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264 18 Modeling Structural Properties of the Frill of Triceratops ANDREW A. FARKE, RALPH E. CHAPMAN, AND ART ANDERSEN Triceratops has an unusual parietosquamosal frill among ceratopsids, with a strongly arched profile both in lateral and rostral views, as well as complete absence of parietal fenestrae. Previous workers have suggested that the frill may have had a defensive role, protecting the neck from the horns of other Triceratops or from attacks by predators . Yet these functions have not been assessed within a biomechanical context. In order to evaluate the structural properties of the frill of Triceratops, a threedimensional digital model was constructed from surface scan data of an original fossil specimen. This model was analyzed using finite element analysis, in which the effects of various loads to the frill were simulated. Greatest overall stresses within the frill occur under loads applied to its most distal portions, as would be expected if the frill behaves at least in part as a cantilevered beam. Strain energy density (reflecting where the forces of loads are absorbed) generally is confined to the element experiencing the loading. In particular, the arched profile of the squamosal seems to be particularly effective at preventing loads to the squamosal from affecting the parietal. This suggests that some structural features of ceratopsian frills, such as the arched profile or thickened medial bars on the squamosals of certain ceratopsians, may have played a role in maintaining the structural integrity of the frill under externally applied loads (such as those from the horns of other Triceratops). Introduction The parietosquamosal frill of the chasmosaurine ceratopsid Triceratops displays unusual morphology relative to the condition in many other neoceratopsians. Whereas most derived neoceratopsians (e.g., Protoceratops, Centrosaurus, and Chasmosaurus ) possess prominent parietal fenestrae, the frill of Triceratops has no such openings. Furthermore, Triceratops has a relatively shorter frill (as compared to basal skull length) than seen in most other chasmosaurines ceratopsids, and the frill displays prominent mediolateral and rostrocaudal arching (Fig. 18.1; Forster 1996). This unique combination of characters has led to speculation that the frill of Triceratops was an effective protective shield against predators or the horns of other Triceratops (e.g., Hatcher et al. 1907; Lull 1908; Lull 1933). Significantly, only one previous study has considered the mechanical properties of the ceratopsian frill (Tyson 1977). Tyson focused in particular on the properties of the frill for counterbalancing the weight of the front half of the skull. She also noted that the frill in some ceratopsids was constructed as a ‘‘perfect frame,’’ and hypothesized that the parietal fenestrae probably formed in areas of relatively low stress under typical loads. Here, we investigate the structural properties of the unusual frill of Triceratops. The present study is not intended as a direct test of hypotheses of frill function, but instead an exploration of how the frill would have behaved under certain me- Modeling Structural Properties of the Frill of Triceratops 265 FIGURE 18.1. Cross sections of the frill (parietal + squamosals) of USNM 2100, Triceratops horridus, in (AB) transverse (taken just caudal to the end of the dorsal temporal fenestrae) and (B) mid-sagittal sections. (C) Right oblique view of the finite element mesh of a Triceratops frill used in this analysis, with load points and directions indicated by arrows. The numbers next to the arrows indicate the load case as discussed in the table and text. The grey unfilled circles bordering the rostral end of the frill indicate the point of constraint. Along the coordinate axes, X indicates dorsal, Y indicates mediolateral, and Z indicates caudal. Scale bar is 500 mm. chanical loads. For instance, what would happen to the frill if the horn of an opponent struck? Were certain parts of the frill better adapted to withstand these (hypothetical) forces, and would the frill be more likely to fracture under certain conditions ? Thus, the structural behavior of the frill itself may help illuminate its evolution and function. Methods Finite element modeling (FEM) is an engineering method that models the physical behavior of complexly shaped objects by breaking them into a number of simpler objects (elements). Equations calculating stress, strain, or other parameters of interest are applied to each of the elements that make up the whole object. Solutions for the equations are iterated across the entire model until the solutions converge, and the results can then be viewed graphically or extracted for statistical analysis (Richmond et al. 2005; Rayfield 2007 present technical and practical...

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