Tsai Hill Criterion⚠ unverified
Mechanical / Composites · Compute the Tsai-Hill failure index for a lamina under plane stress
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| sigma1 | σ1 | Pa | 1.0 | Stress in the fibre (1) direction |
| sigma2 | σ2 | Pa | 1.0 | Stress in the transverse (2) direction |
| tau12 | τ12 | Pa | 1.0 | In-plane shear stress |
| X | X | Pa | 1.0 | Longitudinal strength |
| Y | Y | Pa | 1.0 | Transverse strength |
| S | S | Pa | 1.0 | In-plane shear strength |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | TH | — | Tsai-Hill failure index (dimensionless); failure is predicted when the value reaches or exceeds 1 |
The science & history
Understanding the Parameters
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Anisotropic strengths $X$, $Y$, $S$ — each stress is measured against the strength in its own direction: fiber-direction stress against the high $X$ (Longitudinal Strength), transverse against the low $Y$ (Transverse Strength), shear against $S$. This directionality is what makes it a composite (not isotropic) criterion.
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The interaction term $-\sigma_1\sigma_2/X^2$ — the signature of Tsai-Hill: it recognises that combined fiber and transverse stress interact. Like-signed $\sigma_1,\sigma_2$ reduce the index (mildly protective), opposite-signed increase it. It derives from the distortion-energy (von Mises) form adapted to anisotropy.
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Index vs factor of safety — $TH$ is a utilisation value: $TH < 1$ safe, $TH = 1$ on the failure surface. The strength ratio (factor of safety on load) is $R = 1/\sqrt{TH}$ because the stresses enter quadratically.
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What it does not tell you — Tsai-Hill gives a single yes/no per ply; it does not identify the failure mode (fiber break vs matrix crack vs shear). Mode-based criteria (Hashin, Puck) distinguish these for progressive-failure analysis.
Derivation (Approaching a Proof)
Tsai-Hill adapts Hill's anisotropic yield criterion (a generalisation of von Mises distortion energy to orthotropic materials) to a composite lamina. Hill's criterion for plane stress in the principal material axes has the form
$$F\sigma_1^2 + G\sigma_2^2 - H\,\sigma_1\sigma_2 + N\tau_{12}^2 = 1,$$
with coefficients $F, G, H, N$ set by the material's directional strengths. Tsai identified those coefficients by requiring the criterion to reduce to the correct uniaxial and shear failures: a pure $\sigma_1$ fails at $X$, a pure $\sigma_2$ at $Y$, a pure $\tau_{12}$ at $S$. This fixes $F = 1/X^2$ (with the cross term also scaled by $1/X^2$), $G = 1/Y^2$, $N = 1/S^2$, giving
$$\left(\frac{\sigma_1}{X}\right)^2 - \frac{\sigma_1\sigma_2}{X^2} + \left(\frac{\sigma_2}{Y}\right)^2 + \left(\frac{\tau_{12}}{S}\right)^2 = 1$$
at failure. The single normalisation of the cross term by $X^2$ (rather than $XY$) is the specific Tsai-Hill simplification.
Dimensional check. Each term is (stress/strength)² = (Pa/Pa)² = dimensionless, and the cross term $\sigma_1\sigma_2/X^2 = \text{Pa}^2/\text{Pa}^2$ = dimensionless, so $TH$ is a pure index, as required.
History and Development
Rodney Hill generalised the von Mises yield criterion to anisotropic (rolled) metals in 1948; Stephen Tsai adapted it to unidirectional composites (1965), producing the first widely-used interactive composite failure criterion. It remains a standard first-pass ply-failure check in Composite Laminate Theory, alongside the tensor-polynomial Tsai-Wu criterion and the mode-distinguishing Hashin/Puck criteria used for detailed progressive-failure analysis.
Related Concepts: Longitudinal Strength, Transverse Strength, Von Mises Stress, Static Failure Theories, Composite Laminate Theory, Interlaminar Shear Stress
Notes: Failure at $TH \ge 1$; strength ratio $R = 1/\sqrt{TH}$ (stresses are quadratic). Interactive (cross term $-\sigma_1\sigma_2/X^2$) — from anisotropic Hill/von Mises. Directional strengths $X$ (Longitudinal Strength), $Y$ (Transverse Strength), $S$. Does not give the failure mode — use Hashin/Puck for that.