The section that yields from the outside in

The shape factor is the ratio of a section’s plastic moment to the moment at which its worst fibre first yields. For a rectangle it is 1.50, for a rolled I-section about 1.09, for a tee 1.78. Those numbers are usually presented as a table of bonuses: how much capacity is sitting unused past the elastic limit, waiting to be collected by anyone willing to do a plastic analysis.

Read that way the ranking is unambiguous and it is exactly wrong. The reserve is real, but it is not free and it is not equally accessible. What buys it is curvature, and the sections with the largest reserves are the sections that have to bend furthest to reach them.

Why the reserve exists at all #

Elastic bending puts the stress in proportion to distance from the neutral axis. That means the extreme fibre is working at full capacity while the material at the middle is working at nothing, and everything in between is somewhere on a straight line. The whole argument for putting material far from the middle is a response to that: material near the axis contributes little, so a good section has as little of it there as possible.

First yield arrives when the extreme fibre reaches the yield strain, and at that instant the average stress over the section is well under the yield stress — half of it for a rectangle. Loading further does not increase the extreme fibre’s stress, because there is nowhere for it to go on a flat-topped curve; instead the yielded zone works inward, and each newly yielded fibre steps up from wherever it was on the straight line to the full yield stress.

The reserve is therefore exactly the material that was under-stressed, being brought up. And that immediately explains the ranking: the reserve is largest for the section with the most material near the neutral axis, which is the section that was worst-designed elastically. A rectangle’s 1.50 is a measure of how much of it was doing nothing.

The comparison the whole page turns on is what the same two pictures look like for a section that had nothing idle to recruit.

The price, stated plainly #

Because the strain distribution is a straight line, the strain at the extreme fibre is the curvature times half the depth. Multiplying the curvature by four multiplies the extreme-fibre strain by four. For mild steel with a yield strain of 0.131%, reaching 98% of the rectangle’s plastic moment means straining the outer surface to 0.54% — which is comfortably inside the plateau and is entirely available.

For the tee the demand is 9.0 times the yield curvature, or 1.18% of strain at the extreme fibre, which is right at the end of the plateau and into the beginning of strain hardening. For a material with no plateau at all, a demand of that size is a demand the section cannot meet at constant stress, and the analysis that produced it does not describe the beam.

That agreement is the useful generalisation. The shape factor and the curvature demand are two readings of one quantity, and the section that offers the most reserve is by construction the section that has to travel furthest to collect it. A designer choosing an efficient section is choosing a small shape factor, and getting a small ductility demand with it.

The shape whose neutral axis will not stay put #

Everything so far has been about symmetric sections, where the neutral axis is at mid-depth at every load and the only thing that changes is how much has yielded. A tee is not symmetric, and it does something the others cannot.

Elastically, the neutral axis of a section passes through its centroid, because that is where the first moment of area vanishes and therefore where the linearly varying stresses integrate to no axial force. For this tee the centroid is 149 mm up a 200 mm section, because the flange at the top holds most of the material.

Fully plastic, the axis is somewhere else entirely. Every fibre is at $\pm f_y$, so the condition for zero axial force is no longer about first moments — it is that the areas above and below are equal. That puts the axis at 187.5 mm, inside the flange, 38 mm from where it started.

That migration is the reason the tee’s curvature demand is 9.0 rather than 4.1. The axis has to travel 38 mm while the section bends, and every millimetre of travel is bought by straining the material it passes over. A symmetric section pays nothing for this because its axis does not move.

Which free body produced the number #

The free body is the cross-section itself, cut from the beam, with the stresses on the cut face.

Two statements fix everything. The kinematic one is that plane sections stay plane, so the strain at height $y$ is $\varepsilon_0 - \kappa(y - \bar{y})$ — one straight line with two unknowns in it, the axial strain at the reference height and the curvature. The static one is that the axial force on the cut face is zero for pure bending, so $\int \sigma\,dA = 0$.

Everything else is arithmetic. The section is sliced into four hundred fibres of equal depth; each fibre’s strain follows from the line, its stress from the material’s curve, and its contribution to the axial force from its area. Bisecting on $\varepsilon_0$ until the axial force vanishes locates the neutral axis, and integrating the same stresses against $(y-\bar{y})$ gives the moment. Sweeping $\kappa$ gives the curve.

Nothing in that is specific to plasticity. Feed the same routine a straight-line stress-strain law and it reproduces elastic beam theory, and the check is worth stating because it is the one that makes the rest believable: at small curvature the computed moment divided by $EI\kappa$ comes back as 1.000000, and the neutral axis lands at 100.000 mm on a section 200 mm deep.

The second check is independent of the first and of the whole fibre machinery. The rigid-plastic moment can be computed without any stress-strain law at all: put every fibre at $\pm f_y$ depending on which side of an axis it lies, and slide the axis until the axial force is zero. For a rectangle that gives $f_y bd^2/4$, and the figure’s 41.25 kNm agrees with $275 \times 100 \times 200^2/4$ to within a part in a million. The elastic modulus check and the plastic check come from opposite ends of the curve and neither uses the other.

The same ranking, in shear #

The argument has a twin that is rarely put beside it, and the parallel is exact.

Elastic shear stress is not uniform across a section either: it follows $VQ/Ib$, which for a rectangle is a parabola peaking at 1.5 times the mean. So the first fibre reaches the shear yield stress when the average shear is two thirds of it, and there is a reserve to be collected by letting the yielded zone spread — a shear shape factor, and for a rectangle it is exactly 1.5.

For an I-section it is very nearly 1.0. The web carries almost all the shear and carries it at almost constant stress, because the flanges have already accumulated most of the first moment by the time the flow reaches the web. There is nothing under-stressed to recruit.

The same ranking, for the same reason, from a completely different distribution: the section that wastes material elastically has a reserve, and the one that does not, does not. It is also why plastic shear design is written as the web area times $f_y/\sqrt3$ with no shape factor anywhere in it. The factor was never a bonus; it was a correction for the shape’s own inefficiency, and a shape that is not inefficient does not need one.

The factor is a property of mild steel as well as of the shape #

The tabulated shape factors are geometric — 1.5 for a rectangle, 2.0 for a diamond, $16/3\pi$ for a circle — and they look like properties of the outline. They are properties of the outline and of a flat-topped stress-strain curve, and the second half is doing more work than it appears to.

The whole derivation assumed that a fibre past yield sits at exactly $f_y$ however far it is strained. Take a material with a rounded curve and no plateau — a high-strength steel, an aluminium alloy, a stainless — and the fibres past their proof stress are not at a constant stress; they are climbing. So the “plastic moment” is not a value the section approaches from below. The moment goes on rising with curvature, slowly and without limit until something tears, and the ratio quoted depends entirely on the curvature at which somebody stopped measuring.

Stainless steel is the case where this has been faced squarely. Its curve is markedly rounded and its strain hardening is strong, so a stainless section reaches moments well above its nominal plastic moment at curvatures a mild steel section would still be flattening at. Designing it with a shape factor borrowed from carbon steel is conservative by a wide and unquantified margin, and the response has been to abandon the ratio: the moment is computed by integrating the section at a stated limiting strain rather than by multiplying an elastic modulus by a table value.

Which is the honest general position. A shape factor is a shortcut that compresses one integration into one number, and it can only do that because mild steel’s curve has a flat part to compress. For everything else the integration is the calculation, and the figures on this page are what it looks like.

Where the model stops #

The plastic moment is an asymptote, and no section reaches it. The dashed lines on the plots above are the rigid-plastic values, and every curve approaches without arriving. That is not a numerical artefact: with a flat-topped material law the last fibre to yield is the one at the neutral axis, where the strain is exactly zero, and it yields at infinite curvature. The 98% used throughout is a convention chosen so that a finite number can be quoted.

Strain hardening means the curve does not flatten at all. Push a rectangle far enough and its moment passes the plastic moment and goes on rising, because the outer fibres have left the plateau. That is why a mild steel beam tested to destruction records a moment above the calculated $M_p$, typically by 5–15%. Design ignores it, which is conservative and also the right call, since collecting it requires curvatures that would have destroyed anything attached to the beam.

Everything here assumes the section holds its shape. A thin flange in compression at four times the yield strain may buckle locally long before it reaches the curvature the plot promises, at which point the moment falls away instead of flattening. That is section classification, and it is the reason the plastic moment is not available to every section that can be drawn.

And the whole calculation is for a section rather than a member. Turning a curvature into a rotation needs the length over which the curvature is elevated, which depends on the moment gradient along the beam. A hinge under a sharp moment peak is short and its rotation demand for a given curvature is small; a hinge in a long region of near-constant moment is long and its rotation is large. The collapse mechanism draws hinges as points, and a point has no length at all.

What the picture cannot show #

Three things are absent from every plot on this page and each of them decides something.

The length of the beam is not on it. A moment-curvature curve is a property of a cross-section, and a curvature is not a deflection. Turning one into the other requires integrating along the member, and the integration depends on how the moment varies — a beam under a uniform load has a moment that falls away parabolically from mid-span, so only a short region is anywhere near the plastic moment, while a beam under a constant moment has the whole of it there at once. The same section in the same steel therefore delivers wildly different rotations in the two cases, and neither number is on this page.

Nothing here is drawn at the scale it happens at. The curvature at which the rectangle reaches its plastic moment is $5.4 \times 10^{-5}$ per millimetre, which over a metre of beam is a change of slope of about three degrees. A figure that drew the beam at that curvature would show a straight line. Every plot here has the curvature as an axis precisely because it cannot be shown as a shape, and the same caution applies as to every deflected shape on this site — the picture is a graph, not a portrait.

And the sequence is missing. The curve is drawn as though curvature were increased monotonically from zero. A real beam may reach a curvature, unload, and come back, and it does not retrace the same path — it comes down a straight line of slope $EI$ and arrives somewhere with a permanent set and a residual stress field. The plot is a monotonic loading envelope and says nothing about anything that happens inside it.

The generalisation #

The pattern is that a ratio conceals the axis it was measured along.

The shape factor is a ratio of two moments and it says nothing about the distance between them. The same shape of concealment turns up whenever this site has a number of the form “how much more” — the truss depth’s reciprocal law is a ratio that says nothing about the extra bracing a deeper truss needs; continuity’s third off the mid-span moment is a ratio that says nothing about the rotation the support has to supply to deliver it.

The defence in each case is the same: plot the thing against what it costs rather than quoting it alone. The curvature axis on these figures is not decoration; it is the half of the shape factor that the table left out, and it reverses the reading.

A surprising place this turns up #

The section with the worst shape factor is the section worth building.

That sounds like a paradox and is not. An I-section’s 1.09 says that at first yield it was already working nearly as hard as it can — which is exactly the property wanted, because the elastic section modulus is what most design is done against and a section that has almost nothing in reserve past it is a section that wasted almost nothing before it. The rectangle’s 1.50 is an admission that a third of its material was idle.

So the shape factor is a measure of elastic inefficiency, presented as a plastic bonus. Two sections of the same area with plastic moments of 41.2 and 72.6 kNm are not a section with more reserve and a section with less; they are a poor section and a good one, and the poor one’s larger ratio is the arithmetic of having had further to come.

Where the ladder goes next #

Later rungs on this anchor: moment-curvature under axial load, where the curve stops being symmetric and the shape factor becomes a function of the load. The curve for a reinforced concrete section, which falls rather than flattens once the concrete crushes and whose ductility is decided by how much steel is in it. Cyclic moment-curvature and the loops it draws, which is where seismic energy dissipation is computed. The spread of plasticity along a member, and hinge length as a measurable rather than an assumption. Curvature ductility against displacement ductility, which differ by a factor that depends on the whole structure. And the reverse question: designing a section for a required rotation capacity rather than a required moment.

Historically the shape factor is older than plastic design and was known long before anyone proposed using it. Nineteenth-century testing routinely recorded beams carrying more than the elastic theory allowed, and the usual explanation was that the material was stronger in bending than in tension — a “modulus of rupture” was tabulated as though it were a property, and it is still quoted for concrete and timber. It is not a property of anything; it is the elastic formula applied to a section that stopped obeying it, and the number it reports is the shape factor in disguise.