Last verified 19 August 2026.
The separation that makes it work
Ashby’s performance equation splits into three factors that multiply: one for the functional requirement, one for the geometry, and one for the material. Because they separate, the material term can be optimised without knowing the geometry — which is exactly the situation you are in when the choice still matters and is still cheap to change.
That material term is the index. Everything else on this page is the consequence.
Why the exponents differ
Take three cases with the same objective — minimum mass — and watch the exponent move.
A tie in tension, stiffness specified. Stiffness scales with area. Mass scales with area. Both scale the same way, so the free variable cancels cleanly and you maximise E/ρ.
A beam in bending, stiffness specified. Bending stiffness scales with the second moment of area, so with the square of the section area for a fixed shape. Mass still scales with area. The free variable no longer cancels one-for-one, and you maximise E^(1/2)/ρ.
A beam in bending, strength specified. Bending strength scales with the section modulus, which goes as area to the 3/2 for fixed shape. Mass scales with area. The result is σ^(2/3)/ρ.
The exponent is not a property of the material. It is a record of which dimension the design was allowed to change, and it moves when that changes. Fix the thickness and free the width instead, and you get a different index for the same physical beam.
That is the single most common misuse: quoting a published index without checking that its assumed free variable matches your design.
Common indices
| Function, objective | Index | Maximise |
|---|---|---|
| Tie, stiffness specified, minimum mass | E/ρ | E/ρ |
| Beam in bending, stiffness specified, minimum mass | E^(1/2)/ρ | E^(1/2)/ρ |
| Panel in bending, stiffness specified, minimum mass | E^(1/3)/ρ | E^(1/3)/ρ |
| Tie, strength specified, minimum mass | σ/ρ | σ/ρ |
| Beam, strength specified, minimum mass | σ^(2/3)/ρ | σ^(2/3)/ρ |
| Panel, strength specified, minimum mass | σ^(1/2)/ρ | σ^(1/2)/ρ |
| Any of the above, minimum cost | replace ρ with ρ × cost per kg |
An index of the form M = Eᵃ/ρ plots as a straight line of slope 1/a on log–log axes, which is why property charts and indices were designed together: ranking becomes sliding a line of fixed slope until only a few candidates remain above it.
What an index quietly assumes
Shape is fixed. All of the above assume the section shape does not change — only its scale. Free the shape and a shape factor enters, and a hollow section in a modest material can beat a solid one in a good material.
One objective at a time. Two objectives, such as mass and cost, do not collapse into one index. They give a trade-off surface, and the choice of where to sit on it is a judgement, not a calculation. Combining them with weights hides the judgement inside a number.
One property per axis. Indices are built from bulk properties. Fatigue, fracture toughness, joining, corrosion and availability do not appear, and they are frequently what actually decides.
The constraint is the right one. An index optimises against the constraint you gave it. If the section is really set by deflection under a serviceability limit rather than by strength, an index built on σ ranks the wrong thing very precisely.
The evidence problem nobody mentions
An index is a derived quantity, and under the weakest-link rule it carries the lowest evidence level of its inputs.
Compute E^(1/2)/ρ from a certified modulus and a predicted density and the result is a predicted index — regardless of how exact the arithmetic was. This matters because the arithmetic feels like it adds rigour. It does not. It propagates.
Two habits follow. Record the level of each input beside the index, not just the index. And when two candidates are separated by less than the width the inputs carry, record them as not separated rather than presenting an order the data cannot support. A ranking more precise than its inputs is a fabrication performed by arithmetic — see documenting material decisions.
How to use them without over-trusting them
Use an index to get from hundreds of candidates to a handful, fast, before the geometry is settled. That is what it is for and it is very good at it.
Then stop using it. The final choice among three or four survivors is decided by things no index contains: whether your shop can weld it, whether it is available in the thickness you need in eight weeks, whether there is an approvals history, what it costs this quarter. Those are qualitative requirements, they belong in the record, and they must not be turned into scores so they can be ranked alongside the index.
What this page does not cover
- The derivations. Ashby’s own text does them properly and this page only states the results and the reasoning behind the exponents.
- Shape factors. A substantial topic that changes the conclusions above.
- Multi-objective optimisation and penalty functions.
- Where to get the property values. That is the harder problem, and the reason the framework exists.
- Indices for functions not listed, of which there are many.