Structural Engineering & Statics
where the load goes
Concept

Rayleigh method — where it appears

Getting a critical load or a frequency from an assumed shape, which gives an answer high by the square of the error in the guess.

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

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 a half sine9.870 EI/L²exactits own sag shape9.882 EI/L²0.13% higha mid-span sag10.000 EI/L²1.32% higha parabola12.000 EI/L²21.59% highreference9.8696 EI/L²ten Ritz terms,as an eigenvalue problemevery guess is anupper boundP

Guessing the shape, and getting the load anyway

A column's buckling load can be had from a shape that is wrong everywhere, because the energy criterion is stationary at the true mode. The error in the load is the square of the error in the shape, and it is always high.

stability · stability energy

Named alongside it

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

StiffnessBuckled mode shapeCharacteristic equationConservation of energyCritical loadDeflectionEffective lengthEigenvalueEuler bucklingFlexural rigidityGeometric stiffnessModal mass

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