Composite Laminate Optimization

Material Properties & the On-Axis Reduced Stiffness Matrix

Every composite analysis begins by characterizing the individual ply in its on-axis (1–2) coordinate frame. Four independent elastic constants define an orthotropic ply: fiber-direction modulus E₁₁, transverse modulus E₂₂, major Poisson's ratio ν₁₂, and in-plane shear modulus G₁₂. From these the reduced stiffness matrix [Q] is assembled — the plane-stress version of the 3D constitutive relation.

The stiffness invariants U₁–U₅ (Tsai-Pagano) decouple the angle-dependent terms from the material constants. This allows the off-axis stiffness Q̄(θ) to be written as a simple Fourier series in 2θ and 4θ — enabling fast, closed-form gradient calculations during optimization.

ABD Matrix & Effective Laminate Moduli

Once individual plies are characterized, Classical Lamination Theory (CLT) assembles the full laminate response through the ABD matrix. The extensional stiffness sub-matrix [A] links in-plane loads {N} to mid-plane strains {ε⁰}; the bending stiffness sub-matrix [D] links moments {M} to curvatures {κ}. For a symmetric, balanced laminate, the coupling matrix [B] = 0 and A₁₆ = A₂₆ = 0, decoupling extension from bending and shear from extension.

Applied in-plane loads {N} are converted to mid-plane strains via the A-inverse: {ε⁰} = [A]⁻¹·{N}. These are the laminate-level strains; individual ply strains are identical for a membrane problem, but on-axis ply stresses differ by angle since each ply "sees" a different component of the global strain in its fiber-transverse frame.

Ply Count Optimization Under Strain Constraints

Problem: Determine the minimum number of 0°, ±45°, and 90° plies needed to keep the laminate mid-plane strains below specified limits under a combined in-plane load. The ply thickness is fixed at 0.005 in (glass/epoxy system); only the count per angle is varied.

The 0° plies dominate (24 of 41) because axial load Nₓₓ drives the εₓ constraint. The ±45° balance pair handles the shear load N_xy and provides the shear stiffness needed to satisfy ε_xy without requiring a separate shear ply orientation. The single 90° ply is a minimum health ply — providing just enough transverse stiffness to prevent matrix cracking.

Angle Laminate Optimization — Maximize Shear Modulus

Problem: For a symmetric balanced carbon/epoxy laminate [+θ/−θ]_s, find the ply angle θ that maximizes the laminate shear modulus G_xy. The four-ply laminate has a fixed ply thickness of 0.127 mm (5 mil) per ply — only the angle is free. Excel Solver (GRG Nonlinear method) drives the optimization.

Multi-Material Ply Schedule Optimization

The most comprehensive optimization selects both the material type and the number of plies at each angle simultaneously — a mixed-integer programming problem solved with Excel Solver. Four candidate materials are available; the optimizer assigns binary (AMID) variables to select which material fills each angle-slot and integer ply-count variables for the schedule, subject to multi-axial strain constraints.

IM7/8552 UD was selected for the dominant load-bearing plies due to its superior fiber-direction modulus (22.3 Mpsi vs. 18.7 for AS4). A single AS4/8552 plain-weave fabric ply was included for balanced off-axis stiffness. The final ply schedule minimizes total thickness (and therefore weight) while satisfying all three strain constraints simultaneously under the combined load state.

Plate Bending — Minimizing Deflection via the D-Matrix

In-plane optimization (the A matrix) addresses membrane loads. For structures under transverse (out-of-plane) loading, the bending stiffness matrix [D] governs the response. D_ij depends on ply position through the laminate thickness — plies farthest from the mid-plane contribute most to bending stiffness (z² weighting), exactly analogous to the second moment of area in beam theory.

A 5 × 5 inch simply-supported plate under uniform pressure q₀ = 1000 lb/in² was analyzed using a double Fourier series expansion. The deflection field W(x,y) is expressed as a sum of sinusoidal mode shapes (m, n = 1…5); each mode's amplitude W_mn depends on the effective bending stiffness D_eff(m,n) and the load coefficient Q_mn.

Vibration Frequency Tailoring

Composite laminates offer a unique capability absent in isotropic materials: the natural frequencies of a plate can be tuned independently by varying ply angles, because the anisotropic D-matrix couples bending stiffness differently along different axes. Three optimization problems demonstrate this, all using the same glass/epoxy system (E₁₁ = 18.5 Mpsi).

The fundamental natural frequency of a simply-supported rectangular plate is:

VibEx2 is particularly compelling: by using a two-angle laminate [±20.6° / ±69.7°], the optimizer independently controls D₁₁, D₂₂, and D₁₂+2D₆₆ to simultaneously place the (1,1) mode at exactly 60 Hz and the (1,2) or (2,1) mode at exactly 120 Hz — a 2:1 ratio. This is only possible because the composite's anisotropy provides two free variables (the two ply angles) to satisfy two frequency constraints simultaneously.

Newton's Method for Nonlinear Optimization

Many composite optimization problems yield nonlinear systems — for example, finding the ply angle θ* such that dG_xy/dθ = 0 analytically. Newton's method provides quadratic convergence for smooth objective functions. The workbook implements both the single-variable and two-variable (Jacobian-inverse) versions.

The Newton's method workbook underpins the nonlinear solvers used elsewhere — particularly for the angle optimization in Ex2 and the frequency-targeting problems in VibEx2, where the objective functions are smooth trigonometric functions of the design variables θ.