The axis that moves when the section yields

A neutral axis is defined by what it makes vanish. In an elastic section the stress is proportional to the distance from the axis, so the net force vanishes where the first moment of area does — which is the centroid, and which is why the centroid is where every elastic bending calculation starts.

In a fully plastic section the stress is $\pm f_y$ everywhere, so the net force vanishes where the areas above and below are equal. That is a different definition, and it picks out a different line.

Which free body produced the number #

The section itself, cut through, with the stress on the cut face — twice, at two different stages of the same section’s life.

Elastic. Stress is $\sigma = \kappa E y$ measured from some axis. The net force is $\int \sigma\,dA = \kappa E \int y\,dA$, which vanishes when $\int y\,dA = 0$: the definition of the centroid. Nothing about the material entered except that it is linear, so the elastic neutral axis is a property of the shape and the same for steel, timber and glass.

Plastic. Stress is $-f_y$ above the axis and $+f_y$ below it. The net force is $f_y(A_{tension} - A_{compression})$, which vanishes when the two areas are equal. Again nothing about the material entered except that it is the same in both directions — and that qualification matters, because for concrete it is not, which is why a concrete section’s plastic axis is not an equal-area axis at all.

So the two axes answer the same question — where is there no net force — under two different stress distributions, and there is no reason at all for them to be the same line. They are the same line whenever the section is symmetric about it, which is most of the sections anybody draws, which is why the distinction is met once in a textbook and then forgotten.

The tee, where they are far apart #

A tee with a 300 × 16 flange and a 384 × 12 web has an area of 9,408 mm². Its centroid sits 294 mm above the bottom of the web, because the flange is heavy and high. Its equal-area axis sits at 384 mm — inside the flange, because the web alone is 4,608 mm² and the flange is 4,800, so the halving line lands just above the flange’s underside.

Ninety millimetres apart, on a 400 mm section. Twenty-three per cent of the depth, between two lines that are both called the neutral axis.

Everything downstream follows. The elastic modulus to the bottom fibre is $I/294$; to the top fibre it is $I/106$, which is 2.78 times larger. The plastic modulus is $\int|y - y_{pna}|\,dA$ about the equal-area axis and there is only one of it. The shape factor — plastic modulus over the smaller elastic one, because the smaller one is the one that governs — comes out at 1.79, which is larger than a rectangle’s 1.5 and very much larger than the 1.14 an I-section gives.

That last is worth pausing on, because the usual reading of a shape factor gets it backwards. A large shape factor is not a good section. It means the elastic calculation was very conservative, which means the section had a great deal of material sitting near the axis doing very little until it was fully plastic — and getting there needs curvature, which is deflection. What is left after the first fibre yields is that trade in full: the rectangle’s 1.5 is bought with several times the curvature the I-section’s 1.14 costs.

Two moduli, and only one of them matters — until it does not #

An asymmetric section has two elastic section moduli, and which of them governs depends on which way up the moment is.

Under sagging, the tee above has its flange in compression and its web tip in tension, so the extreme tensile fibre is 294 mm from the axis and the smaller modulus governs. Turn the moment over — a cantilever, a support region of a continuous beam, a wind reversal — and the flange is in tension, the extreme fibre is 106 mm away, and the section is 2.78 times stronger. Two strengths, depending which way up is the whole of that asymmetry.

The plastic section has one modulus and one strength either way up, because a fully yielded section does not care which fibre reached yield first — the areas are the areas whichever way the moment runs. So plasticity removes the asymmetry, and the tee’s factor-of-2.78 disparity becomes no disparity at all.

That is a real design consequence, not a curiosity. It means an asymmetric section under reversing load is very much better used plastically than elastically, and it means the elastic and plastic checks on such a section can disagree about which load case governs. On a rolled tee under wind reversal the elastic check says one direction is nearly three times worse and the plastic check says the two are identical.

An axial force, and the rate the axis moves at #

Put an axial force through the section and the plastic axis must move, because the areas above and below are no longer equal — they must differ by $N/2f_y$.

How far it moves depends entirely on the width it happens to be crossing. Through the tee’s 12 mm web the axis has to travel $N/(2 \times 12 \times f_y)$; through its 300 mm flange it travels a twenty-fifth of that for the same force. Over the whole section the rate changes by a factor of 27.

That produces an N–M interaction curve with a visible kink where the axis crosses the flange–web junction — and the kink is not an artefact of a numerical method or a linearisation. It is the section’s own geometry appearing in its strength envelope, and it is the reason two ways to fail is drawn as a computed curve rather than as the smooth parabola the textbook expression gives.

Where the composite deck gets its 1.80 #

A steel beam acting with a concrete slab is the most asymmetric section most engineers ever design, and it is the case where the two axes are furthest apart in practice.

The elastic centroid of a transformed composite section sits somewhere in the top of the steel web or in the slab, depending on the modular ratio — and the modular ratio depends on whether the load is short or long term, so the elastic axis moves with time. A section made of two materials is the transformation, and the section that changed while it was being loaded is the sequence problem it produces.

The plastic axis does not move with time at all. It sits where the compression capacity of the slab equals the tension capacity of the steel, which is a comparison of two forces and involves no stiffness whatever. That is a large part of why composite design is done plastically wherever the section class allows it: the plastic calculation is independent of creep, of the construction sequence, and of the modular ratio, and the elastic one is a hostage to all three.

Finding it, which is a bisection and not a formula #

There is no closed form for the equal-area axis of a general section, and there does not need to be.

Sweep a trial axis down the section, computing the net force $f_y(A_{below} - A_{above})$ at each position. That function is monotonic — moving the axis down always puts more area into compression — so bisection converges on the zero in forty steps whatever the shape. The plastic modulus is then the first moments of the two halves about that axis, and the moment is $f_y$ times it.

The same routine with a target force other than zero gives the axis under an axial load, and sweeping the target gives the interaction curve. One bisection answers the plastic modulus, the interaction envelope, and the axis position at every load, and it does so for a shape given as any set of rectangles — a rolled section, a plated girder, a section with a hole in it, a composite deck.

That is worth contrasting with the elastic side, where a closed form exists and is a liability. The formula for the centroid of a set of rectangles is easy enough to write down and easy enough to get wrong by a sign, and there is nothing in the answer that says whether it is right. A bisection on a monotonic function has a residual to check.

Where the material is, and where it is worth having #

The two axes give two different answers to the question “where should the material go”, and the disagreement is instructive.

Elastically, material is worth what its distance from the centroid squared is worth. Move a plate from the axis to the extreme fibre and its contribution to the second moment goes as $y^2$; move it half as far again and the contribution rises by a factor of 2.25. That is the material far from the middle, and it is why an I-section exists.

Plastically, material is worth what its distance from the equal-area axis is worth — to the first power. The doubling is a doubling, not a quadrupling. So the plastic calculation values the material near the axis relatively more than the elastic one does, which is the arithmetic behind every shape factor above one.

Put those two together and a useful design statement comes out. Sections whose material is concentrated at the extremes have shape factors near one — the I-section at 1.14, a hollow tube at 1.27 — because there is very little material near the axis for plasticity to recruit. Sections with material spread through the depth have large ones — the rectangle at 1.5, the diamond at 2.0. The shape factor measures how much of the section the elastic calculation was wasting, and the sections that waste least are the sections that were designed elastically in the first place.

Which is why the plastic gain is smallest exactly where it would be most useful, and why plastic design’s real value was never the fifteen per cent on a rolled beam. It was that a structure of such beams could redistribute — the moment that was moved on purpose — and that gain has nothing to do with the section at all.

Why it was ever a surprise #

The distinction is a nineteenth-century one and the delay in noticing it is worth recording, because it says something about how a subject’s default case gets chosen.

Elastic bending theory was complete by the 1820s and its neutral axis was the centroid from the beginning. The plastic bending of a section was worked out properly in the 1910s and 1920s — by Kazinczy in Hungary, Kist in the Netherlands, and then systematically by Baker and his group at Cambridge in the 1930s — and the equal-area axis is in all of it. So the two ideas are a century apart and both are old.

What is striking is how little the difference between them was pressed, and the reason is that the sections everyone was designing were symmetric. Rolled I-sections, plate girders with equal flanges, rectangular timber, round bar: for every one of them the two axes coincide, the shape factor is a single number to be memorised, and a designer can go a whole career without ever meeting a case where the distinction does anything.

The cases where it does something arrived later and from outside structural steel. Composite decks, where the concrete slab makes the section wildly asymmetric. Cold-formed sections, which are asymmetric by construction because they are folded from one sheet. Crane rails and bridge rails, which are tees with heavy heads. Prestressed bridge beams, which are tees or inverted tees by design. In every one of those the two axes are far apart and the elastic shape factor is not 1.15, and every one of them is a twentieth-century structure.

That is a general pattern in this subject and it is worth watching for. A simplification survives not because it is defended but because the cases that would break it are not being built yet — and it then breaks all at once, on a new kind of structure, in the hands of people who learned it as a fact.

Where the model stops #

The material was the same in both directions. For concrete it is not, and the “plastic” neutral axis of a reinforced section is where the concrete’s compression block balances the steel’s tension — an equal-force axis rather than an equal-area one. The whole family of arguments here transfers only in the sense that the axis is wherever the net force vanishes.

The section was assumed able to reach full plasticity. A slender element buckles locally first, and then there is no plastic axis to find because the section never gets there — the section that cannot reach its own strength is that limit, and it is why the section class has to be checked before the plastic modulus is used.

Bending was about one axis. Under biaxial bending the plastic neutral axis is a line at an angle, and it is not perpendicular to the resultant moment: two moments and a neutral axis that obeys neither is that case, and the equal-area condition generalises to a line rather than a level.

And the section was assumed to stay put. A tee or a channel loaded through its centroid twists, because the shear centre is somewhere else — the point that is not in the section is a third axis that has nothing to do with either of these two and that decides whether the section is bending at all.

The generalisation #

The idea worth keeping is that “the neutral axis” is not a property of a section; it is a property of a section and a stress distribution, and changing the distribution moves it.

That is a specific case of something the whole subject keeps doing. A quantity gets a name — the neutral axis, the effective length, the centre of stiffness, the shear centre — and the name suggests it is a fact about the object. It nearly always turns out to be a fact about the object and the question, and the way to find out is to ask what it makes vanish. The neutral axis makes the net force vanish; the shear centre makes the twist vanish; the centre of rotation makes the displacement vanish. Different vanishings, different points, all called centres.

And there is a smaller lesson about checking. If two calculations of the same section disagree by more than the shape factor should allow, the first thing to test is whether they were taken about the same line. On a symmetric section that test never fails and the habit never forms. On the first tee it fails immediately, and the number is not slightly wrong — it is wrong by the ratio of two moduli about axes a fifth of a depth apart.