The section that changed while it was being loaded

Almost every stress in this collection has been computed by dividing a moment by a section modulus. The move is so routine that its assumption is invisible: it assumes the section had that modulus while the moment was arriving.

A great many sections did not. A composite beam was bare steel for a week and a composite section afterwards. A precast unit was a simple span until a stitch was cast over its support. A prestressed beam had a different neutral axis before its slab was on. In every one of those cases the stresses from the two stages add together and the section properties do not, and there is no single division that gives the answer.

Which free body produced the number #

Cut the finished beam at midspan and take the piece to the left. The moment on the cut is the total applied moment, whatever the history, because equilibrium knows nothing about time.

Now ask what stress distribution is on that cut. It is the sum of two distributions: the one that was there when the bare steel carried the wet concrete, and the one added when the composite section carried everything afterwards. The first was computed on the steel’s own modulus about the steel’s own neutral axis. The second was computed on the composite modulus about a neutral axis several hundred millimetres higher.

Adding them is legitimate — the material is elastic and superposition holds. Dividing the total moment by the final modulus is not, because that operation assumes both parts of the moment had the same lever arms available, and they did not.

The consequence has a clean form. Stress superposes; section properties do not. A design that wants a single division has to make the section constant, which is what propping is for.

What propping actually buys #

Propping the steel beam while the slab is cast means the bare steel carries nothing. When the props come out, the whole load is applied at once to the finished composite section, and the single division is correct.

The saving is large: 93 MPa instead of 146, and the deflection smaller by a factor of 1.73. Both improvements have the same source, which is that the first stage of load is being carried on a much better lever arm.

It is not free. The props carry the wet concrete to the floor below, which then carries a load it was not designed for — and on a multi-storey pour that floor is itself props on a floor beneath, so a wet slab is frequently carried by three or four floors at once, none of them complete. The props also have to be there while the concrete gains strength, which is a programme cost; and removing them transfers the whole load in one step, so the stress the beam sees on that day is the design stress rather than a fraction of it. Whether that step happens before or after the finishes are on decides which section carries them, which is another stage.

There is a further asymmetry that catches people. Propping helps the stress by a factor of 1.57 here and the deflection by 1.73, and it does nothing at all for the ultimate capacity, which is the same for both beams. So a member governed by strength gains nothing from propping and a member governed by serviceability gains a great deal — and which of those a composite beam is depends on its span-to-depth ratio rather than on anything about the sequence.

The design decision is therefore about construction and appears in the structural calculation as a change of section properties. That is the shape of every argument in this essay: a decision made on a site, entering the arithmetic as a modulus.

Continuity made afterwards #

The same argument scaled up from a section to a structure gives the more striking result.

Erect two simple spans, cast a stitch over the support, then apply the remaining load to what is now a continuous beam. The dead load’s moment diagram belongs to two simple spans and has no hogging in it at all; everything after the stitch belongs to a continuous beam and does. The result is a support moment of 60% of what a continuous beam would have had, and a span moment of 140%.

Both diagrams are correct. Both satisfy $\sum M = 0$ everywhere. They differ only in the order of two events, and neither the geometry nor the loading records which order was used — the structure that was never complete is the general case, and this is its cross-sectional consequence.

The practical trap is a familiar one: a designer analysing the finished structure gets a hogging moment of 375 and detailed reinforcement for it, over a support where the real hogging moment is 225 and the real span moment is 40% larger than the analysis said.

Where the neutral axis is at each stage #

The two stages have different neutral axes because they have different sections, and the transformed-section construction is what makes each stage computable. One strain profile runs across both materials, because they are stuck together; each material then takes the stress its own modulus gives it at that strain; and widening the softer material by the ratio of the two moduli turns the pair into a single fictitious section whose ordinary properties can be computed in the ordinary way.

A section made of two materials is handled by making it a section of one, and the transformation is compatibility plus Hooke’s law rather than an approximation. What a staged section adds is that the modular ratio itself is not a constant: concrete’s effective modulus falls with time under sustained load, so the transformation for the dead load is done at one ratio and for the imposed load at another.

That is the third stage nobody drew. Over years, the concrete part of a composite section relaxes and sheds stress into the steel, which does not creep — so a beam whose steel stress was computed at handover is carrying more than that a decade later, with no change of load at all.

The arithmetic of two stages, written out #

It is worth doing the sum once explicitly, because the shape of it is what makes the effect so easy to miss.

Let the first stage apply a moment $M_1$ to a section with modulus $Z_1$, and the second apply $M_2$ to a section with modulus $Z_2$. The stress at a fibre that exists in both stages is

$$\sigma = \frac{M_1}{Z_1} + \frac{M_2}{Z_2}$$

and the quantity a single-stage calculation produces is $(M_1 + M_2)/Z_2$. The difference between them is $M_1(1/Z_1 - 1/Z_2)$ — proportional to the first stage’s moment and to the gap between the two moduli, and containing nothing about the second stage at all.

Two things follow. The first is that the error is worst when the first stage is a large fraction of the total, which is exactly the case for a heavy wet slab on a light steel beam. The second is that it is worst when the section changes most, which is the same case again: a shallow steel beam becoming a deep composite one has a modulus ratio of two or three.

The first of those can be watched directly, by moving load from the second stage into the first and leaving everything else alone.

Neither beam has changed shape and neither carries more load, so the whole of that 26 MPa came out of the schedule. It is also the reason the effect is severe in buildings and mild in bridges of the same construction: a building’s wet slab is a large fraction of what the beam will ever carry, while a bridge deck’s superimposed dead load and traffic dwarf the weight of the concrete that made the section composite.

They compound, and the composite beam is where the two maxima coincide. That is why unpropped composite construction is the standard illustration of staging, and it is also why the effect is comparatively mild in the other places it occurs — a precast unit made continuous, a column cast against an earlier lift — where either the first-stage moment or the change of section is small.

A useful sanity check falls out of the same expression. If $Z_1$ and $Z_2$ are within about ten per cent of each other, the whole subject can be ignored, whatever the sequence was.

Camber, which is the same problem stated backwards #

If the stresses are staged, so are the deflections, and the way that is handled in practice is to build the member the wrong shape.

The beam is fabricated with 46.8 mm of hog. It sags to level exactly on the day the slab is poured, and finishes 23.5 mm down at the end — one part in 596 of the span. The largest curvature it ever carries is the camber itself, and it has that with nothing on it at all.

Built to the wrong shape is the essay about that decision. The point here is that the camber is chosen against one stage of a sequence, so it is right on one day and wrong on every other, and choosing which day is a specification rather than a calculation.

The connection between the stages #

A composite section only exists to the extent that its two parts are joined, so the sequence has a fourth quantity in it: how much interaction there is at all.

That quantity does not arrive in proportion to the number of connectors. Composite action against connection stiffness is a steeply rising curve that flattens early, so most of the available benefit is bought by the first few and the remainder buys progressively less — which is why half the studs give most of the beam and why the last few are the expensive ones. Past a point the last per cent cannot be bought at all, because the two parts are already behaving as one piece and there is nothing left to recover.

For staging, the important consequence is that the transition between stages is not instantaneous. The slab does not become structural at a moment; its modulus rises over days, the studs take up their slip, and the section’s properties sweep continuously from one value to another while load is arriving.

Prestress, which is a stage applied on purpose #

Everything above treats staging as a consequence of how a thing was built. Prestressing is the same mechanism used deliberately.

A prestressed member has at least three stages by design — transfer, the arrival of the superimposed load, and the long-term state after losses — and the section properties change between them if the member is later made composite with a slab. The load put on backwards is the essay about the intent, and the zone a tendon has to stay inside is the shape the stages leave behind; what it shares with an unpropped composite beam is the arithmetic, which is a sum of stress distributions computed on different sections.

The stage that is not a construction stage #

There is a version of this problem with no site in it, and it catches people who would never make the composite-beam mistake.

A strengthening scheme is a staged section. A beam already carrying its dead load, plated or bonded or bolted to make it stronger, has the dead load’s stresses locked into the original section and only the subsequent load available to the strengthened one. If the plate is added to a beam already at 60% of its capacity, the plate contributes nothing to that 60%, and the strengthened member’s capacity in the service range is far below what the enlarged section modulus suggests.

The consequence is uncomfortable and general: a strengthening that is not preceded by unloading only strengthens against future load. Propping the member before plating it is the equivalent of propping a composite beam, and it is what turns the calculation back into a single division.

The same reasoning explains why a stiff repair attracts load it was not intended to take. Adding material where the strains are already large gives the new material the strains that are there, so it picks up stress in proportion to its modulus regardless of what anybody intended it to do.

The stage that takes material away #

Every stage so far has added something. The reverse case is commoner than it looks and behaves as the mirror image, with one asymmetry that makes it worse.

Cut material out of a member that is already loaded — a slot in a slab for a new riser, a cope in a beam for a duct, a flange trimmed to clear a new plant base, a prop removed — and the force that was in the removed material has to go somewhere. It goes into what is left, immediately, at a stress that adds to what was already there.

That is the reverse of plating. Adding a plate to a loaded beam gives the plate none of the existing stress; removing part of a loaded beam gives what remains all of the removed part’s share. So the two operations are not symmetric: strengthening only works for future load and weakening works for past load as well.

The arithmetic is a redistribution. If a section carrying a force $N$ loses a fraction $\beta$ of its area, the remaining material picks up the force in proportion to its own stiffness, and the stress in it rises by $1/(1-\beta)$ — before anything new is applied.

Two practical consequences follow, and both are about sequence rather than about capacity. Unload before cutting wherever it can be arranged, which is what propping a beam before coping it achieves. And cut in an order that keeps the remaining section in equilibrium at every step, because the intermediate states are structures too — a member half-cut is a member with a different section and the same load, and it is not covered by either the before or the after calculation.

The order is an input, not a detail #

There is a stronger version of the whole argument worth stating plainly. It is not merely that a member has stages; it is that the order of the stages is a structural input, and reversing two of them gives a different answer.

Two loads applied to one unchanging section superpose, and the order is irrelevant. Two loads applied to a section that changes between them do not, because each was applied to a different member. So a specification that says how many stages there are has said half of it; the sequence is the other half.

The place this bites hardest is a continuous composite deck. The order in which the slab pours proceed along the spans decides which parts of the steel beam were acting alone when each pour’s weight arrived, and therefore the locked-in moments in the finished deck. A pour sequence on a drawing is a structural instruction, and a contractor who reverses it for programme reasons has changed the answer to a calculation nobody re-ran.

The extreme case is a deck that moves. A bridge built by incremental launching is assembled on one bank and pushed out over its piers, so every section of it travels the whole length of the bridge before arriving where it will spend its life. There is no moment during that journey at which the structure has the span arrangement the finished bridge has, and the sections do not merely meet different magnitudes on the way. They meet different signs.

That envelope is why a launched bridge is a constant-depth box with symmetric flanges. A bridge built in place may be deep over its piers and shallow at midspan, and may put its material where the sign of the moment says it belongs; a launched one may not, because the sign at any section is whatever the launch happens to be doing that week. The design case is not a load. It is a history, and the section has to be indifferent to it.

The same history is what a piece of purely temporary steelwork is bought to change.

A launching nose is not a stiffening measure, and it carries almost none of the deck’s weight. What it does is shorten the longest cantilever the sequence ever asks for, and the factor of four between 185,977 and 42,539 kNm is bought entirely by changing when the deck lands rather than anything about what the deck is. That is this essay’s argument in its purest available form: one finished bridge, one set of finished loads, and a governing moment that differs more than fourfold according to a decision about order.

Where the model stops #

Superposition needs elasticity. All of the above adds stress distributions, which is legal while every material is linear. A section that cracks between two stages has changed its properties because of the first stage, and the sum is no longer of independent parts. That is the section stiffer than its cracked value says, and it makes the sequence non-linear as well as ordered.

The stages are not discrete. Concrete gains strength over weeks, props are struck floor by floor, and a building’s dead load arrives as a schedule rather than as an event. Two stages is a model of a continuum, and the model’s error is largest exactly when the two stages are closest together in time.

Ultimate capacity mostly forgets all of it. At failure a ductile section redistributes, the locked-in stresses are relieved by plastic strain, and the plastic moment of the finished section is very nearly what it would have been with no history. So this is a serviceability subject: stresses, deflections and cracking care about the sequence, and collapse largely does not.

What the sequence does leave behind is a set of stresses that add to nothing. Each stage wanted a strain the finished whole could not have, and what survives the disagreement is a self-equilibrating field — a distribution across the section whose resultant force and resultant moment are both zero, so no equilibrium check anywhere can see it, and which nonetheless uses up part of the stress range the material had available. Every staged section carries one.

What the picture cannot show #

A cross-section drawing shows a shape. It cannot show that the shape acquired its parts on different days, and neither can a stress diagram drawn on it, because the diagram is a single distribution and the history is a sum.

Nor does any of it show up in a test. Load a finished composite beam to failure and it fails at the capacity of the finished section, because the plastic redistribution wipes out the difference. The staged stresses are visible only in the service range — as a deflection that does not match, a crack that appears at a load below the calculated one, a strain gauge reading that disagrees with the analysis by exactly the locked-in amount.

The one place they announce themselves is a building where the stages have accumulated up a height.

The generalisation #

The habit worth carrying is that a structure’s internal forces depend on its history, and its equilibrium does not.

Equilibrium is a statement about the present. Compatibility is a statement about a path — how the strains got to where they are — and every quantity in a redundant or a composite structure is fixed by compatibility. So any question whose answer comes from a stiffness is a question whose answer depends on when the stiffness had the value it did.

That is the same reason differential shortening is a sequence problem, why a residual stress has no applied load in it at all, and why the reactions on a structure jacked into position depend on the order the jacks were released. All four are one fact: a structure remembers, and the drawing does not.