The load nobody applied

A rigid top on four legs has four unknown reactions and three equations, and the difference is a direction along which the reactions can be changed without disturbing any balance of forces or moments. That calculation asked what fixes the point along the line, and the answer was elasticity, which is to say the stiffnesses and the tolerances.

This essay asks a different question of the same free direction. If nothing external can excite it, what else can?

A temperature change can. A member held so that it cannot change length, and then warmed, develops a stress of $E\alpha\Delta T$: the product of the stiffness and the expansion coefficient and the temperature change, and nothing else.

Nothing else is the striking part. There is no length in it, no cross-sectional area, no load, and no reference to how the member is held beyond the fact that it is held. A hairpin of steel wire and a bridge girder reach the same stress at the same temperature change, and a hundred and four kelvin puts either one at yield.

Where the length went #

The absence of a length surprises people who have met thermal expansion as a movement, because there a length is the whole of the story: a thirty-seven-metre rail grows by nine millimetres over a forty-kelvin rise and a half-metre one grows by a fifth of a millimetre.

Both of those are the same strain, $\alpha\Delta T$, which is four hundred and eighty parts per million and has no length in it either. The movement differs because a strain multiplied by a length is a movement; the stress needed to cancel the strain is a stress multiplied by nothing. It is the same separation that makes a rigid body a useful fiction in one problem and an impossible one in the next: a strain too small to draw is not a force too small to matter.

So the two facts are consistent and they point in opposite directions for design. A long member needs a large movement joint and a short one needs a small joint — but if either is restrained, both reach the same stress, and the long one is no worse off. That is why continuously welded rail was possible at all. Jointed track had a gap every eighteen metres to take the movement; welding the rails into lines kilometres long removes the joints and the movement with them, and what is left is a stress that a rail eighteen metres long would have had just as badly.

It is also why the rail is laid at a deliberately chosen temperature. A rail stressed to zero at 27°C is in tension in winter and compression in summer, and the compression is the dangerous half: a long member in compression does not fail by crushing but by buckling sideways, where the load that matters is set by the stiffness and the length and can be very small. Continuously welded track buckles laterally in hot weather at a temperature a few tens of kelvin above the stress-free one, and the ballast is what holds it — which is why a heat buckle usually happens where the ballast has been disturbed.

How soft a support has to be before it helps #

The natural response to a stress caused by restraint is to restrain the member less. It is worth seeing how much less.

The member and its supports share the expansion in proportion to their compliances, so the relief depends on the ratio of two stiffnesses and the curve is a plain series combination. What matters is where the interesting part of that curve sits.

Halving the stress needs a support as flexible as the entire member it holds, which is not a bolted end plate or a welded connection or a bearing pad; it is a mechanism. Ordinary structural connections are one to three orders of magnitude stiffer than the members they join, which puts them in the flat right-hand part of the curve where ninety-five to ninety-nine per cent of the stress gets through.

There is no middle course, and that is a genuine design fact rather than a mathematical curiosity. Either a member is deliberately allowed to move — a sliding bearing, a roller, an expansion loop, a bellows — or it carries essentially the whole of $E\alpha\Delta T$ and has to be designed for it. Attempts to split the difference buy a few per cent for a great deal of detailing.

The same arithmetic explains a familiar failure. A pipe run that a designer intended to slide takes up rust, or is clamped by a support installed to stop it vibrating, and its thermal stress goes from nothing to everything with no intermediate stage. The stiffness ratio has moved four decades, and the curve is only steep in the middle.

A structure that loads itself #

The restrained bar has supports, so a sceptic can locate the forces: they are reactions, and something outside is providing them. The more startling case has nothing outside at all.

There is no load on that assembly, no reaction at any support, and no external force of any kind. There are nevertheless forces in it large enough to buckle the aluminium bar. They exist because the three bars are constrained to agree with one another about their length, and they want different lengths.

A state of stress with no external load is called self-equilibrating, and it is exactly what the free direction of the four-legged table was: a pattern of internal forces that satisfies every equilibrium equation identically and is therefore invisible to statics. The four-legged table found that the pattern’s amplitude is set by tolerances. This essay finds that a temperature change sets it too, for free, and keeps resetting it every day and every season.

This is the reason a large steel structure is never in the state its drawings describe. It has residual stresses from rolling and welding, it has locked-in forces from the sequence in which it was erected and bolted, and it has a thermal state that depends on which face the sun has been on. The situation is the one a granular pack is in one level down, where the force network is not reproducible between two identical fillings for exactly the same reason. None of those appears in a load case, all of them are of the same order as the design stresses, and all of them are along the directions the equilibrium equations cannot see.

What the free direction costs a measurement #

There is a consequence for anybody putting a strain gauge on a structure, and it is the same one a silo’s weighing cell meets from the other side.

A gauge reads the strain the member actually has, which is the sum of the strain from the design loads and the strain from every self-equilibrating state present. On a determinate member those are the same thing. On a redundant one they are not, and the difference is not an instrument error — the member really is carrying that force.

The practical method is to read a change rather than a value. Zero the gauges, apply a known load, read the difference, and the self-equilibrating states cancel because they did not change. That works, and it quietly concedes the point: the absolute force in a member of a redundant structure is not measurable without knowing its whole history, and the history includes every temperature it has been.

It also explains why a structure’s measured behaviour under a test load usually agrees with the model much better than its absolute stresses do. The model gets the stiffnesses roughly right, so it predicts increments well; it has no information at all about the locked-in state, so it predicts absolute values badly.

The member that restrains itself #

The three bars need a crosshead to hold them together. A single piece of material needs nothing, because the cold part of it restrains the hot part, and that is where the largest thermal stresses anybody meets actually occur.

Drop a hot glass into cold water. The surface tries to contract and the interior, still hot, will not let it: the surface goes into tension at $E\alpha\Delta T/(1-\nu)$, the factor accounting for the surface being held in two directions at once rather than one. For ordinary soda-lime glass, with a modulus of seventy gigapascals, an expansion coefficient of nine parts per million and a practical tensile strength near fifty megapascals, that stress reaches the strength after about sixty kelvin.

Borosilicate glass has almost the same modulus and the same strength. Its expansion coefficient is three and a third parts per million, a factor of 2.7 smaller, and its survivable temperature difference is correspondingly a factor of 2.7 larger — a hundred and ninety kelvin. That single substitution is the whole reason laboratory glassware and oven dishes are made of it, and it is a change to one factor in $E\alpha\Delta T$ with nothing else touched.

The comparison also says what cannot help. Making the glass thicker does not, because the stress has no thickness in it; it makes matters worse, by slowing the interior’s approach to the surface temperature and so holding the difference longer. Making it stronger helps in proportion, and glass strength is set by surface flaws rather than by composition. What is left is the expansion coefficient, which is why the entire field of thermal-shock-resistant ceramics is a search for materials with small ones — and why fused silica, at half a part per million, survives being taken from a furnace and dropped into water.

Using the mismatch on purpose #

Every effect in this essay is a nuisance in the structures it has been applied to. It is also a mechanism, and it is worth naming because it is where this subject goes next.

Bond a strip of steel to a strip of brass along their length. Heated, the brass wants to grow about half as much again as the steel and cannot, so each is forced to the other’s length — and because the two are side by side rather than in line, the assembly resolves the mismatch by bending. The curvature is proportional to the difference of the expansion coefficients and to the temperature change, and inversely proportional to the thickness, so a thin strip of two metals is a sensitive and entirely passive thermometer.

That is a bimetallic strip, and it is the same self-equilibrating state as the three bars with the geometry rearranged so that the state produces a motion. Thermostats, circuit breakers, the flashing indicator relay in older cars and the temperature compensation in a mechanical watch’s balance wheel are all of them this one effect, deliberately maximised by choosing two metals as different as possible.

The step from nuisance to mechanism is the one that comes next, and it has a name: putting a structure into a state no external load could reach, and doing it on purpose.

The tolerance and the temperature are the same variable #

An earlier calculation gave the reactions under a table one of whose legs is a fraction of a millimetre short. That figure can be read again with a different label on its axis.

A leg one tenth of a millimetre longer than the others and a leg ten kelvin warmer than the others are the same input to this calculation. Both supply a length mismatch, both excite the free direction, and both change the load sharing by an amount proportional to the mismatch and to the stiffness, and not at all to the load.

The comparison is worth making because it settles which effects are large. A steel leg a metre long compresses by a few micrometres under its share of a hundred kilograms. Warming it by one kelvin lengthens it by twelve micrometres — several times its working compression. A one-kelvin temperature difference across a structure does more to the load sharing than the load does, which is why precision instruments are built with kinematic mounts and why a machine tool is left to reach thermal equilibrium before anything is measured on it. The kinematic mount is the determinate answer to this whole essay: six contacts, six constraints, no free direction for a temperature to excite, and therefore no thermal stress at all whatever the instrument is made of.

It is also the reason the same building expands and contracts without its stresses cycling in a simple way. The parts do not reach the same temperature at the same time. A steel frame inside a masonry cladding, a bridge deck in the sun over a shaded girder, a pipe carrying hot fluid inside a cold jacket: each is a differential temperature across a redundant assembly, and each produces exactly the three-bar figure above with one bar warmed instead of all three. It is also the reason a body’s shape decides so much of its behaviour here as elsewhere: what a temperature difference does to a structure depends on which members it reaches and in what order, and that is a question about the arrangement rather than about the material.

Where elasticity and a straight member give out #

Everything here is linear and elastic. Steel’s modulus and expansion coefficient both change with temperature, and once the stress passes the yield point the member relieves itself by flowing — which is why a restrained steel member heated far past a hundred kelvin does not reach a thousand megapascals but yields, and then finds itself in tension on cooling back. That cycle is how residual stresses are put into a weld, and no elastic calculation contains it.

The restraint is taken as axial. A real member held at both ends and heated does not simply develop a stress: past a critical value it buckles, and the post-buckling behaviour is governed by geometry rather than by $E\alpha\Delta T$. The figures compute the force that would be needed to hold the member straight, which is the right quantity for deciding whether it stays straight and the wrong one afterwards.

The strains are small and the response is linear. Everything above adds a thermal strain to an elastic one and subtracts, which is the right first move and hides that both are small quantities — the leading behaviour of any energy near a minimum and nothing beyond it. Materials whose modulus falls sharply with temperature, and temperature ranges wide enough for the expansion coefficient itself to move, need more than this.

The temperature is uniform through each member. A gradient across a section produces a bending moment as well as an axial force, and in a redundant frame that moment is itself restrained, which doubles the number of self-equilibrating states available. A bridge deck warmed on top and cool underneath is the standard case and it needs a design temperature profile, not a design temperature.

And the three-bar assembly assumes rigid crossheads. Real crossheads bend, which lets the bars take slightly different lengths and relieves some of the force — by an amount the second figure’s curve gives, and which for any stiff crosshead is small.

The rest of the structure, which is doing the restraining #

The first figure draws a stress against a temperature change and cannot show what is being restrained. A member is not restrained by the universe; it is restrained by other members, which are themselves being warmed and are themselves trying to move. Every number on that plot is an upper bound reached only when the rest of the structure is far stiffer than the member in question, and deciding whether it is means solving the whole structure rather than the member.

The three-bar figure draws forces and cannot show the displacement, which is the thing the assembly actually does. All three bars grow — by the stiffness-weighted average of what each wanted — and the assembly is longer at the end than at the start. Nothing in a plot of forces says that, and a reader who takes the compression in the aluminium to mean that it got shorter has the picture backwards.

And the table figure draws four reactions as four numbers, which is the free direction seen through its effect. The direction itself is a pattern, not a set of values: plus, minus, plus, minus round the square, with an amplitude nothing determines. Drawing five points along a line is how a one-parameter family of solutions has to be shown, and it makes the family look like five answers rather than like one answer missing.

Still open: how much of a real structure’s stress nobody knows #

The honest summary is that a redundant structure carries a state of stress that is not a function of its loads, and the parts of it that come from temperature are at least computable in principle: a temperature distribution can be measured and the elastic problem solved.

The parts that come from fabrication cannot. Residual stresses from rolling a section are of order half the yield stress and vary through the thickness in a pattern that depends on the mill; welding leaves tensile stresses at yield along the weld line, balanced by compression elsewhere; bolting a flange in a different sequence leaves a different state. Measuring them destroys the thing measured — the standard techniques cut the member and read the relaxation — and predicting them means modelling a thermal and metallurgical history nobody recorded.

The contrast with a determinate assembly is exact and worth stating once. Two pistons connected by a fluid have one equation and one answer, and warming the whole apparatus changes nothing any gauge on it reads. A determinate structure has no self-equilibrating states, and therefore no thermal stresses, no residual stresses and no dependence on the order it was assembled in. Everything in this essay is the price of the fourth leg.

What structural design does about this is instructive. It does not attempt to compute them; it relies on the material yielding locally, which relieves a self-equilibrating stress without any loss of strength, and it restricts that argument to materials ductile enough for it to be true. A structure of cast iron or of high-strength bolts loaded in tension gets no such relief, and the rules for those are much more conservative for exactly this reason.

The habit worth carrying away is the question this whole essay is an answer to. When a calculation has more unknowns than equations, ask what else is allowed to excite the difference. The missing equations do not stay missing; something supplies them, and if the designer does not choose what, the temperature and the tolerances and the erection sequence will. A stiffness deciding an outcome that a force balance could not is the same shape of surprise, and in both cases the quantity that turns out to matter is the one the idealisation discarded first.