The support that moved

Everything statics can determine about a structure comes from the sums cancelling. That is a powerful method and it has a boundary: when there are more unknowns than equations, the sums are satisfied by infinitely many force distributions and something other than equilibrium has to choose between them.

What chooses is compatibility — the requirement that the structure fit together after deforming — and the moment compatibility enters, the structure acquires a sensitivity that statics cannot describe at all. It begins to respond to things that are not loads.

Ten millimetres is a construction tolerance. It is less than the depth of a screed, and it is invisible from the ground.

Why a moved support does nothing to a determinate beam #

Take a simply supported beam and drop one of its supports by any amount. The beam rotates slightly as a rigid body and follows it down. No member changes length, no curvature is imposed, and the bending moments are unchanged to the last decimal place.

That is not a coincidence of the geometry. It is the definition of determinacy stated in the other direction: a determinate structure has exactly enough restraints to be held, so removing or moving one leaves the others free to accommodate it. The reactions are fixed by equilibrium alone, and equilibrium does not know where the supports are, only where the loads are and how far apart the reactions are.

Add a third support and everything changes. Now the beam cannot follow the moving support without bending, because the other two supports are holding the ends where they were. The imposed curvature is real, and curvature times $EI$ is bending moment.

The moments are proportional to EI #

Here is the result that makes settlement genuinely counter-intuitive, and it is worth stating baldly:

$$M_{\text{settlement}} \propto EI \delta$$

A load-induced moment does not contain $EI$ at all — for a determinate structure it is pure statics, and for a redundant one the $EI$ terms cancel between spans of the same section. A settlement-induced moment is directly proportional to it. Double the beam’s stiffness and the settlement moments double, with the load moments unchanged.

This inverts the usual relationship between stiffness and safety. A stiffer structure deflects less under load and is therefore better; a stiffer structure resists imposed movement harder and is therefore worse. Which effect matters depends entirely on whether the action in question is a force or a displacement, and the two kinds of action live in the same structure at the same time.

The proportionality is exact and the figures verify it: doubling $EI$ multiplies the settlement field by 2.0000, which is the check the generator is held to.

Which free body produced the number #

The beam is three 7-metre spans, uniformly loaded at 5 per metre, with the second support pushed down 10 mm.

The settlement field is computed by superposition rather than by teaching the solver about prescribed displacements, and the argument is worth following because it is the standard way this class of problem is handled.

Release the vertical restraint at the settled support, leaving a two-span beam with a free point in the middle. Apply a downward trial force of 1 there and solve; the point deflects by some amount $d$ per unit force, which is a flexibility coefficient. The force that produces exactly 10 mm of deflection is then $F = 0.010/d$, and because the system is linear, every moment in the structure under that force is the moment under the unit force scaled by the same factor.

For this beam that force comes out at 28.0, and the resulting moment field peaks at 73.5 — against a load-induced peak of 24.5. The free body behind the whole calculation is the released structure, and the released structure is determinate: what compatibility supplies is the single number that makes the released displacement match the imposed one.

The check available on it is that the settlement field must be in equilibrium with nothing. Set the load to zero and the beam still has a complete bending-moment diagram, with reactions at every support that sum to zero vertically and to zero in moment. That is a self-equilibrating stress field, and it is the same object as a state of self-stress in a redundant truss or a residual stress in a rolled section.

The assumption the figure rests on is that the beam remains elastic and the settlement is instantaneous. Both matter. A settlement that occurs over years is partly relaxed by creep in a concrete beam, sometimes by most of it; and a beam that yields locally sheds the settlement moment entirely, which is the argument the next section is about.

How big a settlement has to be to matter #

The 10 mm in the figures produces settlement moments three times the load moments, which sounds like a designed-for-failure. The scaling explains why it is not quite as bad as that, and where the real threshold sits.

The settlement moment goes as $EI\delta/L^2$ and the load moment as $wL^2$, so the ratio between them goes as $EI\delta/wL^4$. The fourth power of the span is in the denominator, which is the same $L^4$ that governs deflection under load, and it means the sensitivity to settlement falls very fast as spans get longer. A long-span beam barely notices a support movement; a short stiff one is dominated by it.

That is the practical rule and it is the opposite of what an intuition about “big structures, big problems” would suggest. Settlement is a short-span, stiff-structure problem. Transfer structures, ground beams, stiff shear cores adjacent to flexible frames, and the short end bays of otherwise long-span floors are where it bites — all of them stiff, all of them short, and all of them commonly founded differently from what they connect to.

It has a sign, and both signs are load cases #

Every figure here settles the same support downward, which makes the settlement field look like a fixed property of the beam. It is not. It is a signed action, and the sign decides whether it adds to the load moments or cancels them.

Settling an interior support of a continuous beam downward is, as far as the structure is concerned, partly releasing a prop. The hogging moment over that support falls; the sagging moments in the two spans either side rise. Jack the same support upward and everything reverses: the hogging over it grows and the adjacent sagging shrinks.

So a settlement that is unhelpful for one section is helpful for its neighbour, and the two are the same event. The figures above show the settlement field dominating at 73.5 against 24.5, and what that number does not say is that over the settled support the two fields have opposite signs and partly cancel, while a quarter-span away they add. Reversing the sign is the cleanest way to see it, because nothing else about the beam changes.

Twenty-four and a half kilonewton-metres of difference between the two, from the same ten millimetres of movement pointed the other way, and nothing in the beam or its loading distinguishes them. That is what it means to say the action is signed.

Three consequences follow and they turn a single calculation into a set of them.

Both directions have to be considered. Ground does not only settle — a support can be heaved by swelling clay, lifted by frost, or jacked during construction or repair. And even where movement is one-way, the relative movement of one support against its neighbours takes either sign depending on which of them moved most.

Each support has to be moved independently. Settling all supports equally does nothing at all: the beam translates as a rigid body and no curvature is imposed. What produces moments is one support moving relative to its neighbours, so the load cases are the individual movements, not a uniform allowance.

And the enumeration is the same shape as pattern loading. With $n$ interior supports and two signs there are $2n$ elementary cases, each producing its own self-equilibrating moment field, and the design envelope is assembled from whichever combination is worst at each station — with no single arrangement producing the whole envelope, for exactly the reason an imposed-load envelope never does.

The settlement enumeration has exactly that shape with the loads replaced by support movements, and it has one property the load envelope does not: a load arrangement is a choice about occupancy, and a settlement pattern is a fact about ground nobody has measured. So the number of cases is the same and the confidence in each is lower.

Which is why a specification that says “allow for 25 mm of settlement” has not specified a load case. It has given a magnitude and left out the pattern, and the pattern is where all the structural content is.

What makes it survivable #

The tone of this essay so far has been alarming, and the practice is not. Buildings settle differentially all the time and the great majority are unaffected. Three reasons, and they are the same three that make every imposed-deformation problem tolerable.

It is self-equilibrating. The settlement moments have no net resultant. They cannot cause a collapse mechanism on their own, because a mechanism requires work to be done by external loads and there are none.

It sheds when anything yields. The settlement moment exists because the structure is forced into a curvature. Let a section yield and the curvature is accommodated plastically at a constant moment; the moment stops growing and, as the rest of the structure redistributes, falls away. This is why plastic analysis ignores settlement entirely — at the collapse limit state it has gone, and the collapse load of a ductile structure is genuinely independent of how its supports have moved. That is one of the more remarkable results in the subject and it is a direct consequence of the upper-bound theorem.

It relaxes with time. Concrete creeps, soil consolidates slowly enough that much of the movement occurs during construction while the structure is still being adjusted, and steel connections slip. The sequence matters as much as the total: settlement that happens before a floor is cast is settlement the floor never experiences, which is why construction sequence appears in the calculation at all and why a structure’s history is part of its state.

The residue of concern is therefore in three specific places: at serviceability, where cracking and finishes care about the moment now rather than at collapse; in brittle structures, which cannot shed; and in prestressed or precast structures, where the design is finely balanced and a redistribution of reactions can uplift a support.

That is the mechanism by which a settlement moment disappears, drawn for a load rather than for a settlement, and the drawing is the same because the argument is. A support moment that the structure declines to carry is redistributed to the spans, at the price of a rotation; a settlement moment is a support moment the structure was never asked to carry by anything external, so shedding it costs the same rotation and leaves nothing behind at all. The rotation is the quantity to check, and it is the reason ductility rather than strength is what makes a redundant structure indifferent to its foundations.

The same argument, four other actions #

Settlement is the clearest member of a family, and every member behaves identically because the mechanism is the same: a deformation imposed on a structure that is not free to accommodate it.

Temperature. The load path for a temperature action is the restraint rather than the ground: a restrained member that is heated cannot expand, so it develops a force $EA\alpha\Delta T$ — proportional to stiffness, unrelated to load, and capable of being very large indeed. A 30 metre restrained steel member at 30 degrees of temperature change carries a stress of about 70 N/mm² whatever its size.

Shrinkage. Concrete shrinks and its reinforcement does not, so the bars restrain the concrete and put it into tension with no external action at all.

Prestress. The same phenomenon used deliberately: a deformation imposed on the structure to create a stress field chosen in advance. In a redundant structure this produces secondary moments — the “parasitic” moments — which are precisely settlement moments generated on purpose, and which have to be computed by the same superposition.

Lack of fit. A member fabricated 5 mm too short and forced into place in a redundant frame produces a complete self-equilibrating force system before the structure has been loaded.

Where the model stops #

Linear elastic. Everything superposes only because the structure is linear. Once anything yields or cracks, the load case and the settlement case interact and cannot be added.

Instantaneous. Real settlement is a time history, and the structure’s response to it depends on the creep occurring over the same period. For concrete the effective moments can be half the elastic prediction or less.

Known settlement. The calculation takes 10 mm as given, and the number is the least reliable quantity in the whole exercise. Everything downstream of it is computed to four figures from an input known to perhaps one, which is a common and slightly absurd shape for a structural calculation to have — and the correct response is not more precision in the analysis but a check of what happens across the plausible range. Predicting the actual differential settlement of a foundation is a geotechnical problem of considerably lower precision than anything in this essay, and the honest treatment is usually a range rather than a value.

Elastic supports throughout the rest of the structure. The calculation moves one support and holds the others exactly. Real foundations are all springs, so a settlement at one is accompanied by smaller ones everywhere else, and the differential — which is what matters — is smaller than the absolute movement by however much the others follow.

One direction. Only vertical movement is considered. Rotation of a footing, or horizontal spread — which is what an arch fears most — has the same character and different arithmetic.

The figures share a limitation, and it is the standard one for anything about deflection: the settlement is drawn at a scale that makes it visible, and it is 10 mm on a 21 metre beam. Drawn to scale the settled support would be a fifth of the thickness of the line representing the beam, and the deflected shape would be indistinguishable from a straight line. The moments the figure reports are real and the geometry it draws them on is exaggerated by a factor of several hundred.

The generalisation #

The division that organises all of this is between actions that are forces and actions that are displacements, and it deserves to be a habit rather than a fact.

A force action produces effects proportional to the force and independent of stiffness. A displacement action produces effects proportional to stiffness and independent of any force. Load, self-weight, wind and snow are forces. Settlement, temperature, shrinkage, creep, prestress and lack of fit are displacements. The two families require opposite instincts: to resist a force, add stiffness; to survive a displacement, remove it.

That is why an expansion joint is a reduction in structure, and why articulating a long building into short independent blocks is a stability decision made in reverse. It is why a three-pinned arch is used where the abutments are doubtful: the hinges make it determinate, and a determinate structure cannot feel a settlement at all. Choosing determinacy is choosing insensitivity, and it is bought by giving up the robustness redundancy provides — the alternative load paths that let a continuous beam survive the loss of something a determinate one could not.

There is no free position on that trade, and the interesting engineering is in noticing that the choice is being made. A designer who adds a support to reduce a span has also added a sensitivity to that support’s foundation, and a designer who makes a frame continuous for robustness has made it responsive to temperature. Neither is wrong; both are decisions, and both are frequently made without being noticed.

Navier set out the analysis of redundant structures in 1826, and the settlement problem was well understood by the 1860s — Clapeyron’s three-moment equation handles it directly with a settlement term. What arrived later was the temperament: nineteenth-century practice tended to design out indeterminacy where foundations were uncertain, using hinges liberally, and the modern preference for continuity is a consequence of welding, of better foundation engineering, and of a plastic theory that showed the settlement moments disappear at collapse.

The ladder from here #

Later rungs on this anchor: temperature and the restrained member, where the same argument has a different constant. Prestress secondary moments and the concordant profile that eliminates them. Lack of fit and the deliberate use of it to precamber a truss. Creep relaxation of imposed-deformation effects. Foundation stiffness as a spring rather than a rigid support, which is what a real analysis does and which converts settlement from an imposed displacement into a soil-structure interaction. And the robustness argument for redundancy, which is the counterweight to everything here and is the reason continuity is the default despite all of it.