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

Abstract

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 (s 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 (twotailed p ¼ 0.

Standing on the shoulders of apes: Analyzing the form and function of the hominoid scapula using geometric morphometrics and finite element analysis

“…Finite element validation analyses have shown that both four‐node and eight‐node tetrahedral, and mixed four‐node tetrahedral and eight‐node hexahedral meshes perform well when compared with experimental data (Panagiotopoulou et al, ). Likewise, it has been shown that meshes composed by more than 200,000 elements show negligible stress differences between models with four‐ or ten‐node tetrahedra elements (Brassey et al, ). Because ten‐node tetrahedra are computationally more expensive than those composed by four nodes, the surfaces were meshed using four‐node tetrahedral elements (C3D4) by applying a built‐in Delaunay meshing algorithm in Abaqus v. 6.13.…”

A review of trabecular bone functional adaptation: what have we learned from trabecular analyses in extant hominoids and what can we apply to fossils?

“…; but see Brassey et al. ). Thus, many have suggested caution when inferring functional loading patterns from cross‐sectional shape, especially when one does not know the habitual loading behaviour or how bone may respond, such as in fossil taxa (Demes et al.…”

“…However, because of the denser structure and slower remodelling rate of cortical bone (Eriksen, 1986), functional signals can be more ambiguous than that of trabecular bone. For example, in vivo studies in a variety of animals have shown that limb bone shaft cross-sections are not always reinforced in the planes in which they are habitually loaded Judex et al 1997;Demes et al 1998;Lieberman et al 2004;Wallace et al 2014; but see Brassey et al 2013). Thus, many have suggested caution when inferring functional loading patterns from cross-sectional shape, especially when one does not know the habitual loading behaviour or how bone may respond, such as in fossil taxa (Demes et al 1998;Pearson & Lieberman, 2004;Wallace et al 2014; but see Ruff et al 2006;Brassey et al 2013).…”

More than One Way of Being a Moa: Differences in Leg Bone Robustness Map Divergent Evolutionary Trajectories in Dinornithidae and Emeidae (Dinornithiformes)

“…Previous biomechanical analyses of moa hind limbs have relied upon simplified beam theory models [5], [11], in which complex organic structures are simplified into slender beams. However, in a broad sample of morphologically diverse mammal and bird long bones, the errors introduced into stress calculations resulting from this simplification have been shown to be neither consistent in magnitude nor direction [12]. Factors such as shaft curvature, low values of aspect ratio (length/diameter) and variations in cortical wall thickness are characteristic of organic structures such as long bones, yet these are typically unaccounted for in simple beam equations [12].…”

“…However, in a broad sample of morphologically diverse mammal and bird long bones, the errors introduced into stress calculations resulting from this simplification have been shown to be neither consistent in magnitude nor direction [12]. Factors such as shaft curvature, low values of aspect ratio (length/diameter) and variations in cortical wall thickness are characteristic of organic structures such as long bones, yet these are typically unaccounted for in simple beam equations [12]. However, finite element analysis allows the complex 3D geometry of bones to be incorporated into stress equations, and with access to computed tomography (CT) facilities becoming cheaper and easier, it is now feasible to generate a larger comparative dataset of 3D models on which to perform biomechanical analyses.…”