Structural Engineering & Statics
where the load goes
Concept

Strain energy — where it appears

The work stored in a structure as it deforms, recoverable while it stays elastic, and the quantity every energy method is written in.

Named by 4 essays across 3 fields — 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 05101520253035400246810drop height ÷ static deflectionpeak ÷ static deflectiontwice the static answer — a weight placed, not droppedthe static answera weight dropped from a height

The weight that was dropped

A half-tonne load lowered onto a beam produces 5 kN. The same load dropped one metre onto the same beam produces 160 kN — and onto a beam ten times softer, 54 kN. The stiff structure is the one that suffers, which is the opposite of every other rule on this site.

dynamics · impact factor 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 10^-610^-410^-210^-1610^-1410^-1210^-1010^-8size of the dummy loadrelative errorround-offand nothing elsebest 2e-16∂U/∂P = 1.720635e-2 · unit load = 1.720635e-2

The deflection that is a derivative

A structure's strain energy is one number. Differentiate it with respect to a load and out comes the displacement under that load — and the trick that makes it a method rather than an identity is that the load does not have to be there.

deflection · strain energy

Named alongside it

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

Conservation of energyDynamic amplificationImpact factorImpulseStiffnessBuckled mode shapeCastiglianoComplementary energyCrane loadingCritical loadDeflectionDummy load

All concepts