The building the trough cannot bend

The crack is a strain, not a slope took Burland and Wroth’s view of a masonry building as a deep beam: damage begins when the tensile strain in it reaches a limit, and the strain follows from the building’s deflection ratio — the largest departure of its settled shape from the chord between its ends, over its length. The criterion converts a deflection ratio into a damage category, and the deflection ratio is what has to be predicted.

The usual prediction is the greenfield one. A tunnel or an excavation leaves a settlement trough in ground with nothing on it; the trough’s deflection ratio over the length a building occupies is read off, and the building is assumed to take it. That essay ended by naming the step that assumption skips: the building has a stiffness of its own, and the settlement that matters is the difference between what the ground does and what the building lets it do to it.

A beam on the springs the trough moves #

The building is a beam of length $B$, bending stiffness $EI$ per metre of its width, resting on Winkler springs of modulus $k_s$ — a beam on the ground — and pressing on them with its own weight. The trough moves the springs’ lower ends: it is Gaussian, $s(x) = S\exp(-x^2/2i^2)$, the shape a tunnel leaves, with its inflection points at $\pm i$, sagging between them and hogging outside. Here the trough is 30 mm deep with $i = 15$ m, and the building is 30 m long, so centred on the trough it spans exactly from one inflection point to the other and sags throughout.

The building’s settlement follows from equilibrium between its bending and the springs’ push, $EI\,w'''' + k_s(w - s) = 0$, and one dimensionless number governs it: $\rho = EI/(k_s B^4)$, the building’s bending stiffness against the soil’s over its length.

At $\rho = 10^{-4}$ the building lies on the trough and takes 95% of its deflection ratio. At $10^{-3}$ it settles less in the middle and more at the ends, and takes 66%. At $10^{-2}$ it settles almost as a rigid body, a little more than the trough at its ends and a good deal less in the middle, and takes 17%. The stiff building bridges the trough: it leaves the trough’s curvature to the ground under it and moves down nearly as one piece.

Why one number decides it #

A beam on springs has a length of its own, the distance over which a disturbance at one point dies away along it: $1/\beta$, with $\beta = (k_s/4EI)^{1/4}$, which a beam that sits on the ground found sets everything about it. A building several of those lengths long is a long beam, whose middle cannot feel its ends and follows the ground beneath it; a building shorter than one of them is a short beam, which moves as a whole.

The relative stiffness is that comparison in other clothes. $\beta B = (1/4\rho)^{1/4}$: at $\rho = 10^{-4}$ the building is 7.1 of its own decay lengths long and follows the trough; at $10^{-3}$, 4.0; at $10^{-2}$, 2.2, and it is starting to act as one piece. The two decades of the transition are a factor of about three in $\beta B$, because $\rho$ enters to the quarter power — which is why the transition looks wide on a scale of stiffness and is narrow on a scale of length.

Two decades from following to bridging #

The ratio of the building’s deflection ratio to the ground’s — the modification factor — is a function of $\rho$ alone, for a given trough and position.

The whole change happens across two decades of stiffness, from $10^{-4}$, where the building follows the ground, to $10^{-2}$, where it has stopped following it. A building’s relative stiffness is easy to place on that scale roughly and hard to place on it exactly: a masonry wall’s effective $EI$ depends on its openings and its cracking, and the soil’s modulus under a shallow footing is one of the least well known numbers in the whole calculation. Within the two decades, a factor of three on either moves the answer a long way: a building estimated at $10^{-3}$ takes 66% of the ground’s deflection ratio, and the same building at a third of that stiffness takes about 85%, at three times it 40%. The building and the ground are two paths sharing one deformation, and the share is only as well known as the ratio of their stiffnesses.

In this model sagging and hogging are modified alike. That is a property of Winkler’s springs, which pull as readily as they push and couple nothing along the ground; analyses that treat the soil as a continuum find a hogging building reduced more than a sagging one, because the soil beside a hogging building’s ends carries some of its weight away from the trough. The Winkler model is the simplest that contains the interaction at all, and it is a lower bound on how much the building helps itself.

What bridging costs #

A building that does not take the ground’s curvature has to carry it some other way, and the way is bending.

The soft building’s moment is small because it bends with the ground and carries almost nothing. As the building stiffens, its moment rises: 385 kN·m per metre of width at $10^{-3}$, 967 at $10^{-2}$. It levels at 1,159 kN·m, which is the moment a perfectly rigid beam would carry on springs whose ground has moved — the springs push hardest where the trough has fallen least, at the ends, and least in the middle, and the building spans between those pushes.

So a stiffer building is not a safer building in the same sense. A soft building takes the trough’s curvature as strain in its walls and finishes, which is what Burland and Wroth’s criterion measures. A stiff one takes it as bending in its structure, which is what its foundations and frame are designed for, or are not. The damage criterion moves from cracking to strength as the building stiffens, and a building in the middle of the two decades has a share of each.

The same building, short and long #

The relative stiffness contains the building’s length to the fourth power, and that is where the most consequential part of the answer is.

Hold the building’s stiffness per metre fixed and change its length. At 12 m it takes 4% of the ground’s deflection ratio; at 30 m, 67%; at 60 m, all of it. Doubling the length divides the relative stiffness by sixteen, which moves it more than a decade along the curve. The same construction is stiff when short and soft when long, and a terrace of houses built as one long wall takes the trough’s curvature where each house on its own would have bridged it.

That is the arithmetic of span to the fourth applied to a building lying down: a beam’s flexibility goes as its span to the fourth, and here the trough’s curvature is the load. It is also why movement joints in a long building are a settlement measure as well as a thermal one: cutting a 60 m wall into two 30 m lengths moves each from the ground’s deflection ratio to two-thirds of it, and into four 15 m lengths to a tenth of it.

Where the building stands #

The trough is not uniform, and the building’s position in it decides what it is offered before its stiffness decides how much it takes.

Centred, the building sits in the sagging part of the trough and is offered a deflection ratio of 0.39 thousandths, of which it takes 0.26. Wholly outside the inflection point it sits on the hogging shoulder, offered 0.17 and taking 0.11. Between them, with its centre near the inflection point, it straddles the change from sagging to hogging, the trough beneath it is nearly straight, and the deflection ratio it is offered falls to 0.06 — of which it takes almost none.

The building standing across the inflection point is not moved less; it tilts more than either of the others, because that is where the trough is steepest. But tilt is a rigid-body movement, and a building does not crack from tilting. The damage criterion measures curvature, and the least curved part of a trough is its steepest. A tunnel’s alignment cannot often be chosen to put buildings on its inflection points, but the trough’s width can be influenced by the depth of the tunnel, and a deeper tunnel’s wider trough moves every building towards the gentler part of the curve.

Over the shoulder #

The hogging building deserves its own picture, because hogging is what damages masonry most.

On the shoulder the trough is convex upward beneath the building, which puts the top of a wall in tension — the case Burland and Wroth found twice as damaging as sagging for the same deflection ratio, because a wall’s top has no foundation to hold it together. A soft building follows the convexity. A stiff one tilts with the trough’s slope — 0.58 mm per metre against the ground’s 0.60 — and stays straight. The stiffness that bridges a sagging trough also rides over a hogging one.

It does so for less. A rigid building on the shoulder levels at 505 kN·m per metre against 1,159 in the middle of the trough, because the shoulder’s curvature is gentler — the greenfield deflection ratio there is 0.17 thousandths against 0.39 — and its sign is reversed, so the building carries it as hogging moment over its middle. The hogging building is the more vulnerable one if it is soft and the less loaded one if it is stiff, and which it is depends again on where along the two decades it sits.

Where the factor enters an assessment #

Building damage from tunnelling is assessed in stages, and the modification factor belongs to the last of them. The first screens every building with the greenfield trough alone: a building whose greenfield settlement and slope are below small limits is set aside, whatever its stiffness. The second applies the greenfield deflection ratio and horizontal strain to each remaining building as though it were perfectly flexible, and converts them to a damage category by Burland and Wroth’s deep-beam criterion. Most buildings stop there, because the greenfield assumption is conservative: it gives every building the ground’s curvature in full.

Only the buildings that the second stage puts in a serious category are taken to the third, where their stiffness is estimated, the modification factor is applied, and their own structure — openings, foundations, the continuity of their walls — enters the calculation. The two decades of the factor’s curve are where that third stage earns its cost: a building placed near $10^{-2}$ instead of at the greenfield assumption has its deflection ratio cut by a factor of six, which is often a damage category or two.

The order matters because the factor cannot make a building worse. A soft building is assessed correctly by the greenfield trough; a stiff one is assessed conservatively by it. The risk in skipping the third stage is cost, not safety — with the one exception the moment figure makes: a stiff building assessed as flexible is assessed for the wrong failure.

The relative stiffness, by hand #

For the 30 m building at $\rho = 10^{-3}$ on soil of $k_s = 5{,}000$ kN/m³, the bending stiffness per metre of width is

$$EI = \rho\,k_s B^4 = 10^{-3} \times 5{,}000 \times 30^4 = 4.05 \times 10^6\ \text{kN·m}^2\ \text{per m}.$$

A masonry wall 0.3 m thick and 10 m high, uncracked, with a modulus of 3,000 N/mm², has $EI = 3 \times 10^6 \times 0.3 \times 10^3/12 = 7.5 \times 10^7\ \text{kN·m}^2$ — a wall every 15 m of a building’s width gives $5 \times 10^6\ \text{kN·m}^2$ per metre, a relative stiffness of about $10^{-3}$, in the middle of the transition. Openings and cracking can take a factor of ten off it, and a stiff raft or a concrete frame with infill can add one. That is the range in which real buildings sit, and it is the range in which the answer moves most.

The rigid building’s moment, by hand #

The moment the stiff building levels at can be found without the beam at all. A rigid building centred on the trough settles uniformly, by the mean of the trough under it — the springs must carry its weight and nothing else, so the settlement they average is the building’s. Over 30 m the Gaussian trough’s mean is 25.7 mm, between its 30 mm centre and its 18.2 mm ends.

Each spring then pushes up with $k_s(\bar w - s)$: more than the building’s weight at the ends, where the ground has fallen only 18 mm and the building 25.7, and less in the middle, where the ground has fallen 30. Those pushes are a self-equilibrating load on the building — upward at the ends, downward in the middle relative to its weight — and the moment they make at the centre is

$$M = k_s \int_0^{B/2} (\bar w - s(x))\,x\,dx = 1{,}159\ \text{kN·m per m},$$

the figure’s level to the kilonewton-metre. It is the trough’s own curvature, converted to a moment by the soil’s stiffness. For a trough wide against the building the integral is $k_s S B^4/384\,i^2$ — 1,406 kN·m here, which the Gaussian’s flattening towards the ends brings down to 1,159 — so the price of bridging grows as the fourth power of the building’s length, the same power that decides whether it bridges at all. It is paid in full by the stiffest buildings and in part by everything in the transition.

Springs, a beam, one direction #

The model is the simplest that lets the building and the ground interact, and it leaves out what the full problem has.

Winkler’s springs. The soil is a spring only by approximation: it is a continuum in which a load at one point settles its neighbours, and its stiffness depends on the footing’s width; Winkler’s springs do neither. The continuum spreads the building’s effect along the trough and treats hogging and sagging differently.

No horizontal strain. A trough also stretches the ground horizontally on its shoulders, and a building’s axial stiffness resists that much more effectively than its bending stiffness resists curvature; the horizontal strain the building takes is a second modification factor, and it is often the larger reduction.

A beam. A building is a deep beam that shears as well as bends, which is what the Burland and Wroth criterion was built on; a building soft in shear takes more of the trough’s curvature than its bending stiffness says.

Full contact. The springs here release where the ground falls away from the building, but under a building of ordinary weight over a trough of 30 mm none do: the ends the ground was holding down stay held, and the building’s weight keeps it in contact with a trough it is bridging. A light, stiff structure — a single-storey frame on a stiff slab, or a pipeline — over a deeper trough is the case where contact is lost, and then the building spans between the points it still touches and its moment is set by that span rather than by the springs.

Still open: the trough the building makes #

The trough here is given and the building responds to it. But a building is also a load on the ground through which the tunnel is being driven, and the trough under a building is not the greenfield trough at all: the tunnel’s volume loss is drawn from ground that is already carrying the building’s weight, and a stiff building redistributes that weight towards its ends as the trough develops. Whether the building’s stiffness changes the trough it sits in as much as the trough changes the building — and whether a tunnel driven under a stiff building should be expected to produce a narrower, deeper trough than the one predicted for open ground — is the question of whether the greenfield trough is the right input at all, or only the right input for the soft buildings that need it least.