Structural Engineering & Statics
where the load goes
Generator

The beam-frequency 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.
The first three modes of a simply supported beamThree modes of a simply supported beam of 8 m span, drawn from the general solution with the constants fixed by the support conditions rather than assumed to be sines. Mode 1 is at 4.91 Hz with βL = 3.1416; Mode 2 is at 19.63 Hz with βL = 6.2832; Mode 3 is at 44.18 Hz with βL = 9.4248. The frequencies go as the square of βL, so the threeth mode is 9.0 times the first. The marked points are the nodes.Simply supported, 8 m spaneach shape scaled to its own peakmode 1: 4.91 HzβL = 3.1416mode 2: 19.63 HzβL = 6.2832mode 3: 44.18 HzβL = 9.4248

3 essays call beam-frequency. 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.

 2345678910051015202530floor frequency (Hz)response factor1× pace2× pace3× pace4× pace4 Hz — R = 106 Hz — R = 7 Dynamics

The floor that is strong and unusable

A floor can satisfy every strength check, deflect less than the limit, and still be rejected by the people who work on it — because somebody walking across it at two steps a second happens to be exciting it at exactly the rate it likes to move.

The library, page 1 of 4 — where beam-frequency sits