The science
Ply constants. Rule of mixtures along the fibres, E1 = VfEf + (1−Vf)Em (Voigt bound); across the fibres and in shear the Halpin-Tsai estimate P = Pm(1 + ξηVf)/(1 − ηVf), η = (Pf/Pm − 1)/(Pf/Pm + ξ) with ξ = 2 for E2 and ξ = 1 for G12. The Reuss (series) value is the lower bound.
Off-axis stiffness. A ply rotated by θ has 1/Ex = cos&sup4;θ/E1 + (1/G12 − 2ν12/E1) sin²θ cos²θ + sin&sup4;θ/E2. A few degrees off the fibres already costs a large part of the stiffness.
Classical lamination theory. Ply stiffness Q is rotated to Q̄(θ); the laminate sums A = ∑ Q̄(zk−zk−1), D = ∑ Q̄(zk³−zk−1³)/3. The layups are symmetric so B = 0. Under a root moment per unit width m = PL/b the curvature is κ = D−1(m,0,0); the effective bending stiffness per width is D* = 1/(D−1)11.
Cantilever. δ = P L³ / (3 b D*), deflection curve w(x) = P x²(3L − x)/(6 b D*) (small deflections, shear deformation ignored, which is poor for thick, very stiff-in-bending-but-weak-in-shear stacks).
Failure. Ply strains ε = zκ give ply stresses Q̄ε, rotated to the fibre axes (σ1, σ2, τ12). The Tsai-Hill index (σ1/X)² − σ1σ2/X² + (σ2/Y)² + (τ12/S)² reaches 1 at first-ply failure; since the index is quadratic in load, the failure load is P/√FI. Strengths X = 1500 MPa, Y = 40 MPa, S = 70 MPa are typical carbon/epoxy values, not a datasheet.
Honest limits. Linear elastic, no residual thermal stress, no free-edge stresses or delamination, no progressive damage after first ply failure, no shear deformation, same strength in tension and compression, constant constants independent of Vf. Material constants are typical values for illustration (Ef 230 GPa, Em 3.5 GPa, νf 0.2, νm 0.35); spot-check against a datasheet before real use. Plies are drawn 10 times thicker than real so the layers are visible.
Independent educational model; not a design tool.