The one number the tolerances cannot touch

Three earlier arguments have been about a quantity nobody can compute. Four legs and three equations leave the load sharing undetermined; the stiffnesses that would settle it are known to a factor of two at best, and a leg that is stiff draws load to itself for no better reason; a manufacturing error of a tenth of a millimetre swamps them; and a temperature difference of one kelvin does more than the load.

Buildings nevertheless stand, and they are designed by people who are not troubled by any of this. The reason is that the quantity which decides whether a structure stands is a different quantity, and it does not depend on any of the unknowns.

Why yielding erases the difficulty #

The reason a redundant problem has no answer is a counting one: there are more unknown forces than equilibrium equations, and the missing equations have to come from compatibility — from the requirement that the members’ deformations fit together — which brings the stiffnesses in with them.

A member that has yielded does not obey that requirement. Once a leg is squashing plastically its force is its yield force, whatever its displacement does; the relation between force and deformation, which was the source of the extra equations and of all the trouble in them, has been replaced by a constant.

So each member that yields removes one unknown from the problem rather than adding one. Yield enough of them and the count comes back into balance: the structure becomes determinate, and then over-determinate, and at the moment it can no longer satisfy equilibrium at all it has become a mechanism and it collapses.

At that moment every member that is going to yield has yielded, so every force in the structure is a known capacity rather than an unknown share. The collapse load is a sum of capacities, and a sum of capacities has no stiffness in it, no tolerance, no temperature and no assembly sequence. It is the one quantity in the whole subject that the idealisation does not throw away.

The contrast the two figures make is the whole point. The quantity that is unknowable and the quantity that decides safety are not the same quantity, and a great deal of nineteenth-century anxiety about indeterminate structures was anxiety about the wrong one.

Following a beam through it #

The table collapses when its legs run out, which is simple enough to hide the interesting stage. A beam shows the redistribution happening.

Elastically a propped cantilever with a central load carries three-sixteenths of $WL$ at its fixed end and five thirty-seconds under the load, a ratio of six to five. So the fixed end reaches the section’s plastic moment first, and a design that stopped there would call that the failure load.

What happens instead is that the fixed end yields through its whole depth and can carry no more moment — but it can still rotate, and while it does it holds the plastic moment constant. The beam from that point on behaves as a simply supported one with a fixed moment applied at one end, and every further increment of load goes into the section under the load. That section catches up, reaches the plastic moment in its turn, and with two hinges the beam has a mechanism and deflects without limit.

Twelve and a half per cent is the reserve, and its size is a measure of how unequal the elastic moments were. A beam whose elastic moments are nearly equal has almost no reserve; one whose moments are very unequal has a great deal. The reserve is therefore a property of the elastic distribution — which is to say of the stiffnesses — and the collapse load is not, which the figure checks by computing it a second way.

That second way is a virtual-work calculation on the mechanism alone. Give the midspan a small downward movement, work out how far each hinge rotates from geometry, multiply each rotation by the plastic moment to get the work dissipated, and set it equal to the work the load does. The answer agrees with the incremental calculation to nine figures, and it never mentioned a modulus, a second moment of area or an elastic moment distribution.

The one thing it costs #

The redistribution above requires the fixed end to go on rotating after it has yielded, at constant moment, while the rest of the beam catches up. How far it has to rotate is calculable, and whether the section can rotate that far is a question about the material and the shape rather than about the structure.

A steel section can, comfortably: mild steel strains by twenty times its yield strain before it begins to harden, and by a hundred and fifty times before it breaks, so a compact steel section will rotate several times as far as any redistribution requires. That is the whole reason the method is a steel method.

A material that cannot does not get the reserve. Cast iron, unreinforced concrete in tension, glass, a high-strength bolt loaded into its own plastic range and a slender steel section that buckles locally before it yields fully all reach the first hinge and then break rather than rotate. For those, the first-yield load is the collapse load, and every unknown catalogued earlier comes straight back — which is why a prestressed concrete section is designed to stay uncracked rather than to redistribute, why the design rules for brittle materials are so much more conservative than the rules for ductile ones, and why they stay conservative no matter how well the loads are known.

Ductility is not a virtue of a material; it is what buys the indeterminacy back. A structure of a brittle material has the same free direction, the same unknowable load sharing and none of the relief, so its weakest member decides everything and its weakest member is decided by a tolerance.

Where else a mechanism decides the answer #

The step the beam takes is general enough to be worth recognising elsewhere, because in each case a hard problem is replaced by a count of ways the thing can come apart.

A granular pack is rigid above the isostatic coordination number and flows below it, and the count of contacts against freedoms is exactly the count performed here — with the difference that a pack rearranges continuously and a frame does it in a handful of discrete steps. A soil slope fails along a surface found by searching over candidate slip circles for the least factor of safety, which is the third figure’s scan with a different parameter. And whether a pushed block slides or topples is the same question asked of a determinate body: two mechanisms, each with its own load, and the lower one happens.

The pattern is that strength is a property of the cheapest way out, and the cheapest way out is a geometric object rather than a stress. That is why the collapse calculation is indifferent to everything the elastic one needed, and why it has to be searched for rather than solved.

Two theorems, and only one of them is safe #

The virtual-work calculation above needed a mechanism to be assumed. There is generally more than one, and which is assumed matters in a way that is worth being exact about.

Assume a mechanism and equate work, and the load that comes out is the load at which the structure would collapse if it were forced to collapse that way. A structure given a mechanism it does not want has to be pushed harder to use it, so any assumed mechanism returns a load at least as large as the true one. That is the upper-bound theorem, and the true collapse load is the least value over all mechanisms.

The complementary statement runs the other way. Find any distribution of internal forces that balances the applied load and nowhere exceeds any member’s capacity, and the structure will carry that load. That is the lower-bound theorem, and it is safe: guessing badly gives a load that is too small.

The pair are what make plastic design usable, and the asymmetry between them is the thing to carry away. A mechanism is easy to imagine and gives an unsafe answer; an equilibrium stress field is harder to construct and gives a safe one. Practical design finds several mechanisms, takes the least, and then checks that a corresponding equilibrium field exists — at which point the upper and lower bounds coincide and the answer is exact.

The scan makes the shape of the risk visible. The curve is flat near its minimum, so a hinge position ten per cent wrong costs under one per cent in the answer, which is what makes the method workable by hand and why the classic textbook mechanisms give nearly the right answer. It is also what makes it treacherous: the error is always in the unsafe direction, and flat means there is no warning near the minimum to say whether it has been reached. The mechanism nobody thought of is the one that is not on the curve.

Where the method came from, and what it was resisted for #

Plastic analysis was worked out in the 1930s and 1940s and met considerable opposition, which is instructive because the objection was reasonable.

Elastic design had an intelligible discipline: compute the stresses by resolving every force into components that suit the member, keep them below a fraction of the yield stress, and let nothing anywhere in the structure yield. Plastic design says that parts of the structure will yield in service, deliberately, and that this is fine. To an engineer trained on the first, the second sounds like a licence to accept damage.

What settled it was partly J. F. Baker’s wartime work on air-raid shelters, where the problem was to protect people with a minimum of steel and the elastic method’s answer was unaffordable. The shelters were designed by plastic analysis, they were permitted to deform substantially, and the criterion was that the occupants survive rather than that the steel stay elastic. That is exactly the right use of the method — it answers the question “at what load does this become a mechanism” and not the question “at what load does it stop looking new”.

The modern arrangement keeps both. Strength is checked against collapse, using this method; deflections, vibration and cracking are checked against the elastic behaviour at working load, using the other. Two calculations answering two different questions, and the reason there are two is precisely that the plastic one throws away the information the serviceability check needs.

What the collapse load will not tell anybody #

It is worth being clear about how much is lost along with the stiffnesses, because the method’s own advantage is the source of its limits.

It gives no deflections. At collapse the deflection is undefined; below collapse it depends on the elastic problem with all its unknowns. A structure can be perfectly safe against collapse and unusable because it sags, and the sag has to be computed by the method this one replaced — which needs every stiffness the collapse calculation was so pleased to be rid of, and inherits every uncertainty in them. Whether a contact grips at all is a related difficulty one level down: a quantity that has to be known for the serviceability answer and does not appear in the strength one.

It gives no fatigue life. Fatigue depends on the stress range at working load, which is an elastic quantity, and a member that carries a small share elastically and a full share at collapse is exactly the member a plastic calculation is indifferent to and a fatigue calculation is not.

It assumes the loads are applied once and monotonically. A structure loaded, unloaded and loaded again in a different pattern can accumulate plastic deformation cycle after cycle — incremental collapse, or ratcheting — at loads well below the single-application collapse load. Where a thermal load cycles, which is a self-loading structure’s subject, this is the governing case rather than an exotic one.

It assumes the capacity of a member is a number. A leg’s yield force, a section’s plastic moment: both are treated here as properties known in advance, and both are statistical quantities with their own spread in a real batch of material. The specified value is a lower percentile rather than a mean, which is one of the reasons measured capacities exceed calculated ones.

It has a third load hidden between its two. Under loads that are applied and removed repeatedly rather than once, there is a level — above first yield and below collapse — at which the structure yields during the first few cycles, arrives at a locked-in residual stress state, and from then on responds purely elastically for ever. That is the shakedown load, and the residual state it settles into is exactly the self-equilibrating state a temperature change produces, found by the structure itself rather than put there by a designer. Above the shakedown load and below the collapse load a structure either accumulates deformation every cycle or alternates plastically in one place until it cracks, and neither is visible in a monotonic calculation. Where the repeated load is a temperature, which is the commonest case, shakedown rather than collapse is what governs.

And it says nothing about stability. A mechanism forms at a computable load; a column buckles at a load computed by an entirely different argument, and for a slender member the second can be far lower. Plastic design applies to sections stocky enough to reach their plastic moment before anything buckles, and the section classification rules that decide which those are are most of the length of a steel code.

What perfect plasticity is standing in for #

Yield is taken as perfectly plastic. Real steel strain-hardens, so a yielded section goes on gaining moment slowly rather than holding constant, which makes the true collapse load a little higher than computed — a conservative error, and the reason it is accepted without much comment.

The plastic moment is a section property computed for bending alone. A section also carrying axial force or shear reaches its bending capacity at a lower moment, and the interaction is nonlinear. In a frame where the columns carry substantial axial load, using the pure bending capacity is not conservative.

The mechanism is assumed to form at fixed geometry. A structure deflecting substantially before it collapses carries its load on a deformed shape, and the extra moments that produces can bring the collapse load down by ten or twenty per cent in a tall frame. That effect is second-order in the deflection and first-order in the consequences.

And the four-legged table’s collapse load is four times a leg’s capacity only for a central load. Move the load off centre and the collapse mechanism is a rotation about a line through two legs, the work balance is a different one, and the answer is lower — which is the upper-bound scan of the third figure applied to a different structure, and is the version of the tipping condition that a yielding structure has.

The settlement that tells the three stages apart #

The hero figure draws four leg forces and cannot show the displacement, which is the thing that distinguishes the stages. Before first yield the top settles by micrometres; between first yield and collapse it settles by millimetres as the yielded legs squash; at collapse it moves without limit. A plot of force against load makes the second stage look like a mild change of slope and it is a change of kind. The same omission hides the warning a real structure gives: a steel frame approaching collapse sags visibly and audibly for some time first, which is a large part of why ductile design is trusted, and none of it is in the figures.

The moment figure draws two moments and hides the rest of the beam. A bending moment diagram is a function along the span, and what happens between first hinge and collapse is that its whole shape changes — the point of maximum sagging moment moves, the point of contraflexure moves, and the diagram at collapse is not a scaled version of the elastic one. Two numbers from it are the two that matter and they are not the picture.

And the mechanism scan draws one family of mechanisms, parameterised by one number. The real space of mechanisms for a frame is not a line: it is a set of combinations of beam mechanisms, sway mechanisms and joint mechanisms, combined in ways that cancel hinges, and the number of them grows quickly with the number of members. Drawing the search as a curve with a visible minimum is honest for this beam and flattering to the method in general.

Still open: how much of a real structure’s reserve is this reserve #

The reserve computed here — twelve and a half per cent for the propped cantilever, a factor of up to four for the table — is the redistribution reserve, and it is the part of a structure’s margin that can be calculated.

It is not the whole margin and is usually not most of it. A real structure also has strain hardening, a yield stress typically ten to twenty per cent above the specified minimum, member sizes rounded up to what is manufactured, composite action with floor slabs that nobody counted, and restraint from cladding and partitions that no model contains. Load tests on old buildings routinely find capacities two or three times the calculated one, and the excess is mostly in that list rather than in the redistribution.

Which means the honest description of a structural safety factor is that it covers a set of uncertainties of very different characters: a variability in the loads that is statistical and estimable, a variability in the materials that is controlled and small, and a set of modelling omissions that are systematically in the safe direction and are not quantified at all. Attempts to make design fully probabilistic founder on the third, because a margin nobody has measured cannot be given a distribution.

The habit worth carrying away is the one the first two figures make side by side. When a quantity cannot be computed, ask whether it is the quantity the decision depends on. The elastic load sharing in a redundant structure genuinely cannot be known, and three earlier arguments are about why. The thing anybody actually wants to know — at what load does it fall down — turns out not to need it, and the reason is that the mechanism which destroys the structure is also the mechanism that destroys the ignorance.