The ductility that depends on the ruler

Ductility is presupposed by half the methods on this site and appears in none of their formulae. When it does get a number, the number is almost always elongation after fracture: put the two halves of a broken tensile specimen back together, measure how much longer the marked length is than it was, and quote the increase as a percentage.

That measurement has a property nobody mentions. It depends on how far apart the marks were.

It is a small point stated carefully and a large one stated plainly. Stated plainly: the same steel can honestly be certified at 22 per cent or at 45, and the difference is which two marks were punched on the specimen.

Two different things added together #

A tensile specimen does not stretch uniformly all the way to fracture. It does so up to the ultimate load, and after that the deformation localises: a neck forms somewhere, the section there reduces faster than the material hardens, everything that happens subsequently happens inside a couple of diameters of that neck, and the rest of the specimen unloads slightly and stops moving.

So the measured extension is

$$\Delta L = \varepsilon_u L_0 + \delta_{\text{neck}}$$

— the first term a strain times the gauge length, and the second a length with no gauge length in it. Divide by $L_0$:

$$A = \varepsilon_u + \frac{\delta_{\text{neck}}}{L_0}.$$

The first term is a material property and the second is not. Barba observed in 1880 that $\delta_{\text{neck}}$ scales with the specimen’s cross-sectional dimension rather than with its length, so it goes as $\sqrt{S_0}$, and

$$A = \varepsilon_u + \beta\frac{\sqrt{S_0}}{L_0}.$$

For the 12.5 mm bar drawn, $\varepsilon_u = 0.16$ and $\beta\sqrt{S_0} = 7.2$ mm. Over the short gauge that 7.2 mm is 11.5 per cent of a 62.6 mm length; over the long one it is 5.75 per cent of 125.2. The material contributed the same 16 per cent to both, and the neck contributed twice as much to one as to the other.

The spread a fixed gauge length produces #

The consequences show up hardest where a laboratory has one extensometer and uses it for everything.

A factor of two on a ductility number is not a small error. It is the difference between a steel that satisfies a seismic detailing requirement and one that does not, between a reinforcement class B and a class C, and between a supplier’s certificate that passes and one that is rejected.

And it is entirely removable. Making the gauge length proportional to $\sqrt{S_0}$ puts the same $\sqrt{S_0}$ in the numerator and the denominator of the second term, and the quotient becomes a constant:

$$A = \varepsilon_u + \frac{\beta}{k}, \qquad L_0 = k\sqrt{S_0}.$$

Every specimen of every size then returns the same number, to the last digit the arithmetic carries. That is what 5.65 is for, and the peculiarity of the constant is a historical accident — it is the value that makes $L_0 = 5d$ for a round bar, which is what the old imperial practice used.

Which free body produced the number #

None, and that is unusual enough to be worth saying. Every other quantity on this site comes from cutting something and insisting the sums cancel. This one comes from a ruler.

That is precisely why it behaves the way it does. A quantity obtained by measurement rather than by equilibrium has no theorem protecting it from the measurement’s own arrangement, and the arrangement here — two punch marks a chosen distance apart — is doing part of the work the material is being credited with.

The same shape of problem turns up in strength rather than ductility, and it is worth naming because the diagnosis is identical. A modulus of rupture is a property of the specimen as much as of the material: it depends on how much volume is at high stress and therefore on how likely a flaw is to be found there, so the property depends on the test that measured it in the same way the elongation depends on the marks. Both are laboratory numbers carrying the laboratory in them, and in both cases the fix is to know which part is which rather than to distrust the measurement.

What the number is being used for #

The measurement’s shortcomings would matter less if the quantity were merely descriptive. It is not: elongation after fracture is doing real work in three places.

Material classification. A structural steel’s specification carries a minimum elongation for every grade and thickness, and a bar that misses it is rejected. Reinforcement is classed A, B or C partly on $A_{gt}$ — the elongation at maximum force, which is the uniform part and therefore free of the gauge-length problem — and partly on total elongation, which is not. The two are not interchangeable and the more careful codes have moved to the first.

Weldability and toughness screening. Elongation is used as a proxy for a material being in a ductile condition rather than a brittle one, and for that purpose the gauge length hardly matters because the distinction being drawn is between 25 per cent and 2.

Rotation capacity in a plastic hinge. This is the one where it matters most and fits worst. A plastic hinge needs the section to rotate through an angle, which needs the reinforcement to strain over a hinge length — a third ruler, of the order of the member’s depth, that nobody has calibrated against either of the tensile-test gauges. The curvature the hinge delivers is set by how far into the section the yielding has run, the rotation is that curvature times the hinge length, and the hinge length is an empirical quantity over which the curvature is assumed constant and is not. Comparing that demand with a supply measured over 5.65√S₀ of a 12 mm bar is comparing two lengths that have nothing to do with each other.

What happens after the neck forms #

The gauge-length problem is a consequence of localisation, and localisation is worth a paragraph of its own because it is the same phenomenon that ends several other stories on this site.

Up to the ultimate load the specimen is in a stable state: any section that thins slightly hardens slightly, and the hardening wins, so the thinning stops. That is a competition between two rates — the material’s hardening rate $d\sigma/d\varepsilon$ and the geometric softening from the area reducing — and the ultimate load is exactly where they balance. Considère’s construction is that statement drawn.

Past it the geometry wins, and the deformation runs away into whichever section happened to be weakest. The specimen stops being a member and becomes a mechanism, with a plastic zone a couple of diameters long and the rest of it a rigid body.

The uniform part is the honest one #

If a single number is wanted, the uniform elongation $\varepsilon_u$ — the strain at maximum force — is the one to use, and it is now specified as $A_{gt}$ in the reinforcement standards for exactly this reason.

It is genuinely a material property: it has no gauge length in it, it is the strain at which the material stops hardening fast enough to outrun the reduction in area, and it is the quantity that governs whether a bar can strain across a crack without localising there. For the steel drawn it is 16 per cent, against a total elongation of 27.5 or 21.75 depending on the ruler.

The awkwardness is that $\varepsilon_u$ is harder to measure, because it needs an extensometer on the specimen while the test runs rather than a ruler on the two halves afterwards. The number that is easy to obtain is the one that carries the specimen in it, and the number that is clean requires the equipment.

The rulers a structure actually uses #

Three lengths appear in any argument about ductility in a real structure, and none of them is the tensile test’s.

The bond transfer length. A bar crossing a crack in concrete does not strain over the member’s length; it strains over the distance either side of the crack in which the bond has not yet handed the force back — a few hundred millimetres, set by the bar diameter and the steel ratio. It is the same length that decides the crack spacing, and it is what the bar’s elongation capacity is actually being spent over. What the bar is being asked for across that crack is a total extension, and what it can supply is a strain times the length over which it is free to strain — so a high-ductility bar bonded well enough that it can only strain over 50 mm supplies less extension than a lower-ductility one bonded over 300.

The plastic hinge length. For a member forming a mechanism, the curvature concentrates over something of the order of the effective depth, and the rotation available is that curvature times that length. The empirical expressions for it disagree by a factor of two.

The bolt’s grip. A bolted connection’s ductility is the plate bearing over and the bolt shearing through a few millimetres, and the elongation of the material is not a quantity that enters at all.

So the certified number and the demand are quoted over three different rulers, none of which is the other’s, and the comparison between them is made by a code clause rather than by a calculation. That is not a criticism — the clause is calibrated on tests of members, which is the right way to do it — but it is worth knowing that the chain from a bar’s certificate to a frame’s rotation capacity has an uncalibrated length change in it at every link.

The property that sets the structural ruler #

The essay’s complaint so far is that the demand and the supply are quoted over different lengths and nobody calibrates between them. There is a material property that decides the structural length, it has no ruler in it at all, and it is specified alongside the elongation on every certificate: the ratio of ultimate to yield strength.

Consider a member yielding under a moment gradient — a hinge at a support, with the moment falling linearly to zero at a point of contraflexure a distance $a$ away. The section at the hinge reaches its ultimate moment; sections further along reach only a fraction of it, and any section whose moment is below the yield moment is still elastic. So the yielded length is

$$L_p = a\left(1 - \frac{M_y}{M_u}\right) \approx a\left(1 - \frac{f_y}{f_u}\right)$$

and the rotation available is the curvature capacity times that length.

Put numbers to it. A class B reinforcement with $f_u/f_y = 1.08$ yields over $1 - 1/1.08 = 7.4$ per cent of the distance to contraflexure. Ordinary mild steel at 1.35 yields over 25.9 per cent — three and a half times as long, and therefore delivers three and a half times the rotation from the same curvature capacity and the same elongation.

That is the missing link. The bar’s elongation says how much strain the material can take; the ratio $f_u/f_y$ says over how much of the member it is allowed to take it. A high-elongation bar with a ratio near one is a bar that will spend all of its ductility in a single section, reach its fracture strain there, and break — while the rest of the member is still elastic and has contributed nothing.

Which explains a specification that otherwise reads as belt and braces. The reinforcement classes require a minimum elongation and a minimum $f_u/f_y$ — 1.08 for class B, 1.15 for class C — and a bar can pass one and fail the other. Structural steel carries the same pair, and the yield-to-ultimate requirement is why a very high-strength steel with a short plateau is not simply a stronger version of a mild one.

It also explains why steels that have been cold-worked to raise their yield stress are treated with suspicion in seismic detailing. Cold working raises $f_y$ and barely moves $f_u$, so the ratio falls toward one — the elongation on the certificate may still pass, and the length over which the member can use it has collapsed.

So the honest statement is that ductility in a structure is a product of two numbers on the same certificate: a strain that says how much, and a strength ratio that says over how long. The first is the one everybody quotes and the one this essay has been complaining about. The second is length-free, harder to fake, and doing at least as much of the work.

Why it survived #

A defect this plain, in a measurement this common, has been known since 1880 and is still producing two numbers for one steel. The reason is worth stating because it is not carelessness.

The measurement is very cheap. Two punch marks, a ruler and the two halves of a broken specimen. Nothing else in materials testing is that cheap, and the alternative — an extensometer running through the test to capture $\varepsilon_u$ — needs an instrument, a calibration and an operator who is present at the right moment.

The number is used as a threshold, not as a quantity. Almost every application asks is it above 15 per cent rather than what is it, and a factor of 1.26 between two rulers does not change the answer to that question for a material that is either at 25 or at 3.

And the standards fixed it, in the only way available. They could not make the neck stop being a length, so they made the gauge length proportional to $\sqrt{S_0}$ and required the multiplier to be quoted. The problem is solved by convention rather than by physics, and the convention works exactly as long as everybody follows it.

Where the model stops #

Barba’s law is empirical. $\beta$ is fitted, it is around 0.4 to 0.8 for structural steels, and it is not the same for a material that necks diffusely as for one that necks sharply. The functional form — a strain plus a length over the gauge — is robust; the coefficient is a fit.

The neck’s extension is treated as independent of the gauge. It very nearly is, provided the gauge length is at least a few diameters, and it is not for a very short specimen where the neck’s plastic zone reaches the shoulders.

Where the neck forms is not controlled. If it forms near one end of the gauge length, part of it lies outside the marks and the measured elongation is lower. The standards handle this with a rule for shifting the marks, which is an admission that the measurement is about a location as well as a length.

Nothing here is about rate or temperature. Both change the strength, and both change the ductility more — a steel at its transition temperature has an elongation of a few per cent measured over any gauge at all.

The specimen is round. For a flat product — a plate, a strip, a sheet pile — $\sqrt{S_0}$ is still the right scale and the neck is a different shape, so $\beta$ differs, and the standards carry a separate table of proportional gauge lengths for flats. Comparing a plate’s elongation with a bar’s is a fourth ruler again.

And the drawing shows a curve that stops. Fracture is a separate event from necking, decided by a different mechanism, and the strain at which it arrives is scattered far more widely than any of the numbers on this page. The elongation quoted on a certificate is one specimen’s answer to a question whose distribution is wide, reported to three significant figures.

The check the arithmetic makes available #

Barba’s law is two constants, and two measurements at different gauge lengths determine both. That makes it a genuine test rather than a description.

Take the same steel, measure $A_5$ and $A_{10}$, and solve:

$$\varepsilon_u = 2A_{10} - A_5, \qquad \beta\sqrt{S_0} = 2L_5\,(A_5 - A_{10}).$$

For the numbers drawn — 27.50 and 21.75 per cent — that returns $\varepsilon_u = 16.0$ per cent and a neck extension of 7.2 mm, which are the values the curve was built from. A pair of certificates therefore contains the material’s own ductility and the specimen’s contribution, separately, and neither is on the certificate.

Two consequences follow. A supplier’s $A_5$ and $A_{10}$ that do not satisfy that relation are describing two different heats of steel, or two different test arrangements, and the discrepancy is a signal rather than scatter. And a designer given only one of the two can recover the other, provided $\beta$ is assumed — which is the weakest link and is worth about ten per cent.

The ladder from here #

Later rungs on this anchor: the true stress–strain curve, where the descending branch of the engineering curve disappears and the material turns out to have been hardening the whole time. Bridgman’s correction, which recovers the true stress in a neck from its measured radius and is the only way to get a material’s behaviour past the ultimate load. $A_{gt}$ and the reinforcement classes, and why the codes moved from total elongation to uniform. Rotation capacity and the plastic hinge length, which is the demand side of the same question and is calibrated on beams rather than on bars. Ductility in a connection rather than a member, where the length over which the strain is spread is a bolt’s grip length and can be a few millimetres. And the general lesson, which is worth more than the arithmetic: a quantity measured over a length is a quantity that needs the length quoted with it, and the number of places in structural engineering where that is not done is larger than it should be.