THE UNIVERSITY OF CALGARY Eric Snively A ... - Ohio University

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(nodes), and solving stresslstrain equations for each node and element. The equations can be considered analogs of Hooke's Law (1) F=loc, in which F is force, x is displacement, and k is a spring constant. This relationship is applied ta al1 nodes in a finite element model by matrix equations incorporating these variables for al1 nodes: (2) @}=[kl M. In these equations {a} is the stress matrix. Each number in the matrix refiects the force vector acting upon an element, incorporating vector sums of forces impinging on its nodes from other nodes in the system. The matrix {E) represents the displacement of nodes. [k] is the stiffness matrix for an element, and includes the following material properties: A) Elastic (Young's) modulus: Stress a (forcelarea) divided by strain E (change in lengthlinitial length) parallel to the imposed force. (3) E=ak Units: ~ lm~, in pascals or gigapascals (GPa) B) Poisson's ratio: Lateral strain divided by axial or longitudinal strain. (4) ~=~iateraJ~axia~ Units: dimensionless Because bone is an orthotropic material, elastic modulus and Poisson's ratio Vary according to the direction of loading. Non-linear finite element models, such as that presented in this chapter, incorporate al1 necessary values for E and v. Forces acting upon a structure are deterrnined for the finite element model. These forces depend on the static or kinematic hypotheses to be tested. The
forces are applied to nodes on the surface of the model. Boundary conditions, which are constraints on the movement of elements and the displacement of nodes, are also applied. Some nodes, typically at one end or side of a model, are set to a boundary condition of zero displacement, as though the mode1 is fixed to an immobile surface. Otherwise, little strain would occur at intemal nodes unless tremendous energies were applied, and forces would cause the entire structure to accelerate. Boundary conditions, forces, and material properties are entered into a finite element computer program. The program then solves the resulting systems of Iinear equations, typically by Gaussian elimination. While this algorithm is tenable for the solution of simple matrix equations by hand, the huge number of nodes in a finite element model necessitates intensive use of computer resources. Finite element stress modeling has a nurnber of practical and scientific applications. FEA is a common procedure for investigating material stresses and strains in engineering (Chandrupalta and Belegundu 1997). In modeling 30 solids, a sufficiently large number of elements approximates the continuity of the original object. The finite element method is therefore able to accurately simulate and predict stress-strain relationships in physical structures, if the correct values for material properties are supplied. FEA provides crucially accurate predictions for airfrarne and automotive design (Belytschko et al. 1975), and biomedical engineering (Taylor et ai. 1998). The accuracy af the method thus has imrnediate benefits for vehicle and building safety, as well as for the development of medical prostheses.
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THE UNIVERSITY OF CALGARY Functiona
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Many figures in this thesis use col
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ACKNOWLEûGEM ENTS Many friends and
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Approval page Acknowledgements Tabi
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CONCLUDING COMMENTS Figures for Cha
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CHAPTER 5: Mechanical and phylogene
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LIST OF FIGURES Figure 1 .l. Phylog
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Figure 3.6. Osteological correlates
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AMNH GI HMN IVPP MOR NMC OMNH PIN P
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and constructional morphology revea
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1985, Bryant and Seymour 1990). If
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surfaces of a Tyrannosaurus rex spe
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3. Troodontidae (Figure 1.2): Trwdo
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Hypothesized functions of the tyran
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1. Atomization (Chapters 2 and 3).
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Figure 1.1. P hylogenetic diagram o
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Figure 1.2. Phylogenetic diagram of
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Figure 1 .S. Skeletal restorations
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did not quantify the degree of prox
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qualitative debate over arctometata
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This hypothesis focuses on tyrannos
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Table 2.1. Theropod and Plafeosauru
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specimens in the figures reflects a
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were not attempted. While this limi
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Chapter 3). 1 now address variation
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- Ç6'P 1 68'€tr 1 L'OC 99'66 L'9
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Table 2.3: Log-transformed values f
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experiences a sharp laterally indin
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ecause it can be cursorily dassifie
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arctometatarsalian specimens, the l
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) Shaf€ and region of proximal ar
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a) Ginglymus. Figure 2.14 shows the
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) Shaft and region of proximal arti
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measurements of width pV2S%TU-DE an
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Table 2.4b Variancelmeasurement eac
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the largest sum of linear measureme
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Table 2.5: Specimen statistics Herr
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and Sinraptor dongi specimens. With
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In addition, a hook like proximal a
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oth physical narrowing and elongati
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1s the MT 111 shape segregation fun
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tyrannosaurids, ornithomimid, and t
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proximal constriction, white EImisa
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Figure 2.2. Tyrannosaurid, trwdonti
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Figure 2.4. Dromaeosaurid and carno
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- Anterior (flexor) surface medial
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Figure 2.7. Left MT III of Tyrannos
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Figure 2.8. Left MT III of Gorgosau
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Figure 2.9. Left MT III of an omith
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Figure 2.10. Left metatarsus of Tmd
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Figure 2.1 1. Left MT III of Elmisa
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Figure 2.12. Right MT III of Deinon
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Figure 2.13. Left MT III of Allosau
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Figure 2.14. Left metatarsus of Sin
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Figure 2.15. Plots of MT Ill specir
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Figure 2.16. Plot of third metatars
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tv . ... . Ri. . . .
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Figure 2.19. Plot of third metatars
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CHAPTER 3: Tensile keystone model o
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dynamically transmitted Iommotor fo
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Table 3.1 . Metatarsi examined in t
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interrnetatarsal dynamics in these
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compelled analysis of movement evid
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Once prepared, TMP 94.1 2.602 and i
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Daspletosaurus tomsus (MOR 590), an
Page 145 and 146: 2. Freedom of movement inferred fro
Page 147 and 148: Table 3.2. Surface areas of intemet
Page 149 and 150: phylogenetic bracketing Wtmer 1995)
Page 151 and 152: 4) Forces from anterior displacerne
Page 153 and 154: e expected for tyrannosaurid interm
Page 155 and 156: The preceding discussion derives fr
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Page 159: Figure 3.1. Schematic representatio
Page 163: Figure 3.3. CT reconstruction of ri
Page 167: Figure 3.5. MT II (left element) an
Page 171: Figure 3.7. Osteological correlates
Page 175: Figure 3.9. Osteological correlates
Page 179: Figure 3.1 1 ae. The metatarsal rec
Page 187: Figure 3.13. Ligament contribution
Page 191 and 192: Figure 3.15. Anatorny of the equid
Page 193: Figure 3.16. Cornparison of loading
Page 198 and 199: The practicality of the finite elem
Page 200 and 201: collective density of trabeculae (K
Page 202 and 203: A FORCE INPUTS AND MATERIAL PROPERT
Page 204 and 205: Alexander et al. (1 979) rewrded a
Page 206 and 207: The preceding loading regime entait
Page 208 and 209: detemined by using the photographic
Page 210 and 211: 2. Preparation of data for contour
Page 212 and 213: . - Elastic modulus (GP4 Ex EY Ez P
Page 214 and 215: conversion. Volume-to-nuages facili
Page 216 and 217: maintained. An angle of 2 degrees p
Page 218 and 219: of MT III (Figure 4.4, pinpointed b
Page 220 and 221: anterodorsal rotation of the distal
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Page 224: Figure 4.1. Initial loads and bound
Page 228: Figure 4.3. Finite element mesh of
Page 232: Figure 4.5. Strain distribution in
Page 236 and 237: CHAPTER 5: Mechanical and evolution
Page 238 and 239: y distal intermetatarsal ligaments,
Page 240 and 241: Probable fundion of the tyrannosaur
Page 242 and 243: Ornitholestes, and camosaurs) incre
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discussion elucidates the evolution
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face of the skull met part of the m
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Alternately, a morphological novelt
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outgroup to ail theropods, and the
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convincingly evident in figures of
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parsimonious scenarios are outlined
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dromaeosaurids], and a dade compris
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arctometatarsalian forrns. The conv
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Proximal intermetatarsal ligaments
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witti a surfeit of potentially supp
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Figure 5.2. Phylogeny of the Therop
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Bock W.J. and von Wahlert G. Evolut
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Cume P.J. 1997. Theropoda. In: Fari
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Grenard S. 1991. Handbook of Alliga
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Liem K.F. and Osse J.W.M. 1975. Bio
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Richtsmeier J.T. and Cheverud J.M.
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Woo S., Maynard J., Butler O. 1987.
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Different variables in a PCA will h
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THE UNIVERSITY OF CALGARY Eric Snively A ... - Ohio University

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