The strength that is never used

Concrete is specified by its compressive strength, tested in compression, and designed almost entirely in compression. Its tensile strength appears in no resistance calculation that matters: not in bending, where the concrete below the neutral axis is assumed to have gone; not in columns; not in shear with links, where the truss carries it.

And it decides where nearly every transition in the subject sits. When a section cracks. How much minimum steel it needs. How far a bar has to be lapped. What a member without links carries. How wide a crack opens and how far apart the cracks are. How much of a slab’s stiffness survives. Whether a wall cracks under restrained shrinkage — it does.

A property designed to be ignored, that everything hinges on.

Which free body produced the number #

None, and that is the first thing to say about it. There is no free body that produces a tensile strength for concrete, because there is no stress at which concrete fails in tension in the way a metal has a yield stress.

What happens instead is that a fracture process zone forms — a region of microcracking a few tens of millimetres long — and it grows until it can no longer transfer the force across itself. The “strength” is whatever nominal stress the specimen was carrying when that happened, which depends on how large the specimen was relative to the process zone and on how the stress varied across it.

That is why the value depends on the test.

Direct tension is the honest measurement and is almost never made, because gripping a concrete specimen without cracking it near the grips is difficult.

Splitting — a cylinder crushed across a diameter — is the standard indirect test, and gives about $f_{ctm}/0.9$.

Flexural — a prism broken in bending — gives the largest value of all, because the tension is confined to the extreme fibre and the process zone occupies a large fraction of the depth. On a 100 mm prism it reads about 1.6 times the direct value, and the multiplier falls to one at about 600 mm depth.

A property that reads 1.6, 1.1 and 1.0 on the same concrete depending on how it was measured is not a strength, and treating it as one is where most of the trouble starts.

The real scale, which is a length #

The parameter that makes sense of all of it is not a stress. It is Hillerborg’s characteristic length,

$$\ell_{ch} = \frac{E\,G_F}{f_{ct}^2}$$

where $G_F$ is the fracture energy — the work needed to open a unit area of crack, about 0.08 N/mm for ordinary concrete. For C30 with 16 mm aggregate, $\ell_{ch}$ is about 300 mm.

That length is a property of the material and it separates two regimes. A member much smaller than it behaves plastically, redistributes stress round the process zone, and has a nominal strength near the material’s. A member much larger than it behaves according to linear elastic fracture mechanics, with a nominal strength falling as the inverse square root of size.

Almost every structural concrete member is within a factor of a few of 300 mm, which is exactly the worst place to be: neither limit applies, the transition is where the behaviour is most size-sensitive, and no simple expression covers it. The bigger one is the weaker one is that transition as a curve, and this is the length that positions a member on it.

It also explains why higher-strength concrete is relatively more brittle. $f_{ct}$ rises with the grade and $G_F$ rises much more slowly, so $\ell_{ch}$ falls — from about 400 mm at C20 to about 200 at C80. A high-strength member of a given size sits further toward the fracture end of the curve than a normal-strength one of the same size.

Where the number is actually spent #

It is worth listing the places, because none of them is called a tensile strength calculation.

The cracking moment, $f_{ctm}bh^2/6$, which decides the steel the concrete asks for — a requirement with no load in it at all.

Bond, $f_{bd} = 2.25\eta f_{ctd}$, and therefore every anchorage length and every lap length in the building. The failure mode being defended against is splitting of the cover, which is a tensile failure of the concrete around the bar.

Shear without links, whose expression $0.12k(100\rho f_{ck})^{1/3}$ has a cube root in it that nobody derives — and which is a fracture expression in disguise, with the $k = 1 + \sqrt{200/d}$ term being a size effect and nothing else. The strength with no mechanism in it is that expression’s whole problem.

Punching, which is the same expression on a perimeter — a check made on a perimeter is why the geometry rather than the section is what changes.

Crack spacing, through the transfer length, which puts $f_{ctm}/f_{bd}$ into every crack-width calculation — and since $f_{bd}$ is itself proportional to $f_{ctm}$, the ratio cancels and the crack spacing is nearly independent of the concrete grade. One of the few places the property’s variability does not propagate.

And the whole of tension stiffening, which is the concrete between the cracks still carrying load — stiffer than its cracked section says is what that is worth to a deflection, and it is proportional to the same property.

Restrained shrinkage, which is not a risk #

Put a number to the strain concrete can take before it cracks: $f_{ctm}/E_{cm}$, which for C30 is $2.90/32{,}800 = 88$ microstrain.

Now put a number to what it is going to be asked to take. Free drying shrinkage of an ordinary mix is 300 to 600 microstrain, and thermal contraction from the heat of hydration adds 100 to 300 more.

The ratio is four or more. A fully restrained member is therefore not at risk of cracking; it is certain to crack, several times over, and the design question is only ever how many cracks and how wide.

Two things soften that conclusion and neither removes it. Creep relaxes the stress as it builds, so the actual stress reached is well below $E\varepsilon$ — the strain that was imposed works that out and finds the naive 9.6 N/mm² becoming 2.3, which is below the tensile strength. And restraint is never full: a wall on a foundation is restrained at its base and free at its top, so the strain varies and the cracking is partial.

But the arithmetic is not close. It is the reason water-retaining structures are designed with a crack-width criterion rather than a no-cracking one, the reason movement joints exist, and the reason a long wall poured in one go cracks at intervals of about its own height whatever anybody does.

The one place it is relied on, and why that is defensible #

There is an exception to “never used as a resistance”, and it is instructive because of the conditions attached to it.

A member with no reinforcement at all — a mass concrete gravity dam, a plain concrete foundation, a blinding, an unreinforced ground slab — has nothing else to carry tension with, so its tensile strength is the resistance. Codes permit it, at a heavily reduced value, and the reductions say exactly what the reservations are: a low fractile rather than a mean; a size-effect factor; and a requirement that the failure be non-critical.

That last condition is the real one. A plain concrete member has a shape factor of one and no ductility whatever — what is left after the first fibre yields is nothing at all when the material has no yield — so the load at first crack is the collapse load and the warning is zero. The permission is granted only where a collapse would be tolerable.

A prestressed member is the more interesting case, because it relies on the tensile strength as a limit rather than as a resistance. A Class 1 or Class 2 member is designed so that the concrete’s tensile stress under service load stays below a stated fraction of $f_{ctm}$ — which is to say, the design is arranged so that the member never finds out whether the calculation was right. Four inequalities and a wedge is that arrangement, and two of the four inequalities are tensile-stress limits.

The number nobody measures on site #

There is a practical asymmetry worth pointing out, because it decides how much of this a designer can ever check.

Compressive strength is measured on every pour, several cubes at a time, and the result is on a certificate within a month. It is the number a contract is written around and the number a dispute is settled with.

Tensile strength is measured on almost nothing. It is inferred from the compressive strength through the two-thirds power law, and the inference carries the whole of the scatter of that relationship on top of the scatter of the test — the relationship itself has a coefficient of variation of about 20% around it, on concretes of the same nominal grade with different aggregates and different cements.

So the number that decides the cracking, the bond, the laps, the minimum steel and the shear capacity of a member without links is never measured and is derived from a number that is measured obsessively. That is a defensible arrangement — the correlation is good enough and the direct test is unreliable — but it is worth knowing when reading a calculation that quotes $f_{ctm} = 2.90$ to three figures.

The scatter, and why the mean is used #

Concrete’s tensile strength has a coefficient of variation around 18%, against about 15% for its compressive strength, and it is measured indirectly on top of that.

So which value to use is a real question, and the answer is unusual: the mean, not the characteristic. Every other material property in design is taken at a lower fractile, because a low strength is the unsafe case. Here it is not always.

In the minimum reinforcement check the question is whether the section can survive the moment that cracks it — so the value that matters is the one that will actually crack it, and a high tensile strength is the unsafe case. Using a 5% fractile there would be a mistake in the dangerous direction.

In a crack-control calculation the same logic applies: the force released at a crack is $A_{ct}f_{ctm}$, and the steel has to receive it, so a high value governs. In a shear-without-links check the logic reverses and a low value governs, which is why that expression uses a characteristic strength.

One property, three checks, two different fractiles, and the decision in each case comes from asking which direction is unsafe rather than from a rule about material factors.

Where the model stops #

$f_{ctm} = 0.30f_{ck}^{2/3}$ is a fit. It is a good one across normal-strength concrete and it changes form above C50, where a logarithmic expression takes over — a discontinuity in a design curve that is a confession that the fit had stopped working.

Aggregate was assumed ordinary. Fracture energy depends strongly on the aggregate: a rough crushed rock gives a tortuous crack path and a high $G_F$; a smooth river gravel gives a lower one; lightweight aggregate, where the crack runs through the particles rather than round them, gives a much lower one and a much shorter characteristic length. Lightweight concrete is measurably more brittle for that reason and its size effect is stronger.

Age was ignored. Tensile strength develops more slowly than compressive strength and reaches a smaller fraction of its long-term value at 28 days, which matters for early-age thermal cracking — the case where a member is being restrained hardest and is weakest.

And fibres change the question entirely. Steel or polymer fibres do not raise the tensile strength appreciably, and they raise the fracture energy by an order of magnitude — so they change $\ell_{ch}$ from 300 mm to metres, and move a member from the brittle side of the transition to the ductile side without changing the strength at all.

The generalisation #

The idea worth taking away is that a property can be structurally decisive without appearing in any resistance, and that the way to find such properties is to look at the transitions rather than at the capacities.

Concrete’s tensile strength decides when a member changes from uncracked to cracked, from one crack to many, from bonded to slipping, from ductile to brittle. Every one of those is a change of state, and a change of state is governed by whatever triggers it rather than by whatever carries the load afterwards.

The same shape occurs elsewhere. Steel’s toughness appears in no member check and decides whether a flaw grows — the flaw that sets the strength. A bolt’s preload appears in no ultimate calculation and decides whether a joint slips. A soil’s friction angle appears in a bearing capacity and its stiffness, which appears nowhere, decides the settlement that governs. In each case the property that is specified, tested and factored is not the property that decides what the structure does.

Which suggests a habit. When a design is behaving in a way the resistance calculations do not explain, look for the property that governs the transition rather than the one that governs the capacity — it is usually measured badly, quoted as a single number, and doing most of the work.

And there is a corollary about specification. A material is specified by the property somebody can test, which is not necessarily the property that governs — the strength no specimen had is about the gap between a specified value and a delivered one, and this essay is about a wider gap still: between the property specified and the property that decides. Closing the first is a matter of statistics. Closing the second needs somebody to have noticed which property it is.