🧩 eLaminate Composite Laminate ABD Matrix Calculator

eLaminate is an online tool for composite structural design and teaching, based on Classical Laminate Theory (CLT). It computes ABD stiffness matrices, effective engineering constants, hygrothermal resultants, and ply-level strains/stresses with failure indices for arbitrary stacking sequences.

Composite laminates are built by stacking orthotropic plies at designed orientations. They are widely used in aerospace structures, wind-turbine blades, automotive lightweighting, marine structures, and sports equipment. Unlike isotropic materials, laminate stiffness and strength depend strongly on stacking sequence and ply angles, which must be analyzed through the ABD matrix.

1. Ply Stiffness and Coordinate Transformation

A unidirectional composite ply has four independent engineering constants in material coordinates (fiber direction 1, transverse direction 2): longitudinal modulus E₁, transverse modulus E₂, major Poisson’s ratio ν₁₂, and in-plane shear modulus G₁₂. These form the plane-stress reduced stiffness matrix [Q]:

When a ply is oriented at angle θ, [Q] is transformed into the global reduced stiffness [Q̄]. The transformation involves fourth-power terms in cos θ and sin θ; this tool performs the transformation automatically.

2. Physical Meaning of the ABD Matrix

Integrating [Q̄] through the thickness yields the laminate ABD stiffness matrix, which relates mid-plane strains ε⁰ and curvatures κ to in-plane forces N and moments M:

• [A] Extensional stiffness (lb/in or N/mm): relates in-plane forces to mid-plane strains; controls membrane stiffness.
• [B] Coupling stiffness (lb or N): describes extension–bending and shear–twist coupling. Zero for symmetric laminates, which greatly simplifies design.
• [D] Bending stiffness (lb·in or N·mm): relates moments to curvatures; controls flexural and torsional stiffness.

This tool inverts the full 6×6 ABD matrix by Gaussian elimination, solves for mid-plane strains and curvatures under combined mechanical and hygrothermal loads, then evaluates stresses and strains in each ply.

3. Hygrothermal Effects

Temperature change ΔT and moisture change ΔM produce free expansion strains in each ply. Inter-ply constraint converts these expansions into equivalent hygrothermal forces N^HT and moments M^HT:

Here α are coefficients of thermal expansion and β are coefficients of moisture expansion. The tool automatically adds hygrothermal resultants to the mechanical loads before solving the ABD system.

1. Define the material library: Edit existing materials or click “Add Material”. Each material needs E₁, E₂, ν₁₂, G₁₂, ply thickness t_ply, thermal expansion coefficients α₁/α₂, moisture expansion coefficients β₁/β₂, and strain allowables (for failure assessment).
2. Build the stacking sequence: Add, delete, or reorder plies. Select a material and enter the orientation angle (degrees). Total ply count and thickness update automatically. Prefer symmetric stacks to eliminate B-matrix coupling.
3. Apply loads: Enter in-plane forces Nx, Ny, Nxy (lb/in) and moments Mx, My, Mxy (in·lb/in), plus temperature change ΔT (°F) and moisture change ΔM (%).
4. Run the analysis: Click “Run Full Analysis”. The tool outputs the complete ABD matrix, effective engineering constants, hygrothermal resultants, and ply-level strains/stresses with failure indices.
5. Interpret results: Focus on minimum Margin of Safety (MS) and Tsai-Hill index. MS > 0 means the ply has not reached its strain limit under the current loads; Tsai-Hill values approaching 1 indicate higher failure risk.

All calculations run locally in the browser; no data is uploaded. The unit system is Imperial (psi, lb/in). Convert inputs yourself if you prefer SI units.

Material Inputs

• E1 / E2: Young’s moduli in the fiber and transverse directions (psi). Carbon/epoxy E1 is typically 20–30 Msi; glass/epoxy about 5–10 Msi.
• ν12: Major Poisson’s ratio, typically 0.25–0.40.
• G12: In-plane shear modulus (psi), usually much lower than E1.
• tply: Cured ply thickness (in); common prepreg thicknesses are 0.005–0.010 in.
• α1 / α2: Coefficients of thermal expansion (1/°F). Fiber-direction α1 is often near zero or slightly negative; transverse α2 is larger.
• β1 / β2: Coefficients of moisture expansion (%). Used to model swelling due to moisture absorption.
• Strain allowables (e1tu, e1cu, e2tu, e2cu, e12s): Used for Margin of Safety — longitudinal tension/compression, transverse tension/compression, and shear strain limits.

Output Interpretation

• Effective engineering constants: Equivalent membrane moduli Ex, Ey, Gxy, Poisson ratios, and bending moduli Exb, Eyb derived from A and D — useful for comparison with isotropic materials.
• Hygrothermal resultants N^HT: Equivalent in-plane forces caused by temperature and moisture change; automatically included in the solution.
• Ply strains and stresses: Material-axis strains (ε1, ε2, γ12) and stresses (σ1, σ2, τ12) evaluated at each ply mid-plane.

Margin of Safety (MS)

This tool uses a strain-based definition of Margin of Safety:

The three strain components are evaluated and the minimum is retained. MS ≥ 0 means the ply has not reached its strain limit; larger MS indicates greater reserve. Engineering practice often requires MS ≥ 0.15–0.5 depending on the required factor of safety.

Tsai-Hill Failure Criterion

Tsai-Hill is a quadratic interactive criterion:

X, Y, and S are the longitudinal, transverse, and shear strengths. This tool approximates strengths as “modulus × strain allowable”. Values approaching or exceeding 1 indicate predicted failure. Compared with maximum-strain criteria, Tsai-Hill accounts for stress interaction and is useful for preliminary assessment.

Note: Formal design should also incorporate test data, safety factors, damage-tolerance analysis, and more advanced criteria (Hashin, Puck, LaRC, etc.). Results from this tool are intended for education and conceptual design only.