Structural Engineering & Statics
where the load goes
Concept

Natural period — where it appears

The time one free oscillation takes, and the number every load has to be compared against before it can be called sudden or slow.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

012345-30-20-10102030time (s)displacement (mm)20 kN applied at once and heldthe load arrives in no time at allstatic, 12.67 mmpeak 24.56 mm at 0.250 s

Twice the deflection, for the same load

A weight placed gently on a beam deflects it by one amount. The same weight let go from rest, a millimetre above the same beam, deflects it by twice as much — and the factor of two is exact, for every structure ever built.

dynamics · dynamic amplification 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

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.

dynamics · natural period 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

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 structures · one recordT = 0.3 speak 12.54 mmT = 0.8 speak 46.2 mmT = 1.8 speak 108.95 mmpeak displacementperiod

The spectrum is not a load

A response spectrum looks like a load curve and is not one. Every point on it is the peak of a complete time integration of one particular structure, and the curve is what you get by doing that again for every structure there could be.

dynamics · response spectrum

Named alongside it

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

Mode shapeBase shearCharacteristic equationConservation of energyDampingDeflectionDegrees of freedomDesign spectrumDynamic amplificationEigenvalueFree bodyGround motion

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