Structural Engineering & Statics
where the load goes
Concept

Euler load — where it appears

The load at which a perfectly straight elastic column bifurcates, proportional to the flexural rigidity and to the inverse square of the effective length. It contains no strength at all, so two columns of the same shape in materials of very different strength buckle at loads in the ratio of their moduli.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The one length a section carries into a column. Four profiles of equal area, with the radius of gyration r = √(I/A) drawn as the distance it is — a pair of lines either side of the centroid, at the depth the whole area would have to sit at to give the section the second moment it has. As a 4 m pin-ended column the same 3000 mm² of material carries between 7 and 3146 kN, in the ratio of the squares of those radii and of nothing else.

The one length a section takes into a column

A section has an area, a second moment, two section moduli, a shear centre and a torsion constant. A column has heard of exactly one of them, and it is none of those — it is the length √(I/A), which is where the whole area would have to sit to give the section the stiffness it has.

sections · Radius of gyration A built-up column has a second way to bend. A 12 m column of two chords 300 mm apart, joined by double lacing. On the left it buckles the way a solid column does, by bending; on the right the chords stay straight and the lattice racks, which a solid column cannot do at all. Neither happens alone, and the two flexibilities add rather than the two stiffnesses — so the critical load is 1287 kN against an Euler load of 1341 kN, which is 96% of it, and the column behaves as though its slenderness were 80 rather than 79.

The column made of two columns

A solid column buckles when its bending stiffness runs out. A laced one has a second way to go — the lattice shears, the chords stay straight — and the two flexibilities add rather than the two stiffnesses. A battened column reaches 23% of its own Euler load and behaves as though its slenderness were twice what it is.

stability · Built-up column A built-up column has a second way to bend. A 24 m column of two chords 800 mm apart, joined by single lacing. On the left it buckles the way a solid column does, by bending; on the right the chords stay straight and the lattice racks, which a solid column cannot do at all. Neither happens alone, and the two flexibilities add rather than the two stiffnesses — so the critical load is 3832 kN against an Euler load of 4629 kN, which is 83% of it, and the column behaves as though its slenderness were 66 rather than 60.

The lacing decides the force it has to carry

A perfectly straight column carries no shear, so the diagonals of a built-up column are resisting a force that exists only because the column is crooked. Computing it turns out to be circular — the shear is amplified by the very flexibility the lacing supplies — and the rule of thumb that replaces the calculation is wrong by a factor that grows as the lacing gets worse.

stability · Built-up column Two formulas that agree until the shear is soft. The buckling load over the Euler load, by Engesser (dashed) and Haringx (solid), against the shear stiffness over the Euler load on a logarithmic scale. Marked: double lacing, 23.48: Engesser 0.959, Haringx 0.961; battens, 0.30: Engesser 0.232, Haringx 0.419; rubber bearing, 6.6e-4: Engesser 0.001, Haringx 0.025. They agree within 5 per cent while the shear stiffness is more than about 4 times the Euler load, and for a shear stiffness a thousandth of it Haringx's is 31 times Engesser's.

The bearing Engesser says has buckled

A member that is flexible in shear buckles below its Euler load, and there are two classical formulas for how far below. For a laced column they agree to a fraction of a per cent. For a battened column one gives nearly twice the other. For a laminated rubber bearing, Engesser's says it buckles at a fourteenth of the load it carries every day and Haringx's says it is safe by a factor of nearly three. They are not two theories. Put the axial load into Engesser's shear stiffness and it becomes Haringx's, exactly — so the whole disagreement is about which shear stiffness was measured.

stability · Built-up column

Named alongside it

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

BucklingBuilt-up columnShear stiffnessCritical loadEngesserLacingRadius of gyrationSeries combinationSlendernessAmplificationAsymmetric sectionBase isolation

All concepts