Which of the following is the correct expression for Qbar66 in a 2D laminate transformation?

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

Answer: (B)

In 2D laminate transformation, the transformed reduced stiffness Qbar captures how the lamina responds when rotated by an angle. The component that describes in-plane shear in the global axes, Qbar66, combines the lamina’s shear stiffness Q66 with the interaction of the in-plane normal stiffnesses Q11, Q22, and Q12 through the rotation.

The standard form is Qbar66 = Q66 (s^2 - c^2)^2 + (Q11 + Q22 - 2Q12) s^2 c^2, where c = cos(theta) and s = sin(theta). If we set m = cos(theta) and n = sin(theta), this becomes Qbar66 = (Q11 + Q22 - 2Q12) m^2 n^2 + Q66 (m^2 - n^2)^2. This matches the option that uses the correct combination of terms and the right dependence on m and n.

The other forms misplace terms or use the wrong coefficients (for example, replacing Q66 with Q12, or using (m^2 + n^2)^2, which equals 1 and does not represent the correct rotation behavior, or using m^4 terms that do not reflect the cross-coupling).