Tsai Wu Failure⚠ unverified

Understanding the Parameters

• The five strengths $X_t, X_c, Y_t, Y_c, S$ — a unidirectional ply is strongly orthotropic: it is very strong along the fibres ($X_t$ can be $30$–$50\times$ the transverse $Y_t$) and weak across them. Tension and compression differ too ($X_c$, $Y_c$), because compressive failure is by fibre microbuckling and matrix crushing rather than fibre breakage. These five numbers define the failure envelope's size and shape.

• The stresses $\sigma_1, \sigma_2, \tau_{12}$ — resolved into the material (ply) axes, not the global laminate axes. A laminate analysis first transforms applied loads into each ply's 1–2 frame, then evaluates $FI$ ply-by-ply; the first ply to reach $FI=1$ marks first-ply failure.

• The linear terms $F_1\sigma_1 + F_2\sigma_2$ — these odd-power terms are what let the criterion distinguish tension from compression ($F_1 = 1/X_t - 1/X_c$ is nonzero when the two strengths differ). A purely quadratic criterion cannot; this is Tsai–Wu's key advantage over Tsai–Hill (Tsai Hill Criterion).

• The interaction term $2F_{12}\sigma_1\sigma_2$ — couples fibre and transverse stresses, tilting the failure ellipse. The coefficient $F_{12}$ is hard to measure (it needs a biaxial test), so the implementation uses the common approximation $F_{12} = -\tfrac12\sqrt{F_{11}F_{22}}$ (the Tsai–Hahn / Mises–Hencky form).

• The output $FI$ — a dimensionless index. $FI < 1$ safe, $FI = 1$ onset of failure, $FI > 1$ failed. Note $FI$ is not a margin: because of the linear terms it does not scale linearly with load, so a strength ratio is obtained by solving the quadratic for the load multiplier, not by $1/FI$.

Derivation (Approaching a Proof)

Tsai and Wu (1971) proposed that a failure surface in stress space be written as a general quadratic scalar function of the stress components that is invariant and closes into an ellipsoid:

$$FI = F_i\sigma_i + F_{ij}\sigma_i\sigma_j = 1 \quad (\text{at failure}),$$

with $i,j$ running over the ply stress components. For a unidirectional lamina in plane stress ($\sigma_1,\sigma_2,\tau_{12}$), symmetry arguments (shear sign cannot change strength) eliminate terms linear in $\tau_{12}$ and its cross terms, leaving the six coefficients above. Each is fixed by a simple strength test:

• Uniaxial tension/compression along fibres: $F_1 X_t + F_{11}X_t^2 = 1$ and $-F_1 X_c + F_{11}X_c^2 = 1$ solve to $$F_1 = \frac{1}{X_t} - \frac{1}{X_c}, \qquad F_{11} = \frac{1}{X_t X_c}.$$

• Likewise transverse: $F_2 = \dfrac{1}{Y_t}-\dfrac{1}{Y_c}$, $\;F_{22} = \dfrac{1}{Y_t Y_c}$.

• Pure shear: $F_{66}S^2 = 1 \Rightarrow F_{66} = 1/S^2$.
• The interaction $F_{12}$ needs a biaxial test; absent one, the stability requirement $F_{12}^2 \le F_{11}F_{22}$ (so the ellipsoid stays closed) motivates $F_{12} = -\tfrac12\sqrt{F_{11}F_{22}}$.

Assembling these gives the working failure index. $\blacksquare$ The condition $FI = 1$ is an ellipsoid in $(\sigma_1,\sigma_2,\tau_{12})$ space; being inside it is safe.

Dimensional check. $F_1\sigma_1 \sim (1/\text{Pa})(\text{Pa}) = \text{(–)}$; $F_{11}\sigma_1^2 \sim (1/\text{Pa}^2)(\text{Pa}^2) = \text{(–)}$; $F_{66}\tau^2 \sim (1/\text{Pa}^2)(\text{Pa}^2) = \text{(–)}$. Every term is dimensionless, so $FI$ is a pure number. $\checkmark$

History and Development

• Tsai and Wu, 1971. Stephen W. Tsai and Edward M. Wu proposed the tensor-polynomial criterion as a mathematically rigorous, operationally simple failure surface, superseding the earlier Tsai–Hill (Tsai Hill Criterion) criterion which could not separate tension from compression.

• Interactive vs non-interactive. Tsai–Wu is a single smooth envelope, easy to program and differentiate, which made it the default in laminate-analysis codes. Its weakness is that it does not identify the mode of failure (fibre vs matrix); physically-based criteria like Hashin and Puck were developed to distinguish modes for damage-tolerance work.

• The $F_{12}$ controversy. The interaction coefficient has no unique measurement and materially affects the predicted envelope under biaxial loading. Decades of "World-Wide Failure Exercise" benchmarking have probed how much it matters; the $-\tfrac12\sqrt{F_{11}F_{22}}$ default used here is the pragmatic standard.

Related Concepts: Tsai Hill Criterion, Longitudinal Strength, Transverse Strength, Interlaminar Shear Stress, Composite Longitudinal Modulus, Composite Transverse Modulus, Von Mises Stress 2D

Notes: Registry calculator tsai-wu-failure (unverified). Failure at $FI \ge 1$. Coefficients computed internally from the five strengths; interaction term uses the approximation $F_{12}=-\tfrac12\sqrt{F_{11}F_{22}}$ (no biaxial test needed). Interactive criterion — does not identify failure mode (use Hashin/Puck for that). $FI$ is not a linear margin. All defaults $1.0$.