Joan I. Arimanwa

Closed-form solutions simplify the design of thick orthotropic plates, but their accuracy for plates with a free edge is unknown. This study developed one-term closed-form and multi-term series solutions of Alternative 1 Theory, with three-dimensional (3D) constitutive relations, for uniformly loaded rectangular orthotropic plates with a free edge. Ten support combinations, with the free edge parallel or perpendicular to the stiff direction, were analysed in 1,200 cases (E₁/E₂ = 5–40, a/t = 4–20, b/a = 0.5–3.0). A 3D Ritz solution, with deflections within 0.08% of exact, Lévy-type, and finite element solutions, served as the reference, and the series separated shape error from theory error. The converged series underestimated the 3D deflection by 4–19% at a/t = 4 and by up to 10% at a/t = 20. Its error was largest where the stiff direction spans between a clamped and a simply supported edge; there, it also overestimated the clamped-edge moment by up to 129%. Free-edge orientation governed the response. Plates with the free edge across the stiff direction deflected up to 1,050 times as much as their quarter-turned counterparts, and the one-term solution underestimated their deflection by 11–95% for b/a ≥ 1.0. The one-term formulas were within 10% of 3D elasticity in 319 cases, half of them for four plates with a/t ≤ 5 and b/a ≥ 1.0, partly through a cancellation of shape and theory errors. In thick plates, 3D face stresses at the free edge exceeded the linear values by up to 110%.

Classical plate theory (CPT)¹ is the default model for the bending of thin orthotropic plates, yet the span–thickness ratio beyond which it can be trusted is typically taken from isotropic experience. This study determines this ratio from three-dimensional (3D) elasticity for the 21 distinct orthotropic rectangular plates with clamped, simply supported, and at most one free edge, under uniformly distributed loading. A Ritz solution of 3D elasticity, validated against exact and finite-element solutions, was compared with CPT over 3,360 cases (E₁/E₂ = 5–40, aspect ratios 0.5–3, span–thickness ratios 4–100). The ratio needed for CPT to be within 5% of the 3D deflection ranges from 10 to 90, and the common threshold of 20 fails in 76% of the 420 combinations. For aspect ratios of 1.5 or more and the 15 plates whose fibres end on supported edges, the limit depends only on those two edges and scales with the square root of E₁/G₁₃. These limits follow a Timoshenko beam strip spanning the fibres, which is first-order shear deformation theory in cylindrical bending: the group averages agree within 0.3% and individual plates within 4.5% (9% in one case). This yields a closed-form rule: the square of the limit equals 0.96, 2.65, or 4.80 times E₁/(εG₁₃) for simply supported, propped, and clamped strips, where ε is the tolerance. The rule holds within 1.2% for graphite–epoxy, glass–epoxy, and glulam plates, and soft simple supports increase the limits by at most 3%. Plates with a free edge across the fibres and no clamped edge opposite it carry the load across the fibres and follow √(E₂/G₂₃) instead. Deflection governs the limit in 88% of cases; moments govern mainly when a free edge crosses the fibres.