True Stress vs Engineering Stress: the Necking Criterion and a Worked Check

One tension test produces two stress-strain curves, and structural work uses them in two different rooms. The engineering curve divides the measured force by the original cross-section, σ_eng = P/A₀. It is the curve the code lives in: f_y and f_u are points on it, and a companion article makes the case for why every design clause deliberately keeps that engineering value. The true curve divides the same force by the real cross-section at that instant, σ_true = P/A_i, the area that has already thinned under load. That is the curve the material lives in, and the curve you must hand a nonlinear solver, a forming simulation or a fracture model.

This article is about the second curve: how it relates to the first, the one-line conversion between them, the criterion that predicts exactly where the two part company, and how to build a usable true stress-strain curve for analysis. We anchor it with a worked check on a real A572-50 round tie rod, using the CalcSteel FEM engine to pin the elastic end to the fourth decimal before we walk the curve out to necking by hand. The theme is the opposite of the companion post: not why the gap is safe to ignore in a code check, but what to do the moment you leave the code behind.

Both curves are built from the same measured force P, original length L₀ and original area A₀. Four definitions:

• Engineering stress, σ_eng = P / A₀: the load over every square millimetre the bar started with.
• Engineering strain, ε_eng = ΔL / L₀: the stretch as a fraction of the original length.
• True stress, σ_true = P / A_i: the load over the instantaneous area A_i, which shrinks as the bar stretches (Poisson contraction elastically, then constant-volume plastic flow).
• True strain, ε_true = ln(L_i / L₀) = ln(1 + ε_eng): the sum of every incremental stretch dL/L integrated over the deformation.

In tension A_i is smaller than A₀, so true stress is always the larger of the two, and true strain is always the smaller (because ln(1 + x) < x). The distinction is invisible while the bar barely moves and grows as it deforms. Engineering measures are anchored to the geometry you can put a caliper on before the test; true measures track the geometry that is actually there at each instant.

The two curves are not independent. As long as deformation is uniform along the bar (before any neck forms) and plastic flow conserves volume, A_iL_i = A₀L₀. That gives A_i = A₀ / (1 + ε_eng), and therefore:

σ_true = σ_eng (1 + ε_eng)    and    ε_true = ln(1 + ε_eng)

Read what that means: the conversion factor (1 + ε) depends only on the strain, not on the material or the stress level. So the size of the gap is decided entirely by how far the bar has stretched. That single fact is the whole story of when true and engineering stress matter, laid out across the full range below.

The top rows are why the code keeps engineering stress: at design strains the gap is a rounding error, and its exact value is set by strain alone. The bottom rows are why materials and forming engineers never use anything but true stress: out where steel is deliberately taken, the two numbers differ by tens of percent. Two conditions ride on the formula, and they are the fine print people forget. It holds only while the section stays uniform, so it dies the instant the bar necks, and only while volume is conserved, which is a good model for plastic flow (Poisson's ratio effectively 0.5) but not for the elastic range (about 0.3, where the bar does change volume slightly). In the elastic range the constant-volume form therefore mildly overstates the gap, which makes it the safe one to quote.

Everything above assumed uniform deformation. The criterion that says exactly when that assumption breaks is Considère's, and it is the sharpest idea in this whole subject. A ductile bar in tension is a race between two effects: strain hardening makes the steel stronger as it deforms, while the thinning section makes it carry less load for a given stress. Early on, hardening wins and the load rises. Necking begins at the instant the section loses that race, the point of maximum load, which is also the engineering ultimate strength f_u.

Writing dP = 0 with P = σ_trueA_i and constant volume gives the criterion in true measures:

dσ_true / dε_true = σ_true

Necking starts where the slope of the true stress-strain curve equals its own height. If you model the strain-hardening region with a power law (the Hollomon form) σ_true = K ε_trueⁿ, the criterion collapses to a one-liner: dσ_true/dε_true = nK ε_trueⁿ⁻¹, and setting that equal to K ε_trueⁿ gives ε_true = n at the onset of necking. The strain-hardening exponent n is the true strain at which the bar starts to neck. Converting back, the uniform (engineering) elongation is eⁿ − 1. For a ductile structural steel with n around 0.16, that is a uniform strain of about 17%, which is why coupon tests of structural grades report uniform elongations in that range. This is the criterion the article's title points at: not a clause in the code, but the material law that decides where the code's engineering curve is allowed to turn downhill.

The reason to care about all of this in practice is that a nonlinear finite-element model of steel needs a true stress versus true (or plastic) strain curve as input, and the data you have is an engineering coupon curve. Here is how to turn one into the other.

• Elastic branch. Up to yield, keep the elastic line σ = Eε with E = 200 GPa. The true correction here is under 0.2% (see the worked rod below), so it changes nothing; most solvers take E and f_y directly.
• Uniform plastic branch. From yield up to the ultimate point, convert every engineering pair with σ_true = σ_eng(1 + ε) and ε_true = ln(1 + ε). Then split off the plastic strain the solver wants: ε_pl = ε_true − σ_true/E. This branch is exact and is the workhorse of the model.
• Past necking. The tidy conversion is invalid beyond f_u because deformation is no longer uniform and the stress state at the neck is triaxial. If your analysis goes that far you either measure the neck area directly, apply a Bridgman correction, or extrapolate the power law σ_true = K ε_trueⁿ and accept it as a model, not a measurement.

Skip this conversion and feed the raw engineering curve to the solver, and you tell it the material softens after f_u. It does not: the engineering downturn is an artefact of dividing a falling load by a fixed original area. A solver handed that curve will either predict spurious softening and localization or fail to converge. The single most common serious error in nonlinear steel modelling is exactly this one.

Now measure rather than assert. The most honest object for a pure axial stress is a round tie rod, which is both a real structural member (sag rods, cross-bracing rods, canopy hangers) and the exact geometry of an ASTM E8 round tension coupon. Take a Ø 45.1 mm rod, A = 16.0 cm², 6 m long, in A572-50 (f_y = 345 MPa).

We model it in the CalcSteel FEM engine, the same solver behind the calculators. Under a service tension of N = 276 kN the engine returns the axial force in the member as 276.0 kN (equilibrium, as it must), the section area as A = 16.0 cm², and the elongation of the loaded end as 5.175 mm. That last number is the check on the engine itself: closed-form N L / (A E) = 276 × 600 / (16.0 × 20000) = 0.5175 cm = 5.175 mm, matching to the third decimal. The stiffness is right, so the stress and strain we pull from it are trustworthy, and everything past this point is arithmetic the material actually obeys.

Turn the analysis into the two stresses and then walk the curve out. At the service load, engineering stress is the force over the original area: σ_eng = 276 kN / 16.0 cm² = 17.25 kN/cm² = 172.5 MPa. The strain is that stress over the modulus, ε = 172.5 / 200000 = 0.000863, the same value the engine's 5.175 mm over 6000 mm gives directly. True stress applies the conversion: σ_true = 172.5 × 1.000863 = 172.65 MPa. The material carries 172.65 MPa; your model, your code check and your calculator all say 172.5 MPa. The gap is 0.15 MPa, or 0.086%.

Push the rod to first yield. σ = f_y = 345 MPa needs N = 34.5 kN/cm² × 16.0 cm² = 552 kN. Re-run the engine at N = 552 kN and it returns σ = 345.0 MPa, an elongation of 10.350 mm (so ε = 10.350 / 6000 = 0.001725, exactly A572-50's yield strain), and the axial force back as 552.0 kN. Convert: σ_true = 345.0 × 1.001725 = 345.60 MPa, ε_true = ln(1.001725) = 0.001724. At the one point the code cares about, first yield, the true stress is 0.60 MPa above the tabulated f_y, three tenths of one percent, and in the safe direction.

Now leave the code behind, which is the whole point of this post. Carry the same rod's A572-50 out along its true curve with n around 0.16. Considère puts the onset of necking at ε_true = 0.16, an engineering strain of e^0.16 − 1 = 17.4%, at the engineering ultimate f_u = 450 MPa. The true stress there is σ_true = 450 × 1.174 = 528 MPa, already 17% above the engineering value. Past that the rod necks: engineering stress divides a falling load by the fixed original 16.0 cm² and slides down to fracture, while true stress divides that same load by the collapsing neck area and keeps climbing well past 600 MPa. Same rod, same test, three regimes: identical to three decimals at service, a rounding error at yield, and two completely different numbers at failure.

Reproduce the exercise. In the calculator below, pick a grade (try A572-50, the rod's grade), set a section and a load, and read the stress it reports. Divide that stress by 200,000 MPa to get the strain ε, then multiply the stress by (1 + ε) for the true stress. You will watch the correction sit in the third or fourth decimal for anything short of yield.

One twist worth trying: this is a column in compression, and there the sign of the correction flips. The section fattens under compression, so the real area grows, A_i > A₀, and true stress sits just below the engineering value, σ_true = σ_eng(1 − ε) to first order. At design strains it is again negligible, and compression is governed by buckling long before it could matter, which is exactly what this tool computes.

None of this shows up in a member strength check, and the companion post explains why it should not. But the moment steel is deliberately taken into large plastic strain, the true curve is the only correct description, and reaching for the engineering curve is a real error:

• Nonlinear finite-element analysis. Any run that follows steel past yield, connection ductility, seismic fuses, plastic collapse, blast, crash, must be fed true stress versus true (or plastic) strain, built as in the section above. The engineering curve makes the model soften artificially after f_u.
• Cold forming. Press-braking a plate or roll-forming a cold-formed section drives 10% to 30% strain at the bend. Forming force, springback and residual stress all depend on the flow (true) stress there; the engineering curve, already falling, would mislead badly.
• Fracture and low-cycle fatigue. Reduction of area, true fracture strain and the strain-life relations behind seismic detailing are all written in true measures, because that is where the material's real state lives.

There is one place the two worlds meet inside the code, and it is worth naming. The minimum f_u/f_y ratio and the minimum elongation every structural grade must pass are engineering-stress requirements whose whole purpose is to guarantee a long plastic runway, the very region where the true curve does its work. The code checks strength in engineering stress and buys ductility, measured the same way, to protect a plastic reserve it will describe in true stress the day it has to model that reserve being spent.

Feeding an engineering curve to a nonlinear solver. The signature error. Material models expect true stress versus true (or plastic) strain; the engineering curve tells the solver the steel weakens past f_u, which it never does.

Applying σ_true = σ_eng(1 + ε) after necking. The conversion assumes uniform deformation and dies at the neck. Beyond it the state is triaxial and localized; true stress needs a measured neck area or a Bridgman correction, not the tidy formula.

Thinking f_u is the highest stress the steel ever reaches. f_u is the peak of the engineering curve. True stress at fracture is far higher, often 1.5 to 2 times f_u, because the real neck area is a fraction of the original.

Reading the post-ultimate drop as the material weakening. The engineering curve falls after f_u only because it divides a shrinking load by a fixed original area. The material strain-hardens right to fracture; true stress never decreases.

Which stress does my calculator report? Engineering stress, always, because it computes force over the modelled (original) section, which is the right number to compare against a code f_y or f_u. To get true stress, multiply by (1 + ε).

How is this different from the engineering-first companion post? That one answers the design-range question: why the code keeps the engineering value and loses almost nothing. This one answers the opposite question: what the true curve is, where Considère says it diverges, and how to build it for analysis. Same subject, two rooms.

True stress is what the steel carries and what an analysis past yield needs; engineering stress is what the code and the test report. Keep both, and know which room you are in.

• σ_eng = P/A₀, σ_true = P/A_i; before necking, σ_true = σ_eng(1 + ε) and ε_true = ln(1 + ε). The factor depends on strain alone.
• The Considère criterion dσ_true/dε_true = σ_true locates necking; for a power-law steel it is ε_true = n, uniform elongation eⁿ − 1.
• Worked A572-50 rod (Ø 45.1 mm, A = 16.0 cm2, 6 m): the engine gives 5.175 mm at N = 276 kN, so σ = 172.5 MPa and σ_true = 172.65 MPa (+0.086%).
• At N = 552 kN the engine returns σ = 345.0 MPa = f_y with a 10.350 mm stretch; true stress 345.60 MPa (+0.17%).
• Out at necking (17.4% strain, f_u = 450 MPa) true stress is 528 MPa and climbing; feed a solver the true curve, never the engineering one.