State the Tsai-Hill failure criterion for an orthotropic lamina in plane stress.

State the Tsai-Hill failure criterion for an orthotropic lamina in plane stress.

• A. (σ1/Xt)^2 + (σ2/Yt)^2 - (σ1 σ2)/(Xt Yt) + (τ12/S12)^2 ≤ 1
• B. (σ1/Xt)^2 + (σ2/Yt)^2 + (τ12/S12)^2 ≤ 1
• C. (σ1/Xt)^2 + (σ2/Yt)^2 + (τ12/S12)^2 - (σ1 σ2)/(Xt Yt) ≤ 1
• D. (σ1/Xt) + (σ2/Yt) + (τ12/S12) ≤ 1

Answer: (A)

The Tsai-Hill criterion for an orthotropic lamina in plane stress uses a quadratic, strength-normalized combination of the in-plane stresses to predict failure, capturing how the material’s anisotropy makes the normal stresses interact. The left-hand side combines the normalized normal stresses squared, the interaction between them with a negative cross-term, and the squared shear stress, and it must not exceed one for safe loading. Specifically, weights Xt, Yt, and S12 are the tensile strength along the 1-direction, tensile strength along the 2-direction, and the in-plane shear strength, respectively; plane stress means σ3 and the out-of-plane shear stresses are zero. The negative cross-term (σ1 σ2)/(Xt Yt) embodies the coupling between σ1 and σ2 due to the orthotropic nature, so when both normal stresses are present in the same sense, the interaction lowers the allowable combined loading compared with treating the stresses independently. This form is the correct representation because it reflects both the independent strength limits and their interaction, whereas forms that omit the cross-term or replace the relation with a linear sum do not capture the same failure envelope.

The Tsai-Hill criterion for an orthotropic lamina in plane stress uses a quadratic, strength-normalized combination of the in-plane stresses to predict failure, capturing how the material’s anisotropy makes the normal stresses interact. The left-hand side combines the normalized normal stresses squared, the interaction between them with a negative cross-term, and the squared shear stress, and it must not exceed one for safe loading. Specifically, weights Xt, Yt, and S12 are the tensile strength along the 1-direction, tensile strength along the 2-direction, and the in-plane shear strength, respectively; plane stress means σ3 and the out-of-plane shear stresses are zero. The negative cross-term (σ1 σ2)/(Xt Yt) embodies the coupling between σ1 and σ2 due to the orthotropic nature, so when both normal stresses are present in the same sense, the interaction lowers the allowable combined loading compared with treating the stresses independently. This form is the correct representation because it reflects both the independent strength limits and their interaction, whereas forms that omit the cross-term or replace the relation with a linear sum do not capture the same failure envelope.