Structural Engineering & Statics
where the load goes
Concept

Orthogonality — where it appears

The property that a structure's mode shapes are independent when weighted by its own mass. It is what turns a building with fifty freedoms into fifty separate one-degree problems, and it is the reason every result about one mass on one spring applies to a whole frame — and the reason a damping matrix that will not decompose is expensive.

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

Three modes of a five-storey frame. The first three mode shapes of a five-storey shear frame, from the eigenvalue problem rather than sketched. Mode 1 has a period of 0.6 s, no node and carries 88.0% of the mass; Mode 2 has a period of 0.21 s, one node and carries 8.7% of the mass; Mode 3 has a period of 0.13 s, two nodes and carries 2.4% of the mass. The nth mode crosses the axis n−1 times, which is a theorem rather than a drawing convention.

A structure has more than one period

One mass on one spring has one period. A building with eight floors has eight, each with a shape of its own, and the second one bends the building into a curve nobody drew. They are not harmonics, they do not interfere, and each behaves as though the others were not there.

dynamics · Mode shapes Three quantities converging at three different rates. How much of the exact answer a truncated modal analysis of a ten-storey building reaches, against how many modes it keeps. The mass count is the rule — ninety per cent, reached at two modes. The base shear is ahead of it: one mode carries 85 per cent of the mass and 98 per cent of the base shear. The force in the top storey is behind it, at 80 per cent on one mode and 94 on two. The rule is written in the quantity that converges fastest, and it is checked against a quantity nobody plots.

The modes that were left out

Nobody runs every mode a model has, and the rule for how many is a mass count — enough to account for ninety per cent of the structure. The rule is written in the one quantity that converges fastest. On a twenty-storey frame two modes give the base shear to within one per cent and the force in the top storey to within twenty.

dynamics · Mode shapes

Named alongside it

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

Degrees of freedomEigenvalueModal analysisMode shapeNatural periodSoft storeyStiffness matrixConvergenceModal massResponse spectrumServiceability

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