The same steel, and a wider crack

Concrete cracks. That is not a failure and not a defect; it is the design assumption, and every reinforced member in this collection is analysed on the basis that the concrete below the neutral axis has stopped carrying tension. What is designed is not whether the member cracks but how widely, and the arithmetic of that is one multiplication with two factors in it:

$$w_k = s_{r,max}\,(\varepsilon_{sm} - \varepsilon_{cm})$$

the maximum distance between cracks, times the difference between the average strain in the steel and the average strain in the concrete over that distance. Almost everything interesting about crack control is in the first factor, and almost everything a designer instinctively reaches for is in the second.

Which free body produced the number #

A length of the bar between two cracks, with the concrete around it.

At a crack the concrete carries no tension at all and the bar carries the whole of it. Away from the crack the bar sheds force into the concrete through bond, so the bar’s force falls and the concrete’s rises. A second crack can only form where the concrete has recovered enough tension to reach its tensile strength again — and the distance that takes is the transfer length.

Equate the force the bar can shed to the force the concrete needs to receive: $\pi\phi\,\ell_t\,f_{bd} = A_{c,eff}\,f_{ctm}$, and with $A_s = \pi\phi^2/4$ that rearranges to

$$\ell_t = \frac{\phi}{4}\cdot\frac{f_{ctm}}{f_{bd}}\cdot\frac{1}{\rho_{eff}}$$

proportional to the bar diameter and inversely proportional to the reinforcement ratio in the concrete the bar can control. The force that arrives along a length is the bond mechanism in full; this is the same length, asked from the other end.

The maximum spacing between cracks is twice the transfer length, because a gap of anything less cannot fit another crack in. Which gives the second thing worth carrying: the closest two cracks can be is half the furthest, by construction, so a member has cracks whose spacings differ by a factor of two whatever anybody does.

Why the bar diameter and not the steel area #

Put the transfer length and the effective area together and something falls out that is not obvious from either.

Hold the area of steel fixed and change the bar size. Halving the diameter quarters each bar’s area, so four times as many bars are needed, so the spacing between them falls by four — and the reinforcement ratio in the effective concrete area is unchanged, because the area is unchanged. What has changed is $\phi$ in the numerator, and it has halved.

So the crack width halves for the same steel. The measured version of that arithmetic: 1,340 mm² per metre in 8 mm bars at 38 centres gives 0.17 mm; the same 1,340 in 25 mm bars at 366 centres gives 0.37. A factor of 2.15 for nothing — no extra steel, no extra depth, no change to the bending calculation at all.

That is the whole practical content of crack control, and it is why a code’s crack-width provisions are usually expressed not as a width to calculate but as a table of maximum bar diameters and maximum spacings at a given steel stress. The table is the calculation, done once, and its two columns are the two things that matter.

The cover, which fights the other requirement #

The crack has to get from the bar to the surface, and it fans out on the way. So the width measured where anybody can see it — and where the environment can reach it — is larger than the width at the bar, and the difference grows with the distance travelled. The usual formulation puts about $3.4c$ into the crack spacing directly.

On the section drawn, that cover term is 35% of the total spacing. Raise the cover from 15 mm to 70 and the calculated width nearly trebles.

Now put that beside why the cover is there. Cover protects the reinforcement from carbonation and from chloride ingress; the deeper it is, the longer the front takes to reach the bar. And a crack is a short cut for exactly those fronts. The same clause of the same code asks for deep cover to keep the environment away from the bar and for narrow cracks to keep the environment away from the bar, and the first makes the second harder.

The resolution is the one already given: use smaller bars. That reduces the bar term without touching the cover term, which is the only lever that moves one and not the other. The load that comes from inside is what is being defended against, and it is worth remembering that a corroding bar is a structural action rather than a maintenance problem.

The second factor, and what the concrete is still doing #

The strain term is $(\varepsilon_{sm} - \varepsilon_{cm})$, and the subtraction is the interesting part.

At a crack the steel strain is $\sigma_s/E_s$. Between the cracks it is less, because the concrete is helping — it has recovered some tension through bond and is carrying part of the load. Averaged over the length, the steel does not stretch by as much as the crack-section stress suggests, and the crack is correspondingly narrower.

That is tension stiffening, and stiffer than its cracked section says is the deflection version of the same effect. Here it appears as a subtraction of $k_t f_{ct,eff}(1 + \alpha_e\rho)/\rho$ from the steel stress, with $k_t$ = 0.6 for a short-term load and 0.4 for a long-term one — a distinction that says the effect decays, because sustained loading and repeated loading break down the bond that produced it.

So a member’s cracks widen with time under no change of load at all, and the design value uses the long-term coefficient for that reason. It is one of the few serviceability quantities that gets worse on its own.

What the number is a characteristic of #

The calculated width is not a prediction of any particular crack, and treating it as one leads to arguments on site that nobody can win.

Three sources of scatter are built into the problem. The spacing varies by a factor of two by construction, as above. The bond strength varies with the concrete’s local strength, the bar’s rib pattern, the casting position and how much bleed water settled under the bar — a bar cast in the top of a deep pour has bond of about seventy per cent of a bottom-cast one. And the tensile strength varies, with a coefficient of variation near 18% and a strong size effect on top of it.

Put together, measured crack widths on nominally identical specimens scatter with a coefficient of variation around 40%. The calculated $w_k$ is a characteristic value of that population — roughly the value exceeded by one crack in twenty — so a member designed to 0.3 mm will have cracks at 0.15 and cracks at 0.4, and neither is evidence of anything.

That has a practical consequence worth stating plainly: a single wide crack found on site is not a failed calculation. What would be evidence is a pattern — cracks much more widely spaced than the calculation implies, which means the bond is not developing; or one crack much wider than its neighbours with none between, which means the reinforcement is discontinuous or the section is below its minimum and the first crack was the failure.

Where cracks are not the point at all #

Two situations look like crack-width problems and are not, and both are worth separating out.

A member below its minimum reinforcement does not have a crack-width problem; it has a failure mode. The first crack releases the concrete’s tension into steel that cannot hold it, and what follows is one very wide crack and a collapse. The steel the concrete asks for is the check, and it must be satisfied before a crack-width calculation means anything at all — a width formula applied to a section that fails at first crack computes the width of a crack that is going to open all the way.

A member cracked by restrained deformation has a different arithmetic. The steel stress is not what a bending calculation gives; it is whatever the concrete’s own tensile force becomes when it is released, $A_{ct}f_{ctm}/A_s$, and it does not depend on the applied load at all. That is the calculation that governs water-retaining walls, ground slabs and long façade panels, and it is why the movement nobody applied needs its own set of bars.

The distinction matters because the two respond to more steel in opposite ways. Under load, more steel lowers the stress and narrows the crack. Under restraint, more steel lowers the stress at the crack too — but only because it has taken the same fixed force at a lower stress, so the benefit is real and the total force is fixed by the concrete, exactly as in the minimum reinforcement rule.

Where the limit came from #

The numbers a code prints — 0.3 mm generally, 0.2 for aggressive exposure, 0.1 or a decompression requirement for water-retaining work — have less behind them than their two decimal places suggest, and knowing that is part of using them.

The durability limits were set from mid-twentieth-century exposure trials which mostly failed to find a relationship between crack width and corrosion at all, over the range of widths structures actually have. The mechanism is why: a crack lets the front reach the bar quickly at one point, and corrosion then propagates along the bar under the intact cover rather than being confined to the crack. So the rate of section loss is governed by the cover and the concrete quality, and the crack decides when it starts rather than how fast it goes. The width limits survive because they are a cheap proxy for workmanship and because nobody has produced a better one.

The water-retaining limits are different and are real. A crack through the full thickness of a wall leaks, and the leakage rate goes as roughly the cube of the width, so 0.2 mm passes about eight times what 0.1 does. Even there the number is generous, because a crack narrower than about 0.2 mm will usually autogenously heal — unhydrated cement and dissolved calcium carbonate seal it within a few weeks of first wetting — and the design is written around a self-repair mechanism that appears in no calculation.

The appearance limit is the honest one: 0.3 mm is about the width at which a crack becomes visible to somebody standing near it, and a great deal of crack-width design is being done to that criterion under another name.

The arithmetic on one slab #

A 400 mm slab, C30, 35 mm cover, H16 bars at 150 centres — 1,340 mm² per metre, which is well above the minimum this section needs.

The effective tension area is the width times $2.5(h-d)$, which for a 357 mm effective depth is 1,000 × 108 mm, so the effective reinforcement ratio is 1.25%. The crack spacing is $3.4 imes 35 + 0.17 imes 16/0.0125 = 119 + 218 = 337$ mm. At a quasi-permanent steel stress of 250 N/mm² the strain term is 0.75 millistrain, and the width is 0.25 mm.

Comfortable against 0.3. Now change one thing at a time. Go to H20 at 234 centres — identical steel — and the width is 0.30, exactly at the limit. Go to H12 at 84 and it is 0.21. Raise the cover to 50 mm and the H16 arrangement is at 0.30. Raise the steel stress from 250 to 300 and it is 0.32, over.

Four changes, none of which alters the strength of the member by a measurable amount, and each of which moves the serviceability answer by twenty per cent. That is the signature of a serviceability check, and the reason it is worth doing early: none of its levers is the one strength design uses.

Where the model stops #

Bond was assumed to develop fully. A bar with a splitting failure rather than a pull-out failure does not develop the bond strength the transfer length assumed, and the crack spacing is longer and the cracks wider. Splitting is a cover-controlled failure, so a member with low cover fails to develop bond and gets wider cracks — which is the opposite direction to the cover term above and can dominate at very low cover.

The effective area was a rule. $A_{c,eff}$ — the concrete a bar can control — is given as $2.5(h-d)$ deep by convention, and the convention is a fit to tests rather than a derivation. For a deep member with a single layer of steel it is the weakest step in the calculation, and it is the reason a deep beam’s crack widths are usually underestimated.

Only flexural cracks were considered. Shear cracks run at an angle and are crossed by links rather than by main steel, and their width is controlled by the link spacing and diameter through the same arithmetic with a different geometry. Cracks over a support of a two-way slab run in two directions at once and are controlled by whichever mat is nearer the surface.

And the load was static. A member under repeated load loses bond progressively, so cracks widen and the spacing does not change — which means the calculation’s second factor grows and its first does not. A member under a fatigue-type loading can end at twice its calculated width with no change of load.

The generalisation #

The idea worth carrying is that a quantity can be controlled by a dimension rather than by an amount, and that when it is, the cheap lever and the expensive lever are not the same lever.

Crack width is controlled by bar diameter, not by steel area. Punching shear is controlled by the perimeter, not by the depth of the slab — a check made on a perimeter is exactly that. Local buckling is controlled by the width-to-thickness ratio of a plate, not by how much steel is in the section. Torsional stiffness is controlled by whether a section is open or closed, not by how much material it has — a factor of six hundred for the same steel in the same place.

In every one of those, the instinctive response — add material — works, badly and expensively, while the geometric response works immediately and costs nothing. The way to notice which case is in front of a designer is to look at what appears in the expression: if a length appears that nobody chose deliberately, it is a lever, and it is usually a better one than the area beside it.