LamIO — Classical Laminate Theory

Jones / Kaw — plane stress • Moduli/stresses in ksi; loads in lb/in & lb-in/in; [A] in lb/in, [B] in lb, [D] in lb-in. Strains reported as με.

LamIO assembles the laminate stiffness matrices [A], [B], and [D] from ply reduced stiffness [Q] transformed to [Q̄] at each orientation. Resultant loads {N, M} are related to midplane strains {ε⁰} and curvatures {κ} through the coupled system. Use this to evaluate extension, bending, coupling, and per-ply material strains under applied loads.

Governing Equations

• Constitutive relation

  {N} = [A]{ε⁰} + [B]{κ}, {M} = [B]{ε⁰} + [D]{κ}

• Ply integration

  Aᵢⱼ = Σ Q̄ᵢⱼ⁽ᵏ⁾(zₖ − zₖ₋₁), Bᵢⱼ = ½Σ Q̄ᵢⱼ⁽ᵏ⁾(zₖ² − zₖ₋₁²), Dᵢⱼ = ⅓Σ Q̄ᵢⱼ⁽ᵏ⁾(zₖ³ − zₖ₋₁³). z measured from laminate midplane.

• Ply strains

  {ε}ₓᵧ(z) = {ε⁰} + z{κ}. Material strains {ε}₁₂ obtained via strain transformation at ply angle θ.

Features

• Solid laminate

  Arbitrary ply stack bottom-to-top. Symmetric layups (mirror about midplane) produce B ≈ 0 and decouple extension from bending.

• Sandwich panel

  Bottom face + isotropic core + top face. Core modeled as a 0° isotropic ply (E₁ = E₂ = E). Mirror option duplicates top face plies for the bottom skin.

• Effective moduli

  Ēₓ = 1/(h·a₁₁), Ēᵧ = 1/(h·a₂₂), Ḡₓᵧ = 1/(h·a₆₆) from [A]⁻¹. Quasi-isotropic flag when Ēₓ/Ēᵧ ≈ 1 (±8%).

• Polar plot

  Laminate Ēx, Ēy, Ḡxy vs orientation (0–360°) by rotating the reference axes. A near-circular trace indicates quasi-isotropic extensional response.

• Max-stress criterion

  Independent ratios σ₁/X, σ₂/Y, τ₁₂/S with tension/compression strengths. FI = max(ratios); FI ≤ 1 is PASS.

Linear elastic CLT only. No progressive failure, no interlaminar shear, no transverse shear in sandwich (thin-face assumption). Verify critical plies with test data or more advanced failure theories (Tsai-Wu, Hashin).