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. The error is second-order in the shape, so a crude guess gives a good load; the same insensitivity means a good load is no evidence that the shape was right.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Rayleigh's method: the frequency read off the deflection that was computed anyway. A simply supported beam of 8 m, sagging 13.08 mm under its own weight, sampled at 21 stations. Rayleigh's quotient over those deflections gives 4.91 Hz against the exact 4.91 Hz — 0.119% high, and high rather than low because an assumed shape is a constraint and a constraint stiffens. The rule of thumb, 18 divided by the square root of the deflection in millimetres, gives 4.98 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 Four guesses at one buckling mode. A pin-ended column, with four assumed shapes and the load each of them gives. The reference is a ten-term Ritz expansion solved as an eigenvalue problem, at 9.8696 EI/L² — which is π², as it must be. a half sine gives 9.870, its own sag shape gives 9.882, a mid-span sag gives 10.000, a parabola gives 12.000. Every one of them is high and none of them is low, because an assumed shape is a constraint on the column and a constraint can only stiffen it.

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 Two quotients from one guessed shape. The critical load of a pin-ended column whose outer quarters keep 10% of the middle's flexural rigidity from four guessed shapes, each worked two ways: Rayleigh's quotient, strain energy of the guess's own curvature over the work of the load, and Timoshenko's, which uses the curvature the guess's bending moment would cause instead. The exact load is 3.225 EI₀/L². a half sine: 8.256 by Rayleigh and 3.745 by Timoshenko; its own sag shape: 8.041 by Rayleigh and 3.712 by Timoshenko; a mid-span sag: 8.875 by Rayleigh and 3.847 by Timoshenko; a parabola: 6.600 by Rayleigh and 3.492 by Timoshenko. Both are upper bounds, and from the same shape the second is never the worse of the two.

The bound from underneath

Rayleigh's quotient turns a guessed shape into a buckling load that is always too high. Divide the same guess differently — use the curvature its bending moment would cause instead of its own — and a parabola that was 21.6 per cent high is 1.3 per cent high. Add one more number that needs no guess at all and the load is caught from below as well, which is the only side of it an amplifier can safely use.

stability · Stability energy A soft layer over stiff ground, and a moment neither has. The bending moment down a 20 m pile of bending stiffness 120 MN·m² with a free head, pushed sideways by 150 kN (its characteristic length in ground at 5.00 MN/m³ being 1.89 m), in three grounds: all of it at 5.00 MN/m³; all of it at 0.50 MN/m³; and soft ground 2.0 characteristic lengths (3.8 m) thick over the other (dotted: the interface). The largest moments are 219 kN·m, 346 kN·m and 427 kN·m; the head deflections 20.4 mm, 81.4 mm and 61.6 mm. The layered ground gives a larger moment than either ground on its own: the pile bends as a pile in the soft layer would, and then meets ground that holds it.

The layer a pile feels

A pile pushed sideways in uniform ground has one characteristic length, the fifth root of its stiffness over the ground's, and every answer is a multiple of it. Put a soft layer over stiff ground and that length stops existing. The head feels the top of the ground in proportion to the square of its own deflection, so an average weighted that way gets the head deflection within eight per cent. It gets the moment a quarter too low: with soft ground two characteristic lengths thick over stiff, the pile bends harder than it would in either ground alone, and the peak sits where the stiff ground takes hold.

internal-forces · Lateral pile

Named alongside it

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

Buckled mode shapeCritical loadEigenvalueFlexural rigidityStiffnessUpper boundAmplificationBending momentCharacteristic equationCharacteristic lengthConservation of energyDeflection

All concepts