The steel the concrete asks for

Reinforcement is put into concrete because something is applied to it. Bending steel is sized by a bending moment, links by a shear, ties by a tensile force; in every case the calculation starts with an action and ends with an area.

There is one exception, and it is the reason a great many members are reinforced far more heavily than anything they carry would suggest. Minimum reinforcement is sized by the concrete, and the calculation contains no load at all.

Which free body produced the number #

Two of them, at two different instants in the life of the same section, and the requirement is that the second is stronger than the first.

The first is the uncracked section, an instant before it cracks. The concrete is elastic in both directions, the neutral axis is at the centroid of the gross section, and the moment that takes the bottom fibre to the tensile strength is

$$M_{cr} = f_{ctm}\,\frac{bh^2}{6}$$

which is the tensile strength times the section modulus of a rectangle. There is no steel in it worth speaking of — a lightly reinforced section’s transformed section modulus is within a few per cent of the gross one — and there is certainly no load in it.

The second is the cracked section at ultimate. The concrete below the neutral axis has gone; the steel carries the whole tension; and the moment is the steel force times the lever arm, $A_s f_{yd} z$ with $z$ about $0.95d$. When half the section has given up is that free body in full.

The requirement is $M_u \ge M_{cr}$, and it is a requirement about sequence rather than about strength. If the cracked section is weaker than the uncracked one, then at the instant the concrete cracks the tensile force it was carrying — which is real, and about $\tfrac{1}{2}f_{ctm}bh/2$ — is transferred to a smaller steel force that cannot hold it. The crack runs, the section fails, and it fails at the load that cracked it with no deflection to give warning.

Where the printed constant comes from #

Codes give the requirement as $A_{s,min} = 0.26\,(f_{ctm}/f_{yk})\,b d$, and read as a rule that is a number to look up. Read as the two free bodies above it is a division.

Set the two moments equal and rearrange:

$$A_{s,min} = \frac{f_{ctm}\,bh^2/6}{f_{yd}\,z}$$

For a section whose overall depth is about $1.1d$ and whose lever arm is about $0.95d$, that is $A_s = (1.21/6/0.95)(f_{ctm}/f_{yd})bd = 0.212(f_{ctm}/f_{yd})bd$; putting the partial factor back on the steel gives 0.245 against the printed 0.26. Six per cent, which is the margin the rule carries and not the rule itself.

That matters for a practical reason. A constant that is derived can be re-derived for a section the rule was not written about — a tee, a hollow-core unit, a member with an unusual $h/d$, a section of two materials — and one that is looked up cannot. The tee is the case that bites: its section modulus to the tension face and its lever arm are both quite different from a rectangle’s, and using $0.26bd$ on it is a rule applied outside its derivation with no way of knowing which direction the error runs.

The first consequence, which is backwards #

A deeper member needs more of this steel, for less load.

The cracking moment goes as $h^2$; the ultimate capacity goes as $d$, so the area required goes very nearly as $h$. A 200 mm slab wants 289 mm² per metre and an 800 mm one wants 948 — more than three times as much, in a member three or four times stronger and probably carrying the same load.

The place this bites is not the beam somebody sized for its bending. It is the member that is thick for a reason having nothing to do with strength: a transfer podium at 900 mm, a basement wall at 400, a raft, an architectural upstand, a slab thickened for fire or for acoustics. Each of those is governed by its own thickness, and each ends up with reinforcement whose only justification is that the concrete around it is strong in tension.

The second consequence, which is also backwards #

A stronger concrete needs more of it.

C30 has a mean tensile strength of 2.9 N/mm²; C60 has 4.35, which is 50% more. The minimum steel goes up in proportion, in a member whose bending capacity has hardly changed at all — because the bending capacity of a normally reinforced section is set by the steel, and the concrete is only there to supply the compression block, which was never the limit.

So specifying a higher grade to solve a durability problem, or because the supplier’s standard mix has moved, or because the columns wanted it and the slab shares the pour, brings a reinforcement increase nobody asked for. The strength that is never used is the property behind that, and this is the sharpest of its consequences: the tensile strength appears in no resistance and decides how much steel a member gets.

What happens below the line #

The section that fails at first crack is worth describing, because it does not look like a structural failure of the kind this collection usually draws.

Before the crack the member is stiff, uncracked and deflecting very little. At the cracking moment the concrete below the neutral axis lets go along a line, the whole of its tensile force arrives at the steel in an instant, and if the steel cannot hold it the bar yields immediately and then breaks or pulls out. The deflection between “apparently fine” and “on the floor” is a few millimetres.

Above the line, the same section cracks and carries on. The crack releases the concrete’s force into the steel, the steel stretches, the member gets softer — stiffer than its cracked section says is the transition — and the load can go on rising until the steel yields. The deflection before failure is tens of millimetres and visible, and the crack pattern is a set of many fine cracks instead of one wide one.

That is the whole of the requirement: it buys warning, not capacity. The property that appears in none of the equations is the general version of the same purchase, and it is why the check is made in a limit state that has nothing to do with the loads.

The size effect underneath it #

There is a reason the deeper member’s disadvantage is worse than the $h^2$ against $h$ arithmetic says, and it belongs to fracture rather than to strength.

The tensile strength measured on a small specimen is not the tensile strength of a large member. A flexural test on a 100 mm prism reads about half again what a direct tension test on the same concrete reads, and the ratio falls with the depth of the specimen — because what is being measured is not a stress at which the material fails but the load at which a fracture process zone of a fixed size can no longer be accommodated in the section. Hillerborg’s characteristic length, $E G_F/f_{ct}^2$, is about 300 mm for ordinary concrete, and a member much deeper than that behaves more like a fracture problem and less like a strength one.

So a deep member is doubly disadvantaged: it needs more steel because $M_{cr}$ goes as $h^2$, and its real cracking moment is a smaller fraction of the calculated one because its effective tensile strength is lower than the test that measured it. The bigger one is the weaker one is the whole of that effect, and the minimum reinforcement rule is one of the very few places in design where the two size effects push the same way.

A number from a real slab #

It is worth putting the arithmetic on one member, because the size of the requirement surprises people who have only met it as a rule.

A 400 mm ground-bearing slab in C30, 1 m wide, with 35 mm cover and 16 mm bars. The effective depth is 357 mm. The tensile strength is $0.30 imes 30^{2/3} = 2.90$ N/mm². The cracking moment is $2.90 imes 1000 imes 400^2/6 = 77.2$ kNm per metre.

To beat that, the cracked section needs $A_s imes 435 imes 0.95 imes 357 \ge 77.2 imes 10^6$, which gives 507 mm² per metre — call it H16 at 350 centres, or H12 at 200. The printed rule asks for 538, which is H16 at 350 comfortably.

Now ask what the slab is carrying. A ground-bearing slab spanning nothing at all, or a suspended slab at 4 m spans under a 3 kN/m² imposed load, would want a fraction of that for bending. The reinforcement in a great many slabs is minimum reinforcement, top and bottom, in both directions, and the calculation that produced it never mentioned the loading. That is not waste — it is the price of a member that cracks rather than snaps — but it is worth knowing which calculation is doing the buying.

The other reason the rule exists #

There is a second requirement wearing the same name, and it is worth separating because it produces a different number.

Minimum reinforcement for crack control is about restrained deformation rather than about applied moment. A wall or a slab restrained against shrinkage develops a tensile force it cannot avoid; when it cracks, the force in the concrete goes into the steel, and the steel must be able to carry $A_{ct}f_{ctm}$ without yielding — otherwise the first crack widens indefinitely instead of being followed by a second one somewhere else.

That gives $A_s = k\,k_c\,f_{ct,eff}A_{ct}/\sigma_s$, which looks like the bending rule and is a different calculation: it is about the whole tensile zone’s force rather than about a moment, and the stress allowed in the steel is a serviceability one chosen to limit the crack width rather than the yield strength.

The two are related by the same fact and it is worth stating plainly: restrained concrete always cracks. The strain at which it cracks is $f_{ctm}/E_{cm}$, about 100 microstrain; free shrinkage is 300 to 600. The ratio is about four, so cracking is not a risk to be avoided but a certainty to be detailed for, and the design decision is only ever about crack width. The strain that was imposed is why the naive arithmetic overstates the stress and creep saves the member from cracking as badly as elasticity predicts — and it does not save it from cracking.

Where the model stops #

The tensile strength was a mean value. It is the most variable property concrete has, with a coefficient of variation around 18%, and it is measured indirectly — by splitting or by flexure, which give different answers on the same concrete. Using a mean rather than a characteristic value here is deliberate: the check is that the section will not fail when it cracks, so the value to use is the one that will actually crack it, and a low estimate is the unsafe one. It is one of very few places in design where an upper-bound material strength is the conservative choice.

The section was rectangular. For a tee cracking with the flange in compression, the section modulus to the tension face is much smaller than $bh^2/6$ of the web-plus-flange, and the requirement falls with it. For a tee cracking the other way up — a cantilever, a support region — it rises sharply, and the rule as printed is wrong in the unsafe direction unless the derivation is redone.

The depth was constant. A haunched or tapered member has a different $h$ at every section and therefore a different requirement at every section, and the governing one is not necessarily where the load is worst.

And prestress changes the question entirely. A prestressed section does not crack at $f_{ctm}bh^2/6$; it cracks when the applied moment overcomes both the tensile strength and the precompression, which is four inequalities and a wedge. The requirement is the same in principle — the cracked section must beat the cracking moment — and the cracking moment is several times larger, which is why minimum reinforcement in prestressed members is a much sharper constraint than in reinforced ones.

The generalisation #

The habit worth taking away is to ask, of any design rule, what it is comparing — because a surprising number of them are comparing two states of the same member rather than a demand against a capacity.

This one compares the section before cracking with the section after. The section that cannot reach its own strength compares a plate’s local buckling stress with its own yield stress. A shear-friction check compares a crack that already exists with the reinforcement crossing it. A brittle-fracture check compares the flaw a plate might contain with the flaw its toughness will tolerate. In every case the applied load is absent from at least one side, and a rule with a load missing from one side is a rule about the member’s own behaviour.

Those rules have a family resemblance in how they fail, too. A demand-against-capacity check fails gradually: exceed it by ten per cent and something is ten per cent overstressed. A state-against-state check fails discontinuously, because what is on the far side of it is a different mechanism — and that is exactly why the requirement is a floor with a margin on it rather than a number to be utilised to ninety-nine per cent.