Provide the correct expression for Qbar66 in a rotated lamina: Qbar66 = ?

Provide the correct expression for Qbar66 in a rotated lamina: Qbar66 = ?

• A. Qbar66 = Q11 m^2 n^2 + Q22 m^2 n^2
• B. Qbar66 = Q66 (m^4 + n^4)
• C. Qbar66 = Q11 m^4 + Q22 n^4
• D. Qbar66 = (Q11+Q22-2Q12) m^2 n^2 + Q66 (m^2 - n^2)^2

Answer: (D)

When a lamina is rotated, its in-plane stiffness must be expressed in the new axes, which blends the original normal and shear stiffnesses. The transformed shear stiffness Qbar66 includes the direct contribution from the original shear term Q66 and a coupling term that comes from the normal terms Q11, Q22, and Q12 due to the rotation. The standard compact form is Qbar66 = (Q11 + Q22 − 2Q12) m^2 n^2 + Q66 (m^2 − n^2)^2, where m = cosθ and n = sinθ. This structure encodes how rotation mixes normal and shear responses: the (m^2 − n^2)^2 factor scales the original shear stiffness Q66, while the m^2 n^2 factor brings in the combination (Q11 + Q22 − 2Q12). Using m^2 + n^2 = 1, this can also be written in an equivalent trig form, but the given expression correctly represents the transformed Qbar66. The other expressions don’t capture this mixed dependence—either missing the Q66 term or misplacing the m and n powers—so they don’t reflect the rotation transformation for the shear component.

When a lamina is rotated, its in-plane stiffness must be expressed in the new axes, which blends the original normal and shear stiffnesses. The transformed shear stiffness Qbar66 includes the direct contribution from the original shear term Q66 and a coupling term that comes from the normal terms Q11, Q22, and Q12 due to the rotation. The standard compact form is Qbar66 = (Q11 + Q22 − 2Q12) m^2 n^2 + Q66 (m^2 − n^2)^2, where m = cosθ and n = sinθ. This structure encodes how rotation mixes normal and shear responses: the (m^2 − n^2)^2 factor scales the original shear stiffness Q66, while the m^2 n^2 factor brings in the combination (Q11 + Q22 − 2Q12). Using m^2 + n^2 = 1, this can also be written in an equivalent trig form, but the given expression correctly represents the transformed Qbar66. The other expressions don’t capture this mixed dependence—either missing the Q66 term or misplacing the m and n powers—so they don’t reflect the rotation transformation for the shear component.