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COMPUTATIONAL STUDY OF STRESS CONCENTRATION ABOUT AN OBLIQUE HOLE IN A THICK WALLED TUBE: TOWARD UNDERSTANDING STRUCTURAL IMPROVEMENTS IN BONE By SUSAN M. GROVER A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2004 Copyright 2004 by Susan M. Grover ACKNOWLEDGMENTS I would like to thank my husband, Guy Grover, for his emotional support, technical advice, and proofreading. His encouragement has been vital to maintaining the determination necessary to complete the difficult coursework and research required for this degree. His suggestions on Einite element modeling techniques helped me to evaluate complex geometries efficiently and accurately. I would like to express my gratitude to my advisor, Dr. Andrew Rapoff, for his helpful guidance and the concept for this research proj ect. It has been an interesting and rewarding proj ect for me. I am thankful that I was given the opportunity to work on it. I would like to thank my colleagues in the Applied Biomechanics Laboratory for their support in my research proj ect and in helping to relieve stress. This includes Barbara Garita, Ruxandra Marinescu and Neel Bhatavadekar. Special thanks are in order for Wesley Johnson and Stephanie Dudley, both former students in the Applied Biomechanics Laboratory who convinced me to j oin Dr. Rapoff' s group of graduate students. It has been a very satisfying experience for me and I am forever grateful. I would also like to thank my parents, Julian and Sue Mathison, for encouraging me to go back to school after Hyve years of work and being supportive of me. Their confidence in me and enthusiastic encouragement since childhood have enabled me to accomplish all of my goals thus far. TABLE OF CONTENTS page ACKNOWLEDGMENT S ........._._ ...... .__ .............. iii... LI ST OF T ABLE S ........._._ ...... .... .............. vi... LIST OF FIGURES .............. ....................vii AB STRAC T ................ .............. ix NOT ATION ........._._ ...... .... .............. viii.. CHAPTER 1 INTRODUCTION ................. ...............1.......... ...... 2 BACKGROUND .............. ...............3..... 3 FINITE ELEMENT MODELS................ ...............7. 4 RE SULT S ................. ...............15.......... ..... Generation of Results .............. ...............15.... Listing of Results .................. ............. ...............16...... Validation: Cylinders with Transverse Holes ................. ..............................18 Validation: Flat Plates with Oblique Holes ................. ...............19............... Angle of Obliquity, a ............... ...............20.... Hole Size Ratio, d/ D............ .......__ ...............22. Cylinder Diameter Ratio, d,/D ...._ .................. ...............24. .... Angular Location of Maximum Stress, 8................ ...............25... Depth of Maximum Stress in Hole, h ....._____ ............ ....___ ..........2 Angle of Neutral Axis, ry............... ...............29... 5 DI SCUS SSION ............ ............ ............... 0.... Angle of Obliquity, a ............... ...............30.... Hole Size Ratio, d/ D............ .......__ ...............31. Cylinder Diameter Ratio, d,/D ...._ .................. ...............32. .... Angular Location of Maximum Stress, 8................ ...............32... Depth of Maximum Stress in Hole, h ....._____ ............ ....___ ..........3 Angle of Neutral Axis, ry............... ...............33... 6 CONCLUSIONS .............. ...............35.... LIST OF REFERENCES .........__... ..... .__. ...............38.... BIOGRAPHICAL SKETCH .............. ...............39.... LIST OF TABLES Table pg 31 M odel listing .............. ...............9..... 41 Maximum and minimum stress concentration factors, with location ......................17 LIST OF FIGURES Figure pg 31 Cylinder parameters (forces shown at ry= 00) ................. .............................8 32 Hole parameters............... ............... 33 Sketch of 10node parabolic tetrahedral element ........................... ...............10 34 Volume partitions, volume labels circled............... ...............12 41 Comparison of results from FEA and Pilkey [2] ................. ................. ....... 19 42 Comparison of results from FEA and Stanley and Day [5] ........._.._. ................20 43 Kmax plotted against a............... ...............22... 44 Kmzn plotted against a............... ...............23... 45 Kmax plotted against hole size ratio, d D, at d/D = 0. 75............... ...................2 46 Kmzn plotted against hole size ratio, d D, at d/D = 0. 75............_.__ ........._._ .....24 47 Kmax plotted against d/D for various a at d/D = 0.10O................_ ......................25 48 K,,n plotted against d/D for various a at d/D = 0. 10 ...........__ ... ...._._..........25 49 Angular location of Kma, 8, plotted against a .............. ...............26.... 410 Normalized depth of Kma in the hole interior plotted against a..............................28 NOTATION d hole diameter d, cylinder inner diameter D cylinder outer diameter F applied force (N) h normalized depth in the hole, measured from the midplane Kna maximum stress concentration factor based on the gross cross section, 42ax o;;2 Knm minimum stress concentration factor based on the gross cross section, 42:n Go;;2 r radial location, used to identify the location of Kna and Knzn t cylinder wall thickness z height coordinate (z = 0 is at the fully restrained end of the cylinder) or angle of obliquity, i.e. angle the hole centerline makes with a vector normal to the surface of the cylinder (0) B clock position around the hole, measured in the plane perpendicular to the hole centerline from the point farthest from the direction of applied load (0) O' clock position around the hole, measured in a plane parallel to the cylinder surface from the point farthest from the direction of applied load (0) ry angular location of the neutral axis from a datum line oriented perpendicular to the hole axis (0). (At ry= 00, the region of material around the hole is in tension) 42ax maximum principal stress at a point sufficiently distant from the applied load o,,n minimum principal stress at a point sufficiently distant from the applied load Go,,, nominal stress in a body without a discontinuity Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science COMPUTATIONAL STUDY OF STRESS CONCENTRATION ABOUT AN OBLIQUE HOLE IN A THICK WALLED TUBE: TOWARD UNDERSTANDING STRUCTURAL IMPROVEMENTS IN BONE By Susan M. Grover August 2004 Chair: Andrew J. Rapoff Major Department: Biomedical Engineering To quantify improvements in stress in the canine tibia near the nutrient foramen (a natural hole) the expected stress concentration factor around a similar hole in a homogenous isotropic structure is needed for comparison purposes. The canine tibia is loaded in bending and includes a foramen oriented at an oblique angle to the long axis. This loading and geometry have been somewhat neglected in previous stress concentration factor research, though experimental and computational results exist for flat plates with oblique holes in tension and cylinders with transverse holes in either tension or bending. A series of finite element models have been created to investigate stress concentration factors in homogeneous isotropic hollow cylinders in bending with holes ranging from transverse to oblique. The results of this parametric study indicate that, as the hole becomes more sharply angled, both the maximum and minimum stress concentration factors decrease and shift toward the acutely angled edge created by the intersection of the hole with the cylinder surface. This study includes minimum stress concentration factors, which were largely unreported in prior work, but are very important to this investigation. Due to loading differences, the absolute value of these factors should be compared to maximum stress concentration factors occurring in the canine tibia. This comparison can be used to help quantify the ability of bone to reduce stress around a natural discontinuity. CHAPTER 1 INTTRODUCTION Localized stress around geometric discontinuities such as holes, shoulders, and grooves cannot be predicted using elementary stress formulas. The concentration of stress resulting from these abrupt transitions is frequently too high to be attributed solely to the decrease in net cross sectional area. Stress concentration factors, often determined experimentally or computationally, are used to scale the nominal stress in a continuous structure to account for the effect of the discontinuity. Determining stress concentration factors is of practical importance for many engineered structures because geometric discontinuities are frequently the site of failure, but, in bone, this is not always the case. Although surgically created holes may be a source of failure, natural holes, such as the nutrient foramen, seem to demonstrate superior mechanical performance. Modifications in porosity, mineralization, and fiber orientation can work together to decrease stiffness around a discontinuity to reduce stress. A unique microstructure was previously uncovered at the equine third metacarpus nutrient foramen that results in an elimination of stress concentration at the hole edge, i.e. the structure behaves as if it were continuous [1]. When compared to the expected stress concentration in a homogeneous isotropic structure, the effect of the geometry and microstructure of the hole can be quantified. The nutrient foramen in the equine third metacarpus is oriented normal to the surface of the bone. The hole is elliptically shaped with an aspect ratio of approximately 2: 1 [1]. The maj or axis is parallel to the long axis of the bone and the bone is loaded primarily in uniaxial compression. The expected stress concentration factor for a circular hole in a homogeneous isotropic infinitely large plate with uniaxial stress is 3.00 [2]. Elongating the hole such that the maj or axis of the ellipse is parallel to the line of action of the load and the aspect ratio is 2: 1 results in an expected stress concentration factor of 2.00 in a homogeneous isotropic structure [2]. The microstructure in bone, then, is responsible for reducing the stress concentration factor further to 1.00 [1]. The effect of the geometry and microstructure around a nutrient foramen oriented and shaped differently could provide more insight into the ability of bone to reduce stress. The canine tibia includes a nutrient foramen that enters the bone at a steep angle. This bone is subj ected to bending in the saggital plane and the nutrient foramen is located on the posterior surface, in a region subjected primarily to compression. The stress field for this geometry is more complex than that of a flat plate in uniaxial loading. The maximum stress depends on many factors, including the angle of obliquity, the wall thickness, and the ratio of the hole diameter to the outer diameter of the cylinder. A literature review has not revealed a prior investigation of stress concentration factors for oblique holes in cylinders. The ensuing discussion will provide a set of results that can be used to predict stress concentration factors for homogeneous, isotropic, linearly elastic cylinders with oblique holes. This can later be used to form a comparison with the stress concentration factors predicted through a microstructural examination of the canine tibia to quantify the change in stress concentration achieved by bone. CHAPTER 2 BACKGROUND Considerable prior work has focused on predicting stress concentration factors for flat plates with oblique holes and for cylinders with transverse holes, but the combined effect of an oblique hole in a cylinder seems to be a neglected area of study. An investigation into this overlooked geometry, however, can shed light on variations in stress near the hole when the angle of obliquity, wall thickness, and hole diameter change and provide a set of results to compare to stress concentration factors in various bones near foramina. The microstructural analysis detailed in Goitzen et al. [1] demonstrates a method for determining heterogeneous anisotropic elastic constants in bone using porosity, mineralization, fiber orientation, and osteon orientation measurements. A finite element model was created using a geometry that closely matches that of bone. The elastic constants were mapped to appropriate locations in the model to simulate the mechanical variations that occur in bone. Loading this model in a manner similar to physiological loading revealed the stress pattern that would be expected in vivo. From this, the stress concentration factor was calculated. In the equine third metacarpus study, this factor is 1.00 [1]. In one of the earliest works on oblique holes in flat plates, Ellyin, Lind, and Sherbourne [3] developed a theoretical solution for thin plates with holes between 00 and 450 of obliquity, as measured from a vector normal to the surface of the plate (see a in Figure 32). The solution was obtained by dividing the plate into several thin membranes and enforcing plane elasticity, maintaining compatibility by applying tractions at the boundary surfaces. The results demonstrated that as Poisson's ratio increases, the stress concentration factor decreases. Also, for holes loaded in uniaxial tension parallel to the minor axis of the ellipse created at the hole surface, the maximum stress concentration factor (located near the maj or axis) increases as the obliquity increases. This analytical approach provided a basis for future work in this area. The results of the theoretical work seemed to agree fairly well with Ellyin's later experimental study [4], which involved affixing strain gauges to steel plates with holes of varying obliquity, diameter, and load orientation (parallel to either the maj or or minor axis of the ellipse). A limitation of this method, mentioned by Ellyin [4], results from the difficulty in recording accurate strain measurements in regions with high strain gradients. Since strain gradients tend to increase with hole obliquity, this limits the useful range of obliquities that can be adequately studied with this method. Ellyin' s results demonstrate trends for both load orientations investigated. For holes loaded parallel to the minor axis of the ellipse, K;;m rises rapidly as obliquity steadily increases. The opposite is true for holes loaded parallel to the major axis; i.e., for these holes, Kmx decreases rapidly as obliquity steadily increases. For either hole orientation, as obliquity increases from 00 to 450 (i.e. as the hole becomes steeper), the location of maximum strain (and stress) moves from the midplane of the plate toward the plate surface [4]. Ellyin measured the location of stress inside the hole by applying at least five strain gauges every 45 degrees around the circumference of the hole, spaced at equal distances throughout the thickness. Stanley and Day [5] were among the first to consider applied loads oriented at various angles from 00 to 900 with respect to the maj or axis of the ellipse. As expected based on Ellyin's results [4], the stress concentration factor increased as the angle of load application was varied from parallel to the major axis of the ellipse to parallel to the minor axis. The results also confirm that, when the load is applied parallel to the minor axis, as the obliquity angle increases, the stress concentration factor increases and shifts from midplane (at a 00 angle of obliquity) to the plate surface (at a 600 angle of obliquity). Conversely, when the applied load is parallel to the major axis, as the obliquity angle increases, the stress concentration factor decreases and stays close to the midplane for all obliquity angles less than 600 The most comprehensive work to date on oblique holes in flat plates is Dulieu Barton and Quinn's thermoelastic study [6]. The variable definitions used are the same as those chosen for Stanley and Day's photoelastic study [5], but the plate thicknessto hole diameter ratios were smaller. The results revealed that, similar to the Eindings of Stanley and Day [5], when the applied tensile load is parallel to the minor axis of the surface ellipse, increasing the obliquity angle causes increased maximum stress values. An important observation of the study is that, for the thicknesstohole diameter ratios considered, as this ratio decreases, the stress concentration factor increases. This confirms the findings of Ellyin [4] and Daniel [7], while disputing the weak dependency reported by Stanley and Day [5]. Hollow cylinders with transverse holes through one or both walls have been studied in tension, bending, or torsion in Jessop et al. [8] and results have been summarized in Pilkey [2]. The results of the photoelastic study in Jessop et al. [8] indicate that, for the tensile loading case, stress concentration factors decrease as the hole diametertoouter cylinder diameter ratio increases and wall thickness decreases. For the bending load case, however, Jessop et al. [8] demonstrates that as the hole diametertoouter cylinder diameter ratio increases, the stress concentration factor first decreases slightly and then increases more sharply. The effects of varying the wall thickness and the hole diameter were also uncovered by Jessop et al. [8]. It was also shown that, as wall thickness decreases, maximum stress concentration increases slightly. Varying the hole diameter had a greater effect on the stress concentration factor than changing the cylinder diameter ratio, so the size of the hole has more of an influence than wall thickness on the stress concentration factor [8]. The proj ect documented herein will investigate stress concentration factors in hollow cylinders with oblique holes in bending. Magnitude and location of both minimum and maximum stress concentration factors will be reported at various obliquity angles, hole diameters, and wall thicknesses. The location of the maximum principal stress will be reported in terms of angular location and normalized distance from the mid plane. Plots of all data will be presented and compared to prior work, when applicable. CHAPTER 3 FINITE ELEMENT MODELS Finite element models were created using SDRC IDEAS MS9 software on a Windows 2000 based PC. The parameters used in this study are illustrated in Figure 31 and Figure 32. All parameter combinations used in this investigation are listed in Table 31. The parameters chosen for each model were selected such that hole diameter, angle of obliquity and wall thickness may be studied in isolation of the other parameters. All models were fully restrained in all six degrees of freedom at the z = 0 end and a bending load was applied at the opposite end. Cylinder heights were chosen such that the constraints and loads are located at a distance of at least 3 times the maximum cylinder diameter away from the hole to minimize any effect that the manner of loading may have on the stress at the hole. The bending load was generated using two point forces. A 100 Newton tensile force was applied on a node located directly above the hole center at a radial location, r, of D/2 and a 100 Newton compressive force was applied on a node located 1800 away, also at r = D/2. The angular location of the neutral axis from a datum line oriented perpendicular to the hole axis is designated by ry. For the loading pattern described and used for all models herein, ryis zero. This load puts the hole region in tension. The variable Bwas created to facilitate a direct comparison of the angular location of maximum stresses in each model. This is defined as the angular position around the circular hole and is used in favor of 8', which is defined as the angular position around dd Figure 31 Cylinder parameters (forces shown at r= 00) Direction of  7 applied load BB " Section AA Xd h Section BB Figure 32 Hole parameters Table 31 Model listing Model d (mm) a. de. D (mm) di (mm) dilD 1 1 0 20 5 0.25 2 10 20 10 0.5 3 10 20 15 0.75 4 2 0 20 5 0.25 5 2 0 20 10 0.5 6 2 0 20 15 0.75 7 3 0 20 5 0.25 8 3 0 20 10 0.5 9) 3 0 20 15 0.75 10 1 15 20 5 0.25 11 1 15 20 10 0.5 12 115 20 15 0.75 13 2 15 20 5 0.25 14 2 15 20 10 0.5 15 2 15 20 15 0.75 16 3 15 20 5 0.25 17 3 15 20 10 0.5 18 3 15 20 15 0.75 19 130 20 5 0.25 20 1 30 20 10 0.5 21 1 30 20 15 0.75 22 2 30 20 5 0.25 23 2 30 20 10 0.5 24 2 30 20 15 0.75 25 3 30 20 5 0.25 26 3 30 20 10 0.5 27 3 30 20 15 0.75 28 145 20 5 0.25 29 145 20 10 0.5 30 1 45 20 15 0.75 31 2 45 20 5 0.25 32 2 45 20 10 0.5 33 2 45 20 15 0.75 34 3 45 20 5 0.25 35 3 45 20 10 0.5 36 3 45 20 15 0.75 37 1 60 20 5 0.25 38 160 20 10 0.5 39 160 20 15 0.75 40 2 60 20 5 0.25 41 2 60 20 10 0.5 42 2 60 20 15 0.75 43 3 60 20 5 0.25 44 3 60 20 10 0.5 45 3 60 20 15 0.75 46 1 75 20 5 0.25 47 1 75 20 10 0.5 48 175 20 15 0.75 49 2 75 20 5 0.25 50 2 75 20 10 0.5 51 2 75 20 15 0.75 52 3 75 20 10 0.5 53 3 75 20 15 0.75 the ellipse on the cylinder surface. This notation, illustrated in Figure 32 is consistent with that used by in refs. [5], [6], and [9]. All nodes and elements were created using the free mesh option in IDEAS, which uses solid tetrahedral elements. Quadratic tetrahedral elements were selected over linear tetrahedrons because the former elements provide more accurate displacement results and, hence, more accurate stress results for structures exhibiting a displacement Hield that is not linear. Linear tetrahedrons can only achieve linear displacement and constant stress and strain within the element [10]. Parabolic tetrahedral elements are identified as element type 118 in IDEAS. They include four corner nodes and six midside nodes. Three translational degrees of freedom are allowed at each node, such that parabolic deformation of each edge is permitted [11i]. The displacements of this element are defined by a set of complete quadratic polynomials, specified in Cook et al. [10]. This enables these elements to represent a quadratic displacement Hield within the element. Four integration points are used for these elements. Nodal stresses and strains are extrapolated from the values at the integration points [1l]. Figure 33 is a sketch of this element and the associated nodes. Face 1 Face 4  7 3 10 Face 2 Figure 33 Sketch of 10node parabolic tetrahedral element The automatic mesh checking function in IDEAS was used to enforce the condition that the elements generated have a distortion value greater than 0.1 and a stretch value greater than 0.2. The IDEAS User' s Guide suggests that distortion or stretch above 0.05 is usually acceptable for tetrahedral elements [1l]. An ideal element, i.e. a straightsided tetrahedron with midside nodes located in the geometric center of each line with equilateral triangular faces, would have distortion and stretch values of 1.00. Preventing excessive distortions allows these elements to more accurately represent high order displacement modes [10]. To decrease the number of elements required, each cylinder was partitioned into five volumes prior to meshing. The mesh density generally decreased as the distance from the thruhole increased. The five volumes, illustrated in Figure 34, include two concentric volumes surrounding the hole, with maximum diameters of three and five times the diameter of the hole, labeled as volumes 1 and 2, respectively. A central volume encompasses most of the cylinder and is labeled as volume 3. Upper and lower cylindrical volumes with heights that are 1/5 of the total height are labeled as volumes 4 and 5. Material properties applied to the models were selected to be as general as possible, maintaining linear elastic behavior, isotropy, and homogeneity. Properties provided by SDRC IDEAS for generic isotropic steel were used. The elastic modulus is 206,800 N/mm2, Poisson's ratio is 0.29, and shear modulus is 80155 N/mm2. Model 1 in Table 31 was selected for a convergence study. This geometry was chosen because it will require the maximum number of elements around the hole to attain convergence. All other models were created using the converged mesh density LJ5 Figure 34 Volume partitions, volume labels circled determined from this model. Parameters for this model include an outer diameter of 20 mm, an inner diameter of 15 mm, a hole diameter of 1 mm, and an obliquity angle of 750 This represents the thinnest wall, smallest hole, and largest angle of obliquity. Models with smaller obliquity angles would likely converge sooner and require a less dense mesh because they would be less prone to have geometric issues such as sharp angles and thin walls that generally require smaller elements to achieve convergence. The additional time required to solve all models with a finer mesh was acceptable compared to the time required to perform a convergence study on each model. An initial attempt at stress convergence provided inconclusive results because a large stress gradient is present for large obliquity angles. An inconsistent placement of nodes between models with varying mesh densities was identified as the problem. The free mesh generated in IDEAS results in a nonuniform nodal placement, so the exact location of maximum stress varied slightly from mesh to mesh. To resolve this problem, the convergence study was repeated after investigating the location of maximum stress. The seven meshes created for location investigation used specified element lengths between 0.16 and 0.28 millimeters in volume 1. The average angular location of maximum stress concentration around the ellipse was O' = 7.920 (f 0.79) SD) The percent standard deviation on the location of maximum stress is less than 10 percent, so, in models created after the conclusion of the convergence study, an investigation into the location of maximum stress was not performed. It is assumed that, by using a consistent mesh density equal to the converged mesh density for model 1, the location and magnitude of maximum stress in later models will be accurate. Four finite element meshes were used to test for convergence. In each of the meshes used for the convergence study, two nodes were generated on the elliptical intersection of the hole and the cylinder surface at 8' = f7.920 Figure 35 shows the variation in maximum stress at the two preset node locations as mesh density is increased. Convergence begins to occur when approximately 29000 elements are used for volume 1, corresponding to a specified element length (equivalent to the cube root of the element volume) of 0.22 millimeters. In the other volumes, specified element lengths are progressively larger as the distance from the thruhole increases. The number of elements used in the models ranged from 66997 to 428027 elements. 14 6.7000 6.6500 * 5 6.6000 .E 6.5500 E 6.5000 S6.4500 6.4000 0 10000 20000 30000 40000 50000 60000 Number of Elements in Smallest Volume Figure 35 Maximum stress plotted against number of elements in volume 1 CHAPTER 4 RESULTS Generation of Results Maximum and minimum stress concentration factors are calculated using a ratio of the maximum or minimum principal stress to a nominal value, as shown below. 1700 The variables om and on,,, are the maximum and minimum principal stress, respectively, in volume 1, the smallest volume that encompasses the hole. The nominal stress, ao;;;, may be defined using either the gross cross sectional area or the net cross sectional area. There are advantages to both approaches. To predict stress using the gross cross section approach, a stress value is initially calculated without considering the hole. This value is then scaled by the gross stress concentration factor. A stress concentration factor calculated based on the gross cross section accounts for the combined effects of a reduction in cross sectional area and the stress concentration effect. The net cross sectional area results in a more precise representation of the stress concentration effect of the hole. However, it is less practical, since it requires calculating the stress using the reduced cross section before applying the factor to predict maximum stress. If the hole size is sufficiently small such that the decrease in crosssectional area is minimal, the gross crosssectional stress may be used with little sacrifice to precision. The reference stress, cro;;;, is calculated as shown below using the flexure formula, where S is the section modulus. nom, The section modulus is a ratio of the moment of inertia to the distance of the applied load from the neutral axis. For a continuous hollow cylinder (with no discontinuity), S = iyr D4 d46' Pilkey [2] suggests that for a flat plate, if the hole diametertoplate width ratio is less than 0.5, the gross cross sectional stress may be used. The cylinders used for this study fit within these parameters. In fact, for the geometries used in this study, the cross sectional area decreases a maximum of 8 % after the hole area is removed, so the nominal stress for this study is based on the gross cross section. Listing of Results Maximum and minimum stress concentration factors near the hole are listed in Table 41. Also included are the position (radial, r, and angular, 6) of the maximum and minimum stress concentration factors. Results were obtained using the IDEAS solver and postprocessor. Maximum and minimum principal stresses occur exclusively on the hole surface, located either on the hole interior or at the intersection of the hole with the cylinder surface. There is a center of symmetry for oblique holes, such that for each point on the hole there is a corresponding point at which the stress state is identical. Due to the non uniform placement of nodes in this study, however, the resulting stress values at 17 Table 41 Maximum and minimum stress concentration factors, with location Hole Details Max. Stress Concentration Factor, Km, Min. Stress Concentration Factor, Kmm Model a (0) d/D dilD Value 6 (0) r Value 8 (0)r 1 ) ().)5 ().25 2.94 92.59 9.4() ().956 179.4() 9.62 2 ) ().)5 ().5 2.95 85.53 9.32 ().998 176.45 9.53 3 ) ().)5 ().75 3.()( 94.36 9.71 1.()51 1.()3 9.51 4 ) ().1 ().25 2.96 9().6() 9.69 ().963 179.42 8.89 5 ) ().1 ().5 2.97 92.26 9.71 ().977 178.3() 9.()1 6 ) ().1 ().75 3.()2 88.43 9.54 1.()41 177.51 8.93 7 ) ().15 ().25 2.97 91.()8 9.6() ().943 1.88 8.63 8 ) ().15 ().5 2.96 92.91 9.58 ().966 179.57 8.7() 9 ) ().15 ().75 3.()3 92.26 9.54 1.)5() ).()9 8.62 1() 15 ).()5 ().25 2.91 82.63 9.71 1.()61 6.57 9.83 11 15 ).()5 ().5 2.9() 73.79 9.52 1.()37 3.68 9.6() 12 15 ).()5 ().75 2.86 66.72 9.3()1 1.()87 2.17 9.67 13 15 ().1 ().25 2.88 81.6() 9.6() 1.()4) ().21 9.65 14 15 ().1 ().5 2.86 82.89 9.69 1.()38 2.()1 9.46 15 15 ().1 ().75 2.91 8().84 9.62 1.()67 1.()6 9.24 16 15 ().15 ().25 2.88 8).()7 9.41 1.()31 2.23 9.2() 17 15 ().15 ().5 2.91 81.()7 9.56 1.()46 1.15 9.26 18 15 ().15 ().75 2.95 8().59 9.73 1.186 178.59 7.5() 19 3) ().()5 ().25 2.63 72.63 9.9()1 1.186 4.23 9.93 2() 3) ().)5 ().5 2.64 68.34 9.71 1.()78 7.38 1().()( 21 3) ().()5 ().75 2.69 65.()7 9.7()1 1.141 5.29 1().()( 22 3) ().1 ().25 2.64 74.()5 9.7() 1.161 1.64 9.86 23 3) ().1 ().5 2.65 72. 18 9.7()1 1.197 1.26 9.92 24 3) ().1 ().75 2.7() 72.59 9.69 1.3()9 5.26 7.5() 25 3) ().15 ().25 2.68 72.56 9.7()1 1.137 ().28 9.62 26 3) ().15 ().5 2.69 76.66 9.9()1 1.144 ().86 9.85 27 3) ().15 ().75 2.76 76.35 9.91 1.531 169.6() 7.5() 28 45 ).()5 ().25 2.38 69.()7 1().()() 1.5()2 4.18 1().()( 29 45 ).()5 ().5 2.36 69.()7 1().()( 1.494 4.18 1().()( 3() 45 ).()5 ().75 2.35 69.()9 1().()() 1.541 4.18 1().()( 31 45 ().1 ().25 2.41 66.()9 1().()( 1.554 2.34 1().()( 32 45 ().1 ().5 2.41 66.()9 1().()() 1.558 2.34 1().()( 33 45 ().1 ().75 2.47 66.()8 1().()() 1.736 ().26 7.5() 34 45 ().15 ().25 2.48 63.54 10.00 1.494 0.41 10.00 35 45 ().15 ().5 2.5() 65.83 1().()( 1.499 3.93 1().()( 36 45 ().15 ().75 2.56 68.32 1().()() 1.993 3.67 7.5() 37 6) ().()5 ().25 2.()8 48.86 1().()( 1.8()7 2.93 1().()( 38 6) ().()5 ().5 2.()7 48.86 1().()() 1.794 2.93 1().()( 39 6) ().()5 ().75 2.10 48.86 10.00 1.811 2.93 10.00 4() 6) ().1 ().25 2.14 52.()( 1().()( 1.928 1.87 1().()( 41 6) ().1 ().5 2.16 52.()( 1().()() 1.96() 1.87 1().()( 42 6) ().1 ().75 2.23 52.()( 1().()( 2.196 ().68 7.5() 43 6) ().15 ().25 2.22 48.82 1().()() 1.937 2.21 1().()( 44 6) ().15 ().5 2.25 51.16 1().()() 1.959 2.21 1().()( 45 6) ().15 ().75 2.31 51.17 1().()( 2.688 ().3() 7.5() 46 75 ).()5 ().25 1.73 29.61 1().()() 1.439 ).()1 1().()( 47 75 ).()5 ().5 1.72 3().66 1().()( 1.372 ).()4 1().()( 48 75 ).()5 ().75 1.78 31.22 1().()() 1.273 4.9() 1().()( 49 75 ().1 ().25 1.82 28.42 1().()( 2.()56 2.21 1().()( 5() 75 ().1 ().5 1.84 27.79 1().()() 2.()47 2.21 1().()( 51 75 ().1 ().75 1.93 28.29 1().()( 2.112 1.93 7.5() 52 75 ().15 ().5 1.94 3().42 1().()( 2.383 ).()7 1().()( 53 75 ().15 ().75 2.()2 3().35 1().()() 3.587 1.46 7.5 symmetrical points are not exactly the same. The position indicated in the table is the location where the maximum or minimum principal stress occurs. The angular coordinate 8' listed in Table 41 is calculated directly from the coordinates of the node of interest (see Figure 3.2). This angle is transposed into the B plane for direct comparison of the position of maximum stress between cylinders with varying angles of obliquity. Both angles are equal to zero at the position farthest from the direction of the applied load. The relationship between O' and 6 is tan O' = tan 8 cos a Validation: Cylinders with Transverse Holes A photoelastic study performed by Jessop, Snell and Allison [8], which produced stress concentration factors for cylinders with a transverse hole extending through both walls subj ected to tension, bending, or torsion, is the most applicable prior work for comparison to this study. The results of the bending tests were fitted to a polynomial expression in Pilkey [2]. Figure 41 is a plot ofKna against the hole size ratio, d/D, for a = 0 from the current study and from the photoelastic results expressed in Pilkey [2] for a cylinder in bending. In each configuration, the finite element results exceed the experimental values. The maximum percent difference between the FEA results and the photoelastic results is 8.2 % at d/D = 0.10 and d, D = 0.25. It is important to note that the results from Pilkey [2] are based on a cylinder with a transverse hole extending through the cylinder, whereas the current study considers a hole through the wall only. An additional finite element model was created based on the model geometry identified as having the maximum percent difference, but modified with the addition of a transverse hole in the opposite wall. Kna from this model, identified as 0.05 0.07 0.09 0.11 0.13 0.15 Hole Size Ratio, d/D Figure 41 Comparison of results from FEA and Pilkey [2] 'FEA w/2 Holes' is marked as a solid black square in Figure 41. A comparison of this result to the applicable geometry in Pilkey [2] reveals an 8.8 % difference. Tafreshi and Thorpe [9] compared finite element analysis results of a plate in tension to photoelastic results of a plate in tension and reported 8.9 % difference between photoelastic and finite element results, so the current percent difference is nI ithrin expected levels. The low percent difference is an indication that the finite element analysis results compare well to experimental results. Validation: Flat Plates with Oblique Holes A further investigation was performed to determine the percent difference between FEA results for cylinders with oblique holes and photoelastic results for plates with oblique holes, both subj ected to tensile loading. Tension loads were applied to models with geometries that most closely matched the thickness to hole diameter ratios included in the photoelastic study of Stanley and Day [5]. For the plates in which the applied load is parallel to the maj or axis of the ellipse at the hole surface, the thicknesstohole diameter ratios included in Stanley and Day [5] were 1.33 and 2.00. 'i M .....IIIIIII~II::::IIII;II;;;:; 2.0 2.00  1.50  0.00  3.50 3.00 25 i aFEA Results (di/ D= 0.25) ~AFEA Results (di/ D= 0.50) xFEA Results (di/ D= 0.75)  0 Pilkey [2] (di D= 0.25)  a Pilkey [2] (di D= 0.50)  x Pilkey [2] (di D= 0.75) mFEA w 2 Holes (di/ D= 0.25) Cylinder geometries selected for comparison result in wall thickness to hole diameter ratios of 1.25 and 2.5. This corresponds to geometries with an inner cylinder diameter of 15 mm and a hole diameter of 1 mm or 2 mm. A uniform tension load was applied to the upper surface of the cylinder and the bottom surface was fully constrained in all six degrees of freedom. The results of the finite element models are shown with the results of Stanley and Day [5] in Figure 42. The maximum percent difference is 11.67 %. This occurs at an obliquity angle of 600. At 300 of obliquity, the percent difference in results is 9.87 %. Elsewhere, results are within 3.00 %. 3.50 .. .. FEA cylinders (t/d = 1.25) 3.00 A. FEA cylinders (t/d = 2.5) 0 Stanley and Day [5] (t/d = 1.33) 2.50~ A a Stanley and Day [5] (t/d = 2.0) 1.50 1.00 0.50 0.00 0 20 40 60 80 Angle of obliquity, a (0) Figure 42 Comparison of results from FEA and Stanley and Day [5] Angle of Obliquity, a Figure 43 illustrates the relationship between obliquity angle and Kax for various d/D ratios. For every hole geometry in this study, Kax approaches 3.00 at a = 0 and the value decreases as a increases. At a 600 obliquity angle, Kax is an average of 28% smaller than the value at 00 I  d/D = 0.05 e d/D = 0.10 9 d/D = 0.15 A d/D = 0.05 e d/D = 0.10 9 d/D= 0.15 0 20 40 60 80 Angle of Obliquity, a (0) (a) 0 20 40 60 80 Angle of obliqluity, a (0) (b) . : I I 3.00 I 2.50 2.00 ~C1.50 1.00 0.50 0.00 3.50 3.00 2.50 S2.00 ~C1.50 1.00 0.50 0.00 3.50 3.00 2.50 S2.00 0C 1.50 1.00 0.50 0.00 0 20 40 60 Angle of obliquity, ct (0) The relationship between Kmax and a is similar for every hole diametertoouter cylinder diameter ratio and wall thickness (inversely proportional to the cylinder diameter 3.50 4i~o l e d/D = 0.10 9 d/D = 0.15 (c) Figure 43 Kmax plotted against a, (a) d,/D = 0. 75, (b) d,/D = 0.50, (c) d,/D = 0.25 ratio, d,/D). A slightly larger variation in Kmax values occurs when d,/D is 0.75 (Figure 4 3a). This d,/D ratio represents the thinnest wall. Kmzn is plotted against a for various d,/D ratios in Figure 44. Similar to the Kmax results, more variation in the data occurs for the largest d,/D ratio (Figure 44a). In all of these plots, Kmzn increases in magnitude as obliquity angle increases from 00 to 600, but, for the smallest d/D ratio (i.e. the smallest hole diameter), Kmzn sharply decreases in magnitude as the angle of obliquity approaches 750. This may be due to the combined effects of a large stress gradient near the Kmzn location for this extreme obliquity angle and nonuniform node placement. The node placement may not have captured the absolute minimum stress. Hole Size Ratio, d/D The variation in maximum stress concentration as d/D changes at d,/D = 0.75 is plotted in Figure 44. Plots for other d,/D ratios are similar. The plots demonstrate that Kmax is not strongly dependent on d/D, especially at small obliquity angles. At larger obliquities, increasing d/D causes an increase in Kmax. For an obliquity angle of 750, there is a 12% increase in the maximum stress concentration factor as the hole diameter ratio increases from 0.05 to 0.15. The relationship between Kmzn and d/D is illustrated in Figure 45 at d,/D = 0.75. This relationship is stronger than the relationship between Kmax and d/D. As d/D increases, Kmzn decreases. The rate of decrease is larger for larger angles of obliquity. At a 750 angle of obliquity, the rate of decrease in Kmzn values is very large, but the values 0.50 1 1.00 ~  10 0 20 40 60 81 Angle of obliqluity, a (O) (b) &. 0 20 40 60 8( Angle of obliquity, a (0) 0.00 0.50 1.00 + 1.50 2.00 2.50 3.00 3.50 4.00 0 20 40 Angle of obliqluity, a~ (0) A d/D = 0.05 e d/D = 0.10 9 d/D = 0.15 Ad/D = 0.05 e d/D = 0.15 1.5 2.5 3.0 0.00 0.50 1.00 2.50 3.00 Ad/D= 0.05 e d/D = 0.10 ad/D = 0.15 Figure 44 K,,,,, plotted against a, (a) d, D = 0. 75, (b) d, D = 0.50, (c) d, D = 0.25 0.00 0.05 0.10 0.15 0.20 Hole Size Ratio, d/D Figure 46 Knzn plotted against hole size ratio, d/D, at d, D = 0. 75 are not in keeping with the other values on the chart. This may be due to the previously discussed large stress gradient at the Knzn location and the nonuniform stress gradient for models with a 750 angle of obliquity. Cylinder Diameter Ratio, d/D Figure 47 illustrates the relationship between Kna and the cylinder diameter ratio for various obliquity angles at d/D = 0.10. The relationship at other d/D ratios is very similar. The chart indicates that Kna is not dependent on wall thickness. A comparison of this chart with Figure 45 reveals that Kna is more dependant on d/D than d, D. A plot ofKnm versus the d, D ratio is shown in Figure 48 for a d/D ratio of 0. 10. The behavior of these results is somewhat similar to the behavior of the Kna results, but 0.00 0.05 0.10 0.15 0.20 Hole Size Ratio, d/D Figure 45 Kna plotted against hole size ratio, d/D, at d, D = 0. 75 0.00 0.50 2.00 2.50 3.00a=6 3.50 a =7 4.00 r     3.50 2.00 1.50 1.00 0.50 0.00 e a a _ 3.50 3.00 r:2.00 1.50 1.00 0.50 0.00  = 300 xu = 450 a a = 600  = 750 0.20 0.30 0.40 0.50 0.60 0.70 0.80 Cylinder Diameter Ratio, dilD Figure 47 K;;; plotted against d, D for various a at d/D = 0.10 ~~ ra = 450 1.5() 2.5() ().2 ) ().3) ().4 ) ().5) ().6) ().7) ().8() Cylinder Diameter Ratio, dilD Figure 48 K,,,,, plotted against d, D for various a at d/D = 0.10 the wall thickness tends to have a stronger influence on K,,,,,. As the wall thickness decreases (i.e. d, D increases), K;;,,, decreases 14%. This trend occurs for every obliquity angle, indicating that the effect of the wall thickness on K,,,,, is not influenced by the obliquity angle. Angular Location of Maximum Stress, B The effect of a on the angular location of K;;;a in the hole is illustrated in Figure 49 for each diameter ratio. The angular location is plotted in terms of 6, the angle around the hole in the plane perpendicular to the hole axis. This angle is defined from the 0 20 40 60 8( ar (degrees) (b) 100.00 80.00 60.00 40.00 20.00 0.00 0 20 40 a (degrees)  d/D= 0.05 +d/D= 0.10 ad/D = 0.15 60 80 100.00 80.00 60.00 40.00 20.00 0.00 Ad/D = 0.05 od/D= 0.10 ad/D= 0.15 100.00 80.00 60.00 40.00 20.00 6 d/D= 0.05 e d/D= 0.10 9 d/D = 0.15 0 20 40 a (degrees) 60 80 Figure 49 Angular location of K,no, 8, plotted against a (a) d, D = 0.25, (b) dD = 0. 50, (c) dD = 0. 75 point in the hole furthest from the direction of the applied load. The absolute value of each angular location is plotted for clarity. In general, as a increases from 00 to 750, B shifts from 900 toward 300. For the smallest d/D ratio, 0.05, Kmax tends to plateau to 700 between 150 and 450 of obliquity. The effect is present in all cylinders, but is most pronounced for the thinnest cylinder, i.e. at d/D = 0.75. Depth of Maximum Stress in Hole, h To provide a direct comparison of the position of maximum stress for all thickneses, the radial locations reported in Table 41 have been normalized. The normalized depth of maximum stress in the hole, h, is determined using the following formula, which is a ratio of the distance of the maximum stress as measured from the midplane, to half of the wall thickness: h=4r D d, Normalized depth values of 0.00 and 1.00 represent the midplane of the hole and the surface of the cylinder, respectively. Figure 410 illustrates the relationship between the normalized distance of Kma from the midplane and the angle of obliquity for various cylinder diameter ratios. As the angle of obliquity is increased, the Kmax shifts rapidly toward the plate surface. Kmax is located on the cylinder surface for all obliquity angles over 450. The absolute value of the normalized depth was used to generate the plots, so the surface indicated by the value '1.00' may be either the inner or outer surface. 6 dOD = 0.05 + daD = 0.10 9 dOD =0.15 Ad/D = 0.05 e d/D = 0.10 = d/D =0.15 A d/D = 0.05 + d/D = 0. 10 = d/D 0.15 Angle of Obliquity, ac (degrees) (c) Figure 410 Normalized depth of K,na in the hole interior plotted against a (a) di/D = 0.25, (b) di/D = 0.50, (c) di/D = 0.75degrees (see Figure 3.1). 0 20 40 60 8( Angle of Obliquity, a (degrees) (b) I I 0   0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 1.00 0.50 0.40 0.30 0.20 0.10 0.00 Angle of Neutral Axis, ly To compare the results of this study to the stress concentration factor in the canine tibia, a modification to the results would be necessary. The force couple applied to generate all previous results was oriented such that the neutral axis is at ry= 0. This is the most critical bending load case for this geometry, resulting in the largest maximum principal stresses. The physiologic loading of the canine tibia, however, is such that the neutral axis is closer to ry= 1800. This loading results in compression in the region of the cylinder surrounding the hole. Selected models, chosen for parameters that most closely matched that of the canine tibia, were subj ected to a bending moment with a neutral axis at ry= 1800 (see Figure 3.1). At the nutrient foramen, the canine tibia has an outside wall diameter of approximately 15 millimeters, a wall thickness of approximately 2 millimeters, a nutrient foramen diameter of less than 2 millimeters, and an obliquity angle that approaches 750 The models selected for the loading comparison were 47, 48, 50, and 51 (see Table 3.1). Results indicate that a simple modification of the ry= 00 results yield results comparable to canine tibia stress concentration factors. For each model, the maximum principal stress at ry= 1800 is equivalent to the absolute value of the minimum principal stress at ry= 00. Similarly, the absolute value of the minimum principal stress at ry= 1800 is equivalent to the maximum principal stress at ry=0 0. So, to compare these results to the maximum stress concentration factor in the canine tibia, it would be necessary to choose the absolute value of the minimum stress concentration factor for the most applicable geometry. CHAPTER 5 DISCUSSION Angle of Obliquity, a A photoelastic study performed by Stanley and Day [5] lists results for flat plates with oblique holes subj ected to uniform uniaxial tension parallel to the maj or axis of the ellipse created on the hole surface by the oblique hole (similar to the orientation of the hole in this study). The relationship between Kax and a illustrated in Figure 43 is consistent with the trends reported by Stanley and Day [5], which show that, for this load orientation, an increase in hole obliquity results in a decrease in the maximum stress concentration factor. As the angle of obliquity increases, the hole in the cylinder surface elongates. Results in Pilkey [2] indicate that elongating a hole in this direction in a plate in tension has the effect of reducing the stress concentration factor. As obliquity increases, Kax decreases more in a flat plate in tension than in a cylinder in bending. This effect is due primarily to the geometry of the specimen, rather than the difference in loading. In the work performed by Stanley and Day [5], the maximum stress concentration factor at 600 is approximately 40 % smaller than the value at 00. For cylinders in bending included in the current study, Kax decreases 28 % when a is varied from 00 to 600. Cylinders subj ected to tensile loading, which were compared to plates in tension in Figure 42, result in a Kmax decrease of 28 % between a = 00 and a = 600. As the angle of obliquity increases, minimum stress concentration factors increase in magnitude (see Figure 44). The aspect ratio of the ellipse created on the cylinder surface also increases as obliquity increases. The minimum principal stress is located near the maj or axis of the ellipse. At this location, the radius of curvature decreases. Results indicate that decreasing the radius of curvature causes an increase in the stress gradient, which, in turn, decreases the area over which the stress is distributed. This effect contributes toward increasing the minimum stress concentration factor as obliquity angle increases. Minimum stress concentration factors were rarely reported in prior work on oblique holes in flat plates with uniaxial tension, so the results of this study supply new information to this area of research. The sharp decrease in magnitude of Knun values in Figure 44 for small holes with a 750 angle of obliquity may be attributed to the mesh. An extremely high stress gradient was observed near the location of minimum stress for the smallest hole diameter and the largest hole obliquity. Nonuniform node placement combined with a large stress gradient decreases the probability that a node will be located precisely at the minimum stress location. This is especially important to consider when the stress changes drastically over a small geometric area. The precision with which accurate maximum and minimum stress values can be obtained in this region is influenced by the element size. Hole Size Ratio, d/D The relationship between maximum stress concentration factor and the hole size ratio, d/D, as shown in Figure 45, is comparable to the relationship developed in Jessop et al. [8] for cylinders with transverse holes in bending. As the hole size ratio increases, the maximum stress concentration factor increases slightly. This is expected, since increasing the diameter of the hole would increase the cross sectional area of the discontinuity, which would increase the stress concentration. A graphical comparison of the finite element results to the photoelastic results in Jessop et al. [8] is included in Figure 41. A comparison revealed an 8.2 % difference in results. Cylinder Diameter Ratio, di/D The behavior of the maximum stress concentration factor as the cylinder diameter ratio is varied is also comparable to the results of Jessop et al. [8], which states that the effect of d,/D on the maximum stress concentration factor for a cylinder in bending is not as great as the effect of varying d/D. This indicates that the thickness of the cylinder wall has less of an influence on the maximum stress concentration factor than does the hole diameter. Variations in the diameter or obliquity of the hole tend to have the greatest influence on the maximum stress concentration factor. Angular Location of Maximum Stress, B As was shown in Figure 49, 8, the angular location of maximum stress, shifts from 900 toward 300 as the angle of obliquity increases from 00 to 750. A similar trend was documented in Stanley and Day [5] for a flat plate in uniaxial tension with an oblique hole orientation similar to the orientation of holes in this study. In their photoelastic study, Stanley and Day reported a maximum stress located at 900 for every hole except one [5]. At a 600 obliquity and a thicknesstohole diameter ratio of 2.59 (the largest ratio reported), maximum stress was located at 8= 600 The slices used in the photoelastic study of Stanley and Day [5] were located every 300 around the circumference of the hole, so the accuracy of the locations reported is limited. A trend of decreasing angular location of maximum stress concentration factor may have been present for all thicknesstohole diameter ratios, but was most pronounced when the thicknesstohole diameter ratio was the largest. Depth of Maximum Stress in Hole, h The plots in Figure 410 indicate that the position of Lax shifts toward the plate surface as the angle of obliquity increases. This is comparable to the behavior noted in DulieuBarton and Quinn [6], in which Kax position data were compiled from Stanley and Day [5], McKenzie and White [12], and Leven [13]. All data were from investigations of flat plates with oblique holes loaded in tension parallel to the minor axis of the ellipse (perpendicular to the orientation of holes in this study). Results showed that as the angle of obliquity increases from 00 to 600, the maximum stress moves from the midplane to the plate surface. An additional plot in DulieuBarton and Quinn [6] shows that varying the angle of applied load from parallel to the minor axis of the ellipse to parallel to the maj or axis does little to alter the depth of maximum stress on the hole interior. As the angle of obliquity increases, the location of maximum stress reaches the hole edge more rapidly in cylinders in bending than in flat plates in tension. In the cylinders in this study, the maximum stress is located on the hole edge for all obliquity angles over 450. For flat plates loaded in tension, however, only holes with obliquity angles over 600 exhibit maximum stress located on the hole edge. This may be due to the applied bending load, which causes high tensile stresses to occur near the surface. Angle of Neutral Axis, ry Results in Table 41 were generated using a force couple that results in tension on the region of the cylinder surrounding the hole (ry= 00). Selected models with geometry similar to the canine tibia were loaded such that the region surrounding the hole is in compression. This corresponds to ry= 1800. A comparison of results at different ry 34 angles revealed that the magnitude of maximum principal stress at ry= 180 is equal to the magnitude of minimum principal stress at ry= 00. To use these results as a comparison tool to help quantify mechanical improvements in the canine tibia, the absolute value of the minimum stress concentration factor as reported should be compared to the maximum stress concentration factor in the canine tibia. CHAPTER 6 CONCLUSIONS This study provides a detailed discussion of stress concentration factors in cylinders with either oblique or transverse holes in bending. Results demonstrate a strong relationship between maximum stress concentration factor and the angle of obliquity. For the loading scheme studied, as the obliquity increases, the maximum stress concentration factor decreases. A weak dependency exists between the maximum stress concentration factor and the hole diametertoouter cylinder diameter ratio. An increase in this ratio tends to result in a slight increase in the maximum stress concentration factor. For an obliquity angle of 750, there is a 12% increase in the maximum stress concentration factor as the hole diameter ratio increases from 0.05 to 0.15. The results compare well to previous work. The changes in the maximum stress concentration factor with respect to the angle of obliquity, wall thickness, and hole diameter are supported by the trends of prior studies. The angular location and depth of the maximum stress concentration factor in the hole are also in keeping with earlier findings. As compared to flat plates in tension, though, cylinders in bending tend to result in maximum stress locations that shift more rapidly toward the acute intersection of the hole with the plate as the angle of obliquity increases. In addition, maximum stress concentration factors for plates in tension decrease more than corresponding factors in cylinders in either bending or tension as the obliquity angle is increased from 00 to 600. Minimum stress concentration factors at any angle of obliquity and all data generated from a 750 angle of obliquity have been largely overlooked in previous work, but results are not unexpected. As the obliquity increases from 00 to 600, the magnitude of minimum stress concentration factors tends to increase as the magnitude of maximum stress concentration factors decreases. This may be due to a change in the radius of curvature, which decreases at the location of minimum stress as the obliquity increases. The magnitude and location of maximum stress concentration factors at a 750 obliquity angle, similarly, are consistent with values at other obliquity angles. Somewhat unexpected results, however, occur for the minimum stress concentration factors reported at a 750 angle of obliquity. Nonuniform node placement is responsible for these deviations. At this obliquity angle, the minimum stress occurs near a large stress gradient and the location of minimum stress does not coincide with a node, so the reported stress may be much larger than the absolute minimum, although the two are in close proximity. Data generated by this proj ect may be compared to stress concentration values associated with the nutrient foramen in the canine tibia. The canine tibia is loaded in bending in the saggital plane such that the nutrient foramen is primarily in compression. The bending load applied in this study, however, would put the hole in tension. For directly comparable results, the absolute value of Knun from this report should be considered to be the maximum stress concentration factor. To obtain stress concentration values for the canine tibia, a microstructural analysis of bone samples near the nutrient foramen would be performed to find approximate elastic constants at discrete locations. These values could then be used in a finite element model to simulate the stiffness variations. Stress concentration factors in the canine tibia could then be established for comparison with values explored in detail in this report. If, as expected, a stiffness gradient similar to that discovered around the equine third metacarpus nutrient foramen is present at the canine tibia foramen, the maximum stress concentration factor in bone would be lower than that in a homogeneous, isotropic structure. The results of this study provide comparison data that will help to quantify the ability of bone to reduce stress around geometric discontinuities. LIST OF REFERENCES [1] Goitzen N, Cross AR, Ifju PJ, Rapoff AJ. Understanding stress concentration about a nutrient foramen. JBiomech 2003; 36(10): 15111521. [2] Pilkey WD. Peterson's Stress Concentration Factors, 2nd Edition, John Wiley & Sons, New York, 1997. [3] Ellyin F, Lind NC, Sherbourne AN. Elastic stress field in a plate with a skew hole. JEng MechASCE 1966; 92: 110. [4] Ellyin F. Experimental study of oblique circular cylindrical apertures in plates. Exp M~ech 1970; 10: 195202. [5] Stanley P, Day BV. Photoelastic investigation of stresses at an oblique hole in a thick flat plate under uniform uniaxial tension. J Strain Anal Eng 1990; 25(3): 157 175. [6] DulieuBarton JM, Quinn S. Thermoelastic stress analysis of oblique holes in flat plates. Int J2~ech Sci 1999; 41 : 527546. [7] Daniel IM. Photoelastic analysis of stresses around oblique holes. Exp M~ech 1970; 10: 467473. [8] Jessop HT, Snell C, Allison IM. The stress concentration factors in cylindrical tubes with transverse cylindrical holes. Aeronaut Q 1959; 10: 326344. [9] Tafreshi A, Thorpe TE. Numerical analysis of stresses at oblique holes in plates subjected to tension and bending. J Strain Anal Eng 1995; 30(4): 317323. [10] Cook R.D, Malkus DS, Plesha ME, Witt RJ. Concepts and Applications of Finite Element Analysis, 4th Edition, John Wiley & Sons, New York, 2002. [1l] IDEAS 9.0 Structural Dynamics Research Corporation. Online Help Bookshelf, Simulation: Element Library, 2001. [12] McKenzie HW, White DJ. Stress concentration caused by an oblique round hole in a flat plate under uniaxial tension. JStrain AnalEng 1968; 3(2): 98102. [13] Leven MM. Photoelastic determination of the stresses at oblique openings in plates and shells. Weld'ing Research Council Bulletin l970; 153: 5280. BIOGRAPHICAL SKETCH Susan Andrea Mathison was born on September 30, 1974, in Huntsville, Alabama, to Julian and Sue Mathison. She has one older brother, Michael. She attended Huntsville High School from 1988 to 1992, where she was a member of the concert band and co captain of the color guard. After graduation, she moved to Knoxville, Tennessee, and became a student in radiological engineering at the University of Tennessee. It was there that she met Guy Grover, another student in the same department. They married on August 6, 1994, in Huntsville and she changed her name to Susan Mathison Grover. A year later, she decided to switch her major. She graduated in May 1997 with a Bachelor of Science degree in engineering science and mechanics with a concentration in biomedical engineering. On June 23, 1997, she j oined USBI, Inc., a division of United Technologies, performing structural analyses of reusable Space Shuttle Solid Rocket Booster hardware. She worked there for 5 years, before resigning in January 2003 to attend the University of Florida fulltime to pursue a Master of Science degree in biomedical engineering. Since then, she has been working in the Applied Biomechanics Laboratory at UF. She is also currently working as an Intern at Regeneration Technologies, Inc., in Alachua, Florida. After graduation, she plans to find a job in the medical device industry. 