Structural Engineering & Statics
where the load goes
Concept

Relative stiffness — where it appears

A structure's bending stiffness compared with the ground's, made dimensionless with its length. For a building on soil it is its stiffness over the soil's modulus times its length to the fourth: when small the building follows the ground, and when large it bridges the ground and carries the difference in bending.

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

The building does not care how far it went down; it cares how much it tilted. Five footings on soil that is 35% as stiff under one of them, carrying 60 kN/m. They settle between 12 and 54 mm, and the number that matters is neither of those: it is the angular distortion between neighbours, 4.32 per thousand, or one in 231 — against a limit of one in 500 for cracking in finishes, which this does not. A building that went down half a metre uniformly would be undamaged and would need a new front step; this one has moved a twentieth as far and has cracked.

The settlement that matters is the difference

A building that goes down half a metre uniformly is undamaged and needs a new front step. One that goes down a twentieth as far, unevenly, has cracked. The superstructure can even the difference out — and the only way it can do so is by carrying the difference itself, as a force.

deflection · Differential settlement 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

Named alongside it

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

Differential settlementAngular distortionBeam on springsDamage criterionDeflection ratioElastic foundationFoundationHoggingLoad pathRaftRedistributionSagging

All concepts