Engineering & Science
Tsai-Wu Tensor Composite Failure Criterion (Tension vs Compression Asymmetry)
Calculate the Tsai-Wu orthotropic lamina failure index FI = F1*s1 + F2*s2 + F11*s1^2 + F22*s2^2 + F66*t12^2 + 2*F12*s1*s2 accounting for unequal tensile (Xt, Yt) and compressive (Xc, Yc) composite strengths.
Why carbon-epoxy needs Tsai-Wu when transverse compression is 4x stronger than tension!
Quadratic Tsai-Hill lacks linear stress terms, forcing you to swap allowables manually between tension and compression—yet unidirectional carbon-epoxy has transverse compressive strength Yc = 200 MPa that is FOUR TIMES higher than transverse tensile strength Yt = 50 MPa (and longitudinal Xt = 1,500 MPa vs microbuckling Xc = 1,200 MPa)! Stephen Tsai and Edward Wu (1971) introduced linear odd-order strength tensors F1 = (1/Xt) - (1/Xc) and F2 = (1/Yt) - (1/Yc) alongside quadratic F11 = 1/(Xt*Xc), F22 = 1/(Yt*Yc), F66 = 1/(S^2), and Tsai-Hahn interaction F12 = -0.5 * SQRT(F11 * F22)!
Under plane stress (signed sigma_1, signed sigma_2, shear tau_12), a single unified polynomial handles BOTH tension (+) and compression (-) automatically: FI = F1*sigma_1 + F2*sigma_2 + F11*sigma_1^2 + F22*sigma_2^2 + F66*tau_12^2 + 2*F12*sigma_1*sigma_2 (< 1.0 Safe, >= 1.0 First-Ply Failure)! Watch the dramatic sign effect at sigma_1 = +600 MPa, tau_12 = 35 MPa: under transverse COMPRESSION sigma_2 = -80 MPa, FI = 0.1478 (Safe!), whereas under +45 MPa transverse TENSION, FI jumps to 1.0263 (Matrix Cracking Failure!)!
| Signed Stress State (s1, s2, t12) | Transverse Mode | Tsai-Wu Index FI | Structural Verdict |
|---|---|---|---|
| (+600, -80, 35) MPa | Transverse Compression (Yc = 200) | FI = 0.1478 (< 1.0) | Safe (Linear F2*s2 = -1.20 helps!) |
| (+600, +45, 35) MPa | Transverse Tension (Yt = 50) | FI = 1.0263 (>= 1.0) | First-Ply Matrix Failure! |
| Zero/Negative Strength Input | Yt <= 0 or Yc <= 0 | Guarded Input Error | "Invalid Strength Allowables" |
Tsai-Wu Tensor Polynomial Failure Index FI (Signed Tension + / Compression -)
Computes F1, F2, F11, F22, F66, and Tsai-Hahn F12 = -0.5*SQRT(F11*F22) from positive magnitudes Xt (B5), Xc (B6), Yt (B7), Yc (B8), and S (B9).
=IF(OR(B5 <= 0, B6 <= 0, B7 <= 0, B8 <= 0, B9 <= 0), "Invalid Strength Allowables", ROUND((((1 / B5) - (1 / B6)) * B2) + (((1 / B7) - (1 / B8)) * B3) + ((B2 ^ 2) / (B5 * B6)) + ((B3 ^ 2) / (B7 * B8)) + ((B4 / B9) ^ 2) - (SQRT(1 / (B5 * B6 * B7 * B8)) * B2 * B3), 4))Tsai-Wu Ply Failure Verdict (< 1.0 Safe vs >= 1.0 Failure)
Evaluates whether the Tsai-Wu failure envelope has been breached.
=IF((((1 / B5) - (1 / B6)) * B2) + (((1 / B7) - (1 / B8)) * B3) + ((B2 ^ 2) / (B5 * B6)) + ((B3 ^ 2) / (B7 * B8)) + ((B4 / B9) ^ 2) - (SQRT(1 / (B5 * B6 * B7 * B8)) * B2 * B3) >= 1, "First-Ply Failure (FI >= 1)", "Safe Lamina (FI < 1)")Try the example
Use the original values or change them to test your understanding.