Yield Strain: the Criterion Behind the Code, With a Worked Check

Every steel design check you run, whether it is bending, axial load, shear or the interaction equation, ends in a comparison against f_y, the yield strength. That is a stress. But steel does not carry a stress sensor; it carries deformation. What actually decides whether a fibre has yielded is whether it has stretched past a certain strain, the yield strain ε_y. Stress is the bookkeeping; strain is the physics.

For structural steel that strain is small and remarkably consistent: about 0.00125 (0.125%) for a 250 MPa steel, and it barely passes 0.0018 even for the strongest common grades. Everything the code layers on top of it, the safety factors, the slenderness limits, the plastic-design rules, assumes that number. This article unpacks ε_y = f_y/E from first principles, shows where the 0.2% offset comes from, and then runs a real IPE 300 beam through the CalcSteel FEM engine to check, to the fourth decimal, exactly when its outer fibre reaches yield strain.

Here is a definition you can quote: yield strain is the tensile or compressive strain at which a steel fibre stops behaving elastically and starts to yield. For a linear-elastic material it is ε_y = f_y/E, the yield stress divided by the modulus of elasticity.

It is a pure number, dimensionless, because it is a length change divided by a length (millimetres per millimetre). Engineers write it three equivalent ways: as a decimal (0.00125), as a percentage (0.125%), or in microstrain (1250 µε), the unit a strain gauge reports. All three describe the same event: a one-metre bar of 250 MPa steel has stretched 1.25 mm at the instant its fibre yields.

Two things make ε_y special. First, it is a material property, not a member property: it does not care whether the steel is a beam, a column or a bolt. Second, because E is nearly identical for every structural steel, ε_y is set almost entirely by the grade, which is exactly why the code can tabulate f_y and never once mention strain.

If strain is the real trigger, why does no design clause mention it? Because of a convenient accident of steel: its stiffness barely changes. Mild steel, high-strength low-alloy steel and quenched-and-tempered steel all share a modulus of elasticity within a couple of percent of E = 200 GPa. (AISC and NBR 8800 adopt 200 GPa; Eurocode uses 210 GPa.) One slope, for every grade.

That single fact collapses strain and stress into interchangeable quantities. On a stress-strain plot, all structural steels ride up the same elastic line from the origin; the only thing the grade changes is how far up that line the steel yields. A 250 MPa steel steps off at ε_y = 0.125%; a 345 MPa steel rides the same line to 0.173%; a 690 MPa steel to 0.345%. The code prints the vertical coordinate (f_y, a stress) because it is easy to measure and easy to tabulate, but the horizontal coordinate (ε_y, a strain) is doing the actual work. Change E and the whole mapping shifts; hold E fixed and f_y is just ε_y in disguise.

The relationship is Hooke's law rearranged. In the elastic range σ = E·ε, so at the yield point σ = f_y gives ε_y = f_y/E. Nothing more. The table runs that division for the grades you meet most, using E = 200 GPa.

Notice the spread: the strongest structural steel yields at under three times the strain of the mildest. Yield strain lives in a narrow band, roughly 0.12% to 0.35%, and for ordinary building steel it is essentially 0.12% to 0.18%.

One nuance worth carrying: Eurocode 3 adopts E = 210 GPa rather than 200, which nudges every ε_y down by about 5% (S355 becomes 0.00169 instead of 0.00178). It changes the strain, not the design stress, a reminder that ε_y is a derived quantity that inherits whatever E your code declares.

Everything so far assumed a clean yield point, the sharp knee that mild steel shows. But quenched-and-tempered steels, cold-formed sections and most high-strength grades have no knee; their curve rounds over smoothly with no obvious instant of yield. For those, the code has to invent a yield point, and it does so with a strain.

The rule is the 0.2% offset: draw a line parallel to the elastic slope but shifted 0.2% (0.002) along the strain axis; where it cuts the stress-strain curve, that stress is f_y. Read that again: the code's strength is defined by a strain construction. The total strain at that point is ε = 0.002 + f_y/E, the 0.2% permanent (plastic) set plus the elastic part that springs back. For a 690 MPa steel that is 0.002 + 0.00345 = 0.00545, or 0.545% total strain.

This is the deepest sense in which yield strain is the criterion behind the code: for a large class of steels, f_y is not a stress the material announces, it is a stress we back-calculate from a strain we chose.

Reaching ε_y is not failure; it is the start of steel's best trick. A mild-steel fibre that hits ε_y does not keep climbing the elastic line, it flattens onto a yield plateau, flowing at roughly constant stress f_y while the strain runs on. That plateau typically extends to a strain of order 1.5% to 2% before strain hardening begins, roughly 10 to 15 times ε_y. Only after the plateau does the curve rise again toward the ultimate stress f_u, then fall to fracture near 20% to 25% elongation.

That long, flat reserve is what the whole edifice of ductile design spends. Plastic analysis, moment redistribution, seismic energy dissipation, the very idea of a plastic hinge, all of them assume a fibre can sit at ε_y and keep straining without shedding load. It is why the codes demand a minimum f_u/f_y ratio (about 1.1 to 1.25) and a minimum elongation: they are buying plateau length. A steel that yielded and immediately fractured would meet the same f_y and be lethal to design with. Yield strain marks the door; the plateau is the room you get to use behind it.

Time to make it concrete. Take a simply supported IPE 300 beam spanning L = 6 m under a uniform load w = 20 kN/m, a routine floor beam. We model it in the CalcSteel FEM engine, the same solver behind the calculators, and read the results rather than trust a formula.

The engine returns a maximum bending moment M_max = 90.0 kN·m at midspan, matching the hand value wL²/8 = 20·6²/8 = 90.0 kN·m to the decimal. For the cross-section it computes an elastic section modulus S_x = 539.8 cm³ and second moment I_x = 8097 cm⁴. (S_x sits about 3% under the handbook 557 cm³ because the engine builds properties from the rolled dimensions with simplified fillets, a conservative and self-consistent value.) Those three numbers, the moment from the analysis and the section modulus from the geometry, are all we need to turn a load into a strain.

The extreme-fibre stress is σ = M/S_x. In consistent units, σ = 9000 kN·cm / 539.8 cm³ = 16.67 kN/cm² = 166.7 MPa. Divide by the modulus to get the strain the fibre is actually feeling: ε = σ/E = 166.7 / 200000 = 0.000834, that is 0.0834%.

Now the comparison the code makes for you, done explicitly. Against MR250 (ε_y = 0.00125) the fibre sits at 0.000834 / 0.00125 = 67% of yield strain. Against A572 Gr.50 (ε_y = 0.001725) it sits at 48%. Same beam, same load, same deflection: the only thing the grade changed is how much of the yield strain you have spent. That ratio ε/ε_y is identical to the familiar stress utilisation σ/f_y, which is the whole point: the utilisation figure your software prints is a yield-strain fraction wearing a stress costume.

A footnote for the practitioner: the engine also reports a midspan deflection of 2.084 cm, or L/288. This beam is comfortable on strain yet would flag against an L/360 deflection target, a reminder that yield strain governs strength, not serviceability.

You can watch this happen. In the beam calculator below, set up a span and a load and read the bending stress it reports; divide that stress by 200,000 MPa in your head and you have the outer-fibre strain. Raise the load and watch the stress climb toward f_y. The moment it touches f_y, the outer fibre is sitting exactly at yield strain ε_y, and the section has just begun to yield.

It is the same engine that produced the worked numbers above, free, with no login for the maths.

So at what load does this beam actually reach yield strain? First yield happens when M_max equals the yield moment M_y = f_y·S_x. For MR250, M_y = 25 kN/cm² · 539.8 cm³ = 13495 kN·cm = 135.0 kN·m, which for wL²/8 needs w_y = 8·M_y/L² = 8·135.0/6² = 29.99 kN/m.

We do not have to trust that algebra; we can ask the engine. Re-run the same beam at w = 29.99 kN/m and it returns M_max = 134.95 kN·m, σ = 250.0 MPa, ε = 0.001250. The strain matches ε_y for MR250 to the fourth decimal: this is first yield, confirmed by the solver and not asserted by a formula. The service load of 20 kN/m therefore sits at exactly two-thirds of the first-yield load, the 1.5 margin you would expect from the 67% strain utilisation.

And first yield is not collapse. Only the single outermost fibre has reached ε_y; every fibre below it is still elastic. Load can keep rising as yield spreads inward, which is the plastic reserve the plateau makes available.

Because ε_y = f_y/E and E is fixed, swapping steel grade moves the yield strain and nothing else about the elastic behaviour. Re-roll the worked beam in A572 Gr.50 instead of MR250 and follow what changes.

Yield strain rises from 0.00125 to 0.001725, a 38% jump. The first-yield load rises in lockstep: w_y goes from 29.99 to 41.39 kN/m (the engine's yield moment climbs to 186.2 kN·m), also 38%. But the stiffness, I_x and therefore the 2.084 cm deflection at 20 kN/m, does not move at all, because E is unchanged. This is the single most useful consequence of yield strain being f_y/E: a higher grade buys strength (more yield strain to spend) but not stiffness (the same slope E). It is why upgrading a floor from MR250 to A572 stops it yielding sooner yet does nothing for a beam that was failing on deflection or vibration, a trap worth remembering before you specify a premium grade.

Confusing yield strain with yield strength. ε_y is a strain (0.00125); f_y is a stress (250 MPa). They are linked by E, but they are not the same quantity, and only one of them is what the material actually senses.

Assuming a stronger grade is stiffer. The most common misread of the grades chart. Higher f_y means higher ε_y, not a steeper line. Every structural steel shares the same E, so elastic deflection is grade-independent.

Using engineering strain where true strain matters. At yield the difference is negligible (0.125% is 0.125% either way), so ε_y is safely an engineering strain. The distinction only bites far out on the strain-hardening range, near fracture.

Reading the 0.2% offset as 0.2% total strain. The offset is the plastic part only; the total strain at f_y is 0.002 plus the elastic f_y/E, for example 0.545% for a 690 MPa steel.

Forgetting ε_y is the same in tension, compression and bending. It is a fibre property. A bent flange, a compressed column fibre and a tension coupon all yield at the same ε_y for a given grade.

Is yield strain the same for all steel? Nearly. It lands between about 0.12% and 0.18% for common building grades because E is shared and f_y varies only from 235 to 355 MPa; high-strength steels push it toward 0.35%.

Why 0.2% and not another offset? It is a long-standing testing convention (ASTM A370 and E8, ISO 6892), chosen as a small, repeatable permanent set that is easy to construct on a chart and reflects a practically negligible residual deformation.

Yield strain is the quiet quantity every steel clause is built on: reach it, and the material stops being elastic.

• ε_y = f_y/E: about 0.00125 (0.125%) for 250 MPa steel and 0.001725 for 345 MPa, both on the same modulus E = 200 GPa.
• The code writes limits in stress only because stress is easy to tabulate; the physical event is always the extreme fibre reaching ε_y.
• For steels with no sharp knee, f_y is defined at the 0.2% offset, so the code's strength is literally a strain measurement.
• Worked IPE 300 (L = 6 m, w = 20 kN/m): the engine gives M_max = 90.0 kN·m, so the outer fibre is at ε = 0.000834, 67% of MR250's yield strain.
• Push it to w = 29.99 kN/m and the engine returns σ = 250.0 MPa, ε = 0.001250 = ε_y: first yield, confirmed to the fourth decimal.