Concept

Imperfection

4 essays name this object, across one field. What follows is each of them, and the objects they name alongside it.
5010015020000.20.40.60.811.2slenderness (effective length ÷ radius of gyration)they cross at λ = 75squashingEuler bucklingreal columns, which are neither

Strong enough and still falls over

A column can fail at a fraction of the load its material could carry, by going sideways. Buckling is a failure of stability rather than of strength, and it is decided by geometry.

stability · buckling
00.20.40.60.80246810applied load ÷ buckling load1.1×1.3×1.4×2.0×3.3×10.0×first-order analysis says the answer is always 1×one over one minus the ratio

The load that makes itself worse

A structure that has leaned carries its weight off the axis, which makes it lean further. The amplification is one over one minus the load ratio, and it runs away long before the buckling load.

stability · second order
00.0050.010.0150.020.02500.20.40.60.81lateral deflection at mid-heightload ÷ P꜀ᵣP ÷ P꜀ᵣ = 1.00, approached and never reachedinitial bow: δ₀ = 0.002

The column that was never straight

Euler's load is the load at which a perfectly straight column becomes indifferent to being bent. No column is perfectly straight, so no column ever reaches it — and the load it never reaches can still be measured.

stability · buckling
05010015020025001020304050brace stiffness (units of EI/L³)critical load (units of EI/L²)ideal stiffness ≈ 159 EI/L³21.839.5 — braced9.87 — unbraced

The brace that need not be strong

A brace holding a column at mid-height carries almost no force. What it has to be is stiff — and the stiffness required is exact, large, and reached at a knee past which more buys nothing at all.

stability · effective length

Named alongside it

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

Critical loadEffective lengthAmplification factorEuler bucklingResidual stressSecond order effectsBifurcationBracingBuckled mode shapeEccentricityEigenvalueLateral restraint

All concepts