Composite Laminate Notation and ABD Matrix
Laminate notation, ply orientation conventions, and the ABD matrix relationships for classical laminate analysis.
Laminate Notation
A laminate is described by listing ply angles from the tool side (bottom) to the bag side (top). Subscripts denote repetition and symmetry.
[0/±45/90]₂s Reading: 0°, +45°, −45°, 90°, 90°, −45°, +45°, 0° Subscript ₂ = repeat twice Subscript s = symmetric about midplane Other notations: [0/90]₃ = 0/90/0/90/0/90 [0₂/45₂]s = 0/0/45/45/45/45/0/0 [0/90]ns = n sub-laminates, symmetric
ABD Matrix
Constitutive equation (CLT):
⎧ N ⎫ ⎡ A B ⎤ ⎧ ε⁰ ⎫
⎨ ⎬ = ⎢ ⎥ ⎨ ⎬
⎩ M ⎭ ⎣ B D ⎦ ⎩ κ ⎭
N = membrane force resultants (N/m)
M = moment resultants (N·m/m)
ε⁰ = midplane strains
κ = curvatures
Extensional stiffness:
A_ij = Σ_k Q̄_ij^(k) · t_k
Bending-extension coupling:
B_ij = ½ Σ_k Q̄_ij^(k) · (z_k² − z_{k-1}²)
Bending stiffness:
D_ij = (1/3) Σ_k Q̄_ij^(k) · (z_k³ − z_{k-1}³)
where Q̄_ij = transformed reduced stiffness of ply k
t_k = thickness of ply k
z_k = distance to top of ply k from midplaneKey Relationships
- Symmetric laminate: B = 0 (no bending-extension coupling)
- Balanced laminate: A₁₆ = A₂₆ = 0 (equal +θ and −θ plies)
- Isotropic equivalent: A₁₁ = A₂₂, A₁₆ = A₂₆ = B = 0 (quasi-isotropic, e.g. [0/±45/90]s)
- For symmetric laminates, the in-plane and bending responses decouple
Variables
| Symbol | Definition | Units |
|---|---|---|
| A_ij | Extensional stiffness | N/m |
| B_ij | Coupling stiffness | N |
| D_ij | Bending stiffness | N·m |
| Q̄_ij | Transformed reduced stiffness of ply | Pa |
| N | Membrane force resultant | N/m |
| M | Moment resultant | N·m/m |
| ε⁰ | Midplane strain | — |
| κ | Curvature | 1/m |
| t_k | Ply thickness | mm |
| z_k | Distance to ply boundary from midplane | mm |
Related Knowledge
See the Composite Structures Knowledge category for classical laminate theory and composite FEA.