The building the trough twists

The building the trough cannot bend put a building square across a tunnel’s settlement trough and asked how much of the trough’s curvature the building takes. The answer was a single number, the building’s bending stiffness against the soil’s over its length to the fourth power, and over two decades of that number a building goes from following the trough to bridging it. Every building in that essay had its long axis at right angles to the tunnel.

Few do. A tunnel is driven along a route chosen for reasons of its own, and the buildings above it stand where streets put them. A building whose long axis crosses the tunnel at forty-five degrees is the ordinary case rather than the exception, and the ground under it does not just sag along its length. A Gaussian trough is curved across its width and flat along the tunnel’s axis, so a footprint laid diagonally across it sees a surface that is bent along one diagonal and not the other. Its four corners do not settle into a plane. The ground twists a skewed building as well as bending it, and the twist is a different load, answered by a different stiffness, which the bending essay never needed.

The same trough, turned #

The trough is the one the bending essay used: 30 mm deep at the tunnel’s axis, with its inflection points 15 m either side, Gaussian across its width, $s = S\,e^{-n^2/2i^2}$ in the distance $n$ from the tunnel’s axis. The building is 30 m long and 12 m wide, three storeys and 10 m high, with masonry walls 300 mm thick, and it stands on soil whose subgrade modulus is 5,000 kN/m³ — a beam on the ground in both of the senses that follow. Its centre is over the tunnel’s axis.

Square across the trough, the building spans the sagging zone between the inflection points and its corners all settle alike: 18.2 mm each, a plane. Turned 45°, the building’s long axis runs diagonally across the trough. Two of its corners are now closer to the tunnel’s axis than the other two, and they settle 27.4 mm against 18.4. The corners close to the axis are diagonally opposite each other, A and C, and so are the ones further away, B and D. A surface with that pattern — high at one pair of opposite corners, low at the other — is the shape a flat sheet takes when it is twisted, and the measure of it is the warp: how far one corner sits off the plane through the other three, which is $s_A + s_C - s_B - s_D$. Here it is 18 mm.

Sag along it, warp across it #

The two shapes trade off against each other as the building turns. The sag along the building is the trough’s curvature along the building’s length, and the curvature of a Gaussian trough along a line at an angle $\beta$ to its transverse direction is its full transverse curvature times $\cos^2\beta$. At 45° that is a half, and the sag along the building has nearly halved, from 11.8 mm to 6.4.

The warp goes the other way. It is the trough’s curvature taken once along the building and once across it, and that mixed curvature goes as $\sin\beta\cos\beta$ — nothing at either end of the range and largest between them. At 45° it is 18 mm, and it stays above the sag for every angle from about 21° to the building lying almost along the tunnel. For most of the angles a building can make with a tunnel, the larger of the two shapes the ground hands it is the one a deflection-ratio check does not read.

That check was the subject of the crack is a strain, not a slope: Burland and Wroth’s reading of a building as a deep beam, whose tensile strain follows from its deflection ratio along its length. Applied to a skewed building it reads the sag and nothing else. What the warp does to the building depends on whether the building can follow it.

A building in two actions #

A building on springs under a warped surface can be split into two problems, provided its cross-sections keep their shape — which they do if its floors are stiff in their own planes. At each section along its length, the ground’s settlement across the building’s width has a mean and a slope. The mean settles the section and is answered by the building’s bending, exactly as in the square case: $EI\,w'''' + k_s W (w - \bar s) = 0$. The slope tilts the section about the building’s long axis and is answered by the building’s torsion: $EI_w\,\theta'''' - GJ\,\theta'' + (k_s W^3/12)(\theta - \varphi) = 0$, with $\varphi$ the ground’s slope across the width and $k_s W^3/12$ the rotational stiffness the soil under a strip of width $W$ supplies. The first term is the cost of a twist that varies along the building: if one section turns more than the next, the two long walls must bend in their own planes in opposite senses, and $EI_w = EI\,W^2/4$ is the two walls’ bending stiffness at their distance from the building’s axis. Each has its own relative stiffness. For bending it is the bending essay’s $\rho = EI/(k_s W B^4)$; for torsion it is $\tau = GJ/(k_s W^3 B^2/12)$, the building’s torsional stiffness against the soil’s over its length squared.

The second number is where this building’s form enters. Its bending stiffness is its two long walls acting as deep beams, 300 mm thick and 10 m deep, and it hardly depends on anything but the walls. Its torsional stiffness depends on whether those walls form a closed section with the floors and the roof — whether, cut across, the building is a tube.

The slit in the building #

A closed section and an open one of the same material differ in torsion by a factor that has nothing to do with how much material either uses. The slit that costs a factor of six hundred found it in a steel tube cut along its length; the same factor appears in a building. Cut across, a building whose floors and roof are fixed to its walls in shear — floors that act as a beam lying down — is a rectangular tube, 12 m wide and 10 m high with 300 mm walls, and Bredt’s formula gives its torsional constant as $4A^2 t/\oint ds$ — four times the enclosed area squared times the thickness over the perimeter, about 393 m⁴. A building whose floors bear on the walls without being tied to them in shear is the same walls with no tube, and its torsional constant is the open-section sum $\sum b t^3/3$: about 0.40 m⁴.

The same walls, with and without floors that close them, differ in torsional stiffness by a factor of about a thousand. Their bending stiffness does not change at all. Against this building’s soil, the closed box has $\tau = 0.47$ and the open one $\tau = 0.00047$, while its bending relative stiffness, the same for both, is $\rho = 0.0021$.

The two curves have the same shape and are not in the same place. In bending, the building is in the middle of its transition, taking about half of the ground’s sag along it. In torsion, the closed box is well into the stiff side of its own transition and takes under a fifth of the ground’s peak twist rate, while the open box sits on the flat at the soft end and takes 87 per cent of it. The flat is not at one because of the walls’ warping stiffness, which no change in $GJ$ removes: it is what the next figure shows. A building can be soft in bending and stiff in torsion, or the reverse, and which it is in torsion is decided by its floors.

It is worth seeing why the torsion curve sits further right. The soil’s resistance to the section tilting goes as $W^3$ — a strip’s rotational stiffness is its width cubed — while its resistance to the section settling goes as $W$. A building 12 m wide on a 30 m length is held against tilting by a soil far stiffer, in proportion to its length, than the soil holding it against bending, so the building’s own torsional stiffness has to be correspondingly larger before it counts for anything.

Where the twist goes #

Along the building, the ground’s twist rate is largest at the centre, where the building crosses the tunnel’s axis and the trough is most strongly curved, and falls away toward its ends. The open building does not follow that shape. It twists at a nearly uniform rate, about 5.6 × 10⁻⁵ per metre, which is the ground’s twist averaged along the building: its end sections finish turned 1.67 milliradians from each other, a little more than the ground’s 1.53. A uniform twist costs an open section almost nothing, since each long wall simply tilts in its own plane. A twist that varies costs it the opposite bending of its two long walls, and that is a stiffness the walls do have: it is the section that cannot stay flat, warping torsion, at the scale of a building.

The closed building smooths the ground’s twist into a gentle arc under a fifth of its height: its sections rotate together, and the difference between their rotation and the ground’s is taken by the soil springs pushing and pulling across the width.

Both are correct responses, and they move the problem to different places. The open building takes the twist’s average by letting its walls go separately. Each long wall tilts in its own plane, each end wall tilts in its own, and at the top, where nothing ties them, they twist a little out of their planes to meet at the corners. Every one of those movements is one that a free-standing masonry wall makes easily, and the strains it puts into the walls are very small. The open box is not damaged by the twist; it simply stops being a box at the top, which is not a problem until a floor or roof that depends on the walls staying square finds them out of square by the warp’s share at its height.

The closed building cannot do that, because its floors keep its sections square. It holds the twist off, and what it holds off it carries.

Off the tunnel’s axis #

A building centred over the tunnel’s axis is the symmetric case, and in it the two long facades are mirror images: whatever the twist does, it does to each of them alike. Move the building off the axis and that symmetry goes. Its two long facades now stand at different distances from the tunnel, and the ground under them is bent differently — 6 m off the axis, at 45°, the deflection ratio of the ground under one facade is 1.1 × 10⁻⁴ and under the other 2.2.

Neither building passes that difference on in full. Both are bent by the mean of the two, through their bending stiffness, and both halve it. But the open box, which takes the ground’s twist without resisting it, also takes the part of the twist that varies along it, and a twist that varies along an open box is exactly the thing that bends its two facades in opposite senses. So its facades separate: 0.96 against 0.67 at 6 m, and 0.51 against 0.14 at 12 m. The closed box holds the twist off and keeps its facades together — 0.84 against 0.80, and 0.35 against 0.30.

The difference is modest in absolute terms here, because the trough is a mild one. But its direction is the useful part. In the open box, the worse of the two facades is worse than the building’s own bending says; an assessment that reads the building as one deep beam averages the two facades and misses the one that is carrying more. In the closed box, the averaged reading is close to right for both, and the twist shows up somewhere else.

The torque the closed box carries #

What the closed box does not take of the ground’s twist, the soil under it pushes against: one side of each section is pressed harder and the other relieved, and the couple of those pressures is a torque on the building. At 45° it reaches about 3,500 kN·m at the building’s centre. In a closed section a torque travels round the walls as a shear flow, the same at every point of the circuit, as it does round a box girder’s cell, of $q = T/2A$ — here about 14.5 kN per metre of the building’s length. That flow passes through each long wall in its own plane, as a shear stress of 48 kPa in a 300 mm wall, and through each floor and the roof. And it passes from each floor into each wall through whatever joins them.

The 48 kPa is not small for masonry. The initial shear strength of a bed joint with no compression across it — the value that governs the upper part of a wall, where little weight bears on the joints — is between 100 and 200 kPa for ordinary mortars in EN 1996-1-1, and old lime mortars are at or below the lower end. A third storey’s bed joints carrying half their strength in shear from a settlement that is a tolerable 30 mm deep are not cracked, but they are not untouched either.

The joints between floor and wall are the weaker link. A timber floor bearing on a masonry wall through joists in pockets — a floor that one code calls rigid and another does not — is held to it by friction and by whatever ties a builder fitted, and the 14.5 kN/m that a closed box needs to pass across that joint is a demand those connections were never designed for. If they slip, the box opens, and the building becomes the open box, which takes the twist and is barely stressed by it — and whose facades, off the tunnel’s axis, start to bend apart. The thing that makes the building stiff against twist is the thing the twist loads, and its failure is not a collapse but a change of state: a building that was a tube becomes a set of walls.

The stress falls less than the thickness rises. A 450 mm wall carries 34 kPa where a 300 mm wall carries 48, a reduction to 0.70 for a thickness one and a half times as great, because the thicker walls make a stiffer tube, which takes less of the twist and so holds more of it off. It is the same trade the bending essay found for a stiffer building’s moment: stiffness is not free, because it is paid for in force.

Bredt’s box and the trough’s twist, by hand #

The torsional constants are Bredt’s and Saint-Venant’s. Closed: an enclosed area of 12 × 10 = 120 m², a perimeter of 44 m and a wall 0.3 m thick give 4 × 120² × 0.3 / 44, about 393 m⁴. Open: four plates totalling 44 m of length, 0.3 m thick, give 44 × 0.3³/3, about 0.40 m⁴. Their ratio is about a thousand.

The ground’s twist rate at the centre of a building at 45° over the tunnel’s axis is the mixed curvature of the trough there, $\sin\beta\cos\beta \cdot S/i^2$, which is a half of 0.03/225, or 6.7 × 10⁻⁵ per metre; the figure’s peak is 6.5, a little less because the building’s width averages the slope over 12 m of a curving surface.

The relative stiffness of the closed box in torsion: with masonry at E = 2,000 MPa and G = E/2.6, $GJ$ is about $3.0 \times 10^8\ \mathrm{kN\,m^2}$; the soil’s rotational stiffness under a 12 m strip is 5,000 × 12³/12, or 720,000 kN·m per radian per metre; over a 30 m length that gives 3.0 × 10⁸ / (720,000 × 900), about 0.47.

And the wall stress follows from the torque: 3,500 kN·m over twice the enclosed area, 240 m², is 14.5 kN/m, which over a 0.3 m wall is 48 kPa. The open box’s warping stiffness, the two long walls’ EI at 6 m either side of the axis, is $EI\,W^2/4$, about $3.6 \times 10^9\ \mathrm{kN\,m^4}$. Against its Saint-Venant stiffness of $3 \times 10^5\ \mathrm{kN\,m^2}$, warping is the larger for any variation of twist shorter than the square root of their ratio, about 110 m — so over a 30 m building it is the only torsional stiffness the open box has, and it is why the box twists evenly rather than following the ground.

Where the picture stops #

The sections keep their shape. The decoupling into bending and torsion needs floors stiff in their own planes, so that a cross-section rotates as a whole. That is exactly the assumption that defines the closed box. For the open box it stands in for the end walls and cross walls, which tie the two long walls together at intervals; between those walls an open box’s sections can distort, and what its walls do at the top, where they twist slightly out of plane, is outside this model and is described in words rather than computed.

The springs are bilateral and the soil is Winkler. A building pressed harder on one side of each section and relieved on the other is relieved by less than its own weight here, so its springs keep their contact. A deeper trough, or a lighter building, would lift a strip along one edge, and the torque the soil can apply would be capped.

The trough is the greenfield one. A building that resists the ground’s twist also changes it, as the settlement that matters is the difference warned for bending, and nothing here lets the building’s stiffness feed back into the trough.

And the walls have no openings. A facade with windows has a shear stiffness well below a solid wall’s, which makes a real tube softer in torsion and the shear in its piers higher than the averaged figure here.

The corner where the walls meet #

They cannot show the open box’s top. A box without floors tied to its walls takes the twist by its walls tilting separately, and at the top, where the walls meet, it is out of square by an amount that depends on the warp and the building’s height — which is where old buildings over tunnels are found cracked at their corners. And they cannot show a floor-to-wall connection slipping, or a building that was closed when the tunnel started and open by the time it finished.

Still open: the building that is half a tube #

Most masonry buildings are closed in part: a concrete ground slab tied to the walls, a roof that is a diaphragm, timber floors between that are not. Cut across, such a building is a tube at its base and at its top and an open section between them, and its torsion is carried by the closed parts and passed between them by the walls. Whether a building closed only at its base takes the ground’s twist as a tube or as a set of walls — and whether the twist then concentrates in the storeys whose floors do not close it, so that the damage appears at the levels that were not strengthened — is the question a partly closed building puts to the settlement of a skewed trough.