| 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, S 22, 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 S 11, 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, S &thetas;&thetas;) and the shear stress (Sr&thetas; ), 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 and 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, S&thetas;&thetas;, and Sr&thetas; stresses are analytic functions of the complex variable z. It is also proven that the Srr, S&thetas;&thetas;, and Sr&thetas; 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, S&thetas;&thetas;, and S r&thetas; stresses are zero throughout the element of a tree bole. |
