The beam stressed one bay at a time

A tendon draped through a continuous beam is, read as a load, a set of upward pushes where it curves and inward pulls where it is anchored. On a beam with more supports than it needs, those pushes try to lift the beam off its interior supports, the supports refuse, and the prestress pushes back: reactions with no load applied, and the straight-line moment they make between supports, which is the secondary moment. The tendon that can be moved showed that the split between primary and secondary is a choice of bookkeeping, and the relief that belongs to the pressure line found the term that the split hides in a shear check — a constant secondary shear in each span, which beside the end support of a two-span beam was half the shear left for the links.

Every one of those beams was stressed in one operation. The whole tendon reached its full force on the whole beam before anything else happened, and the secondary moment was a property of the finished structure. That is not how long continuous beams are usually built. A viaduct, a long transfer beam or a post-tensioned slab running past six columns is cast a bay at a time, and each bay’s length of tendon is coupled to the one before it and stressed when its own concrete is strong enough. The first bay is stressed while it is the whole structure. The second is stressed onto a beam of two spans, the third onto three. At every stage the beam is a different indeterminate structure, with its own supports and its own secondary reactions.

The question is what the finished beam carries. Superposition still holds — each stage is linear, and the moments simply add — but what is added is each stage’s effect on the beam that existed then, not on the beam that exists now. Whether the sum is larger or smaller than the one-stage answer, by how much, and whether the creep that follows moves it back, is what the rest of this essay works out.

Two spans stressed in two pulls keep exactly half #

Start with the beam from the shear essay: two 16 m spans, a tendon carrying 2,400 kN, running from the centroid at each end to 320 mm below it at midspan and 300 mm above it over the middle support. Stressed in one operation, the tendon tries to lift the beam off the middle support, and holding it down produces a secondary moment of 409 kN·m there, sagging — positive in the sense the tendon wants, opposite to the hogging the applied load will put over that support.

Now build it in two stages. The first bay is cast on its two supports and its tendon stressed. A beam on two supports is statically determinate: the tendon can camber it upward freely, nothing resists, and no reaction is provoked. The first bay’s tendon makes no secondary moment at all. The second bay is then cast against the first, its length of tendon coupled on at the joint over the middle support, and stressed. Now the beam has three supports. The second bay’s curvature tries to lift it off the middle one, and the middle support holds it down.

The answer for the second stage is 204 kN·m at the middle support — exactly half of 409. In the one-stage analysis the two spans are mirror images, so each span’s curvature contributes the same amount to the moment over the middle support, and the 409 is two equal halves. Built in stages, the first span’s half never arrives, because its curvature was spent on a beam with nothing to resist it. The first bay simply cambered, and by the time the second bay was attached, the first was already in its deformed position and the new continuity locked that position in as the zero of everything that followed.

This is the whole of the effect in its simplest form. A span’s tendon makes secondary moment only at supports that are interior when it is stressed. A support that is the end of the beam when a bay is stressed is a support the bay’s curvature cannot load, because an end support of a beam is one the beam can rotate on freely. The one-stage analysis credits every span with the effect it would have had if every support had been there; the staged beam credits each span only with the effect it had on the supports behind it.

The beam as it is built #

With more bays, the sum is less obvious, because each new stage also changes the moments the earlier stages made. Take four spans of the same 16 m and the same tendon, and build from the left. After the second bay is stressed, the beam is the two-span beam above, with 204 kN·m over its first interior support. After the third, there are two interior supports. After the fourth, three.

The moment at the first interior support does not stay at 204. When the third bay is stressed it falls to 150; when the fourth is stressed it rises back to 164. The newest support always carries the largest moment, 218 and then 219 kN·m, and the supports behind it settle at values that no single stage put there. The diagram after the last stage — 164, 160 and 219 over the three interior supports — is the beam’s actual secondary moment, and it is lopsided. A four-span beam of identical spans, identical tendon and identical section, symmetric in every respect except the order in which it was built, carries a secondary moment a third larger over one support than over its mirror image.

Read from the one-stage side, that lopsidedness is impossible. The one-stage analysis of a symmetric beam gives a symmetric answer — 350, 234, 350 — and nothing in it knows that the builder started at the left. The staged answer remembers. A designer who checks a long beam against the one-stage diagram is checking a structure that was never built, in the sense the structure that was never complete argued for composite beams: the order of construction is a load case.

Each new bay pushes on the beam behind it #

The accumulation becomes readable when each stage is drawn alone — the moment the new bay’s curvature adds, imposed on the beam that exists when it is stressed.

Every stage after the first has the same shape. The new bay’s curvature tries to lift the beam off the support at the joint — the support that has just become interior — and the moment it provokes there is about 218 kN·m whatever the stage. Behind that support the moment changes sign: stage 3 adds −54 kN·m at the first support, stage 4 adds −58 at the second, and the effect one support further back is small again and positive. This is the alternation every continuous beam shows when one span is disturbed, and its ratio is fixed by the three-moment equation. Across unloaded equal spans the support moments obey $M_{i-1} + 4M_i + M_{i+1} = 0$, so each is the next multiplied by a root of $r^2 + 4r + 1 = 0$: $r = \sqrt{3} - 2 \approx -0.27$. A moment at one support reappears at the next with a little over a quarter of its size and the opposite sign — the beam’s own version of the carry-over that moment distribution passes from joint to joint. Here −54 against 218 is −0.25 and −58 against 219 is −0.26, a short beam with an end nearby falling just under the ratio of an endless one.

So the secondary moment over any support of the finished beam is built from three pieces: the large positive moment its own next bay put there when the support first became interior, a negative carry-over from the bay after that, and diminishing alternating terms from the bays beyond. What it never receives is a contribution from the bay before it — from the span whose far end it is — because that span was stressed when the support was still the end of the beam. In the one-stage analysis both neighbouring spans contribute, and they contribute about equally. Losing one of the two is why the staged answer is close to half, and the carry-overs from later stages, which subtract, are why it is slightly less than half anywhere but in the last bay.

The last support is the exception and the lopsided diagram says why. Over the third interior support, nothing comes later to subtract. The 219 kN·m the fourth bay put there stays, against the one-stage 350 — 63 per cent, against 47 per cent over the first support. The beam keeps more of its secondary moment at the end where it was finished than at the end where it was started.

The four spans together #

Laid over the one-stage diagram, the staged secondary moment is a flatter shape well inside it.

Stressed at once, the secondary moment over the three interior supports is 350, 234 and 350 kN·m. Bay by bay it is 164, 160 and 219. At the first interior support the difference is 186 kN·m, which against a primary moment of $P \times e = 2400 \times 0.30 = 720$ kN·m is a quarter of everything the tendon does at that section. That is not a refinement.

What the difference means for a design depends on the sign of what it is added to. Over an interior support, the applied load hogs the beam and the secondary moment, being sagging, relieves it. A secondary moment of 164 rather than 350 is 186 kN·m less relief, so the support section carries 186 kN·m more hogging than the one-stage analysis claims. In the spans, the applied load sags the beam and the secondary moment adds to it; there the staged beam is the less loaded one. So the one-stage analysis errs on opposite sides at the two places a continuous beam is designed: it understates the support moment and overstates the span moment.

The support is where the error is unsafe, and it is unsafe at the worst time. The staged diagram is the one the beam has on the day the last bay is stressed, when the concrete of the newest bay is youngest, its strength lowest and the stresses in the tendon highest — before any of the losses that the member’s own drying takes back have arrived. A check made at transfer against the one-stage secondary moment is a check of the wrong beam at the moment the right one matters most.

The secondary shear, and the end it is read at #

The relief essay asked whether the secondary shear beside an end support — the term that doubled the shear left for the links — would be larger or smaller in a beam built in stages. Secondary shear is the slope of the secondary moment diagram, so it is constant in each span and equal to the change in secondary moment across the span divided by its length. The staged diagram has less moment and different slopes, so it has different shears.

It is smaller, and by about as much as the moment. In the first span of the four-span beam the secondary shear is 21.9 kN stressed at once and 10.3 kN bay by bay. In the last span it is 21.9 and 13.7, with the sign reversed — the same lopsidedness the moments showed, seen through their slopes. The second span barely carries any: −0.3 kN, against −7.3 for the one-stage beam, because the first two interior supports finished with nearly equal moments and a straight line between nearly equal values is nearly flat. And in the two-span beam from the shear essay the secondary shear halves exactly, from 25.5 kN to 12.8, as the moment did.

That answers the question as it was put, and the answer is the opposite of the one that would trouble a designer. In the shear essay the shear left for the links 0.8 m inside the end support was 25 kN after the tendon’s slope had taken its share, and the secondary shear brought it to 51. Stressed bay by bay, the same beam on the day of the last stressing carries $25 + 13 = 38$ kN there. The one-stage number is larger, so the one-stage check of the end shear is conservative. Where the support moment was understated, the end shear is overstated; the two errors run in opposite directions for the same reason, because a smaller secondary moment over the interior support means both less relief there and a gentler slope towards the end.

Creep takes most of it back #

None of these numbers stays put. The staged secondary moments are what the beam has when the last bay is stressed, and the concrete then creeps under the stresses the prestress maintains. A beam whose secondary moment differs from the one-stage value is a beam whose continuity was made after part of its curvature had already happened. The first bay cambered freely, was joined to the second, and its tendon is still in it, still trying to camber it further. Creep grows that curvature, and the growth is now resisted by supports that were not there when the original camber happened. The moments creep provokes are the moments the one-stage beam had from the start, being slowly paid in.

This is the problem worked through in the support that had no moment when it was cast, where beams loaded as simple spans were later made continuous and the support moment grew over twenty years towards the value a monolithic beam would have had. It is the same mechanism with a different action — a held curvature from the tendon rather than a held load — and it has the same one-line answer by the age-adjusted modulus. The moment after creep is the staged moment plus a fraction of the difference between the one-stage and the staged:

$$M_{\infty} = M_{\text{staged}} + \frac{\varphi}{1 + \chi\,\varphi}\left(M_{\text{one stage}} - M_{\text{staged}}\right),$$

where $\varphi$ is the creep coefficient and $\chi$ the ageing coefficient — the number that makes this shortcut agree with the full superposition integral. That essay found it to be about 0.8 for this class of problem, a restraint that grows while the action is held, against 1.33 for a strain held constant; the class matters, and this is the first one.

The secondary shear in the first span rises from 10.3 kN with the curve. At a creep coefficient of 1 it is 16.7; at 2, 19.2; at 3, 20.5. The fraction recovered is $\varphi/(1 + 0.8\varphi)$, which is 68 per cent at a coefficient of 1.5, 77 per cent at 2 and 88 per cent at 3. Over the first interior support the moment goes from 164 kN·m to 307, against the one-stage 350. Creep does most of the work and stops short of finishing it — for an ageing coefficient of 0.8 the fraction reaches 1 only at a creep coefficient of 5, which no structural concrete carries in any member.

This is what turns the two errors of the one-stage analysis into a design rule. There are two beams to check, not one: the staged beam at the day the last bay is stressed, and the crept beam years later. The support moment is worst in the first: least secondary relief, youngest concrete, full tendon force. The end shear and the span moments are worst in the second, where the secondary moment has grown back towards the one-stage values. The one-stage analysis is never exactly either of them, but it sits just above the second, which is why designing the spans and the end shear to it works in practice — it is a slightly conservative estimate of a beam that will exist in a decade. Over the supports it is an unconservative estimate of the beam that exists on the first day.

However many bays #

It is fair to ask whether the factor of a half is a peculiarity of short beams. The fraction kept at the first span, as the number of equal spans grows:

Two spans keep exactly 0.50 of the one-stage secondary shear in the first span. Three keep 0.46, four 0.47, and from there the curve is flat: eight spans keep 0.47. The carry-overs that subtract from the earlier supports decay by a factor of about four at each further support, so beyond the third bay a new stage changes almost nothing near the start of the beam. The last span keeps more — 0.62 to 0.66 for any number past two — because it has the full moment of its own stage and nothing after it. After creep at a coefficient of 2, the first span is back to 0.88 of the one-stage value at every length.

So the result is not a feature of one beam’s proportions. A long continuous beam built from one end carries roughly half its one-stage secondary moment over most of its length on the day it is finished, somewhat more towards the end where it was closed, and eight or nine tenths of it once the concrete has crept. The numbers move with unequal spans — three spans of 16, 24 and 16 m keep 191 and 180 kN·m against the one-stage 314 over the two supports, 61 and 57 per cent — but the reason does not.

Why the beam remembers its own history #

It can seem odd that a structure with identical members, identical tendon and identical supports could carry different moments depending on the order it was assembled in. A linear elastic structure has one state of stress for each set of applied actions, and the finished beam has the same supports and the same tendon however it was built.

The resolution is that the actions are not the same. A tendon stressed on a bay that is free to camber does its work on a determinate structure, and the camber it produces is not a stress — it is a change of shape. When the next bay is cast against the cambered one, the joint is made in that deformed position, and continuity is established around it. The cambered shape becomes the beam’s reference geometry. In the one-stage beam no such shape ever exists: the camber is imposed on a beam that is already continuous, the supports refuse it, and the refusal is the secondary moment. The staged beam has absorbed part of its tendon’s curvature as geometry, and the one-stage beam has absorbed all of it as stress. They are the same beam with different stress-free shapes.

This is the same reason a support that moved produces moments in an indeterminate beam and none in a determinate one: an imposed deformation is resisted only where there is redundancy to resist it, and the redundancy here arrives in stages. Every bay added is one more redundant support, and each redundancy can only resist deformations that happen after it exists.

Creep weakens that separation, and the reason is that creep is a deformation that happens after. The tendon in the first bay goes on trying to curve its concrete for decades, and the curvature it adds now is resisted by every support that has been built since. That is why creep moves the moments towards the one-stage values and not away from them, and why it can never overshoot in practice: what it adds is a fraction of the original curvature, resisted by the finished structure, and the fraction of a creep coefficient that the ageing coefficient lets through is below one.

What the model leaves out #

The joints here sit over the supports, which makes each stage’s arithmetic clean. In practice the joint between bays is put about a fifth of a span past the support, where the dead-load moment is small, and each pour includes a short cantilever into the next bay. That moves the numbers — part of the next bay’s curvature is then stressed on the earlier stage — without changing the reason: each length of tendon still loads only the supports that are interior when it is stressed.

Every stage here has the same creep coefficient and the same modulus. A real beam built over months has bays of different ages, so the oldest bay has crept furthest by the time the last is stressed and will creep least afterwards. The first span — the one with the largest shortfall — is also the one with the least creep left to recover it, so the 77 per cent above is an upper estimate for that span. The newest bays recover more of a smaller deficit.

The tendon force here does not change with time, but every tendon loses force as the concrete shrinks and creeps and the steel relaxes, and the secondary moments fall in proportion. The comparison between staged and one-stage survives a uniform loss, since both diagrams scale together, but the absolute moments after creep are smaller than the figures above by the long-term loss, commonly a tenth to a fifth.

And at collapse, provided the sections over the supports can rotate, the construction history matters much less. A secondary moment is self-equilibrated — it is made by reactions that sum to nothing — so a beam that can form hinges and redistribute will reach the same collapse load whatever secondary moment it started with, which is the same conclusion the composite beams reached about the order of construction. The staged moments matter for the stresses in service, for cracking over the supports at transfer, and for any section that cannot rotate far enough to make the history irrelevant.

The shape of the argument #

The earlier essays on secondary prestress found the secondary moment as a property of a finished continuous beam, and then found that its split from the primary was a matter of where the tendon was described from. This one finds that the secondary moment is also a matter of when the tendon was stressed, and that the finished beam does not carry the number its one-stage analysis gives.

Two things follow that a designer of a staged beam can use. The first is a rule of thumb that is not merely approximate but derived: a continuous beam stressed one bay at a time from one end carries about half its one-stage secondary moment over its earlier supports, exactly half in the two-span case, and somewhat more over its last one. The second is that the beam must be checked twice — at the day of the last stressing, when the supports carry the least secondary relief, and after creep, when the secondary moment has returned to eight or nine tenths of the one-stage value — and that the one-stage analysis on its own is the right number for neither check, though it sits close enough to the second to be used for it.

Still open: the tendon stressed from both ends of the beam #

Every beam here was built from one end, and the lopsided diagrams record that. A long viaduct is often built from both abutments towards a closure bay in the middle, or outwards from a central pier in both directions, and the closure bay is the one stage stressed onto a beam with interior supports on both sides of it. Whether a beam built towards a central closure is symmetric, whether the closure bay then carries the largest secondary moment on the beam, and whether the order of construction can be chosen to place the secondary moment where the applied load needs relief — making the construction sequence a design variable rather than an inconvenience to be checked — is the question that a closure pour asks of this one.