Classical laminate theory and first-ply failure

From plies to a plate

Classical laminate theory stacks anisotropic plies into a single plate stiffness. Each ply contributes its transformed stiffness Q̄(θ) weighted by position: the A matrix (membrane) integrates Q̄ through the thickness, D (bending) weights it by z², and B (coupling) by z.

The B matrix is the interesting one: it couples stretching to bending. A laminate with B ≠ 0 curls when you pull it and stretches when you bend it — usually unwanted, which is why practical layups are symmetric about the midplane, forcing B = 0 identically.

Quasi-isotropic layups

Stacking equal fractions of 0°, ±45°, and 90° plies (e.g. [0/±45/90]s) makes the in-plane stiffness direction-independent — the laminate satisfies G = E/2(1+ν) like an isotropic metal, while its strength remains strongly directional. This is the workhorse layup when load directions are uncertain.

First-ply failure

With midplane strains solved from the loads, each ply's stresses rotate back into its material axes and feed a failure criterion. Max stress checks each mode independently and tells you which mode governs; Tsai–Wu blends all stress components into one quadratic interaction, capturing the biaxial strengthening and weakening the separate checks miss.

The strength ratio R reported by the calculator is the multiplier on the entire load state that brings the critical ply to failure: R = 1.8 means 80% margin. First-ply failure is conservative for laminates that tolerate matrix cracking — transverse cracks in 90° plies typically appear long before fibers break — so ultimate strength predictions need progressive-failure analysis, planned for a later milestone.