Each ply's own stiffness comes from the rule of mixtures: fiber and matrix moduli combine as E₁ = V᷇·E᷇ + (1−V᷇)·E᷇ₓ along the fiber (Voigt, iso-strain) and 1/E₂ = V᷇/E᷇ + (1−V᷇)/E᷇ₓ across it (Reuss, iso-stress) — the two textbook bounds, not a single interpolated curve. That gives every ply an orthotropic in-plane stiffness matrix Q, which is then rotated into the beam axis by the standard tensor transform to get Q̄ for whatever angle you set.
The laminate's bending stiffness follows classical laminate theory (CLT): each ply's rotated stiffness is weighted by (zᵈ³−zᵈ₋₁³)/3 — its distance from the mid-plane, cubed — and summed into the laminate bending-stiffness term D₁₁. A ply far from the mid-plane (near the outer faces) contributes far more bending stiffness than the same ply near the center, exactly like the flanges of an I-beam matter more than its web. This is a genuinely different calculation from a simple orientation-averaged modulus: two laminates with the same plies in a different stacking order can have the same axial modulus but different D₁₁, and this model shows that.
The beam is then treated as a simply-supported span under a center point load, deflecting by the standard δ = FL³/(48·EI) with EI = width·D₁₁, and each ply's own bending stress recovered from its distance off the mid-plane and its own (rotated) modulus — so a stiff 0° ply and a soft 90° ply at the same height carry very different stress under the same curvature.
- Eᵇ (flexural) — effective modulus recovered from D₁₁ via Eᵇ=12D₁₁/h³; depends on stacking order, not just ply fractions.
- Eₓ (axial) — simple thickness-weighted average of every ply's rotated modulus; independent of stacking order.
- Peak ply stress — the largest |σ| among all plies at the current load, evaluated at each ply's own mid-height and its own rotated modulus.
Real-world relevance: this is exactly why aircraft wing skins and wind-turbine blades are laid up as a deliberate stacking sequence (e.g. [0/45/−45/90]) rather than just "the right fraction of 0° plies" — ply order changes bending stiffness and where peak stress lands, even when the total fiber content is identical.