Transformation of Elastic Properties for Lumber with Cross Grain

Abstract

Application of orthotropic constitutive models to lumber with cross grain requires transformation of coordinates when material axes are not aligned with board axes. This paper presents a procedure to construct the coordinate transformation matrix using angles measured on the surface of a board for any orientation of wood fibers and growth rings within the board. Procedures in the literature are limited to special orientations of the wood material, although their authors have not always recognized the limitations. The key to the present method is the recognition that for each set of three independent surface angles there corresponds a set of Euler angles that can be used in the coordinate transformation matrix. Numerical examples show that the procedure identifies significant variations in board elastic properties for changes in the grain and ring angles. The examples also reveal potentially critical material shear stresses in uniaxial situations following stress transformations associated with combinations of seemingly small angles.