Structural Engineering & Statics
where the load goes
Sections and stress

The dimension nobody can measure

Every flexural capacity in reinforced concrete is proportional to the effective depth, and the effective depth is not on the drawing. It is what is left of the thickness after a cover, a link and half a bar diameter have been taken off it — and each of those is a site tolerance. In a slab, ten millimetres of workmanship is six per cent of the strength.

Assumes Bending is a pair of forces, pushing and pulling, Where the steel is, not how much of it and Plane sections stay plane, and what the assumption costs.

13 min read 8 figures Geometry beats materialWhich failure arrives firstScale changes everything

The capacity of a reinforced concrete section in bending is

M=Asfyd z,z≈0.9dM = A_s f_{yd}\,z, \qquad z \approx 0.9d

and every part of it is known to two or three figures except one. The area of steel is a catalogue number. The design strength of the steel is a characteristic value divided by a partial factor. The lever arm is a computed fraction of dd.

And dd, the effective depth, is not a dimension anybody draws. It is what is left over:

d=h−c−ϕlink−12ϕbard = h - c - \phi_{link} - \tfrac{1}{2}\phi_{bar}

The overall depth hh is on the drawing. The other three are a specified cover, a link diameter and half a bar diameter, and once the concrete has gone in, none of them can be measured at all.

The dimension the capacity rides on is not a drawn dimension. Moment capacity of a 225 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: 225 − 30 − 0 − 8 = 187 mm here. Every one of those three is a site tolerance rather than a design decision. Ten millimetres of bar position is 5.7% of this slab's capacity and 1.8% of a 600 mm beam's — the same workmanship costs 3.0 times as much in the shallow member, and the shallow member is the one whose steel is walked on before the pour.
Fig. 1 Moment capacity of a slab against the cover to its reinforcement. The line is very nearly straight, because capacity is proportional to a dimension the drawing does not contain.

Which free body produced the number

The couple. Cut the section, and what crosses the cut is a compression in the concrete near the top face and a tension in the steel, equal and opposite because there is no axial force. The moment is the pair times the distance between them, and that distance is the lever arm.

The compression sits at about 0.4x0.4x from the compression face, where xx is the neutral axis depth, and the tension sits at the centroid of the bars. So

z=d−0.4xz = d - 0.4x

and the whole capacity rides on where the steel is, measured from the compression face. Not from the soffit, not from the centre of the section, and not from anything a tape measure can reach after the pour.

Differentiate and the sensitivity comes out slightly above one:

dM/Mdd/d=dd−0.4x>1\frac{dM/M}{dd/d} = \frac{d}{d - 0.4x} > 1

because losing depth costs the lever arm the full amount while xx stays where it is. On a lightly reinforced slab the elasticity is a little above 1.05 — capacity falls slightly faster than in proportion. On a heavily reinforced beam it is larger, because xx is a larger fraction of dd.

Why the slab is the problem and the beam is not

The arithmetic is a proportion, so it is the ratio of the error to the depth that matters, and the depth varies across the structures on one project by a factor of three or four.

A 225 mm slab with 30 mm cover and 16 mm bars has d=187d = 187 mm. Ten millimetres of misplacement is 5.3% of it.

A 600 mm beam with 35 mm cover and 25 mm bars has d=552d = 552 mm. The same ten millimetres is 1.8%.

The same workmanship costs the slab three times what it costs the beam. And the direction of the practical problem runs the same way: a beam’s steel sits in a cage in a narrow, deep form and is difficult to displace, while a slab’s top steel is a mat spread over an entire floor plate that people walk on to place the concrete.

A scatter that belongs to the site, not to the material. The as-built effective depth of a 225 mm slab, and the capacity that follows from it. The drawing says 187 mm. The distribution is centred 5 mm below it, because reinforcement in a slab is walked on and pushed down far more often than it is lifted, and it has a standard deviation of 9 mm. Two losses follow and they are separate: the bias takes 2.8% off the capacity before any scatter is considered, and the 5% fractile takes 11.2%. Neither appears in any material partial factor, because neither is about the material — the concrete and the steel are exactly as strong as they were tested to be, and the member is 11.2% weaker than the drawing says because a dimension nobody can measure after the pour came out different.
Fig. 2 The as-built effective depth of a slab, and the capacity that follows. Two losses, and they are separate: a bias, which is a loss before any scatter is considered, and a fractile, which is the scatter.

There is a further asymmetry that makes it worse. Slabs are usually designed close to their limit — they are repetitive, the material saving of getting them right is multiplied by the whole floor area, and there is rarely a reason to make one thicker than it needs to be. Beams are frequently governed by something else entirely: a depth chosen for services, a section chosen from a list, a stiffness limit rather than a strength one. So the member whose capacity is most sensitive to the error is also the member with the least margin to absorb it.

The bias, which is the part with a known sign

Measurements of as-built bar position are not symmetric about the drawn value. Top reinforcement in slabs comes out low, consistently and by several millimetres, and the reasons are mechanical rather than careless: the mat is stood on, the chairs sink into the mesh below, the pour pushes it down, and nothing pushes it up.

That gives the distribution two features, and they are worth separating because they are different arguments.

The bias is a loss with a known sign. It is not covered by any statistical treatment of scatter, because it is not scatter — a five-millimetre bias on a 187 mm effective depth is 2.7% of the capacity, gone, before any distribution is considered.

The scatter about the biased mean is the ordinary statistical question, and it is answered the way the strength no specimen had answers it: take a lower fractile. With a standard deviation of 9 mm the 5% fractile sits about 15 mm below the biased mean and 20 mm below the drawing, which is a ninth of the capacity.

Neither of those numbers is in any material partial factor, because neither is about a material. The concrete is exactly as strong as it was tested to be. The steel is exactly as strong as its certificate says. The member is a tenth weaker than the calculation because a dimension came out different, and the calculation has no term for it.

Where the same dimension appears again

dd is not only in the bending calculation. It appears, with the same sensitivity or worse, in most of the other checks made on the same member.

Shear. The strength of a member with no links is proportional to dd through the area and inversely through the size term, so the net effect is milder — but the shear force is carried on bwdb_wd and the reduction is direct.

A strength with no mechanism in it, made of four. The shear a member carries with no links in it, split into the mechanisms that carry it, against the member's effective depth on a logarithmic axis. The three bands are calibrated to Taylor's measured shares at one 1000 mm × 500 mm member and are then evaluated everywhere else, so the shape of the total is a prediction. Aggregate interlock is the band that dies: it depends on how tightly the crack faces are held together, crack width grows with member depth, and it falls from 70% of a shallow member's strength to 29% of a deep one's. That decay is the whole of the size effect, and the dashed line is the code's fitted k = 1 + √(200/d), which knows nothing about interlock and falls by a factor of 1.52 where the model falls by 2.39 over the same twentyfold range. Dowel action is why the expression contains the flexural reinforcement ratio, which nothing in a truss analogy would predict.
Fig. 3 The mechanisms carrying shear in a member with no links. All of them are computed on the effective depth, and a slab’s is the shallowest in the building.

Punching. A check made on a perimeter is made on a perimeter drawn 2d2d from the column face and on a depth dd, so the resisting area goes very nearly as d2d^2. Ten millimetres off a 187 mm slab is about 11% of the punching capacity — twice the flexural loss, on the failure that gives the least warning.

A check made on a perimeter, not on a section. One bay of a flat slab, 7.5 m square, on a 400 × 400 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 3950 mm long against 1600 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 661 kN across 3950 × 187 mm, a shear stress of 0.894 N/mm² against a resistance of 0.678.
Fig. 4 The perimeter a punching check is made on. Both the length of the perimeter and the depth it acts over come from the effective depth, so the capacity goes as its square.

Crack width and deflection. Both are computed on the cracked section’s stiffness, which depends on dd through the neutral axis position and the steel’s lever arm — and deflection is the check where a small loss of stiffness is amplified by span to the fourth into a visible one.

Anchorage. The tension the bar has to develop is the tension the calculation gives it, and where a bar is lapped or curtailed is set out from a shift distance that is itself a function of dd.

The tension a bar must carry is the moment from further along. A beam with a diagonal crack at 40 degrees, and the tension the bottom bar is asked for. Beam theory takes the tension at a section from the moment at that section; the truss model does not, because the compression in the diagonal above the crack has to be balanced by tension at the bar's far end. The tension diagram is therefore the moment diagram displaced by a_l = z(cot θ − cot α)/2, which for z = 380 mm and vertical links is 226 mm — 11 bar diameters. The development length starts from there, so a curtailment computed from the moment diagram alone stops the bar 226 mm too early at every point it is cut off.
Fig. 5 The shift rule. A diagonal crack means the bar at a section carries the tension belonging to a section further along, and the distance is proportional to the lever arm.

So a member built with its steel 10 mm low is not a member with one check 5% short. It is a member with four checks short by between 2% and 11%, all correlated, all from the same cause, and all invisible.

The correlation is the part that should be uncomfortable. Structural reliability is built on the assumption that the many small uncertainties in a calculation are largely independent, so that they combine as a root-sum-square rather than as a sum — which is the same argument the strength no specimen had makes about a material’s scatter and the same one a load combination makes about actions. A single misplaced mat breaks that assumption completely: one cause, four consequences, all in the same direction, on the same member, on the same day. It is the structural equivalent of a common-mode failure, and it is produced by a boot.

Why the tolerance is where it is

The obvious response is to specify a tighter tolerance, and it is worth asking why nobody does.

The cover is not a free variable. It is set by durability — the depth of concrete the chlorides or the carbonation front has to travel through before it reaches the steel — and by fire, and by the bond length the bar needs. Reducing it to gain effective depth trades a structural quantity for a durability one, on a member whose design life is fifty or a hundred years.

The link diameter is set by the shear design. Half the bar diameter is set by the bar. Neither is a tolerance.

What is left is the accuracy with which the specified cover is achieved, and here the trade is different: a tighter tolerance is a matter of chairs, spacers, inspection and rework, all of which cost money on every square metre of every floor. The profession’s answer has been to accept the scatter and to build the consequence into the design in the crudest possible way — by rounding the effective depth down and by using generous minimum reinforcement.

The characteristic strength, which nothing was measured at. A lognormal population of strengths with a mean of 34 N/mm² and a coefficient of variation of 0.13. The characteristic value is the 5% fractile — 27.2 N/mm², which is 80% of the mean, and which need not be the strength of any specimen that was tested. Dividing it by 1.50 gives 18.2, and the shaded sliver below that is the fraction of the population that would fail to reach it: 8.9e-7, or one in 1,122,229. A factor applied to a fractile is not covering the scatter, because the scatter has already been spent getting to the fractile.
Fig. 6 How a material’s own scatter is handled: a characteristic value at a lower fractile, then a partial factor. The geometry gets neither, which is the asymmetry this essay is about.

There is a third response that is used more often than either and is worth naming because it looks like neither: put the check somewhere the tolerance does not reach. A slab whose thickness is governed by deflection rather than by strength has a flexural capacity with a large margin in it, and a ten-millimetre error eats margin rather than capacity. That is not a fudge; it is a deliberate choice about which limit state governs, and it is one of the reasons stiffness is not strength is a useful thing to know about a member as well as about a material.

That is a defensible engineering answer and it has one uncomfortable property: it is not stated anywhere. A reader of a calculation sheet sees d=187d = 187 and has no way to know whether the author intended that as a nominal value, a mean, or a lower bound.

What the picture cannot show

The figures above draw one member’s capacity against one dimension, and a floor is not one member.

A slab spanning between beams has hundreds of square metres of top steel over the supports, and the bars are not all misplaced by the same amount. The floor’s capacity is not the capacity of the worst strip; it is the capacity of a two-dimensional plate that redistributes, so a locally low bar mat is carried by its neighbours the way the slab that spans both ways describes. The correlation length of the error matters as much as its size, and nothing here measures it.

A two-way slab is a one-way slab as soon as it is not square. The share of the load carried by the strips spanning the short way, against the ratio of the sides. The two families of strips cross at the centre and must deflect equally there, and a strip's deflection goes as the fourth power of its span — so at a ratio of 1.33 the short strips already take 76% and at 2 they take 94%. The panel drawn here is 6 × 7.5 m, a ratio of 1.25, and its short strips take 70.9%. Two-way action is worth having at a ratio of one and worth almost nothing by two.
Fig. 7 A plate does not fail where one strip is weak. Load sheds into the stiffer direction and into the neighbouring strips, so a local shortfall is not a local failure — which is the reason floors survive workmanship that a beam calculation would not.

Nor does anything here show the case where the error is in the other direction and the bar is too high in a member where it belongs at the bottom, which is the same loss with the sign of the eccentricity reversed and is much rarer.

Where the model stops

The reinforcement was assumed to be where its centroid says. A single layer of bars has a centroid at the bar centre and that is what the formula uses. Two layers have a centroid between them, and a bar detailed as a second layer and placed as a third takes the whole bundle down — which is a much larger error than a misplaced chair and is a detailing decision rather than a workmanship one. Where the steel is, not how much of it is the argument that position beats area, and this is that argument arriving as a tolerance.

The bias and the scatter are quoted, not derived. They come from surveys of built structures and they vary enormously with the type of member, the type of formwork and the supervision. Four millimetres and eight are representative of slab top steel in ordinary construction, and they are not a property of concrete.

The lever arm was taken as computed rather than as capped. Real design caps zz at about 0.95d0.95d to avoid relying on a very small compression zone, which flattens the sensitivity slightly at the lightly reinforced end.

The lower bar was assumed to be the one that matters. In a continuous slab the critical section is over the support and the critical steel is the top mat, which is the one that gets walked on — so the sensitive check and the vulnerable bar are the same bar, which is worse than it would be if they were different ones. In a simply supported slab the bottom steel is held up off the formwork by chairs and comes out very nearly where it was drawn.

And the whole argument assumes plane sections. The assumption underneath is what makes a lever arm meaningful at all, and near a support or a concentrated load it is not true — at which point the effective depth stops being a lever arm and becomes a geometrical input to a strut-and-tie model, where its sensitivity is different again.

Wrong in shape, right in two integrals. The compression zone of a C40 section with its neutral axis 120 mm down, drawn twice. The curved outline is the real parabolic-rectangular stress distribution — the material's own law read off the linear strain profile plane sections supplies. The rectangle over it is what every design office uses instead: intensity η f_cd = 22.1 MPa over a depth λx = 100 mm. The two shapes are visibly different and give the same answer, because a bending calculation asks a stress distribution only two questions — how much compression there is, and where its resultant acts. Both are 771 kN at 49.9 mm from the face. The factors are α = 0.8095 and β = 0.4160, and λ = 2β follows from wanting the same centroid. A triangle and a full rectangle match neither integral and are nowhere near.
Fig. 8 The compression the lever arm is measured to. The block is a deliberate simplification of a curved stress distribution, chosen so that its resultant and its position are both right — and its position is what the effective depth is measured from.

The generalisation

The habit worth having is to identify, for any capacity, which of its terms is a dimension and which is a property, and then to ask which of the dimensions is actually controlled.

Properties are tested. Every material in the calculation has a certificate, a characteristic value and a factor covering the difference between the specimen and the structure. Dimensions get a tolerance on a drawing and, very often, no factor at all — and the ones that matter are not always the ones that are easy to check. A slab’s thickness is measured on site to within a few millimetres and it is the least important of the four terms; the position of the steel inside it is the most important and is checked, if at all, by looking.

One test of whether a term is genuinely controlled is to ask what would happen if it were wrong and nobody noticed. A concrete strength that comes out low is caught by a cube result, and there is a procedure. A bar diameter that is wrong is caught by looking at the steel. An effective depth that is 15 mm short is caught by nothing at all: the member is finished, it looks exactly as it should, it passes every visual inspection ever devised, and its capacity is a tenth lower than the drawing says. That is the definition of an uncontrolled variable, and it sits in the numerator of every equation on the sheet.

Geometry beats material is usually a cheerful observation about how much a designer can buy with a shape. Here it is the same fact read as a warning: if the geometry decides the strength, then the strength is only as reliable as the geometry, and a beautifully specified concrete placed around badly positioned steel is a strong material in a weak member.

Where this goes next

The neutral axis is wherever the first moment vanishes. A 300 by 600 section with 1800 mm² of steel at a depth of 540, carrying 250 kNm after the tension side has cracked. The neutral axis is no longer at mid-depth: it has risen to 234.5 mm from the top, which is where the first moment of the compression zone plus the transformed steel vanishes. The compression is 15.4 N/mm² at the top fibre and the steel carries 301 N/mm²; the resulting couple is 541 kN on a lever arm of 462 mm, which multiplies back to the 250 kNm applied. The uncracked section would have had 6673×10⁶ mm⁴ against the cracked 3809×10⁶ — a loss of 43% of the stiffness. Where the steel is, not how much of it A section in bending resists a moment with a couple, and a couple is a force times a distance. The force is bought — it is an area of steel at a stress. The distance is free, decided by where the bars were put, and it is the variable almost nobody optimises because it does not appear on an order. The characteristic strength, which nothing was measured at. A lognormal population of strengths with a mean of 30 N/mm² and a coefficient of variation of 0.15. The characteristic value is the 5% fractile — 23.2 N/mm², which is 77% of the mean, and which need not be the strength of any specimen that was tested. Dividing it by 1.50 gives 15.5, and the shaded sliver below that is the fraction of the population that would fail to reach it: 6.4e-6, or one in 155,818. A factor applied to a fractile is not covering the scatter, because the scatter has already been spent getting to the fractile. The strength no specimen had A material property is written into a calculation as a number, and a material does not have one. It has a population of strengths with a mean and a spread, and the number used is a low fractile of that population — a value that need not have been measured, that most of the material exceeds, and whose distance below the mean is decided entirely by the scatter. Where plane sections stop staying plane. Strain across a cut face at four span-to-depth ratios, with the straight line the theory assumes drawn faintly behind. For a slender beam the two coincide; for a beam as deep as its span the real distribution is nothing like a straight line, and beam theory has no claim on it. Plane sections stay plane, and what the assumption costs Beam theory rests on one sentence about geometry. It is very nearly true for a slender member, wrong for a deep one, and everything in the subject that fails does so where it stops holding. A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 400 × 400 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 4427 mm long against 1600 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 604 kN across 4427 × 225 mm, a shear stress of 0.606 N/mm² against a resistance of 0.658. A check made on a perimeter, not on a section Every shear check in this collection is made on a plane cut through a member. A slab sitting on a column has no such plane, because the shear leaves in every direction at once — so the check is made on a closed line, and a line grows with the column while the load grows with the square of the bay.

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.

Bending momentBondCharacteristic strengthCoverCracked sectionDurabilityEffective depthLever armPartial factorPlane sectionsPunching shearReinforcement ratioSection modulusSize effectTolerance