Concept

Hydrostatic stress — where it appears

The average of the three principal stresses, the part of a stress state that changes a material's volume rather than its shape. Yielding in a metal is insensitive to it, which is why yield criteria depend only on the deviatoric part and why confinement raises a brittle material's strength.

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

Also named here as tresca — the same set of essays touches all of them, so they are one junction rather than several.

One criterion inside the other, touching at six points. The two yield criteria in principal stress space with the third principal stress zero, both normalised by the yield stress. Von Mises is the ellipse — σ₁² − σ₁σ₂ + σ₂² = f_y², which is a circle seen at an angle — and Tresca is the hexagon inscribed in it, touching at the six points where one principal stress is zero or the two are equal. Everywhere else Tresca is the smaller, by up to 15.5 per cent, and the widest gap is at pure shear, where σ₁ = −σ₂ and the two answers are 205 and 178 N/mm². The ratio there is exactly 2/√3, computed rather than quoted, and it is the whole reason a web is checked against f_y over root three.

The shear strength nobody measured

Every web on this site is checked against the yield stress divided by the square root of three, and no test produced that number. It is a consequence of a decision about what makes a metal yield, and the alternative decision gives a different answer by fifteen per cent.

materials · Yield criterion

One point, every plane through it, one circle. A point carrying 180 N/mm² across one face, 90 across the other and 40 of shear. As the plane is turned, the pair (σ, τ) runs round a circle of radius 60.2 centred at 135.0 — and it goes round at twice the rate the plane does, which is the part always misremembered and the part that makes the picture work. The principal stresses are 195.2 and 74.8, on planes 20.8° from the face the 180 acts on; the largest shear on any plane is 60.2, exactly the radius, and it sits 45° from those — which is 90° round the circle. The von Mises stress that ranks this state against any other is 170.6.

The circle nobody draws

A plane stress state has three principal stresses and the third is zero. When the two on the drawing share a sign, the largest shear in the state involves the one that is not there — and the circle a designer has drawn is not the circle that governs.

sections · Principal stress

Named alongside it

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

Confinement Mohr circle Principal stress Tresca Von mises Yield criterion Deviatoric stress Ductility Equivalent stress Free body Interaction diagram Plane stress

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