The stress at which nothing in particular happens

Every figure on this site before this one rests on a sentence that has never been written down: stress is Young’s modulus times strain, in both directions, without limit, forever. It is the reason a truss can be solved by arithmetic, the reason a moment diagram is the integral of a load, and the reason a deflection is a fourth power of a span. It is also false, and the interesting question is not whether it is false but where the number that replaces it comes from.

That number is the yield stress. It appears in the plastic hinge, in the column curve, in the plate’s effective width, in every comparison of one section against another. It is quoted to three significant figures and treated as a property of steel in the way that density is a property of steel. It is not. For five of the six materials below there is no moment in a tensile test at which anything identifiable happens, and the yield stress is a line somebody agreed to draw.

What the plateau is, and why almost nothing has one #

The flat part of the mild steel curve is not a plateau in the sense of a gentle top. It is genuinely flat, and it is long: yielding begins at a strain of 0.131% and the stress does not rise again until about 1.5%, which is eleven times as far. Between those two points the specimen extends by about one part in seventy at no increase in load at all.

What is happening in that interval is not that every atom in the bar has reached its limit at once. It is that yielding is spreading. A band of yielded material — a Lüders band — nucleates somewhere, usually at a shoulder or a scratch, and travels along the specimen at constant load, converting elastic material to plastic material as it goes. The plateau ends when the bands have covered the whole gauge length and there is nothing left to convert. It is a propagation, and the flat stress is the stress at which the front travels.

That mechanism needs something specific: interstitial carbon and nitrogen atoms sitting in the iron lattice, pinning dislocations in place so that a higher stress is needed to start them moving than to keep them moving. Take the carbon out, or work the steel cold so that the dislocations are already free, and the plateau disappears. Which is why the high-strength grades have none — they are strengthened by processes that destroy exactly the thing that produces the yield point.

So the sharp yield point is not the normal case that other materials fail to achieve. It is a peculiarity of one family of alloys, and structural engineering inherited its vocabulary from a period when that family was almost the only structural metal there was.

The construction, and the arbitrariness in it #

For a material with no yield point, the number is made as follows. Take the initial straight part of the curve, whose slope is the elastic modulus. Draw a line of that slope starting not at the origin but at a strain of 0.002. Where it cuts the curve is the 0.2% proof stress, and that is the number that goes into every calculation.

The construction has a defensible meaning: it is the stress at which the specimen has acquired 0.2% of permanent strain, so that unloading from there would leave the bar 0.2% longer than it started. That is a real physical statement about the specimen. What is not physical is the 0.002.

Move the offset and the number moves with it. At 0.1% the same specimen reads 438 N/mm². At 0.5% it reads 495. That is a spread of 13% on a quantity that later gets multiplied by a section modulus and compared against a bending moment to three figures. The 0.2% is a convention, chosen because it is small enough to be a negligible permanent set in most applications and large enough to be measurable with an extensometer of ordinary quality. It is a good convention. It is not a measurement.

The arbitrariness is worth dwelling on because of where the number ends up. A designer who reads 460 from a mill certificate does not treat it as the answer to “what stress leaves 0.2% of permanent strain”. It is treated as the stress at which the material stops working elastically — which is a claim about a curve that has no such point on it. The two readings agree closely enough for a member in uniform tension, and they come apart the moment the stress varies across a section, because then the question is not when the worst fibre yields but how much of the section has.

What the curve becomes once a section is made of it #

A tensile specimen is at one stress throughout. A beam is not: the stress varies linearly with distance from the neutral axis, so the extreme fibre reaches the material’s limit while everything nearer the middle is still well inside it. Yielding therefore does not arrive all at once. It starts at the surface and works inward, and how it works inward depends entirely on the shape of the curve that this essay has been about.

The plateau is what makes that stress diagram flat-topped. A material with a rounded curve produces a rounded stress block, and a material with a falling branch — concrete in compression — produces one that peaks somewhere inside the section rather than at the face. Each of those is a different arithmetic for the same question, and all of them are the same computation with a different function plugged in.

The one part of the curve that is a property #

There is exactly one number on a stress-strain diagram that behaves the way a material property is supposed to behave, and it is the slope of the initial straight part.

The elastic modulus of steel is about 210,000 N/mm², and it is 210,000 N/mm² for mild steel, for high-strength steel, for weathering steel, for stainless steel to within a few per cent, and for a bar that has been cold-worked to twice its original strength. It cannot be alloyed upward in any useful amount. It is set by the stiffness of the interatomic bonds and the spacing between atoms, and metallurgy has almost no purchase on either.

That is a fact with consequences that reach into every other field on this site, and it gets its own essay. What matters here is the contrast: the strength of a steel is a number that processing can move by a factor of three, and the stiffness is a number that processing cannot move at all. One of them is a property of the alloy and the other is a property of the element.

Which free body produced the number #

Every number on these curves comes from one free body, and it is worth naming because it is smaller and simpler than any other free body on this site: a cylindrical bar of known cross-sectional area, gripped at both ends, with a measured force applied along its axis and a measured extension over a gauge length in the middle.

Cut the bar anywhere in the gauge length. The only action crossing the cut is an axial force, equal to the applied one, because nothing else is applied and the bar has no weight worth counting. Divide it by the area and the result is called the stress. Divide the extension by the gauge length and the result is called the strain. Both are averages over the cut and over the length, and both are exact statements about the specimen.

The subtlety is in the word area. The stress plotted on every one of these curves is the force divided by the original area, measured before the test — which is a convenient bookkeeping fiction, because the bar gets thinner as it stretches. At a strain of 2% the difference is under 2% and nobody cares. Past the ultimate stress it matters entirely: the specimen forms a neck, the area in the neck falls rapidly, and the engineering stress curve turns downward.

That downturn is one of the most persistently misread features in the subject. It looks like a material losing strength, and the material is doing nothing of the kind — the true stress in the neck goes on rising monotonically until fracture. The falling branch is an artefact of dividing by an area that is no longer there. What it does record honestly is the onset of instability: the point where the specimen stops deforming uniformly and starts concentrating everything into one place, which is the same class of event as a column deciding to go sideways and is analysed the same way.

Where the model stops #

The curve describes a specimen. Whether it describes the structure is a separate question with several separate answers.

A specimen is pulled slowly. Yield stress rises with strain rate — for mild steel, by something like 10% per decade of rate at ordinary temperatures. A tensile test takes minutes; an impact takes milliseconds. Six decades of rate is not nothing, and it is the reason a structure struck by a vehicle behaves less ductilely than the same structure loaded by hand.

A specimen is at room temperature. Heat it and the curve collapses, with the modulus going before the strength. Cool it and mild steel becomes capable of failing with no plastic deformation whatever, at a stress it passes every test at.

A specimen is pulled one way. Yield it in tension and it will yield in compression at a lower stress than before — an asymmetry that does not appear anywhere on a monotonic curve and cannot be inferred from it.

A specimen is one direction of one piece. Rolled steel is measurably weaker and much less ductile across the thickness than along the rolling direction, because the inclusions are flattened into planes. Timber differs along and across the grain by a factor of twenty. Concrete differs in tension and compression by a factor of eleven, and cast iron by four — and both of those are the reason those materials are used in the shapes they are used in.

The generalisation, which is uncomfortable #

The pattern behind all of this is that a material property is a summary of a test, and the test was designed by somebody with a purpose.

The tensile test was standardised to compare batches of steel for acceptance. It is superbly suited to that: it is cheap, repeatable, and sensitive to the things that go wrong in a rolling mill. It was never designed to supply the constants of a theory of structures, and the constants it supplies have the shape of an acceptance test rather than the shape of a mechanics.

The consequence is visible whenever a calculation needs something the test does not measure. The rotation a plastic hinge can deliver before the section falls apart is not on the curve. The stress a crack will tolerate is not on the curve — it requires an entirely different specimen with a deliberate notch in it. The range of stress a welded detail survives ten million times is not on the curve, and cannot be inferred from it at all: two steels whose curves differ by a factor of two lie on the same fatigue line.

So the honest position is that the stress-strain curve is one measurement among several, that it happens to be the one everybody was taught first, and that a surprising amount of structural failure lives in the gap between what it reports and what a structure needs.

A surprising place this turns up #

The most useful consequence of the plateau has nothing to do with strength. It is that mild steel makes its own factor of safety visible.

A member that has yielded has a permanent set. It sags, or it is out of line, or a painted surface has flaked along a band. Somebody notices. A structure of mild steel that has been overloaded generally announces it, and there is a long tradition of reading old ironwork for exactly these signs.

A high-strength steel member loaded to its proof stress has 0.2% of permanent strain in it, which over a two-metre length is four millimetres and over a bolt is invisible. The strengthening that removed the plateau also removed the warning. That is a real trade, and it is not a trade the strength number describes — it is a consequence of the shape of a curve, which no single number can carry.

Where the ladder goes next #

Later rungs on this anchor: the true stress curve and where necking really begins, which is Considère’s construction and is a stability argument rather than a strength one. The upper and lower yield points, and why the upper one is not usable. Strain ageing, in which a steel left alone after being strained recovers a yield point it had lost, and the embrittlement that comes with it. The statistical side — a characteristic strength is a fractile of a distribution rather than a value, and the scatter in concrete is many times the scatter in steel. Hardness as a proxy for strength and the empirical relations between them. And the anisotropy of rolled plate, where the through-thickness ductility is the property that decides whether a welded joint tears.

Historically, the tensile test is older than the theory it now feeds. Musschenbroek was pulling specimens in the 1720s; Hodgkinson’s tests on cast iron in the 1820s and 1830s established the tension-compression asymmetry and led directly to the asymmetric beam sections used for the next fifty years. The 0.2% offset convention is a twentieth-century arrival, standardised as aluminium and the alloy steels became structural materials and it became clear that the yield point everybody had been relying on was a property of one particular kind of iron.