Structural Engineering & Statics
where the load goes
Concept

Transfer function — where it appears

The ratio between what comes out of a system and what went in, as a function of frequency. A building is a transfer function between the ground and its floors, with a sharp peak at each of its own periods, and a component mounted on a floor experiences the output rather than the input.

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

The ground is a structure, and it has a period. The transfer function of 38 m of soil at 75 m/s over bedrock at 800 m/s: how much the surface moves for a given motion in the rock, at every frequency. It is a column fixed at the bottom and free at the top, so its resonances are the odd harmonics — the peaks stand at 1, 3, 5, 7 times the first, which is a fixed-free column and nothing else. The fundamental is at 0.493 Hz, a period of 2.03 s, and it is 4H/v_s exactly. The peak amplification is 7.5 against the bound 1/(α + πξ/2) = 7.5, which agrees to 0.2% — and the α in it is the impedance ratio, 0.0554 here. That is the term that keeps the answer finite: assume rigid bedrock and α is zero, the bound becomes 13 and the model is predicting an amplification set by damping alone. What limits the surface motion is that the energy can leave downward.

The ground has a period of its own

An earthquake is measured on rock and felt on soil, and between the two is a layer that behaves exactly like a structure — a column fixed at bedrock, free at the surface, with a fundamental period of four times its depth over its shear wave velocity and a set of odd harmonics above it. The motion a building receives is the rock motion through that filter, and the filter is sharp.

dynamics · Site response The building is a filter with one very sharp tooth. What a component mounted on the roof of a eight-storey building feels, divided by what the same component would have felt on the ground. A rigid item sees the floor's peak acceleration rather than the ground's, which is already 2.29 times as much. An item whose own period matches the building's sees 6.1 times as much, at 0.94 s — a resonance inside a resonance, and the two amplifications multiply rather than add. Far above the building's period the ratio falls back toward one, because a component much softer than its support simply follows the ground. The shape of this curve is a property of the building and not of the earthquake, which is why equipment is qualified against a floor spectrum rather than a ground one.

The spectrum a floor hands on

Most of what breaks in an earthquake is not the structure. It is a transformer, a chiller, a rack, a ceiling, a pipe — and every one of them is bolted to a floor rather than to the ground. The motion it feels has already been through a filter with one very sharp tooth in it.

dynamics · Floor spectrum

Named alongside it

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

AmplificationDampingResonanceResponse spectrumBase isolationBedrockDuctility demandImpedanceModal participationMode shapeNatural periodSecondary system

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