Tsai–Hill Failure Index for a Lamina

An isotropic material has one strength, and checking it is a division. A lamina has at least five — longitudinal tension and compression, transverse tension and compression, and in-plane shear — and they differ by more than an order of magnitude. Checking each stress against its own allowable one at a time misses the fact that they interact: a ply can fail with every individual stress comfortably below its own limit. Hill adapted von Mises's yield criterion to anisotropic metals in 1948, and Tsai reduced it to plane stress for a lamina, giving $FI = (\sigma_1/X)^2 - \sigma_1\sigma_2/X^2 + (\sigma_2/Y)^2 + (\tau_{12}/S)^2$, with failure predicted at $FI \ge 1$.

The stresses and strengths must all be in the ply's own material axes: 1 along the fibres, 2 across them, 12 the in-plane shear. Stresses computed in the laminate's global axes have to be rotated into each ply's frame first, which is what classical lamination theory spends most of its algebra doing, and the criterion is then applied ply by ply. Every term is quadratic, so the criterion is an ellipse in stress space and the shear sign never matters.

Now the four things it does not tell you, each of which has caught somebody out.

First: it gives an index, not a margin. An index of 0.64 does not mean 36% of the load is left. Because every term is quadratic, the stresses can all rise by $1/\sqrt{0.64} = 1.25$ before the criterion is reached — 25% of headroom, not 36%. The quantity worth quoting is the strength ratio $R = 1/\sqrt{FI}$, the factor by which the whole stress state may be multiplied. Quoting the index as though it were a reserve factor is conservative at high values and dangerously optimistic at low ones, and it is a common error in reports.

Second: it does not say how the ply failed. Fibre rupture, fibre microbuckling, matrix cracking parallel to the fibres and interface debonding are four unrelated physical events, and this criterion adds them into one scalar. When the transverse and shear terms dominate the sum, what the index is really predicting is a matrix crack — and in a multi-directional laminate that is first-ply failure, not final failure. The cracked ply sheds its transverse load to its neighbours and the laminate carries on, sometimes to twice the load. Whether first-ply failure is your design limit is a decision about the structure, not about this equation. Hashin's criterion and Puck's exist precisely because they separate the modes and tell you which one is running out; Tsai–Hill does not.

Third: it is blind to the sign of the stress. $X$ and $Y$ appear as single strengths, and a real lamina has different tensile and compressive values in both directions — longitudinal compression is 50 to 70 percent of tension because the fibres microbuckle, while transverse compression is several times transverse tension because a matrix crack cannot open under a squeeze. The usual patch is to enter whichever strength matches the sign of the applied stress, which works but is a patch. The Tsai–Wu criterion handles it properly by carrying separate tensile and compressive terms.

The interaction term $-\sigma_1\sigma_2/X^2$ deserves its own warning. It is inherited from plasticity theory, where it describes the coupling of principal stresses in a yielding metal, and it has no clear physical justification for a brittle fibre composite. It has the effect that a modest tensile $\sigma_2$ slightly raises the $\sigma_1$ the criterion allows. Treat that as an artefact rather than as capacity — the mode-separating criteria drop the term entirely.

Fourth, and most important: none of this covers delamination. Plies separating from one another is the failure mode that actually retires composite structures. It is driven by interlaminar stresses — through-thickness tension and interlaminar shear — that do not appear in a plane-stress criterion at all, and it is most often initiated by impact rather than by design loads. A dropped tool can leave a delamination that halves compressive strength while every in-plane index on the part still reads well below 1. That is a fracture-mechanics problem in mode I and mode II interlaminar toughness, and it belongs with strain energy release rate and the fracture pages rather than with any strength criterion here.

Used with those four limits in mind, Tsai–Hill is a good tool: it is one line, it is interactive, it needs only three strengths, and it reliably finds the transverse direction, which is where a unidirectional ply nearly always runs out first. The design answer to a transverse problem is almost never a better resin. It is plies at other angles, so that no single ply is asked to carry a significant load across its own fibres.