Concept

Poisson ratio — where it appears

The ratio of sideways contraction to lengthwise extension in a material pulled along one axis, about 0.3 for steel and 0.2 for concrete. It sets how much a restrained plate stiffens over a free beam of the same material, by one over one minus its square, and it enters shear modulus and plate rigidity alike.

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

The corner is where the plate twists, and it has to be held down. The twisting moment M_xy over a 6 × 6 m simply-supported panel under 10 kN/m², from Navier's double series. It is zero along both centrelines and largest at the corners, which is the opposite of the bending moments and is why no strip reading contains it: 31% of the load crosses this panel in twist, and a strip can only bend. At a free corner the twisting moment is statically equivalent to a downward point force of 2M_xy — 26.7 kN here, 7.4% of the whole load on the panel, applied at a point — so the corner has to be held down. A panel whose corners are free lifts there, deflects more, and cracks diagonally across them. The deflection at the centre is 2.53 mm, against 4.05 mm for the strip that ignores all this — 60% more, which is the size of what the twist is carrying.

A third of the load crosses sideways

A slab spanning both ways is usually explained as two beams sharing a load by a fourth power, and the explanation is not merely approximate — it is missing a mechanism. A real plate carries load three ways, and the third one has no beam strip in it: it is twisting, it accounts for a third of the load on a square panel, and it is why the corners lift.

deflection · Plate torsion

The modulus a strip shows depends on how far it is bent. The moment a strip needs over the moment E times the beam's second moment would need, against Searle's number, width squared over radius times thickness, at a Poisson's ratio of 0.30. It is 1.003 at 1, 1.063 at 10, 1.078 at 30 and 1.095 at 1,000, against 1.099 for a plate.

The strip that becomes a plate

A beam bends with stiffness E and a plate with stiffness E/(1 − ν²), and an engineer picks whichever matches the shape on the drawing. A long strip of metal does not make that choice by its shape. Bent gently, it is a beam, however wide it is, and its cross-section curls into a shallow saddle. Bent further, the saddle can no longer form, because curling it would stretch the strip along its length, the middle goes flat, and the same strip is now a plate with a tenth more stiffness. The number that decides is width squared over radius times thickness, and most of the plates in structures are on the wrong side of it for the formula they are given.

materials · Elastic modulus

Named alongside it

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

Anticlastic curvature Corner uplift Curvature Deflection Elastic modulus Equilibrium Flexural rigidity Lower-bound Navier series Plate bending Plate stiffness Plate torsion

All concepts