Stress functions for a heterogeneous section of a tree

Abstract

Consider a tree that is composed of wood, which is orthotropic with respect to the cylindrical coordinate axes, where the z-axis is directed up the center of the tree. When a section of a tree is considered as a Relaxed Saint-Venant's Problem the stresses in the plane of a transverse cross section will equal zero. By allowing some of, or all of the compliance coefficients to be functions of the radial coordinate r, the structure of the stress functions can be fundamentally altered. If the compliance coefficient in the z-direction ( S 3333) is allowed to remain constant, the S rz and S θz shear stresses will be functions of the coordinates, dimensions, applied loads, and compliance coefficients, while the other stresses will only be functions of the coordinates, dimensions and applied loads. If S 3333 is also allowed to be a function of r, then all the nonzero stresses will be functions of the coordinates, dimensions, applied loads, and compliance coefficients.