The structure that was never complete

A structural analysis takes a complete structure and applies a load to it. That is a fiction about how buildings come into existence, and for most structures it is a harmless one: the sequence of erection changes nothing that matters and the finished frame carries what the finished frame carries.

For a large minority it is not harmless at all, and the reason is simple enough to state in a sentence. A member carries what was present when it became structural, and if half the load arrived before the structure was finished, half the stresses belong to a structure that no longer exists.

Nothing in that figure is a modelling subtlety. It is two different structures analysed correctly, and which one exists is decided by whether somebody put props under the beam for a fortnight.

Superposition, applied to structures rather than loads #

The arithmetic is the ordinary superposition this collection uses everywhere, with one change: each load case is applied to a different structure.

Unpropped, the sequence is: bare steel section, second moment 4.8 × 10⁸ mm⁴, carries the 216 kNm of wet concrete; then the composite section, 1.35 × 10⁹ mm⁴, carries the 324 kNm that comes afterwards. The stresses add because both are elastic; the section properties do not, because they were different when each was applied.

$$\sigma = \frac{M_1}{Z_{\text{steel}}} + \frac{M_2}{Z_{\text{comp}}} = 180.0 + 111.7 = 291.7\ \text{MPa}$$

Propped, the steel carries nothing until the props come out, and the whole 540 kNm arrives on the composite section: 186.2 MPa.

The deflections behave the same way and by a larger margin — 49.3 mm unpropped against 28.6 mm propped, a ratio of 1.73, because the stage carried on the bare steel is carried at a third of the finished stiffness.

Which one to build is not obvious #

Propping gives lower stresses and less deflection, so it ought to win. It usually loses, and the reasons are not structural.

Props have to be installed, they have to bear on something — usually the floor below, which must be checked for a load nobody designed it for — they obstruct the site for weeks, and they have to be struck in a controlled sequence. Unpropped construction lets the steel frame be erected and the slab poured with nothing underneath, which is faster and cheaper by more than the extra steel costs.

So the ordinary answer is: build unpropped, and design the beam for the higher stress. What makes that answer defensible is the observation the figures make — the total is what matters, and the unpropped beam is perfectly safe if it was designed as an unpropped beam.

What must not happen is the mixture: a beam designed on the assumption of propping and built without props is overstressed by 57% with nothing on any drawing to say so. That is a communication failure rather than an engineering one, and it is the reason drawings for composite floors carry a note about propping in large letters.

Continuity made later is not continuity #

The second case is more general and stranger. Erect two spans as simple beams, put some load on them, then make the joint over the middle support continuous and add the rest.

The first stage’s moments are stuck. The beam was simply supported when that load arrived, so it carries a sagging moment of $wL^2/8$ with nothing over the support, and making the joint continuous afterwards does not undo it — the load is already there and the deformation has already happened. Only the second stage sees a continuous beam. The redundancy that continuity creates arrived after part of the load did, and a redundancy cannot act on a load that is already resting somewhere.

The consequence is a redistribution in the opposite direction from the usual one. Continuity normally buys a reduction in the peak moment by moving sagging into hogging; here it only gets to move the part of the load that arrived late, so the beam has a larger midspan moment and a smaller support moment than the finished analysis says.

How far the staged answer sits from the finished one is set by one number, and it is the number nobody records: how much of the load was on before the joint was made.

So the finished-structure calculation is not a conservative approximation of the staged one; it is wrong in both directions, and how wrong depends on a construction decision. It is unsafe at midspan and wasteful at the support, and the error grows as the joint is made later.

Both errors matter and they matter differently. The support is where the reinforcement or the splice is, and over-providing there is a waste. Midspan is where the beam is checked for the moment it will actually see, and under-providing there is the other kind of mistake.

At collapse the sequence disappears entirely #

There is one check on which none of this matters, and knowing which one it is resolves what would otherwise be an awkward question: how a code can permit unpropped construction at all, when the stress is 57 per cent higher than the drawing’s own analysis says.

The answer is that the difference between the staged structure and the finished one is a self-stress state — a set of internal forces in equilibrium with no external load. That is exactly what the locked-in stage is: the 216 kNm sitting in the bare steel is balanced within the section, and the composite beam carries it with nothing applied. And the static theorem of plastic collapse says a structure collapses at a load for which some equilibrium stress field exists nowhere exceeding the yield condition — so adding a residual field that is itself in equilibrium changes nothing about the load at which one can be found.

The plastic moment of the composite section is the same number either way. Propped and unpropped, the beam above has the same $M_{pl}$, the same collapse load, and the same margin against it. The 291.7 MPa is real, it is elastic, and it is spent before the section reaches its plastic capacity, at which point the strain distribution is set by the plastic neutral axis and has forgotten which fibres got there first.

The same holds for the two-span case. The staged diagram — 324 at the support and 378 at midspan — and the finished one differ by a self-stress state, so both structures form the same three hinges at the same load.

Three conditions are attached and each of them is a real check rather than a formality.

The section must be able to reach its plastic moment, which means class 1 or 2 — a slender web or flange puts the capacity out of reach and the elastic stress is then the whole of the answer.

The rotation must be available. Redistributing the locked-in field means the first hinge has to turn while the rest of the structure catches up, and a composite section with the slab in tension over a support has limited rotation capacity for exactly this reason.

And nothing about serviceability is covered. Deflection, cracking, vibration and the stress under working load are all elastic quantities, and every one of them keeps its subscript for the stage it belongs to. So the sequence vanishes from the strength check and never vanishes from the rest — which is the reverse of the usual expectation that the ultimate check is the demanding one.

The concrete fills the sag it caused #

The 32 mm the bare steel deflects under wet concrete has a consequence the load case does not contain: the slab is poured to a level, so the extra depth in the middle is extra concrete, and the extra concrete is more load on the same bare steel.

Take the sag as parabolic, so its mean depth is two-thirds of the maximum. On beams at 3 m centres:

$$\Delta w = \tfrac{2}{3}\,\delta\,\gamma_c\,s = \tfrac{2}{3} \times 0.032 \times 24 \times 3 = 1.54\ \text{kN/m}$$

which is 12.8 per cent of the 12 kN/m assumed. That deflects the beam further, which calls for more concrete again, and the series is geometric with ratio 0.128:

$$\delta_{final} = \frac{32}{1 - 0.128} = 36.7\ \text{mm}$$

— 15 per cent more sag and 15 per cent more concrete than the nominal calculation, both of them permanent, and the concrete showing up as dead load on every subsequent check.

The ratio is the interesting part, because it is a stability criterion rather than a correction. Writing $\delta = k w$ with $k = 5L^4/384EI$, the loop closes on itself when

$$\tfrac{2}{3}\gamma_c\, s\, k \;\ge\; 1$$

For this beam $k = 2.68$ mm per kN/m against a limit of 20.8, so it is a factor of 7.8 clear. But $k$ carries $L^4$, so a beam of the same section at 20 m span would be past it, and the pour would not converge — the concrete would keep finding somewhere to go. That is ponding, met here on a beam rather than on a flat roof, and it is the same shape as every other place a load grows with the deflection it produces: a geometric series that sums to something finite until it does not.

The design response is the one that also fixes the appearance, and it is why it is done: precamber the beam by the calculated sag. The steel is fabricated bowed upward by 32 mm, the wet concrete brings it to level, the slab is its nominal thickness everywhere, and the feedback above never starts. What precamber cannot do is change any stress on this page — it moves the geometry and leaves the staged stress field exactly where it was.

The same idea, three fields along #

This site has already met staged construction twice without calling it that.

And a support that settles is the same category: an imposed deformation applied to a structure in whatever state it was in when the settlement happened, producing moments that depend on when rather than on how much.

The general form is worth stating once. Superposition requires the same structure for every case being superposed, and construction sequence is the ordinary way that requirement is broken. Nothing about the arithmetic changes; what changes is that the section properties, the support conditions or the connectivity carry a subscript for the stage they belong to.

Creep moves the answer back, slowly and only partly #

There is one mechanism that acts to undo all of this, and it is worth knowing how far it goes because the answer is “some of the way”.

Concrete’s creep coefficient rises steeply for the first year and continues for a decade, reaching two or three times the elastic strain under a stress held from twenty-eight days. Sustained stress produces strain that goes on arriving, and in a staged structure that strain acts to relax the difference between the state the structure was built in and the state it would have been in if built at once. The redistribution is real, it takes a decade, and it is partial.

For a composite beam, creep in the slab softens the composite section, which shifts stress back toward the steel — the wrong direction, making the unpropped case slightly worse over time rather than better. For a beam made continuous later, creep under the locked-in simple-span moments produces a slow redistribution toward the finished-structure answer, so the support moment grows over years and the midspan moment falls.

That second effect is genuinely useful and it is why the staged and finished analyses are often treated as bounds rather than as alternatives: the structure starts at one and creeps some fraction of the way to the other, and designing for the envelope of the two is the honest response. The fraction depends on the age of the concrete when the joint was made, which is a construction record rather than a design parameter — the same information a long-term deflection calculation needs and rarely has.

Where else the order decides the answer #

Four cases, all ordinary, and each one is the same argument in different clothing.

Column shortening in a tall building. Columns and cores shorten elastically under the load of the floors above and creep further over years. A perimeter column carrying mostly gravity shortens more than a core carrying gravity and lateral load, and the difference accumulates up the height — 30 mm over forty storeys is unremarkable. Floors are cast level, so the difference has to be compensated as the building rises, by casting each floor slightly high. Getting it wrong tilts the floors and pushes load into whichever element was left long.

The two curves peaking in different places is why the compensation cannot be a single number applied everywhere. A builder correcting for the load-driven part alone leaves the roof out by the shrinkage; one correcting for the total at the roof over-corrects the middle of the building, which is where the load-driven difference actually lives.

Cable structures and tensioned roofs. The geometry a cable net takes depends on the order in which it was stressed, and the final force distribution is not unique for a given set of lengths — a state of self-stress can be added anywhere, and the stressing sequence chooses which one.

Arches and long-span bridges built as cantilevers, or pushed out over their piers. Each segment carries its own weight from the moment it is placed and finds itself part of a very different structure when the two halves meet. A launched deck is the extreme version: the whole bridge is assembled behind one abutment and slid out, so every section passes over every pier on its way to its final position.

That envelope is the general shape of everything on this page. A staged structure does not have a set of internal forces; it has one per stage, and what has to be provided for is the outer boundary of all of them. A section designed for its finished sign is a section that will be hogged at some point during construction and has nothing in the top face to say so — which is why a launched bridge is a constant-depth box with symmetric flanges, top and bottom alike, in a structure whose service moments are anything but symmetric.

Demolition, which is construction run backwards. Every argument above applies in reverse and with less information, since the sequence that built the structure is usually unknown and the locked-in forces are unmeasured.

What the propped beam is really buying #

It is worth being exact about what propping does, because the usual description — “the props carry the wet concrete” — is not quite it.

The props do carry the wet concrete, briefly. What matters is what happens when they are struck: the load they were carrying is released onto the beam, and by then the beam is the composite section. So the propping has not removed a load; it has postponed it until the structure that receives it is stiffer. Nothing is carried by anything less than the finished section at any point.

That reading makes the arithmetic obvious and it makes the risk obvious too. The moment of striking is a load application, it happens quickly, and it happens to a slab whose concrete may be four days old. Striking too early applies the full moment to a section whose concrete has a fraction of its final modulus, which both overstresses the slab and produces a deflection that does not recover — so the propping schedule is a structural document, and “strike when convenient” is not an instruction anybody should write.

The same logic explains why partial propping exists: a single prop at midspan cuts the bare steel’s moment to exactly a quarter — 216 kNm becomes 54.0 — and props at the third points cut it to 19.2, which is a little under a twelfth. Each prop costs something and each one postpones more of the load, so the design variable is not whether to prop but how much of the early load to postpone, and it is continuous rather than binary.

The one case where the sequence is used on purpose #

Everything so far treats the sequence as something that happens to a structure. It can also be a design tool, and the clearest example is a jacking operation.

Building a long-span bridge as two cantilevers and closing it at midspan leaves a structure whose moments are largely those of two cantilevers. Jack the two halves apart before closing — pushing them with a stated force, then welding the joint — and the closure locks in a moment field of the designer’s choosing, which can be set to cancel a large part of the cantilever moments. The structure is then in a state no analysis of the finished bridge under its loads would ever produce.

That is prestressing by a different route: instead of a tendon, the imposed action is a jack and a weld, and the self-stress state it creates is permanent, invisible and entirely intentional. The same technique straightens sagging arches, redistributes load between piles, and pre-loads temporary props so that a structure does not have to move before its support starts working.

Two things make it worth naming rather than filing as a trick. It is one of the few places where an engineer chooses which of the infinitely many self-stress states a redundant structure carries — usually that choice is made for them by fit and sequence, as the wobbling table showed. And it is entirely undetectable afterwards: the jacking force is a number in a construction record, and a structure that has been jacked looks exactly like one that has not.

What the picture cannot show #

Everything here is elastic. A structure loaded past first yield in one stage carries a permanent set into the next, and the superposition above stops applying entirely — the honest treatment is an incremental nonlinear analysis in which the sequence is part of the model.

The stages are two. Real sequences have dozens, each with its own structure, and the arithmetic is the same one repeated — which is why staged analysis is a routine facility in structural software and a substantial modelling effort in practice.

Nothing here has a temperature in it. A steel frame erected in winter and completed in summer has locked-in forces from thermal movement at whatever temperature each connection was made — an imposed strain applied at a moment in the sequence, which is this essay’s subject and that one’s mechanism at the same time.

Where the ladder goes #

The first rung is the creep redistribution above, computed rather than described: the fraction of the way from the staged answer to the finished one depends on the age at which the structure was made continuous, and it can be worked out from the same creep coefficient the deflection calculation uses.

The second is the temporary structure in its own right. The frame during erection is a structure with different geometry, different restraints, no cladding, no floors and its own loads — and it is the state in which a large share of structural collapses happen, for the obvious reason that it is the state nobody analysed.

The third is the record. Everything above says that a structure’s stresses depend on facts about its history, which means the drawings are not a complete description of the object — a fact that becomes acute the moment somebody proposes to alter it. What was locked in during construction is still there forty years later, and the only way to know it is to have written it down.