{"@context":{"@language":"en","Affiliation":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","AggregatedSourceRepository":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","Campus":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","Creator":"http:\/\/purl.org\/dc\/terms\/creator","DateAvailable":"http:\/\/purl.org\/dc\/terms\/issued","DateIssued":"http:\/\/purl.org\/dc\/terms\/issued","Degree":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","DegreeGrantor":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","Description":"http:\/\/purl.org\/dc\/terms\/description","DigitalResourceOriginalRecord":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","FullText":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","Genre":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","IsShownAt":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","Language":"http:\/\/purl.org\/dc\/terms\/language","Program":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","Provider":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","Publisher":"http:\/\/purl.org\/dc\/terms\/publisher","Rights":"http:\/\/purl.org\/dc\/terms\/rights","ScholarlyLevel":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","Title":"http:\/\/purl.org\/dc\/terms\/title","Type":"http:\/\/purl.org\/dc\/terms\/type","URI":"https:\/\/open.library.ubc.ca\/terms#identifierURI","SortDate":"http:\/\/purl.org\/dc\/terms\/date"},"Affiliation":[{"@value":"Applied Science, Faculty of","@language":"en"},{"@value":"Civil Engineering, Department of","@language":"en"}],"AggregatedSourceRepository":[{"@value":"DSpace","@language":"en"}],"Campus":[{"@value":"UBCV","@language":"en"}],"Creator":[{"@value":"Wong, Vanessa","@language":"en"}],"DateAvailable":[{"@value":"2011-02-22T21:24:22Z","@language":"en"}],"DateIssued":[{"@value":"2006","@language":"en"}],"Degree":[{"@value":"Master of Applied Science - MASc","@language":"en"}],"DegreeGrantor":[{"@value":"University of British Columbia","@language":"en"}],"Description":[{"@value":"With the rapid advancements in telescope technology, the next generation telescopes will be\r\non the order of 20 - 50 [m] in diameter. Since traditional telescopes have been substantially\r\nsmaller, this huge increase in telescope size necessitates a study into new solutions for\r\ntelescope enclosure geometry.\r\nIn an attempt to come up with new geometries for large telescope enclosures, a study was\r\ncarried out on Platonic and Archimedean spheres. After careful consideration, the Platonic\r\nicosahedral sphere (a type of geodesic dome) and the Archimedean rhombicosidodecahedral\r\nsphere was selected for further analysis.\r\nSensitivity analysis was performed on the two selected configurations by varying the\r\nenclosure radius (R), and the member cross-section size (CSS), thickness (T), and type (CT).\r\nA 3-dimensional model of each case was generated in a finite element analysis program and\r\nthe corresponding nodal deflections and member forces were obtained. After plotting the\r\nresults, it was discovered that for both the rhombicosidodecahedron and icosahedron\r\nconfiguration, the optimal R is 25 [m]. The optimal CSS for the rhombicosidodecahedron\r\nconfiguration is 0.35 [m] while the optimal CSS for the icosahedron configuration is 0.3 [m].\r\nThe optimal T for both configurations is 0.02 [m]. In addition, the optimal CT for the\r\nrhombicosidodecahedron configuration is circular while the optimal CT for the icosahedron\r\nconfiguration is square. The two configurations were also compared against one another and\r\n\r\nit was discovered that the icosahedron configuration generally performs better than the\r\nrhombicosidodecahedron configuration.\r\nIn addition to exploring new structural geometries for telescope enclosures, one must not\r\nforget all the expertise which has been put in older telescope enclosures. PhotoModeler is a\r\nphotogrammetry software which allows its user to take pictures of an existing structure then\r\ngenerate 3-dimenional models using various functions in the software. Using this program,\r\none can combine structural attributes from older telescope enclosures with new geometries to\r\ncreate a hybrid enclosure suitable for next generation telescopes.\r\nFinally, as a supplementary to the structural geometries suggested in this report, several\r\ncomposite materials which may be suitable for use as cladding are also presented. These\r\ncomposite panels include: metal composites, polymer composites, and honeycomb core\r\ncomposites.","@language":"en"}],"DigitalResourceOriginalRecord":[{"@value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/31608?expand=metadata","@language":"en"}],"FullText":[{"@value":"Design of Structural Geometries for Large Telescope Enclosures B y Vanessa W o n g B.A.Sc Universi ty of Br i t i sh Co lumb ia , 2005 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R OF APPLIED SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH C O L U M B I A December 2006 \u00a9 Vanessa Wong, 2006 Abstract With the rapid advancements in telescope technology, the next generation telescopes wi l l be on the order of 20 - 50 [m] in diameter. Since traditional telescopes have been substantially smaller, this huge increase in telescope size necessitates a study into new solutions for telescope enclosure geometry. In an attempt to come up with new geometries for large telescope enclosures, a study was carried out on Platonic and Archimedean spheres. After careful consideration, the Platonic icosahedral sphere (a type of geodesic dome) and the Archimedean rhombicosidodecahedral sphere was selected for further analysis. Sensitivity analysis was performed on the two selected configurations by varying the enclosure radius (R), and the member cross-section size (CSS), thickness (T), and type (CT). A 3-dimensional model of each case was generated in a finite element analysis program and the corresponding nodal deflections and member forces were obtained. After plotting the results, it was discovered that for both the rhombicosidodecahedron and icosahedron configuration, the optimal R is 25 [m]. The optimal CSS for the rhombicosidodecahedron configuration is 0.35 [m] while the optimal CSS for the icosahedron configuration is 0.3 [m]. The optimal T for both configurations is 0.02 [m]. In addition, the optimal C T for the rhombicosidodecahedron configuration is circular while the optimal C T for the icosahedron configuration is square. The two configurations were also compared against one another and it was discovered that the icosahedron configuration generally performs better than the rhombicosidodecahedron configuration. In addition to exploring new structural geometries for telescope enclosures, one must not forget all the expertise which has been put in older telescope enclosures. PhotoModeler is a photogrammetry software which allows its user to take pictures of an existing structure then generate 3-dimenional models using various functions in the software. Using this program, one can combine structural attributes from older telescope enclosures with new geometries to create a hybrid enclosure suitable for next generation telescopes. Final ly, as a supplementary to the structural geometries suggested in this report, several composite materials which may be suitable for use as cladding are also presented. These composite panels include: metal composites, polymer composites, and honeycomb core composites. -111-Table of Contents Abstract \u2022 i i Table of Contents iv List of Tables \u2022 - v i i List of Figures \u2022\u2022 ix List of Equations \u2022 x i i i List of Symbols and Abbreviations \u2022 xiv Glossary xvi Acknowledgements xv i i i 1. Overview and Summary 1 1.1. Literature Review 1 1.2. Overview : . 2 2. The Historical Evolution of Telescopes and Telescope Enclosures 4 2.1. History of Telescopes 4 2.2. Types of Telescopes 6 2.2.1. Radio Telescopes 6 2.2.2. Optical Telescopes \u2022 7 2.3. Evolution of Telescope Enclosures 14 3. Methods of Generating Enclosure Geometries 21 3.1. Platonic and Archimedean Solids 21 3.1.1. Platonic Solids 22 -iv-3.1.2. Archimedean Solids .....26 3.2. Geodesic domes 30 3.2.1. Advantages and Disadvantages of Geodesic Domes 31 3.2.2. Geodesic Dome Classification..... 32 3.3. Spherical tiling programs... 34 4. Geometric Solutions to the Enclosure Problem 37 4.1. Generation of Icosahedron Geodesic Dome 37 4.2. Generation of Rhombicosidodecahedron Dome 44 5. Formulation of Model and Loads for Finite Element Analysis 46 5.1. Case Definition and Load Combination 46 5.2. Type of Structure 47 5.3. Material Properties , 48 5.4. Dead and Live Load 48 5.5. Snow and Ice Loads 49 5.6. Wind Loads : 53 5.7. Earthquake Loads \u2022\u2022\u2022\u2022 66 5.8. Summary of Loads 72 6. Finite Element Analysis Results 74 6.1. Optimal Cases 75 6.2. Optimal Configuration 92 7. Use of Photogrammetry to Assess Existing Structures 94 7.1. PhotoModeler Pro 95 8. Cladding Options for Enclosure 106 9. Conclusions and Recommendations 112 10. Bibliography 115 A . Appendix A - Finite Element Analysis Results 118 A . l . Icosahedron Results 119 A.2 . Rhombicosidodecahedron Results... 132 j -v i -List of Tables Table 3.1 Properties of Platonic Solids 24 Table 3.2 Summary of dihedral angle and angular deficiency for each Platonic solid 25 Table 3.3 Archimedean Solids Properties.... 29 Table 3.4 Classification of Geodesic Domes 33 Table 4.1 Syntax for Importing Objects in A u t o C A D 39 Table 4.2 Summary of Projected Point 43 Table 5.1 Case Definitions : 46 Table 5.2 Control Case ; 47 Table 5.3 Load Combinations for Ultimate Limit States 47 Table 5.4 Member Properties ;. 48 Table 5.5 Snow and Ice Loading as provided by A M E C Dynamic Structures 49 Table 5.6 S A P output of vibration frequencies for first 12 modes (Rhombicosidodecahedron) 58 Table 5.7 S A P output of vibration frequencies for first 12 modes (Icosahedron) 59 Table 5.8 Fundamental Lateral Period of Vibration for Rhombicosidodecahedron..... 68 Table 5.9 Fundamental Lateral Period of Vibration for Icosahedron 68 Table 5.10 M v Values from N B C C 2005 Table 4.1.8.11 69 Table 5.11 Linear Acceleration for Rhombicosidodecahedron ...71 Table 5.12 Linear Acceleration for Icosahedron 71 Table 5.13 Summary of Loads on Rhombicosidodecahedron 72 Table 5.14 Summary of Loads on Icosahedron 73 -vi i -Table 6.1 Summary of Masses for Rhombicosidodecahedron Structure 74 Table 6.2 Summary of Masses for Icosahedron Structure 75 Table 6.3 Optimal Cases 88 Table 6.4 Optimization Formulas 89 Table 6.5 Comparison of Rhombicosidodecahedron and Icosahedron Configurations for Different Cases... 92 Table 7.1 Camera Calibration Adequacy Check 98 Table 8.1 Summary of Panel Attributes 110 Table 8.2 Summary of Cladding Material Properties.... I l l List of Figures Figure 2.1 Focal length and magnification 8 Figure 2.2 Refracting Telescopes 9 Figure 2.3 Spherical Aberration.... 10 Figure 2.4 Coma in parabolic mirrors 12 Figure 2.5 Common types of reflecting telescopes 12 Figure 2.6 Cassegrain-Hybrid Telescopes 14 Figure 2.7 Range of motion of telescope 16 Figure 2.8 Gemini Telescope in Hawaii 16 Figure 2.9 Arch Girder Design ..' 17 Figure 2.10 Calotte Telescope Enclosure Configuration. 18 Figure 2.11 Cylindrical Telescope Enclosure Configuration 19 Figure 2.12 Faceted Telescope Enclosure Configuration 20 Figure 2.13 Rectangular Telescope Enclosure Configuration 20 Figure 3.1 Mayan Pyramid located in the ancient city of Uxmal 21 Figure 3.2 Platonic Solids 23 Figure 3.3 Archimedean Solid Classifications 28 Figure 3.4 Montreal Biosphere 31 Figure 3.5 Different Types of Geodesic Dome Triangulation 34 Figure 3.6 Screen capture from Gmsh 35 Figure 3.7 Screen capture from Makros-A 35 Figure 3.8 Screen capture from N e t G E N 36 -ix-Figure 4.1 Coordinates of Vertices for Icosahedron 38 Figure 4.2 Points, lines, and texts for icosahedron imported from Excel to A u t o C A D ......... 40 Figure 4.3 Points, lines, and texts for rhombicosidodecahedron imported from Excel to A u t o C A D 40 Figure 4.4 Icosahedron Face Numbering System 41 Figure 4.5 Point Projection Diagram 42 Figure 4.6 Transformation of Icosahedron to Geodesic Sphere 43 Figure 4.7 Coordinates for Vertices of Rhombicosidodecahedron............ 44 Figure 4.8 Triangulation of Rhombisocidodecahedron : : 45 Figure 5.1 Different Triangles on the Rhombicosidodecahedron Configuration 50 Figure 5.2. F low around a sphere 55 Figure 5.3 Deformed Shape (Mode 1) of Rhombicosidodecahedron 59 Figure 5.4. Deformed Shape (Mode 1) of Icosahedron.. 60 Figure 5.5. Lift and Drag Force on Blunt Bodies 61 Figure 5.6 Drag Coefficients for F low around Blunt Bodies (Potter and Wiggert) 62 Figure 5.7 Projected Area of Icosahedron Configuration, Case 1 65 Figure 5.8 Uniform Hazard Spectrum for Vancouver 66 Figure 6.1 Average X-deflection for Rhombicosidodecahedron 77 Figure 6.2 Average Y-deflection for Rhombicosidodecahedron 77 Figure 6.3 Average Z-deflection for Rhombicosidodecahedron 78 Figure 6.4 Average Structural Force (X) for Rhombicosidodecahedron 78 Figure 6.5 Average Structural Force (Y) for Rhombicosidodecahedron 79 Figure 6.6 Average Structural Force (Z) for Rhombicosidodecahedron 79 -x-Figure 6.7 Average Structural Moment (X) for Rhombicosidodecahedron 80 Figure 6.8 Average Structural Moment (Y) for Rhombicosidodecahedron 81 Figure 6.9 Average Structural Moment (Z) for Rhombicosidodecahedron 81 Figure 6.10 Average X-deflection for Icosahedron 83 Figure 6.11 Average Y-deflection for Icosahedron 83 Figure 6.12 Average Z-deflection for Icosahedron 84 Figure 6.13 Average Structural Force (X) for Icosahedron 85 Figure 6.14 Average Structural Force (Y) for Icosahedron 85 Figure 6.15 Average Structural Force (Z) for Icosahedron 86 Figure 6.16 Average Structural Moment (X) for Icosahedron 87 Figure 6.17 Average Structural Moment (Y) for Icosahedron 87 Figure 6.18 Average Structural Moment (Z) for Icosahedron 88 Figure 7.1 Calibration Gr id for PhotoModeler 98 Figure 7.2. Plan V iew of Camera Angles 99 Figure 7.3 Marking Points on a Cylinder.. 100 Figure 7.4. Referencing curves and lines from pictures 101 Figure 7.5. 3-dimensional view of generated bottle with texture added 101 Figure 7.6. Model exported into A u t o C A D 102 Figure 7.7 Lego Structure 103 Figure 7.8 3-dimensional model of Lego structure (Poor quality camera) 104 Figure 7.9 3-dimensional model of Lego structure (Higher quality camera) 105 Figure 8.1 The Subaru Telescope in Hawaii 106 Figure 8.2 The Gemini Telescope in Hawaii 107 -x i -Figure 8.3 The Hobby-Eberly Telescope in Texas 107 -x i i -List of Equations Equation 3.1 Dihedral Angle Expression .'. 24 Equation 3.2 Angular Deficiency : 25 Equation 3.3 Golden Ratio 26 Equation 4.1 Distance Equation.. 38 Equation 4.2 Unit Vector for X-axis 42 Equation 4.3 Unit Vector for Y-axis 42 Equation 4.4 Unit Vector for Z-axis 43 Equation 4.5 Projected Coordinates 43 Equation 5.1 The Strouhal Number 56 Equation 5.2 Drag Force 61 Equation 5.3 External Pressure Caused by Wind on Structure 64 Equation 5.4 Min imum Lateral Acceleration due to Earthquake 67 - x i i i -List of Symbols and Abbreviations a R = resolution limit [radians] X - wavelength 5 = angular deficiency ()) = Golden Ratio p = density of fluid u = kinematic viscosity A = projected area normal to direction of flow, lateral acceleration due to earthquake G = center point of lens C a = shape factor (snow load calculation) Cb = basic roof snow load factor (snow load calculation) C D = drag coefficient C c = exposure factor C g = gust factor C p = external pressure coefficient C s = slope factor (snow load calculation) C w = wind exposure factor (snow load calculation) CSS = member cross-section size C T = member cross-section type D = aperture (width of the objective), object diameter F = focal point of central rays F ' = focal points of edge rays f = vortex shedding frequency FD = drag force F S = force due to factored snow and ice load IE = importance factor for earthquake I W = importance factor for wind load I S = importance factor for snow load (snow load calculation) i = unit vector along X-axis -xiv-j = unit vector along Y-axis . k = unit vector along Z-axis M v = factor to account for higher mode effect on base shear N B C C = National Bui lding Code of Canada p = number of sides in each face of a Platonic solid, external pressure caused by wind on structure q = number of faces meeting at each vertex in a Platonic solid, reference velocity pressure R = enclosure radius Rd = ductility related force modification factor Re = Reynolds number Ro = overstrength related force modification factor S(T a) = spectral acceleration at time T a according to U H S S r = 1 in 50 year rain load (snow load calculation) S s = snow and ice load S = factored snow and ice load St = Strouhal number T = member cross-section thickness T a = fundamental lateral period of vibration of structure U H S = Uniform hazard spectrum V = wind velocity W = wind load -xv-Glossary Angular Resolution: the ability of an optical instrument to measure angular separations between points on an object Boundary layer: a thin layer attached to the boundary in which viscous effects are concentrated Chromatic aberration: caused by the lens having a different refraction index fro different wavelengths of light, resulting in a fringe of color around an image Coma: a monochromatic aberration of a lens where a point source image cannot be focused, the resulting image has a comet-shaped appearance Dual polyhedron: pairs of polyhedra where the vertices of one correspond to the faces of its dual F-ratio: ratio of focal length to aperture Focal length: a measure of how an optical system focuses or diverges light Golden Ratio (
Eye Eyepiece Lens (Convex) Objective Lens (Convex) Galileo's Telescope Eye Eyepiece Lens (Concave) Figure 2.2 Refracting Telescopes Refracting telescopes are prone to residual chromatic and spherical aberrations. Chromatic aberration is caused by the lens having a different refraction index for different wavelengths of light, resulting in a fringe of color around an image. There are two types of chromatic aberrations, longitudinal and transverse. Longitudinal chromatic aberration occurs when different wavelengths are focused at different focal lengths and transverse chromatic aberration occurs when the wavelengths focus at different positions on the focal plane. However, chromatic aberration can be minimized by using two lenses of different chemical composition; thus reducing the amount of chromatic aberration over a certain range of wavelengths (Nave 2000). 2. The Historical Evolution of Telescopes and Telescope Enclosures Spherical aberration occurs when light rays hitting near the edges of a lens refract differently than light rays hitting the center (See Figure 2.3 below). The result is a blurred image due to failure of the light rays to focus at a common point (Hendersen 2004). Spherical aberration may be minimized by using a system of convex and concave lenses or using aspheric lenses. C = Center Point of Lens F = Focal Point of Central Rays F' = Focal Point of Edge Rays Figure 2.3 Spherical Aberration Over the years, refracting telescopes have become less popular as a choice for research telescopes due to its numerous problems. As mentioned above, refracting telescopes are prone to chromatic and spherical aberrations. In addition, the volume of the lens used in a refracting telescope must be entirely free of defects or air bubbles. This can be difficult to achieve when the lens exceeds 30 - 50 [cm] in diameter. Even i f it were possible to manufacture perfects lenses, it would be difficult to provide support for the lens so that it would not sag due to gravity. Since support can only be provided around the perimeter of the -10-2. The Historical Evolution of Telescopes and Telescope Enclosures lens, this is a major issue for large lenses. On the other hand, the requirements for reflecting telescopes are less stringent (more details on reflecting telescopes wi l l be presented in the following section). For reflecting telescopes, only one side of the mirror needs to be perfect and support can be administered on the entire backside of the mirror. Reflecting Telescope -Reflecting telescopes employ a system of curved and plane mirrors to produce an image (\"Reflecting Telescopes\", Wikipedia). The reflecting telescope was originally created by Sir Isaac Newton to overcome chromatic aberration which is a large problem in refracting telescopes. Because the image is reflected rather than refracted, the problem of different color having different refraction indices is eliminated. In a reflecting telescope, the image is magnified and reflected by the primary mirror(s) then reflected by a secondary mirror to a focal plane where the image can be viewed. Although chromatic aberration is absent in reflecting telescopes, other types of image distortion such as spherical aberration and coma exists. If spherical mirrors are used, spherical aberration may become an issue for the telescope. However, this problem can be eliminated by adding a corrector lens or using a different shaped mirror. Parabolic mirrors are sometimes used to replace spherical mirrors to prevent spherical aberration. Unfortunately, parabolic mirrors are often associated with another type of image distortion - coma. Coma occurs when light rays from a point source (a star) are not parallel to the optical axis. Similar to spherical aberration, light rays which are further away from the optical axis are not reflected to the same point as central rays, resulting in a wedge-shaped image (see figure2.4 below). 2. The Historical Evolution of Telescopes and Telescope Enclosures Figure 2.4 Coma in parabolic mirrors There are several different types of reflecting telescope designs (see figure 2.5 below). A Newtonian Focus Cassegrain Focus Figure 2.5 Common types of reflecting telescopes The Newtonian focus consists of a paraboloid primary mirror and a flat secondary mirror which reflects the light rays to a focal plane located on the side of the telescope. This type of design is simple and inexpensive; hence, it is most popular amongst amateur astronomers. One drawback of the Newtonian focus design is that the secondary mirror is supported by struts which reduce the quality of the viewed image. -12-2. The Historical Evolution of Telescopes and Telescope Enclosures Similar to the Newtonian focus, the Cassegrain focus uses a paraboloid primary mirror; however, it employs a. hyperboloid secondary mirror which reflects the light through an opening in the primary mirror. The secondary mirror is usually supported by a clear glass plate which seals the telescope tube, eliminating the problem of image obstruction affecting all Newtonian telescopes. The Ritchey-Chretien focus has the same mirror configuration as the Cassegrain focus, the only difference is that the primary mirror is hyperbolic rather than parabolic. This simple change results in the elimination of coma and spherical aberration. Cassegrain-Hybrid telescopes -The Schmidt-Cassegrain focus and the Maksutov-Cassegrain focus are two common types of hybrid telescopes whose principles are based on the Cassegrain focus. Both telescopes have a similar mirror configuration as the classical Cassegrain focus (see figure 2.6 below). Spherical aberration is eradicated in the Schmidt-Cassegrain focus where a Schmidt corrector plate is added behind the secondary mirror. The Schmidt corrector plate is formed by producing a vacuum environment on one side of the plate and grinding the plate such that spherical aberration created by the primary mirror is abolished. Lastly, the Maksutov-Cassegrain focus is similar to a Schmidt-Cassegrain focus except that all lenses are spherical in shape. In addition, the secondary mirror is commonly a mirrored section of the corrector lens which results in less image obstruction. -13-2. The Historical Evolution of Telescopes and Telescope Enclosures Schmidt - Cassegrain Focus Maksutov - Cassegrain Focus Figure 2.6 Cassegrain-Hybrid Telescopes 2.3. Evolution of Telescope Enclosures Modern research telescopes have become increasingly expensive and powerful over the last decades. However, these telescopes are only capable of generating high quality images under very specific conditions; hence the need for an enclosure. In addition to sheltering expensive equipments, telescope enclosures protect the telescope from natural elements (rain, snow, dust, and wind) and temperature gradients. With the size of modern research telescopes reaching diameters of 30 - 50 [m], the size of telescope enclosures have increased substantially as well. In order to understand the implication of this sudden increase in size demands, it is important to first become familiar with the different types of telescope enclosures which exist today. -14-2. The Historical Evolution of Telescopes and Telescope Enclosures The requirements for a typical telescope enclosure are as follows (Quattri et al. 2000): \u2022 Minimize enclosed volume and surface area (infrastructure cost usually proportional to developed surface) \u2022 Protect telescope from solar degradation, snow, rain, dust, and wind \u2022 Eliminate wind disturbance to the telescope during viewing \u2022 Regulate ambient temperature within the enclosure to match external temperatures In accordance with the first requirement, it is easy to understand why most telescope enclosures are spherical in shape. Spheres have a lesser volume and surface area than any cubic or prismatic solids. In addition, since telescopes usually rotate in a circular path, using a spherical enclosure would be the most space efficient approach (see figure 2.7 below). One of the most conventional telescope enclosure designs is the rotating circular dome design with shutters which open laterally. The dome shaped enclosure allows for a complete range of motion for the telescope inside. Although this design is economical to build, it is better suited for smaller telescopes (Desroches 2003). As the size of the telescope increases, the enclosure size must increase accordingly. Since this dome shaped design consists of an \"orange-peel\" like structure, stability issues may pose as a problem when the diameter of the enclosure becomes sufficiently large. -15-2. The Historical Evolution of Telescopes and Telescope Enclosures Figure 2.7 Range of motion of telescope A small change was later made to the aforementioned design to give the opening more freedom. Instead of having the shutters opening laterally, the shutters now slide over the enclosure vertically (see figure 2.8 below). Figure 2.8 Gemini Telescope in Hawaii Courtesy of the Gemini Observatory Website 3,-1,-1) (-cb3,+1,-1) (-cb3, - 1 , -1) (+l ,+cb 3 ,+l) (+l ,+cb 3 , - l ) (+i,-4>3,+i) (-1, + ), 0, +cb2) (+(2+d>), 0, -cp2) (-(2+cb), 0, +(b2) (-(2+cb), 0, -cb2) m2, +(2+(t>), o) (+cp2, -(2+ Until* w*f*. vm- Ssbm c * B y \/ \u2022 > B \u00bb > > < \u00bb<*>\u2022 * ! ; B . I \u2022 ' . n h H - -Figure 5.3 Deformed Shape (Mode 1) of Rhombicosidodecahedron Table 5.7 SAP output of vibration frequencies for first 12 modes (Icosahedron) Mode Period [s] Frequency [cyc\/s] 1 0.572764 1.7459 2 0.546453 1.83 3 0.169415 5.9027 4 0.143683 6.9597 5 0.085347 11.717 6 0.082647 12.1 7 0.07876 12.697 8 0.07801 12.819 9 0.075862 13.182 10 0.075439 13.256 11 0.070794 14.125 12 0.070434 14.198 -59-5. Formulation of Model and Loads for Finite Element Analysis SSfifiHI ff*. fe* *sm C**i\u00ab Sni j . ttw fow *>*w &**f\u00ab a* n (\u2022 U U \/ S > B a \u00ab \u00ab \u00bb (lit' \u2022 > ! ; \u00ab I \u2022 - n f7M \u2022 , J2\u00ab iz^ - \u2022 Figure 5.4. Deformed Shape (Mode 1) of Icosahedron Looking at the vortex shedding frequency and natural frequency, one can conclude that resonance effects are negligible. For the other cases proposed with a smaller diameter, vibration frequencies are larger and even further away from the St frequency of 0.16 Hz . Achieving a vortex shedding frequency of 0.16 H z is nearly impossible because the St number remains relatively constant after Re reaches 300. The vortex shedding frequency does not rise rapidly until Re reaches 5 x 10 4. For a structure with a diameter of 50 m, this would be equivalent to an average wind velocity of approximately 700 m\/s. Hence, it is safe to say that vortex shedding would not be an issue for this particular telescope enclosure. -60-5. Formulation of Model and Loads for Finite Element Analysis Drag Forces on Blunt Bodies Direction of Flow \u2022 Drag Force, F D Figure 5.5. Lift and Drag Force on Blunt Bodies The term drag may be defined as \"the force the flow exerts on the body in the direction of flow\". For the purpose of this report, one can visualize it as the force exerted on the structure by a lateral acting force, such as wind. The calculation of the force exerted on the enclosure by wind may be calculated using the following equation. Equation 5.2 Drag Force I - D ^ C f \/ p V - A where C D = drag coefficient [dimensionless] p = density of fluid [M\/L 3] V -velocity of fluid [L\/T] A = projected area normal to direction of flow [L ] The only variables above which depend on the actual structure are the projected area, A and the drag coefficient, C D . The projected area can be easily calculated using simple geometry; however, the drag coefficient requires more thought and understanding of basic fluid dynamics principles. -61-5. Formulation of Model and Loads for Finite Element Analysis 1 < i i i i i i i i ill i \u2022 l t 11 i'l I i I i 11111 i i I 1 1 1 hi 1\u2014i I I,I i h i ; 2 4 6 8 lO2 2 4 6 X 103 2 4 ' 6 8 10* 2 4 6 8 10s 2 4 6 8 10\" 2 4 6 8 107 Re = VDh Figure 5.6 Drag Coefficients for Flow around Blunt Bodies (Potter and Wiggert) From Figure 5.6 above, one can se that at 10 < Re < 2 x 10 , the drag coefficient remains relatively constant for both smooth spheres and cylinders. It is also important to note that at Re ~ 2 x 10 5, boundary layer for smooth surface blunt bodies undergoes transition to a turbulent state and pushes separation back, resulting in a decrease in drag. Comparing the smooth bodies against rough bodies, one can also see that surface roughness results in a slight drop in drag at 5 x 10 4 < Re < 2 x 105 (as shown by the dashed line above). From the calculations and Figure 5.6 above, one can conclude that a drag coefficient of approximately 0.7 is acceptable. -62-5. Formulation of Model and Loads for Finite Element Analysis Sample calculation for reference wind velocity, Case 1 Standard wind velocity value used at A M E C Dynamic Structures m H:= 45.2531n A := D H A = 2.263x 10 3 m 2 Re:=V \u2014 \u2022 o Re= 2.583x 103 C D :=0 .7 At Re = 2583 q - ^ C D - p V 2 q = 2.564kPa Reference velocity pressure After obtaining the reference velocity pressure, one can use the N B C C 2005 code to factor this value to obtain an appropriate wind loading on the structure of interest. The specified external pressure caused by wind on the structure may be calculated using the following equation: Wind Load 2 D:=50m u:=1.51\u2014 V:=78\u2014 s s p : - 1 .204^ -63-5. Formulation of Model and Loads for Finite Element Analysis Equation 5.3 External Pressure Caused by Wind on Structure p : = I w q - C e - C g - C p [ N B C C 2005 4.1.7.1(1)] Where q Importance factor for wind load : reference velocity pressure (calculated above) exposure factor gust factor external pressure coefficient A l l the values are specified in the code with the exception of q which was calculated above and C p which is described in the commentary section of the code. The external pressure coefficient is a dimensionless ratio of wind-induced pressures on a building to the dynamic pressure of the wind speed at a particular reference height. For simplification purposes, it was assumed that the wind loading acts parallel to the ground surface. Under this assumption, the vale of the external pressure coefficient is 1.0. Sample calculation for wind loading according to NBCC 2005, Case 1 Importance factor for wind load Exposure factor Gust factor C e = 1.352 External pressure coefficient P : = T w q C e - C g ' C P p = 6.935kPa -64-5. Formulation of Model and Loads for Finite Element Analysis The external pressure calculated above was multiplied by the average projected area of the structure. This yields the average force exerted on the enclosure. B y dividing this average force by the number of keypoints within that projected area, the average force exerted on each keypoint may be obtained. Figure 5.7 Projected Area of Icosahedron Configuration, Case 1 The number of keypoints on the right half of the enclosure is 259. Hence, the average wind force exerted on each keypoint is calculated as follows: A := D H N k p : = 2 5 9 A W:=p N k p W = 60.585kl< A force of 60.6 k N is exerted on each keypoint in direct contact with lateral wind forces. -65-5. Formulation of Model and Loads for Finite Element Analysis 5.7. Earthquake Loads In order to achieve unobstructed viewing, telescope enclosures are almost always situated on top of remote and high mountain ranges. If these mountain ranges are located near volcanoes or active seismic locations, then the telescope enclosure must be designed to resist seismic forces. To simulate reality as much as possible the earthquake load specifications in this project were obtained from the N B C C 2005 code. The appropriate site characteristics and location must be considered in the selection of spectral acceleration values; however, for the purposes of this report, the Uniform Hazard Spectrum (UHS) for Vancouver was used (see Figure 5.8 below). N B C C 2005 UHS (Vancouver. C lass A Site) 0.8 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) Figure 5.8 Uniform Hazard Spectrum for Vancouver -66-5. Formulation of Model and Loads for Finite Element Analysis Since it was assumed that the telescope would be situated on top of a mountain range, a Class A site was used for analysis. Class A sites consist of hard rock and have an average shear wave velocity of greater than 1500 [m\/s]. The Vancouver U H S above was adjusted to reflect a Class A site. This adjustment is done by multiplying spectral accelerations at Ta = 0.2, 0.5, 1.0, and 2.0 [s] by corresponding F a (acceleration-based site coefficient) and F v (velocity-based site coefficient) values. These values may be found from Table 4.1.8.4B and Table 4.1.8.4A from N B C C 2005. After find these values, intermediate spectral acceleration points were linearly interpolated to obtain the above graph. The base shears suggested by N B C C 2005 was calculated as a function o f the weight o f the building to obtain lateral accelerations. Equation 5.4 below illustrates how the lateral accelerations for each case were obtained. Equation 5.4 Minimum Lateral Acceleration due to Earthquake R d ' R \u00b0 [ N B C C 2005 4.1.8.11(2)] where S(T a) = spectral acceleration at time T a T a = fundamental lateral period of vibration of structure M v = factor to account for higher mode effect on base shear IE = earthquake importance factor for structure Rd = ductility related force modification factor Ro = overstrength related force modification factor A :=\u2022 -67-5. Formulation of Model and Loads for Finite Element Analysis In order to calculate the lateral acceleration specified above, the fundamental lateral period of vibration for each case had to be obtained. This was done using SAP2000 and the results are presented again for convenience: Table 5.8 Fundamental Lateral Period of Vibration for Rhombicosidodecahedron Case Fundamental Period, Ta(s) 1 0.396 2 0.334 3 0.214 4 0.139 5 0.395 6 0.399 7 . 0.398 8 0.396 9 0.396 10 0.395 11 0.397 Table 5.9 Fundamental Lateral Period of Vibration for Icosahedron Case Fundamental Period, T a(s) 1 0.573 2 0.455 3 0.337 4 0.218 5 0.571 6 0.577 7 0.575 8 0.572 9 0.572 10 0.572 11 0.574 -68-5. Formulation of Model and Loads for Finite Element Analysis After obtaining the fundamental lateral period of vibration, the spectral accelerations for each particular case was obtained from the Vancouver U H S . One can see that the stiffer a structure is, the higher the acceleration. As a result, when comparing Case 1 to 4, one can see that the enclosure with the smallest radius (Case 4), due to its shorter and stiffer members, would be subjected to the highest earthquake load. The M v factor accounts for higher mode effects of the structure. According to the N B C C 2005 code, M v factors are found in Table 4.1.8.11 and are dependent on the type of lateral resisting system of the structure. The ratio of S a(0.)\/S a(2.0) usually indicates the approximate location of the site; the west coast of Canada usually has a value of < 8.0 while the east coast of Canada usually has a value of > 8.0. For the purpose of this analysis, it was decided that a moment resisting frame system would be chosen as the lateral force resisting system. Hence, according to Table 5.10 below, the value of M v is 1.0. Table 5.10 M v Values from NBCC 2005 Table 4.1.8.11 Sa(0.2)\/Sa(2.0) Type of Lateral Resisting System Mvfor Ta<1.0 Mv for Ta > 2.0 <8.0 Moment resisting frames or \"coupled walls\" 1.0 1.0 Braced frames 1.0 1.0 Wall, wall-frame systems, other systems 1.0 1.2 > 8.0 Moment resisting frames or \"coupled walls\" 1.0 1.2 Braced frames 1.0 1.5 Wall, wall-frame systems, other systems 1.0 2.5 -69-5. Formulation of Model and Loads for Finite Element Analysis The IE factor accounts for how important the structure is. The structure under consideration in this analysis is extremely expensive and costly to repair; hence it has been classified under the high importance category with a value of 1.3. Values for the IE factor may be obtained from Table 4.1.8.5 in the N B C C 2005. In order to account for the inelastic properties of a structure, N B C C divides the base shear by a force reduction factor (Rd) and a system overstrength factor (RQ). The force reduction factor is related to the amount of ductility capacity in the structure while the overstrength factor accounts for dependable forms of overstrength (cross-section size rounded up, strain hardening...etc). For this analysis, an elastic beam was chosen; as a result, both Rd and Ro have values of 1.0. Sample Calculations for Lateral Acceleration Rhombicosidodecahedron, Case I T \u2022= 0.396s a At Ta = 0.396s, s(Ta):=0.56g M v : =1 .0 IE := 1.3 R d :=1.0 R o :=1.0 , (s(T a)-M v- I E) A := R d Rc> A = 0.728g Icosahedron, Case 1 -70-5. Formulation of Model and Loads for Finite Element Analysis T a := 0.573s a A t T a = 0.573s, s(Ta):=0.41g M y := 1.0 IE:=1.3 R d :=1.0 V = 1 . 0 (s(T a ) -M v . I E ) A := Rd , Ro A = 0.533g Table 5.11 Linear Acceleration for Rhombicosidodecahedron Case Linear Acceleration (g's) 1 0.73 2 0.80 3 0.96 4 0.98 5 0.73 6 0.73 7 \u2022 0.73 8 0.73 9 0.73 10 0.73 11 0.73 5.12 Linear Acceleration for Icosah Case Linear Acceleration (g's) 1 0.53 2 0.65 3 0.80 4 0.96 5 0.53 6 0.53 7 0.53 8 0.53 9 0.53 10 0.53 11 0.53 -71-5. Formulation of Model and Loads for Finite Element Analysis Table 5.11 and Table 5.12 above summarize the linear acceleration calculated for each of the cases specified in section 5.1. 5.8. Summary of Loads The following tables are summaries of the loads applied on each structure. Table 5.13 Summary of Loads on Rhombicosidodecahedron Case Load Unfactorcd Load Load Factor Factored Load 1,5-11 Dead (m\/s2) 9.81 1.25 12.26 Snow\/Ice (N\/keypoint) 41270 1.50 61905 Wind (N\/keypoint) 60580 1.40 84812 Earthquake (m\/s2) 7.11 1.00 7.11 2 Dead (m\/s2) 9.81 1.25 12.26 Snow\/Ice (N\/keypoint) 26274 1.50 39411 Wind (N\/keypoint) 37080 1.40 51912 Earthquake (m\/s2) 7.89 1.00 7.89 3 Dead (m\/s2) 9.81 1.25 12.26 Snow\/Ice (N\/keypoint) 14780 1.50 22170 Wind (N\/keypoint) 22510 1.40 31514 Earthquake (m\/s2) 9.46 1.00 9.46 4 Dead (m\/s2) 9.81 1.25 12.26 Snow\/Ice (N\/keypoint) 6618 1.50 9927 Wind (N\/keypoint) 9224 1.4.0 12914 Earthquake (m\/s2) 9.59 1.00 9.59 -72-5. Formulation of Model and Loads for Finite Element Analysis Table 5.14 Summary of Loads on Icosahedron Case Load Unfactored Load Load Factor Factored Loadi 1,5-11 Dead (mis2) 9.81 1.25 12.26 Snow\/Ice (N\/keypoint) 80766 1.50 121149 Wind (N\/keypoint) 61380 1.40 85932 Earthquake (m\/s2) 5.20 1.00 5.20 2 Dead (m\/s2) 9.81 1.25 12.26 Snow\/Ice (N\/keypoint) 60010 1.50 90015 Wind (N\/keypoint) 38080 1.40 53312 Earthquake (m\/s2j 6.33 1.00 6.33 3 Dead (m\/s2) 9.81 1.25 12.26 Snow\/Ice (N\/keypoint) 33750 1.50 50625 Wind (N\/keypoint) 23110 1.40 32354 Earthquake (m\/s2) 7.89 1.00 7.89 4 Dead (m\/s2) 9.81 1.25 12.26 Snow\/Ice (N\/keypoint) 11902 1.50 17853 Wind (N\/keypoint) 9472 1.40 13261 Earthquake (m\/s2) 9.46 1.00 9.46 6. Finite Element Analysis Results 6. Finite Element Analysis Results After the different cases and load combinations were defined, the models were analyzed using A N S Y S . For each of the case defined, the model was subjected to three different load combinations (dead + snow & ice, dead + wind, and dead + earthquake). Hence, a total of 66 models were analyzed. In order to allow for a more efficient process, input script files were written such that for each difference case, only a small variable had to be change and not the entire model. The results which are of interest to this report are the maximum deflection at each node and the maximum structural force and moment of each element. Another parameter which was of interest was the structural mass of the model. This was considered an important factor in comparing the models against one another because huge savings in mass may result in more lenient deflection tolerances. Table 6.1 Summary of Masses for Rhombicosidodecahedron Structure Case Mass (1000 kg) 1 2423.54 2 1938.83 3 1454.12 4 969.42 5 2769.76 6 1731.1 7 2077.32 8 1994.23 9 1537.22 10 1052.51 11 1903.44 -74-6. Finite Element Analysis Results Table 6.2 Summary of Masses for Icosahedron Structure Case Mass ( 1 0 0 0 kg) 1 1844.24 2 1475.39 3 1106.54 4 737.7 5 2107.7 6 1317.31 7 1580.78 8 1517.55 9 1169.78 10 800.93 11 1448.46 Table 6. land Table 6.2 above summarize the mass of each structure for each particular case. From the above values, one can see that the rhombicosidodecahedron is approximately 24% heavier than the icosahedron option. This is because the rhombicosidodecahedron structure has more members and divisions. The member lengths in the icosahedron are relatively longer and hence more prone to buckling and other types of failure. If configuration selection was strictly based on nodal deflection, element forces, and element moments, the rhombicosidodecahedron would be the optimal choice. However, once the above masses were considered in the decision making process, the results were radically different. 6.1. Optimal Cases The results obtained from each case were plotted on graphs in order to compare them against one another. A n optimal case would have minimal nodal deflection and structural element force and moment. With deflection, it is obvious that smaller structures wi l l have shorter member lengths; i f cross-section size and thickness is kept constant, then the structure wi l l experience less deflection than larger structures. However, one must keep in mind that a smaller structure, although lighter in weight, w i l l have a smaller enclosed area. A s a result, it -75-6. Finite Element Analysis Results is important to find a balance point by minimizing nodal deflection and element forces while maximizing enclosed area. The following graphs are average results from analysis of the different cases of rhombicosidodecahedron and icosahedron under different loading combinations. Please note that the series in each graph is an average value of the three load combinations specified in section 5.1. Each graph represents a different response parameter ( U x , U y . . .etc). Within each graph are three series; each series represent changes in the response parameter when a certain variable (radius of enclosure, cross-section size\/type) is changed. For graphs which reflect each load combinations separately, please refer to Appendix A . Deflection (Rhombicosidodecahedron) One can see from the graphs below that as the radius of the telescope enclosure increases in size, the maximum nodal deflection increases proportionally. After a radius of approximately 20 [m], there is a sharp increase in the slope of the curve, this means that the rhombicosidodecahedron is perhaps better suited for a radius of less than 20 [m]. Comparing the change in response due to changes in cross-section size and thickness, it is apparent that a change in thickness results in a more drastic change in maximum deflection. After reviewing the changes in mass of the structure at different cross-section size and thickness, it can be concluded that increasing the cross-section thickness to obtain a lower deflection is a better option than increasing the cross-section size in terms of weight. -76-6 . Finite Element Analysis Results Average X-deflection 0.05 0.1 30.00 25.00 _ 20.00 E. I 15.00 u Q 10.00 5.00 0.00 Cross-section size\/thickness (m) 0.15 0.2 0.25 0.3 0.35 10 15 Radius (m) 20 0.4 0.45 0.5 25 \\ \\ \\ \\ 30 -Average Ux (Radius) Average Ux (Cross section size) Average Ux (Cross section thickness) Figure 6.1 Average X-deflection for Rhombicosidodecahedron Average Y-deflection 0.05 0.1 18.00 16.00 14.00 _ 12.00 E \u2022r 10.00 o | 8.00 B V 6.00 4.00 2.00 0.00 Cross-section size\/thickness (m) 0.15 0.2 0.25 0.3 0.35 10 15 Radius (m) 20 0.4 0.45 25 0.5 I I I l \\ \\ \\ N I I | 30 -Average Uy (Radius) -Average (Cross-section size) Average Uy (Cross-section thickness) Figure 6.2 Average Y-deflection for Rhombicosidodecahedron 6. Finite Element Analysis Results 0.05 0.1 Average Z-deflection Cross-section size\/thickness (m) 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 5.00 4.00 3.00 | 2.00 1.00 0.00 L. , 1 , 1 0 5 10 15 20 25 30 Radius (m) Average Uz (Radius) Average Uz (Cross-section size) Average Uz (Cross-section thickness) Figure 6.3 Average Z-deflection for Rhombicosidodecahedron Average Structural Force (X) 1000.00 0.00 15 Radius (m) 30 -Average Fx (Radius) Average Fy (Cross-section size) \u2014 \u2014Average Fy (Cross-section thickness) Figure 6.4 Average Structural Force (X) for Rhombicosidodecahedron 6. Finite Element Analysis Results Average Structural Force (Y) Cross-section size\/thickness (m) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 120.00 -100.00 z 80.00 0) u o \u00b1 60.00 n *_ 3 U is 40.00 55 20.00 0.00 0 5 10 15 20 25 30 Radius (m) Average Fy (Radius) Average Fy (Cross-section size) Average Fy (Cross-section thickness) Figure 6.5 Average Structural Force (Y) for Rhombicosidodecahedron Average Structural Force (Z) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 80.00 70.00 60.00 \u2014 50.00 \u2022 u o \u00b1 40.00 \u2022 p ts a 30.00 tn 20.00 10.00 0.00 0 5 10 15 20 25 30 Radius (m) Average Fz (Radius) Average Fz (Cross-section size) Average Fz (Cross-section thickness) Figure 6.6 Average Structural Force (Z) for Rhombicosidodecahedron - 7 9 -6. Finite Element Analysis Results Structural Force (Rhombicosidodecahedron) For all three structural force graphs, there is a sharp increase in element force between a radius of 15 [m] and 20 [m] followed by a decrease after a radius of 20 [m]. This increase is especially significant in the axial (F x) component of the element force. For the major and minor shear forces (F y and F z ) , the increase is less significant. Both the axial and shear forces retain an approximately linear shape when subjected to changes in the cross-section size and thickness. Looking at the axial force graph (Figure 6.4 above), one can see that changing the cross-section thickness has more impact than changing the cross-section size due to the flat slope o f the cross-section size curve. However, in terms o f major and minor shear forces Figure 6.5 and Figure 6.6), changing the cross-section size or thickness wi l l have similar impact on the magnitude of the force. Average Structural Moment (X) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 40.00 i 1 1 1 1 1 1 1 1 1 1 10.00 5.00 0.00 -I 1 1 0 5 10 15 20 25 30 Radius (m) Average Mx (Radius) Average Mx (Cross-section size) Average Mx (Cross-section thickness) Figure 6.7 Average Structural Moment (X) for Rhombicosidodecahedron -80-6. Finite Element Analysis Results Average Structural Moment (Y) 0.05 0.1 0.45 0.5 180.00 -Average My (Radius) -Average My (Cross-section size) Average My (Cross-section thickness) Figure 6.8 Average Structural Moment (Y) for Rhombicosidodecahedron Average Structural Moment (Z) 0.15 0.2 0.25 0.3 300.00 250.00 200.00 a B S 150.00 3 = 100.00 O CO 50.00 0.00 0.05 0.1 \u2014I\u2014 0.35 \u2014I\u2014 10 15 Radius (m) 20 0.4 0.45 0.5 25 1 \/ 30 -Average Mz (Radius) Average Mz (Cross-section size) Average Mz (Cross-section thickness) Figure 6.9 Average Structural Moment (Z) for Rhombicosidodecahedron -81-6. Finite Element Analysis Results Structural Moment (Rhombicosidodecahedron) The structural moments displays a similar trend as the structural forces above. Both torsional and bending moments display a linear trend when the cross-section size and thickness is altered. Similar to the graphs for structural forces, the torsional and bending moments experience a sudden increase in magnitude between a radius of 15 [m] and 20 [m] with a decrease after 20 [m]. Deflection (Icosahedron) The trends in the deflection vs. radius for the icosahedron appear to be quite different from that of the rhombicosidodecahedron above. However, please keep in mind that these graphs are obtained by taking the average of the response from different load combinations. The plateau behaviour seen below is characteristic of the second load combination (dead + wind). This trend was less significant in the rhombicosidodecahedron configuration above. What this implies is that for the icosahedron configuration, savings in deflection w i l l not be obtained by decreasing the radius size within the range of 15 - 20 [m]. -82-6. Finite Element Analysis Results Average X-deflection Cross-section size\/thickness (m) 0.15 0.2 0.25 0.3 0.35 H 1 h 15 Radius (m) -Average Ux (Radius) Average Ux (Cross-section size) Average Ux (Cross-section thickness) 35.00 30.00 25.00 \u00a3 20.00 c o \u00ab 15.00 a 10.00 5.00 0.00 0.05 Figure 6.10 Average X-deflection for Icosahedron Average Y-deflection 0.1 Cross-section size\/thickness (m) 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 \\ \\ \\ 10 15 Radius (m) 20 25 30 -Average Uy (Radius) Average Uy (Cross-section size) Average Uy (Cross-section thickness) Figure 6.11 Average Y-deflection for Icosahedron -83-6. Finite Element Analysis Results Average Z-deflection 0.05 0.1 6.00 5.00 ~ 4.00 E_ | 3.00 o a> 5= a) O 2.00 1.00 0.00 Cross-section size\/thickness (m) 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ' ' 1 ft \\ \\ s I I 10 15 Radius (m) 20 25 30 \u2022Average Uz (Radius) Average Uz (Cross-section size) Average Uz (Cross-section thickness) | Figure 6.12 Average Z-deflection for Icosahedron Structural Force (Icosahedron) Similar to the deflection, the element axial and shear forces experience a plateau between a radius of 15 [m] to 20 [m]. However, there is an increasing trend (as the radius increases) with the axial force while the shear forces display a decreasing trend. From the sudden increase in axial force at a radius of 25 [m], one may postulate that the structure has buckled (failed); however, one must bear in mind that the 25 [m] radius is the \"control case\" and also appears in the cross-section size and thickness series. The cross-section size and thickness series do not display any abrupt increases; hence the possibility of the control case failing is unlikely. - 8 4 -6. Finite Element Analysis Results 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0 Average Structural Force (X) Cross-section size\/thickness (m) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 1 1 1 1\u2014I 1 1 1 1 1 ; \/ #\u2022 '\u2014\u20141 ; 1 7 i 1\u2014\u2022 ,**t*z..,^*?T \u2014 J \/ , , ; . i \u2014 \/ | 10 15 Radius (m) 20 25 30 -Average Fx (Radius) -Average Fx (Cross-section size) \u2014 - Average Fx (Cross-section thickness) OT 80 00 70 00 60 00 -50 00 -40 00 -30 00 20 00 -10 00 -0.00 Figure 6.13 Average Structural Force (X) for Icosahedron Average Structural Force (Y) Cross-section size\/thickness (m) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 1 1 1 1\u2014 0.4 0.45 0.5 10 15 Radius (m) 20 25 30 -Average Fy (Radius) -Average Fy (Cross-section size) - Average Fy (Cross-section thickness) Figure 6.14 Average Structural Force (Y) for Icosahedron -85-6. Finite Element Analysis Results o.oo Average Structural Force (Z) Cross-section size\/thickness (m) 0.15 0.2 0.25 0.3 0.35 H ! 1 h 10 15 Radius (m) 20 25 30 -Average Fz (Radius) -Average Fz (Cross-section size) Average Fz (Cross-section thickness) Figure 6.15 Average Structural Force (Z) for Icosahedron Structural Moment (Icosahedron) Compared to the rhombicosidodecahedron, the icosahedron configuration had a smaller range in torsional and bending moment magnitude when the radius was changed. In addition, the average torsional and bending moment in the icosahedron did not experience the sudden spike in magnitude at a radius of 20 [m] as observed in the rhombicosidodecahedron configuration. This means that the icosahedron configuration is more stable and less influenced by the radius of the structure. The moment vs. cross-section size\/thickness plots displayed a similar trend as the rhombicosidodecahedron plots, increasing linearly with increasing size or thickness. -86-6. Finite Element Analysis Results Average Structural Moment (X) Cross-section size\/thickness (m) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 35.00 -i i 1 1 1 1 1 ! Figure 6.16 Average Structural Moment (X) for Icosahedron Average Structural Moment (Y) Cross-section size\/thickness (m) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 250.00 i 1 \u2022 1 50.00 0.00 -I 1 1 0 5 10 15 20 25 30 Radius (m) |^^\u2014Average My (Radius) \u2014\u2014\u2014Average My (Cross-section size) Average My (Cross-section thickness) Figure 6.17 Average Structural Moment (Y) for Icosahedron -87-6. Finite Element Analysis Results 300.00 250.00 1. 200.00 c | 150.00 CD i I 100.00 p s w 50.00 0.00 0.05 0.1 Average Structural Moment IZ) Cross-section size\/thickness (m) 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 10 15 Radius (m) 20 25 30 -Average Mz (Radius) -Average (Cross-section size) -Average Mz (Cross-section thickness) Figure 6.18 Average Structural Moment (Z) for Icosahedron The following table summarizes the results of optimal cases for each variable and load combination with the consideration of mass included in the calculation. Table 6.3 Optimal Cases Case Variable Load Combo Rhombicos idodecahedron Icosahedron Optimal Radius (m) 1,2, 3 1 1 Average 1 1 Optimal Cross-sect ion Size (m) 1,2,3 6 6 Average 7 6 Optimal Cross-sect ion Thickness (m) 1,2, 3 10 10 Average 10 10 Optimal Cross-sect ion Type 1,2,3 11 11 Average 11 1 -88-6. Finite Element Analysis Results Referring to Table 6.3 above, one can see that for both the rhombicosidodecahedron and the icosahedron configuration, case 1 (25 [m] radius) is the optimal case when choosing the optimal radius. The optimal cross-section size is case 6 (0.3 [m] cross-section size) when load combinations 1, 2, and 3 were considered separately. However, when the average of the three load combination results was considered, case 7 (0.35 [m] cross-section size) had a more optimal cross-section size than case 6 for the rhombicosidodecahedron configuration. The optimal cross-section thickness is case 10 (0.02 [m] cross-section thickness) for both configurations. Comparing the results of using a square cross-section vs. a circular cross-section, it appeared that the circular cross-section yielded more optimal results. The above decisions were arrived at using the following formulas: Table 6.4 Optimization Formulas Variable Formula Radius P m V Cross-section size P m C S S Cross-section thickness P-m-t Cross-section shape Pir. where P = parameter ( U x , U y , U z , F x , F y , Fz , M x , M y , M z ) m = mass of structure V = volume of enclosed structure CSS = cross-section size t = cross-section thickness Each of the parameter or results obtained from analysis was first multiplied by the mass of the corresponding structure in order to factor in the importance of mass. Mass is an -89-6. Finite Element Analysis Results important issue in construction and the lighter a structure is, the more cost effective it w i l l be to construct. However, a lighter structure is often smaller and smaller structures wi l l always have smaller deflections and element forces. As a result, an ideal structure wi l l have a radius of zero [m]. Clearly, one cannot have such a structure; hence, the enclosed volume must also be taken into consideration. In the selection of cases in Table 6.3, the case with the lowest value from each variable group (ie. Case 1-4, case 1, 5, 6, 7, Case 1, 8, 9, 10, and Case 1, 11) was deemed the \"best\" case for that particular parameter. The case which is deemed the \"best\" case for the most parameter is then selected as the optimal case. Referring to Table 6.4 above, one can see the optimization formula for \"radius\" is P*m\/V. The goal was to minimize the magnitude of the parameter (deflection and element forces) and mass while maximizing the enclosed volume; this is achieved when minimizing the formula. The optimization formula for the cross-section size variable is P * m * C S S . This formula aims at minimizing the magnitude of the parameter, mass, and cross-section size. It is known that smaller cross-sections have larger deflections but lesser element forces. Hence, a balance point must be reached. B y multiplying the optimization formula by the cross-section size, one is essentially compensating for its deficiencies. For example, a structure with a smaller element size w i l l have a larger deflection than a structure with a larger element size. The large deflection w i l l be scaled down when it is multiplied by the small value of the cross-section size. The small deflection in the large element structure wi l l be scaled up when it is multiplied by the large value of the cross-section size. One problem with this method is that -90-6. Finite Element Analysis Results the user must carefully consider the correlation between cross-section size and deflection\/element forces. This wi l l be further discussed in section 9. The optimization formula for cross-section thickness is P*m*t. Similar to the two optimization formulas above, both the parameter and mass magnitude is minimized in this formula. The cross-section thickness works in a similar manner as the cross-section size. A smaller thickness wi l l result in larger deflections but smaller element forces. Hence, by multiplying the parameters by the cross-section thickness, an optimized point may be reached. -91-6. Finite Element Analysis Results 6.2. Optimal Configuration Table 6.5 Comparison of Rhombicosidodecahedron and Icosahedron Configurations for Different Cases Variable Value Load Combo Optimal Choice | Radius (m) 50 1,2,3 I Average I 40 1,2,3 I Average I 30 1,2,3 Average R 20 1,2,3 I Average I Cross-section Size (m) 0.4 1,2,3 I Average I 0.45 1,2,3 I Average I 0.3 1,2,3 I Average I 0.35 1,2,3 I Average R Radius (m) 0.05 1,2,3 I Average I 0.04 1,2,3 I Average I 0.03 1,2,3 I Average R 0.02 1,2,3 I Average I Cross-section Type Square 1,2,3 I Average I Circle 1,2,3 I Average I After comparing the individual cases within each configuration, the performance of each configuration was compared against one another. Table 6.5 above summarizes the optimal configuration for each individual case. It is evident that for a majority of the cases, the icosahedron configuration is the optimal case. This is determined by comparing the value given by the above optimization formulas. For each parameter (ie. U x ) , the configuration with the lower optimized value was given a point. At the end, the configuration with the -92-6. Finite Element Analysis Results most points for each individual case was deemed the \"opt imal\" configuration, yielding the above table. 7. Use of Photogrammetry to Assess Existing Structures 7. Use of Photogrammetry to Assess Exist ing Structures Photogrammetry, as its name implies, is the combination of photography and metrology. Using photogrammetry, one can obtain accurate 3-dimensional coordinates of any given object or structure from a set of 2-dimensional photographs. Similar to most conventional surveying techniques, photogrammetry is based on the principles of tr iangulat ion. Triangulation is a technique used to determine the distance between two points or the relative position of two or more points in space. In photogrammetry, by taking two or more photographs of an object, points which are visible in both photographs may be triangulated to obtain their relative positions. Photogrammetry can be easily applied to the assessment of existing structures. Bui lding plans and specifications for structures may be difficult to obtain at times, especially when the structure in question is an older building. Existing land surveying tools involve the use of Cumbersome and expensive equipment; hence, photogrammetry may be an ideal alternative in such cases. In the design of new telescope enclosures, a company often takes one of two paths: 1. Recycle a previous design 2. Generate an entirely new design -94-7. Use of Photogrammetry to Assess Existing Structures However, there are currently many interesting and innovative telescope enclosure designs available and it would be quite wasteful to not examines and learn the benefits of each. If one could study the pros and cons of existing telescope enclosure designs, then a \"super\" or \"hybr id\" enclosure encompassing the advantages of several conventional designs may be created. This process may be aided by the use of photogrammetry. With photogrammetry, 3-dimensional representations of different existing telescope enclosures may be created quickly and efficiently. Once these models are loaded into any C A D programs, merging and combining individual features from each design w i l l be quite simple. 7.1. PhotoModeler Pro There is a software developed by Eos Systems Inc. called \"PhotoModeler Pro\" which aids the user in extracting measurements and 3-dimensional models from photographs. Operation of the software is relatively simple and the only equipment required is a camera and a computer. Similar to any photogrammetry software available on the market, the accuracy and precision of the resulting model is dependent on the quality of the camera used. The generation of 3-dimensional models in PhotoModeler is based on 3 principles (PhotoModeler User Manual 2004): 1. PhotoModeler assumes that a ray of light coming from some point through the focal node of the lens of a camera and hitting the film can be described by a straight line. -95-7. Use of Photogrammetry to Assess Existing Structures 2. B y knowing where the camera was at the time of exposure, PhotoModeler can use the location of where the ray of light hit the f i lm to calculate the equation of that ray pf light in three dimensions. 3. Each point that is to be measured is imaged in at least two photographs, and preferable in three of more. These points are used to compute the light ray positions and their intersections for determining the positions in 3-dimensional space. The above assumptions are based on a perfect case scenario and may be affected by the following factors: Air effects A i r effects are only a concern for aerial photos since air may cause distortion in photographs. Objects which are less than 1000 [m] in size are usually not prone to air distortion problems. Lens Distortion Lens distortion is virtually unavoidable because all lens are prone to some degree of distortion. However, PhotoModeler has the capability to compensate for lens distortion through a calibration procedure; as a result, regular consumer cameras may be used to obtain fairly accurate measurements. Imperfect Imaging and Point Location Both of these problems may be alleviated by using digital cameras. PhotoModeler assumes that the imaging media (the photograph) is perfectly flat. Photographs produced by film -96-7. Use of Photogrammetry to Assess Existing Structures cameras need to be developed then scanned into the computer for processing. This procedure may cause the photograph to bend or the image to blur. Another problem which may cause imperfect imaging is the limits of resolution. Objects which are smaller require a higher resolution camera to obtain accurate measurements. Imperfect point location refers to the uncertainty in the distance between the imaging medium and the camera lens and body. In digital cameras, this is fixed and is not a problem; however, in film cameras, this distance may be difficult to obtain. Changes in Camera Characteristics The last factor which affects measurement accuracy is the changes in the internals of the camera between photographs. For a single measurement project, the focal length, lens distortion characteristics, and lens position must remain constant in order for the software to calculate the position of the camera and generate 3-dimensional coordinates. Hence, the user must refrain from using the zoom feature or changing the camera lens. Once a suitable camera is selected for a certain project, the camera must be calibrated for PhotoModeler to obtain its focal length, imaging scale, image center, and lens distortion. This information is used to generate relationships between points on a photograph and the location of 3-dimensional points. The procedures for calibrating a camera for PhotoModeler are as follows: -97-7. Use of Photogrammetry to Assess Existing Structures 1. Take pictures of the calibration grid below. \u2022 \u2022* . \u2022 . . . . r . - . **9 Figure 7.1 Calibration Gr id for PhotoModeler 7 2. Take 2 photos from each side (total of 8), one with the camera in regular orientation then another with the camera rotated 90\u00b0 C W . 3. Load photos into PhotoModeler by clicking on New Project -> Calibration Project 4. Execute calibration then check for adequacy of calibration. Table 7.1 Camera Calibration Adequacy Check Criteria Max. Suggested Value Actual Value Max. Marking Residual 1 0.322 Max RMS Residual 0.5 0.151 Total Error 0.02 0.012 7 Courtesy of PhotoModeler User Manual -98-7. Use of Photogrammetry to Assess Existing Structures 5. Check marking residual display to ensure there are no patterns in the error as this w i l l indicate that the camera lens has high distortion. Common patterns to look for include: error bars pointing in one direction, point towards the center, or pointing away from the center. Creating Simple Drawings 1. Take photographs of the object. For higher accuracy, 2 or more pictures should be taken. In addition, photograph should be taken at an angle greater than 90\u00b0 from one other. The accuracy of the resulting model w i l l depend on the number of photographs taken; it is generally recommended that each point on the object appear on 3 or more photographs. 2. Start a new project. Load pictures into program. 3. Mark points on all pictures. L ink pictures by clicking on points which appear in both pictures then click on the \"quick reference\" button. 4. Once 8 or more points are referenced, cl ick on the process button. 5. Mark out the lines and curves on the object then reference the lines and curves from each picture. 6. Add texture and surface where needed. Object Camera Camera Figure 7.2. Plan View of Camera Angles -99-7. Use of Photogrammetry to Assess Existing Structures Example 1. For relatively simple and symmetrical objects such as the water bottle below, two photographs may be sufficient. The water bottle was placed on the calibration grid in order to obtain more reference points. Once the photographs are loaded into the program, the user marks points and lines which appear in more than one photograph. These points and lines are then referenced between the photographs so that the software w i l l recognize that they are the same. In addition, there are other features in PhotoModeler such as automatic cylinder and surface generation which aid in the generation of the 3-dimensional model. For example, when generating cylinders one only needs to mark four points on the surface of the cylinder. The only requirement is that the first two points must be on one edge and the second two points on the other edge. Figure 7.4 below shows the process of referencing curves and lines which are appear in both photographs. 1 s t point point 2 n d point (X) Figure 7.3 Marking Points on a Cylinder -100-7. Use of Photogrammetry to Assess Existing Structures i H c* S5**\u00ab ifcto* art* il s* B P ww.IS- 8 \"k *, \"\\ -8 b \u00abf \"\u2022\u00bb\u00bb tt ft, x V \/ n e* *>ta\u00bb. \/? \u00ab \u2022 \u2022 * . -