Faculty of Life Sciences, University of Manchester, Manchester, UK.
J R Soc Interface. 2013 Feb;10(79):20120823. doi: 10.1098/rsif.2012.0823.
Classic beam theory is frequently used 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), while much more time-consuming to perform, require no such assumptions. This study 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 is essential for reasonable biomechanical predictions. High-resolution computed tomographic scans of eight vertebrate long bones were used to calculate diaphyseal stress owing to various loading regimes. Under compression, FEA values of minimum principal stress (σ(min)) were on average 142 per cent (±28% s.e.) larger than those predicted by beam theory, with deviation between the two models correlated to shaft curvature (two-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 per cent (±4% s.e.), with deviation between the models significantly correlated to cross-sectional asymmetry at midshaft (two-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 per cent (±5% s.e.) 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.
经典梁理论在生物力学中常用于模拟脊椎动物长骨的应力行为，特别是在创建种内比例模型时。尽管方法简单，但经典梁理论要求将复杂的不规则骨骼近似为细长梁，而将复杂的有机结构简化到这种程度所带来的误差尚不清楚。替代方法，如有限元分析（FEA），虽然执行起来耗时更多，但不需要进行这种假设。本研究比较了使用经典梁理论和 FEA 获得的结果，以量化梁理论的误差，并就何时需要进行全面的 FEA 以进行合理的生物力学预测提供建议。使用 8 根脊椎动物长骨的高分辨率计算机断层扫描来计算由于各种加载情况引起的骨干的骨小梁的应力。在压缩下，FEA 计算的最小主应力（σ（min））值平均比梁理论预测值高 142%（±28%标准差），两个模型之间的偏差与轴曲率相关（双侧 p = 0.03，r² = 0.56）。在弯曲下，FEA 计算的最大主应力（σ（max））和梁理论值的平均差异为 12%（±4%标准差），两个模型之间的偏差与中轴处的横截面不对称性显著相关（双侧 p = 0.02，r² = 0.62）。在扭转中，假设最大应力值出现在最小皮质厚度的位置使梁理论和 FEA 值最接近，在这种情况下，FEA 计算的τ（扭转）值平均比梁理论高 14%（±5%标准差）。因此，当需要估计绝对骨干应力时，FEA 是首选的建模解决方案，尽管在某些情况下，梁理论计算的弯曲值可能是可以接受的。
