The order the loads arrived in

A bar leaning against a wall, with friction at both ends, has four unknown forces and three equations to find them with. The treatment of friction as an inequality drew the consequence as a chord across a triangle: a whole family of equilibrium states, every one of them satisfying every equation and every friction bound, and nothing in statics that prefers one.

A drawing can leave the choice open. A real bar cannot. It is carrying one set of forces and not the others, and something decided which. The answer is not a hidden equation, because there is none to find. It is the route the structure took to where it is.

The cleanest place to watch that happen is a bridge bearing. A deck sits on a sliding bearing over a pier with a lateral stiffness of 20 kN/mm, and the bearing’s coefficient is 0.03. Two things happen to it on a given day: the deck moves 4 mm over the pier as it warms, and the load on the bearing rises from 2000 kN to 4000 kN as traffic arrives. At the end of the day both have happened. The only question is which happened first.

Same deck position, same bearing load, same coefficient, same pier. Sixty kilonewtons in one pier and eighty in the other, and both are right.

A spring in series with a slider whose strength moves #

Everything in that figure follows from a model with two parts. The pier is a spring. The bearing is a slider that holds any force up to $\mu N$ and none above it. They are in series, so they carry the same force $H$, and the deck’s movement $u$ is shared between the pier’s deflection and the bearing’s slip.

While $|H| < \mu N$ the slider is locked and the whole of any movement goes into the pier, whose force changes by $k$ for every millimetre. When $|H|$ reaches $\mu N$ the slider gives, $H$ stays pinned at the limit, and further movement goes into slip. That is all there is.

The one feature that makes this a structure with a memory is that the limit is not fixed. It is $\mu N$, and $N$ is the load on the bearing, which is a separate thing that happens on its own schedule. Raise $N$ while the bearing is gripping and the limit moves away from $H$; nothing moves, and the pier keeps the force it had. Lower $N$ while the bearing is gripping and the limit moves toward $H$, and if it arrives, it drags $H$ down with it and the bearing slips back.

So in the first order, the deck moved while the limit was 60 kN, the pier force rose 20 kN a millimetre, reached 60 kN after 3 mm and stayed there while the bearing slid the last millimetre. Then the load doubled, the limit rose to 120 kN, and the pier went on carrying 60, because doubling the limit is not a force and cannot push anything.

In the second order the limit was 120 kN before anything moved. The pier force rose the whole 4 mm to 80 kN without reaching it. The bearing never slid.

With more movement the two orders diverge further, and the figure shows why: in both of them the pier ends at a limit, but at the limit that was in force when it last slid. A bearing that slides records the load it was carrying at that moment in the force it leaves behind.

The present does not determine the force #

The unsettling thing about the two figures is not the size of the difference. It is what the difference means for a calculation.

An elastic structure has no memory. Its forces are a function of its present loads and its present displacements, and the order in which they arrived is irrelevant. That is what licenses the addition everything else rests on: superposition, influence lines, load combinations, the whole habit of analysing load cases separately and adding the results. That essay records the conditions under which superposition fails, and a support with a gap in it is one of them. A support with friction in it is another, and a stronger one, because a gap closes and reopens in the same place while a friction contact can grip anywhere.

The forces a friction contact leaves behind are locked in: they are carried with no load holding them, they balance themselves across the structure, and no free body drawn in the present can reveal them. They belong to the same family as the residual stresses in a rolled section and the forces in a member built to the wrong length, both of which are invisible to statics for the same reason — they are states of self-stress, and statics can only say that such states are possible. The difference is how they got there. A misfit is fixed in the workshop and a residual stress in the mill. A friction contact renews its locked-in force every time it slides, and it slides every time the loads and movements on it take the route that makes it.

The same point has been made about a structure built in stages, where a beam made continuous late carries a different moment from one cast continuous, and the order of building is a load case. Friction reaches the same conclusion with less. Nothing about the bridge is built or changed between the two orders above; the same pier, bearing and deck are there throughout. The only thing that differs is the sequence of two ordinary loads.

Three things a support can do under repetition #

One order of two events is a curiosity. A bridge does not live through two events; it lives through a daily thermal cycle for a century. So take the simplest repeated history — the deck going out and back by the same amount, again and again, under a constant bearing load of 2000 kN and a bound of ±60 kN — and see where the pier ends up.

When the whole elastic swing the movement would put into the pier, $k\delta$, is inside the bound, nothing slides and nothing is remembered. The support behaves like a spring and the history is irrelevant. That is the only regime in which a friction contact is as simple as the drawing suggests.

This is the middle case, and it is the one with the memory in it. The bearing slides once, and in sliding it shifts the pier’s zero. After that the whole cycle fits inside the bound, so it never slides again, and the pier at the deck’s starting position carries 40 kN that no load on the bridge is putting there. The support has shaken down: it has found a locked-in force that lets it carry the repeated movement elastically for ever.

In the third case no amount of history helps. The movement is too large for the bound’s width, so the bearing slides at both ends of every cycle and the pier runs round the same loop between the two limits. This is the case whose force range is the easiest to design for, because it is fixed at twice the bound; and the case whose wear is the worst, because the bearing is sliding a fixed distance every day for the life of the bridge.

Two thresholds, and what Melan says about them #

The three cases are three regions of a single ratio, the elastic swing over the bound, $k\delta/\mu N$.

The first threshold, at one, says whether the support slides at all. The second, at two, says whether it ever stops. That second one has a name. It is Melan’s shakedown theorem in its smallest possible setting: a structure under a repeated load shakes down if and only if there is some locked-in state that keeps the whole of the elastic variation inside the limit. Here the elastic variation is a swing of $k\delta$ and the limit has a width of $2\mu N$, so the condition is exactly $k\delta \le 2\mu N$.

The resemblance to plasticity is not an analogy. A slider in series with a spring is precisely the element from which elastic-perfectly-plastic behaviour is built, and the bearing is one of those elements with its yield force set by a separate load. The map with three regions that shakedown draws for a pressure vessel or a heated beam — elastic, shaking down, and cycling plastically — is this chart, and the requirement recorded there, that the steady load and the cycling one have to be of different kinds, is met here by construction. The bearing’s load sets the bound. The deck’s movement is imposed. One is a force and the other is a displacement, and a sliding bearing is the simplest structure there is in which both are present at once.

What the theorem does not say is the part this essay is about. It says a shaken-down state exists. It does not say which one the structure reaches, and between the two thresholds there are many. In the middle figure the pier ended with 40 kN locked in because it started from zero and the first movement was outward. Start it from a different force, or begin with the deck contracting, and it shakes down to a different locked-in force, just as permanently, and just as invisibly.

Which supports remember #

The chart has a consequence that is easy to miss because it reads as a detail of the middle band. Only the middle band remembers.

Below the first threshold there is nothing to remember: the bearing never slides, the pier behaves as a spring, and its force at any moment is the movement times the stiffness. Above the second threshold the memory is real and short. The bearing slides at both ends of every cycle, so the force is pinned to one limit or the other twice a day, and whatever the history had put into the pier is overwritten within a single cycle. Two bridges that started from different locked-in forces are indistinguishable by the second evening.

Between the two thresholds the first slide is kept for ever. The support shakes down to a locked-in force set by the first movement large enough to slide it, and every later cycle, being smaller than the bound’s width, leaves that force exactly where it is.

Which band a real pier sits in depends on both the stiffness and the size of the movement, and the answer is not the intuitive one. A short stiff pier of 100 kN/mm under a bearing carrying 4000 kN at a coefficient of 0.03 has a bound of ±120 kN; a daily thermal movement of 10 mm would put 1000 kN into it, far past twice the bound, so it slides every day and forgets every day. A tall slender pier of 5 kN/mm under the same bearing sees 50 kN from the same 10 mm — inside the bound, so the daily cycle never slides it at all. The seasonal movement of perhaps 40 mm puts 200 kN into it, which is between one bound and two. So the tall pier slides in its first summer, locks a force in, and carries that force through every summer after, because no later season is larger than the bound’s width.

The flexible pier is the one with the permanent memory. The stiff pier, which jumps often and forgets, has none; the slender one, which barely feels the daily cycle, is carrying a record of the first warm season after its bearings were released.

Thirty mornings #

A real bearing’s history is not a clean repeated cycle. The deck goes out a different amount each day, depending on the weather, and the load on the bearing is whatever the traffic happens to be at the moments that matter. Take a month of it: each day the deck expands by up to 8 mm and comes back, and the bearing load sits somewhere between 2000 and 4000 kN, changing twice a day.

Every one of those readings is at the same deck position. An engineer measuring the pier force on the fourth morning and the eleventh would record two different numbers for what looks like the same bridge in the same state, and neither reading would say anything about whether the design was right. A measured force in a frictional support is one member of the admissible range, chosen by the weather. What can be checked against it is the bound, and the bound is the design statement, because it is the only one the history cannot move.

That is also the answer to the practical question the essay began with. A pier under a sliding bearing is designed for $\mu N$ in either direction — not because the force is usually that large, but because the force is usually unknowable, and the bound is the smallest statement about it that is true on every morning of the bridge’s life. The force a sliding bearing delivers is already a pair of load cases because friction opposes whichever way the deck is going; the history adds that, in between, the pier may be carrying anything in the range, locked there by a day nobody recorded.

Where the order is decided on site #

The first day of the history is the one that is chosen, and it is usually chosen by somebody who is not thinking about pier forces at all.

A sliding bearing is installed with its sliding plate fixed temporarily to its lower part so the deck can be placed without the bearing moving. When the fixing is released, the pier is carrying whatever force the construction put into it, and that is the starting point of every history after it. Release the bearings at dawn, with the deck cold and short, and the first warm afternoon slides them one way; release them in the heat of the afternoon and the first night slides them the other. The bearing’s preset — the offset it is installed at so that its travel is centred on the expected range of a movement nobody applied — is an adjustment for the same fact: that where a sliding surface starts is a decision, and the structure keeps it.

What the slider in series leaves out #

One coefficient does all the work. The bearing grips and slides at the same $\mu$ here. A real one grips at a higher coefficient than it slides at, which is what makes a slow movement arrive in jumps and adds a dynamic overshoot to every locked-in force the history produces.

The bearing is alone. A deck on several piers has one sliding bearing on each, and the deck ties them together. The history then selects not a force in one pier but a distribution of forces across all of them, and a slip at one bearing moves the others — the indeterminate version of every figure above.

The load and the movement are independent. They are here, and that is what licenses the shakedown condition. On a pier that sways under its own horizontal force, or a bearing whose load depends on which way the deck is rocking, the normal force and the tangential one are coupled, and the clean threshold at $k\delta = 2\mu N$ is no longer guaranteed.

The coefficient is constant. PTFE’s coefficient rises with cold and falls with contact pressure, so the bound itself moves with the weather and the traffic independently of $N$, and the locked-in forces move with it.

Nothing creeps. The pier is elastic for ever. A concrete pier carrying a locked-in force for months relaxes some of it, which moves every history back toward zero at a rate its own creep coefficient sets.

Still open: whether a guarantee survives when load and friction are coupled #

The shakedown threshold above is exact because the bearing’s load does not depend on the force in the pier. Couple them and the question changes character. The bound then moves with the very force it bounds, a structure can in principle cycle without settling even when a locked-in state exists that would have held it, and the neat equivalence between “some admissible state exists” and “the structure will find one” stops being a theorem.

That is the frictional face of a difficulty that runs through the whole of limit analysis. The bound theorems need a structure’s movement at its limit to be normal to its limit surface, and friction’s movement is not — a sliding contact moves along its surface, not away from it. For a single bearing with an independent load the difficulty never arises. For a real structure on many contacts, with loads that shift as it moves, which of its admissible states it is actually in remains a question about its history, answered one history at a time.