Structural Engineering & Statics
where the load goes
Concept

Deflection ratio — where it appears

The largest departure of a building's settled shape from the straight line between its ends, divided by its length. Damage criteria for masonry convert it into a tensile strain, and a building's own stiffness decides how much of the ground's deflection ratio it actually takes.

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

The deflection a building can take, and which way it is bending. The limiting deflection ratio — the sag or hog of a building divided by its length — at a critical tensile strain of 0.075 per cent, against how long the building is compared with its height. Two curves, and the gap between them is the whole finding: hogging is worse than sagging by exactly 2.0 times, because the neutral axis of a hogging building sits near the bottom and the tension face is the full height of the wall. The minimum is at L/H = 1.6, where the two mechanisms cross: a squat building cracks diagonally in shear and a long low one cracks in bending at the extreme fibre.

The crack is a strain, not a slope

Angular distortion is what every table of settlement limits is written in, and it is a proxy. What cracks a building is a tensile strain in it, and reading the building as a deep beam says which of two mechanisms supplies it, why hogging is exactly twice as damaging as sagging, and why a horizontal stretch of a few millimetres takes half the capacity away before the settlement starts.

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 distortionCrack widthDamage categoryDeep beamElastic foundationHoggingHorizontal strainMasonryRelative stiffnessSaggingServiceability

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