What is the ABD matrix in Classical Laminate Theory (CLT)?

The ABD matrix is a 6×6 stiffness matrix that governs the constitutive behavior of composite laminates. The A submatrix (in-plane stiffness) relates membrane forces to in-plane strains, the B submatrix (coupling stiffness) couples bending to membrane response, and the D submatrix (bending stiffness) relates bending moments to curvatures. Each submatrix is assembled analytically from the ply thickness, fiber orientation, and elastic constants. For a symmetric laminate, B = 0 and bending-extension coupling vanishes.

How are equivalent elastic moduli Ex and Ey computed for a laminate?

The equivalent elastic moduli are derived from the inverse of the A matrix (compliance matrix a = A⁻¹): Ex = 1/(h·a₁₁), Ey = 1/(h·a₂₂), Gxy = 1/(h·a₆₆), νxy = −a₁₂/a₁₁, where h is the total laminate thickness. These are smeared-property constants that treat the laminate as a single equivalent orthotropic plate, and they differ from the individual ply properties.

What happens when the Tsai-Wu failure index exceeds 1.0?

A Tsai-Wu failure index (FI) greater than 1.0 predicts failure initiation in that ply. The criterion evaluates (σ₁/F₁t)² + (σ₂/F₂t)² + 2F₁₂σ₁σ₂ + (τ₁₂/F₆)² ≤ 1. The ply that first reaches FI = 1 defines the First Ply Failure (FPF) load. For ultimate load capacity, a progressive failure analysis is required, where the failed ply's stiffness is reduced (or zeroed) and the remaining laminate is re-analyzed.

Does ply stacking order affect laminate stiffness in [0/90/±45]s vs [±45/0/90]s?

For symmetric laminates (the 's' suffix denotes symmetry), the in-plane stiffness A matrix is independent of stacking order. However, the bending stiffness D matrix depends on the cube of each ply's distance from the mid-plane, so ply order has a significant effect on bending and buckling behavior. Placing high-stiffness plies at the outermost positions maximizes the D matrix. Asymmetric laminates also see changes in the coupling matrix B with reordering.

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Composite Analysis Tool

Composite Laminate Analysis (CLT / ABD Matrix)

Compute ABD matrix, equivalent elastic moduli, fiber-angle-dependent stiffness, and ply failure indices in real time using Classical Laminate Theory.

$$A_{ij}=\sum_{k=1}^{N}\bar{Q}_{ij}^{(k)}t_k,\quad E_x=\frac{A_{11}A_{22}-A_{12}^2}{A_{22}\cdot h}$$

Material & Laminate Parameters

Material Preset

E1, E2, G12 [GPa], ν12, t_ply [mm]

Stacking Sequence

e.g. [0/90/±45]s → symmetric / [0/45/-45/90]T → total

Failed to parse stacking sequence

Ply count: 8

Applied Load Nx 100 N/mm

In-plane force per unit width (tension direction)

Applied Load Ny 0 N/mm

Sweep fiber angle 0°→180° to display Ex(θ)

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GPa

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GPa

Gxy

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GPa

νxy

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Total Thickness h

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Max FI (Ply#)

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Max Failure Index

Ex(θ) — Stiffness Anisotropy

Ply Failure Index FI

Theory — Classical Laminate Theory (CLT)

ABD Matrix

$$A_{ij}=\sum_{k=1}^{N}\bar{Q}_{ij}^{(k)}t_k$$

$$B_{ij}=\frac{1}{2}\sum_{k=1}^{N}\bar{Q}_{ij}^{(k)}(z_k^2-z_{k-1}^2)$$

$$D_{ij}=\frac{1}{3}\sum_{k=1}^{N}\bar{Q}_{ij}^{(k)}(z_k^3-z_{k-1}^3)$$

Equivalent Elastic Moduli

$$E_x=\frac{A_{11}A_{22}-A_{12}^2}{A_{22}\cdot h}$$

$$E_y=\frac{A_{11}A_{22}-A_{12}^2}{A_{11}\cdot h}$$

$$G_{xy}=\frac{A_{66}}{h},\quad \nu_{xy}=\frac{A_{12}}{A_{22}}$$

Transformed Stiffness

With $m=\cos\theta,\ n=\sin\theta$:

$$\bar{Q}_{11}=Q_{11}m^4+2(Q_{12}+2Q_{66})m^2n^2+Q_{22}n^4$$

Maximum Stress Failure Index

$$FI=\max\!\left(\frac{|\sigma_1|}{X},\frac{|\sigma_2|}{Y},\frac{|\tau_{12}|}{S_{12}}\right)$$

Failure when FI ≥ 1.0. Evaluated in each ply fiber coordinate system.