Structural Engineering & Statics
where the load goes
Generator

The mode-shapes generator

Rendered here at the parameters it defaults to, with every essay that calls it — which is the same list as the blast radius of changing it.
Three modes of a five-storey frameThe 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.five floors · 300 t eachmode 10.6 s · 1.65 Hz88.0% of the massno nodemode 20.21 s · 4.83 Hz8.7% of the massone nodemode 30.13 s · 7.61 Hz2.4% of the masstwo nodes

3 essays call mode-shapes. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

8 m simple span · sag 13.08 mmthe static shape, not an eigenvectorω² = g · Σ(w·δ) ⁄ Σ(w·δ²)Σ(w·δ) = 197.3 · Σ(w·δ²) = 2.030Rayleigh, from the static shape: 4.915 Hzexact, from the characteristic equation: 4.909 Hz18/√δ with δ in mm: 4.977 Hz Dynamics

The period nobody chose

Every structure has a natural period, it decides the answer to every question in this field, and no drawing anywhere records it. It is a consequence of a mass picked for one reason and a stiffness picked for another — and it has already been computed, by the serviceability check.

 five floors · 300 t eachmode 10.6 s · 1.65 Hz88.0% of the massno nodemode 20.21 s · 4.83 Hz8.7% of the massone nodemode 30.13 s · 7.61 Hz2.4% of the masstwo nodes Dynamics

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.

 eight modes · 2400 ttwo modes reach 90% of itmode 1 · 0.93 s85.6%85.6% cumulativemode 2 · 0.31 s9.1%94.7% cumulativemode 3 · 0.19 s3.0%97.7% cumulativemode 4 · 0.14 s1.3%99.0% cumulativemode 5 · 0.12 s0.6%99.6% cumulativemode 6 · 0.1 s0.3%99.9% cumulativemode 7 · 0.09 s0.1%100.0% cumulativemode 8 · 0.09 s0.0%100.0% cumulative Dynamics

Most of the mass moves together

A sixteen-storey frame has sixteen modes, and the first one carries eighty-four per cent of the mass. That is why a hand calculation on one mode gets the base shear right to within a tenth — and why the same calculation gets the acceleration at roof level wrong by a factor of two.

The library, page 2 of 4 — where mode-shapes sits