Two ways of being wrong

Almost every argument in this collection about a structure’s strength has quietly used one of two theorems, and they are worth separating, because they are not two ways of getting the same answer. They are two ways of getting a different answer, in opposite directions, and only one of the two errors is on the safe side.

The static, or lower-bound, theorem. If a distribution of internal forces can be found that is in equilibrium with the applied load and nowhere exceeds the strength of any member, the structure will carry that load. It does not matter whether the distribution is the real one.

The kinematic, or upper-bound, theorem. If a mechanism can be found, the load computed from its work equation is at or above the collapse load. It does not matter whether it is the mechanism that actually forms.

Which free body produced the number #

The frame has two pinned bases, so it is once redundant. Release the horizontal reaction at the right base and the primary structure is determinate: the moments at the two eaves and at mid-span follow from statics, and the redundant adds a moment of $X h$ to each of them with a sign that depends on which side of the frame the section is.

Every value of $X$ therefore gives a complete equilibrium field for a given load factor. The lower-bound theorem says the frame will carry any load factor for which some $X$ keeps all three moments inside $M_p$ — so the best lower bound is found by asking, for each $X$, how large the load factor can be, and taking the largest.

That is a one-parameter search with a closed answer: each section imposes an interval on $X$, and the field exists while the three intervals overlap. The largest load factor for which they do is 0.71429.

The three mechanisms, and the two that are wrong #

The upper bounds come from work equations, and each is a line of arithmetic.

Beam mechanism — hinges at the two eaves and mid-span. Internal work $4M_p\theta$, external $\lambda W L\theta/2$, so $\lambda = 8M_p/WL = 1.000$.

Sway mechanism — hinges at the two eaves only, the bases being pinned. Internal $2M_p\theta$, external $\lambda H h\theta$, so $\lambda = 2M_p/Hh = 1.250$.

Combined — the two added, with the hinge at one eaves cancelling. Internal $4M_p\theta$, external $\lambda(WL/2 + Hh)\theta$, so $\lambda = 4M_p/(WL/2 + Hh) = 0.714$.

The lowest of the three is the answer, and the other two are not approximations to it — they are claims that the frame is 40 and 75 per cent stronger than it is, produced by work equations with no error in them.

That is the whole danger of the kinematic method. It is easy, it is quick, its arithmetic is elementary, and every mistake it makes is unsafe. A mechanism that has been overlooked does not produce a warning; it produces a higher number.

Why the two meet, and what it means when they do #

At $\lambda = 0.714$ the equilibrium field has $|M| = M_p$ at mid-span and at one eaves, and 0.143 $M_p$ at the other. Those two sections are exactly the two the combined mechanism puts hinges at.

That is not a coincidence; it is the reason the theorems are worth having. When a load factor is both an upper bound and a lower bound it is the collapse load exactly, and the collapse load has then been proved rather than computed — by two arguments that share no arithmetic and no assumptions beyond ductility.

It also says what a plastic analysis is for. There is no differential equation to solve, no stiffness to know, no sequence of yielding to follow. The answer is settled by a statement about equilibrium and a statement about work, and neither of them mentions how the structure got there.

Which way each method is wrong, and which one designers use #

Put simply: the lower bound is the design method and the upper bound is the check.

A designer choosing reinforcement, sizing a bolt group, or laying out a strut-and-tie model is constructing an equilibrium field and satisfying it everywhere — which is a lower-bound calculation whose answer is safe whether or not it is the real distribution. That is why this collection has so many of them.

A thrust line inside the masonry is a lower bound: any line that stays inside proves the arch stands. A strut-and-tie model is a lower bound: any admissible truss proves the region works. Redistributing moments in a continuous beam is a lower bound: any moment diagram in equilibrium with the load may be designed for. Three quite different design methods, all of them the same theorem.

The upper bound is what an assessor uses. Given a structure that already exists, the fastest way to find out what it will carry is to look for the mechanism it will form — and the discipline that goes with the method is to look for all of them, because the one that was missed is the one that governs.

The price of admission is ductility #

Both theorems assume the same thing, and it is not stated in either.

They assume a section that reaches its capacity and holds it while the rest of the structure catches up. The lower-bound theorem needs it because the equilibrium field it constructs is not the elastic one — getting from the elastic distribution to the assumed one requires some sections to yield and go on yielding while others load up. The upper-bound theorem needs it because a mechanism requires every hinge to be turning at once at full moment.

A structure that cannot do that is entitled to neither theorem. Three cases matter:

A brittle failure mode anywhere voids both. A shear failure, a bolt in tension, a weld, an anchorage — each reaches a capacity and then loses it, so the redistribution the theorems assume never happens. That is why plastic design of steelwork carries a requirement that shear and connections be stronger than the members they join: not for their own sake, but to keep the mechanism in the ductile parts.

A slender section voids both. A section that buckles locally before it reaches its plastic moment cannot form a hinge that holds, which is what the whole classification system exists to identify.

And a limited rotation capacity limits how far the redistribution may be taken, which is why codes cap moment redistribution at a percentage rather than allowing any equilibrium field at all.

Counting the mechanisms, which is where the method actually fails #

The kinematic method’s difficulty is not arithmetic. It is completeness.

A frame with $n$ critical sections and $r$ redundancies has $n - r$ independent mechanisms, and every other mechanism is a combination of those. The portal here has three critical sections and one redundancy, so there are two independent mechanisms — beam and sway — and the third is their sum.

That count is the discipline the method needs. Enumerate the independent mechanisms, combine them in every way that cancels a hinge, and minimise over the lot. Miss one and the answer is too high, with nothing to say so.

And the combinations are where the answer usually is. On this frame the two independent mechanisms give 1.00 and 1.25 and their combination gives 0.714 — the combination is 29 per cent below the lower of the two it is made of, because cancelling a hinge removes internal work while the external work of both is retained. A designer who checks the obvious mechanisms and stops has checked the two that are not the answer.

Masonry is the interesting case #

An arch of dry stone has no tensile strength at all, no ductility in the usual sense, and the lower-bound theorem applies to it perfectly.

The reason is that its limit state is not a material strength. The stones do not crush — the stresses in a masonry arch are a small fraction of the stone’s strength — so what limits the structure is geometry: whether a line of thrust in equilibrium with the load can be found that stays inside the masonry.

That is a lower-bound statement with a purely geometric admissibility condition, and it is exactly Heyman’s insight: masonry needs no ductility because it is never asked for any. Its collapse is a mechanism of rigid blocks rotating about hinges at the surfaces, which makes the upper-bound theorem available too, and the two bracket the answer in the ordinary way.

Which is a good place to see what the theorems really need. Not ductility as such, but the ability to reach an assumed distribution without anything failing on the way. Stone gets there by having enormous reserve in the only mode it can fail in; steel gets there by yielding; and concrete gets there by having reinforcement that yields, which is the same argument with the ductility supplied by an inclusion.

What a plastic analysis buys, and what it does not #

It is worth being precise about the size of the prize, because the theorems are often presented as though they were mainly an economy.

For a simply supported beam, plastic analysis buys the shape factor and nothing else — 13 per cent for an I-section, 50 for a rectangle. There is one critical section, one hinge turns it into a mechanism, and no redistribution is available.

For a continuous or framed structure it buys the shape factor and the redistribution, and the second is usually the larger. A fixed-ended beam under a uniform load has an elastic moment ratio of 2:1 between support and span; at collapse the two are equal, and the collapse load is $16M_p/L^2$ against the $12M_p/L^2$ at which the first hinge forms — a third more, before the shape factor is counted at all.

Which locates the economy exactly. It comes from redundancy, and a structure with none gets none of it. That is the same trade one support too many is about, priced in strength rather than in the effort of an analysis.

Why the theorems arrived so late #

Both theorems are twentieth-century results, which is startling for statements this simple about structures people had been building for millennia.

The reason is that they are about materials rather than about statics. A lower-bound theorem is worthless unless a section can hold its capacity while its neighbours load up, and until steel was made ductile and reinforced concrete was made to yield in its steel rather than crush in its concrete, no structural material reliably could. Masonry is the exception that proves it: its limit state is geometric rather than material, so the theorem was available all along and was used by anyone who drew a thrust line, several centuries before anybody proved it.

That also explains the order in which the two were adopted. The kinematic method came first, in the 1930s and 1940s, because a work equation is a calculation a designer can do; the static method’s dominance came later, with the strut-and-tie and yield-line methods, once it was clear that the safe side of the answer was worth the extra construction.

Where the model stops #

Everything above is first-order. A frame that sways significantly under load has second-order moments the mechanism’s work equation does not contain, and the whole analysis becomes an upper bound on something that is itself falling. Plastic design of sway frames therefore carries an explicit limit on slenderness, and beyond it the collapse load is a stability question rather than a strength one.

The plastic moment is treated as a constant. It is reduced by axial force and by shear, and in a frame’s columns the axial force is not small — so a real analysis iterates: find the collapse mechanism, compute the axial forces, reduce $M_p$, and repeat.

Only three critical sections were considered. That is right for a prismatic frame with point loads, and a member under a distributed load has its hinge at a position that has to be searched for, which is what the single-span figure above is doing.

And nothing here says how much deflection the collapse mechanism needs. A load factor of 0.714 is reached at a deflection that may be far outside anything serviceable, and a structure can be plastically adequate and unusable — which is the standing reason strength and stiffness are separate checks.

What the pictures cannot show #

The bracket in the first figure closes to a point, which makes the answer look inevitable. It closed because a one-redundant frame with three critical sections has just enough freedom for the two bounds to meet; with more redundants and more sections the search is a linear program, and a hand calculation that stops early leaves a genuine gap between the two bounds rather than a point.

Nor can any figure show what a collapse mechanism looks like at the moment it forms, because it does not form at a moment. Hinges appear one at a time as the load rises, the structure softens each time one does, and the “mechanism” is the arrangement present at the end of a sequence that took the whole loading history to produce.

The assumption the figure rests on #

The load is assumed to be applied proportionally — every component growing together, in a fixed ratio, characterised by a single factor $\lambda$.

Real loads do not arrive that way, and the consequence is more interesting than it sounds. Under varying repeated loads a frame can fail at a lower factor than either theorem gives, by ratcheting or alternating plasticity — accumulating deflection cycle by cycle at a load that would be perfectly safe if applied once and held.

So the collapse load computed here is a bound for one loading path, and the shakedown load, which is the honest limit for a structure under variable loading, is a different and smaller number obtained from a different pair of theorems. The two pairs are close relatives — one is a lower bound on a stress field and the other is a lower bound on a residual stress field — and knowing which one a design is relying on is the difference between a structure that stands and one that walks.