In a 2D laminate transformation, which of the following is the correct expression for Qbar11, given m = cos θ and n = sin θ?

In a 2D laminate transformation, which of the following is the correct expression for Qbar11, given m = cos θ and n = sin θ?

• A. Qbar11 = Q11 m^2 n^2 + Q12(m^4+n^4)
• B. Qbar11 = Q11 m^4 + 2(Q12+2Q66)m^2 n^2 + Q22 n^4
• C. Qbar11 = Q11 n^4 + 2(Q12+2Q66)m^2 n^2 + Q22 m^4
• D. Qbar11 = (Q11+Q22-4Q66)m^2 n^2 + Q12(m^4+n^4)

Answer: (B)

The main idea is how in-plane stiffness of a lamina changes when you rotate the material axes by an angle θ. The in-plane reduced stiffness along the x’ direction (Qbar11) is obtained by projecting the original stiffness components onto the rotated axes. The correct form combines contributions from Q11 and Q22 along with the coupling terms Q12 and Q66, all modulated by the orientation factors m = cosθ and n = sinθ. Qbar11 = Q11 m^4 + 2(Q12 + 2Q66) m^2 n^2 + Q22 n^4. Here, the m^4 term comes from the portion of the original Q11 that lies along x’ when you rotate, the n^4 term comes from the portion of Q22, and the cross term 2(Q12 + 2Q66) m^2 n^2 captures how Poisson coupling (Q12) and in-plane shear stiffness (Q66) contribute to stiffness along x’ through the interaction of the x and y directions after rotation. The factor 2 reflects the symmetric contributions from the two directions in the rotation. This expression also behaves correctly in limiting cases: at θ = 0, Qbar11 reduces to Q11; at θ = 90°, it reduces to Q22, illustrating how the rotated stiffness blends the original components.

The main idea is how in-plane stiffness of a lamina changes when you rotate the material axes by an angle θ. The in-plane reduced stiffness along the x’ direction (Qbar11) is obtained by projecting the original stiffness components onto the rotated axes. The correct form combines contributions from Q11 and Q22 along with the coupling terms Q12 and Q66, all modulated by the orientation factors m = cosθ and n = sinθ.

Qbar11 = Q11 m^4 + 2(Q12 + 2Q66) m^2 n^2 + Q22 n^4.

Here, the m^4 term comes from the portion of the original Q11 that lies along x’ when you rotate, the n^4 term comes from the portion of Q22, and the cross term 2(Q12 + 2Q66) m^2 n^2 captures how Poisson coupling (Q12) and in-plane shear stiffness (Q66) contribute to stiffness along x’ through the interaction of the x and y directions after rotation. The factor 2 reflects the symmetric contributions from the two directions in the rotation.

This expression also behaves correctly in limiting cases: at θ = 0, Qbar11 reduces to Q11; at θ = 90°, it reduces to Q22, illustrating how the rotated stiffness blends the original components.