Structural Engineering & Statics
where the load goes
Concept

Settlement trough — where it appears

The dish-shaped depression the ground surface takes over a tunnel or a deep excavation, usually drawn as a Gaussian curve. It is deepest over the cut and falls away on either side, and a building on it is bent, stretched and, if it is not square to the trough, twisted.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A stiff building bridges the trough it stands in. The settlement of a building 30 m long centred on the trough, over a trough 30 mm deep whose inflection points are 15 m from its centre (dashed: the ground with no building), measured from the building's settlement under its own weight, for relative stiffnesses EI/(kₛ·B⁴) of 10⁻⁴, 10⁻³ and 10⁻². At 10⁻⁴ its deflection ratio is 95% of the ground's; at 10⁻³ its deflection ratio is 66% of the ground's; at 10⁻² its deflection ratio is 17% of the ground's. The stiff building settles almost as a rigid body and leaves the trough's curvature to the ground beneath it.

The building the trough cannot bend

A tunnel leaves a settlement trough, and damage criteria are applied to the trough's deflection ratio over the length of each building above it. But a building is not the ground: it has a bending stiffness, and it takes only as much of the ground's curvature as that stiffness lets it. One number decides how much — the building's stiffness over the soil's modulus times its length to the fourth — and over two decades of it a building goes from following the trough to bridging it. Because the length enters to the fourth power, the same building is stiff when short and soft when long.

deflection · Differential settlement A building that crosses the trough at an angle stands on a warped surface. Plan of a 30 × 12 m building 10 m high, its walls 300 mm thick, over a trough 30 mm deep with its inflection 15 m from the tunnel's axis, its long axis 45° from square across the tunnel. The lines are the greenfield trough's settlement contours, 5, 10, 15, 20, 25 mm, parallel to the tunnel's axis (dashed). The corners settle A 27.4, B 18.4, C 27.4, D 18.4 mm, so they do not lie in a plane: A and C together settle 18.1 mm more than B and D. A building spanning square across the trough has no such warp, and one lying along it has none either; between the two, the trough twists the building as well as bending it.

The building the trough twists

A building that crosses a tunnel's settlement trough at an angle stands on ground that is warped as well as sagged. Its four corners do not settle into a plane, and beyond about twenty degrees of skew the warp is larger than the sag. Whether the building follows that warp or resists it depends on a property its bending never needed — whether its walls and floors close into a tube — and the same walls, closed or open, differ a thousandfold in their stiffness against it.

deflection · Differential settlement

Named alongside it

The objects these essays reach for when they reach for this one.

Deflection ratioDifferential settlementElastic foundationRelative stiffnessDiaphragmHoggingSaggingShear flowStiffnessTorsional constant

All concepts