Which expression correctly defines Qbar11 for a rotated lamina in terms of Q11, Q12, Q66, Q22 and m = cos theta, n = sin theta?

Which expression correctly defines Qbar11 for a rotated lamina in terms of Q11, Q12, Q66, Q22 and m = cos theta, n = sin theta?

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

Answer: (B)

When a lamina is rotated, the stiffness seen along the global axes is a transformed version of the in-material stiffness, built from how the material directions project onto the rotated axes. The terms m and n are the direction cosines, with m = cos theta and n = sin theta. For the global x-direction, the transformed reduced stiffness Qbar11 collects contributions from the material-stiffness along its 1-direction (Q11), along its 2-direction (Q22), and from the coupling terms Q12 and Q66 that mix normal and shear responses. The correct expression weights Q11 by m^4, Q22 by n^4, and the cross-coupled part by m^2 n^2, specifically 2(Q12 + 2Q66) m^2 n^2. This combination arises because projecting the material axes into the global x-direction involves fourth-order products of the direction cosines, and the shear term Q66 contributes to both normal and shear coupling in the rotated frame. The factor of 2 accounts for the symmetric contributions in the transformation. A quick check helps build intuition: if theta is 0, m = 1 and n = 0, so Qbar11 reduces to Q11, as expected when the global x-axis aligns with the material 1-direction. If theta is 90 degrees, m = 0 and n = 1, and Qbar11 reduces to Q22, which again makes sense. The other forms would place the same terms with m and n swapped or omit the shear-coupling contribution, which would correspond to a different transformed component (not Qbar11) or ignore part of the rotated stiffness.

When a lamina is rotated, the stiffness seen along the global axes is a transformed version of the in-material stiffness, built from how the material directions project onto the rotated axes. The terms m and n are the direction cosines, with m = cos theta and n = sin theta. For the global x-direction, the transformed reduced stiffness Qbar11 collects contributions from the material-stiffness along its 1-direction (Q11), along its 2-direction (Q22), and from the coupling terms Q12 and Q66 that mix normal and shear responses.

The correct expression weights Q11 by m^4, Q22 by n^4, and the cross-coupled part by m^2 n^2, specifically 2(Q12 + 2Q66) m^2 n^2. This combination arises because projecting the material axes into the global x-direction involves fourth-order products of the direction cosines, and the shear term Q66 contributes to both normal and shear coupling in the rotated frame. The factor of 2 accounts for the symmetric contributions in the transformation.

A quick check helps build intuition: if theta is 0, m = 1 and n = 0, so Qbar11 reduces to Q11, as expected when the global x-axis aligns with the material 1-direction. If theta is 90 degrees, m = 0 and n = 1, and Qbar11 reduces to Q22, which again makes sense.

The other forms would place the same terms with m and n swapped or omit the shear-coupling contribution, which would correspond to a different transformed component (not Qbar11) or ignore part of the rotated stiffness.