Structural Engineering & Statics
where the load goes
Deflection

The curvature nobody applied

Concrete shrinks as it dries, by about half a millimetre in every metre. In a symmetrically reinforced member that is a shortening and nothing else. In a member with more steel in one face than the other — which is every beam and every slab — the steel holds one side back and the section bends, with no load on it at all.

Assumes The deflection that arrives three years late and The movement nobody applied.

16 min read 8 figures The statics of things that do not moveWhich failure arrives firstThe load must go somewhere

A deflection calculation is a division: a moment by a stiffness, integrated twice. Everything in it comes from the load.

A concrete member’s deflection contains several contributions that came from nowhere near the load, and one of them is a curvature the member acquired by drying out. It has no moment behind it, it satisfies equilibrium with nothing on the section, and it adds to the deflection exactly as though somebody had applied a bending moment along the whole span.

The least reliable number in the material decides the answer, briefly. What a twenty per cent error in the concrete's tensile strength does to a computed deflection, against how far past cracking the beam is. Well past the cracking moment it does almost nothing — at 1.9 times M_cr the spread is 56 per cent — because the section is nearly fully cracked and the interpolation has run out. Just above cracking it does everything: at 1.19 times M_cr the same twenty per cent moves the deflection by 3658 per cent. Tensile strength is the property with the widest scatter and the least direct test, and a beam designed to sit near its cracking moment has put the answer on it.
Fig. 1 What a twenty per cent error in the concrete’s tensile strength does to a computed deflection. Well past cracking it does almost nothing; just above cracking it does everything.

Which free body produced the number

Take a slice of a reinforced section and let the concrete shrink freely by a strain εcs\varepsilon_{cs}. The steel does not shrink. So the two cannot both do what they want, and the section settles at a compromise fixed by compatibility.

Write the compromise as a uniform strain plus a curvature — ε0+κy\varepsilon_0 + \kappa y, which is all a plane section can take. Two conditions close it: the axial forces sum to zero, because nothing is applied, and the moments sum to zero for the same reason.

If the steel is symmetric about the centroid, the second condition is satisfied at κ=0\kappa = 0 and the answer is a pure shortening. If it is not, the steel’s restraint has a lever arm about the centroid and the curvature is

κcs=εcs αe SI\kappa_{cs} = \varepsilon_{cs}\,\alpha_e\,\frac{S}{I}

where SS is the first moment of the reinforcement about the section’s own centroid and αe\alpha_e the modular ratio. A first moment of the steel — an asymmetry — is the entire driver, and where the steel is symmetric it is zero.

Two things follow immediately, and both are counter-intuitive.

Why the lightly reinforced member is worse

The first is that more steel does not mean more curvature.

Adding steel to the tension face increases SS and increases II, and past a point the second wins. Adding steel to the compression face reduces SS directly. So the worst case is a member with a modest amount of steel, all of it in one face — which is a lightly reinforced slab, the member nobody expects to have a problem.

The second is that cracking makes it worse rather than better. A cracked section has lost its tension zone, so its centroid has risen toward the compression face and the steel’s lever arm about that centroid has grown, while II has fallen by 40% or more. Both changes increase κcs\kappa_{cs}.

The neutral axis is wherever the first moment vanishes. A 350 by 556 section with 1600 mm² of steel at a depth of 500, carrying 150 kNm after the tension side has cracked. The neutral axis is no longer at mid-depth: it has risen to 154.0 mm from the top, which is where the first moment of the compression zone plus the transformed steel vanishes. The compression is 12.4 N/mm² at the top fibre and the steel carries 209 N/mm²; the resulting couple is 334 kN on a lever arm of 449 mm, which multiplies back to the 150 kNm applied. The uncracked section would have had 5489×10⁶ mm⁴ against the cracked 1863×10⁶ — a loss of 66% of the stiffness.
Fig. 2 A 350 mm wide beam with 1,600 mm² of steel at an effective depth of 500, drawn after the tension side has cracked. The neutral axis has risen to 154.0 mm from the top of a 556 mm section — far above mid-depth, because the concrete below it has gone — and the second moment has fallen from 5,489 × 10⁶ mm⁴ to 1,863 × 10⁶, a loss of 66 per cent. Both movements enlarge the shrinkage curvature, since the steel’s lever arm is measured from a centroid that has climbed and is then divided by an I that has collapsed.

Neither of those is a small correction, and they compound. The lever arm grows by roughly the distance the neutral axis moved and the divisor falls by two thirds, so a section that cracks acquires a shrinkage curvature several times the one the gross section would have given it — from a load that changed nothing about the shrinkage.

The beam is stiffer than its cracked section and softer than its gross one. Moment against mid-span deflection for a 350 × 550 mm beam spanning 8.0 m, with the two bounds it lies between. The uncracked line is what the gross transformed section gives; the cracked line is what the section at a crack gives; and the curve between them is the member, because between the cracks the concrete is still carrying tension and the average curvature is not either section's. At the service load the deflection is 24.8 mm — span over 323 — against 8.5 uncracked and 27.0 fully cracked, a factor of 3.18 between the bounds. The interpolation ζ = 1 − β(M_cr/M)² sits it 88 per cent of the way across, and β falls from one to a half under sustained or repeated load because the bond that does the dragging deteriorates.
Fig. 3 A member between its two bounds. The uncracked line is what the gross section gives, the cracked line what the section at a crack gives, and the member sits between them because the concrete between the cracks is still carrying tension.

Which of those two bounds a member sits near is decided by how far past its cracking moment it is, and a beam is stiffer than its cracked section says by an amount that depends on the concrete’s tensile strength — the property with the widest scatter and the least direct test.

That sensitivity is the figure at the top of this essay, and it is worth reading as a warning about the whole subject: a member designed to sit just above its cracking moment has put its computed deflection on a number nobody can measure to better than twenty per cent, and a twenty per cent change there moves the answer by a factor of thirty-seven.

Creep, which multiplies a different thing

The other time-dependent effect is creep, and the two are frequently conflated because both take years and both make deflections worse.

The deflection that arrives years late. The 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 3.76, after five years by 4.15, and it approaches 4.27. 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.
Fig. 4 The multiplier on a member’s deflection under a sustained load. After a year it has been multiplied by 3.76 and it approaches 4.27, with nothing added to the load.

Creep is a multiplier on what the load did. The deflection that arrives three years late is the essay about it, and its arithmetic is a reduced effective modulus applied to the same moment.

Shrinkage is an addition that the load had no part in. A member carrying nothing at all still acquires its shrinkage curvature.

The two interact, because the shrinkage curvature itself develops slowly and is relieved by creep as it develops, so the usual treatment applies an age-adjusted modulus to both. But the distinction matters for where each is largest: creep is worst on a heavily and permanently loaded member, and shrinkage curvature is worst on a lightly loaded one. A lightly loaded slab can have more of its long-term deflection from shrinkage than from creep.

What it is worth on a real slab

The arithmetic is short enough to do once.

Take a 200 mm slab spanning 6 m, reinforced with 1,100 mm²/m in the bottom face only, at an effective depth of 165 mm. Cracked, the neutral axis sits about 45 mm from the top, so the steel’s lever arm about the cracked section’s centroid is around 120 mm. With a free shrinkage strain of 400 microstrain, a modular ratio of 15 and a cracked second moment of about 210×10⁶ mm⁴ per metre, the curvature comes to roughly 0.35 per kilometre.

A constant curvature over a simply supported span deflects it by κL2/8\kappa L^2/8, which for 6 m is 1.6 mm. On its own that is nothing — span over 3,800.

Now put the same slab at a span of 9 m and thin it to 175 mm, which is what a designer working to a span-to-depth table would do. The curvature roughly doubles, because the section is shallower and the steel more one-sided; the L2L^2 term more than doubles again; and the shrinkage deflection reaches something like 8 mm, against a total limit of 9 m over 250, which is 36. It has taken a fifth of the allowance before any load has been applied.

The dimension the capacity rides on is not a drawn dimension. Moment capacity of a 175 mm slab against the cover to the reinforcement. The line is very nearly straight, because capacity is A_s·f_yd·z and z is about 0.9d — so the capacity is proportional to a dimension that is not on the drawing. What is on the drawing is the overall depth, and the effective depth is what the cover, the link and half a bar diameter leave of it: 175 − 25 − 0 − 6 = 144 mm here. Every one of those three is a site tolerance rather than a design decision. Ten millimetres of bar position is 7.4% of this slab's capacity and 1.8% of a 600 mm beam's — the same workmanship costs 3.8 times as much in the shallow member, and the shallow member is the one whose steel is walked on before the pour.
Fig. 5 Moment capacity of a 175 mm slab against the cover to its reinforcement. The line is very nearly straight, because the capacity rides on the effective depth, and the effective depth is whatever the cover, the link and half a bar diameter leave of the slab: 175 − 25 − 0 − 6 = 144 mm here. Ten millimetres of bar position is 7.4 per cent of this slab and 1.8 per cent of a 600 mm beam — the same workmanship costs 3.8 times as much in the thin member.

Everything that punishes a thin slab punishes it twice over, and that figure is the strength half of it. The shrinkage half runs the same way for the same reason: a shallower section puts its steel proportionally further from the centroid and gives the curvature a smaller second moment to divide by, so the term that was 1.6 mm at 200 mm and 6 m is several times that at 175 mm and 9.

That is the shape of the whole effect. It is negligible on short, thick, symmetrically reinforced members and a substantial fraction of the allowance on long, thin, singly reinforced ones — which are exactly the members a span-to-depth rule produces, because the rule was calibrated on the total and not on its parts.

The cheapest fix is compression steel. Putting a modest area in the top face reduces SS directly, and unlike almost every other intervention in this collection it costs nothing structurally: the top bars are doing no work in a sagging region and are there purely to make the section shrink straight.

The restraint that is outside the member, which cracks it

Everything above is internal restraint: the reinforcement inside the section holding the concrete back, with a lever arm about the centroid, producing a curvature. There is a second kind, it is usually larger, and it does something else entirely.

A floor slab is attached to columns, to cores, to walls and to the floor below. None of those shrinks with it, so the slab is held axially as well as bent internally. A fully restrained member that wants to shrink by 400 microstrain and is not allowed to develops

σ=εcsEc=400×10−6×30,000=12 N/mm2\sigma = \varepsilon_{cs} E_c = 400\times10^{-6} \times 30{,}000 = 12\ \text{N/mm}^2

of tension, against a tensile strength of about 2.9. Creep relieves perhaps half of it, because the strain arrives slowly, leaving 6 — still more than twice what the concrete can take.

Full restraint is rare. A floor slab held by ordinary columns and a core has a restraint factor of perhaps 0.3 to 0.5, which puts the sustained tension at 1.8 to 3.0 N/mm². That straddles the tensile strength, which is the honest description of why some floors crack from restrained shrinkage and some do not, and why nobody can say in advance which.

The consequence loops back into everything in this essay. A slab that has cracked from restraint has spent its cracking moment before any load arrives, so it behaves as the fully cracked section from day one — and the cracked section is precisely the one with the larger shrinkage curvature, because its centroid has risen and the steel’s lever arm about it has grown. External restraint cracks the member, and cracking makes the internal-restraint curvature worse.

The design responses are all about reducing the restraint rather than resisting it, which is what an imposed deformation always asks for: movement joints, a pour sequence that lets each bay shrink before the next is cast against it, delay strips left open for weeks and closed late, and slip membranes under ground slabs. Where the restraint cannot be removed, the reinforcement is sized by a crack-width calculation rather than by a strength one — steel provided not to carry a force but to divide one large crack into many small ones.

The steel decides how many cracks, and never how much movement. Crack width and crack count against the reinforcement ratio, for a restrained wall whose total movement is fixed at 3.3 mm by the strain it was not allowed to make. The two curves are reciprocals of one another by construction, because their product is that total and the total contains no steel. At 1.11 per cent the wall makes 32 cracks 0.10 mm wide; at twice the steel it makes about twice as many, half as wide. Reinforcement in a restrained member is not resisting anything. It is dividing a movement that was going to happen into pieces small enough to be acceptable.
Fig. 6 Crack width and crack count against the reinforcement ratio, for a restrained wall whose total movement is fixed at 3.3 mm by the strain it was not allowed to make. The two curves are reciprocals of one another by construction, because their product is that movement and the movement contains no steel at all: at 1.11 per cent the wall makes 32 cracks 0.10 mm wide, and at twice the steel about twice as many, half as wide.

That is the whole difference between a restraint calculation and a strength one. Steel in a restrained member is not resisting anything, because the movement it is arguing with was fixed before any reinforcement was chosen; it is dividing that movement into pieces small enough to be acceptable. Which is also why doubling the steel halves the crack and never removes it.

Which gives the two mechanisms in one sentence. Internal restraint bends the member; external restraint cracks it; and the crack makes the bend worse.

The general shape of the effect

Shrinkage curvature is one member of a family, and the family is easier to recognise than any of its members.

Whenever a member is given a strain distribution it cannot take — because plane sections have to stay plane — the difference between what it wants and what it is allowed becomes a self-equilibrating stress field: a distribution of stress across the section whose resultant force is zero and whose resultant moment is zero, which is to say one that is invisible to every equilibrium check that could be made on it. A thermal gradient does it, differential shrinkage does it, a welding pass does it, and a section made of two materials with different time-dependent behaviour does it continuously.

Warmer on top: the same two limits, one field along. A 30 m deck 12 °C warmer on top than underneath, over a depth of 600 mm. Free to move, it takes a curvature of 0.240 per km and lifts 27.0 mm at midspan, carrying no stress at all. Held down, it carries 18 kNm and ±15.1 MPa at the extreme fibres and does not move. Every real deck is somewhere between, and where it sits is decided by the bearings rather than by the deck.
Fig. 7 A deck warmer on top than underneath. Free to move it takes a curvature and lifts 27 mm at midspan with no stress at all; held down it carries 18 kNm and ±15.1 MPa and does not move.

The pair in that figure is the whole logic. A curvature and a moment are alternatives: a member free to bend takes the curvature and carries no stress, a member restrained carries the stress and does not move, and every real member is somewhere between with the position decided by its supports rather than by itself. The movement nobody applied is the essay about the restrained end of that spectrum; this one is about the free end, where the price is a deflection instead of a stress.

Where it accumulates

In a single member the effect is a deflection. In a building it is a difference between one part and another.

The two parts peak in different places for a reason worth keeping. The load-driven difference is largest halfway up, because a floor near the top has almost nothing above it to shorten what is beneath, and a floor near the bottom has almost nothing beneath it. The shrinkage-driven difference does not care what is above it at all — every metre of column shrinks whether or not it is loaded — so it accumulates monotonically and their sum is worst at the roof, at 56 mm across a 9 m bay, a floor out of level by one in 160.

That is the columns being shorter than the core with the shrinkage half separated out, and the separation matters because the two are corrected by different means: the load part by a construction-stage adjustment, the shrinkage part by making the two elements shrink alike.

The deflection as a sum

Putting it together, the deflection of a concrete member at the end of its life is a sum with at least five terms.

The elastic deflection under the load that is present. The creep multiplier on the permanent part of it. The shrinkage curvature, from no load at all. The cracking, which reduced the stiffness the first two terms divide by. And the shear deflection, which for most beams is small.

Only the first of those five is what a deflection calculation classically computes, and stiffness is not strength is the essay about why the sum matters at all, and the others are between them frequently larger. That is why concrete deflection prediction is unreliable in a way steel deflection prediction is not: the steel beam’s answer is a division and the concrete beam’s is a sum of five terms, four of which depend on time, humidity, cracking history and construction sequence.

Where the model stops

The free shrinkage strain is an estimate. It depends on the humidity, the member’s surface-to-volume ratio, the cement content and the aggregate, and it is quoted as a range of two to one. Everything computed from it inherits that range.

Shrinkage is not uniform across the section. The surfaces dry first and the core last, so a thick member has its own internal gradient — a self-equilibrating field within the section, on top of the curvature the reinforcement causes. The arithmetic here treats the shrinkage as a single number applied everywhere.

Cracking and shrinkage are not independent. The shrinkage curvature above was computed on a cracked section, and part of what cracked it may have been the restrained shrinkage itself — so a slab restrained at its ends cracks earlier than its applied moment predicts and then has a larger curvature for the same load. The sequence closes on itself and there is no closed form for it.

Restraint outside the member has been ignored. A slab cast between stiff walls cannot shorten, so it cracks instead, and a cracked slab has a different stiffness and a different shrinkage curvature. That coupling is not in any of the expressions above.

The steel that is sized by the concrete. Ultimate moment of a 1000 × 400 mm section against the area of tension steel in it, with the moment that cracks the section drawn across. The cracking moment is 77.2 kNm and contains no steel at all — it is f_ctm times the gross section modulus, 2.90 N/mm² times bh²/6 — so it is a horizontal line, and every section to the left of where the two meet is one whose first crack is its failure. The crossing is at 507 mm², and rearranging the two expressions gives 0.245·(f_ctm/f_yk)·bd against the 0.26 the codes print — the constant is a section modulus divided by a lever arm and not a fitted number. The rule as printed asks for 538 mm² here, which is 6% more than the derivation needs, and that margin is the whole of the safety in a check whose failure mode is sudden.
Fig. 8 Ultimate moment against the area of tension steel for a 1,000 × 400 section, with the moment that cracks the section drawn across it as a horizontal line. The cracking moment is 77.2 kNm and contains no steel at all — it is f_ctm times the gross section modulus, 2.90 N/mm² times bh²/6 — so nothing a designer does to the reinforcement moves it. That is why a restraint appearing in no load case can spend it, and why the section a deflection calculation ought to be using is not always the one the applied moment implies.

Why it is nearly always found late

There is a characteristic history to a shrinkage-curvature problem, and it is worth recognising because every stage of it looks like something else.

The slab is designed to a span-to-depth ratio and passes. It is built and it is flat. Over the first winter a line of cracking appears over the supports, which is attributed to the construction sequence. Over two or three years the mid-span sags visibly — visibly, because a person can see about one part in 250 of a span when there is a straight line such as a partition head or a window reveal to compare it against.

By then the investigation has three candidates and no way to separate them. The concrete may have been weaker than specified; the slab may have been loaded early; the props may have been struck before it had strength. All three are plausible, all three are unprovable years later, and none of them is what happened.

What settles it, when anything does, is the shape. A deflection caused by load is proportional to the load’s own moment diagram and is largest where the load is. A shrinkage deflection is proportional to a curvature that is nearly constant along the span, so it produces a circular arc — and a bay that has sagged uniformly, including the parts carrying nothing, has sagged for a reason the loading cannot explain.

The second tell is the neighbours. Shrinkage curvature depends on the reinforcement layout, so two bays with the same load and different bar arrangements deflect differently. A load-driven problem does not do that.

What the picture cannot show

The curvature has no moment diagram. There is nothing to draw for it in the place where a structural drawing would put the cause of a deflection, and no analysis output contains a member force corresponding to it.

That absence is the practical difficulty. A deflection that exceeds a limit is investigated by looking at the loads and the stiffnesses, and the term responsible may be neither — it may be a curvature that arrived because the top steel was curtailed and the bottom steel was not, in a member carrying nothing unusual at all.

Nor does any figure show that the effect is a rate. Half of the shrinkage happens in the first few months and the rest over years, so a slab that passed its deflection check at handover fails it three years later with nothing having changed on it.

The generalisation

The habit worth carrying is to treat a strain the member cannot take as a load case in its own right.

Every one of the effects in this essay begins the same way: some part of a section wants a strain that a plane section cannot accommodate, and the difference has to go somewhere. If the member is free, it goes into a curvature and produces a movement with no stress. If it is held, it goes into a self-equilibrating stress field with no movement. Real members are partly held and get some of each.

That is the same statement as a residual stress, as a thermal gradient, as differential creep between two materials, and as the locked-in stresses of a section built in stages. Five subjects, one mechanism, and the useful diagnostic is a question rather than a calculation: what strain does this member want, and what is it allowed?

Where this goes next

The deflection that arrives years late. The 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 3.00, after five years by 3.29, and it approaches 3.38. 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. The deflection that arrives three years late A concrete beam that passes every check on the day it is built goes on deflecting for a decade, and ends up three times where it started. Nothing about the load changed, and nothing about the strength was ever in question. It moves, or it pushes. Never both, and never neither. A 30 m steel member 30 °C warmer than it was built, in three conditions. Free, it grows 10.8 mm and carries nothing. Held, it moves nothing and carries 75.6 MPa in compression — which is E·α·ΔT and contains neither the length nor the area of the member, so the identical stress arises in a two-metre strut. Held by a spring it does some of each: 3.2 mm of movement and 53.2 MPa, and the split is decided by the spring rather than by the member. The movement nobody applied A temperature change is the only load in this collection that a structure can decline. Let it move and it produces a movement with no stress; hold it and it produces a stress with no movement — and that stress contains no length, no area and no second moment, so a bracket and a bridge girder carry exactly the same one. The beam is stiffer than its cracked section and softer than its gross one. Moment against mid-span deflection for a 300 × 550 mm beam spanning 8.0 m, with the two bounds it lies between. The uncracked line is what the gross transformed section gives; the cracked line is what the section at a crack gives; and the curve between them is the member, because between the cracks the concrete is still carrying tension and the average curvature is not either section's. At the service load the deflection is 23.2 mm — span over 345 — against 8.3 uncracked and 25.2 fully cracked, a factor of 3.03 between the bounds. The interpolation ζ = 1 − β(M_cr/M)² sits it 88 per cent of the way across, and β falls from one to a half under sustained or repeated load because the bond that does the dragging deteriorates. Stiffer than its cracked section says At a crack the concrete below the neutral axis has gone and the steel carries the tension alone. Between the cracks it has not gone — bond drags it back into tension, the steel strain drops, and the curvature averaged over a length of beam is neither section's. Two differences up the same building, peaking in different places. Differential shortening between a perimeter column and the core of a 40-storey building, plotted up the height. The part driven by load peaks at level 20 — 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 40, at 43 mm, which across a 9 m bay is a floor out of level by one in 208. The columns are shorter than the core Every column in a tall building gets shorter as the building is built on top of it, and the core beside it gets shorter by a different amount. The floors between them tilt by the difference — and the difference is largest exactly half way up, because a floor near the top has almost nothing built above it and a floor near the bottom has almost nothing beneath it.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

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

What links here

Every essay whose body links to this one.

The objects this essay names

Each one links to every other essay that touches it.

CamberCompatibilityCracked sectionCreepCurvatureDeflectionDifferential shorteningModular ratioReinforcementRestraintSelf-equilibratingServiceabilityShrinkageTension stiffeningThermal gradient