Structural Engineering & Statics
where the load goes
Sections and stress

The section that changed while it was being loaded

A stress is computed from a moment and a section modulus. When part of the moment arrived while the section was a different shape, there is no single section modulus to divide by — the stresses add and the properties do not, and two identical finished beams can differ by half again in stress with nothing on the drawing to say which is which.

Assumes Two beams, or one beam four times as stiff, A section made of two materials, one of them pretended away and The structure that was never complete.

19 min read 10 figures The load must go somewhereWhich failure arrives firstThe statics of things that do not move

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.

The props decide where the stress ends upBottom-fibre stress in the steel of a 12 m composite beam carrying 6 kN/m of wet concrete and 9 kN/m afterwards. Unpropped, the bare steel takes the first stage alone and reaches 146 MPa; propped, the finished composite section takes everything and reaches 93 MPa — a ratio of 1.57. 62% of the unpropped beam's final stress was locked in before the slab was structural at all. The deflections differ by 1.73 times for the same reason, and no drawing of the finished beam distinguishes the two.146 MPaunpropped24.6 mm at midspansteel alonecomposite93 MPapropped14.3 mm at midspancomposite
Fig. 1 Bottom-fibre stress in the steel of a composite beam, propped and unpropped. Same beam, same total load, same finished section — and a ratio of 1.57 between the two answers.

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.

Built as two beams, used as oneBending moments in a two-span beam erected as simple spans under 12 kN/m and made continuous before the remaining 18 kN/m arrived, against the same beam built continuous from the start. The support moment is 225 kNm rather than 375 — 60% of it — and the midspan moment is 263 rather than 188, which is 140%. Both diagrams are in equilibrium with the same total load; they differ only in when the joint was made, which appears nowhere on the drawing.375 kNm built continuous225 staged263188same beam, same load, different history
Fig. 2 Bending moments in a two-span beam erected as simple spans and made continuous before the rest of the load arrived, against one built continuous from the start. Both are in equilibrium with the same total load.

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 ∑M=0\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.

The same strain, two moduli, and a width multiplied to say soA timber section with a steel plate in it, carrying 32.0 kNm. Plane sections stay plane, so the strain at a height is the same in both materials; Hooke's law then puts the stresses in the ratio of the moduli, which here is 19.09. Multiplying the stiffer material's WIDTH by that ratio gives a fictitious section of one material with the same neutral axis and the same forces — 595.2×10⁶ mm⁴ of it, against 351.0 for the same shape with the moduli ignored. The steel plate is 3.8% of the area and carries 43% of the moment, at 154 N/mm² against the timber's 8.1. The transform is not an approximation: it is compatibility and Hooke's law written down.timbersteel plateas builttransformed to one materialas it is× 19.1154 N/mm²8.1 in the timberstressstiffness 1.70× the same shape with the moduli ignored
Fig. 3 Two materials, one strain profile, and a width multiplied by the modular ratio to say so. The stiff material is 3.8% of the area and carries 43% of the moment.

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.

The deflection that arrives years lateThe multiplier on a concrete member's deflection under a sustained load, against time. The elastic deflection arrives on the day the load does and is the 1.0 at the left. After a year it has been multiplied by 3.11, after five years by 3.41, and it approaches 3.50. Nothing has been added to the load and nothing about the strength has changed: this is a serviceability failure arriving on a structure that passed every strength check on the day it was built.1 d10 d100 d2.7 yr27 yr0123time under loaddeflection ÷ the deflection on day one1 year: ×3.115 years: ×3.41the deflection the calculation gives
Fig. 4 The multiplier on a concrete member’s deflection under a sustained load. Nothing has been added to the load; the material’s effective stiffness has fallen.

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 M1M_1 to a section with modulus Z1Z_1, and the second apply M2M_2 to a section with modulus Z2Z_2. The stress at a fibre that exists in both stages is

σ=M1Z1+M2Z2\sigma = \frac{M_1}{Z_1} + \frac{M_2}{Z_2}

and the quantity a single-stage calculation produces is (M1+M2)/Z2(M_1 + M_2)/Z_2. The difference between them is M1(1/Z1−1/Z2)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.

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 Z1Z_1 and Z2Z_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.

Cambered against the wet loadA 14 m composite beam whose flexural rigidity rises from 94 to 260 kN·m² when the slab sets, so the first two loads are carried by the bare steel and the rest by the composite section. Fabricated with 46.8 mm of camber, it moves through -38.3, 0.0, 13.9, 23.5 mm as the four stages arrive — 0.0 mm on the day the slab is poured, and 23.5 mm at the end, which is one part in 596 of the span. The largest curvature it ever has is 46.8 mm of hog, and it has that with nothing on it. Every shape is drawn at the same exaggeration and the drawing is a diagram of a proportion: the vertical scale is 75243 times the horizontal.levelas fabricated: 46.8 mm of camber-38.30.013.923.5self-weight of the steel: 8.5 mm on EI = 94wet concrete: 38.3 mm on EI = 94finishes and services: 13.9 mm on EI = 260imposed load: 9.6 mm on EI = 260
Fig. 5 A composite beam whose rigidity rises from 94 to 260 kN·m² when the slab sets, moving through four stages from a fabricated camber to its final position. The vertical scale is exaggerated seventy-five thousand times.

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.

Half the benefit arrives for a small fraction of the connectionHow composite a beam is, against the stiffness of what joins its two halves. Zero is two loose planks and one is a solid section, and the curve between them is Newmark's partial-interaction equation solved for this load case. Half of the available stiffness has arrived by k = 5 and nine tenths of it by k = 80, which is 16 times as much connection for the second half of the benefit as for the first. That shape is why a floor with studs at a spacing a person could step over behaves very nearly as though it were glued, and why the last few studs are the expensive ones.020004000600080001000000.20.40.60.81connector stiffness per unit lengthhow composite it isbondedloose
Fig. 6 How composite a beam is, against the stiffness of what joins its two halves. Half the available benefit has arrived at a connection stiffness of 5 and nine tenths at 80.

The curve is very steep at the left, which is why half the studs give most of the beam and why the last few are the expensive ones.

Between two beams and one, and much nearer oneHow composite a beam is, against the one dimensionless group that decides it: αL, where α² = K·EI∞/(EA*·EI₀). At αL = 0 the layers slide freely and the beam is two beams; past about 20 the connection is stiff enough that the last per cent is unbuyable. The beam drawn sits at αL = 19.8 and is 98% composite, deflecting 122.1 mm against 115.9 for full interaction and 375 for none. The curve is steep where a real design sits, which is why halving the number of studs does not halve anything.051015202530354000.20.40.60.81αLdegree of interaction98% at αL = 19.8one beamtwo beamsEI∞/EI₀ = 3.24 · the whole range is a factor of 3.24 in deflection
Fig. 7 The same question against the one dimensionless group that decides it. The beam drawn sits at αL = 19.8 and is 98% composite; past about 20, the last per cent cannot be bought.

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.

Two triangles that cross zero, and a block that does notStress across a 300 × 700 mm section at each stage, compression positive. The prestress alone gives -5.90 MPa at the top and 19.24 at the bottom; at transfer, with only self-weight on it, the top is at -0.65 MPa and in service the section runs from 12.53 to -1.86 MPa — compression everywhere. The same beam with no prestress reaches -17.25 MPa at the bottom fibre, which is 5.8 times what the concrete can hold.prestress alone-5.9019.24tensionat transfer-0.6513.99tensionin service12.53-1.86tensionwith no prestress17.25-17.25tension
Fig. 8 Stress across a section at each stage. The prestress alone gives −5.90 MPa at the top and 19.24 at the bottom; in service the section runs from 12.53 to −1.86 and is in compression everywhere.

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.

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.

The thinner the heated layer, the more stress it leaves behindSelf-equilibrating stress against the depth the heat reaches, at a fixed surface temperature of 24 °C. At the right-hand edge the whole section is heated on a straight line and the stress is **exactly zero** — a plane section can follow a straight profile with no stress at all. Everywhere left of it the profile is bent, the section cannot follow it, and the difference is a stress field with no resultant force and no resultant moment. The surface compression climbs without limit as the layer thins; the tension underneath peaks at 3.34 N/mm² when the heat reaches 24% of the depth, which is about where a summer afternoon puts it.0%20%40%60%80%100%0123456depth the heat reaches, as a fraction of the sectionself-equilibrating stress (N/mm²)compression at the facetension below ita straight profileleaves nothing
Fig. 9 A self-equilibrating stress field, produced by a strain the section cannot follow. Every staged section carries one of these, left over from the difference between what each stage wanted and what the whole could do.

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.

Two differences up the same building, peaking in different placesDifferential shortening between a perimeter column and the core of a 52-storey building, plotted up the height. The part driven by load peaks at level 26 — exactly half way up, because a floor near the top has almost nothing built above it to shorten what is beneath, and a floor near the bottom has almost nothing beneath it to shorten. The part driven by shrinkage does not care what is above it at all and accumulates all the way to the roof. Their sum is worst at level 52, at 56 mm, which across a 9 m bay is a floor out of level by one in 160.-60-40-20020020406080100120140160180column shorter than core (mm)height (m)from loadfrom shrinkagethe sumworst 56 mmat level 52one in 160
Fig. 10 Differential shortening up a tall building. The load-driven part peaks halfway up and the shrinkage-driven part accumulates to the roof, and their sum is worst at the top.

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.

Where this goes next

loose: two beams, and the faces slide16.2 mmbonded: one beam, and the faces cannot4.0 mmI × 4with connectors at k = 200: 5.4 mm, 89% composite Two beams, or one beam four times as stiff Stack two planks and they bend as two beams whose faces slide past one another. Bond the faces and the pair has one neutral axis, four times the second moment and half the stress. Nothing was added but a restraint on slip. 292 MPaunpropped49.3 mm at midspansteel alonecomposite186 MPapropped28.6 mm at midspancomposite The structure that was never complete Every analysis in this collection is of a finished structure loaded once. Real ones are built in pieces, and each piece carries whatever was present at the moment it became structural — so the stress in a member depends on when it arrived, which appears nowhere on any drawing. 1234403020100-10-20-30-40stage in the beam's lifeshape (mm, sag positive)against noneagainst wetagainst deadagainst total Built to the wrong shape on purpose A cambered beam is fabricated curved upward so that load bends it down to something like straight. Nothing in the analysis changes, no stress anywhere is altered, and almost every mistake made with it is a bookkeeping mistake about which loads count. timbersteel plateas builttransformed to one materialas it is× 19.196 N/mm²5.0 in the timberstressstiffness 1.70× the same shape with the moduli ignored A section made of two materials, one of them pretended away Multiplying a material's width by the ratio of the moduli produces a fictitious section of one material with the right neutral axis and the right forces. It is not a trick — it is compatibility and Hooke's law written down — and it says a stiff material takes what its modulus asks for.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

CamberComposite actionConstruction sequenceCreepDifferential shorteningLocked in stressModular ratioPartial interactionPlane sectionsPrestressSection modulusSelf equilibratingStaged constructionSuperpositionTransformed section