Bending and torsion induced stresses in cylindrically orthotropic and inhomogeneous timber beams

The structural design of timber beams subject to bending often relies on the application of the simple Euler–Bernoulli beam theory. However, the simplistic formulas for stress calculations overlook the inherent characteristics of the wood material and the true distribution of the annual rings within the cross-sectional area. This paper introduces a method for determining all six stress components for a cantilever-type beam that is subjected to concentrated end loads. The method considers an inhomogeneous cross-section and employs cylindrically orthotropic material properties. Notably, this approach does not necessitate prior knowledge of elastic and shear centres. It is founded on the formulation of a displacement field incorporating unknown in-plane distortion and warping functions. These are then solved through a straightforward finite element procedure. The efficacy of the method is validated by a series of numerical examples that align with analytical results. Furthermore, a benchmark example for cylindrically orthotropic cross-sections is proposed. As a practical demonstration, an analysis is performed on a real sawn timber cross-section with material parameters representative of Norway spruce. The findings reveal significant disparities in the maximum stresses when compared to conventional engineering approaches. In this specific instance, the maximum longitudinal normal stress resulting from bending is approximately 20 % higher than the outcomes of typical engineering methods. This emphasizes the critical role played by the actual distribution of annual rings across the specific beam cross-section.