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.

受弯曲木梁的结构设计通常依赖于简单的欧拉-伯努利梁理论的应用。然而，简单化的应力计算公式忽略了木质材料的固有特性以及年轮在横截面积内的真实分布。本文介绍了一种确定承受集中端荷载的悬臂梁的所有六个应力分量的方法。该方法考虑不均匀横截面并采用圆柱正交各向异性材料属性。值得注意的是，这种方法不需要事先了解弹性中心和剪切中心。它建立在包含未知面内畸变和扭曲函数的位移场的公式基础上。然后通过简单的有限元程序来解决这些问题。该方法的有效性通过一系列与分析结果相符的数值例子得到了验证。此外，还提出了圆柱正交各向异性横截面的基准示例。作为实际演示，我们对具有代表挪威云杉材料参数的真实锯材横截面进行了分析。研究结果表明，与传统工程方法相比，最大应力存在显着差异。在此特定实例中，弯曲产生的最大纵向正应力比典型工程方法的结果高出约 20%。这强调了年轮在特定梁横截面上的实际分布所发挥的关键作用。