Elastic equations for a cylindrical section of a tree

Considering a cylindrical section of a tree subjected to loads independent of x3 as a relaxed Saint-Venant’s problem, it was shown that plane sections remain plane. Since plane sections remain plane, the displacement equations for the neutral fiber derived using either the relaxed Saint-Venant’s problem or elementary beam theory are equivalent. The stresses in the plane of the transverse cross-section were found to equal to zero. Therefore, it is appropriate to use elementary beam theory to estimate the three-dimensional stress functions when the wood is considered to be homogeneous. In addition the three-dimensional displacement equations allow the required elastic coefficients in cylindrical coordinates to be measured from full size samples