative of the straight-line diagram above and is often the term that decides the links near a support. Secondary moments at collapse, and how much of them a code allows to be redistributed. The same effect in a frame, where prestressing a beam pushes its columns sideways and the secondary field includes a sway. Prestress in a two-way slab, where the equivalent loads run in two directions and the secondary field is a surface. And the case where the parasitic moment is used deliberately: a profile chosen so that its secondary moment relieves the support the applied load overloads, which is the only place in this subject where a designer gets to choose a moment diagram.

Linear transformation was published by Guyon in the 1950s and is one of the very few theorems in structural engineering that gives a designer a genuinely free choice. It is also one of the least used, because the freedom it grants is over a detail — where a duct sits at a support — and the person who benefits from it is usually not the person who proved that it costs nothing.