Compute laminate [A], [B], and [D] stiffness matrices using Classical Lamination Theory for a unidirectional orthotropic lamina.
What is an ABD matrix?
In composite laminate mechanics, the ABD matrix links in-plane loads and bending moments to mid-plane strains and curvatures. It is the core stiffness model used in Classical Lamination Theory (CLT), especially for carbon fiber and glass fiber laminate design.
- A matrix: in-plane extensional stiffness
- B matrix: bending-extension coupling stiffness
- D matrix: bending stiffness
How to use this ABD matrix calculator
1) Enter lamina engineering constants
Provide E1, E2, G12, and ν12 for one ply material. This tool assumes the same material properties for all plies.
2) Enter thickness and stacking sequence
Input a uniform ply thickness and a list of ply angles in degrees. Angles are interpreted from bottom ply to top ply. A symmetric sequence such as [0/45/-45/90]s typically yields a near-zero B matrix.
3) Click Calculate
The calculator builds transformed reduced stiffness for each ply, integrates through thickness, and reports A, B, D in SI units.
Equations behind the calculator
This page uses plane-stress orthotropic lamina relations. For each ply angle, the reduced stiffness matrix is transformed into global laminate axes, then integrated by thickness coordinates zk:
- A = Σ Q̄k(zk - zk-1)
- B = 1/2 Σ Q̄k(zk2 - zk-12)
- D = 1/3 Σ Q̄k(zk3 - zk-13)
Here, Q̄k is the transformed reduced stiffness matrix of ply k.
Interpreting your results
If B is near zero
The laminate is approximately uncoupled in bending-extension response, which is often preferred in structural panels and aerospace skins.
If B is large
The laminate is unsymmetric or intentionally coupled. In-plane loads can induce bending curvature and moments can induce in-plane strain.
A and D trends
Larger 0° content usually increases axial A11. More off-axis plies can improve shear-related terms and damage tolerance. D terms scale strongly with laminate thickness, so small thickness changes can significantly affect bending behavior.
Practical design notes
- Keep units consistent. This calculator expects GPa and mm, then converts internally to SI.
- Use balanced layups (e.g., +45/-45 pairs) to reduce in-plane shear coupling.
- Use symmetric layups to suppress B-matrix coupling when required.
- Validate final designs with detailed finite element analysis and failure criteria.
Limitations
This tool is intended for preliminary laminate stiffness estimation. It does not include thermal expansion, moisture effects, nonlinear behavior, progressive failure, interlaminar stresses, or ply-by-ply strength checks such as Tsai-Wu or Hashin.