The bigger one is the weaker one

Make three beams of the same concrete, geometrically similar in every dimension — depth, span, notch, aggregate spacing if it were possible — and load each until it fails. Divide each failure load by the appropriate area to get a nominal stress, which is supposed to remove the size from the comparison.

It does not. The three nominal stresses come out 3.34, 2.98 and 1.81 N/mm², falling steadily with size, and the largest specimen is 46 per cent weaker than the smallest.

Nothing about the material changed. What changed is that a strength was being treated as a material property and it is behaving as a property of the specimen.

Which free body produced the number #

The energetic explanation needs no free body and two areas.

When a crack of length $a$ extends by $da$ in a body of thickness $b$, two things happen. It creates surface: $b\,da$ of new crack, costing $G_f\,b\,da$ of fracture energy, where $G_f$ is a property of the material with units of energy per unit area.

And it unloads a region around itself. The size of that region scales with the crack length — it is roughly a wedge of dimensions proportional to $a$ — so the energy released is proportional to $\sigma^2 a\,da\,b/E$.

Set the two equal:

$$\frac{\sigma^2 a}{E} \sim G_f \quad\Longrightarrow\quad \sigma \sim \sqrt{\frac{EG_f}{a}}$$

The released energy grows with size and the consumed energy does not. In a geometrically similar family the crack length at failure is proportional to the specimen size $D$, so nominal strength falls as $D^{-1/2}$ — for a large enough specimen.

For a small one it cannot, because the whole specimen is smaller than the region a crack needs to unload. There the strength is a plastic plateau. Bažant’s law is the interpolation between the two:

$$\sigma_N = \frac{Bf_t}{\sqrt{1 + D/D_0}}$$

with $D_0$ the size at which the two mechanisms are comparable.

The specimen and the structure are on opposite sides #

$D_0$ for a normal-strength concrete is of the order of a hundred millimetres, and that is exactly the awkward value.

A laboratory specimen — a 100 mm cube, a 150 mm cylinder, a small notched beam — sits on or below the transition, in the region where the plateau is a decent approximation. A real structural member — a 1,500 mm deep beam, a slab band, a foundation — sits well above it, in the region where the −½ slope has taken hold.

Measured on the law above: 3.10 N/mm² at 100 mm, 1.14 at 1,500. The test overestimates the structure by a factor of 2.71.

That is not a small conservatism buried in a partial factor. It is a systematic bias whose size depends on the ratio of the member to the specimen, which varies from structure to structure and is not accounted for by any factor applied uniformly.

It is worth setting beside the other scaling laws this collection is built on, because it is the odd one out. Deflection grows as the fourth power of span and self-weight as the cube of a linear dimension while cross-sectional strength grows as the square, and both of those are statements about geometry that hold for any material whatever. The size effect is a third relationship of the same shape and it is the only one that is about the material — which is why the nominal stress, an operation designed to divide the geometry out, does not remove it.

The other explanation, and why it is not enough #

Weibull’s account is statistical and much older. A brittle material contains flaws; failure is decided by the worst one; a larger specimen contains more of them; so the expected strength falls with volume as

$$\sigma_N \propto D^{-n/m}$$

with $n$ the number of dimensions being scaled and $m$ the Weibull modulus. It is a straight line on log axes, and it has no length scale in it anywhere.

Both laws predict that bigger is weaker, and they disagree about everything else. Over the range drawn here they differ by up to 251 per cent. More importantly, Weibull’s predicts a strength that falls for ever with the same slope, and it has nothing to say about where a transition occurs — because there is no transition in it to say anything about.

The physical distinction is about whether failure is decided before or during the crack’s growth. Weibull’s mechanism applies where a structure fails as soon as a crack starts, so the answer depends on the worst flaw. The energetic mechanism applies where the crack grows stably first, so the structure fails when the growth becomes unstable rather than when it begins — and the material’s fracture energy rather than its worst flaw is then the deciding property.

Concrete does the second, because it has a large fracture process zone in which aggregate interlock and bridging keep the crack stable for a while. Glass and fine ceramics do the first.

Where it shows up in structures #

The size effect is not a laboratory curiosity, and the places it appears are the places design has had the most trouble.

Shear in beams without stirrups. The classic case, and the one the whole subject was developed for. A shallow beam’s shear strength per unit area is much higher than a deep one’s, and the design expressions carry an explicit depth term — $(1 + \sqrt{200/d})$ and its relatives — which is a size effect written as a formula and rarely named as one.

Punching shear. The check made on a perimeter has the same depth term for the same reason, and it is the case where it bites hardest, because a flat slab’s resistance already goes as the 1.49 power of the effective depth before any size effect is applied.

Plain concrete and unreinforced members. A mass concrete dam, a plain footing, a concrete pavement: no reinforcement means no stable crack growth after cracking, and the size effect is at its strongest.

Bond and anchorage. Splitting failures scale with the same argument, which is why bond strengths measured on short pull-out specimens overstate what a long lapped bar achieves — and why the anchorage zone behind a prestressing plate is reinforced against a splitting tension rather than checked against a tensile strength.

The first of those is the one place a code writes the effect down, so it is worth drawing at the depths a design actually spans.

The two lines are the whole difficulty in one picture. A deeper member carries a great deal more shear and is worse at carrying it per unit of area, and only one of those two numbers is the one the check is written in. That is also why the depth term looks like an empirical correction rather than a theory: it is the interpolation above, evaluated over the range of depths a designer meets, and then written as $(1 + \sqrt{200/d})$ because a formula is easier to legislate than a curve.

Where the size effect is absent is equally worth knowing: any member whose failure is a yielding one. A steel section reaching its plastic moment does not care how big it is, because the mechanism is plastic and has no crack in it. The size effect is a brittleness effect, and reinforcing a member is one of the ways of removing it.

A stronger concrete has a worse size effect #

$D_0$ is the position of the transition and it is proportional to a material length — Hillerborg’s characteristic length,

$$\ell_{ch} = \frac{E\,G_f}{f_t^2}$$

which is the one combination of concrete’s properties with the dimensions of a distance. Everything about where a member sits on the curve is decided by it, and it moves in a direction that reverses the ordinary reading of “a better concrete”.

For a normal 30 N/mm² mix — $E$ around 32,000, $G_f$ around 0.1 N/mm, $f_t$ around 3.0 — it is about 356 mm. For an 80 N/mm² mix at $E = 42{,}000$, $G_f = 0.14$ and $f_t = 5.0$ it is about 235 mm. A third shorter, so the transition arrives at a smaller size, so a member of any given depth sits further along the falling branch.

The mechanism is visible on a broken surface. A crack in ordinary concrete runs round the aggregate, because the paste is the weak phase — so the surface is tortuous, the crack is long for the distance it advances, and the two faces interlock behind the tip and go on carrying stress across it. A crack in a high-strength mix runs through the aggregate, because the paste is now as strong as the stone. The surface is flat, the path is short, the bridging is gone, and both the fracture energy and the interlock behind the tip fall away.

Two consequences follow, and both run against intuition.

A stronger concrete is more brittle at every scale, and the size effect is the measure of it. Specifying a higher grade to solve a shear problem in a deep member raises $f_t$ by less than the size-effect penalty takes back, and on a large enough member it is a net loss.

Read the two curves together and the trade is exact. The stronger mix’s plateau is higher by two thirds, which is what was bought, and its transition arrives at two thirds of the size, which is what was not noticed. At 1,500 mm the two members carry 1.57 and 1.14 N/mm² — the stronger mix is still ahead, but by 38 per cent rather than by the 67 per cent its tensile strengths would suggest, and more than a third of the improvement has been spent on being further along a falling curve.

And the aggregate size is a lever nobody thinks of as structural. $G_f$ scales roughly with the maximum aggregate size, so a mix with 40 mm aggregate has a longer characteristic length than the same strength with 10 mm, and a large plain-concrete member is genuinely better off with the coarser mix. That is a rare instance of a mix-design decision made for a reason that is neither strength nor durability nor workability, and it explains why mass concrete has always been specified with the largest aggregate that will go through the pump.

What a model test is worth #

The practical version of all this is a rule about testing.

A scale model of a structure that fails by yielding is trustworthy. The mechanism has no length scale, and a model at any scale reproduces it, which is why plastic collapse tests on small steel frames have been useful for eighty years.

A scale model of a structure that fails by cracking is not. It fails at a nominal stress above the prototype’s, by a factor that depends on the scale ratio and on $D_0$, and the model has to be interpreted through the size-effect law rather than read directly.

There is a third case that is worse than either: a model whose failure mode changes with scale. A member that is shear-critical at full size can be flexure-critical at model scale, because shear strength falls with size and flexural strength does not. The model then fails by a different mechanism from the prototype and gives an answer to a question nobody asked.

A hundred years of not believing it #

The size effect has been observed, denied, rediscovered and resisted more than once, and the shape of the resistance is instructive.

Galileo has it, in the Two New Sciences of 1638, as the reason a large animal cannot have the same proportions as a small one — though his version is the square-cube law about self-weight rather than this one about strength. Leonardo has a note about a long wire being weaker than a short one, which is Weibull’s mechanism three centuries early.

What made the modern version hard to accept is that it contradicts a very deep habit. The nominal stress exists precisely to remove size from a comparison, and a subject that has spent two centuries learning to compare structures by stress finds it uncomfortable to be told that the comparison does not work. The response for most of the twentieth century was to treat the observed variation as scatter, as a testing artefact, or as a consequence of some secondary difference between the specimens.

The argument that settled it was not a better experiment but a dimensional one: a plastic limit gives a strength independent of size, linear elastic fracture mechanics gives one falling as $D^{-1/2}$, and any real material must transition between them somewhere. Once the question is where rather than whether, the data has an obvious shape and the shape has a parameter with units of length.

It is a good example of a general pattern in this collection. A quantity that looks like a material property turns out to be a property of the object — an effective length, a shape factor, a stiffness in an imposed-deformation problem — and the mistake in each case is the same one: treating a computed ratio as though it were something the material carries around with it.

Where the model stops #

$D_0$ and $B$ are fitted, not derived. Both come from regression on a set of tests of one geometry, and the same material tested in a different geometry gives different values. The law is a two-parameter fit to a physical argument, not a derivation from material constants.

It is a law for one failure mode. A structure whose failure changes character over the size range — from crushing to shear to flexure — is not described by a single curve at all.

And it assumes geometric similarity. Real structures are not similar to laboratory specimens: aggregate size stays the same while everything else grows, reinforcement bar sizes come in steps, and cover does not scale. Each of those breaks the similitude the law is written for, and the aggregate one is fundamental — the fracture process zone is a few aggregate diameters across whatever the member’s size, which is where $D_0$ comes from in the first place.

Reinforcement removes it, which is why it is rarely met #

A designer working on ordinary reinforced concrete meets the size effect once — in the shear expression’s depth term — and otherwise never, and it is worth being clear about why.

A reinforced member in bending does not fail by cracking. It cracks at a low load, goes on carrying, and fails when the steel yields and the concrete crushes — a mechanism with no crack propagation in it and therefore no size effect. The reinforcement has converted a brittle failure into a plastic one, and the conversion removes the phenomenon along with the brittleness.

What reinforcement does not convert is any mode it does not cross. Shear in a member with no stirrups; punching around a column; splitting along a bar; torsion before the links engage; failure of plain concrete anywhere. Those keep the size effect, and they are exactly the modes that reinforced concrete design treats with the most caution and the largest margins.

So the practical rule is short: wherever the design relies on the tensile strength of unreinforced concrete, the strength is a function of size. That single sentence covers every case above and explains why the subject can be nearly ignored in one part of concrete design and cannot be ignored at all in another.

The nearest thing to it elsewhere in the subject is the load that never came near failing anything, where a fatigue detail category turns out to be a strength that depends on the geometry of a weld rather than on the steel — quoted as a stress range because that is the convention, and describing a joint rather than a material. Both are quantities that look like properties, are tabulated like properties, and belong to the object.

What the pictures cannot show #

The three specimens are drawn as rectangles, and the thing that differs between them is not visible: the crack’s process zone is the same physical size in all three, so it occupies a large fraction of the small one and a negligible fraction of the large one. That ratio is the entire mechanism, and it cannot be drawn at three scales on one page.

Nor can the figures show the scatter. Every point on these curves is a mean of a distribution whose coefficient of variation is 10 to 20 per cent, and the size effect over a factor of two in size is comparable with the scatter within one size — which is why the effect took so long to be accepted and why it needs a wide size range to demonstrate.

A third thing outside the figures is time. Everything drawn is a monotonic test to failure at a standard loading rate, and concrete’s fracture energy is rate-dependent: loaded slowly it is lower, so a sustained load produces a size effect stronger than the one measured. A structure under permanent load is further along the curve than any test of it.

The assumption the figure rests on #

The whole curve is drawn for one material with one fracture energy, and $G_f$ is the least standardised quantity in concrete testing: the value depends on the specimen, on the notch, on the loading rate and on the method of extracting it, with a spread between methods larger than the differences between mixes. The transitional size $D_0$ is proportional to it, so the position of the transition — the most useful thing the theory provides — inherits that uncertainty directly.

The ladder from here #

Later rungs on this anchor: the fracture process zone measured rather than inferred, and the characteristic length $EG_f/f_t^2$ that sets $D_0$. Shear in beams without stirrups, where the size effect is the whole of the design expression’s depth term. The energetic size effect in compression, which is why a tall concrete cylinder is weaker than a cube. Type I against Type II size effects — failure at crack initiation against failure after stable growth — which are the two mechanisms this essay has treated as competitors and which apply to different structures. And the same argument at the other extreme of scale, where a specimen smaller than the process zone shows no size effect at all and behaves as the plastic plateau says it should.