After the first yield, which is not the end

The elastic calculation stops at first yield. A section whose extreme fibre has reached the yield stress has, by that calculation, reached its capacity, and everything beyond it is outside the theory.

Nothing happens at that moment. The outer fibres stop taking more stress and go on taking more strain; the material behind them, which was under-stressed, loads up; and the section carries a rising moment while its stress block spreads from a triangle toward a rectangle. Only when the whole section is at yield can it take no more, and by then it is carrying about half as much again as first yield allowed.

What a hinge is, and what it is not #

A plastic hinge is a length of beam in which the whole section has yielded. It rotates freely, like a pin — and it carries a constant moment while doing so, which a pin does not.

That combination is the whole of plastic analysis. Once a hinge has formed, the section cannot resist any more moment, so any further load applied to the structure has to be carried by everything else. The hinge does not break; it stops arguing.

Two properties follow that make the analysis unusually clean. First, the moment at a hinge is known exactly: it is the plastic moment $M_p$ of the section, a property of the geometry and the yield stress and of nothing else. Second, the hinge is a release, so it changes the structure’s degree of redundancy — the counting rule applies to a structure with hinges in it, and each hinge takes it one step toward being a mechanism.

$M_p$ comes from the same free body as everything else. The fully plastic stress block is two rectangles, one in compression and one in tension, and the pair is a couple:

$$M_p = \sigma_y \times \text{(area of one half)} \times \text{(distance between the two centroids)}.$$

For a rectangle that gives $\sigma_y bd^2/4$ against the elastic $\sigma_y bd^2/6$: a ratio of $1.5$. For an I-section it is about $1.14$, because there was hardly any under-stressed material near the neutral axis waiting to be recruited.

Collapse is a counting problem #

For a determinate beam, one hinge is enough. A simply supported beam with a hinge at mid-span has three pins in a line and is a mechanism, so first hinge and collapse are the same event.

That figure is the baseline the rest of the essay is measured against. One hinge, at the obvious place, and a collapse load that is the elastic first-yield load multiplied by nothing but the shape factor of the section. There is no reserve here because there is no redundancy, and every gain below comes from adding restraint rather than from adding steel.

For a redundant structure first hinge and collapse are not the same event, and the gap between them is what plastic design is for. A propped cantilever is once redundant, so it takes two hinges: one at the fixed end and one somewhere in the span. A fixed-ended beam takes three. In general,

$$\text{hinges needed} = \text{degree of redundancy} + 1.$$

That is the arithmetic that makes redundancy worth having in a ductile material. The structure does not fail when its worst-stressed section reaches capacity; it fails when enough sections have reached capacity to leave no route to the ground at all.

Between first hinge and collapse, the moment diagram is changing shape. Each new hinge caps the moment at that section, and further load redistributes toward sections with capacity left. Moment redistribution is that process, and it is the mechanism by which a plastic structure reaches a strength its elastic analysis said it did not have.

The work equation, which needs no elastic analysis at all #

The collapse load is found without tracking any of that history, which is what makes the method practical.

Assume a mechanism. Give it a small displacement. Then equate the work the loads do as the mechanism moves to the work absorbed at the hinges:

$$\Sigma\,(\text{load} \times \text{its displacement}) = \Sigma\,(M_p \times \text{hinge rotation}).$$

For a uniformly loaded beam whose mechanism has a peak displacement $\delta$, the load rides a triangular displacement diagram, so the external work is $wL\delta/2$. The internal work is $M_p$ times the sum of the rotations, which for a hinge at distance $a$ from the left support is $\delta/a + \delta/(L-a)$ at the sagging hinge, plus $\delta/a$ again at a fixed left end, and so on.

Setting them equal gives the collapse load for that assumed mechanism. Different assumed hinge positions give different answers, and the curve of those answers is drawn beneath the mechanism in the figures on this page. Its minimum is the collapse load, and where the minimum falls is where the hinge really forms.

The three standard results come straight out of that search:

The middle row is the interesting one. The hinge is not at mid-span, and the coefficient is not a round number, because the position had to be found by minimising and the minimum falls at $\sqrt{2}-1$ of the way in from the propped end. The generator behind these figures searches eight hundred positions and reports the lowest; the result agrees with the algebra to four figures, which is a check that the search is finding the real minimum rather than a plateau.

Watching the diagram move #

The work equation jumps straight to the answer. Following the loading in order is slower and shows where the extra capacity actually came from, and the propped cantilever is small enough to do it on.

Elastically, that beam has a hogging moment of $wL^2/8$ at the wall and a sagging peak of $9wL^2/128$ — about $0.070wL^2$ — at $0.375L$ from the prop. The wall is working nearly twice as hard as the span, which is the usual situation in a redundant beam: the stiffer route attracts the load, and a built-in end is a very stiff route.

The first hinge therefore forms at the wall, when $wL^2/8 = M_p$, which is a load of $8M_p/L^2$. At that instant the span is carrying $0.5625M_p$ — a little over half its capacity, entirely unused.

After it, the wall can take no more moment but goes on delivering the $M_p$ it has. So the structure behaves from then on as a simply supported beam with a constant moment applied at one end, and every further increment of load goes into the span. The sagging peak climbs, the hogging one does not, and the diagram changes shape while the structure carries load it had no elastic capacity for.

Collapse arrives when the span reaches $M_p$ as well, at $11.66M_p/L^2$. That is forty-six per cent more load than the first hinge — and the whole of it was bought by letting one section stop resisting.

Two things in that sequence are worth keeping. The reserve came from the unevenness of the elastic distribution: a structure whose sections all reach capacity together has no redistribution available, so an optimised elastic design is a design with nothing left. And the redistribution needed the wall hinge to rotate through a real angle while the span caught up — which is the ductility demand, arriving as a number rather than a principle.

The mechanism nobody thought of #

The upper-bound property is a genuine hazard rather than a technicality, and frames are where it bites.

A portal frame has three independent mechanisms. A beam mechanism, with hinges in the beam and at the two corners. A sway mechanism, with hinges at the four column ends. And a combined mechanism, obtained by adding the two and cancelling the hinge that appears in both — which is the point at which the arithmetic stops being obvious, because combining mechanisms removes internal work while keeping the external work of both.

The combined mechanism almost always governs, and it is the one an analyst working from a picture is least likely to draw. Checking the beam mechanism and the sway mechanism and taking the lower of the two returns a number that is too high, safely-looking, and wrong.

A pitched frame makes the difficulty visible, because its combined mechanism cannot be drawn as a hinge diagram at all without first finding where each rigid part turns.

The centre is not a fixed feature of the frame. It is a consequence of the geometry the mechanism was assumed to have, and moving that geometry moves it — which is the same statement as the hinge search on the propped cantilever, made in two dimensions instead of one.

Twelve per cent of collapse capacity for six-tenths of a metre of roof pitch is not a detail, and no inspection of a moment diagram produces it. It comes from the mechanism, and the mechanism had to be enumerated.

The professional discipline that follows is to enumerate rather than to inspect: count the independent mechanisms as the number of possible hinge positions minus the degree of redundancy, generate every combination, and take the minimum. A frame’s redundancy count turns up again here, doing a different job — it says how many mechanisms have to be searched.

And the lower-bound theorem provides the confirmation. Once a candidate collapse load is in hand, constructing an equilibrium moment distribution that nowhere exceeds $M_p$ at that load proves the answer from below. When the two bounds meet, no mechanism was missed. That is the only way to be certain, and it is why serious plastic analysis always does both.

Upper bound, lower bound, and why that matters #

The work equation has a property that has to be understood before it can be used safely: it always overestimates.

Assuming a mechanism is imposing a shape on the collapse, and a structure forced into a particular shape is a stronger structure than one free to choose. So every mechanism gives an upper bound on the true collapse load, and the true value is the lowest bound over all possible mechanisms. Missing a mechanism means returning a number that is too high — unsafe, with no warning in the arithmetic.

The counterpart is the static or lower-bound theorem, which works the other way. Guess a distribution of moments that is in equilibrium with the loads and nowhere exceeds $M_p$; then the structure will not collapse under those loads. That guess gives a lower bound — safe, and possibly conservative.

The two theorems bracket the answer, and when a mechanism and an equilibrium distribution give the same number, that number is exact. This is the same pair of bounds that appears whenever a structural approximation is made: guessing a displacement shape overestimates a buckling load and guessing a force system underestimates a collapse load, and knowing which of the two a method belongs to decides whether its errors are dangerous.

The size of the hazard is on that picture rather than in the argument about it. An analyst who checked the beam mechanism and the sway mechanism and took the lower of the two would have reported 1.000 against a true capacity of 0.714 — a claim of 1.40 times what the frame has — and one who happened to check only the sway mechanism would have claimed 1.75 times it. Both numbers look like answers. Neither has anything in it that says it is too high.

The lower-bound curve is what closes that gap, and it costs one further calculation rather than a new method: pick the redundant, draw the moment field it implies, and find the largest load factor that keeps every ordinate inside Mp. Its peak touching the lowest mechanism is the proof that no mechanism was missed, because a load factor cannot be simultaneously above the true collapse load and supported by an equilibrium field within capacity.

For beams the mechanism list is short and being sure of it is easy. For frames it is not: a portal has a beam mechanism, a sway mechanism, and combinations of the two, and the combined one usually governs while being the one least likely to be thought of. The standard remedy is to enumerate the independent mechanisms systematically and take every combination, which is what the profession’s hand method does and what a computer does exhaustively.

What it buys, and what it takes on trust #

Plastic design gives a materially lighter structure, and it does so by using the redundancy that elastic design pays for and leaves idle.

The gains are largest where redundancy is greatest. A fixed-ended beam’s collapse load is $16M_p/L^2$ against an elastic first-yield capacity of $12M_y/L^2$ — and with $M_p$ some fourteen per cent above $M_y$ for an I-section, the plastic method delivers around fifty per cent more load for the same section. For a continuous multi-span beam the gain is similar and arrives at every internal support.

The second gain is subtler and worth more: the collapse load does not depend on the elastic distribution. Support settlement, temperature, fabrication misfit and locked-in stresses all change the elastic moments and change nothing about the collapse load, because the structure yields its way past them on the road to a mechanism. That is why a plastic calculation is robust to precisely the assumptions an elastic redundant analysis is fragile to.

What it takes on trust is ductility, and the demand is quantitative rather than a matter of principle. The first hinge to form has to keep rotating, at full moment, while the remaining hinges develop — a rotation capacity of several times the yield rotation. That requires a section whose plates will not buckle locally while it is happening, which is why plastic design is permitted only for the stockiest section classes, and a material with a substantial plateau past yield, which is why it is a steel method and not a general one.

The hinge that also has to carry shear #

The work equation uses one value of $M_p$ at every hinge. The section does not necessarily have one.

A hinge is a section entirely at yield in bending, and yielding is a statement about a combination of stresses. Where the section is also carrying shear, part of the material’s yield capacity is committed to the shear stress and is not available for the direct stress: by von Mises, a fibre carrying a shear $\tau$ can only reach a direct stress of $\sqrt{\sigma_y^2 - 3\tau^2}$. At half the shear yield stress that is $0.87\sigma_y$, and at the shear yield stress it is nothing at all.

The practical rule follows the same shape. Below about half the section’s plastic shear capacity the interaction is negligible and is ignored. Above it, the web’s contribution to the plastic moment is reduced by a factor $(2V/V_{pl}-1)^2$ — a quarter of it gone at $0.75V_{pl}$, all of it at $V_{pl}$. For a rolled I-section the web supplies perhaps a fifth of $M_p$, so those are reductions of about 5% and 20% in the hinge’s capacity.

The place this bites is not random. The two hinges of a propped cantilever sit at opposite extremes of shear. The span hinge is where the moment peaks, which is where the shear passes through zero — so it has no interaction at all, by construction. The support hinge is where the shear is largest. So the wall hinge is the weaker of the two and the work equation, which takes $M_p$ as one number, cannot say so.

The consequence is a small and systematic overestimate of the collapse load on short, heavily loaded, deep members — exactly the ones where the shear is high relative to the moment. On an ordinary floor beam it is nothing. On a stocky transfer member it is real, and it is one of the few places where the neat separation between the shear diagram and the moment diagram stops being a separation.

Where the model stops #

Ductility, quantitatively. As above. A brittle material, a fatigue-loaded member, or a section that buckles locally cannot reach the state the analysis assumes, and the calculated collapse load is then a number describing a structure that does not exist.

No instability. The whole method assumes the structure stays in the geometry it was drawn in. A frame close to its critical load amplifies its sway on the way to collapse, and second-order effects reduce the collapse load below the plastic value — sometimes far below.

Proportional loading. The work equation assumes all loads grow together. A structure loaded in one pattern and then another can shake down or ratchet, and the collapse load under repeated variable loading is a different and lower quantity.

No serviceability information. Plastic analysis says what load causes collapse. It says nothing about deflection under working load, and a beam sized plastically may fail its deflection check comfortably — the two calculations answer different questions and both have to be done.

Hinges are points. A real plastic hinge occupies a length of beam over which the moment exceeds $M_y$, and that length can be a substantial fraction of the span. Treating it as a point is accurate for the collapse load and misleading about the deformed shape.

The figures on this page have a distortion that matters. The mechanism is drawn as two straight segments with a sharp kink, which is what the analysis assumes and not what a beam does — the real deformed shape is curved everywhere, with a region of high curvature rather than a point of infinite curvature. The straight-line mechanism is a kinematic device for computing work, and reading it as a picture of a collapsing beam overstates how localised the damage is.

The ladder from here #

Later rungs on this anchor: the plastic moment and shape factors derived. Mechanisms in portal frames — beam, sway and combined. The combination of mechanisms and the systematic enumeration. Upper and lower bound theorems proved. Shakedown and ratcheting under variable load. Rotation capacity and section classification. Moment redistribution limits in codes. Plastic hinges in reinforced concrete, where the rotation capacity is the whole question. And yield-line analysis, which is this argument applied to slabs and where the mechanism search becomes genuinely hard.

Plastic theory was developed at Cambridge under Baker from the mid-1930s, in part from the observation that steel-framed buildings damaged in bombing had survived deformations no elastic analysis could account for. The theorems were put on a rigorous footing by Greenberg and Prager around 1950, and the method was in British design codes within a decade of that.