In Classical Lamination Theory (CLT) for aerospace composite plates governed by the constitutive relation $\begin{bmatrix} N \\ M \end{bmatrix} = \begin{bmatrix} A & B \\ B & D \end{bmatrix} \begin{bmatrix} \epsilon^0 \\ \kappa \end{bmatrix}$, which of the following statements are TRUE?

(A) For a symmetric laminate, the extension-bending coupling matrix is identically zero ($[B] = [0]$).

(B) When $[B] = [0]$, in-plane mechanical forces ($N$) produce zero out-of-plane bending curvatures ($\kappa$).

(C) The extensional stiffness matrix $[A]$ depends on the individual ply stiffnesses and ply thicknesses.

(D) A quasi-isotropic laminate behaves elastically anisotropic in in-plane extensional stiffness.

Answer Key:

A, B, C

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Coupling Matrix Definition:** $B_{ij} = \frac{1}{2} \sum_{k=1}^N (\bar{Q}_{ij})_k (z_k^2 - z_{k-1}^2)$. For laminates symmetric about the midplane ($z = 0$), plies with identical stiffness and orientation at $\pm z$ cancel out, yielding $[B] = [0]$ (Statement A is TRUE). 2. **Stage 2: Extension-Bending Decoupling:** When $[B] = 0$, $N = A \epsilon^0$ and $M = D \kappa$. In-plane loads do not induce warping or bending curvature (Statement B is TRUE). 3. **Stage 3: Extensional Stiffness Matrix:** $A_{ij} = \sum_{k=1}^N (\bar{Q}_{ij})_k (z_k - z_{k-1}) = \sum (\bar{Q}_{ij})_k t_k$ (Statement C is TRUE). 4. **Stage 4: Quasi-Isotropy:** By definition, quasi-isotropic laminates have in-plane stiffness invariant under coordinate rotation (Statement D is FALSE). 5. **Stage 5: Official Reference:** Robert M. Jones, *Mechanics of Composite Materials*, Chapter 4.

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