Data from: Finite element modelling vs. classic beam theory: comparing methods for stress estimation in a morphologically diverse sample of vertebrate long bones

Classic beam theory is frequently employed in biomechanics to model the stress behaviour of vertebrate long bones, particularly when creating intraspecific scaling models. Although methodologically straightforward, classic beam theory requires complex irregular bones to be approximated as slender beams, and the errors associated with simplifying complex organic structures to such an extent are unknown. Alternative approaches, such as Finite Element Analysis (FEA), whilst much more time-consuming to perform, require no such assumptions. This paper compares the results obtained using classic beam theory with those from FEA to quantify the beam theory errors and to provide recommendations about when a full FEA analysis is essential for reasonable biomechanical predictions. High-resolution computed tomographic (CT) scans of eight vertebrate long bones were used to calculate diaphyseal stress due to various loading regimes. Under compression, FEA values of minimum principal stress (σ_min) were on average 142%(±28% SE) larger than those predicted by beam theory, with deviation between the two models correlated to shaft curvature (2-tailed p=0.03, r^2=0.56). Under bending, FEA values of maximum principal stress (σ_max) and beam theory values differed on average by 12% (±4% SE), with deviation between the models significantly correlated to cross-sectional asymmetry at midshaft (2-tailed p=0.02, r^2=0.62). In torsion, assuming maximum stress values occurred at the location of minimum cortical thickness brought beam theory and FEA values closest in line, and in this case FEA values of τ_torsion were on average 14% (±5% SE) higher than beam theory. Therefore, FEA is the preferred modelling solution when estimates of absolute diaphyseal stress are required, although values calculated by beam theory for bending may be acceptable in some situations.

经典梁理论（classic beam theory）在生物力学领域被广泛应用于模拟脊椎动物长骨的应力行为，尤其在构建种内缩放模型时更为常用。尽管该方法在方法论上较为简洁，但经典梁理论要求将复杂不规则的骨骼近似为细长梁，而将复杂的生物组织结构简化至该程度所带来的误差目前尚不明确。诸如有限元分析（Finite Element Analysis, FEA）这类替代方法，尽管计算耗时显著更长，却无需做出此类假设。本研究将经典梁理论的计算结果与有限元分析的结果进行对比，旨在量化经典梁理论的误差，并针对何时需开展完整的有限元分析以获得合理的生物力学预测结果给出建议。本研究通过对8块脊椎动物长骨的高分辨率计算机断层扫描（computed tomographic, CT）结果，计算了不同载荷工况下的骨干应力。在压缩工况下，有限元分析得到的最小主应力（σ_min）平均值比经典梁理论的预测值高出142%（±28% 标准误），且两种模型的结果偏差与骨轴曲率显著相关（双侧检验p=0.03，r²=0.56）。在弯曲工况下，有限元分析得到的最大主应力（σ_max）与经典梁理论的预测值平均相差12%（±4% 标准误），且两种模型的结果偏差与骨干部中位截面的不对称性显著相关（双侧检验p=0.02，r²=0.62）。在扭转工况下，假设最大应力出现在骨皮质厚度最小的位置时，经典梁理论与有限元分析的结果最为吻合；此时有限元分析得到的扭转切应力（τ_torsion）平均值比经典梁理论高出14%（±5% 标准误）。因此，当需要获取绝对骨干应力的估算值时，有限元分析是更优选的建模方法，不过在部分场景下，经典梁理论针对弯曲工况计算得到的结果仍可接受。