### Abstract:

Consider the bole of a tree to consist of a linear elastic material that is
orthotropic with respect to the cylindrical coordinates. When the bole of a tree is
subjected to resultant loads in the directions of the Cartesian base vectors, the S₁₁, S₂₂, S₃₃, and S₁₂ stresses in Cartesian coordinates are coupled. It is desirable to use
beam elements to analyze the structural behavior of trees because of the ease with
which these can be incorporated into Finite Element Models. However, elementary
beam theory is not able to consider the problem where the S₁₁, S₂₂, S₃₃, and S₁₂ stresses are coupled. The objective of this study was to determine the magnitudes
of the normal stresses in the radial and tangential directions (Srr, Sθθ) and the shear stress (Srθ), relative to the normal stress in the x₃ direction for an element of a tree
bole.
In cylindrical coordinates the strains are not unique at r = 0. Therefore, a
constitutive equation was adopted in cylindrical coordinates where the elastic
coefficients are dependent on r. An element of a tree bole was considered as a
cantilever beam and posed as a Relaxed Saint-Venant's Problem in Cartesian
coordinates. It was found if the strains resulting from the generalized plane strain
part of the problem were considered linear functions of the x₁ and x₂ coordinates,
then the strain compatibility conditions arid equilibrium equations could be
satisfied.
Given the assumption that the generalized plane strains are linear in x₁ and x₂,
it was proven that the Srr, Sθθ, and Srθ stresses are analytic functions of the complex
variable z. It is also proven that the Srr, Sθθ, and Srθ stresses are equal to zero on the
lateral surface of the element of the tree bole. Therefore, using the analyticity of the
stress functions and the fact that they are zero on the lateral surface it is possible to
show that the Srr, Sθθ, and Srθ stresses are zero throughout the element of a tree bole.