Cross-Sectional Distortion in Polar and Nonpolar Orthotropic Lumber : A Unified Analytical and Numerical Approach

Wood exhibits cross-sectional distortions such as cupping, diamonding, skewing, edgewise cupping, and ovalization when moisture loss below the fibre saturation point induces anisotropic radial–tangential shrinkage. Because classical cupping models use different coordinate systems, displacement gauges, and distortion measures, their relations are difficult to compare directly. This study reformulates cross-sectional distortion as a moisture-induced eigenstrain–compatibility problem and compares Hsu, Cowin, Booker, least-squares reconstruction, and small-strain finite-element verification within a common pith-centred coordinate system and chord-sag metric. Seven Wood Handbook-type distortion modes are reproduced numerically, showing that different forms can arise from the same shrinkage anisotropy through different pith-relative geometries. For constant shrinkage coefficients, the Hsu, Cowin, and Booker-linear forms give identical leading-order chord-sag predictions. The least-squares reconstruction provided a compatibility-consistent numerical reference that agreed closely with the analytical solution, and finite-element verification gave nearly identical chord-sag predictions. Reanalysis of Booker’s dataset showed that the published theoretical values are most consistently interpreted using pointwise $R$-dependent shrinkage input, while residual discrepancies indicate the importance of board-specific ring geometry. The framework unifies classical wood-science intuition, analytical cupping models, and numerical reconstruction, and provides a basis for graded shrinkage fields, image-based ring-field modelling, and three-dimensional lumber distortion.