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In the laminate ABD constitutive equation, the COUPLING matrix B is computed as:
A{'text': '$B_{ij} = \\sum_k \\bar{Q}_{ij}^k (z_k - z_{k-1})$', 'label': 'A'}
B{'text': '$B_{ij} = (1/2) \\sum_k \\bar{Q}_{ij}^k (z_k^2 - z_{k-1}^2)$', 'label': 'B'}
C{'text': '$B_{ij} = 0$ always', 'label': 'D'}
D{'text': '$B_{ij} = (1/3) \\sum_k \\bar{Q}_{ij}^k (z_k^3 - z_{k-1}^3)$', 'label': 'C'}
Answer & Solution
Correct answer: B. {'text': '$B_{ij} = (1/2) \\sum_k \\bar{Q}_{ij}^k (z_k^2 - z_{k-1}^2)$', 'label': 'B'}
ABD entries: A is linear in z (∝ thickness), B is quadratic in z (extension-bending coupling via first moment of area), D is cubic in z (bending). The factor 1/2 in B comes from the z² integral.
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