State the Tsai-Wu failure criterion for a lamina and give the standard coefficients F1, F2, F11, F22, F66, F12.

State the Tsai-Wu failure criterion for a lamina and give the standard coefficients F1, F2, F11, F22, F66, F12.

• A. F1 σ1^2 + F2 σ2^2 + F11 σ1 + F22 σ2 + F66 τ12 ≤ 1
• B. F1 σ1 + F2 σ2 + F11 σ1^2 + F22 σ2^2 + F66 τ12^2 + 2 F12 σ1 σ2 ≤ 1
• C. F1 σ1^2 + F2 σ2^2 + F11 σ1 + F22 σ2 + F66 τ12 ≤ 1
• D. F1 σ1 + F2 σ2 + F11 σ1^2 + F22 σ2^2 + F66 τ12^2 + F12 σ1 σ2 ≤ 1

Answer: (B)

Tsai-Wu treats lamina failure as a quadratic surface in the in-plane stress components, combining how the material behaves differently in tension and compression with how it responds to shear. The failure criterion is written as a quadratic inequality in the in-plane stresses: F1 σ1 + F2 σ2 + F11 σ1^2 + F22 σ2^2 + F66 τ12^2 + 2 F12 σ1 σ2 ≤ 1. This form correctly includes a linear part in the normal stresses, squared terms for σ1 and σ2, a squared shear term, and a cross-term that couples σ1 and σ2 through 2 F12. The key is that the shear term is τ12^2 and the cross-term carries a factor of 2, which together reproduce the shape of the failure envelope derived from the material strengths in tension, compression, and shear. The standard coefficients are expressed in terms of the lamina’s strength data. Typically, F1 = 1/σ1t − 1/σ1c, F2 = 1/σ2t − 1/σ2c, F11 = 1/(σ1t σ1c), F22 = 1/(σ2t σ2c), F66 = 1/(τ12t τ12c), and F12 is often taken as −√(F11 F22) when shear strengths in tension and compression are not separately specified. When τ12t = τ12c, this simplifies to F66 = 1/τ12^2 and F12 = −√(F11 F22).

Tsai-Wu treats lamina failure as a quadratic surface in the in-plane stress components, combining how the material behaves differently in tension and compression with how it responds to shear. The failure criterion is written as a quadratic inequality in the in-plane stresses: F1 σ1 + F2 σ2 + F11 σ1^2 + F22 σ2^2 + F66 τ12^2 + 2 F12 σ1 σ2 ≤ 1. This form correctly includes a linear part in the normal stresses, squared terms for σ1 and σ2, a squared shear term, and a cross-term that couples σ1 and σ2 through 2 F12. The key is that the shear term is τ12^2 and the cross-term carries a factor of 2, which together reproduce the shape of the failure envelope derived from the material strengths in tension, compression, and shear.

The standard coefficients are expressed in terms of the lamina’s strength data. Typically, F1 = 1/σ1t − 1/σ1c, F2 = 1/σ2t − 1/σ2c, F11 = 1/(σ1t σ1c), F22 = 1/(σ2t σ2c), F66 = 1/(τ12t τ12c), and F12 is often taken as −√(F11 F22) when shear strengths in tension and compression are not separately specified. When τ12t = τ12c, this simplifies to F66 = 1/τ12^2 and F12 = −√(F11 F22).