The settlement that matters is the difference

Every building settles. The interesting question is not how far.

A structure that goes down uniformly is undamaged: no member changes length relative to its neighbours, no beam picks up a rotation it did not have, nothing cracks. It has a step at the front door that needs rebuilding and a drain that runs the wrong way, and structurally it is the building it was.

What damages it is the difference between one foundation and the next. And the measure that predicts damage is not the difference either — it is the difference divided by the distance over which it occurs.

$$\beta = \frac{\delta_{i+1} - \delta_i}{L}$$

the angular distortion, which is a rotation the structure has been given without being asked.

Which free body produced the number #

The superstructure, taken as a beam sitting on springs.

The frame spans between its own foundations, so it is not a passenger. Each footing pushes up with $k_i v_i$, the frame bends between them, and the whole thing is an elastic beam on discrete elastic supports — which is the machinery this site already has for a beam on bearings, applied one level down.

Three quantities come out and they behave very differently.

The mean settlement is fixed by statics. Vertical equilibrium says $\sum k_i v_i = W$. If every $k_i$ is the same, the mean of the $v_i$ is $W/nk$ and no arrangement of the frame’s stiffness can move it. Computed here: 24.0 mm for a frame at a thousandth of the soil’s stiffness and 24.0 mm for one at a thousand times it, agreeing to $3\times10^{-10}$.

The differential settlement falls as the frame stiffens. The tilt across the building goes from 66.5 mm to 0.04 mm across the same range.

And the bending moment rises to meet it. 237 kNm to 1,665 kNm — a factor of seven, and still climbing at the stiff end.

The trade, drawn #

Sweep the frame’s bending stiffness against the soil’s, holding everything else, and both curves are monotone in opposite directions:

There is no stiffness at which both are small. The frame either accepts the movement or carries the force that prevents it, and the choice between them is what “designing for settlement” actually means.

The right-hand column shows the mechanism. A limp frame gives the soft support 22.8% of the total load — near its tributary share of 25% — while a rigid one gives it 8.1%. The frame is bridging over the soft patch, and the load it takes off that footing has gone onto the neighbours, along with the moment required to carry it there.

That column is the stiffest path takes the load with the soil as one of the paths. A stiff frame on soft ground routes load away from the soft support and onto the firm ones, and the routing is free in the sense that nobody chose it and expensive in the sense that a moment came with it. Nothing in the design intended a bridging action; it appeared because one of the two stiffnesses in series happened to be the larger.

Why the criterion is a rotation #

Buildings do not crack because they moved. They crack because something in them was rotated past what it could take.

A partition sitting between two floors that differ in level by $\Delta$ over a length $L$ has been sheared by $\Delta/L$. A brittle finish accommodates about one part in five hundred before it cracks visibly and about one in three hundred before the crack is structural. A frame’s members are more tolerant — cladding distortion and door frames generally complain first.

Which gives the numbers the whole subject is written in:

The building drawn is at 1 in 231, which fails the first and is approaching the second. Its mean settlement of 31 mm is entirely unremarkable — a building can settle 100 mm uniformly and nobody will know.

Two buildings with the same settlement can be in completely different conditions, and the number that separates them is a slope.

None of that is a strength check. A settlement calculation and a strength calculation are two entirely separate criteria applied to the same frame, and on a building of ordinary proportions on ordinary ground it is the first that binds — which is the ordinary situation described in stiffness is not strength, arriving here from the foundations rather than from the span.

Sagging and hogging are not the same damage #

The profile drawn dishes downward in the middle, which is the commonest shape and the more forgiving one. A building that hogs — settling more at its ends than at its centre, which happens over a stiff lens or where the ends are more heavily loaded — is in worse trouble for the same distortion.

The reason is where the tension is. A sagging profile puts the building’s top in compression and its bottom in tension, and the bottom of a building is generally a stiff, well-reinforced ground floor with a raft or a heavy beam in it. A hogging one reverses that: the tension is at the top, where there is nothing but the roof and the cladding, and a masonry facade in tension cracks at a much smaller distortion than one in compression.

The usual allowance is a factor of two: a hogging profile is treated as twice as damaging as a sagging one at the same angular distortion. That is a rule rather than a derivation, and what is worth carrying is why a rule is needed at all — the damage criterion depends on where the building’s own tension capacity is, which is a property of the structure and not of the ground.

The timing matters as much as the magnitude #

A settlement that happens before the finishes are installed does no damage at all. One that happens afterwards does all of it.

On sand, most of the settlement occurs as the load is applied — during construction, before the partitions are up — so the damaging fraction is small. On clay, consolidation takes years, and nearly all of it happens after the building is occupied.

That single distinction is worth more than any refinement of the calculation. The same 40 mm of differential settlement is harmless on one soil and cracks the building on the other, and the difference is entirely in the timing.

It is also the reason a construction sequence is a settlement variable. Building the heavy core first and the light perimeter later lets the core’s settlement happen before the connection between them is made — the same manoeuvre that delayed connections use for differential shortening, applied to the ground instead of to creep.

The same geometry appears one level up in the columns are shorter than the core: two vertical elements changing length by different amounts, connected by floors that have to accommodate the difference. Settlement and shortening are one problem with the movement happening at opposite ends of the column, and neither of them is visible in a frame analysis that applies its loads all at once.

And what a partition actually sees is not the settlement but the settlement after the day it was built, which is a different curve with a different peak — the running total that built to the wrong shape is written about, with consolidation supplying the increments instead of casting stages.

The raft, which changes the question #

Everything above assumes separate footings. Put the building on a raft and the problem changes shape, because the raft is a continuous beam on a continuous elastic foundation rather than a frame on discrete springs.

The mechanism is the same and the governing number becomes a relative stiffness of the classic kind:

$$K_r = \frac{E_s I_s}{E_{soil} B^3}$$

with the structure’s own stiffness over the soil’s. Below about 0.01 the raft is flexible and follows the ground; above about 0.5 it is rigid and settles as a plane. In between it does both, and the moments in it are a function of that ratio rather than of the load.

The uncomfortable consequence is the same one: a rigid raft has almost no differential settlement and enormous bending moments in it, and a flexible one is the other way round. The raft’s thickness is chosen on that trade, not on the pressure under it.

The characteristic length in that figure is the raft’s own property, and the ground supplies only one of the two numbers that make it.

A settlement is a support that moved #

There is a shorter way to hold all of this, and it connects to something this site established long ago.

A support that moves in a determinate structure causes no forces at all — the structure simply takes up a new position, and every internal force is exactly what it was. In an indeterminate one it causes a complete field of moments in equilibrium with no applied load whatever, proportional to $EI$.

Differential settlement is precisely that, with the movement supplied by the ground rather than by a jack. So the whole trade above is a restatement of the redundancy result: a stiffer structure attracts more force from an imposed displacement, and a determinate one attracts none.

Which gives the design move the last section was circling. A building that cannot be made stiff enough to bridge the soft patch can instead be made less redundant — a movement joint through the full height, articulated foundations, a determinate arrangement of transfer beams — and it will then follow the ground with no force at all. The two strategies are the two ends of the same curve and there is nothing in between except compromise.

What follows once the frame starts to crack is the redistribution nobody wrote down working in the structure’s favour for once. A settlement moment is an imposed-deformation effect rather than an applied load, so the first hinge that forms relieves it instead of passing it on — the deformation is satisfied by the hinge and the moment simply stops being required. That is why these effects are so much less dangerous in practice than a linear analysis says, and it is the same asymmetry that makes a thermal moment safer than a wind moment of the same magnitude.

The settlement caused by somebody else’s building #

Everything above is a building settling under its own weight. The commonest way a building is actually damaged by settlement is not that at all: it is settling because of something built beside it.

Three mechanisms, and all three reach sideways.

Load spread. A new building’s weight spreads laterally through the ground, so it raises the vertical stress under its neighbour as well as under itself. The increment falls away with distance but reaches a good fraction of the new building’s own width.

Excavation. Digging beside an existing structure removes the lateral support the ground had, and the soil moves toward the hole. What appears at the surface is a settlement trough, deepest at the excavation and dying away over a few times the excavation’s depth. Tunnelling produces the same trough from below.

Dewatering. Lowering a water table raises the effective stress in everything above it, over an area far larger than the works — and consolidates it. This one has no boundary at all except the extent of the drawdown.

How far any of them reaches is the same characteristic-length question the raft section asked, read the other way round.

What makes all three worse than self-weight settlement is the timing argument two sections above, with the fraction set to one. An existing building has already consolidated under its own load. Its settlement history is complete, its finishes were installed against the shape it has now, and every subsequent millimetre is post-installation. Where a new building on clay might deliver half its differential settlement before the partitions go up, a neighbour delivers all of it afterwards, to a structure with no remaining capacity to absorb it.

The profile is unfavourable too. A settlement trough is deepest at the near face and shallow at the far one, so the existing building is tilted and hogged rather than dished — and the previous section priced hogging at twice the damage of sagging at the same distortion, because the tension goes into the top of the facade where there is nothing to carry it.

Which is why this is the one settlement problem with a whole industry of procedure around it: condition surveys before the work, levelling and crack monitoring during it, trigger levels agreed in advance, and a contractual mechanism for stopping. The observational method described above exists mostly for this case, because it is the case where the trigger has to be read by somebody who can halt the excavation.

It is also the one place in this collection where a movement is undone while it is happening. Compensation grouting injects grout into the ground between the works and the building, in measured quantities, at the rate the settlement is arriving — jacking the structure back up as fast as the excavation lets it down. It is controlled by the monitoring rather than by a design, in a loop with a person in it, and it has been used to hold buildings within a millimetre or two while a tunnel was driven beneath them. Everywhere else in this subject a deformation is predicted, accommodated or resisted; here it is cancelled, continuously, by an operator watching an instrument.

And the shape of it is one this collection has met before in another field. The load case belongs to somebody else’s project, arrives on a programme the owner does not control, and is not present in any calculation the building was designed against — which is exactly the position a low roof beside a tall building is in when a neighbour builds tall and starts collecting its drifted snow. The two hazards are unrelated mechanically and identical in their governance: a structure correctly designed for an open site, made wrong by a development next door, with the damage arriving through a route the original design had no reason to consider.

Where the model stops #

The soil is a set of independent springs. It is not: pressing one footing down pulls its neighbours with it, because the soil is a continuum. That coupling makes the real settlement bowl wider and shallower than a Winkler model gives — so the model above overstates the local differential and understates the overall dish. Getting it right needs an elastic half-space rather than springs.

The spring stiffnesses are known. The subgrade modulus is the least reliable number in foundation engineering: it is not a soil property at all, it depends on the size of the loaded area, and quoting it to two significant figures is an act of optimism. Everything on this page should be read as a sensitivity study rather than a prediction.

And the frame is linear and elastic. A real frame cracks, its joints yield a little, and a settlement applied over ten years is resisted with a creep-reduced stiffness rather than a short-term one — all of which move it toward the flexible end of the trade and reduce the moments it actually carries. Settlement effects are self-relieving, which is why buildings survive differential settlements that a linear analysis says should have destroyed them.

What the pictures cannot show #

The settlement profile is drawn as a smooth line through five points. Real ground varies over metres, so the true profile has features between the footings that no calculation on a five-spring model contains.

Nor can the figures show time. Every diagram here is a final state, and the thing that damages a building is a rate as much as a magnitude — a settlement delivered over a decade is accommodated by creep and one delivered in a week is not.

The assumption the figure rests on #

The soft patch is one support at 35% of the others’ stiffness, and the answer depends on that entirely. Real variability is not a patch under one footing; it is a field, and the worst arrangement is rarely the one somebody guessed. What is drawn is a load case rather than a prediction, and the honest use of it is to run several — a soft patch under each footing in turn, and a smooth tilt across the whole plan — and design for the envelope.

The measurement that settles it #

None of this is predictable to better than a factor of two, and the profession’s response is worth recording because it is unusual: it measures.

Levelling points are installed on the columns before the frame is complete and read at intervals for years. What comes back is a set of settlement histories, one per column, from which the differential and its rate are read directly — and if the readings are diverging from the predictions, the response is available while there is still something to be done about it.

That is the observational method, and it is one of the few places in structural engineering where a design is allowed to be provisional. The calculation sets the trigger levels; the measurements decide whether the contingency is needed. It exists here rather than elsewhere because the input — the ground — is the one material a designer cannot specify, test in advance, or replace.

The habit worth taking from it is narrower than the method. When the governing input has a factor of two of genuine uncertainty in it, the useful output is a trigger level rather than an answer, and a number quoted to three figures from such an input is a number pretending to be a different kind of number.

The ladder from here #

Later rungs on this anchor: the elastic continuum in place of springs, and how much the coupling changes the answer. Consolidation settlement against immediate settlement, and the timing question made quantitative. The relative-stiffness parameter $K_r$ derived rather than quoted, and the two thresholds that bound it. Damage criteria from the other direction — deflection ratio and horizontal strain, which predict cracking better than angular distortion does for a sagging profile. Piled rafts, where the piles and the raft share the load in a proportion that is itself a stiffness ratio. And the case this page keeps circling: a building whose stiffness is deliberately reduced — movement joints, articulated foundations — so that it follows the ground rather than fighting it.