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A case study in the conceptual design of an anodized aluminium cladding system for a steel torus structure Fitch, Devan Carless 2006

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A CASE STUDY IN THE CONCEPTUAL DESIGN OF AN ANODIZED ALUMINIUM CLADDING SYSTEM FOR A STEEL TORUS STRUCTURE by D E V A N C A R L E S S F I T C H M . Eng., Imperial College of Science, Technology and Medicine, 2003 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF M A S T E R OF A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Civi l Engineering) T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A December 2006 © Devan Carless Fitch, 2006 A B S T R A C T Typically, structural systems, such as cladding panels are used to achieve a desired architectural appearance, enclose building envelopes and act as barriers to ingress of the external environment. Due to the large surface area of such applications, efficient design of these panels is of paramount importance, i f a cost effective solution is to be achieved. Existing structures utilizing cladding systems typically consist of flat rectangular panels enclosing a regular and rectangular building. For this design project, the structure to be enclosed is a torus structure, and in order to achieve an aesthetic appearance which is acceptable to the client, both curved and flat panels are investigated. Through careful review of available literature, parasitic cladding panels (these do not resist global loads) are selected over a stressed skin (architectural fabric) structural solution due to the highly non-linear and hence complex behaviour of the latter coupled with the fact that the particular geometry of the structure considered in this project does not readily lend itself to implementation of a stressed skin solution. Anodized aluminum is selected over composite materials for the cladding panels, due to the higher confidence level in achieving the desired aesthetic appearance of the panels, and the long-term durability of this appearance with this material. The design of flat cladding panels is relatively simple and there are many analytical solutions available in the literature. However, there is little information that can be found in the literature regarding the conceptual design of curved cladding panels. The structural design of curved panels subject to environmental loading such as uniform normal pressure and thermal gradients is complex due to their nonlinear behavior and susceptibility to buckling. A finite-element (FE) investigation into the influence of panel parameters including; geometry, support conditions and stiffening elements on design efficiency is conducted. From the results of the investigation, a reasonable approach to the design of curved aluminum panels is outlined; starting with analytical methods o f structural analysis o f un-stiffened panels and progressing to finite element analysis of longitudinally and radially stiffened panels with discrete supports. Throughout this thesis, charts are developed which wi l l simplify and direct future analysis. They might be helpful for the conceptual design of these panels. Areas of concern and of particular importance are identified, allowing these aspects to be considered at the outset of a project, when ii A B S T R A C T decisions and changes can be made more readily and with fewer consequences in terms of budget and schedule over-runs. TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS , iv LIST OF TABLES v i i LIST OF FIGURES ix LIST OF SYMBOLS AND ABBREVIATIONS x i i i ACKNOWLEDGEMENTS xv i i NOTES xv i i i CHAPTER 1: INTRODUCTION 1 1.1 Objectives and outline: 1 1.2 Literature review: 4 1.3 AMEC Dynamic Structures design project background: 6 CHAPTER 2: CLADDING STRUCTURAL CONCEPTS 12 2.1 Parasitic panels (curtain walls): 13 2.2 Stressed skin panels (tension/membrane structures): 15 2.3 Structural concept selection: 17 CHAPTER 3: CLADDING MATERIAL OPTIONS 19 3.1 Metallic panels: 19 3.2 Composite panels: 20 3.3 Material selection: 23 CHAPTER 4: STRUCTURAL LOADING 28 4.1 Gravity load: 28 4.2 Wind load: 29 4.3 Thermal load: 33 4.3.1 Thermal conductivity: 33 4.3.2 Thermal radiation: : 34 4.3.3 Thermal convection: 36 4.3.4 Total response 37 4.4 Seismic Load: 43 CHAPTER 5: PRELIMINARY STRUCTURAL ANALYSIS 45 iv TABLE OF CONTENTS 5.1 Flat plates: 46 5.1.1 Theory 46 5.1.2 Application 50 5.2 Curved plates: 54 5.2.1 Theory 56 5.2.2 Application 58 5.3 Panel sizes determined from typical sections: 64 5.4 Conclusions from preliminary structural analysis: 74 CHAPTER 6: FINITE ELEMENT (FE) ANALYSIS 75 6.1 FE Validation: 76 6.1.1 Flat plates 76 6.1.2 Curved plates 81 6.2 Effect of longitudinal stiffeners: 93 6.2.1 Flat panels 94 6.2.2 Curved panels 94 6.3 Effect of radial stiffeners (in addition to longitudinal stiffeners): 107 6.4 Effect of initial imperfections: .114 6.5 Influence of panel alignment relative to gravity: 114 6.6 Thermal effects: 115 6.7 Additional loading Scenarios: 115 CHAPTER 7: CONNECTION DESIGN... 116 7.1 Stiffener to plate connection: 116 7.1.1 Adhesive bonding: 116 7.1.2 Riveting of stiffeners 118 7.2 Stiffener to main structure connection: 119 CHAPTER 8: INTERACTION BETWEEN PANELS 120 8.1 Butting of adjacent panels: 120 8.2 Overlapping of adjacent panels: 120 CHAPTER 9: AESTHETICS 121 CHAPTER 10: CONCLUSIONS AND RECOMENDATIONS 123 TABLE OF CONTENTS REFERENCES 126 APPENDIX A: TABULATED DATA 127 APPENDIX B: VISUAL BASIC PROGRAM CODES AND FORMATTED SPREADSHEETS 250 APPENDIX C: ANSYS INPUT FILES 293 LIST OF TABLES Table 3.1 Comparison of bending stiffness and weight per unit area of sandwich and monolithic panels 22 Table 3.2 Property and cost comparison of metals versus composites 24 Table 3.3 Labour statistics for composite material construction 25 Table 3.4 Representative properties of Anodized aluminum alloy and steel 27 Table 5.1 Coefficients for flat plate bending action (Long plate AR>1) 47 Table 5.2 Coefficients for flat plate bending action (Square plate AR=1) 48 Table 5.3 Maximum plate areas (mm 2) according to analytical buckling analysis 68 Table 5.4 Required plate geometry determined from representative sections 68 Table 5.5 Required plate geometry determined from representative sections (curved edge span reduced by a factor of 2) 70 Table 5.6 Required plate geometry determined from representative sections (curved edge span reduced by a factor of 4) 72 Table 6.1 Comparison of analytical and F E predictions of central deflection and stress (t = 1/16", SS edges) 77 Table 6.2 Comparison of analytical and F E predictions of central deflection and stress (t = 3/16", SS edges) 77 Table 6.3 Comparison of analytical and F E predictions of central deflection and stress [t = 2/16", F ix . edges] 80 Tables 6.4 Comparison of analytical and F E eigenvalue buckling values (SS straight edges, A = 100,000 mm 2 ) 84 Table 6.5 Comparison of analytical and F E eigenvalue buckling values (SS straight edges, A = 40,000,000 mm 2 ) 85 Table 6.6 A N S Y S Design optimization terminology 99 Table 6.7 Optimization variables (longitudinally stiffened plates)..... 103 Table 6.8 Variation in optimization variables (longitudinally stiffened plates) with number of support points 104 Table 6.9 Optimization variables (longitudinally stiffened plate) with b/R = 1.0 106 Table 6.10 Optimization variables (longitudinally stiffened plate with A l stiffeners) 106 vii LIST OF TABLES Table 6.11 Optimization variables for longitudinally and radially stiffened plate 108 Table 6.12 State variables omitted from the design optimization, to reduce run time 108 Table 6.13 Optimized variable values (longitudinally and radially stiffened plate) with b/R = 0.1 109 Table 6.14 Optimized variable values (longitudinally and radially stiffened plate) 110 LIST OF FIGURES Some of the figures in this report are taken from third party, sources and are referenced appropriately. It should be noted that a number of these figures have been digitally traced and enlarged/enhanced in order to improve their clarity. Figure 1.1 Visual summary of work completed 3 Figure 1.2 Photograph of the "London Eye" 6 Figure 1.3 Artist 's impression of the appearance of the final structure 7 Figure 1.4 Typical structural segment with a 15° angular span 8 Figure 1.5 Variation of structural sections 9 Figure 1.6 Rendering of the structural framing, not all spokes shown 10 Figure 2.1 Typical construction of a honeycomb sandwich panel 14 Figure 2.2 Strength, stiffness and weight for sandwich and solid construction 15 Figure 2.3 Examples of pre-stressed membranes 17 Figure 2.4 The Mil lennium Dome in London, England 18 Figure 3.1 Comparison of specific properties of fibers versus polymers and metals 21 Figure 3.2 Comparison of specific properties of composite materials versus polymers and metals 22 Figure 3.3 Variation in specific strength with temperature for C F R P composites versus aluminium 25 Figure 4.1 Example gravitational loading on individual panels 28 Figure 4.2 Velocity profiles over terrain with three different roughness characteristics for uniform gradient wind velocity of 100 mph 30 Figure 4.3 Typical C F D model for wind analysis 31 Figure 4.4 Wind flow lines around three typical building shapes 32 Figure 4.5 1-D heat transfer model 38 Figure 4.6 Heat transfer model formatted spreadsheet (1 of 2) 40 Figure 4.6 Heat transfer model formatted spreadsheet (2 of 2) 41 Figure 4.7 Heat transfer model equilibrium temperatures 42 Figure 4.8 Seismic connections 44 ix LIST OF FIGURES Figure 5.1 Approximation of the Wheel surface geometry using discrete panels 45 Figure 5.2 Flat plate subject to uniform normal pressure (nomenclature also shown) 46 Figure 5.3 Analytical analysis of flat plate stress and deflection (1 of 2) 51 Figure 5.3 Analytical analysis of flat plate stress and deflection (2 of 2) 52 Figure 5.4 Variation in max. tensile stress with plate aspect ratio [t = 2/16", SS edges] ....53 Figure 5.5 Variation of max. tensile stress with plate aspect ratio [SS edges] 53 Figure 5.6 Variation of max. tensile stress with plate aspect ratio [Fix. edges] 54 Figure 5.7 Single curvature plates 55 Figure 5.8 Buckling with a point of inflection 56 Figure 5.9 Snap-through buckling 57 Figure 5.10 Lateral buckling 58 Figure 5.11 Analytical analysis of curved plate buckling 59 Figure 5.12 Variation in buckling ratio with plate area for several different aspect ratios (t = 2/16", b/R =0.5) 60 Figure 5.13 Plate area at which buckling occurs for t = 1.5875mm (1/16") 61 Figure 5.14 Plate area at which buckling occurs for t = 3.175mm (2/16") 61 Figure 5.15 Plate area at which buckling occurs for t = 4.7625mm (3/16") 62 Figure 5.16 Plate area at which buckling occurs for t = 1.5875mm (1/16") 63 Figure 5.17 Plate area at which buckling occurs for t = 3.175mm (2/16") 63 Figure 5.18 Plate area at which buckling occurs for t = 4.7625mm (3/16") 64 Figure 5.19 12:00 C L O C K POSITION (Segment radius shown in brackets) 65 Figure 5.20 3:00 C L O C K POSITION (Segment radius shown in brackets) 66 Figure 5.21 5:00 C L O C K POSITION (Segment radius shown in brackets,) 67 Figure 5.22 Graphical comparison of required and analytical buckling limits of plate geometries (8m straight edge plate length) 69 Figure 5.23 Graphical comparison of required and analytical buckling limits of plate geometries (curved edge span reduced by a factor of 2, 4m straight edge length) 71 Figure 5.24 Graphical comparison of required and analytical buckling limits of plate geometries (curved edge span reduced by a factor o f 4, 4000mm straight edge length) 73 X LIST OF FIGURES Figure 6.1 ' S H E L L 6 3 ' Geometry 75 Figure 6.2 Typical flat plate finite element model [Units of Nmrn] 76 Figure 6.3 Comparison of analytical and F E central deflection prediction (SS flat plate) [A= 10,000,000 mm 2 1= 1/16"] 78 Figure 6.4 Comparison of analytical and F E central tensile stress (SS flat plate) [A=l0,000,000 m m 2 t = 1/16"] 78 Figure 6.5 Comparison of analytical and F E central deflection prediction [A=l,000,000 mm 2 , t = 3/16", SS edges] 79 Figure 6.6 Comparison of analytical and F E central tensile stress [A = 1,000,000 mm , t = 3/16", SS edges] 79 Figure 6.7 Comparison of analytical and F E central deflection prediction [A = 10,000,000 mm 2 , t = 2/16", F ix . Edges] 80 Figure 6.8 Comparison of analytical and F E central tensile stress [A = 10,000,000 mm 2 , t = 2/16", F ix . Edges] 81 Figure 6.9 Typical curved plate Finite Element model (Units of Nmm) 82 Figures 6.10 Buckling mode shapes for a curved plate (SS straight edges, 0.8m x 5.5524m, 2/16" thick) .83 Figure 6.11 Comparison of analytical and F E buckling rations [t = 1/16", b/R = 0.1] 86 Figure 6.12 Comparison of analytical and F E buckling ratios [t = 1/16", b/R = 2.0] 86 Figure 6.13 Comparison of analytical and F E buckling ratios [t = 3/16", b/R = 0.1] 87 Figure 6.14 Comparison of analytical and F E buckling ratios [t = 3/16", b/R = 2.0] 87 Figure 6.15 Nonlinear load-deflection curve (a) and linear (Eigenvalue) buckling curves (b). 88 Figure 6.16 Newton-Raphson (left) and Arc-Length (right) methods 89 Figure 6.17 A N S Y S large deflection non linear buckling time history plot for nodes of 1 maximum displacement [ A R = 10, b/R = 0.1, A = 100,000 mm 2 , t = 1/16", esize = 5, No . substep = 100, Pmax = 40x10"6 kNmm" 2 , Arclength method] 90 Figure 6.18 A N S Y S large deflection non linear buckling time history plot for nodes of maximum displacement [AR = 10, b/R = 2.0, A = 100,000 mm 2 , t = 1/16", esize = 5, No . substep = 100, Pmax = 4500 x 10"6 kNmm" 2 , Arclength method] 91 xi LIST OF FIGURES Figure 6.19 A N S Y S large deflection non linear buckling time history plot for nodes o f maximum displacement [AR = 2.85, b/R = 0.5, A = 50,580,000 mm 2 , t = l /16"mm, esize = 200, No . substep = 100, Pmax = 0.002 x 10"6 kNmm" 2 , Arclength method] 92 Figure 6.20 Typical F E model of longitudinally stiffened curved plate (units of Nmm) 95 Figures 6.21 Buckling mode shapes for a curved plate (SS along longitudinal stiffeners edges), curved edge length 800mm, straight edge length 5524mm [eigenvalue buckling analysis] 96 Figures 6.22 Buckling mode shapes for a curved plate (SS along longitudinal stiffener edges), curved edge length 1600mm, straight edge length 5524mm [eigenvalue buckling analysis] 97 Figures 6.23 Linear deflection and V o n Misses effective stress in plate subject to normal pressure ..98 Figure 6.24 Typical structural optimization model 100 Figures 6.25 Convergence to local minima for subproblem approximation method, and convergence to global minima when random design runs performed first 102 Figure 6.26 Normalized variation in optimized variables with number of support points (b/R = 0.1) 105 Figure 6.27 Normalized variation in optimization variables with the number of radial stiffeners (SS along longitudinal stiffener length) I l l Figures 6.28(a) 3D graphical comparison of optimal stiffened plate and required geometries .112 Figure 6.28 (b) 3D graphical comparison of optimal stiffened plate and required geometries .113 Figure 7.1 Adhesive bond failure categorization 117 Figure 7.2 Adhesive design stresses 117 Figure 7.3 Rivet failure modes 118 Figure 10.1 Proposed methodology for the design of curved aluminum cladding panels...125 xii LIST OF SYMBOLS AND ABBREVIATIONS Chapter 1 N / A Chapter 2 N / A Chapter 3 F E F R P Finite Element Fiber Reinforced Polymer Chapter 4 C F D Computational Fluid Dynamics qcon Heat transfer through the material k Thermal conductivity of the material T Temperature within the conductor at a given distance x x Measure of distance through the thickness of the material A Cross sectional area of the conductor Ic Radiation insolation at the earth's surface n Turbidity factor ams Average molecular scattering coefficient over all wavelengths a. Total attenuation coefficient over all wave lengths Average particulate scattering coefficient over all wavelengths Average absorption coefficient over all wavelengths Molecular scattering coefficient for air Relative thickness of the air mass (the cosecant o f the solar altitude a, the angle rays make with the horizontal) Insolation at the outer edge of the atmosphere Solar constant - 1395 W / m 2 Solar altitude (angle rays make with the horizontal) Absorptivity for solar radiation cc/owtemp Apsorptivity for low temperature radiation cr Stefan Boltzman constant dps a m Io Ebo a Xlll LIST OF SYMBOLS AND ABBREVIATIONS T T Gr P a g q k v x Pr h AT q solar (fconvl (Iradl (j[conv2 <Jrad2 Chapter 5 P E t L b R A R b/R 8b Mb <?b S Temperature of the material [K] Temperature of the surroundings [K] Modified Grashof number Inverse of the air temperature [K" 1] Stefan Boltzman constant Gravitational acceleration Constant heat flux [W] Conductive resistance of air (a function of temperature) Viscosity of air (a function of temperature) Representative dimension of the plate Prandtl number for air (a function of temperature) Heat transfer coefficient Temperature rise Heat transfer from the solar radiation [W] Convection heat transfer loss from the external surface of the plate [W] Radiation heat transfer loss from the external surface o f the plate [W] Convection heat transfer loss from the interior surface of the plate [W] Radiation heat transfer loss from the interior surface of the plate [W] Pressure Young's Modulus Plate thickness Long length of rectangular plate Short length of rectangular plate, square plate edge length Radius of curvature Aspect ratio Included angle Central deflection due to bending action Edge moment due to bending action Maximum central tensile stress due to bending action Section Modulus xiv LIST OF SYMBOLS AND ABBREVIATIONS E Wb Pb A b L t sm Wm Pm Cc, Cd qcr R a v E h a I A a Ix C Chapter 6 bhor A K S Young's Modulus of the plate Total load taken in bending Effective pressure taken in bending Area of the plate Short width of the plate Length of the plate Thickness of the plate Coefficients Central deflection due to membrane action Maximum central tensile stress due to membrane action Total load taken in bending Effective pressure taken in bending Coefficients Buckling pressure Radius of curvature Ha l f o f the included angle Material Poisson's ratio Young's Modulus Plate thickness Rise of the plate Section 2 N D moment of area Sectional area Included angle of the plate Section 2 n d moment of area Warping constant Projected horizontal length of curved edge b Central deflection Stiffness matrix Stress stiffness matrix XV LIST OF SYMBOLS AND ABBREVIATIONS X\ i eigenvalue and (j)j i t h eigen-vector of displacements Esize Element edge length in mm Pmax Maximum load for nonlinear analysis Substep Increment inload during nonlinear analysis Pb Plate curved edge length pL Plate straight edge length pth Plate thickness lonth Longitudinal angle stiffener leg thickness lonw Longitudinal angle stiffener leg width maxseqv Linear max von mses effective stress modef Eigenvalue buckling ratio lslen Longitudinal angle stiffener leg slenderness ratio weightA Plate weight to area ratio Thick2 Plate thickness in sixteenths of an inch Noradst2 Number of radial stiffeners Rslen Radial angle stiffener leg slenderness ratio Rbenst Radial stiffener bending stress Theta Included angle Chapter 7 N/A Chapter 8 N/A Chapter 9 N/A Chapter 10 N/A xvi ACKNOWLEDGEMENTS I would like to thank my academic supervisor Dr. Stiemer, who provided me with endless support and guidance throughout the course of my research, and who convinced me to do a thesis in the first place. I would like to thank Prof. Halliday for giving me the opportunity to work on such an interesting and challenging project. I would like to thank Mike Gedig and Y e Zhou who provided me with technical guidance and advice during my time spent at A M E C Dynamic Structures Ltd. Finally, I would like to thank N S E R C for the funding awarded to me through an Industrial Postgraduate Scholarship and A M E C Dynamic Structures Ltd. for their contribution to this funding. xvii NOTES Officially Canada follows the metric (SI) system for units, first developed in Europe. In this system dimensions are given in meters (m). The official system of units in the United States is imperial and in this system dimensions are given in feet (ft) or inches ("). Due to the close proximity with the United States, many materials used in Canada are sourced from American fabricators, which has resulted in Canadian engineers becoming equally adept at using both systems of units. Personally my preference is for the metric system; due to the consist base of the units, however as this project has involved collaboration and information from fabricators including those in the United States, both systems of units are used in this report. Contrary to what many may perceive as common logic it is felt that this is the best approach, as many dimensions are easier to express in imperial units, such as the thickness of aluminum plates, which typically are available in sixteenths of an inch, which gives an 'unpleasant' number when converted to the metric equivalent. In fact, with the exception of plate thickness, all other data is presented in metric (SI) units. xviii CHAPTER 1: INTRODUCTION C H A P T E R 1: I N T R O D U C T I O N 1.1 Objectives and outline: This thesis report details an investigation into the preliminary design of a structural system for enclosing a c iv i l engineering structure. In order to avoid a purely academic exercise and to ensure a practical investigation which w i l l be beneficial to designers, this investigation was conducted as part of a larger conceptual design project by a large steel engineering and fabricating consultant in British Columbia, Canada. These ensured practical aspects of fabrication and erection were considered throughout the thesis project. Through a review of available literature, allowing a comparison and evaluation of the advantages of parasitic panels versus a stressed skin structural solution (and associated materials) for enclosing the torus structure considered in this project, parasitic aluminum cladding panels are selected as the most appropriate structural solution. Chapters 2 and 3 of this report contain discussions and summaries of information obtained, which directly lead to selection of this structural and material combination for this project. Aluminium cladding is commonly used in order to enclose the envelope of a building and to provide a barrier to ingress of the external environment, both for user comfort and structural durability. For standard rectangular buildings, the design of these panels is relatively simple, due in part to the ease of structural analysis of flat plates but mostly due to the presence of prescriptive design methodologies developed by manufacturers (for proprietary systems) and governing/regulatory committees around the world. However, for non-standard structures, such as buildings with irregular and curved shapes, there are no such prescriptive guidelines, and the designer must rely on engineering judgment, research findings and innovative original analysis, in order to ensure confidence in a conservative and efficient design. Relatively little information could be found in the literature, with regard to the design of curved aluminum panels subject to thermal and pressure loading. There exists no closed form solution for the design o f these panels, and there is a lack of investigation into the influence of design parameters on the structural efficiency of these panels. The main objective of this thesis is to 1 CHAPTER 1: INTRODUCTION perform a parametric investigation into the stability of curved aluminum panels, subject to environmental loading, with particular emphasis on the resistance to buckling subject to uniform normal pressure. Through careful literature review and finite element analysis, an investigation into the influence of panel parameters including; geometry, support configuration and stiffening elements, is completed. The aim of this investigation is to allow the development of a guideline for the design of curved panels, and to provide designers with a better understanding of the influence of panel parameters at the outset of design. A visual summary of the work completed in this thesis investigation is shown in Figure 1.1. 2 CHAPTER 1: INTRODUCTION Structural analysis; plates subject to uniform normal pressure of 4.2 kNmm-2 Maximum stress and deflection as a function Buckling pressure and maximum area as a function of support type (SS or Fixed), plate aspect of panel aspect ratio, curvature and thickness. (SS ratio, area and thickness. (Analytical) straight edges. Analytical) Development of visual design aids Validation of F E models with Analyt ical results Design optimisation of longitudinally stiffened panels with a number of different radial stiffeners. (FE) Design optimisation of longitudinally stiffened panels for a number of support configurations. (FE) Thermal analysis External air conditions T.i < Interior air conditions • CJeonv! 1 -D heat transfer model, determination of cladding temperature as a function of external air temperature and interior air temperature. Figure 1.1 Visual summary of work completed 3 CHAPTER 1: INTRODUCTION 1.2 Literature review A n extensive literature review was conducted at the outset of this thesis project. Unfortunately, relatively little information regarding the design of curved panels subject to uniform normal pressure could be found. The references of note are described in the following pages. The aluminum manufacturer; ' A L C A N ' produced a design book entitled 'Strength of Aluminum' by Cedric Marsh 1 , for use by design engineers in 1989. In addition to providing tabulated data related to properties of the various aluminum alloys produced by this company, this reference outlines some simple structural analysis methods. This includes analytical equations for calculating the maximum tensile stress and deflection of flat aluminum plates with varying boundary conditions (simply supported or fixed), and of rectangular or circular geometry, subjected to uniform normal pressure (over the entire plate or a small central area). Warren Young's 'Roarke's formulas for stress and strain' , provides a method for calculating the maximum tensile stress and deflection o f flat isotropic plates subjected to uniform normal pressure for several plate geometries, including; rectangular, triangular and polygonal for simply supported or fixed edges. Methods of analysis are also provided for uniform normal pressure over a small central area or for triangular pressure distributions on rectangular plates. However the methods of analysis generally involve the combined use of equations and tabulated coefficients provided for specific cases, with the tables requiring linear interpolation and covering only a limited range of geometries. Timoshenko and Gere's 'Theory of elastic stability' gives an in-depth analytical analysis of the buckling resistance of isotropic plates of single curvature subject to uniform normal pressure for simply supported or fixed un-curved edges. They identify three possible buckling modes; 'snap-through' buckling (in-plane), buckling with a point of inflection (in-plane) or lateral buckling (out-of-plane), and provide appropriate equations for evaluation of the buckling pressures. N o closed form solution to the maximum stress or deflection in isotropic plates of single curvature could be found in the literature. With the apparent lack of analytical solutions for 4 CHAPTER 1: INTRODUCTION stresses and deflections of curved plates subject to uniform normal pressure, research in this area has focused on 'Finite Element' (FE) analysis. 'Finite-Element analysis of cylindrical panels with random initial imperfections' by V Papadopoulos and M . Papadrakakis 4 presents results of a stochastic finite element analysis of curved panels subject to normal pressure. They suggest a methodology for including the random nature of imperfections associated with plate geometry and material properties, allowing lower bounds of buckling stability and upper limits of acceptable imperfections to be determined. They found reductions in buckling pressure of up to 50% compared with perfect panels, with an associated coefficient of variation of 2. However the results presented are for a very small set of specific plate geometries. Much interest has been shown in the buckling and post-buckling behaviour of cylindrical shells. Shen 5 has published numerous articles related to buckling analysis of cylindrical shells. O f particular interest are his articles on post-buckling analysis of imperfect stiffened laminated cylindrical shells under combined external pressure and thermal loading. He utilizes a numerical procedure based on a boundary layer theory and a single perturbation analysis. The results presented are mostly for fiber-reinforced-polymer (FRP) laminated cylinders, as opposed to isotropic materials and the geometries considered are limited. While the behaviour of cylindrically curved panels is different from that of complete cylinders it may be possible to apply some of his methods to the analysis of cylindrically curved panels. CHAPTER 1: INTRODUCTION 1.3 AMEC Dynamic Structures design project background: A concept for a major tourist attraction tentatively entitled; 'Panasonic Mega Wheel ' (the Wheel), has been developed by a team of architects for construction in the near future. With a similar purpose to the 'London Eye ' (a world famous tourist attraction located in London, England), this attraction consists of a rotating track fastened to a torus structure, with several viewing gondolas attached to allow the public to take advantage of elevated views o f the surrounding area. Figure 1.2 shows a photograph of the 'London Eye ' . It is an ambitious project, with an anticipated wheel height of 152m, surface area of 16 000 m 2 and a total weight of 5600 ton (approximate values). Figure 1.3 shows an artist's impression of what the final structure w i l l look like, based on the initial concepts developed by the architects. 6 CHAPTER 1: INTRODUCTION This figure also shows the presence of a hotel, a structure at the base of the wheel (an entertainment complex), and a suspended circular disc which represents an advertising screen. Figure 1.3 Artist impression of the appearance of the final structure6 The main Wheel structure wi l l consist of a steel truss structure, comprised of welded steel box sections or piles running longitudinally and lighter standard hot rolled structural shapes such as tubes or wide flanges acting as bracing. 6 A typical structural segment with a 15° angular span is shown in Figure 1.4 (some bracing elements have been omitted from this figure for clarity). 7 CHAPTER 1: INTRODUCTION Figure 1.4 Typical structural segment with a 15° angular span Due to the high cost of field labour, the wheel structure will be designed for ease of construction, in order to minimize the erection cost. The proposed erection scheme involves the pre-assembly of segments of the wheel in a fabrication shop environment, with the segments bolted together during erection in the field. Figure 1.5 shows the cross sections at various locations along the wheel circumference. 8 CHAPTER 1: INTRODUCTION 5 00 CLOCK POSmON Figure 1.5 Variation of structural sections Pretension spokes wi l l be used to minimize deflection and stresses during erection and also in the final configuration where gravity w i l l act to flatten the wheel. Once completed the main structure wi l l essentially form a closed ring. Closed rings are very efficient in resisting in-plane loading due to the development of hoop stresses. However before completion, the main structure wi l l 9 CHAPTER 1: INTRODUCTION form an open section, which can not develop hoop stresses, and is significantly less resistive to in-plane loading. Spokes wi l l be installed concurrently with the main structure erection in order to control deflections (these spokes wi l l be designed to be adjustable), by developing tensile forces which w i l l resist lateral deflection due to gravity. The spokes wi l l not be designed to meet at the wheel center, as this would create space problems. Figure 1.6 shows a rendering of the structural framing, including some of the pretension spokes. Final closing of the wheel structure w i l l require accurate alignment of structural elements at the apex. This is a very complex and risky procedure and requires in depth analysis and planning. O f particular importance is a method of accurately moving and adjusting the elements at the apex. To achieve the desired architectural appearance of the attraction, a cladding system wi l l be used to enclose the entire wheel. The cladding system w i l l be designed to resist self weight, wind 10 CHAPTER 1: INTRODUCTION pressure and thermal loading. The structural design of the cladding system, while at first appearing to be secondary to the design of the main structure, may in fact dominate the entire design. Slight variations in cladding design have significant consequences in terms of weight imposed on the main structure and cost of the entire project due to the large surface area to be covered. 11 CHAPTER 2: CLADDING STRUCTURAL CONCEPTS CHAPTER 2: CLADDING STRUCTURAL CONCEPTS Decisions are made continuously through out a project time-line; starting with the initial concept and continuing until final completion. The scope and influence of decisions on the final project cost and schedule are generally at a maximum at the start of the project and are progressively diminished as the project is guided through to completion. It is extremely important to make informed and 'the right' decisions during the conceptual design stage as these have the potential for much larger influence on project success than decisions made during the detailed design stage. There are many different conceptual variations which can yield a feasible design for any given structural problem. The ultimate selection of any concept to be brought forward to the detailed design stage depends on several factors unique to each project and the needs of the client, these may include; cost, appearance, political implications, factor of safety, environmental impact, schedule, material usage and ease of fabrication/erection (repeatability and risk). A s a general rule of thumb, 5-10% of the total cost of a c iv i l structure w i l l be engineering fees for design, with the remaining 90-95% fees for materials and labor during fabrication/erection. One could question this fee split, as poor or uninformed decisions made during the design stage can easily double or triple the final cost of a structure. Ideally, without time and budgetary constraint, all feasible options would be explored at the conceptual stage to a level which would allow a confident and defendable choice of a final concept with which to proceed to the detailed design stage. However generally speaking, once the client has decided to proceed with a project and has obtained sufficient funding, there is enormous pressure to obtain the shortest possible time to completion. It is commonly observed by experienced engineers that a client w i l l take some time in officially starting a project and retaining an engineering firm, but once this has occurred the client wants the project completed 'yesterday'. Due to time and budgetary constraint, the conceptual design stage is generally shortened as much as possible with the aim of 'fast-tracking' the project and saving costs. This generally leads to 12 CHAPTER 2: CLADDING STRUCTURAL CONCEPTS decisions in the early stages made from engineering experience and intuition, with the result that individual personnel experience and preference can play a large role in the final decision of a concept for a given structural problem. However the behavior of complex structures and systems is not always intuitive and in many cases can be counter intuitive, hence it is of vital importance that adequate time be spent during the conceptual design phase for projects such as this to explore as many conceptual options as possible. It must also be considered that due to the large scale of this structure, that there may not be one single concept which will be most suitable for the entire Wheel. The magnitude and form of loading will vary considerably along the perimeter of the Wheel and hence several different conceptual options may be most suitable for different portions of the Wheel and may lead to considerable cost savings. It is not sufficient simply to take a generic cross section of the Wheel and determine an optimum concept from this single cross section, as the loading on any cladding structure will vary significantly around this cross section and indeed around the entire wheel as a whole. It is extremely difficult to separate structural and materials options for the cladding of the Wheel. The effectiveness and properties of certain structural concepts change significantly with different materials, and in fact some concepts are not feasible with certain materials and vice versa. Therefore the following discussion of structural concepts also includes some discussion of materials. At the end of this chapter, a single structural concept is selected with which to proceed and in the following chapter, possible materials are discussed in detail and a material for the particular concept chosen. 2.1 Parasitic panels (curtain walls): Cladding panels which do not aid the main structure in resisting loads such as those due to self-weight and earthquake are termed parasitic. The panels are designed purely to take the local loads applied to each individual panel, with the main structure designed to resist the reaction forces at points of connection with the panels. The design of such panels is somewhat simplified, as the interaction between panels and main structures is not accounted for in resisting global 13 C H A P T E R 2: C L A D D I N G S T R U C T U R A L CONCEPTS loads. However the final design is not as structurally efficient as a stressed skin panel design, where panel and main structure interaction is accounted for.. The efficiency of parasitic panels may be increased through the use of a composite section such as 'sandwich panel' construction as opposed to monolithic construction. Sandwich panels generally consist of two layers of a material; the 'facings', separated by and bonded to a third material (or structure); the 'core'. An analogy can be drawn between a sandwich panel and a steel I-beam, where the facings of the sandwich panel are equivalent to the flanges of the I-beam and carry the direct tensile and compressive stresses, and with the core of the sandwich panel equivalent to the web, which carries mostly shear stress. A typical sandwich panel construction is shown in Figure 2.1. Figure 2.1 Typical construction of a honeycomb sandwich panel The core provides a lightweight means of increasing the distance between the direct stress carrying areas of the panel and thus increases the stiffness with little weight penalty, this is shown in Figure 2.2. 14 CHAPTER 2: CLADDING STRUCTURAL CONCEPTS I it I I I * 1 Relative Stiffness 100 700 3700 Relative Strength 100 350 925 Relative Weight 100 103 106 Figure 2.2 Strength, stiffness and weight for sandwich and solid construction' 2.2 Stressed skin panels (tension/membrane structures): Stressed skin design has been used extensively in the aeronautics industry since the 1930's, however interest in stressed skin design for c iv i l buildings dates back only to the 1950's when tests on steel portal framed structures revealed measured stresses and deflections that were considerably smaller than those predicted by the usual design calculations, with the only explanation for the reduced stresses being the profiled steel sheet cladding. 9 Cladding panels in a building help resist applied loads through resisting displacements in the joints of the supporting framework which act in the plane of the panels. If this effect is accounted for, the supporting framework members may be reduced in size according to the reduction in load carried by these members through sharing of load with the cladding panels, which can lead to considerable weight and cost savings. Stressed skin action makes considerable demands on the fastening system and, indeed, the ultimate strength of most diaphragms is dependent on the strength of the fastenings.9 The design of such systems is also considerably more complex compared with parasitic panels. Architectural fabrics are a popular example of stressed skin construction used in c iv i l engineering buildings. The term 'architectural fabric' refers to the use of pliable fabric supported over a structural framework to define a space and/or create separation between the interior of a structure and the environment. Traditionally, humans have used membrane structures (skins or 15 CHAPTER 2: CLADDING STRUCTURAL CONCEPTS fabrics) with the fabrics primarily used as an outer skin that drapes over a skeleton structure (timber), with the skeleton providing most of the structural performance. A recent advance in the use of fabric for structures has been the development of tensioned fabric structures. Tension fabric structures use fabrics as the primary structural element. There is no draping over skeletons, rather the tension built up in the fabric membrane surface defines the space while providing shelter and other structural aspects. The guiding principle behind these structures is the minimal surface principle, such as that displayed by hyperbolic parabloids observed in materials such as soap bubbles and other minimal surface materials. Tensioned fabric structures are being increasingly used around the world due to their lightweight, water resistance and low maintenance, compared with heavier structural materials.10 The behaviour of tensioned fabric structures differs from that of conventional structures. Conventional structures depend on internal rigidity (stiffness), as well as tensile, compressive and shear strength for load-carrying capability and stability. In contrast, tensioned fabrics, act like membranes and possess little or no bending or shear stiffness, they rely on their form, internal prestress, and tensile strength for load-carrying capacity and stability. Under load significant changes in geometry of the structure may occur, resulting in highly nonlinear behaviour. With proper design, the load capacity will tend to increase with slight increases in deformation. Membrane structures depend on two-dimensional planar pre-stressing to maintain their shape and to support loads. This pre-stressing can be provided by either; pressurization, dead weight, or edge tractions and requires a reactive medium to maintain the desired shape, such as foundations, compression members, pressurized gases, fluids or counter-stressing members." Figure 2.3 shows some typical examples of pre-stressed membranes. 16 CHAPTER 2: CLADDING STRUCTURAL CONCEPTS Figure 2.3 Examples of pre-stressed membranes 1 1 For this particular project, a tension fabric structure could be used to envelope the main Wheel structural elements. The anti-clastic surface provided by the geometry of the inner perimeter of the wheel, is ideally suited to the use o f a pre-stressed membrane. However the doubly curved outer perimeter of the Wheel would require the use of pressurization and a second interior surface (to provide a seal) in order to allow the use of a pre-stressed membrane. 2.3 Structural concept selection Architectural fabric may be a viable alternative to aluminum panels for this application. However, as noted previously, the behaviour of tensioned fabric structures is highly nonlinear and complex. This added complexity, increases the required time for structural analysis, and greatly increases the risk, as proven by the 'Mil lennium Dome' in London, England (See Figure 2.4). Errors were made in modeling and analysis of this high profile tension structure, resulting 17 CHAPTER 2: CLADDING STRUCTURAL CONCEPTS in the required expenditure of millions of dollars above the original budget, in order to correct design mistakes. Figure 2.4 The Mil lennium Dome in London, England The scope of the analysis required for evaluation of this structural system for use on the Wheel would be sufficient for a thesis project in-of-itself, and is not further considered in this thesis. For this reason, parasitic cladding panels are selected as the preferred structural concept for this thesis project. 18 CHAPTER 3: CLADDING MATERIAL OPTIONS CHAPTER 3: CLADDING MATERIAL OPTIONS The subsequent discussion is related to materials selection for parasitic cladding panels only. There are several different materials available for use in stressed skin panels, however for the reasons discussed in chapter 2 this later structural option is not considered further in this report. 3.1 Metallic panels: Both steel and aluminum panels have been used extensively in practice for the cladding of buildings around the world. These traditional structural materials continue to be used despite the onset of recent advances in composite materials and plastics technology. This is due in part to the wealth of experience accumulated over years of use, which garners confidence in engineers, and also to the expanding array of treatments and alloying available which continues to improve the properties of these materials. The extensive use of aluminum and steel panels for applications similar to the requirements of this project, proves that metallic panels are a viable option for this project. In fact, at this stage of the design, aluminum panels are the preferred design solution for the cladding of the Mega Wheel (aluminum panels are preferred over steel panels due to their lower specific gravity). Aluminum panels may be anodized or powder coated in order to achieve the desired color for aesthetic purposes. This also serves to protect the metal from oxidation. Powder coating of aluminum panels is achieved by propelling small solid dry particles of 'paint' charged electrostatically and propelled towards the earthed aluminum. The powder builds up evenly on the surface to which it is firmly held through electrostatic attraction, once the desired thickness is reached, the powder is fused under heat into a continuous layer. The different constituents comprising the powder react with each other and with the underlying aluminum to form a coherent and firmly bonded coating. The anodizing of aluminum panels is achieved through a series of steps, including emulsion cleaning, inhibited alkaline cleaning, chemical etching, desmutting, anodizing and finally 19 CHAPTER 3: CLADDING MATERIAL OPTIONS 12 coloring . The anodizing step is achieved by suspending the panels in an electrolytic bath, a cathode (typically stainless steel or titanium) is placed in the bath and a current is passed through the solution at a predetermined voltage. Oxygen is formed at the surface of the panel, which is acting as an anode, this oxygen reacts with the aluminum at the surface to from a layer of aluminum oxide, which continues to 'grow' to a set thickness, which is a function of the applied voltage. The porosity of this oxide layer allows it to be colored during the anodization process. Anodization is preferred over powder coating for this application due to the improved appearance of anodized panels. It is currently not possible to achieve a uniform coating thickness with powder coating, which tends to result in an 'orange-peel' f inish 1 3 , which is unacceptable to the client for this project. Anodized panels are also typically more durable than powder coated panels. 3.2 Composite panels: A n adequate definition of a composite material, which fully envelopes all the permutations found in practice today is very difficult. However, the following definition is sufficient for the materials considered in this report; 'a composite material is any combination of two or more chemically distinct materials with a physical interface between the two, for the purpose of achieving material properties which can not be obtained through either constituent material on its own' . This definition envelopes materials such as concrete to advanced fiber reinforced plastics (FRP) and metal matrix composites ( M M C ) . Advanced composite materials such as fiber reinforced polymers, have been used in the aeronautics industry for many years. These composites essentially consist of small, high aspect ratio fibers of a material such as glass or carbon, embedded in polymer or metal matrices, such as epoxy or aluminum. Figure 3.1 shows a graphical comparison of the specific modulus and specific strength (properties as a ratio of the unit weight of the material) of some typical fibers versus those of metals and polymers. 20 CHAPTER 3: CLADDING MATERIAL OPTIONS w O 14 r— 12 10 \— en c 0) c 8 o <D 6 CL (0 4 h-2 r Polyethylene •  Kevlar49 • — S-Glass • 0 High-strength carbon SiC • * B o r o n High-modulus carbon 0 • E-glass • Al 2 0 3 I I I I I I 3 4 5 Specific modulus (x 10s in.) Figure 3.1 Comparison of specific properties of fibers versus polymers and metals7 The small, high aspect ratio fibers are typically brittle and have low durability when exposed directly to environmental factors such as moisture and ultraviolet radiation. When the fibers are combined with polymer or metal matrices, they form composite materials with a very wide range of properties. The matrices provide protection from environmental factors and in conjunction with the interface present between fiber and matrix, introduce more ductile and energy absorbing failure mechanisms. For fiber reinforced polymers, the fibers are often embedded uni-directionally to create sheets with a typical thickness of 1.25mm. These sheets are bonded together to create 'laminates', where the individual fiber orientation within each sheet may vary to create the desired mechanical properties of the material. A comparison of typical specific properties of common composite materials versus polymers and metals is shown in Figure 3.2. 21 CHAPTER 3: C L A D D I N G MATER IAL OPTIONS 0) 1 o Q. W Fiberglass -E-glass epoxy Polymers Fiber/polymer matrix Kevlar-epoxy Metals HS carbon-epoxy Boron-epoxy Fiber/metal matrix £ Boron-aluminum Carbon-magnesium Q 0 Boron-titanium 2 3 Specific modulus (x 10Bin.) Figure 3.2 Comparison of specific properties of composite materials versus polymers and metals7 A s discussed previously, the efficiency of parasitic panels may be increased through the use of sandwich panel construction. This form of construction is particularly suited to composite materials, and results in greater increases in efficiency relative to metallic materials. The facings and/or core may be constructed from monolithic or composite materials to obtain the desired properties. Table 3.1 shows a comparison between beam stiffness and weight per unit area of two different sandwich panels; F-board (glass reinforced polymer skin with aluminum honeycomb core) and M-board (aluminum skin with aluminum core), with an equivalent monolithic aluminum panel. Thickness (mm) Bending stiffness per unit width (Nm2/m) Weight per unit area of sandwich beam (kg/m2) Weight per unit area of monolithic Al, with same bending stiffness (kg/m2) F-board 13.7 1,100 3.08 15 26.4 4,500 4.21 25 52.3 205,000 7.54 41 M-board 13.9 3,500 4.67 23 26.6 13,500 5.73 36 52.0 52,500 7.84 56 Table 3.1 Comparison of bending stiffness and weight per unit area of sandwich and monolithic panels 22 CHAPTER 3: CLADDING MATERIAL OPTIONS 3.3 Material selection It is clear that the high specific properties (ratio of property value to density) of composite materials and/or structures makes them appear ideal for this type of application from a purely structural view point, as the weight savings will be highly beneficial in terms of reduced loading on the main structure (hence reduced member sizes and cost) and reduced transportation and erection costs. The danger of galvanic corrosion between panels and substructure if of different metals would also be alleviated if fiber reinforced plastics were used. However, composite materials are generally significantly more expensive than traditional materials such as steel or aluminium, and there are also questions surrounding the durability of composite materials. It is extremely difficult to accurately asses the expected cost of a composite material or structural system in the conceptual design stage. As an illustration of this statement, the leading airframe producers in the United States are engaged in a major cooperative effort to develop software to reduce the cost of composite manufacturing. The Composites Affordability Initiative (CAI) is a cooperative effort between government and industry to reduce the cost of aerospace composite structures so that design engineers can take advantage of their unique benefits. The group has identified a major obstacle indicating that designers in the early concept stage, when most important decisions occur, have no way to determine the impact of their choices on program cost. They are currently focusing efforts on methods of adequately predicting the costs associated with composite materials and structures. Tables 3.2 and 3.3 may serve as general indications of material and labour costs associated with the use of composite materials versus metallic materials. However it should be noted that these are taken from experience garnered in the marine industry and may not be strictly applicable to this project. They are included here due to the lack of alternative information for structures similar to the Wheel. 23 CHAPTER 3: CLADDING MATERIAL OPTIONS Material Density Tensile Strength Tensile Modulus Ultimate Elongation % 1995 Cost $/lb lbs/ft3 qm/cm 3 psix 10 3 Mpa psix 106 Gpa Resins Orthophthalic Polyester 76.7 1.23 7 48.3 0.59 4.07 1 1.05 Resins Isopthalic Polyester 75.5 1.21 10.3 71.1 0.57 3.9 2 1.19 Resins Vinyl Ester 69.9 1.12 11-12 76-83 0.49 3.38 4-5 1.74 Resins Epoxy (Gougeon Proset) 74.9 1.2 7-11 48-76 0.53 3.66 5-6 3.9 Resins Phenolic 71.8 1.15 5.1 35.2 0.53 3.66 2 1.1 Fibers E-Glass (24 oz WR) 162.4 2.6 500 3450 10.5 72.45 4.8 1.14 Fibers S-Glass 155.5 2.49 665. 4589 12.6 86.94 5.7 5 Fibers Kevlar ® 49 90 1.44 525 3623 18 124.2 2.9 20 Fibers Carbon-PAN 109.7 1.76 350-700 2415-4830 33-57 227-393 0.38-2.0 12 Cores End Grain Balsa 7 0.11 1.32 9.11 0.37 2.55 n/a 3.7 Cores Linear PVC (Alrex R62.80) 5-6 .08-0.1 0.2 1.38 0.0092 0.06 • 30 5.2 Cores Cross-linked PVC (Dlab H-100) 6 0.1 0.45 3.11 0.0174 0.12 n/a 5.95 Cores Honeycomb (Nomex® HRH-78) 6 0.1 n/a n/a 0.06 0.41 n/a 13.25 Cores Honeycomb (Ndaplast H8PP) 4.8 0.08 n/a n/a n/a n/a n/a 0.8 Laminates Solid Glass Polyester hand lay-up 96 1.54 20 138 1.4 9.66 n/a 2.5 Laminates Glass/Polyester Balsa Sandwich Vacuum assist 24 0.38 6 41 0.4 2.76 n/a 4 Laminates Glass/Vinyl Ester PVC Sandwich SCRIMP® 18 0.29 6 41 0.4 2.76 n/a 5 Laminates Solid Carbon/Epoxy filament wound 97 1.55 88 607 8.7 60 n/a 10 Laminates Carbon/Epoxy Nomex Sandwich prepreg 9 0.14 9 62 0.5 3.45 n/a 20 Metals ABS Grd A (ASTM 131) 490.7 7.86 58 400 29.6 204 21 0.29 Metals ABS Grd AH (ASTM A242) 490.7 7.86 71 490 29.6 204 19 0.34 Metals Aluminum (6061-T6) 169.3 2.71 45 310 10 69 10 2.86 Metals Aluminum (5086-H34) 165.9 2.66 44 304 10 69 9 1.65 Wood Douglas Fir 24.4 0.39 13.1 90 1.95 13.46 n/a 1.97 Wood White Oak 39.3 0.63 14.7 101 1.78 12.28 n/a 1.07 Wood Western Red Cedar 21.2 0.34 7.5 52 1.11 7.66 n/a 2.26 Wood Sitka Spruce 21.2 0.34 13 90 1.57 10.83 n/a 4.48 Note: The values used in this table are for illustration only and should not be used for design purposes. In general, strength is desifined as yield strength and modulus will refer to the material's initial modulus. A core thickness of 1" with appropriate skins was assumed for the sandwich laminates listed. Table 3.2 Property and cost comparison of metals versus composites' 24 CHAPTER 3: CLADDING MATERIAL OPTIONS Source Type of Construction Application Lbs/Hour* Ft 2/Hour # Hours/Ft 2 + Scott Fiberglass boat construction Single skin with frames Recreational 20* 33* 03 + Scott Fiberglass boat construction Military 12* 20* .05+ Scott Fiberglass boat construction Sandwich Construction Recreational 10* 17* .06+ Scott Fiberglass boat construction Military 6* 10* .10+ BLA Combat Feasibility Study Single skin with frames Flat panel (Hull) 13** 22** .05** BLA Combat Feasibility Study Stiffeners and frames 5** 9** .12** BLA Combat Feasibility Study Core Preparation for Sandwich Construction Flat panel (Hull) 26** 43** .02** BLA Combat Feasibility Study Stiffeners 26** 43** .02** BLA Combat Feasibility Study Vaccuum assisted Resin Transfer Molding (VARTM) Flat panel (Hull) 10 § 43 § .02 § BLA Combat Feasibility Study Stiffeners 14§ ,07 s * Based on mat/woven roving laminate ** Based on one WR or UD layer * Single ply of mat/woven roving laminate * Time to laminate one ply of mat/woven roving (reciprocal of Ft2/hr) § Finished single ply based on weight of moderately thick single-skin laminate Table 3.3 Labour statistics for composite material construction' Durability issues associated with F R P composites include property degradation due to U V and laminate debonding due to moisture. The properties of F R P composites are also generally more temperature dependent than those of aluminum as shown in Figure 3.3. 1.6 -1.4 -1.2 -1.0 -c ffi. W 0.8 & 0.6 0.4 h 0.2 h 0.0 Carbon-epoxy Aluminum _L _L 50 100 150 200 Temperature (°C) 250 300 Figure 3.3 Variation in specific strength with temperature for F R P composites versus aluminium 7 25 CHAPTER 3: C L A D D I N G MATER IAL OPTIONS Damage and failure modes for composites differ from metals. Whereas a metal grillage wi l l transition from elastic to plastic behaviour and collapse in its entirety, composite panels w i l l fail one ply at a time, causing a change in strength and stiffness, leading ultimately to catastrophic failure. Because composite laminates do not exhibit the classic elastic to plastic stress-strain behaviour that metals do, safety factors based on ultimate strength are generally higher, especially for compressive failure modes. Composite materials and structures have been used extensively in the aeronautics industry, where weight savings in the aircraft structure leads to considerable cost savings in terms of fuel, and is beneficial to the overall performance of the aircraft. However, their use in c iv i l engineering has not been nearly as extensive, where the high capital cost of such panels and problems associated with their durability are not offset by a reduction in lifecycle costs. The design of anisotropic composite materials is also considerably more complex than the design of monolithic isotropic materials such as aluminium, and while the design of sandwich panels for stiffness is relatively straightforward, design for strength is more complex. This is because sandwich structures exhibit a range of failure modes, depending on materials, geometry and loading, the most problematic are usually debonding of the skins and core shear failure 1 4 . For this particular project, the aesthetics of composite panels may not be adequate. It is common for F R P composites to become bleached and exhibit surface defects such as 'crazing' and 'bubbling' due to U V exposure. It is not certain how effective methods of protection for F R P panels would be over long time periods. Composite materials and structures are certainly viable options for this project, however, concerns associated with long term durability and aesthetics must be addressed before any time or budgetary expenditure on conceptual designs using these materials are performed. Assuming adequate durability and aesthetics can be achieved, it is ultimately the cost of the system which wi l l determine i f it is the most suitable for this project. Similarly to the discussion of architectural fabric as a material option, the evaluation of composite materials or structures is of large enough scope to be a thesis in-of-itself, and is not considered further in this thesis project. 26 CHAPTER 3: CLADD ING MATER IAL OPTIONS Therefore, through a process of elimination, anodized aluminum appears to be the most appropriate choice of cladding material at this stage. For the remainder of this report, only anodized aluminum parasitic cladding panels w i l l be considered. Table 3 .4 shows the material properties assumed for anodized aluminum for the structural analysis conducted (assumed properties of steel are also shown for stiffening elements). These values are obtained through discussion with aluminum fabricators (such as Alcan ) as representative of the properties of the aluminum alloys which are most likely to be used for the intended application (the expected structural demands and finishing requirements). Material Property Aluminum Alloy Steel Density [kNs mm ] Poisson's ratio Young's Modulus [kNmm"2] Shear Modulus [kNmm"'1] 2.71E-12 0.34 70 26.96 7.85E-12 0.3 200 79.29 Table 3 .4 Representative properties of Anodized aluminum alloy and steel 27 CHAPTER 4: STRUCTURAL LOADING CHAPTER 4: STRUCTURAL LOADING 4.1 Gravity load: The cladding panels w i l l be loaded by their self weight due to the action of gravity. Depending on the particular location of a given panel on the Wheel structure, the loading direction wi l l vary considerably, as shown in Figure 4.1. Figure 4.1 Example gravitational loading on individual panels Depending on the position of the panel around the wheel, the self weight of the panel can act to stiffen the panel against buckling due to uniform normal pressure, such as in the case of a panel aligned with gravity, or can decrease the resistance of the panel to buckling as in the case of a panel aligned perpendicular to gravity. 28 CHAPTER 4: STRUCTURAL LOADING This variation in loading around the wheel due to gravity will allow some freedom in modifying the design of the panels, and may allow some different conceptual options to be utilized, significantly increasing the efficiency of the structure. 4.2 Wind load: The cladding or exterior surface of any structure must resist thousands of cycles of wind loading every day.15 During storms, the structure will be buffeted by gusts from various directions, and over a long period of time the building and cladding structure must continue to resist these forces without failing. The wind, as it strikes the building, is affected by the climatic conditions for the geographical area, the surrounding terrain, and the effects of other buildings adjacent to the site.15 The term 'wind profile' is used to describe the shape and gradient of the wind acting on the structure, which also includes turbulence and eddies in the boundary layer. The boundary layer separates the region of laminar flow from uniform flow, which occurs due to the 'friction' from the earth's surface. This boundary may be as high as several thousand feet above the ground surface terrain. The wind profile acting on the structure is heavily dependant on the terrain over which the wind travels before reaching the structure (the length of influence can be up to several kilometers).15 Figure 4.2 shows typical wind velocity profiles for three distinct extents of ground terrain. 29 C H A P T E R 4: S T R U C T U R A L L O A D I N G 1800 1600 1400 --1200 -— o 1000 m o > 800 o ,5 ea £ 600 "cb "3 ffi 400 200 0 100 M P H 95 M P H 90 M P H 84 M P H 78 M P H 72 M P H • V , a Z l ; ! 62 M P H 48 M P H 31 M P H 100 M P H % M P H 90 M P H 84 M P H 76 M P H VraZ1"-1 100 M P H 97 M P H 92 M P H 86 M P H 72 M P H Centre of large city Rough wooded country, towns, city outskirts Flat open country, open flat coastal belts Figure 4.2 Velocity profiles over terrain with three different roughness characteristics for uniform gradient wind velocity of 100 mph 1 5 In design codes, wind loading is typically analyzed as statistical variations, which include events that can be expected over a period of time. This means the design load must anticipate the statistical recurrence of an event over a 50 or 100 year period, which may represent the life of the building. These design loads are deduced from meteorological data in the local area. 1 5 Due to the size and unconventional shape of the Mega Wheel, advanced wind analysis either in the form of Computational Fluid Dynamic (CFD) or experimental wind tunnel testing is required. The presence of a large hotel very close to the Mega Wheel further complicates the wind analysis. The design pressure loading wi l l vary significantly with the panel position around the Wheel structure. A typical C F D model for wind analysis is shown in Figure 4.3. 30 CHAPTER 4: STRUCTURAL LOADING Figure 4.3 Typical CFD model for wind analysis6 Wind tunnel testing typically compromises two stages; tests on a small-scale model of the building and its environment and tests on full-scale portions of the building cladding to measure the actual performance. A third test, a dynamic wind tunnel test, may also be conducted to measure stiffness and motion characteristics, such as oscillation.15 Some consideration of impact loading may also be warranted, as debris contained within a significant storm may result in extensive damage to cladding systems. The Australian design code, explicitly considers this 'missile loading'. Dynamic effects, such as vibration can be catastrophic and must be considered. Serious vibration, generally implying vibration at one of the natural frequencies of the structure can arise in two ways. Self-excitation, in which the form of the structural element is such that the deflection to an applied wind force actually increases the wind force (aerodynamic instability, 31 CHAPTER 4: STRUCTURAL LOADING e.g. the Tacoma Narrows bridge). Forced vibration due to instability in the flow pattern, induced by the structure itself or due to a natural periodicity in the airflow itself. Figure 4.4 shows flow patterns around three typical building shapes, in general there are areas of both positive and negative pressure (suction). (c) Figure 4.4 Wind flow lines around three typical building shapes15 As a preliminary step, the National Building Code is used to determine the design wind pressure which the Wheel and hence cladding panels will be subjected to. Taking the maximum height of the Wheel as 152m, the design pressure according to the National building code for the geographical location is 4.2 kNm" . The pressure load will act transversely to each of the panels. This leads to several areas of concern, which will form the limiting cases for design of these panels. Firstly the central deflection of the panels must be kept to within acceptable limits from an aesthetic point of view. Secondly, instability in the form of buckling must be prevented. 32 CHAPTER 4: STRUCTURAL LOADING 4.3 Thermal load: The entire structure will be situated in an area where there will be large swings in ambient temperature. The temperature of the structure itself will vary considerably more than this due to radiation absorbed from the sun, either directly or through reflection from surrounding buildings. The aluminum panels will be linked to steel elements and hence the difference in thermal expansion between the two metals must be provided for, either through flexible connections and expansion joints or by taking account of the stresses induced due to constrained expansion/contraction in subsequent buckling and yielding analyses. The use of epoxy adhesive to join stiffeners to the aluminum panels gives rise to the possibility of using steel stiffeners to strengthen the panels, while minimizing the risk of galvanic action and eventual galvanic corrosion through the separation layer provided by the epoxy adhesive. If this is the case, thermal analysis is required to determine the loads on the bond material. There is the possibility of extremely high temperatures within the Wheel. The hollow structure may act as a large chimney, with heated air rising to the top of the Wheel. This aspect needs to be analyzed further with the possible use of venting holes around the wheel and particularly at the apex investigated. The thermal analysis of the structure is a highly complex problem requiring much analysis. As a preliminary step a simple heat transfer model can be created through application of first principles of heat transfer; conduction, convection and radiation. 4.3.1 Thermal conductivity: When a temperature gradient exists in a body, there is an energy transfer from the high temperature region to the low temperature region. Equation 4.1 is an expression for determining the heat transfer through conduction (Fourier's law of heat conduction). 33 CHAPTER 4: STRUCTURAL LOADING = -kA— Eq.4.1' 6 dx Where: qco„ is the heat transfer through the material k is the thermal conductivity of the material T is the temperature within the conductor at a given distance x x is a measure of distance through the thickness of the material A is the cross sectional area of the conductor 4.3.2 Thermal radiation: Radiation incident on the earth from the sun may be absorbed, scattered or reflected to some degree by the earth's atmosphere, before reaching any man made structures on the surface. The radiation insolation at the earth's surface, Ic, can be expressed as a function of coefficients which describe the earth's atmosphere. Eq. 4.31 6 Eq. 4.416 Where: Ic is the radiation insolation at the earth's surface n is the turbidity factor ams is the average molecular scattering coefficient over all wavelengths a, is the total attenuation coefficient over all wave lengths aps is the average particulate scattering coefficient over all wavelengths a is the average absorption coefficient over all wavelengths Eq. 4.2 n = «/ = ams + a + a 34 C H A P T E R 4: S T R U C T U R A L L O A D I N G The turbidity factor, n, is a convenient means of specifying atmospheric purity and clarity; its value ranges from about 2.0 for very clear air to 4 or 5 for very smoggy, industrial environments.16 The molecular scattering coefficient for air at atmospheric pressure is given as a m =0.128-0.054logm Eq. 4.51 6 Where: ams is the molecular scattering coefficient for air m is the relative thickness of the air mass (the cosecant of the solar altitude a, the angle rays make with the horizontal) The insolation at the outer edge of the atmosphere is expressed in terms of the solar constant Eb0 by I0=Eh0 sin a Eq.4.6 1 6 Where: Io is the insolation at the outer edge of the atmosphere Ebo is the solar constant - 1395 W/m a is the solar altitude (angle rays make with the horizontal) At radiation equilibrium, the net energy absorbed from the sun must equal the long-wavelength radiation exchange with the surroundings, resulting in the expression given in equation 4.7. ^] ocxlin=alowlemp(7(T4-Tlr) Eq.4.7 , 6[f] Where: asun is the absorptivity for solar radiation Biowtemp is the absorptivity for low temperature radiation cr is the Stefan Boltzman constant 35 CHAPTER 4: STRUCTURAL LOADING T is the temperature of the material [K] T^r is the temperature of the surroundings [K] For highly polished aluminum a s u n = 0.15 and aiowtemp = 0.04. 1 6 These values should be increased due to the anticipated dark color of the cladding panels, a conservative value would be to take values which halve the difference between flat black lacquer and polished aluminum. For flat black lacquer, a s u n = 0.96, aiowtemp = 0.95. 1 6 4.3.3 Thermal convection: Heat transfer to the environment in the form of convection is a function of both the temperature difference and the fluid dynamics at the boundary between the two. Similar to fluid dynamics, analytical solutions only exist for very simplified problems. Ho lman 1 6 cites research findings for free convection from vertical and inclined surfaces to water under constant-heat-flux conditions. He presents empirical equations which allow the calculation of the temperature of a plate with a constant heat flux source (such as solar radiation), the relevant expressions are shown in equations 4 .8 -4 .10 . kv2 Where: Gr is a modified Grashof number P is the inverse of the air temperature [K" 1] a is the Stefan Boltzman constant g is gravitational acceleration q is the constant heat flux [W] k is the conductive resistance of air (a function of temperature) v is the viscosity of air (a function of temperature) x is a representative dimension of the plate The assumed expression for the convective heat transfer coefficient, h, is given in equation 4.9 (turbulent flow). Eq . 4 .8 1 6 36 CHAPTER 4: STRUCTURAL LOADING h = -0A7(GrPr)U4 x h Where: Pr is the Prandtl number for air (a function of temperature) h is the heat transfer coefficient AT is the temperature rise 4.3.4 Total response There are two unknowns in the total response; the cladding temperature and the temperature of the interior of the Wheel structure. A l l other variables are either dependent on these two values or can be assumed reasonable values. Using the equations listed previously in this chapter, a one-dimensional heat transfer model was generated. Heat transfer equilibrium is enforced for a control volume centered around a l m length of cladding; the heat transfer into the cladding panel from solar radiation must be balanced by heat transfer due to radiation and convection from the exterior surface of the panel to the outside ambient environment, and from the interior surface of the panel to the interior of the wheel structure. The 1-D heat transfer model is shown schematically in Figure 4.5. This model is essentially describing equation 4.11, hence there is no unique solution. Eq . 4 .9 1 6 Eq. 4.10 1 6 37 C H A P T E R 4: S T R U C T U R A L L O A D I N G Solar rays External air conditions T.i T, Cjsolar q r a d l Interior air conditions T 2 T a Figure 4.5 1-D heat transfer model olar \rud\ t conv2 Where: Eq. 4.11 qsoiar is the heat transfer from the solar radiation [W] qconvi is the convection heat transfer loss from the external surface of the plate [W] qradi is the radiation heat transfer loss from the external surface of the plate [W] qConv2 is the convection heat transfer loss from the interior surface of the plate [W] qrad2 is the radiation heat transfer loss from the interior surface of the plate [W] A 'formatted spreadsheet' was created which allows the computation of one of the unknown temperatures given the other. Proprietary spreadsheet programs such as Microsoft Excel, allow repetitive calculations to be performed quickly through a referencing network between cells which contain data. This referencing allows functions to be performed on the data presented in a number of cells and the solution presented in another separate cell. However the functions themselves are typically hidden from the user when in use, and are impossible to view from a 38 CHAPTER 4: STRUCTURAL LOADING hardcopy printout. Formatted spreadsheets retain the advantages of traditional spreadsheet programs, while maintaining visibility of formulas. A formatted spreadsheet contains two distinct sections; an input section and a calculation/output section, with each section divided into 6 major columns (with some columns interspaced between these for formatting purposes) and with each row dedicated to a single variable. The user inputs information into the first three major columns of both sections, once this is completed, the user executes a 'Macro' (a computer program embedded into the formatted spreadsheet written in Visual Basic), which then performs the desired calculations and places the results in the fourth major column. In the input section, the first column contains a description of the input variables, the second column contains the variable name and the third column is the value of the input variable. Once the 'Macro' is executed, the input variable names are automatically defined as parameters within Excel, and assigned their input values, these values are then stored numerically in the fourth major column. In the output section, the first column contains a description of the variable to be evaluated, the second column contains the variable name and the third column contains the formula for evaluating the numeric value of the design variable, which is expressed in text format and in terms of the input variables or any output variables which have been defined previously (i.e. in a row above the row corresponding to the current design variable). Once the 'Macro' is executed, the variable names are automatically defined as parameters, the text formula is converted to a formula referencing the previously defined parameters, and the fourth column stores the cell references and the numeric value of the design variable. The fifth and sixth columns in both sections, are dedicated to user input of the appropriate units for each of the variables and reference notes respectively. Figure 4.6 shows the formatted spreadsheet used to implement the heat transfer model. 39 C H A P T E R 4: S T R U C T U R A L LOADrNG PROJECT Mega Wheel Cladding Panels DATE 8/1/2006 FILE 1D heat model.xls TIME 4:04 PM REF INPUT 1 Outside surface cladding temp Te1 = Inside surface cloadding temp Te2 = Te1 83 [ C'C] Outside air temp Tail = 40 40 [°C] Inside air temp Ta2 = BflHHH = f l f l k s ° i c ' Incident solar flux qA = 1000 1000 [W/m2] Plate thickness t = 0.0015875 1.5875E-03 [m] Plate surface area Ax = 1 1.00 [m2] thermal conductivity of cladding k 202 202 [W/m°C] absorptivity for solar radiation alpsol = (0.96+0.16)/2 0.56 absorptivity for low temp radiation alplt = (0.04+0.95)/2 0.50 Stefan boltzman constant sigma = 5.669E-08 5.669E-08 [W/m2K] gravitational acceleration g = 9.81 9.81 [m/s2] Run Solve Te1 Solve Ta2 COMPUTATIONS 1 Miscelaneous initial cafes Temperatures in kelvin Te1k = Te1+273 356 [K] Te2k - Te2+273 356 [K] Talk = Ta1+273 313 [K] Ta2k = Ta2+273 323 [K] Properties of air at temp Te1 Viscosity of air nu1 = VLOOKUP(Te1 k,air!A3:D353,2) 2.14E-05 [m2/s] 1 Conductive resistance of air kaih = VLOOKUP(Te1 k,air!A3:D353,3) = 0.03046 [W/(m °C] 1 Prandtl number of air Pr1 = VLOOKUP(Te1k,air!A3:D353,4) 0.69604 1 Properties of air at temp Te2 Viscosity of air nu2 = VLOOKUP(Te2k,air!A3:D353,2) 2.14E-05 [m2/s] 1 conductive resistance of air kair2 = VLOOKUP(Te2k,air!A3:D353,3) 0.03046 [W/(m°C] 1 Prandtl number of air Pr2 = VLOOKUP(Te2k,air!A3:D353,4) 0.69604 1 Incident Solar radiation Solar energy transfer qsolar - qA'Ax 1000 [W] 1 Figure 4.6 Heat transfer model formatted spreadsheet (1 of 2) 40 CHAPTER 4: STRUCTURAL LOADING Conduction Conductive resistance Rcond = t/(k*Ax) 7.859E-06 [°C/W] 1 conductive energy transfer qcond (Te1-Te2)/Rcond 0.00 [W] 1 Convection Cladding outer surface beta of air at temp Te betal = 1/Te1k 0.003 [1/K] 1 Grashof number Gr1 = g*beta1 *qA*(AxA0.5)A4/(kair1 *nu1 A2) 1.98.E+12 1 Convection heat transfer coef hcondl = 0.17*(Gr1 *Pr1 )A0.25*kair1/(AxA0.5) 5.611 [W/(m2 oC] 1 Convective resistance Rconvl = 1/(hcond1*Ax) 0.178 [°C/W] 1 Convective energy transfer qconvl = (Te1-Ta1)/Rconv1 241 [W] 1 Cladding inner surface beta of air at temp Te beta2 = 1/Te2k 0.003 [1/K] 1 Grashof number Gr2 = g*beta2*qA*(AxA0.5)A4/(kair2*nu2A2) 1.98.E+12 1 Convection heat transfer coef hcond2 = 0.17*(Gr2*Pr2)A0.25*kair2/(AxA0.5) , 5.611 [W/(m 2 °C] 1 Convective resistance Rconv2 = 1/(hcond2*Ax) 0.178 [°C/W] 1 Convective energy transfer qconv2 = (Te2-Ta2)/Rconv2 185 [W] 1 Total convective energy transfer qconv = qconvl+qconv2 = 426 [W] Radiation Cladding outer surface resistance factor hr1 = sigma*(Te1 kA2+Ta1 kA2)*(Te1 k+Ta1 k)*alplt/alpsol 7.5329 [W/(m2K] 1 radiative resistance Rradl = 1/(hr1*Ax) = 0.1328 [°C/W] 1 radiative energy transfer qradl (Te1-Ta1)/Rrad1 324 IW] 1 Cladding inner surface resistance factor hr2 - sigma*(Te2kA2+Ta2kA2)*(Te2k+Ta2k)*alplt/alpsol = 7.8619 [W/(m2K] 1 radiative resistance Rrad2 = 1/(hr2*Ax) 0.1272 [°C/W] 1 radiative energy transfer qrad2 = (Te2-Ta2)/Rrad2 259 rw] 1 Total radiation energy transfer qrad qrad1+qrad2 = 583 [W] Total energy transfer qtotal = qconv+qrad = 1010 rw] Energy discrepancy qdiff = (qsolar-qtotal)/(qsolar)*100 = -0.98 [%] 1 Heat transfer, J.P. Holman, Fifth edition,1981 Figure 4.6 Heat transfer model formatted spreadsheet (2 of 2) A visual basic ( V B A ) program was created which systematically varies the ambient outside air temperature and interior temperature of the wheel and solves for the equilibrium cladding temperature (to the nearest degree) using the formatted spreadsheet in figure 4.6. The resulting data is plotted in Figure 4.7. 41 C H A P T E R 4: S T R U C T U R A L L O A D I N G 55 ] 50 -I— 1 , 1 , , , 20 25 30 35 40 45 50 Inner wheel air temperature [°C] Figure 4.7 Heat transfer model equilibrium temperatures As can be seen from Figure 4.7, the response is very linearized, which is not surprising given the simplified heat transfer model used. This plot is in terms of three variables, the ambient air temperature, cladding temperature and the air temperature of the interior of the wheel structure, providing a quick means of solving for one of the unknowns given the other two. For example, assuming an ambient air temperature of 40 °C and an inner wheel temperature of 50 °C yields a cladding temperature of 80 °C. Conversely, if a reliable estimate of the likely cladding temperature can be ascertained, for example from similar cladding materials in a similar location, an estimate of the air temperature within wheel structure can be obtained. This plot clearly shows that a reduction in the inner wheel air temperature will significantly lower the cladding panel temperature. This may be achieved through the introduction of strategically placed venting holes (such as at the apex), to allow significant air flow between the interior and exterior of the structure. 42 CHAPTER 4: STRUCTURAL LOADING The minimum cladding temperature is likely to be equal to the minimum ambient air temperature, which is available from meteorological data for the particular geographic location of the project (most likely to occur on a winters night), similarly so for the maximum ambient air temperature (most likely to occur on a summers day). With an educated assumption of the inner wheel temperature, an estimate of the cladding temperature can be obtained using Figure 4.8. This then gives a reasonable estimate of the likely temperature variation which the structure must be designed to withstand. The preceding analysis is a very simplified approximation to the actual behavior and response of the structure, and is intended to enable an order of magnitude analysis of the temperature of the cladding allowing computation of resulting structural stresses due to restraint of the expansion/contraction associated with these changes in temperature. More advanced computational methods are required in order to achieve more reliable predictions of the anticipated thermal loading on the structure. The computational models must include the effects of surrounding buildings (solar reflection), the position of the sun in the sky, weather, and also the structural configuration during the various stages of construction. 4.4 Seismic Load: Although the principles of cladding design for earthquake resistance are fairly consistent, the actual design criteria and detailing for mitigation of cladding failure vary widely from country to country, and even deviate to some extent among buildings in the same city.15 Tall buildings are generally designed to be ductile in order to limit effective seismic lateral forces, with resulting large displacements within the structural frame, which must be accommodated by the cladding system. The cladding connections must be designed to allow for movement of the main structural frame without generating significant stresses in the panels or connections (this obviously is not the case for stressed skin panels). The connections are generally designed to allow for in-plane translation between the panels and the frame (swaying motion) or in-plane rotation (rocking motion). Examples of these connections are shown in Figure 4.8. 43 C H A P T E R 4: S T R U C T U R A L L O A D I N G ! Top connections accomodate ! ( D ® i 1 • lateral, in-plane, story drift, by use of slotted holes (as shown) or long flexible rods. The lower connections as shown are • • • • relatively fixed and are load mm bearing, however should be somewhat ductile . ' : 1 J L : ^ Swaying (U.S.) '• '1 f 1 Connections are designed with ® tmmam • © slots or oversized holes to allow rocking motion as shown, to accomodate story drift. The lower connections are bearing, ® " -1 1 o however, should they fail, the 1 upper ones can also support panel 1 L dead load. Rocking (Japan) Figure 4.8 Seismic connections13 44 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S CHAPTER 5: PRELIMINARY STRUCTURAL ANALYSIS The Wheel can be coarsely generalized as a circular section extruded along a circular path. In order to fully capture the exact geometry of the Wheel would require panels curved in two perpendicular directions. However the radius of curvature of the circular path is significantly larger than the radius of curvature of the any portion of the Wheel section, suggesting that panels with single curvature would provide an adequate approximation, provided the panel lengths are 'small enough', such that the aesthetics are not compromised. In order to further reduce costs, flat panels may be used to approximate the Wheel geometry again provided that the panels are of 'small enough' dimension. Figure 5.1 is a schematic representation of the geometric options. Flat panels Figure 5.1 Approximation of the Wheel surface geometry using discrete panels 45 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S 5.1 Flat plates: From a structural analysis standpoint, simply supported flat rectangular panels are the easiest to design, due to the large volume of available analytical solutions for the stress and deflection in such plates subjected to uniform normal pressure. Flat plates are also not subject to buckling instability provided the edges are restrained against horizontal movement (if the edges are not horizontally restrained, the resulting circumferential compression at large deflection may cause buckling). 5.1.1 Theory A flat plate subject to a uniform normal pressure will carry the load primarily in bending, provided the deflection does not exceed half the plate thickness. However if the plate is relatively thin and deflects appreciably under load it will act as a membrane and will carry the load primarily in direct tension. If this membrane action is not accounted for, and traditional bending theory used to design the cladding panels, overly thick plates for a given area will result. Analytical solutions exist for the maximum stresses and deflections in plates resisting normal pressure through bending action and membrane action respectively, for a series of different boundary conditions. Fig 5.2 Flat plate subject to uniform normal pressure (nomenclature also shown) 46 CHAPTER 5: PREL IM INARY STRUCTURAL ANALYS I S Long Plate Bending action (AR>1): ELt3 Wb = PhA Mh=Ch Wbb s = Eq. 5.1 1 Eq. 5.2 Eq. 5.31 Eq. 5.4 Eq. 5.5 Where: 5b is the central deflection due to bending action Mb is the edge moment due to bending action Ob is the maximum central tensile stress due to bending action S is the section Modulus E is the Young's Modulus of the plate Wb is the total load taken in bending Pb is the effective pressure taken in bending A is the area of the plate b is the short width of the plate L is the length of the plate t is the thickness of the plate Ca, Cb are coefficients The values o f coefficients C a and Cb depend on the boundary conditions of the plate (simply supported or fixed) and the aspect ratio of the plate. Boundary Condition C a c b Simply supported 0.125' 0.14 Fixed 0.08 0.028 Table 5.1 Coefficients for flat plate bending action (Long plate AR>1) ' 47 CHAPTER 5: PREL IMINARY STRUCTURAL ANALYS I S Long Plate Membrane action (AR>1): 5. = 0.34 / , \ l / 3 ELt Wm=pmA CT„, = 0.37 yLt j -1/3 Where: 8m is the central deflection due to membrane action cym is the maximum central tensile stress due to membrane action Wm is the total load taken in bending pm is the effective pressure taken in bending Square Plate Bending action (AR=1): sh=cc Et' Mh=CdWh M, Where: Cc, Cd are coefficients Eq . 5.61 Eq. 5.7 Eq . 5.81 Eq. 5.91 Eq. 5.101 Eq. 5.11 The values of coefficients C c and Cd depend on the boundary conditions of the plate (simply supported or fixed) and the aspect ratio of the plate. Boundary Condition c c c d Simply supported (corners held down) 0.043 0.05 Fixed 0.014 0.052 Table 5.2 Coefficients for flat plate bending action (Square plate AR=1) ' 48 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S Square Plate Membrane action ( A R = 1): 8„ = 0.28 cr„, = 0.28 ELt J W } m 2 / 3 Lt) -1 /3 Eq. 5.121 Eq. 5.131 In reality any given plate w i l l resist the normal pressure through a combination of both membrane and bending action. B y equating the deflection due to membrane action and due to bending action, the portion of the applied loading taken in bending and in membrane action can be determined as shown below: (Whb3^ y ELt3 j = k. / , \ i / 3 v ELt j ELt 3 " b (ELt) 1/3 m AWh3^BWm ,W = Wh+Wm Wh3^-^ h c c This involves the solution of a cubic equation of the form given in equation 5.14. x3 + mx = n Eq. 5.14 1 7 A solution to cubic equations of this form was found in an online reference. The solution procedure is shown immediately following this text, and is included in the formatted spreadsheets used for analysis of flat plates. Notice that (a-b)3 + 3ab(a -b) = a3 -b3 Therefore, i f a and b satisfy lab = m and a3 - b3 = n then a - b is a solution of equation 5.14, substituting these expressions for m and n, into equation 5.14 yields equation 5.15. a"-no.3-m3121 = 0 Eq. 5.15 49 CHAPTER 5: PREL IM INARY STRUCTURAL ANALYS I S - J Equation 5.15 is a quadratic in a and is solved using equation 5.16 with the substitution g = a n±4n1 + An? Ill „ _ . r g = Eq. 5.16 5.1.2 Application The use o f these relatively simple analytical equations permits the development of computer programs which can systematically vary all input variables to fully graph the entire design space. This was achieved through the use of a combination of formatted spreadsheets and V B A macros in Microsoft Excel . Initially a formatted spreadsheet is created which calculates the maximum stress and deflection for given parameter values (E, W , t, L , b e t c . ) . Once this was completed, a V B A program was created which references this formatted spreadsheet. This program systematically varies the input parameter values and records the maximum stress and deflection corresponding to the particular combination of design variables. The program then creates a database from which all subsequent plots are generated, such as those shown in Figures 5.4 and 5.5, which show the maximum central tensile stress in flat plates of differing thickness and support conditions, subject to the design pressure of 4.2 kNm" . The formatted spreadsheet used is shown in Figure 5.3. The V B A program code can be found in Appendix B and tabulated results can be found in Appendix A . 50 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S PROJECT MegaWheel Cladding Panels DATE 8/1/2006 FILE flatplate.xls TIME 4:50 PM REF INPUT 1 Plate Width b = 2138 2138 [mm] Plate length L = 4677 4677 [mm] Plate thickness t = 3.175 3.18 [mm] Plate boundary conditions be = "F" F [SS,F] Normal pressure P = 0.0042 0.0042 [Nmm'2] Young's Modulus E = 70000 70000 [MPa} Runt Simply Supported (SS) Fixed (F) COMPUTATIONS 1 Geometric Properties Aspect ratio AR = L/b 2.19 Area A = L*b 9999426 [mm2] S = tA2/6 1.68 [mm2] Representative length bav = (L+b)/2 3407.50 [mm] High aspect ratio (>2) Solution of cubic governing load distribution Total load W = P*A 41998 [N] Deflection coefficient k_b_ss = 0.14 0.14 1 Deflection coefficient k_b_f = 0.028 0.028 1 Deflection coefficient k_b = IF(bc="SS",k_b_ss,k_b_f) 0.028 Deflection coefficient k_m = 0.34 0.34 1 Moment coefficient k2_b_ss = 0.125 0.125 1 Moment coefficient k2_b_f = 0.08 0.080 1 Moment coefficient k2_b = IF(bc="SS",k2_b_ss,k2_b_f) 0.080 Cubic coefficient A' = k_b*bA3/(E*L*tA3) 0.0261 Cubic coefficient B' = k_m*(b"3/(E*L*t))A(1/3) 0.7176 Cubic coefficient C = (A7B')A3 4.82E-05 Cubic coefficient m' = 1/C 2.07E+04 Cubic coefficient n' = W/C" 8.71 E+08 Cubic coefficient g' = n72+(SQRT(n'A2+4/27*m'A3))/2 8.71 E+08 2 Cubic coefficient g" = n72-(SQRT(n'A2+4/27*m'A3))/2 -3.80E+02 2 Cubic coefficient g = IF(g'>=0.0,g',g") 8.71 E+08 2 Cubic coefficient a" = gA(1/3) 9.55E+02 2 Cubic coefficient b" = m'/(3*a") 7.24E+00 2 Load taken by bending W_b = a"-b" 947.91 [N] 2 Load taken by membrane W_m = W-W_b 41049.68 [N] Figure 5.3 Analytical analysis of flat plate stress and deflection (1 of 2) 51 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S Bending action Moment/unit width M_b = k2_b*W_b*b/L 34.67 [Nmm"1] 1 Stress s_b = M_b/S 20.63 [MPa] 1 Deflection d_b = k_b*W_b*bA3/(E*L*tA3) 24.75 [mm] 1 Membrane action Tension stress s_m = 0.37*((W_m/(L*t))A(2/3))*EA(1/3) 30.04 [MPa] 1 Deflection d_m = k_m*(W_m*bA3/(E*L*t))A(1/3) 24.75 [mm] 1 Max stress high AR s_max_har = s_m+s_b = 50.67 [MPa] 1 Low aspect ratio (<2) Solution of cubic governing load distribution Deflection coefficient k'_b_ss = 0.043 0.04 1 Deflection coefficient k'_b_f = 0.014 0.01 1 Deflection coefficient k'_b = IF(bc="SS",k'_b_ss,k ,_b_f) 0.01 Deflection coefficient k'_m = 0.28 0.28 1 Moment coefficient k2'_b_ss = 0.05 0.050 1 Moment coefficient k2'_b_f = 0.052 0.052 1 Moment coefficient k2'_b = IF(bc="SS",k2'_b_ss,k2'_b_f) 0.052 Cubic coefficient AA' k'_b*bavA2/(E*tA3) 0.0726 Cubic coefficient BB' = k'_m*(bavA2/(E*t))A(1/3) 1.0467 Cubic coefficient C C = (AA7BB')A3 3.33E-04 Cubic coefficient mm' = 1/CC = 3.00E+03 Cubic coefficient nn' = W / C C "• 1.26E+08 Cubic coefficient 99' = nnV2+(SQRT(nn,A2+4/27*mm'A3))/2 1.26E+08 2 Cubic coefficient 99" = nn72-(SQRT(nn'A2+4/27*mm'A3))/2 -7.95E+00 2 Cubic coefficient 99 = IF(gg'>=0.0,gg',gg") 1.26E+08 2 Cubic coefficient aa" = ggA(1/3) 5.01 E+02 2 Cubic coefficient bb" = mm'/(3*aa") = 2.00E+00 2 Load taken by bending W_b' = aa"-bb" = 499.47 [N] 2 Load taken by membrane W_m' = W-W_b' 41498.12 [N] Bending action Moment/unit width M_b' = k2'_b*W_b' 25.97 [Nmm 1] 1 Stress s_b' = M_b7S 15.46 [MPa] 1 Deflection d_b' = k'_b*W_b'*bavA2/(E*tA3) 36.24 [mm] 1 Membrane action Tension stress s_m' = 0.28*((W_m7(bav*t))A(2/3))*EA(1/3) 28.28 [MPa] 1 Deflection d_m' = k'_m*(W_m*bavA2/(E*t))A(1/3) 36.11 [mm] 1 Max stress low AR s_max_lar = s_m'+s_b' = 43.74 [MPa] 1 Deflection defl = IF(AR<1.999,(d_b'+d_m')/2,d_b) 24.75 [mm] Max stress s_max = IF(AR<1,999,s_max_lar,s_max_har) = 50.67 [MPa] 1 Strength of Aluminum, Cedric Marsh, Alcan, 1983 2 http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Quadratic_etc_equations.html Figure 5.3 Analytical analysis of flat plate stress and deflection (2 of 2) 52 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S 50 0 a o 00 1.0 2.0 30 40 5.0 6.0 7.0 8.0 90 10.0 11 0 12.0 Aspect Ratio (L/b) Fig. 5.4 Variation in max. tensile stress with plate aspect ratio [t = 2/16", SS edges] 80.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 Plate aspect ratio AR (L/b) Fig. 5.5 Variation of max. tensile stress with plate aspect ratio [SS edges] 53 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S 80 0 | Plate aspect ratio AR (L/b) j Fig. 5.6 Variation of max. tensile stress with plate aspect ratio [Fix. edges] The maximum stress in the plate may not govern the design, as it is important to ensure the deflection of the plate upon loading does not degrade the aesthetics of the structure. By taking limiting values of central deflection and tensile stress, it is possible using the formatted spreadsheets, V B A programs and plots created, to interpolate the maximum plate area for a given aspect ratio. From the preceding analysis, very large plate areas are acceptable from a yielding standpoint, and hence it is likely that central deflection will be the limiting case for plate area (due to aesthetic considerations). 5.2 Curved plates: In order to satisfy the aesthetic demands, the panels may need to be curved, increasing the complexity of the structural model. The plates may be curved in one or two perpendicular directions. No analytical equations could be found which give the maximum stress or deflection of singularly or doubly curved plates subject to uniform normal pressure. Due to the geometry of 54 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S these plates they are subject to buckling instability, which is very likely to be the limiting case for design, as thin large area plates are desired in order to achieve light-weight panels. However analytical equations which predict the buckling pressure of singularly curved plates, simply supported or clamped along the un-curved edges and free at the curved edges were found in Timoshenko and Gere's 'Theory of Elastic Stability'. 3 They consider a strip of plate and idealize this strip as a curved beam or bar simply supported at either end. Unfortunately no analytical equations for buckling of doubly curved plates could be found in the literature, however this form of plate is unlikely to be a feasible option for this project, due to the complexity and cost of fabricating such panels. While it is intuitive that a curved plate which is curved in the short dimension w i l l be stiffer than a plate curved in the long direction, a plate which is curved in the long dimension has an aesthetic advantage over a plate which is curved in the short direction for a given plate area, as it can more closely approximate curvature in two directions due to the smaller straight dimension length. These wi l l be considered as separate cases with aspect ratios ranging from 1 to infinity, to avoid problems with scale (when plotting graphs with aspect ratio as one of the axes) i f considering one general plate form and varying the aspect ratio from 0 to infinity. Short edge curved Long edge curved Figure 5 .7 Single curvature plates 55 CHAPTER 5: PREL IM INARY STRUCTURAL ANALYS I S 5.2.1 Theory A curved bar with hinged ends and with a centre-line in the form of an arc of a circle subjected to a uniformly distributed pressure q w i l l buckle once the critical buckling pressure is reached. This buckling pressure is directly related to the form of the buckled shape which is governed by the geometric properties of the bar and the boundary conditions of support. Timoshenko and Gere identify two possible in-plane buckling mode shapes for such a bar; a shape with a point of inflection i f the rise ' a ' is large and symmetrical or 'snap-through' buckling i f the rise ' a ' is small (shallow arch). Snap-through buckling requires some axial strain to develop, whereas buckling with a point of inflection is the first mode of extensionless buckling. Equations are given in this reference for both forms of buckling, allowing each to be evaluated and the lowest buckling pressure taken as the limiting pressure. A third form of buckling was also considered; out of plane lateral buckling of the bar, however this is unlikely to dominate for all but the most extreme of plate dimensions. Buckling with a point of inflection : buckled shape A Figure 5.8 Buckling with a point of inflection <icr = 1 2 ( l - v 2 ) i v 3 l a Eh3 Eq . 5.17 3 Where: qcr is the buckling pressure 56 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S R is the radius of curvature a is half o f the included angle v i s the material Poisson's ratio E is Young's Modulus h is the plate thickness Snap-through buckling rise a Buckled shape Surface pressure q Figure 5.9 Snap-through buckling qcr =• 384 Elau ~5 ZT u = 1 + . f 4 ( l-ffl) 3 27 m2 m = 41 Aa2 Where: a is the rise of the plate L is the curved length of the plate I is the section 2 n d moment of area A is the sectional area Eq .5 .18 J Eq. 5.19 j Eq. 5.20 3 57 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S "3 Lateral buckling : /- Surface pressure q A \ qR \ ^ // q R R / Figure 5.10 Lateral buckling R2 a2[rc2 +a2(EIx/C)] Where: a is the included angle of the plate Ix is the section 2 n d moment of area C is the warping constant 5.2.2 Application Again, the use of these relatively simple analytical equations permits the development of computer programs which can systematically vary all.input variables to fully graph the entire design space. This was achieved through the use of a combination of formatted spreadsheets and V B A macros in Microsoft Excel . From the tabulated results, several graphical plots similar to that shown in Figure 5.12 were produced for different plate thickness and included angles (b/R). These figures show buckling ratio against plate area, where a ratio of less than 1.0 corresponds to no buckling instability for the design pressure of 4.2 kNm" . The formatted spreadsheet used is shown in Figure 5.11. The V B A program code is shown in Appendix B and tabulated results can be found in Appendix A . 58 Eq. 5.21 3 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S PROJECT PANASONIC MEGAWHEEL AL CLADDING DATE 8/1/2006 FILE Short edge curved.xls TIME 4:10 PM IREF INPUT 1 Plate Width b = 1600 = 1600 [mm] Plate length L = 5524 = 5524 [mm] Plate thickness t = 4.7625 = 4.76 [mm] Angle (b/R) ang = 1 = 1.00 [rad] Normal pressure P = 0.0042 0.0042 [Nmm2] Young's Modulus E = 70000 = 70000 [MPa] Shear Modulus G = 26000 = 26000 [MPa] Poisson's ratio nu = 0.3 = 0.3 Run1 1 Theory of Elastic Stability, Timoshenko and Gere, 2nd edition, 1961 COMPUTATIONS 1 Geometric Properties Aspect ratio AR = L/b 3.45 Plan area A = L*b 8838400 [mm2] Sectional area Asec = L*t 26308 [mm2] Second moment of area I = L*t*3/12 4.97E+04 [mm4] Torsional stiffness J = L*tA3/3 1.99E+05 [mm4] Torsional Rigidity C = G*J 5.17E+09 Representative length bav = (L+b)/2 3562.00 [mm] Radius R = b/ang = 1600 [mm] alpha alpha = ang/2 = 0.50 [rad] Arch rise arise = R*(1-COS(alpha)) 195.87 [mm] Projected horizontal width bho 2*R*SIN(alpha) 1534.16 [mm] Buckling with inflection point dum1 dum1 = 12*RA3*(1-nuA2) 4.47E+10 Buckling pressure q' E*tA3*(PI()A2/alphaA2-1 )/dum1 6.50E-03 [Nmm2] Buckling ratio bu_rat = P/q' 0.65 Buckling without inflection (snap-through) m m = 4*l/(Asec*ariseA2) = 1 97E-04 dum22 dum22 = (4/27)*(((1-m)A3)/mA2) 3.81 E+06 dum2 dum2 = IF(dum22<0,1 E99,dum22) 3.81 E+06 Buckling pressure 2 q'2 = (E*l"arise/bhoA4)-(1 +(dum2)A0.5)*384/5 = 1.85E+04 [Nmm"1 Buckling ratio2 bu_rat2 = P/q'2 0.000000227 Lateral buckling dum3 dum3 = PI()A2+angA2*(E'l/C) 10.54 dum4 dum4 = (PI()A2-angA2)A2 78.67 Buckling pressure 3 q'3 = E*l*dum4/(RA3*angA2*dum3) 6.34E+00 Buckling ratio3 bu_rat3 = P/q'3 0.000662 Buckling pressure q MIN(q',q'2,q'3) 0.00650486 [Nmm2] Buckling ratio bu_rat_m = P/q 0.65 Buckling check bu_check IF(bu_rat_m<1.0,"Ok !!","Buckling H") Ok !! Figure 5.11 Analytical analysis of curved plate buckling 59 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S Figure 5.12 Variation in buckling ratio with plate area for several different aspect ratios (t = 2/16". b/R = 0.5] From Figure 5.12 and similar plots it is possible to interpolate the plate area which will have a buckling ratio of 1.0 for a given plate thickness, plate aspect ratio and included angle. The resulting plots of allowable plate area against aspect ratio for different included angles and plate thickness' are shown in Figures 5.13-5.15 (short edge curved). 60 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S -b/R = 0.1 - b/R = 0.5 b/R = 2.0 p = 4.2 kNmm | < 3.0E+06 -1.0E+06 4 5 6 Aspect Ratio (L/b) 8 9 10 Figure 5.13 Plate area at which buckling occurs for t = 1.5875mm (1/16") -b/R = 0.1 -b/R = 0.5 b/R = 2.0 p = 4.2 kNmm 2 1.5E+07 -< 5.0E+06 Figure 5.14 Plate area at which buckling occurs for t = 3.175mm (2/16") 61 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S 3.0E+07 i -b/R = 0.1 - b/R = 0.5 b/R = 2.0 p • 4.2 kNmm 2 4 5 6 Aspect Ratio (L/b) 10 Figure 5.15 Plate area at which buckling occurs for t = 4.7625mm (3/16") From figures 5.13 - 5.15, it can be seen that as the plate aspect ratio increases and curvature increases, the plate is able to withstand a higher pressure without buckling for a given plate area, hence a larger plate area is feasible. This is intuitive as the larger the curvature, the more arching action and the higher the aspect ratio for a given area the shorter the width of the panel increasing it's stiffness. A similar process was followed for a plate, where the curved edges are longer than the un-curved edges, where as expected a smaller plate area is required for a given plate thickness and included angle. Figures 5.16 - 5.18 show the plate area at which buckling occurs for the three plate thicknesses and included angles considered. For some combinations of included angle and aspect ratio, buckling ratios could not be calculated due to restrictions, as noted in the theory (R/t less than 10). 62 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S —•—L/R = 0.1 = 0.5 L/R = 2.0 p = 4.2 k N m m - 2 • 3 4 Aspect Ratio (L/b) Figure 5.16 Plate area at which buckling occurs for t = 1.5875mm (1/16") 2.0E+06 -£ to < 4 5 Aspect Ratio (L/b) - • - L / R = 0.1 - » - L / R = 0.5 L/R = 2.0 p a 4.2 k N m m - 2 Figure 5.17 Plate area at which buckling occurs for t = 3.175mm (2/16") 63 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S 6.0E+06 5.0E+06 4.0E+06 E "2 3.0E+06 1.0E+06 O.OE+00 - • - L / R = 0.1 * L/R • 05 L/R = 2.0 p = 4.2 kNmnv 2 if* • \ \ • 1 • 0 1 2 3 4 5 6 7 8 9 10 Aspect Ratio (L/b) Figure 5.18 Plate area at which buckling occurs for t = 4.7625mm (3/16") A s mentioned previously, while the plate curved in the long dimension has a structural disadvantage compared with a plate curved in the short dimension, it has an aesthetic advantage. Without more in depth analysis of the aesthetics of the Wheel and a method of quantifying the aesthetic quality of discretization of a curved line by straight line segments it is difficult to determine which plate form provides the best solution. 5.3 Panel sizes determined from typical sections: Figures 5.19 - 5.21, show typical sections taken from the preliminary main structure design A u t o C A D drawings (electronic drawings to scale). The straight lines show the outline of the geometric shape desired, originally scaled from architectural drawings. A continuous spline was drawn using a computer aided design ( C A D ) package, through all vertices, subsequently; discrete curved lines were drawn from vertex to vertex, which fit as closely as possible the continuous spline, in order to allow determination of representative values of width and radius of curvature. 64 CHAPTER 5: PREL IM INARY STRUCTURAL ANALYS I S (3164) M a i n structural framing Continuous spline — Curved panel segments • Segment vertices 65 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S Figure 5.19 12:00 C L O C K POSITION (Segment radius shown in brackets) (4688) •••••• — — Main structural framing Continuous spline Curved panel segments • Segment vertices Figure 5.20 3:00 C L O C K POSITION (Segment radius shown in brackets) C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S Main structural framing Continuous spline «——-———• Curved panel segments • Segment vertices Figure 5.21 5:00 C L O C K POSITION (Segment radius shown in brackets,) 67 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S Taking the segment lengths and radii of curvature from each of the three representative sections, it is possible to get an indication of the resulting panel areas, aspect ratios and included angles for comparison with the indicated sizes from the preliminary structural analysis (shown in Table 5.3). This also requires an assumption of the plate length (long un-curved edge L ) , two values are assumed for the subsequent tables; 8000mm and 4000mm. t = 1.5875mm t = 3.175mm t= 4.7625mm 1 AR b/R = 0.1 b/R = 0.5 b/R = 2.0 b/R = 0.1 b/R = 0.5 b/R = 2.0 b/R = 0.1 b/R = 0.5 b/R = 2.0 1 7.5E+04 2.8E+05 5.8E+05 3.2E+05 9.9E+05 2.3E+06 7.5E+05 2.2E+06 5.0E+06 2 1.6E+05 4.9E+05 1.1 E+06 6.8E+05 2.0E+06 4.5E+06 1.5E+06 4.4E+06 1.0E+07 2.9 2.4E+05 7.0E+05 1.7E+06 9.6E+05 2.8E+06 6.5E+06 2.2E+06 6.2E+06 1.5E+07 3.8 3.2E+05 9.3E+05 2.2E+06 1.3E+06 3.8E+06 8.8E+06 2.9E+06 8.4E+06 2.0E+07 4.5 3.8E+05 1.1 E+06 2.6E+06 1.5E+06 4.3E+06 1.0E+07 3.4E+06 9.6E+06 2.3E+07 5.3 4.5E+05 1.3E+06 3.1 E+06 1.8E+06 5.1 E+06 1.2E+07 4.0E+06 1.2E+07 2.7E+07 6.4 5.3E+05 1.6E+06 3.7E+06 2.2 E+06 6.2E+06 1.5E+07 4.8E+06 1.4E+07 3.3E+07 7.9 6.6E+05 1.9E+06 4.5E+06 2.7E+06 7.8E+06 1.8E+07 5.9E+06 1.8E+07 4.1E+07 10 8.4E+05 2.5E+06 5.7E+06 3.5E+06 9.8E+06 2.3E+07 7.6E+06 2.2E+07 5.4E+07 Table 5.3 Maximum plate areas (mm 2) according to analytical buckling analysis Table 5.4 show the required plate geometries by assuming the panels span directly between the vertices created by the longitudinal members of the main structure (with a plate length of 4m or 8m). If the resulting combinations of plate area, aspect ratio and included angle are compared with the maximum values given by the preliminary analytical buckling analysis (Table 5.3), it becomes apparent that the required areas are much larger than the upper limit for buckling. LT = 8000 [mm] L 2 = 4000 [mm] Segment bhor Radius b b/R ART A2 AR2 [mm] [mm] [mm] [mm2] [mm2] O c 1 2755 3818 2819 0.74 2.26E+07 2.84 1.13E+07 1.42 _o o o 2 2750 3164 2845 0.90 2.28E+07 2.81 1.14E+07 1.41 o o '35 o ^ 3 3149 2982 3317 1.11 2.65E+07 2.41 1.33E+07 1.21 cj 4 7549 37965 7561 0.20 6.05E+07 1.06 3.02E+07 0.53 J£ 1 4180 9281 4216 0.45 3.37E+07 1.90 1.69E+07 0.95 o 0 c o 2 4960 8640 5031 0.58 4.02E+07 1.59 2.01 E+07 0.80 u !_ '35 3 4950 4688 5215 1.11 4.17E+07 1.53 2.09E+07 0.77 © o Q. 4 5668 7959 5795 0.73 4.64E+07 1.38 2.32E+07 0.69 CO 5 7710 28582 7733 0.27 6.19E+07 1.03 3.09E+07 0.52 J£ 1 9805 9197 10341 1.12 8.27E+07 0.77 4.14E+07 0.39 u o C O 2 7497 7741 7826 1.01 6.26E+07 1.02 3.13E+07 0.51 o '55 3 7495 7110 7895 1.11 6.32E+07 1.01 3.16E+07 0.51 3 O a 4 8599 13946 8741 0.63 6.99E+07 0.92 3.50E+07 0.46 5 11284 45862 11312 0.25 9.05E+07 0.71 4.52E+07 0.35 Table 5.4 Required plate geometry determined from representative sections (for different assumed plate lengths L ) 68 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S A graphical comparison of the required plate areas according to tables 5.4 and limits provided by analytical buckling analysis is shown in Figure 5.22 for a plate length of 8m. The main plot in Figure 5.22 is 3 dimensional, with the axes corresponding to; aspect ratio (AR) , included angle (b/R), and plate area (A) . The smaller plots are 2 dimensional projections of the 3 dimensional main plot, taken perpendicularly to each other. Area /mm A2 Analytical limits + Segments tabulated * le+08 8e+07 6e+07 4e+07 2e+07 Area !mm*2 le+08 9eK>7 8e*07 7e*07 6e+07 5e-<-07 4e*07 3e+07 2e+07 le+07 0 Area /mmA2 10 5 4 AR Figure 5.22 Graphical comparison of required (8m straight edge plate length) and analytical buckling limits of plate geometries 69 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S From Figure 5.22, it is obvious that there is a large difference between the allowable plate sizes and the required plate sizes as calculated for a length of 8000 mm. A similar conclusion is drawn when considering a plate length of 4m (further analysis shows no reasonable plate length can be accommodated). This means that it is not possible to span between the main longitudinal members of the Wheel structure with the current main structure design. There are two options; the use of an intermediary supporting system to reduce the distances any given plate is required to span and/or the use of longitudinal and/or radial stiffeners to increase the allowable plate dimensions (by increasing plate resistance to buckling). To investigate the possibility of using an intermediary supporting system, similar tables to those shown in Tables 5.5 are constructed, assuming that there is a supporting grillage that reduces the spanning distance for each segment. Table 5.5 shows the required plate dimensions if it is assumed that there is an intermediary supporting grillage which reduces the spanning distance of the curved edges by half. The result is that for all sections, the included angle range of segments (b/R) is reduced to values approximately in the range 0.1 - 0.5, with the required plate areas of similar magnitude to the allowable plate areas according to analytical structural analysis. Li = 8000 [mm] L2 = 4000 [mm] Segment bhor [mm] Radius [mm] b [mm] b/R A [mm2] AR V A 2 [ran2] AR2 o c 1 1378 3818 1385 0.36 1.11E+07 5.78 5.54E+06 2.89 o o o 2 1375 3164 1386 0.44 1.11E+07 5.77 5.54E+06 2.89 o © in o ^ 3 1574 2982 1593 0.53 1.27E+07 5.02 6.37E+06 2.51 4 3775 37965 3776 0.10 3.02E+07 2.12 1.51E+07 1.06 1 2090 9281 2094 0.23 1.68E+07 3.82 8.38E+06 1.91 o o c o 2 2480 8640 2489 0.29 1.99E+07 3.21 9.95E+06 1.61 o "a> 3 2475 4688 2505 0.53 2.00E+07 . 3.19 1.00E+07 1.60 o C O O a. 4 2834 7959 2849 0.36 2.28E+07 2.81 1.14E+07 1.40 5 3855 28582 3858 0.13 3.09E+07 2.07 1.54E+07 1.04 J£ 1 4903 9197 4963 0.54 3.97E+07 1.61 1.99E+07 0.81 o o c o 2 3749 7741 3786 0.49 3.03E+07 2.11 1.51E+07 1.06 u Q '.s '3> 3 3748 7110 3792 0.53 3.03E+07 2.11 1.52E+07 1.05 o I O o Q . 4 4299 13946 4317 0.31 3.45E+07 1.85 1.73E+07 0.93 5 5642 45862 5645 0.12 4.52E+07 1.42 2.26E+07 0.71 Table 5.5 Required plate geometry determined from representative sections (curved edge span reduced by a factor of 2) 70 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S Figure 5.23 shows the graphical comparison between the limit of panel sizes according to analytical structural analysis and the required sizes according to tables 5.5 for a panel length of 4m. As can clearly be seen, none of the panels are feasible, while the panel areas are achievable, the required included angle and aspect ratio results in buckling of these plates subject to the design normal pressure. Analytical limits + Segments tabulated * Area JrnmA2 Figure 5.23 Graphical comparison of required (curved edge span reduced by a factor of 2, 4m straight edge length) and analytical buckling limits of plate geometries If an intermediary supporting framework is created which wi l l introduce connection points which quarter the required spanning distance of the curved edges of the plates (b), compared with spanning directly between the longitudinal members in the present main framework structure, even with plates only 2000mm in length, only a few of the proposed plates are feasible from a structural standpoint (see Table 5.6 and Figure 5.24). The remainder of the plates would need to be stiffened. For those plates which do not require stiffening in order to resist the environmental loading, some form of connection to the main structure is required. Stiffeners bonded/fastened to the underside of the plates would provide ideal locations for connection to the main structure. 71 CHAPTER 5: PREL IM INARY STRUCTURAL ANALYS I S L, = 4000 [mm] L 2 = 2000 [mm] Segment Radius b b/R ART A 2 A R 2 [mm] [mm] [mm] [mm2] [mm2] O c 1 689 3818 690 0.18 2.76E+06 5.80 1.38E+06 2.90 _o u o 2 688 3164 689 0.22 2.76E+06 5.81 1.38E+06 2.90 o o o 3 787 2982 789 0.26 3.16E+06 5.07 1.58E+06 2.53 4 1887 37965 1887 0.05 7.55E+06 2.12 3.77E+06 1.06 JC 1 1045 9281 1046 0.11 4.18E+06 3.83 2.09E+06 1.91 u o c o 2 1240 8640 1241 0.14 4.96E+06 3.22 2.48E+06 1.61 o '« 3 1238 4688 1241 0.26 4.96E+06 3.22 2.48E+06 1.61 o o a 4 1417 7959 1419 0.18 5.68E+06 2.82 2.84E+06 1.41 CO 5 1927 28582 1928 0.07 7.71 E+06 2.07 3.86E+06 1.04 1 2451 9197 2459 0.27 9.83E+06 1.63 4.92E+06 0.81 u o c o 2 1874 7741 1879 0.24 7.52E+06 2.13 3.76E+06 1.06 u '3 3 1874 7110 1879 0.26 7.52E+06 2.13 3.76E+06 1.06 © o a 4 2150 13946 2152 0.15 8.61 E+06 1.86 4.30E+06 0.93 5 2821 45862 2821 0.06 1.13E+07 1.42 5.64E+06 0.71 Table 5.6 Required plate geometry determined from representative sections (curved edge span reduced by a factor of 4) 72 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S Analytical limits + Segments tabulated * 2.5e+07 2e+07 1.5e+07 le+07 5e+0G 0 Area *nmA2 2.5e+07 2e+07 1.5e+07 le+07 5e+06 0 0 0.2 0 4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2 <biR) Figure 5.24 Graphical comparison of required (curved edge span reduced by a factor of 4, 4m straight edge length) and analytical buckling limits of plate geometries 73 C H A P T E R 5: P R E L I M I N A R Y S T R U C T U R A L A N A L Y S I S 5.4 Conclusions from preliminary structural analysis: From analytical buckling analysis, maximum allowable plate areas are achieved with thick plates of high aspect ratio. In the absence of any quantifiable information regarding the relative aesthetics of the two competing plate configurations (long side or short side curved), subsequent analysis will focus on plates curved in the short direction only, due to the significant structural advantage this plate configuration possesses. With the current structural configuration of the main structure, the cladding panels will require an intermediate supporting structure, which will reduce the required panel area of the individual panels used to enclose the main structure. This supporting structure must reduce the required spanning distance of segments by approximately 75% (curved edge length b) and will also serve as the connection points for the panels. It may be possible to reduce or remove the need for intermediary supporting structure, through stiffening of individual panels. Due to the added complexity of introducing a supporting structure in the form of stiffeners to each panel, analytical solutions are no longer adequate, and subsequent analysis must include the use of finite elements. 74 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S CHAPTER 6: FINITE ELEMENT (FE) ANALYSIS A s the complexity of the structural model is increased, analytical equations are no longer available which can give accurate predictions of buckling pressure or stress upon loading. The analysis of these models requires the use of F E software such as ANSYS®. It is well known that the use of F E can lead to significant errors in design, due to the tendency of treating such analytical tools as 'black boxes'. In order to safeguard against this, all previous results obtained using proven analytical/empirical derivations were used to validate the ANSYS® models and analysis techniques used for further investigation. In addition, convergence checks of all F E models were performed (through mesh density variation). In order to ensure repeatability and allow others to fully follow the steps taken during the F E analysis conducted as part of this thesis project, ANSYS® input files, written in ANSYS® programmers design language ( A P D L ) were created in order to define model geometry, meshing and analysis type. This also allows design optimization to be implemented during advanced analysis. A l l ANSYS® input files developed during the course of this thesis can be found in Appendix C. For all finite element models, the aluminum cladding panels are meshed using ANSYS® ' S H E L L 6 3 ' elements. The geometry, node locations, and the coordinate system for this element are shown in Figure 6.1. Figure 6.1 ' S H E L L 6 3 ' Geometry 1 75 CHAPTER 6: FINITE E LEMENT ANALYS I S The 4 S H E L L 6 3 ' element has bending and membrane capabilities and allows both in-plane and normal loads. The element has six degrees of freedom at each node: translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z-axes. Stress stiffening (sstiff) and large deflection capabilities are included, and a consistent tangent stiffness matrix option is available for use in large deflection (finite rotation) analyses. The element is defined by four nodes, four thicknesses, an elastic foundation stiffness, and the orthotropic material properties. Orthotropic material directions correspond to the element coordinate directions. 6.1 FE Validation: 6.1.1 Flat plates A typical finite element model of a flat plate subject to uniform normal pressure is shown in Figure 6.2. The mesh density o f this model has been reduced for clarity. ELEMENTS * *• m f I A U G 2 200 6 1 6 : 5 1 : 0 8 PRE 3 MEGAWHEEL 05 Figure 6.2 Typical flat plate finite element model [Units of Nmm] 76 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S A comparison of A N S Y S nonlinear large deflection values for central deflection and tensile stress in simply supported flat plates with values calculated previously using analytical equations is shown in Tables 6.1 and 6.2. A [mm2] 10,000,000 t [mm] 1.5875 (1/16") Analytical Ansys FE L b AR A a A a [mm] [mm] [mm] [MPa] [mm] [MPa] 10000.0 1000.0 10.0 11.4 35.8 11.651 34.97 8899.9 1123.6 7.9 13.3 37.7 8017.9 1247.2 6.4 15.3 39.6 15.529 38.865 7294.9 1370.8 5.3 17.4 41.4 6691.5 1494.4 4.5 . 19.5 43.3 19.772 42.703 6180.3 1618.0 3.8 21.7 45.1 5741.7 1741.6 3.3 23.9 46.9 24.31 46.44 5361.2 1865.2 2.9 26.2 48.7 5028.0 1988.9 2.5 28.5 50.4 28.932 50.284 4733.8 2112.5 2.2 30.9 52.1 4472.1 2236.1 2.0 33.4 53.8 33.541 54.37 3162.3 3162.3 1.0 43.6 50.9 44.038 49.32 Table 6.1 Comparison of analytical and F E predictions of central deflection and stress (t = 1/16", SS edges) [* for AR=1.0 actual max stress is 57.635 M P a , does not occur at centre] A [mm2] 1,000,000 t [mm] 4.7625 (3/16") Analytical Ansys FE L b A R A a A [mm] [mm] [mm] [MPa] [mm] [MPa] 3162.3 316.2 10.0 0.7 14.0 0.6979 13.78 2814.4 355.3 7.9 1.1 16.9 2535.5 394.4 6.4 1.4 19.3 1.424 19.3 2306.9 433.5 5.3 1.8 21.1 2116.0 472.6 4.5 2.2 22.5 2.158 22.12 1954.4 511.7 3.8 2.6 23.4 1815.7 550.8 3.3 3.0 24.1 3.103 24.45 1695.4 589.8 2.9 3.4 24.7 1590.0 628.9 2.5 3.8 25.1 3.842 24.98 1497.0 668.0 2.2 4.2 25.4 1414.2 707.1 2.0 4.6 25.6 4.73 25.65 1000.0 1000.0 1.0 6.1 22.6 6.376 21.76 Table 6.2 Comparison of analytical and F E predictions of central deflection and stress (t = 3/16", SS edges) [* for AR=1.0 actual max stress is 22.557 M P a , does not occur at centre] The correlation between A N S Y S large deflection analysis and values obtained using the analytical equations in chapter 5 is excellent for both the central deflection and stress (the central values are maximum for plates with an aspect ratio greater than or equal to 2.0). The comparison is shown graphically in Figures 6.3 and 6.4 for plate thicknesses of 1/16" and 3/16". 77 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S 50.0 40.0 E E, f 30.0 c c .2 20.0 <p o 0) 3= <D •a x 10.0 n 0 0 -•— Analytical Ansys FE p • 4.2 kNm ; ^ 1 1 b \ 0.0 2.0 4.0 6.0 8.0 Aspect Ratio AR (L/b) 10.0 12.0 Figure 6.3 Comparison of analytical and F E central deflection prediction (SS flat plate) [A= 10,000,000 mm 2 1 = 1/16"] 70.0 i 60.0 [MP 50.0 -</> ( A <B £ 40.0 '35 30.0 c a 2 20.0 -c a> O 10.0 0.0 0 0 * • p • 4.2 kNnv : L 2.0 4.0 6.0 8.0 Aspect Ratio AR (L/b) -•—Analytical Ansys FE • Ansys FE' 10.0 12.0 Figure 6.4 Comparison of analytical and F E central tensile stress (SS flat plate) [A= 10,000,000 m m 2 1 = 1/16"] * For an aspect ratio of 1.0 the maximum stress does not occur at the centre of the plate. The circular data point on Figure 6.4 represents the maximum tensile stress in the plate according to A N S Y S Nonlinear large deflection analysis (similarly for Figure 6.6). 78 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S 8.0 0.0 -I 1 1 1 1 1 1 0.0 2.0 4.0 6.0 8.0 10.0 12.0 Aspect Ratio AR (L/b) 2 Figure 6.5 Comparison of analytical and F E central deflection prediction [A=l,000,000 mm , t = 3/16", SS edges] 35.0 0.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 Aspect Ratio AR (L/b) Figure 6.6 Comparison of analytical and F E central tensile stress [A = 1,000,000 mm , t = 3/16", SS edges] 79 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S For plates with fixed edges, the comparison between A N S Y S F E and the analytical solutions is again excellent with regard to the central deflection, however the central tensile stress prediction is poor, with the A N S Y S F E values significantly lower than those predicted by the theory (see Table 6.3 and Figures 6.7 - 6.8). A [mm2] 10,000,000 t [mm] 3.175 Analytical Ansys FE L b AR A a A a (edge) a (center) [mm] [mm] [mm] [MPa] [mm] [MPa] [MPa] 10000 1000 10.00 8.54 48.85 8.7 46.1 25.9 8978 1114 8.06 10.01 48.81 8146 1228 6.64 11.51 48.78 11.6 53.6 28.7 6872 1455 4.72 14.62 48.91 5942 1683 3.53 17.87 49.29 4677 2138 2.19 24.76 50.67 24.9 67.3 34.4 4441 2252 1.97 35.76 44.45 4033 2479 1.63 35.14 45.55 3694 2707 1.36 34.75 46.26 3408 2935 1.16 34.55 46.65 3162 3162 1.00 34.50 46.77 34.7 72.5 37.1 Table 6.3 Comparison of analytical and F E predictions of central deflection and stress [t Fix . edges] 2/16" 40.0 35.0 E 30.0 -£ 25.0 (cen 20.0 c o o o 15.0 -*— (t> T3 Max 10.0 Max 5.0 0.0 0.0 . p = 4.2 k N m : -•—Analytical Ansys FE 2.0 4.0 6.0 8.0 A s p e c t Ratio AR (L/b) 10.0 i2.q Figure 6.7 Comparison of analytical and F E central deflection prediction [A = 10,000,000 mm , t = 2/16", F ix . Edges] 80 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S 80.0 70.0 „ 60.0 (S Q. s " 50.0 o c S 40.0 * -•—Analytical Ansys FE (centre) • Ansys FE (edge) 4.0 6.0 8.0 Aspect Ratio AR (L/b) 10.0 12.0 Figure 6.8 Comparison of analytical and F E central tensile stress [A = 10,000,000 mm 2 , t = 2/16", Fix . Edges] The significantly lower values of central tensile stress as predicted by A N S Y S © nonlinear analysis are a cause of concern. The analytical solution for determining the deflection and stress stems from the determination of the proportion of the load taken in bending and membrane action. Considering the fact that the deflection predictions are so close and the F E values represent converged models, suggests that the discrepancy does not arise due to poor F E predictions, but rather from errors in the analytical coefficients used to determine the maximum stress in the plate. 6.1.2 Curved plates For the curved plates, no analytical equations could be found which gave the deflection or stress in the plate subject to a uniform normal pressure. The predicted buckling pressure calculated from the theory presented in chapter 5, was used to validate ANSYS® eigenvalue buckling predictions and large deflection non linear buckling analysis. For eigenvalue buckling analysis, the buckling problem is formulated as the eigenvalue problem shown in equation 6.1. 8! C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S ( M + >US"]M=(Q) E q . 6 . 1 , S Where [K] is the stiffness matrix, [S] is the stress stiffness matrix, A,J is the i eigenvalue and is the i t h eigen-vector of displacements. This eigenvalue problem is solved using one of several available methods such as the reduced, subspace, block Lanczos, un-symmetric, damped and Q R damped methods. Figure 6.9 shows a typical F E model of a curved plate (single curvature, mesh density reduced for clarity). MEGAWHEEL 05 Figure 6.9 Typical curved plate Finite Element model (Units of Nmm) Figures 6.10 show the first four buckled mode shapes of a curved plate 800mm wide (curved edge), 5524 mm long, 2/16" thick and with an included angle (b/R) of 0.1. The first buckled mode shape obtained from the A N S Y S © eigenvalue analysis is consistent with the predicted mode shape of extensionless buckling with a point of inflection obtained from the analytical 82 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S equations. The 'frequency' (freq) listed in Figures 6.10 and subsequent figures showing eigenvalue buckling mode shapes, actually refers to the ratio of buckling pressure to the design pressure of 4.2 kNmm"", with a value less than 1.0 corresponding to buckling. AN NODAL S O L U T I O N *>t : \ A U G 9 2005 Mode 1 ' • *j SKN =-.999993 y NODAL S O L U T I O N ffc. /VN 5F-36U, Mode 2 \ -i -.999993 -.53555 -.111107 .33333 6 -,777771 -.333329 .111114 .335537 1 M E G A W H E E L * 05 -.999994 - . ' 5555] - . 11110B .333335 . ' -.777772 -.333329 .111114 .353557 1 M E G A H H E E L * OS AN NODAL SOLUTION -«8fc A U G 3 ZD05 3TEe=i mk •- 10 = 34 = 14 SUB =3 a 2 « ,AVS' Mode 3 V DMX =1.001 / ; V 3KN =-.99999 3KX =1 ' AN NODAL S O L U T I O N , « K - \ AUG 9 Z005 7^T\ \ ^,34:40 F1-",™., Mode 4 A * - J DMI = 1 . 0 0 1 • ; V 3MN =-.999899 EfKX =1 / . ,:• i 'S^:-M .11111 . .333337 .777779 -.7777G9 -.333327 .111115 .55555B 1 MEGAWHEEL*OS -.999899 -.555477 -.1UQ55 .333367 .777789 ' -.777688 -.333266 .111156 .555578 1 M E G A W H E E L * O S Figures 6.10 Buckling mode shapes for a curved plate (SS straight edges, 0.8m x 5.5524m, 2/16" thick) Tables 6.4 and 6.5 compare the predicted buckling pressures of curved plates simply supported along the un-curved edges according to A N S Y S eigenvalue buckling analysis and the analytical equations presented previously. Plate geometries used for comparison were chosen in order to cover the entire design space. Not all o f the comparisons are shown, however the data presented in Tables 6.4 and 6.5 are representative of the all the data. From these tables it can be seen that the A N S Y S eigenvalue predictions are consistently conservative relative to the analytical prediction. However there is no trend with regard to the 83 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S magnitude of the difference with aspect ratio, included angle, thickness or area of the plates considered. |A = 100,000 [mm'] b/R = 0.1 | Analytical Eigenvalue analysis L b AR Radius R/t buck, press buck, ratio buck, press buck, ratio % diff [mm] [mm] [mm] [Nmm"'] (Nmm"z] " A •',-<• t = I .SiWfmm 1000.0 100.0 10.0 1000.0 629.9 0.101 0.0 0.130 0.032 -28.6 729.5 137.1 5.3 1370.8 863.5 0.039 0.1 0.046 0.092 -15.8 502.8 198.9 2.5 1988.9 1252.8 0.013 0.3 0.014 , 0.303 -7.9 316.2 316.2 1.0 3162.3 1992.0 0.003 1.3 0.003 1.282 -2.4 t = 4 76251mm H8ilililllif!!!ti 1000.0 100.0 10.0 1000.0 210.0 2.733 0.0 3.302 0.001 -20.8 729.5 137.1 5.3 1370.8 287.8 1.061 0.0 1.082 0.004 -2.0 502.8 198.9 2.5 1988.9 417.6 0.347 0.0 0.369 0.011 -6.3 316.2 316.2 1.0 3162.3 664.0 0.086 0.0 0.106 0.040 -22.5 |A= 100,000 [mm'] b/R = 0.5 | Analytical Eigenvalue analysis L b AR Radius R/t buck, press buck, ratio buck, press buck, ratio % diff [mm] [mm] [mm] [Nmm*] [Nmm'z] t = 1.5875 [mm 1000.0 100.0 10.0 200.0 126.0 0.503 0.008 0.524 0.008 -4.2 729.5 137.1 5.3 274.2 172.7 0.195 0.022 0.202 0.021 -3.5 316.2 316.2 1.0 632.5 398.4 0.016 0.264 0.016 0.259 -1.9 t = 4.7625 [mm 1000.0 100.0 10.0 200.0 42.0 13.582 0.000 . 15.255 0.000 -12.3 729.5 137.1 5.3 274.2 57.6 5.272 0.001 5.674 0.001 -7.6 316.2 316.2 1.0 632.5 132.8 0.429 0.010 0.437 0.010 -1.8 |A= 100,000 [mm'] b/R = 2.0 | Analytical Eigenvalue analysis L b AR Radius R/t buck, press buck, ratio buck, press buck, ratio % diff [mm] [mr 1 [mm] [Nmm2] [Nmm'l • = 1.5875 [mm 1000 0 100.0 10.0 50.0 31.5 1.820 0.0 2.046 0.002 -12.4 729.5 137.1 5.3 68.5 43.2 0.706 0.0 0.790 0.005 -11.8 316.2 316.2 1.0 158.1 99.6 0.058 0.1 0.064 0.066 -10.8 t = 4.7625 [mm 1000.0 100.0 10.0 50.0 10.5 49.133 0.0 55.403 0.000 -12.8 729.5 137.1 5.3 68.5 14.4 19.074 0.0 21.335 0.000 -11.9 316.2 316.2 1.0 158.1 33.2 1.554 0.0 1.712 0.002 -10.2 Tables 6.4 Comparison of analytical and FE eigenvalue buckling values (SS straight edges, A = 100,000 mm2) 84 CHAPTER 6: FINITE ELEMENT ANALYSIS |A = 40,000,000 [mrrT] b/R = 0.1 Empirical/Analytical eq. Eigenvalue analysis L b AR Radius R/t buck, press buck, ratio buck, press buck, ratio % diff [mm] [mm] [mm] [Nmm-2] [Nmm-2] t' =H.58f5j[rnm 20000.0 2000.0 10.0 20000.0 12598.4 0.000 331.950 0.000 322.953 -2.8 14589.8 2741.6 5.3 27416.4 17270.2 0.000 855.096 0.000 832.838 -2.7 6324.6 6324.6 1.0 63245.6 39839.7 0.000 10497.188 0.000 10294.118 -2.0 t = 4 7625 [mm 20000.0 2000.0 10.0 20000.0 4199.5 0.000 12.294 0.000 11.917 -3.2 14589.8 2741.6 5.3 27416.4 5756.7 0.000 31.670 0.000 30.817 -2.8 6324.6 6324.6 1.0 63245.6 13279.9 0.000 388.785 0.000 383.352 -1.4 |A = 40,000,000 [mm'] b/R = 0.5 Empirical/Analytical eq. Eigenvalue analysis L b AR Radius R/t buck, press buck, ratio buck, press buck, ratio % diff [mm] [mm] [mm] [Nmm-2] [Nmm-2] • ='i;587ltrnm 20000.0 2000.0 10.0 1000.0 629.9 0.000 18.464 0.000 16.535 -11.7 14589.8 2741.6 5.3 1370.8 863.5 0.000 47.563 0.000 42.583 -11.7 6324.6 6324.6 1.0 3162.3 1992.0 0.000 583.887 0.000 523.821 -11.5 t = 4.7625 [mm JilllfefitllHP 20000.0 2000.0 10.0 1000.0 210.0 0.006 0.684 0.007 0.612 -11.8 14589.8 2741.6 5.3 1370.8 287.8 0.002 1.762 0.003 1.578 -11.7 6324.6 6324.6 1.0 3162.3 664.0 0.000 21.625 0.000 19.428 -11.3 Table 6.5 Comparison of analytical and F E eigenvalue buckling values (SS straight edges, A = 40,000,000 mm 2 ) Figures 6.11 - 6.14 shows the data presented in Tables 6.4 and 6.5 graphically and includes additional data not shown in Tables 6.4 and 6.5 (this data can be found in Appendix A ) . A s can be seen from these Figures,, the correlation is very good for the entire range of plate geometries considered, with the A N S Y S values being marginally lower. 85 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S Figure 6.11 Comparison of analytical and F E buckling rations [t = 1/16", b/R = 0.1] 2.0 i 1.8 1.4 3 °-£ 1 . 2 o "*-» re 0)10 o 5 0 8 0.6 A = 100,000 mm2 s*'" A = 500,000 mm2 -•- A = 4,000,000 mm2 * Ansys eigenvalue A = 100,000 mm2 Ansys eigenvalue A • 500,000 mm2 A Ansys eigenvalue A = 4,000,000 mm2 04 0 2 00 0 0 tx 1.0 2 0 3 0 4.0 5.0 6.0 Aspect Ratio Ub 7.0 8 0 9 0 10.0 Figure 6.12 Comparison of analytical and F E buckling ratios [t = 1/16", b/R = 2.0] 86 CHAPTER 6: FINITE ELEMENT ANALYSIS 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Aspect Ratio L/b Figure 6.13 Comparison of analytical and F E buckling ratios [t = 3/16", b/R = 0.1] 87 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S It is well established from experimental testing; that buckling values obtained from eigenvalue analysis and analytical equations are un-conservative, as imperfections in material, geometry, boundary conditions and loading are not taken account of. In order to obtain a more realistic value of buckling pressure, A N S Y S large deflection non linear buckling analysis using both the Newton-Raphson and Arc-Length methods was performed. Large deflection non linear buckling analysis also allows post-buckling strength of the panels to be evaluated. While buckling is undesirable from an aesthetic point of view and can not be tolerated on a Serviceability limit state (due to the compromising of architectural designs, and the possibility of looking 'unsafe' to patrons), at the Ultimate limit state, the additional strength of the panels in the post buckling range may allow a more efficient and hence less costly design. Figure 6.15 shows a graphical comparison of nonlinear and linear (eigenvalue) load deflection curves, demonstrating the un-conservative nature of eigenvalue predictions. Figure 6.15 Nonlinear load-deflection curve (a) and linear (Eigenvalue) buckling curves (b) A N S Y S uses the 'Newton-Raphson' method to solving nonlinear problems. In this method, the applied load is subdivided into a series of load increments, which can be applied oyer several load steps as shown in Figure 6.16. Before each converged solution point, the Newton-Raphson method evaluates the out-of-balance load vector which is the difference between the restoring forces (the loads corresponding to the element stresses) and the applied loads. ANSYS® then performs a linear solution, using the out-of-balance loads, and checks for convergence. If convergence criteria are not satisfied, the out-of-balance load vector is reevaluated, the stiffness matrix is updated, and a new solution is obtained. This iterative procedure continues until the 88 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S problem converges. A number of convergence-enhancement and recovery features are available within A N S Y S O , such as line search, automatic load stepping, and bisection, which can be activated to help the problem to converge. If convergence cannot be achieved, then the program 18 attempts to solve with a smaller load increment. For the curved plates considered in this analysis, the tangent stiffness matrix can become singular, causing severe convergence difficulties using the Newton-Raphson method alone (due to zero or negative stiffness) as the plate buckles into a stable configuration. A N S Y S allows the selection of an alternative scheme termed the 'Arc-Length ' method, to help avoid bifurcation points and track unloading. The Arc-Length method forces the Newton-Raphson equilibrium iterations to converge along an arc as shown in Figure 6.16, thereby often preventing divergence. Figure 6.16 Newton-Raphson (left) and Arc-Length (right) methods1 Figure 6.17 shows a pressure deflection plot for several nodes of a plate model, as the pressure is increased during a large deflection non linear buckling analysis using A N S Y S . It should be noted that this particular plate has curved free edges longer than the straight supported edges. The nodes plotted represent the nodes which exhibit the largest deflection, they correspond to the midpoint of the two curved free edges (the time history plot of node 431 is plotted with and without stress stiffening effects included in the F E analysis). From this figure, the buckling pressure for this particular plate geometry is approximately 32 xlO" kNmm" which is -ft 9 considerably lower than the buckling pressure of 130 x 10 kNmm" obtained from the eigenvalue analysis (which is very close to the value obtained using the analytical equations). There are several areas of concern which arise from this result. The significant decrease in 89 CHAPTER 6: FPNITE E LEMENT ANALYS I S apparent buckling pressure between the A N S Y S 0 nonlinear and eigenvalue buckling analysis is surprising (a factor of 4). There are also concerns with the validity of the ANSYS® nonlinear buckling analysis as the analysis terminates abruptly at a pressure of approximately 26 x 10" kNmm" . Attempts at tracking through this point were unsuccessful, which casts doubt on whether this corresponds to buckling or numerical instability. The deflected shape of the plate at different time steps is shown in the same plot. Upon evaluation of the deflected shape at the point of analysis termination, the shape is inconsistent with any of the buckled mode shapes as predicted using eigenvalue buckling analysis, furthermore, the shape does not seem intuitive as a buckled shape. (The predicted buckling shape for this plate is a single point of inflection.) —•—node 431 -"•—node 212 node 431 ss tif off Sub step 18 Deflection /mm Figure 6.17 A N S Y S large deflection non linear buckling time history plot for nodes of maximum displacement [ A R = 10, b/R = 0.1, A = 100,000 mm 2 , t = 1/16", esize = 5, No . substep = 100, Pmax = 40x10" 6 kNmm" 2 , Arclength method] A second A N S Y S nonlinear buckling analysis was conducted for a plate of the same area and aspect ratio as analyzed in Figure 6.16, with the exception that the included angle (b/R) was increased to 2.0 and with the curved free edges as the shorter dimension. Figure 6.18 shows the 90 C H A P T E R 6: F P N I T E E L E M E N T A N A L Y S I S time history plot for three nodes. Nodes 12 and 232 exhibit the largest deflection and correspond to the intersection between the curved free edges and the centre line of the plate, while node 2332 is at the center of the plate. Again, attempts at tracking through the point of analysis termination were unsuccessful. The maximum pressure resisted by the plate before termination 6 2 of the analysis is determined as approximately 4300 x 10" kNmm" , which is significantly larger than the buckling pressure of 2000 x 10" kNmm" obtained from eigenvalue buckling analysis which is even more surprising than the significantly lower apparent buckling pressure as determined for the plate geometry considered in figure 6.16. Similar to the concerns with the results shown in Figure 6.15, the displaced shape at the point of analysis termination does not correspond to any of the buckled mode shapes predicted using eigenvalue buckling analysis, and does not appear to be consistent with any intuitive buckled shape. 4500 • H i — • • a * -3.0 Deflection /mm -node 232 -node 12 node 2332 Figure 6.18 A N S Y S large deflection non linear buckling time history plot for nodes of maximum displacement [ A R = 10, b/R = 2.0, A = 100,000 mm 2 , t = 1/16", esize = 5, No . substep = 100, Pmax = 4500 x 10"6 kNmm" 2 , Arclength method] A third plate geometry was analyzed using A N S Y S c nonlinear buckling analysis. The previous two plates considered both had an aspect ratio of 10 and were very small in area (only 100,000 91 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S 2 • • mm ). This third plate has an aspect ratio of 2.85 (short free edges curved), included angle (b/R) of 0.5, thickness of 1/16" and a plate area of 50,580,000 mm . The time history plot of the three nodes giving rise to the largest deflection at the point of analysis termination is shown in Figure 6.19. The apparent buckling pressure according to this figure is 0.0155 x 10"6 kNmm" 2 which is significantly larger than the buckling pressure of 0.006 x 10" kNmm" as predicted using eigenvalue buckling analysis. The displaced shape at the analysis termination is again not consistent with the predicted buckled mode shape according to eigenvalue buckling analysis (single point of inflection), however unlike the previous two cases, the shape is an intuitive buckled mode shape. This shape consists of two points of inflection, suggesting that the buckled mode shape predicted using A N S Y S non linear analysis in this case, corresponds to a higher buckling mode which is consistent with the larger buckling pressure compared with the eigenvalue buckling pressure for the first mode shape. r / -—•••I .•• * I - 7 8 4 l o.ooo + 1 1 •—, 1 1 1 1 1 1 ! , , . 1 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 140 Deflec t ion / m m Figure 6.19 A N S Y S large deflection non linear buckling time history plot for nodes of maximum displacement [ A R = 2.85, b/R = 0.5, A = 50,580,000 mm 2 , t = l/16"mm, esize = 200, No . substep = 100, Pmax = 0.002 x 10"6 kNmm" 2 , Arclength method] 92 CHAPTER 6: FINITE ELEMENT ANALYSIS Unfortunately 'tracking through' the point of buckling was not achieved using either the Netwon-Raphson or Arc-Length methods (with variation in the analysis parameters) for any of the geometries considered. This would normally be a cause of concern, as without the ability to track through the point of buckling, it is not possible to rule out numerical instability as the cause of the divergence indicating buckling in figures 6.17- 6.19. However i f the apparent buckling point is actually caused by numerical instability, this implies with better selection of convergence criteria and solution controls that the actual buckling stress predicted wi l l be higher. This does not change the unsettling fact that the non linear large deflection analysis is giving an apparent higher buckling pressure than the eigenvalue analysis for two of the three plate geometries considered, which is counter-intuitive. There is some evidence to suggest a higher buckling pressure is predicted, as the mode shapes approached in the non-linear analysis do not correspond to the first mode shape predicted by the analytical equations and the eigenvalue buckling analysis (single point of inflection) and correspond to a higher mode. This may be overcome by altering the support conditions slightly or applying some perturbations such as a moment along each supported edge in order to force the first mode shape to be that of a single point of inflection. Tracking through the point of divergence may be achieved through better selection of analysis parameters used in defining the Arc-Length and Newton-Raphson methods. Finite element nonlinear buckling analysis is notoriously sensitive to input and model parameters, with many describing the analysis as an art as opposed to an exact science. Irrespective of the cause of the inconsistent results of the A N S Y S nonlinear buckling analysis, the possible remedies all require a large investment of time and require numerous iterations per analysis, which does not lend itself to automated design optimization. Therefore the decision was taken to continue with the eigenvalue buckling analysis and to determine justifiable methods of reducing the eigenvalue buckling predictions in order to alleviate the un-conservative nature of such predictions. 6.2 Effect of longitudinal stiffeners: Provision needs to be made for the plates to be connected to the main structure of the Wheel. This can be achieved through the use of stiffeners connected to the underside of the panels. Due 93 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S to the additional cost of curving or torching stiffeners to follow a curved profile, these stiffeners wi l l follow the un-curved edges of the single curvature panels. 6.2.1 Flat panels Angle stiffeners effectively shorten the free span distance of the plate by a distance which is a function of the angle stiffener width and stiffness. Additional longitudinal and transverse stiffeners could be used to create an orthotropic plate, greatly increasing the allowable maximum plate size. However this option is not investigated due to aesthetic considerations. From a yielding standpoint un-stiffened plates of large dimensions are feasible. A s the plate dimensions are increased, the discretization of the curved Wheel becomes coarser. If flat plates are to be used as cladding, it would be desirable to keep the dimensions of the plate smaller in order to achieve a better approximation of the curved surfaces of the Wheel. For this reason, stiffened flat plates are not investigated further in this report. 6.2.2 Curved panels The inclusion of angle stiffeners along both un-curved edges greatly increases the capacity of a given panel as it emphasizes the arching action of the plate. Figure 6.20 shows a typical F E model of a longitudinally stiffened plate, the lighter elements represent the curved plate, while the darker elements represent the vertical leg of the angle stiffeners (horizontal leg bonded to underside of the plate not shown). 94 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S E L E M E N T S AN PRE 9 .420E-05 A U G 2 200 6 1 6 : 0 5 : 5 6 MEGAWHEEL 05 Figure 6.20 Typical FE model of longitudinally stiffened curved plate (units of Nmm) The influence of angle stiffeners on the buckling mode shape of the curved panels is shown in Figures 6.21. The first buckled mode shape is consistent with the first buckled mode shape for the un-stiffened plate, however the subsequent modes are different. The angle stiffeners are only participating in the first buckled mode shape, and for the subsequent three modes, the stiffeners act to provide fixed supports to the edges of the plate spanning the distance between the stiffeners. 95 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S NOD AX SOLUTION rilEO*15.35 9 Mode 1 MEGAWHEEL' 05 SUB "3 PREQ-32. 10 9 UZ (AVG) R3YO»0 Mode 3 AN A N MEGAWHEEL' 05 FABO-32.057 UZ (AW R3if8»0 3MN =-.008923 Mode 2 MEGAWHEEL' 05 NOD At, SOLUTION Mode 4 MEGAWHEEL* 05 AN A N US * 2003 Figures 6.21 Buckling mode shapes for a curved plate (SS along longitudinal stiffeners edges), curved edge length 800mm, straight edge length 5524mm [eigenvalue buckling analysis] Similar buckling mode shapes for a stiffened plate of the same included angle, straight edge length and plate thickness, but with the width of the curved edges increased by a factor of two (and therefore a lower aspect ratio), are shown in Figures 6.22 for comparison purposes. O f interest is the swapping of modes 2 and 3 relative to the stiffened plate with a higher aspect ratio. However the first buckling mode is again consistent with the un-stiffened case. 96 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S „ « * X . O ^ X O K A N . . _, j AUG 4 2005 Mode 1 M i JO* 14 TRECFl .14 8 UZ (AVO) fl.3YS=0 ' j| ^ : „ » „ , , , „ m , o » A N S 3 Mode 2 AUV3o™5 PRECFl. 719 SMX =i ^ H B K 3KN =-.999992 • -.999992 -.555549 .111106 .3'JJ3> J -.77777 -.333328 .111115 .555557 1 MEGAWHEEL'05 -.999639 -.5552 7 5 -.110911 .3 33 454 .777818 -.777437 -.333093 .111271 .55553 6 1 MEGAWHEEL* D5 A N NODAL SOLUTION " " » W S 3 Mode 3 "mJSL PAECF1.743 DMX * i -_^^BgraffjB9a a 1^^^^H 3KM =-.oi7976 HHHHHHH|HHHHHk SMX =1 1 ) NODAL SOLUTION AN-." S 3 Mode 4 ""m'"°l PREQ=2.205 UE (AVG) ™II ~~:~ZIA 3MN 696264 ™ l ' ........ - " 1 '"" = 1 • M B B M • • -.01797E .2M241 .434438 1 i067S .886892 .•95132 .321349 .547566 .773783 1 MEGAWHEEL'05 -.696264 -.319317"' .057631 ' .4",<!!V/U .811526 -.50779 -.130843 .246105 .623032 1 M E G A W H E E L ' 05 Figures 6.22 Buckling mode shapes for a curved plate (SS along longitudinal stiffener edges), curved edge length 1600mm, straight edge length 5524mm [eigenvalue buckling analysis] According to the eigenvalue analysis, the plate analyzed in Figures 6.22 has a small safety margin with respect to buckling for the first mode. Linear analysis is used in order to obtain an indication of the stress and deflection of this plate subject to the design pressure as attempts at using nonlinear analysis gave inconsistent results with regard to buckling. The linear analysis w i l l most likely result in decreased predictions of deflection and increased predictions of stress compared with values obtained from a converged non-linear large deflection analysis. The linear deflection and V o n Misses stress plots for the same plate shown in Figures 6.21 subject to the design pressure of 4.2 kNm" 2 are shown in Figures 6.23. The stress is well within reasonable limits at 130 M P a and the deflection at the centre of the plate is 50mm. It is of interest to note the location of maximum stress occurs parallel to the stiffeners with a small offset from the stiffener edge, this suggests that careful detailing of the transition from stiffener to plate is required. 97 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S A N 3TBP=1 SUB = 1 TIME=1 U3 (AVG; R3Y3-0 DM7 =51.566 9MM =-51.566 3MX •.. 935901 Deflection [mm] NODAL 3TEP-1 BUB » 1 TIMH=1 aSOV (AVG) DMX =51.5 66 3MN --423B03 3MX =.131599 A N Von-Mises Stress [GPa] KEGAWHEEL'OS Figures 6.23 Linear deflection and V o n Misses effective stress in plate subject to normal pressure Without information regarding allowable deflection, it appears that buckling is the limiting case for design, for this particular geometry. 6.2.3 Optimization ANSYS® provides well developed optimization algorithms and hence ANSYS® is used to perform the design optimization. The A N S Y S L program offers two optimization methods; the subproblem approximation method, which is an advanced zero-order method and the first order method, which is based on design sensitivities. The subproblem approximation method can be efficiently applied to most engineering problems, while the first order is more suitable for problems that require high accuracy. For both the subproblem approximation and first order methods, the program performs a series of analysis-evaluation-modification cycles. That is, an analysis of the initial design is performed, the results are evaluated against specified design criteria, and the design is modified as necessary. This process is repeated until all specified criteria are met. 1 8 Figure 6.24 shows a typical structural optimization model, and Table 6.6 defines some common A N S Y S " J terminology used when performing design optimization. 98 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S Design | Optimization Term ( Description design variables j (DVs) | Independent quantities, varied to achieve the optimum design. Upper and lower limits are | specified to serve as "constraints" on the design variables. These limits define the range o f variation for the D V . state variables (SVs) j Quantities that constrain the design. A l s o known as "dependent variables," they are typically response quantities that are functions o f the design variables. A state variable may have a maximum and minimum limit, or it may be "single sided," having only one limit. objective function | The dependent variable that is to be minimized. It is a function o f the D V s (that is, changing the values o f the D V s should change the value o f the objective function). optimization j variables ] Collect ively, the design variables, state variables, and the objective function. In an A N S Y S optimization, these variables are represented by user-named variables called parameters. Y o u must identify which parameters in the model are D V s , which are S V s , and which is the objective function. design set or design \ i S 1 A unique set o f parameter values representing a given model configuration. Typical ly, a design set is characterized by the optimization variable values; however, all model parameters (including those not identified as optimization variables) are included in the set. feasible design | A design that satisfies a l l specified constraints (those on the S V s as wel l as on the D V s . If any one o f the constraints is not satisfied, the design is considered infeasible. The best design is the one which satisfies all constraints and produces the minimum objective function value. (If al l design sets are infeasible, the best design set is the one closest to being feasible, irrespective o f its objective function value.) analysis file A n A N S Y S input file containing a complete analysis sequence (preprocessing, solution, and postprocessing). The file must contain a parametrically defined model, using parameters to represent all inputs and outputs to be used as D V s , S V s , and the objective function. | loop file | A n optimization file (named Jobname.LOOP), created automatically via the analysis file. | The design optimizer uses the loop file to perform analysis loops. A single pass through the analysis file. Output for the last loop performed is saved in file Jobname.OPO. A n (or simply iteration) is [optimization iteration^ One or more analysis loops which result in a new design set. Typical ly, an iteration equates to one loop; however, for the first order method, one iteration represents more than one loop. optimization j | database j j Contains the current optimization environment, which includes optimization variable definitions, parameters, a l l optimization specifications, and accumulated design sets. This database can be saved (to Jobname.OPT) or resumed at any time in the optimizer. Table 6.6 ANSYS® Design optimization terminology 1 99 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S File.DB \ ANSYS Database File /Analysis File \ . (parametrically 1 Vdefined model)y' OPEXE ANSYS UJ Model Database Optimization Database File.OPT Optimization Data File OPEXE File.LOOP \ Loop File J OPEXE' ^ X ^ ^ N , I ' 7 : File.OPO \ Last Loop j Output / Figure 6.24 Typical structural optimization model It was decided to use the subproblem approximation method for the design optimization analysis. The subproblem approximation method is described as an advanced zero-order method, as it only requires the values of the dependent variables, and not their derivatives. There are two concepts that play a key role in the subproblem approximation method: the use of approximations for the objective function and state variables, and the conversion of the constrained optimization problem to an unconstrained problem.18 Initially, the program establishes the relationship between the objective function and the design variables by curve fitting. This is done by calculating the objective function for several sets of design variable values (that is, for several designs) and performing a least squares fit between the data points. The resulting curve (or surface) is called an approximation. Each optimization loop generates a new data point, and the objective function approximation is updated. It is this approximation that is minimized instead of the actual objective function. State variables are handled in the same manner. An approximation is generated for each state variable and updated at the end of each loop.1 8 100 C H A P T E R 6: F P N I T E E L E M E N T A N A L Y S I S ANSYS allows control over the curve fitting for the optimization approximations. A linear fit, quadratic fit, or quadratic plus cross terms fit can be selected. The default settings were used for this analysis; a quadratic plus cross terms fit is used for the objective function, and a quadratic fit is used for the state variables. State variables and limits on design variables are used to constrain the design and make the optimization problem a constrained one. The ANSYS program converts this problem to an unconstrained optimization problem because minimization techniques for the latter are more efficient. The conversion is done by adding penalties to the objective function approximation to account for the imposed constraints. The search for a minimum of the unconstrained objective function approximation is then carried out by applying a Sequential Unconstrained Minimization Technique (SUMT) at each iteration.18 At the end of each loop, a check for convergence (or termination) is made. The problem is said to be converged if the current, previous, or best design is feasible and any of the following conditions are satisfied: • The change in objective function from the best feasible design to the current design is less than the objective function tolerance. • The change in objective function between the last two designs is less than the objective function tolerance. • The changes in all design variables from the current design to the best feasible design are less then their respective tolerances. • The changes in all design variables between the last two designs are less than their respective tolerances. Convergence does not necessarily indicate that a true global minimum has been obtained. It only means that one of the four criteria mentioned above has been satisfied. For subproblem approximation, the optimizer initially generates a small number of random designs to establish the state variable and objective function approximations; however these may not be adequate to ensure the global minimum is being reached. In order to speed up convergence and to increase the chance that the global minimum is being achieved, a larger number of random designs is run first, and all infeasible designs discarded before initiating the subproblem approximation method. 101 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S This is illustrated graphically in Figures 6.25, however there is no guarantee that convergence at the global minimum is being achieved (2-D space shown, in reality the design space is multidimensional). Optimization function Subproblem approximation method Convergence Design space Optimization function Subproblem approximation method after random design runs. Convergence Design space Figure 6.25 Convergence to local minima for subproblem approximation method, and convergence to global minima when random design runs performed first After the sub approximation method has converged to an 'optimal' design set, an additional optimization function; the sweep tool is used to perform a global sweep of design space. It performs several loops, during each loop a single design variable is incremented through the design range for that variable, while the remaining design variables are held fixed at the converged 'optimal' values. This is performed as a check to ensure the minimum has in fact been found. 102 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S Table 6.7 lists the different variables used in the optimization runs. Limits are placed on design variables in order to minimize the design space and hence increase the probability of convergence to global minima and to reduce the run time of the analysis. The upper limits for plate width, pb, and length, pL , were obtained as approximately 2000 and 8000 mm from fabricators. The plate thickness, pth, is allowed to range from 1 to 4 mm, as preliminary optimization runs showed the optimal thickness to be within this range. A lower bound of 1.3 for the buckling ratio is chosen in order to provide a safety margin due to the typically un-conservative nature of eigenvalue buckling analysis. A n upper limit of 150 M P a was chosen for the maximum V o n Mises effective stress in order to provide a large safety margin against plate yielding and with the knowledge that this value w i l l most probably increase when other factors such as thermal gradients and non linear behaviour among others are considered in the analysis. The objective function driving the optimization analysis is defined as the plate weight divided by the plate area (this is minimized). For simplicity, the longitudinal stiffeners are assumed to consist of angle stiffeners of equal leg length and thickness. There are limits on stiffener slenderness ratios according to C A N / C S A SI6.1 (Canadian steel design handbook), the parameter, lslen, expresses the slenderness of the longitudinal stiffeners as a ratio to the slenderness allowed by the code, hence an upper limit of 1 is chosen for this parameter. Parameter Description Lower limit Upper limit Design Variable pb Curved edge length /mm 400 2500 u PL Straight edge length /mm 400 8500 II pth Plate thickness /mm 1 5 II lonth Longitudinal angle stiffener thickness /mm 2 25 II lonw Longitudinal angle stiffener leg width /mm 25 250 State Variable maxseqv Linear Max Von Misses effective stress /GPa 0 0.15 II modef Eigenvalue buckling ratio 1.3 2.1 II lslen Longitudinal angle stiffener leg slenderness ratio 0 1 Objective Function weightA Plate weight to area ratio - - • Table 6.7 Optimization variables (longitudinally stiffened plates) Initially during the optimization runs, the longitudinal stiffeners were assumed to be simply supported along their entire length, however it soon became apparent that this was driving the optimization loops to solutions where the plate aspect ratio was extremely large. This is not representative of the support configuration in the field, as these plates w i l l be supported at discrete points. It was found that the number of support locations and support width greatly influences the behavior of the plate and gives rise to significantly different optimized panels. To 103 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S take account of this, separate optimization runs are completed for a number of different support configurations. Unfortunately, it is not feasible from a computational point of view, to include support configuration as a design variable in the optimization runs as this dramatically increases the run time of optimization runs (this statement applies to the computers and resulting computing power accessible at the time of conducting this thesis project). There w i l l be some form of physical connection between the longitudinal stiffeners and the curved plates, either in the form of epoxy adhesive or through fastening devices such as rivets. For the purposes of this investigation it is assumed that the stiffener is bonded to the plate with the use of a bonding agent which allows no relative movement between the stiffener and the plate. For the majority of the analyses conducted, it is assumed that the longitudinal stiffener w i l l be made of steel, with some investigation of the effect of using aluminum stiffeners conducted. Table 6.8 shows the variation in optimized variable values with the number of support points per longitudinal stiffener. For each model, the support width is 100 mm per support point, with a simply supported boundary condition. Optimization runs are performed for 2, 3, 4 and 6 support points per stiffener and for an included angle, (b/R), of 0.1. This curvature is chosen due to the results from the section geometry analysis (Tables 4.3), which suggests that the plates used for the Wheel w i l l have included angles of this magnitude. A value of 0.1 is at the lower range of values expected, a larger curvature (corresponding to a smaller included angle) results in a increased structural resistance to buckling, hence the optimized plates with an included angle of 0.1 represent the lower end of efficiency. Constants Theta = 0.10 (included angle b/R) Support point width = 100 [mm] No. of support points pb PL pth lonth lonw weightA modef maxseqv (per Ion. st.) [mm] [mm] [mm] [mm] [mm] [gmm 2 ] [GPa] 2 874.38 875.7 3.1745 13.64 30.68 0.024 1.32 0.057 3 956.17 1610.5 3.0587 6.11 64.70 0.021 1.51 0.15 4 893.15 1289.6 2.9727 4.66 41.43 0.015 1.30 0.14 6 723.71 2171.4 2.5751 4.35 26.22 0.012 1.36 0.15 Table 6.8 Variation in optimization variables (longitudinally stiffened plates) with number of support points Figure 6.26 shows the normalized variation in key optimization variables as the number o f discrete support points per longitudinal stiffener is increased from 2 to 6 (along each longitudinal 104 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S stiffener edge). It can be seen that as the number of support points increase the width of the optimized panel decreases from 1100 mm to 700 mm and the length increases from 1600 to 2200 i.e. the plate aspect ratio is increasing. It is important to note that the weight of the structure follows very closely the curves for longitudinal stiffener width and longitudinal stiffener thickness suggesting that it is these parameters that have the greatest influence. Hence it is important to select a stiffener with a highly efficient cross section, such as an I or Z shaped section. 2.70 0.20 I 1 1 1 , — , 1 1 2 3 4 5 6 7 Number of support points per longitudinal stiffener Figure 6.26 Normalized variation in optimized variables with number of support points (b/R = 0.1) For comparison purposes the optimization was redone for a plate with an included angle (b/R) of 1.0 (see Table 6.9), and with two support points per longitudinal stiffener. A s predicted, the resulting plate is considerably more efficient compared with the optimized panel with an included angle of 0.1. 105 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S Constants Theta = 1.00 (included angle b/R) Support point width = 100 [mm] No. of support points (per Ion. st.) pb [mm] PL [mm] pth [mm] lonth [mm] lonw [mm] weightA [gmm 2 ] modef maxseqv [GPa] 2 505.19 7674.6 1.8645 3.32 25.59 0.010 1.92 0.055 Table 6.9 Optimization variables (longitudinally stiffened plate) with b/R =1.0 It was previously assumed that the longitudinal stiffeners were made of steel. For comparison purposes, the optimization analysis was redone for two of the cases previously considered, this time with aluminum rather than steel stiffeners. A s can be seen from the results presented in Table 6.10, this results in a much reduced weight and although the plate dimensions are smaller, a much reduced weight to area ratio (increased efficiency). Constants Theta = 0.10 (included angle b/R) Support point width = 100 [mm] No. of support points pb PL pth lonth lonw weightA modef maxseqv (per Ion. st.) [mm] [mm] [mm] [mm] [mm] [gmm 2 ] [GPa] 2 947 1264 2.96 8.91 69.38 0.015 1.40 0.106 6 630 2999 2.17 4.96 25.72 0.008 1.30 0.122 Table 6.10 Optimization variables (longitudinally stiffened plate with A l stiffeners) The optimization function of weight divided by area as a measure of structural efficiency for the design of the cladding panels is inadequate. This is due to the need to include other aspects such as material cost and labour, which are also dependant on many of the variables in the model, such as number of support points, as this w i l l dictate the number of bolts and time required to bolt panels in place. However it is very difficult to quantify such costs in a way that lends itself to a merit function which can be minimized. There is also the drawback, that as the complexity of the merit function increases, the less use it serves to the engineer in terms of taking a given merit function and being able to understand the effect of variables on this function conceptually. The derivation of a more comprehensive and appropriate optimization function is an aspect which requires further attention in the future and should be clearly defined before further in-depth F E analysis is conducted. 106 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S 6.3 Effect of radial stiffeners (in addition to longitudinal stiffeners): While it is generally accepted that curving or torching of stiffeners to achieve a curved profile is costly, the use of radial stiffeners in conjunction with longitudinal stiffeners for connection may give rise to a much more efficient plate which wi l l greatly reduce the number of panels and the number of and time spent on installing connective bolts. The larger plate allowed through the use of radial stiffeners may not necessarily create an undesirable aesthetic characteristic, due to the curvature of the plate, which follows the smaller radius of curvature of the Wheel. For the subsequent analyses, steel stiffeners are assumed, however this can be easily altered to aluminum in the A N S Y S input files. Similar to the longitudinal stiffener, the radial stiffeners are assumed to be angle stiffeners. However in order to simplify the analysis models, the stiffening effect of radial stiffeners is accounted for by 'smearing' the stiffeners to create a locally thicker plate across the stiffener leg width at each radial stiffener location. The increase in thickness is calculated by enforcing equivalence of bending stiffness (EI). Similarly, the curved leg of the longitudinal stiffeners bonded to the plate is also smeared. The vertical leg of the longitudinal stiffeners remain in order to provide the support location points and the appropriate arching action. Initially the number and thickness of radial stiffeners were used as design variables in the optimization, with the longitudinal stiffener width and thickness kept constant. A plate thickness of 2/16" was chosen, as this is the maximum plate thickness according to fabricators contacted at the time of performing the analysis (for anodized aluminum panels). A value of 2/16" is also reasonably close to the optimized plate thicknesses as determined from the analysis conducted in section 6.2 of this thesis. Table 6.11 shows the design variables used during the optimization runs. Comparing Tables 6.11 and 6.10, some of the limits for common variables have changed. The changes are made subsequent to preliminary optimization runs which allow a reduction in range of the variables in an,attempt to ensure global convergence and reduce run time. 107 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S Parameter Description Lower limit Upper limit Design Variable pb Curved edge length /mm 1200 6000 PL Straight edge length /mm 1200 6000 Thick2 Plate thickness /16ths of an inch 1 3 noradst2 Number of radial stiffeners 2 10 radth Radial angle stiffener leg thickness /mm 2 25 lonth Longitudinal angle stiffener leg width 2 25 radw Radial angle stiffener leg width /mm 25 250 II lonw Longitudinal angle stiffener leg angle width /mm 25 250 State Variable maxseqv Linear Max Von Misses effective stress /Gpa 0 0.15 n modef Eigenvalue buckling ratio 1.3 2.1 Objective Function weightA Plate weight to area ratio • - -Table 6.11 Optimization variables for longitudinally and radially stiffened plate For these models, it was found that the optimization run time could only be brought to within reasonable limits, i f some of the state variables were excluded. The state variables which were omitted during the analysis are shown in Table 6.12. The parameters; lslen and rslen, which relate to the slenderness ratios of the longitudinal and radial stiffeners according to C A N / C S A SI6.1 guidelines respectively, were omitted, however the range of allowable widths and thicknesses was limited to try and ensure slenderness ratios would not likely be exceeded. In future analyses it may be prudent to apply a penalty in terms of 'apparent' increased weight (i.e. have a statement in the code which w i l l increase the weight of the plate arbitrarily) i f the slenderness limits are exceeded, however the effect of this on the convergence of the design optimization is unknown. A third parameter, rbenst, which relates to the expected stress in the radial stiffeners i f they are fabricated straight and then curved to match the plate curvature before bonding to the plate, is also omitted. This is due partly to decrease run time, but also to allow flexibility in design, as torching o f curved stiffeners is also possible. The bending stress is easily calculated once the curvature of the plate is known, again, it may be possible in future analyses to include a penalty in terms of 'apparent' increased weight i f the stress in the stiffener is such that it is not feasible to bend and torching is required (to reflect the increased cost of torching the stiffener). Other parameters, such as a parameter associated with the bond stresses present at the interface between stiffener and plate may also be included in the analyses. This is particularly important i f epoxy bonding is to be used. Parameter Description Lower limit Upper limit State Variable II lslen rslen rbenst Longitudinal angle stiffener leg slenderness ratio Radial angle stiffener leg slenderness ratio Radial stiffener bending stress 0 0 0 1 1 0.25 Table 6.12 State variables omitted from the design optimization, to reduce run time 108 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S Table 6.13 shows the optimized variable values for a case of a plate with two supports points per longitudinal stiffener and for a similar case where the longitudinal stiffener leg length (lonw) has been decreased by 50%. The plate is assumed to be simply supported along a length of 100mm at either end of the longitudinal stiffeners (2 support points). A s expected, the introduction of radial stiffeners allows for a much increased plate area relative to the plate stiffened with longitudinal stiffeners only, the weight is also significantly decreased. A s the longitudinal stiffener leg length (therefore stiffening effect) is reduced, the number of radial stiffeners increases significantly. The high number of radial stiffeners at the optimal design is another indication that the optimization function of weight divided by area needs to be redefined to include aspects associated with the cost incurred with adding radial stiffeners. It is of interest to note, that for this particular case of two support points per longitudinal stiffener, the optimal structural system consists of plates with the long direction aligned with the radial stiffener direction, i.e. the direction of curvature (which is intuitive). 2 support points of 100mm per longitudinal stiffener Constants Optimised variables lonth lonw Thick2 pb PL noradst2 radth radw weightA [mm] [mm] [16ths inch] [mm] [mm] [mm] [mm] [gmm"2] 10.6 120.4 2 4317 4483 10 3.1 30.8 0.021 10.6 60.2 2 2958.2 6995.1 14 3.7 32.6 0.019 Table 6.13 Optimized variable values (longitudinally and radially stiffened plate, b/R = 0.1) It was decided that the large number of radial stiffeners obtained for the 'optimal' designs, was unrealistic. Without a more comprehensive and appropriate optimization function, subsequent optimization runs were conducted for set number of radial stiffeners, with stiffener dimensions as design variables. Table 6.14 shows the resulting design variable values for a plate thickness of 2/16" and with each longitudinal stiffener simply supported along a width of 100mm at either end. A plate thickness of 2/16" is selected as this is the maximum thickness achievable according to anodized aluminum fabricators contacted at the time of conducting this research. The minimum allowable stiffener thickness was reset to 5mm for this analysis to prevent overly slender stiffeners. 109 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S Constants Theta = 0.10 (included angle b/R) Thick2 = 2.00 (16ths of an inch) Two support points of 100mm per longitudinal stiffener No. of radial pb PL radth lonth radw lonw weightA maxseqv stiffeners [mm] [mm] [mm] [mm] [mm] [mm] [gmm 2 ] [GPa] 0 873.3 872.46 - 13.64 - 30.86 0.024 0.057 2 1254.9 4148.3 5.16 16.53 81.81 215.66 0.100 0.076 3 3308.9 1235.5 7.79 10.45 58.79 74.36 0.033 0.158 4 2769.4 1273.1 5.19 8.73 36.33 33.71 0.021 0.149 6 2918.3 1395.6 5.07 8.67 35.93 32.98 0.024 0.146 Table 6.14 Optimized variable values (longitudinally and radially stiffened plate) It is of interest to note, the large increase in weight to area ratio for the case of two radial stiffeners versus no radial stiffeners. The radially stiffened F E model is created such that there are two or more radial stiffeners, with the first two stiffeners assumed to support the curved edges of the plate, the remaining radial stiffeners are placed to subdivide the plate equally. In this configuration two radial stiffeners do not significantly reduce the free span distance of the plate, hence their addition is in fact detrimental to the efficiency of the plate. A s the number of radial stiffeners is increased to four, the weight to area ratio is reduced to an efficiency greater than that achieved using longitudinal stiffeners alone. However further increasing the number o f radial stiffeners to 6 results in a decrease in efficiency, such that the system has the same efficiency as a plate longitudinally stiffened alone. This may be due to the fact that the true optimal solution is not being achieved. If one inspects Table 6.13 it can be seen that the use of 14 radial stiffeners does in fact further increase the system efficiency (past the mark set by 4 radial stiffeners). This was achieved by allowing a stiffener thickness of only 3mm, suggesting that a reduction in the allowable minimum stiffener thickness would result in a greater efficiency for the radially stiffened systems presented in Table 6.14. However the optimization variables associated with stiffener slenderness would need to be reintroduced. Figure 6.27 is a graphical representation of the data tabulated in Table 6.14, normalized to the values for two radial stiffeners. 110 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S Again, the plate weight is being heavily influenced by stiffener dimensions, suggesting that stiffener design, in terms of selection of an efficient sectional shape will be highly beneficial. Figures 6.28 show the position of the current optimized plate designs in b/R, AR and area space, in comparison to the plate geometries required as determined from looking at the three typical Wheel sections (see section 5.3). 'Segments tabulated 2' refers to the plate geometries determined without any intermediary supporting framework (see Table 5.4 L 2 ) , and segments tabulated 1 refers to the plate geometries determined assuming a supporting framework which quarters the required curved length span distance (see Table 5.6 L 2 ) . As can be seen, for the included angle value evaluated (0.1), the plates stiffened either longitudinally or by a combination of longitudinal and radial stiffeners meet the requirements of segment size, if an intermediary supporting framework is used. C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S Figure 6.28(b) 3D graphical comparison of optimal stiffened plate and required geometries C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S It should be considered during future analysis, that the inclusion of radial stiffeners introduces the possibility of providing support points along the curved edges, this would undoubtedly increase the allowable plate area from a structural point of view, and may drive optimization analyses to designs where the plate aspect ratio is closer to 1. This will be increasingly important if panel interaction is a concern, as a square plate minimizes the edge length for a plate of a given area. 6.4 Effect of initial imperfections Initial imperfections will serve to lower the buckling resistance of any given plate, and hence the presence of imperfections must be accounted for in the analysis. Through a systematic approach of inputting several different imperfections, such as a shallower curve, irregular curve sympathetic with the buckling mode shape etc... the reduction in buckling resistance can be related to the original imperfections. This can be used to determine allowable tolerances during fabrication. If a relationship between the expected plate imperfections (size and distribution) and their effect on the buckling pressure of plates of differing geometry can be obtained, this reduction can be coded into optimization runs based on eigenvalue buckling analysis. This is especially important due to the un-conservative nature of the eigenvalue buckling analyses performed. 6.5 Influence of panel alignment relative to gravity: Depending on the position of the panel around the wheel gravity will be acting at a different angle relative to the panel. The self weight of the panel can act to stiffen the panel against buckling due to uniform normal pressure, such as in the case of a panel aligned with gravity, or can decrease the resistance of the panel to buckling as in the case of a panel aligned perpendicular to gravity. This variation in loading around the wheel due to gravity will allow some freedom in modifying the design of the panels, and may allow some different conceptual options to be utilized, significantly increasing the efficiency of the structure. 114 C H A P T E R 6: F I N I T E E L E M E N T A N A L Y S I S 6.6 Thermal effects: The effects of differential and restrained thermal expansion/contraction between the dissimilar metals and different portions of the Wheel, will lead to strain and stress distributions within each panel, which may lead to a reduced buckling and/or yielding resistance. Once an indication of the stress level in the optimized plates due to thermal effects is achieved, a reduction factor to the buckling resistance of these plates can be applied in future optimization runs to ensure conservative solutions. This method of including combined thermal and pressure loading is not very refined, however the problem of analyzing the effects simultaneously is very complex and due to the problems encountered during this thesis project in terms of analyzing the effects of pressure alone, the time and effort required would not allow the completion of the designs in a timely fashion. 6.7 Additional loading Scenarios: Loading scenarios in addition to those considered in the preceeding chapters, need to be considered, and may in fact govern for the design of these panels. This includes panel loading during seismic events, assembly, transportation and final erection. 115 C H A P T E R 7 C O N N E C T I O N D E S I G N CHAPTER 7: CONNECTION DESIGN 7.1 Stiffener to plate connection: 7.1.1 Adhesive bonding: The use of an adhesive to bond stiffeners to the underside of the anodized aluminum panels is the most desirable solution from an aesthetics view point, as this leaves no evidence on the top surface of the panels, which are visible to the public. Mechanical joints of the type produced with rivets, bolts and similar fasteners provide only localized joining, whereas adhesive bonding results in a continuous joint over the entire contact area, yielding a more efficient solution. The use of adhesive would also allow the possibility of using steel stiffeners to strengthen the aluminum panels, while minimizing the risk of galvanic action and eventual galvanic corrosion through the separation layer provided by the epoxy adhesive. Through discussion with 3 M ™ representatives, structural adhesives which w i l l cure at room temperature are available for this application. The performance post-cure depends on external temperature with degrading performance with increasing temperatures; however it is claimed that temperatures of -40 to + 40 °C are easily manageable with only a small variation in performance. The typical thickness of bond to ensure max strength is approximately 5/1000 of an inch. There are two main options; two-part adhesive or tape adhesive. The double sided tape adhesive is much easier to install and to ensure a uniform bond thickness, however it can only withstand approximately lA lb of dead shear per square inch. The two-part adhesive can withstand 1000 lbs of dead shear per square inch, however the application is much more rigorous and viscosity may be an issue as this can lead to problems ensuring adequate bond thickness. A work shop would be required to allow application of clamping forces to ensure that the surfaces are in contact during cure as the adhesives have no initial tackiness. 116 C H A P T E R 7 C O N N E C T I O N D E S I G N There is some concern over the ability to achieve an adequate bond between the anodized aluminum panels and any stiffeners (steel or aluminum) if used. If adequate bonding with anodized aluminum is not achievable, cleaning of anodized plates subsequent to anodizing to remove anodized material in bond areas or covering of bond areas prior to anodizing to prevent anodized layer from being formed, can be performed to allow the use of epoxy bonding. Adhesive bond failure can be categorized into three general groups; adhesive failure, adhesive-cohesive failure and cohesive failure, as illustrated in Figure 6.1. Adhesive failure Adhesive-cohesive failure Cohesive failure Figure 7.1 Adhesive bond failure categorization Typically, the adhesive is weaker than the substrate, which is most likely to be the case in joining two metallic components together as is the expected possible use for this project. Generally, the adhesive is selected to offer adequate resistance to tensile, shear, cleavage and peeling stresses, as shown in Figure 7.2. Figure 7.2 Adhesive design stresses 117 C H A P T E R 7 C O N N E C T I O N D E S I G N 7.1.2 Riveting of stiffeners Riveting provides a viable alternative to epoxy bonding stiffeners to the aluminum plates. Rivets have been used extensively to great success in the aircraft industry for similar applications. There are many options in terms of size and color of rivet, and also whether to countersink and to what level. Figure 7.3 shows typical rivet failure modes which must be designed against. A - Shear in rivet B - Tension in plate C - Tearing or shearing of plate D - Crushing of the plate or rivet Figure 7.3 Rivet failure modes 7.1.3 Cold forming of plate edges Cold forming the plates along either un-curved edge end to form the longitudinal stiffeners to be used for connection to the main Wheel structure is a viable option. Similarly to adhesives, this solution provides no visible evidence on the exposed surface of the panels. Cold forming can only be used for the longitudinal edge stiffeners, any internal stiffeners would have to be riveted or bonded. Economically it does not make sense to use more than 1 method of providing 118 C H A P T E R 7 C O N N E C T I O N DESIGN stiffeners, as each method requires capital investment in terms of machinery and space. Hence this solution is only viable if no further stiffening of the plates is required (i.e. no additional radial stiffeners are required). 7.2 Stiffener to main structure connection: In order to minimize the installation time and hence cost, the panels will be bolted to the main structure via the supporting longitudinal stiffeners. This will require much effort and care in the use of C A D software in order to ensure the accurate geometric positioning of holes in order to achieve the desired profiles. Once the optimal cladding panel geometries and loads are determined, the loads at each of the support locations can be determined, allowing detailed design of the panel to main structure connections. These connections will almost certainly be bolted connections, to allow for ease of erection in the field. 7.3 Summary: The connection design for this project, whether achieved through the use of adhesive, bolts or rivets, is not a critical aspect. Connection design is well developed and prescriptive, with manufacturers and suppliers providing all material information and design guidance required. For this reason, the connection design is not considered further in this report, allowing more time to be spent on the analysis and design of the panels. 119 C H A P T E R 8: I N T E R A C T I O N B E T W E E N P A N E L S CHAPTER 8: INTERACTION BETWEEN PANELS Welding of panels to form larger panels is not an option either before or after anodizing as this compromises the protection afforded by anodization in the area of the welds, and also is not aesthetically acceptable. This effectively limits the allowable panel dimensions to approximately 2m x 8m [dimensions obtained from fabricators]. With a surface area of approximately 16 m per panel, there will be a large edge length where separate panels are required to interact over the surface of the Wheel. For a given area, the edge length is at a minimum for a square plate and increases with increasing aspect ratio, suggesting that a square panel is more desirable than a rectangular panel. This aspect has not been included in any of the optimization schemes and should be considered. 8.1 Butting of adjacent panels: Butting of adjacent panels is undesirable due to the additional stresses induced upon constrained thermal expansion and the poor aesthetics of gaps between panels upon thermal contraction. A flexible medium could be used between panels to alleviate this concern, however the labour costs and aesthetics associated with this option make it undesirable. 8.2 Overlapping of adjacent panels: Overlapping of panels appears to be the most appropriate solution at this stage. This option would involve the use of tapered stiffeners to allow for overlaps. 120 C H A P T E R 9: A E S T H E T I C S CHAPTER 9: AESTHETICS While a structural engineer would typically wish that form would follow function for projects such as these, it is not always so. Due to the desire of the Client to create a high profile tourist attraction, aesthetics of the structure is the primary concern, with structural efficiency and hence cost coming second (within reason). There are many unanswered questions with regard to the final ' look' of the Mega Wheel. Due to the large scale of the project it is difficult to get a grasp of what the aesthetics w i l l be like in terms of approximating the curved structure with discrete panels, which may be of only single curvature or entirely flat. The question of how large the panels can be before the aesthetics are compromised is difficult to answer. Who w i l l decide when the aesthetics have been compromised? The temptation to refer these questions and decisions to the original architect should be avoided. Modern science is more than capable of providing defendable and objective evaluations of aesthetics. For example, there is a physical limit to the smallest angle which the human eye can detect, with this value and a set viewing position, simple geometry can be used to determine what length of straight panels would give the appearance of a completely curved surface. Similar knowledge could be used to determine the smallest size of rivet which would not be visible to the naked eye. This knowledge is available but is not common to most engineers and would require research and interaction with third party scientists and researchers. Information listed online by Wikipedia states that in the limit, the human eye can resolve distances of approximately 0.93mm at a distance of l m . If a viewing distance of 100m is assumed, these values suggest a fastener of 9.3cm in diameter w i l l not be visible (provided the colors match). However this does not take into account reflectivity, the fastener head w i l l reflect light at a different angle than the panel surface, which may make it visible as a defect. There is a need to quantify the aesthetics of the structure such that there is a tangible numeric value(s) which can be achieved and included in any contract for fabrication. Factors such as color consistency, reflectivity, geometrical accuracy etc... can be included to derive a form of 121 C H A P T E R 9: A E S T H E T I C S aesthetic merit function. This will allow different surface discretization and panel-stiffener connection schemes to be compared and evaluated in an objective manner. 122 C H A P T E R 10: CONCLUSIONS A N D R E C O M E N D A T I O N S CHAPTER 10: CONCLUSIONS AND RECOMENDATIONS From discussions with fabricators; the maximum panel dimensions in terms of fabrication is 3.175mm (2/16") thick and 2m wide by 8m in length. The anodization quality in terms of grain and color consistency can be guaranteed to match panel to panel. From the structural analysis performed thus far the following conclusions can be drawn: 1. Flat plates have no buckling instability, and allow very large panel sizes; up to the limit provided due to manufacturing considerations. . 2. For single curvature plates, plates curved in their short dimension have a significant structural advantage over plates curved in their long direction, with increasing allowable panel sizes with increasing aspect ratio. 3. In order to achieve reasonable panel dimensions and weight/area ratios, stiffening of the panels in the form of longitudinal and possibly radial stiffeners is required. 4. The use of longitudinal stiffeners greatly increases the allowable panel dimensions, with the resulting weight/area ratio of the panels heavily dependent on the number of supporting points per stiffener and hence the stiffener weight. 5. The use of aluminum longitudinal stiffeners results in an approximate increase in structural efficiency (defined as weight/area) of a factor of 2 relative to the use of steel stiffeners for the plate geometries considered. 6. Radial stiffeners further increase the allowable panel dimensions, with the resulting weight/area ratio of the panels heavily dependent on the stiffener dimensions and hence weight. 7. Stiffening elements drive the optimization function of weight divided by area, suggesting attention to efficient design of stiffener is merited. Areas of stress concentration are formed parallel to stiffeners, which in some cases govern the maximum stress in the plate, suggesting adequate time be spent designing the transition from plate to stiffener. 8. Preliminary investigation suggests the use of adhesive to bond stiffeners to panels is conceivable. 123 C H A P T E R 10: C O N C L U S I O N S A N D R E C O M E N D A T I O N S Areas of concern and requiring further research and investigation: 1. ANSYS Nonlinear buckling analysis was found to be unreliable and in most cases un-conservative. Resulting in the use of eigenvalue buckling analysis for the stability investigation. This is unsatisfactory due to the un-conservative nature of eigenvalue analysis. Further attempts at achieving reliable results using ANSYS® Nonlinear buckling analysis is warranted, at the same time, initial imperfections should be included in the FE models to obtain reduction factors to apply to eigenvalue buckling analyses. 2. Questions surrounding the aesthetics of the structure must be answered before structural design of the panels can continue. Of particular importance is a decision regarding allowable deflections and discretization of the curved surfaces of the Wheel. This will allow upper bounds and relative merits of structural systems to be evaluated. This in turn, will allow some of the design variables to be reduced either in range, or removed as variables completely. For example if the aspect ratio of panels can be limited to a smaller range or even a single value, this would dramatically reduce run time during optimization and will also allow more confidence that the global minimum is being reached (in terms of the optimization function). 3. The use of the efficiency function panel weight divided by area, to drive optimization of panel variables is questionable. This basic function does not include aspects associated with material cost, fabrication, erection and labor. A more comprehensive optimization function is warranted. 4. In depth CFD analysis and/or wind tunnel testing is required in order to investigate the effect of dynamic pressure application on the panels. Dynamic instability such as flutter may be a concern. 5. Thermal analysis is required in order to determine the implications associated with the use of differential metals for the panels and supporting structure, the induced stresses need to be quantified, and methods of alleviation investigated. This will allow detailed design of connections. 6. Further loading scenarios, such as seismic, transport and erection need to be evaluated. 124 3 OQ' 3 >-> 1-1 o T3 O co n Cu 3 CD ft-cr o 0Q 8 s a. ft CO tfq' 3 o o a &. p 3 * 3 5 3 3 o ET a. D -5' OQ g 1 Con'ti. nuts 2 Panel loading 3. Preliminary Sizing: Determine al'oa.ible s t r e s s [Me-miiK- slructur.ii ;ind Analysis of.unstiffened panels I rief.ectiors and panei geometry cnviionmenldl loading • w using formatted spreadsheets and design chans 4. Merit funitioii Determine ci suitable mt-rii f-jnttion fo ' optimisation of pare design moods 5. Panel optimization: Ana<yS'S s->d optimisation of FE models using eigenvalue buckling'analysis and>linea'r ' suevs.'ri.;f oi-tion analysis 5 1 Buckling reduction racto-s Determine DJCkiinq redact on factors to account for imperfections. 5 2 Thermal Analysis Determine additional strr".ses f c inclusion, in FE-pptimisation. f b Wind tiiniicliCF-D' 7. Additional Loading Scenarios. Determine wind load distributions Cnnck alternate loading during fabrication, transportation, erection Investigate dynarriu effects 8. Design checking-Non - l i n c a ' FE a n a k S ' S Experimental testing. >3 3 O 3 cr 3 5' OQ co o >-+> cr co' cr a CO_ co' i-i £ . n> o 3 3 o c 3 ft CO P O T3 O CO CO D -C L n CO Oq' 3 ft! cr o o OQ <^ S5 cr n 3 -ft CO Oq" 3 O '-*> o 3 < ft O-P_ 5" 3 5 3 3 o. ET a. S* 5' OQ o > -o H m 70 O O O r-O Z CO I O ?o m O O 2 m Z D > H O Z co REFERENCES 1. Cedric Marsh. (1983). Strength of Aluminum. Alcan Canada Products Ltd. 2. Young, W . (1989). Roarke's Formulas for Stress and Strain. M c G r a w - H i l l . 3. Timoshenko, S. P. and J. M . Gere. (1961). Theory of Elastic Stability (Engineering Society Monographs), 2nd. M c G r a w - H i l l Education. 4. Papadopoulos, V . and M . Papadrakakis. (2004). "Finite-Element Analysis of Cylindrical Panels with Random Initial Imperfections." Journal of Engineering Mechanics 130(8), August, 867-876. 5. Shen, Hui-Shen. (2003). "Postbuckling of Pressure-Loaded Functionally Graded Cylindrical Panels in Thermal Environments" Journal of Engineering Mechanics 129(4), April,414-425.. 6. Zhou, Y . (2005). Preliminary Structural Design, Panasonic Mega Wheel, Amec Dynamic Structures Ltd. 7. Askeland, D . R. and P. P. Phule. (2004). The Science and Engineering of Materials, Chapter 16 - Composites: Teamwork and Synergy in Materials, http://www.ccm.udel.edu/Personnel/homepage/class_web/Lecture%20Notes/2004/Askela ndPhuleNotes-CH16Printable.ppt#l, 4th. 10/05/06. 8. Marine Composites. (2003), . http://www.marinecomposites.com/PDF_Files/G_Composite_Materials.pdf, 2nd, Eric Greene Associates, ed. 10/05/2006. 9. Davies, J. M . and E . R. Bryan. (1982). Manual of Stressed Skin Diaphragm Design. London: Granada. 10. Merritt, F. S. (1996). "Tensioned Fabric Structures - A Practical Introduction edited by R. E . Schaeffer." Journal of Architectural Engineering 2(3), September, YlXTensioned Fabric Structures - A Practical Introduction, R. E . Schaeffer, ed. 11. Leonard, J. W . (1987). Tension Structures: Behaviour and Analysis. M c G r a w - H i l l . 12. N . A . Tiner. "Anodizing influences," http://www.p2pays.org/ref/06/05117/.. 22 July 2005. 13. K ing , R. G . (1988). Surface Treatment and Finishing of Aluminium. Pergamon Press. 14. Mil ton, S. and S. M . Grove. (1997). "Composite Sandwich Panel Manufacturing Concepts for a Lightweight Vehicle Chassis.", 30th International Symposium on Automative Technology and Automation (IS A T A ) . University of Plymouth. 15. Council on Tall Buildings and Urban Habitat, C . 1. (1992). Cladding, B . L . Bassler, ed. New York: M c G r a w - H i l l . 16. Holman, J. P. (1981). Heat Transfer, 5th. M c G r a w - H i l l . 17. O'Connor, J. J. and E . F. Robertson. (1996). "Quadratic, cubic and quartic equations," http://www-history.mcs.st-andrews.ac.uk/HistTopics/Quadratic_etc_equations.html. 14/05/2005. 18. A N S Y S Inc. (2005). ANSYS 9.0 Help Files. 19. 3 M . Adhesive Technology Designer's Reference Guide. 2005. 20. Wikipedia, http://en.wikipedia.org/wiki/Naked_eye, August. 22/08/2005 126 A P P E N D I X A: T A B U L A T E D D A T A APPENDIX A: TABULATED DATA A l Flat plates: p = 4.2 kNrrr2 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 100,000 t [mm] . 1.5875 S imply suppor ted e d g e s L b AR A a [mm] [mm] [mm] [MPa] 1000.0 100.0 10.0 0.2 12.7 890.0 112.4 7.9 0.3 15.6 801.8 124.7 6.4 0.4 18.1 729.5 137.1 5.3 0.5 20.2 669.2 149.4 4.5 0.7 21.7 618.0 161.8 3.8 0.8 22.8 574.2 174.2 3.3 0.9 23.7 536.1 186.5 2.9 1.0 24.3 502.8 198.9 2.5 1.1 24.7 473.4 211.2 2.2 1.3 25.1 447.2 223.6 2.0 1.4 25.4 316.2 316.2 1.0 1.9 22.3 A [mm2] 100,000 t [mm] 3.1750 S imply suppor ted edges L b AR A CT [mm] [mm] [mm] [MPa] 1000.0 100.0 10.0 0.0 3.1 890.0 112.4 7.9 0.0 4.0 801.8 124.7 6.4 0.1 4.9 729.5 137.1 5.3 0.1 6.0 669.2 149.4 4.5 0.1 7.1 618.0 161.8 3.8 0.2 8.4 574.2 174.2 3.3 0.2 9.7 536.1 186.5 2.9 0.3 11.1 502.8 198.9 2.5 0.4 12.6 473.4 211.2 2.2 0.5 14.1 447.2 223.6 2.0 0.6 15.5 316.2 316.2 1.0 1.0 12.7 A [mm2] 100,000 t [mm] 4.7625 S imply suppor ted e d g e s L b AR A a [mm] [mm] [mm] [MPa] 1000.0 100.0 10.0 0.0 1.4 890.0 112.4 7.9 0.0 1.8 801.8 124.7 6.4 0.0 2.2 729.5 137.1 5.3 0.0 2.6 669.2 149.4 4.5 0.0 3.1 618.0 161.8 3.8 0.1 3.7 574.2 174.2 3.3 0.1 4.2 536.1 . 186.5 2.9 0.1 4.9 502.8 198.9 2.5 0.1 5.6 473.4 211.2 2.2 0.2 6.3 447.2 223.6 2.0 0.2 7.1 316.2 316.2 1.0 0.4 5.7 128 APPENDIX A : T A B U L A T E D D A T A A [mm2] 200,000 t [mm] 1.5875 Simply supported edges L b A R A CT [mm] [mm] [mm] [MPa] 1414.2 141.4 10.0 0.6 20.8 1258.6 158.9 7.9 0.7 22.6 1133.9 176.4 6.4 0.9 23.8 1031.7 193.9 5.3 1.1 24.6 946.3 211.3 4.5 1.3 25.1 874.0 228.8 3.8 1.4 25.5 812.0 246.3 3.3 1.6 25.8 758.2 263.8 2.9 1-8 26.0 711.1 281.3 2.5 2.0 26.2 669.5 298.7 2.2 2.2 26.4 632.5 316.2 2.0 2.4 26.6 447.2 447.2 1.0 3.1 23.8 A [mm2] 200,000 t [mm] 3.1750 Simply supported edges L b A R A cr [mm] [mm] [mm] [MPa] 1414.2 . 141.4 10.0 0.1 6.3 1258.6 158.9 7.9 0.2 8.1 1133.9 176.4 6.4 0.2 1.0.0 1031.7 193.9 5.3 0.4 12.0 946.3 211.3 4.5 0.5 14.1 874.0 228.8 3.8 0.6 16.0 812.0 246.3 3.3 0.8 17.8 758.2 263.8 2.9 1.0 19.4 711.1 281.3 2.5 1.1 20.6 669.5 298.7 2.2 1.3 21.7 632.5 316.2 2.0 1.5 22.5 447.2 447.2 1.0 2.1 19.2 A [mm2] 200,000 t [mm] 4.7625 Simply supported edges L b A R A a [mm] [mm] [mm] [MPa] 1414.2 141.4 10.0 0.0 2.8 1258.6 158.9 7.9 0.0 3.5 1133.9 176.4 6.4 0.1 4.4 1031.7 193.9 5.3 0.1 5.3 946.3 211.3 4.5 0.2 6.3 874.0 228.8 3.8 0.2 7.4 812.0 246.3 3.3 0.3 8.6 758.2 263.8 2.9 0.4 9.9 711.1 281.3 2.5 . 0.5 11.2 669.5 298.7 2.2 0.6 12.6 632.5 316.2 2.0 0.7 14.0 447.2 447.2 1.0 1.2 11.4 129 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 300,000 t [mm] 1.5875 Simply supported edges L b A R A [mm] [mm] [mm] [MPa] 1732.1 173.2 10.0 0.9 23.6 1541.5 194.6 7.9 1.1 24.6 1388.7 216.0 6.4 1.3 25.2 1263.5 237.4 5.3 1.5 25.6 1159.0 258.8 4.5 1.8 25.9 1070.5 280.3 3.8 2.0 26.2 994.5 301.7 3.3 2.2 26.4 928.6 323.1 2.9 2.4 26.6 870.9 344.5 2.5 2.7 26.9 819.9 365.9 2.2 2.9 27.1 774.6 387.3 2.0 3.1 27.3 547.7 547.7 1.0 4.1 24.7 A [mm2] 300,000 t [mm] 3.1750 L b A R A cr [mm] [mm] [mm] [MPa] 1732.1 173.2 10.0 0.2 9.6 1541.5 194.6 7.9 0.4 1.2.1 1388.7 216.0 6.4 0.5 14.6 1263.5 237.4 5.3 0.7 17.0 1159.0 258.8 4.5 0.9 19.0 1070.5 280.3 3.8 1.1 20.6 994.5 301.7 3.3 1-3 21.8 928.6 323.1 2.9 1.5 22.8 870.9 344.5 2.5 1.8 23.5 819.9 365.9 2.2 2.0 24.1 774.6 387.3 2.0 2.2 24.6 547.7 547.7 1.0 3.0 21.3 A [mm2] 300,000 t [mm] 4.7625 Simply supported edges L b A R A a [mm] [mm] [mm] [MPa] 1732.1 173.2 10.0 0.1 4.2 1541.5 194.6 7.9 0.1 5.3 1388.7 216.0 6.4 0.2 6.6 1263.5 237.4 5.3 0.2 8.0 1159.0 258.8 4.5 0.3 9.5 1070.5 280.3 3.8 0.5 11.2 994.5 301.7 3.3 0.6 12.9 928.6 323.1 2.9 0.8 14.5 870.9 344.5 2.5 1.0 16.1 819.9 365.9 2.2 1.2 17.6 774.6 387.3 2.0 1.4 18.9 547.7 547.7 1.0 2.1 15.7 130 APPENDIX A : T A B U L A T E D D A T A A [mm2] 500,000 . t [mm] 1.5875 L b A R A cr [mm] [mm] [mm] [MPa] 2236.1 223.6 10.0 1.4 25.4 1990.1 251.2 7.9 1.7 25.8 1792.9 278.9 6.4 2.0 26.2 1631.2 306.5 5.3 2.3 26.5 1496.3 334.2 4.5 2.6 26.7 1382.0 361.8 3.8 2.9 27.0 1283.9 389.4 3.3 3.2 27.3 1198.8 417.1 2.9 3.5 27.6 1124.3 444.7 25 3.8 27.9 1058.5 472.4 2.2 4.1 28.3 1000.0 500.0 2.0 4.5 28.6 707.1 707.1 1.0 5.9 26.2 A [mm 2] 500,000 t [mm] 3.1750 Simply supported edges L b A R A a [mm] [mm] [mm] [MPa] 2236.1 223.6 10.0 0.6 15.5 1990.1 251.2 7.9 0.8 18.3 1792.9 278.9 6.4 1.1 20.5 1631.2 306.5 5.3 1.4 22.1 1496.3 334.2 4.5 1.7 23.2 1382.0 361.8 3.8 1.9 24.0 1283.9 389.4 3.3 2.2 24.6 1198.8 417.1 2.9 2.5 25.0 1124.3 444.7 2.5 2.8 25.3 1058.5 472.4 2.2 3.0 25.6 1000.0 500.0 2.0 3.3 25.8 707.1 707.1 1.0 4.4 22.8 A [mm2] 500,000 . t [mm] 4.7625 Simply supported edges L b A R A [mm] [mm] [mm] [MPa] 2236.1 223.6 10.0 0.2 7.1 1990.1 251.2 7.9 0.3 9.0 1792.9 278.9 6.4 0.5 11.1 1631.2 306.5 5.3 0.6 13.2 1496.3 334.2 4.5 0.9 15.4 1382.0 361.8 3.8 1.1 17.3 1283.9 389.4 3.3 1.4 19.0 1198.8 417.1 2.9 1.6 20.4 1124.3 444.7 2.5 1.9 21.6 1058.5 472.4 2.2 2.2 22.5 1000.0 500.0 2.0 2.5 23.2 707.1 707.1 1.0 3.5 19.9 131 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 1,000,000 t [mm] 1.5875 Simply supported edges L b A R A a [mm] [mm] [mm] [MPa] 3162.3 316.2 10.0 2.4 26.6 2814.4 355.3 7.9 2.8 27.0 2535.5 394.4 6.4 3.2 27.4 2306.9 433.5 5.3 3.7 27.8 2116.0 472.6 4.5 4.1 28.3 1954.4 511.7 3.8 4.6 28.8 1815.7 550.8 3.3 5.1 29.3 1695.4 589.8 2.9 5.6 29.8 1590.0 628.9 2.5 6.1 30.3 1497.0 668.0 2.2 6.6 30.9 1414.2 707.1 2.0 7.2 31.5 1000.0 1000.0 1.0 9.4 29.1 A [mm ] 1,000,000 t [mm] 3.1750 Simply supported edges L b A R A CT [mm] [mm] [mm] [MPa] 3162.3 316.2 10.0 1.5 22.5 2814.4 355.3 7.9 1.9 23.8 2535.5 394.4 6.4 2.3 24.7 2306.9 433.5 5.3 2.7 25.2 2116.0 472.6 4.5 3.1 25.6 1954.4 511.7 3.8 3.5 25.9 1815.7 550.8 3.3 3.9 26.1 1695.4 589.8 2.9 4.3 26.3 1590.0 628.9 2.5 4.7 26.5 1497.0 668.0 2.2 5.1 26.7 1414.2 707.1 2.0 5.5 26.9 1000.0 1000.0 1.0 7.3 24.3 A [mm2] 1,000,000 t [mm] 4.7625 Simply supported edges L b A R A CT [mm] [mm] [mm] [MPa] 3162.3 316.2 10.0 °-7 14.0 2814.4 355.3 7.9 1.1 16.9 2535.5 394.4 6.4 1-4 19.3 2306.9 433.5 5.3 1.8 21.1 2116.0 472.6 4.5 2.2 22.5 1954.4 511.7 3.8 . 2.6 23.4 1815.7 550.8 3.3 3.0 24.1 1695.4 589.8 2.9 3.4 24.7 1590.0 628.9 2.5 3.8 25.1 1497.0 668.0 2.2 4.2 25.4 1414.2 707.1 2.0 4.6 25.6 1000.0 1000.0 1.0 6.1 22.6 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 2,000,000 t [mm] 1.5875 L b A R A a [mm] [mm] [mm] [MPa] 4472.1 447.2 10.0 3.8 28.0 3980.2 502.5 7.9 4.5 28.6 3585.7 557.8 6.4 5.2 29.4 3262.4 613.0 5.3 5.9 30.1 2992.5 668.3 4.5 6.6 30.9 2763.9 723.6 3.8 7.4 31.7 2567.8 778.9 3.3 8.1 32.5 2397.6 834.2 2.9 8.9 33.3 2248.6 889.4 2.5 9.7 34.2 2117.0 944.7 2.2 10.6 35.0 2000.0 1000.0 2.0 11.4 35.8 1414.2 1414.2 1.0 14.9 33.5 A [mm2] 2,000,000 t [mm] 3.1750 Simply supported edges L b A R A [mm] [mm] [mm] [MPa] 4472.1 447.2 10.0 2.8 25.4 3980.2 502.5 7.9 3.4 25.8 3585.7 557.8 6.4 3.9 26.2 3262.4 613.0 5.3 4.5 26.5 2992.5 668.3 4.5 5.1 26.7 2763.9 723.6 3.8 5.7 27.0 2567.8 778.9 3.3 6.3 27.3 2397.6 834.2 2.9 7.0 27.6 2248.6 889.4 2.5 7.6 27.9 2117.0 944.7 2.2 8.3 28.3 2000.0 1000.0 2.0 9.0 28.6 1414.2 1414.2 1.0 11.7 26.2 A [mm2] 2,000,000 t [mm] 4.7625 Simply supported edges L b A R A a [mm] [mm] [mm] [MPa] 4472.1 447.2 10.0 1.9 21.7 3980.2 502.5 7.9 2.5 23.2 3585.7 557.8 6.4 3.1 24.2 3262.4 613.0 5.3 3.6 24.9 2992.5 668.3 4.5 4.2 25.4 2763.9 723.6 3.8 4.7 25.7 2567.8 778.9 3.3 5.3 25.9 2397.6 834.2 2.9 5.9 26.2 2248.6 889.4 2.5 6.5 26.4 2117.0 944.7 2.2 7.0 26.6 2000.0 1000.0 2.0 7.6 26.7 1414.2 1414.2 1.0 10.1 24.0 APPENDIX A : T A B U L A T E D D A T A A [mm2] 4,000,000 t [mm] 1.5875 Simply supported edges L b A R A cr [mm] [mm] [mm] [MPa] 6324.6 632.5 10.0 6.2 30.4 5628.8 . 710.6 7.9 7.2 31.5 5070.9 788.8 6.4 8.3 32.7 4613.7 867.0 5.3 9.4 33.8 4232.1 945.2 4.5 10.6 35.0 3908.8 1023.3 3.8 11.7 36.2 3631.4 1101.5 3.3 13.0 37.4 3390.7 1179.7 2.9 14.2 38.6 3180.0 1257.9 2.5 15.5 39.7 2993.9 . 1336.0 2.2 16.8 40.9 2828.4 1414.2 2.0 18.1 42.1 2000.0 2000.0 1.0 23.7 39.6 A [mm2] 4,000,000 t [mm] 3.1750 Simply supported edges L b A R A rj [mm] [mm] [mm] [MPa] 6324.6 632.5 10.0 4.7 26.6 5628.8 710.6 7.9 5.6 27.0 5070.9 788.8 6.4 6.5 27.4 4613.7 867.0 5.3 7.4 . 27.8 4232.1 945.2 4.5 8.3 28.3 3908.8 1023.3 3.8 9.2 28.8 3631.4 1101.5 3.3 10.2 29.3 3390.7 1179.7 2.9 11.2 29.8 3180.0 1257.9 2.5 12.2 30.3 2993.9 1336.0 2.2 13.3 30.9 2828.4 1414.2 2.0 14.3 31.5 2000.0 2000.0 1.0 18.7 29.1 A [mm2] 4,000,000 t [mm] 4.7625 Simply supported edges L b A R A rj [mm] [mm] [mm] [MPa] 6324.6 632.5 10.0 3.8 25.1 5628.8 710.6 7.9 4.6 25.6 5070.9 788.8 6.4 5.4 26.0 4613.7 867.0 5.3 6.2 26.3 4232.1 945.2 4.5 7.0 26.6 3908.8 1023.3 3.8 7.9 26.8 3631.4 1101.5 3.3 8.8 27.1 3390.7 1179.7 2.9 9.6 ' 27.4 3180.0 1257.9 2.5 10.5 27.6 2993.9 . 1336.0 2.2 11.5 27.9 2828.4 1414.2 2.0 12.4 28.3 2000.0 2000.0 1.0 16.3 25.8 134 APPENDIX A : T A B U L A T E D D A T A A [mm2] 10,000,000 t [mm] 1.5875 Simply supported edges L b A R A a [mm] [mm] [mm] [MPa] 10000.0 1000.0 10.0 11.4 35.8 8899.9 1123.6 7.9 13.3 37.7 8017.9 1247.2 6.4 15.3 39.6 7294.9 ' 1370.8 5.3 17.4 41.4 6691.5 1494.4 4.5 19.5 43.3 6180.3 1618.0 3.8 21.7 45.1 5741.7 1741.6 3.3 23.9 46.9 5361.2 1865.2 2.9 26.2 48.7 5028.0 1988.9 2.5 28.5 50.4 4733.8 2112.5 2.2 30.9 52.1 4472.1 2236.1 2.0 33.4 53.8 3162.3 3162.3 1.0 43.6 50.9 A [mm2] 10,000,000 t [mm] 3.1750 Simply supported edges L b A R A a [mm] [mm] [mm] [MPa] 10000.0 1000.0 10.0 9.0 28.6 8899.9 1123.6- 7.9 10.5 29.4 8017.9 1247.2 6.4 12.1 30.3 7294.9 1370.8 5.3 13.7 31.1 6691.5 1494.4 4.5 15.4 32.0 6180.3 1618.0 3.8 17.1 33.0 5741.7 1741.6 3.3 18.9 33.9 5361.2 1865.2 2.9 20.7 34.8 5028.0 1988.9 2.5 22.6 35.8 4733.8 2112.5 2.2 24.5 36.7 4472.1 2236.1 2.0 26.4 37.6 3162.3 3162.3 1.0 34.6 35.2 A [mm2] 10,000,000 t [mm] 4.7625 Simply supported edges L b A R A fj [mm] [mm] [mm] [MPa] 10000.0 1000.0 10.0 7.6 26.7 8899.9 1123.6 7.9 9.0 27.2 8017.9 1247.2 6.4 10.4 27.6 7294.9 1370.8 5.3 11.9 28.1 6691.5 1494.4 4.5 13.4 28.6 6180.3 1618.0 3.8 14.9 29.1 5741.7 1741.6 3.3 16.4 29.7 5361.2 1865.2 2.9 18.0 30.2 5028.0 1988.9 2.5 19.7 30.8 4733.8 2112.5 2.2 21.3 31.4 4472.1 2236.1 2.0 23.0 32.0 3162.3 3162.3 1.0 30.2 29.7 135 APPENDIX A : T A B U L A T E D D A T A A [mm2] 20,000,000 t [mm] 1.5875 Simply supported edges L b A R A rj [mm] [mm] [mm] [MPa] 14142.1 1414.2 10.0 18.1 42.1 12586.4 1589.0 7.9 21.1 44.7 11339.0 1763.8 6.4 24.3 47.2 10316.5 1938.6 5.3 27.6 49.7 9463.2 2113.4 4.5 30.9 52.1 8740.3 2288.2 3.8 34.4 54.5 8120.0 2463.1 3.3 37.9 56.9 7581.9 2637.9 2.9 41.6 59.2 7110.7 2812.7 2.5 45.3 61.5 6694.6 2987.5 2.2 49.1 63.8 6324.6 3162.3 2.0 53.0 66.0 4472.1 4472.1 1.0 69.2 62.6 A [mm ] 20,000,000 t [mm] 3.1750 Simply supported edges L b A R A a [mm] [mm] [mm] [MPa] 14142.1 1414.2 10.0 14.3 31.5 12586.4 1589.0 7.9 16.7 32.7 11339.0 1763.8 6.4 19.3 34.1 10316.5 1938.6 5.3 21.8 35.4 9463.2 2113.4 4.5 24.5 36.7 8740.3 2288.2 3.8 27.3 38.0 8120.0 2463.1 3.3 30.1 39.3 7581.9 2637.9 2.9 33.0 40.7 7110.7 2812.7 2.5 35.9 42.0 6694.6 2987.5 2.2 38.9 43.3 6324.6 3162.3 2.0 42.0 44.5 4472.1 4472.1 1.0 54.9 42.0 A [mm2] 20,000,000 t [mm] 4.7625 Simply supported edges L b A R A c? [mm] [mm] [mm] [MPa] 14142.1 1414.2 10.0 12.4 28.3 12586.4 1589.0 7.9 14.5 29.0 11339.0 1763.8 6.4 16.7 29.8 10316.5 1938.6 5.3 19.0 30.6 9463.2 2113.4 4.5 21.4 31.4 8740.3 2288.2 3.8 23.8 32.3 8120.0 2463.1 3.3 26.2 33.1 7581.9 2637.9 2.9 28.8 34.0 7110.7 2812.7 2.5 31.3 34.9 6694.6 2987.5 2.2 34.0 35.8 6324.6 3162.3 2.0 36.7 36.7 4472.1 4472.1 1.0 47.9 34.3 136 APPENDIX A : T A B U L A T E D D A T A A [mm2] 40,000,000 t [mm] 1.5875 Simply supported edges L b A R A CT [mm] [mm] [mm] [MPa] 20000.0 2000.0 10.0 28.7 50.6 17799.8 2247.2 7.9 33.6 54.0 16035.7 2494.4 6.4 38.6 57.3 14589.8 2741.6 5.3 43.8 60.6 13383.1 2988.9 4.5 49.1 63.8 12360.7 3236.1 3.8 54.6 66.9 11483.4 3483.3 3.3 60.2 70.0 10722.4 3730.5 2.9 66.0 73.0 10056.0 3977.7 2.5 71.9 75.9 9467.6 4224.9 2.2 77.9 78.8 8944.3 4472.1 2.0 84.1 81.7 6324.6 6324.6 1.0 109.9 77.6 A [mm2] 40,000,000 t [mm] 3.1750 Simply supported edges L b A R A CT [mm] [mm] [mm] [MPa] 20000.0 2000.0 10.0 22.8 35.8 17799.8 2247.2 7.9 26.6 37.7 16035.7 2494.4 6.4 30.6 39.6 14589.8 2741.6 5.3 34.7 41.4 13383.1 2988.9 4.5 39.0 43.3 12360.7 3236.1 3.8 43.3 45.1 11483.4 3483.3 3.3 47.8 46.9 10722.4 3730.5 2.9 52.4 48.7 10056.0 3977.7 2.5 57.1 50.4 9467.6 4224.9 2.2 61.8 52.1 8944.3 4472.1 2.0 66.7 53.8 6324.6 6324.6 1.0 87.2 50.9 A [mm2] 40,000,000 t [mm] 4.7625 Simply supported edges L b A R A CT [mm] [mm] [mm] [MPa] 20000.0 2000.0 10.0 19.8 30.9 17799.8 2247.2 7.9 23.2 32.1 16035.7 2494.4 6.4 26.7 33.3 14589.8 2741.6 5.3 30.3 34.5 13383.1 2988.9 4.5 34.0 35.8 12360.7 3236.1 3.8 37.8 37.0 11483.4 3483.3 3.3 41.7 38.3 10722.4 3730.5 2.9 45.7 39.5 10056.0 3977.7 2.5 49.8 40.8 9467.6 4224.9 2.2 54.0 42.0 8944.3 4472.1 2.0 58.3 43.2 6324.6 6324.6 1.0 76.2 40.7 137 APPENDIX A: T A B U L A T E D D A T A 142 APPENDIX A : T A B U L A T E D D A T A A [nW] 100,000 t [mm] 3.175 Fixed edges L b AR A a [mm] [mm] [mm] [MPa] 1000.0 100.0 10.0 0.005 2.0 897.8 111.4 8.1 0.008 2.5 814.6 122.8 6.6 0.012 3.0 745.5 134.1 5.6 0.017 3.6 687.2 145.5 4.7 0.024 4.2 637.3 156.9 4.1 0.032 4.9 594.2 168.3 3.5 0.042 5.7 556.6 179.7 3.1 0.055 6.5 523.4 191.0 2.7 0.070 7.3 494.0 202.4 2.4 0.088 8.2 467.7 213.8 2.2 0.11 9.2 444.1 225.2 2.0 0.24 13.1 422.7 236.6 1.8 0.24 13.1 403.3 247.9 1.6 0.25 13.1 385.6 259.3 1.5 0.26 13.1 369.4 270.7 1.4 0.28 13.1 354.5 282.1 1.3 0.29 13.1 340.8 293.5 1.2 0.31 13.1 328.0 304.8 1.1 0.32 13.1 316:2 316.2 1.0 0.34 13.1 A [mm'] 100,000 t [mm] 1.5875 Fixed edges L b AR A o [mm] [mm] [mm] [MPa] 1000.0 100.0 10.0 0.042 8.0 897.8 111.4 8.1 0.065 10.0 814.6 122.8 6.6 0.10 12.2 745.5 134.1 5.6 0.14 14.5 687.2 145.5 4.7 0.19 17.1 637.3 156.9 4.1 0.25 19.9 594.2 168.3 3.5 0.33 22.8 556.6 179.7 3.1 0.42 25.8 523.4 191.0 2.7 0.52 28.8 494.0 202.4 2.4 0.63 31.6 467.7 213.8 2.2 0.76 34.3 444.1 225.2 2.0 1.34 38.2 422.7 236.6 1.8 1.37 38.6 403.3 247.9 1.6 1.40 38.9 385.6 259.3 1.5 1.42 39.1 369.4 270.7 1.4 1.44 39.3 354.5 282.1 1.3 1.46 39.5 340.8 293.5 1.2 1.48 39.6 328.0 304.8 1.1 1.50 39.6 316.2 316.2 1.0 1.52 39.6 A [mm'] 100,000 t [mm] 4.7625 Fixed edges L b AR A a [mm] [mm] [mm] [MPa] 1000.0 100.0 10.0 0.0016 0.9 897.8 111.4 8.1 0.0024 1.1 814.6 122.8 6.6 0.0035 1.3 745.5 134.1 5.6 0.0050 1.6 687.2 145.5 4.7 0.0070 1.9 637.3 156.9 4.1 0.0094 2.2 594,2 168.3 3.5 0.012 2.5 556.6 179.7 3.1 0.016 2.9 523.4 191.0 2.7 0.021 3.2 494.0' 202.4 2.4 0.026 3.6 467.7 213.8 2.2 0.032 4.1 444.1 225.2 2.0 0.070 5.8 422.7 236.6 1.8 0.073 5.8 403.3 247.9 1.6 0.075 5.8 385.6 259.3 1.5 0.079 5.8 369.4 270.7 1.4 0.082 5.8 354.5 282.1 1.3 0.087 5.8 340.8 293.5 1.2 0.092 5.8 328.0 304.8 1.1 0.10 5.8 316.2 316.2 1.0 0.10 5.8 APPENDIX A : T A B U L A T E D D A T A A [mm2] 200,000 t [mm] 3.175 Fixed edges L b AR A a [mm] [mm] [mm] [MPa] 1414.2 141.4 10.0- 0.02 4.0 1269.7 157.5 8.1 0.03 5.0 1152.0 173.6 6.6 0.05 6.0 1054.3 189.7 5.6 0.07 7.2 971.8 205.8 4.7 0.09 8.5 901.3 221.9 4.1 0.13 9.9 840.4 238.0 3.5 0.2 11.4 787.1 254.1 3.1 0.2 13.0 740.3 270.2 2.7 0.3 14.8 698.6 286.3 2.4 0.3 16.6 661.5 302.4 2.2 0.4 18.5 628.0 318.5 2.0 0.9 25.5 597.8 334.6 1.8 0.9 25.6 570.4 350.6 1.6 1.0 25.6 545.3 366.7 1.5 1.0 25.6 522.4 382.8 1.4 1.0 25.7 501.3 398.9 1.3 1.1 25.7 481.9 415.0 1.2 1.1 25.7 463.9 431.1 1.1 1.2 25.7 447.2 447.2 1.0 1.2 25.7 A [mm2] 200,000 t [mm] 1.5875 Fixed edges L b AR A a [mm] [mm] [mm] [MPa] 1414.2 141.4 10.0 0.2 16.2 1269.7 157.5 8.1 0.3 20.1 1152.0 173.6 6.6 0.4 24.2 1054.3 189.7 5.6 0.5 28.4 971.8 205.8 4.7 0.7 32.4 901.3 221.9 4.1 0.8 36.0 840.4 238.0 3.5 1.0 39.0 787.1 254.1 3.1 1.2 41.4 740.3 270.2 2.7 1.4 43.4 698.6 286.3 2.4 1.6 44.8 661.5 302.4 2.2 1.8 45.9 628.0 318.5 2.0 2.8 42.2 597.8 334.6 1.8 2.8 42.9 570.4 350.6 1.6 2.8 43.4 545.3 366.7 1.5 2.8 43.8 522.4 382.8 1.4 2.8 44.1 501.3 398.9 1.3 2.9 44.3 481.9 415.0 1.2 2.9 44.5 463.9 431.1 1.1 2.9 44.6 447.2 447.2 1.0 2.9 44.6 144 APPENDIX A : T A B U L A T E D D A T A A [mm2] 300,000 t [mm] 3.175 Fixed edges L b AR A a [mm] [mm] [mm] [MPa] 1732.1 173.2 10.0 0.05 6.0 1555.1 192.9 8.1 0.07 7.5 1410.9 212.6 6.6 0.11 9.1 1291.2 232.3 5.6 0.15 10.9 1190.2 252.1 4.7 0.21 12.8 1103.9 271.8 4.1 0.28 14.9 1029.3 291.5 3.5 0.38 17.2 964.1 311.2 3.1 0.48 19.6 906.6 330.9 2.7 0.61 22.1 855.7 350.6 2.4 0.76 24.7 810.1 370.3 2.2 0.93 27.3 769.2 390.0. 2.0 1.8 33.8 732.2 409.7 1.8 1.8 34.1 698.6 429.5 1.6 1.9 34.3 667.9 449.2 1.5 1.9 34.4 639.8 468.9 1.4 2.0 34.5 614.0 488.6 1.3 2.0 34.6 590.2 508.3 1.2 2.1 34.7 568.2 528.0 1.1 2.1 34.7 547.7 547.7 1.0 2.2 34.7 A 300,000 t 1.5875 Fixed edges L b AR A <T [mm] [mm] [mm] [MPa] 1732.1 173.2 10.0 0.36 24.1 1555.1 • 192.9 8.1 0.54 29.3 1410.9 212.6 6.6 0.74 34.0 1291.2 232.3 5.6 1.0 38.0 1190.2 252.1 4.7 1.2 41.2 1103.9 271.8 4.1 1.4 43.5 1029.3 291.5 3.5 1.7 45.2 964.1 311.2 3.1 1.9 46.4 906.6 330.9 2.7 2.2 47.3 855.7 350.6 2.4 2.4 47.8 810.1 370.3 2.2 2.6 48.2 769.2 390.0 2.0 4.0 42.4 732.2 409.7 1.8 4.0 43.0 698.6 429.5 1.6 4.0 43.5 667.9 449.2 1.5 3.9 44.0 639.8 468.9 1-4 3.9 44.3 614.0 488.6 1.3 3.9 44.6 590.2 508.3 1.2 3.9 44.7 568.2 528.0 1.1 4.0 44.8 547.7 547.7 1.0 4.0 44.9 145 APPENDIX A : T A B U L A T E D D A T A A [mm2] 500,000 t [mm] 3.175 Fixed edges L b A R A [mm] [mm] [mm] [MPa] 2236.1 223.6 10.0 0.13 10.1 2007.6 249.1 8.1 0.20 12.5 1821.5 274.5 6.6 0.30 15.2 1667.0 299.9 5.6 0.42 18.2 1536.6 325.4 4.7 0.58 21.4 1425.1 350.8 4.1 0.76 24.7 1328.8 376.3 3.5 1.0 28.0 1244.6 401.7 3.1 1.2 31.3 1170.5 427.2 2.7 1.5 34.2 1104.6 452.6 2.4 1.8 36.9 1045.8 478.1 2.2 2.1 39.2 993.0 503.5 2.0 3.5 40.3 945.2 529.0 1.8 3.6 40.8 901.8 554.4 1.6 3.6 41.2 862.3 579.9 1.5 3.6 41.5 826.0 605.3 1.4 3.7 41.8 792.7 630.8 1.3 3.7 42.0 761.9 656.2 1.2 3.7 42.1 733.5 681.7 1.1 3.8 42.2 707.1 707.1 1.0 3.8 42.2 A 500,000 t 1.5875 Fixed edges L b A R A a [mm] [mm] [mm] [MPa] 2236.1 223.6 10.0 0.87 36.4 2007.6 249.1 8.1 . 1.2 40.7 1821.5 274.5 6.6 1.5 43.8 1667.0 299.9 5.6 1.8 45.8 1536.6 325.4 4.7 2.1 47.1 1425.1 350.8 4.1 2.4 47.9 1328.8 376.3 3.5 2.7 48.3 1244.6 401.7 3.1 3.0 48.6 1170.5 427.2 2.7 3.4 48.8 1104.6 452.6 2.4 3.7 48.8 1045.8 478.1 2.2 4.0 48.9 993.0 503.5 2.0 5.9 42.0 945.2 529.0 1.8 5.8 42.6 901.8 554.4 1.6 • 5.8 43.1 862.3 579.9 1.5 5.8 43.6 826.0 605.3 1.4 5.8 43.9 792.7 630.8 1.3 5.7 44.2 761.9 656.2 1.2 5.7 44.3 733.5 681.7 1.1 5.7 44.4 707.1 707.1 1.0 5.7 44.4 146 APPENDIX A : T A B U L A T E D D A T A A 1,000,000 t 3.175 Fixed edges L b A R A o [mm] [mm] [mm] [MPa] 3162.3 316.2 10.0 0.52 20.2 2839.2 352.2 8.1 0.77 24.9 2576.0 388.2 6.6 1.1 29.6 2357.4 424.2 5.6 1.5 33.9 2173.1 460.2 4.7 1.9 37.6 2015.4 496.2 4.1 2.3 40.6 1879.1 532.2 3.5 2.7 42.9 1760.1 568.1 3.1 3.2 44.6 1655.3 604.1 . 2.7 3.6 45.9 1562.2 640.1 2.4 4.1 46.8 1479.1 676.1 2.2 4.5 47.5 1404.3 712.1 2.0 6.9 42.4 1336.7 748.1 1.8 6.8 43.0 1275.4 784.1 1.6 6.8 43.6 1219.4 820.1 1.5 6.8 44.0 1168.2 856.0 1.4 6.8 44.3 1121.0 892.0 1.3 6.8 44.6 1077.6 928.0 1.2 6.9 44.7 1037.3 964.0 1.1 6.9 44.8 1000.0 1000.0 1.0 6.9 44.9 A [mm2] 1,000,000 t [mm] 1.5875 Fixed edges L b A R A CT [mm] [mm] [mm] . [MPa] 3162.3 316.2 10.0 2.0 46.7 2839.2 352.2 8.1 2.4 47.9 2576.0 388.2 6.6 2.9 48.5 2357.4 424.2 5.6 3.3 48.7 2173.1 460.2 4.7 3.8 48.8 2015.4 496.2 4.1 4.2 48.8 1879.1 532.2 3.5 4.7 48.8 1760.1 568.1 3.1 5.1 48.8 1655.3 604.1 2.7 5.6 48.8 1562.2 640.1 2.4 6.1 48.8 1479.1 676.1 2.2 6.6 48.8 1404.3 712.1 2.0 9.6 42.0 1336.7 748.1 1.8 9.5 42.6 1275.4 784.1 1.6 9.4 43.1 1219.4 820.1 1.5 9.4 43.5 1168.2 856.0 1-4 9.3 43.8 1121.0 892.0 1.3 9.3 44.1 1077.6 928.0 1.2 9.3 44.2 1037.3 964.0 1.1 9.3 44.3 1000.0 1000.0 1.0 9.3 44.3 147 APPENDIX A: T A B U L A T E D D A T A A [mm2] 2,000,000 t [mm] 3.175 Fixed edges L b AR A a [mm] [mm] [mm] [MPa] 4472.1 447.2 10.0 1.7 36.4 4015.2 498.1 8.1 2.3 40.7 3643.0 549.0 6.6 3.0 43.8 3333.9 599.9 5.6 3.6 45.8 3073.2 650.8 4.7 4.2 47.1 2850.3 701.7 4.1 4.8 47.9 2657.5 752.6 3.5 5.4 48.3 2489.2 803.5 3.1 6.1 48.6 2340.9 854.4 2.7 6.7 48.8 2209.3 905.3 2.4 7.3 48.8 2091.7 956.2 2.2 8.0 48.9 1986.0 1007.1 2.0 11.7 42.0 1890.4 1058.0 1.8 11.7 42.6 1803.7 1108.8 1.6 11.6 43.1 1724.5 1159.7 1.5 11.5 43.6 1652.0 1210.6 1.4 11.5 43.9 1585.4 1261.5 1.3 11.5 44.2 1523.9 1312.4 1.2 11.5 44.3 1467.0 1363.3 .1.1 11.5 44.4 1414.2 1414.2 1.0 11.5 44.4 A [mm2] 2,000,000 t [mm] 1.5875 L b AR A a [mm] [mm] [mm] [MPa] 4472.1 447.2 10.0 3.6 48.8 4015.2 498.1 8.1 4.2 48.8 3643.0 549.0 6.6 4.9 48.8 3333.9 599.9 5.6 5.6 48.8 3073.2 650.8 4.7 • 6.3 48.8 2850.3 701.7 4.1 6.9 48.9 2657.5 752.6 3.5 7.7 49.0 2489.2 803.5 3.1 8.4 49.1 2340.9 854.4 2.7 9.1 49.4 2209.3 905.3 2.4 9.9 49.6 2091.7 956.2 2.2 10.6 49.9 1986.0 1007.1 2.0 15.4 43.5 1890.4 1058.0 1.8 15.2 44.1 1803.7 1108.8 1.6 15.1 44.6 1724.5 1159.7 1.5 15.0 45.0 1652.0 1210.6 1.4 15.0 45.3 1585.4 1261.5 1.3 14.9 45.6 1523.9 1312.4 1.2 14.9 45.7 1467.0 . 1363.3 1.1 14.8 45.8 1414.2 1414.2 1.0 14.8 45.8 APPENDIX A : T A B U L A T E D D A T A A [mm2] 4,000,000 t [mm] 3.175 Fixed edges L b A R A a [mm] [mm] [mm] [MPa] 6324.6 632.5 10.0 4.0 46.7 5678.3 704.4 8.1 4.9 47.9 5151.9 776.4 6.6 5.7 48.5 4714.8 848.4 5.6 6.6 48.7 4346.1 920.4 4.7 7.5 48.8 4030.9 992.3 4.1 8.4 48.8 3758.3 1064.3 3.5 9.4 48.8 3520.2 1136.3 3.1 10.3 48.8 3310.5 1208.3 2.7 11.3 48.8 3124.4 1280.2 2.4 12.2 48.8 2958.1 1352.2 2.2 13.2 48.8 2808.6 1424.2 2.0 19.2 42.0 2673.5 1496.2 1.8 19.0 42.6 2550.8 1568.1 1.6 18.9 43.1 2438.8 1640.1 1.5 18.8 43.5 2336.3 1712.1 1.4 18.7 43.8 2242.1 1784.1 1.3 18.6 44.1 2155.1 1856.0 1.2 18.6 44.2 2074.7 1928.0 1.1 18.6 44.3 2000.0 2000.0 1.0 18.6 44.3 A [mm2] 4,000,000 t [mm] 1.5875 Fixed edges L b A R A CT [mm] [mm] [mm] [MPa] 6324.6 632.5 10.0 6.0 48.8 5678.3 704.4 8.1 7.0 48.9 5151.9 776.4 6.6 8.0 49.0 4714.8 848.4 5.6 9.0 49.3 4346.1 920.4 4.7 10.1 49.7 4030.9 992.3 4.1 11.2 50.1 3758.3 1064.3 3.5 . 12.3 50.6 3520.2 1136.3 3.1 13.4 51.2 3310.5 1208.3 2.7 14.6 51.8 3124.4 1280.2 2.4 15.8 52.4 2958.1 1352.2 2.2 17.0 53.1 2808.6 1424.2 2.0 24.5 47.2 2673.5 1496.2 1.8 24.3 47.8 2550.8 1568.1 1.6 24.1 48.3 2438.8 1640.1 1.5 23.9 48.7 2336.3 1712.1 1.4 23.8 49.0 2242.1 1784.1 1.3 23.7 49.2 2155.1 1856.0 1.2 23.7 49.4 2074.7 1928.0 1.1 23.6 49.5 2000.0 2000.0 1.0 23.6 49.5 149 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 10,000,000 t [mm] 3.175 Fixed edges L b AR A a [mm] [mm] [mm] [MPa] 10000.0 1000.0 10.0 8.5 48.8 8978.2 1113.8 8.1 10.0 48.8 8145.9 1227.6 6.6 11.5 48.8 7454.8 1341.4 5.6 13.0 48.8 6871.8 1455.2 4.7 14.6 48.9 6373.4 1569.0 4.1 16.2 49.1 5942.4 1682.8 3.5 17.9 49.3 5566.0 1796.6 3.1 19.5 49.6 5234.4 1910.4 2.7 21.2 49.9 4940.1 2024.2 2.4 23.0 50.3 4677.2 2138.0 2.2 24.8 50.7 4440.8 2251.8 2.0 35.8 44.5 4227.2 2365.6 1.8 35.4 45.1 4033.1 2479.5 1.6 35.1 45.6 3856.2 2593.3 1.5 34.9 46.0 3694.0 2707.1 1.4 34.8 46.3 3545.0 2820.9 1.3 34.6 46.5 3407.5 2934.7 1.2 34.5 46.6 3280.3 3048.5 1.1 34.5 46.7 3162.3 3162.3 1.0 34.5 46.8 A 10,000,000 t 1.5875 Fixed edges L b AR A <7 [mm] [mm] [mm] [MPa] 10000.0 1000.0 10.0 11.3 50.2 8978.2 1113.8 8.1 13.1 51.0 8145.9 1227.6 6.6 14.9 52.0 7454.8 1341.4 5.6 16.8 53.0 6871.8 1455.2 4.7 18.8 54.1 6373.4 1569.0 4.1 20.7 55.2 5942.4 1682.8 3.5 22.8 56.4 5566.0 1796.6 3.1 24.9 57.6 5234.4 1910.4 2.7 27.0 58.9 4940.1 2024.2 2.4 29.2 60.1 4677.2 2138.0 2.2 31.4 61.4 4440.8 2251.8 2.0 45.3 55.7 4227.2 2365.6 1.8 44.8 56.4 4033.1 2479.5 1.6 44.4 57.0 3856.2 2593.3 1.5 44.2 57.4 3694.0 2707.1 1.4 43.9 57.7 3545.0 2820.9 1.3 43.8 58.0 3407.5 2934.7 1.2 43.7 58.2 3280.3 3048.5 1.1 43.6 58.3 3162.3 3162.3 1.0 43.6 58.3 150 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 20,000,000 t [mm] 3.175 Fixed edges L b AR A o [mm] [mm] [mm] [MPa] 14142.1 1414.2 10.0 14.0 48.9 12697.1 1575.2 8.1 16.3 49.1 11520.1 1736.1 6.6 18.6 49.4 10542.7 1897.0 5.6 21.0 49.9 . 9718.2 2058.0 4.7 23.5 50.4 9013.4 2218.9 4.1 26.0 51.0 8403.8 2379.9 3.5 28.6 51.6 7871.5 2540.8 . 3.1 31.2 52.3 7402.6 2701.8 27 33.9 53.1 6986.4 2862.7 2.4 36.7 53.8 6614.5 3023.6 2.2 39.5 54.6 6280.2 3184.6 2.0 56.9 48.8 5978.1 3345.5 1.8 56.4 49.4 5703.7 3506.5 1.6 55.9 50.0 5453.4 3667.4 1.5 55.6 50.4 5224.2 3828.4 1.4 55.3 50.7 5013.4 3989.3 1.3 55.1 50.9 4819.0 4150.2 1.2 55.0. 51.1 4639.1 4311.2 1.1 54.9 51.2 4472.1 4472.1 1.0 54.9 51.2 A [mml 20,000.000 t [mm] 1.5875 Fixed edges L b AR A a [mm] [mm] [mm] [MPa] 14142.1 1414.2 10.0 18.0 53.7 12697.1 1575.2 8.1 20.9 55.3 11520.1 1736.1 6.6 23.8 57.0 10542.7 1897.0 5.6 26.8 58.7 9718.2 2058.0 4.7 29.8 60.5 9013.4 2218.9 4.1 33.0 62.3 8403.8 2379.9 3.5 36.2 64.1 7871.5 2540.8 3.1 39.5 65.9 7402.6 2701.8 2.7 42.9 677 6986.4 28627 2.4 46.4 69.5 6614.5 3023.6 2.2 49.9 71.3 6280.2 3184.6 2.0 71.9 65.6 5978.1 3345.5 1.8 71.2 66.4 5703.7 3506.5 1.6 70.6 67.0 5453.4 3667.4 1.5 70.1 67.5 5224.2 3828.4 1.4 69.8 67.8 5013.4 3989.3 1.3 69.5 68.1 4819.0 4150.2 1.2 69.3 68.3 4639.1 4311.2 1.1 69.2 68.4 4472.1 4472.1 1.0 69.2 68.5 A [mm2] 20,000,000 t [mm] 4.7625 Fixed edges L b AR A a [mm] [mm] [mm] [MPa] 14142.1 1414.2 10.0 11.7 48.8 12697.1 1575.2 8.1 13.8 48.8 11520.1 1736.1 6.6 15.9 48.8 10542.7 1897.0 5.6 18.0 48.8 9718.2 2058,0 4.7 20.2 48.8 9013.4 2218.9 4.1 22.4 48.9 8403.8 2379.9 3.5 24.7 49.1 7871.5 2540.8 3.1 27.0 49.3 7402.6 2701.8 2.7 29.4 49.6 6986.4 2862.7 2.4 31.8 49.9 6614.5 3023.6 2.2 34.3 50.2 6280.2 3184.6 2.0 49.5 43.9 5978.1 3345.5 1.8 49.1 44.5 5703.7 3506.5 1.6 48.7 45.0 5453.4 3667.4 1.5 48.4 45.4 5224.2 3828.4 1.4 48.2 45.7 5013.4 3989.3 1.3 48.0 46.0 4819.0 4150.2 1.2 47.9 46.1 4639.1 4311.2 1.1 47.8 46.2 4472.1 4472.1 1.0 47.8 46.3 151 A P P E N D I X A : T A B U L A T E D D A T A A2 Curved plates: APPENDIX A: T A B U L A T E D D A T A A [mm2] 100,000 t [mm] 1.5875 b/R [rad] 0.1 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm2] 1000.0 100.0 10.0 1000.0 629.9 0.101 0.0 890.0 112.4 7.9 1123.6 707.8 0.071 0.1 801.8 124.7 6.4 1247.2 785.6 0.052 0.1 729.5 137.1 5.3 1370.8 863.5 0.039 0.1 669.2 149.4 4.5 1494.4 941.4 0.030 0.1 618.0 161.8 3.8 1618.0 1019.2 0.024 0.2 574.2 .174.2 3.3 1741.6 1097.1 0.019 0.2 536.1 186.5 2.9 1865.2 1175.0 0.016 0.3 502.8 198.9 2.5 1988.9 1252.8 0.013 0.3 473.4 211.2 2.2 2112.5 1330.7 0.011 0.4 447.2 223.6 2.0 2236.1 1408.5 0.009 0.5 316.2 316.2 1.0 3162.3 1992.0 0.003 1.3 A [mm2] 100,000 t [mm] 3.1750 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1000.0 100.0 10.0 1000.0 315.0 0.810 0.0 890.0 112.4 7.9 1123.6 353.9 0.571 0.0 801.8 124.7 6.4 1247.2 392.8 0.417 0.0 729.5 137.1 5.3 1370.8 431.8 0.314 0.0 669.2 149.4 4.5 1494.4 470.7 0.243 0.0 618.0 161.8 3.8 1618.0 509.6 0.191 0.0 574.2 174.2 3.3 1741.6 548.5 0.153 0.0 536.1 186.5 2.9 1865.2 587.5 0.125 0.0 502.8 198.9 2.5 1988.9 626.4 0.103 0.0 473.4 211.2 2.2 2112.5 665.3 0.086 0.0 447.2 223.6 2.0 2236.1 704.3 0.072 0.1 316.2 316.2 1 0 3162.3 996.0 0.026 0.2 A [mm2] 100,000 t [mm] 4.7625 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1000.0 100.0 10.0 1000.0 210.0 2.733 0.0 890.0 112.4 7.9 1123.6 235.9 1.927 0.0 801.8 124.7 6.4 1247.2 261.9 1.409 0.0 729.5 137.1 5.3 1370.8 287.8 1.061 0.0 669.2 149.4 4.5 1494.4 313.8 0.819 0.0 618.0 161.8 3.8 1618.0 339.7 0.645 0.0 574.2 174.2 3.3 1741.6 365.7 0.517 0.0 536.1 186.5 2.9 1865.2 391.7 0.421 0.0 502.8 198.9 2.5 1988.9 417.6 0.347 0.0 473.4 211.2 2.2 2112.5 443.6 0.290 0.0 447.2 223.6 2.0 2236.1 469.5 0.244 0.0 316.2 316.2 1.0 3162.3 664.0 0.086 0.0 156 APPENDIX A: TABULATED DATA A [mm2] 100,000 t [mm] 1.5875 b/R [rad] 0.5 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm -2] 1000.0 100.0 10.0 200.0 126.0 0.503 0.0 890.0 112.4 7.9 224.7 141.6 0.355 0.0 801.8 124.7 6.4 249.4 157.1 0.259 0.0 729.5 137.1 5.3 274.2 172.7 0.195 0.0 669.2 149.4 4.5 298.9 188.3 0.151 0.0 618.0 161.8 3.8 323.6 203.8 0.119 0.0 574.2 174.2 3.3 348.3 219.4 0.095 0.0 536.1 186.5 2.9 373.0 235.0 0.078 0.1 502.8 198.9 2.5 397.8 250.6 0.064 0.1 473.4 211.2 2.2 422.5 266.1 0.053 0.1 447.2 223.6 2.0 447.2 281.7 0.045 0.1 316.2 316.2 1.0 632.5 398.4 0.016 0.3 A [mm'] 100,000 t [mm] 3.1750 b/R [rad] 0.5 L b AR Radius R/t buckling press [Nmm'2] buckling ratio [mm] [mm] [mm] 1000.0 100.0 10.0 200.0 63.0 4.024 0.0 890.0 112.4 7.9 224.7 70.8 2.837 0.0 801.8 124.7 6.4 249.4 78.6 2.074 0.0 729.5 137.1 5.3 274.2 86.4 1.562 0.0 669.2 , • 149.4 4.5 298,9 94.1 1.206 0.0 618.0 161.8 3.8 323.6 101.9 0.950 0.0 574.2 174.2 3.3 348.3 109.7 0.762 0.0 536.1 186.5 2.9 373.0 117.5 0.620 0.0 502.8 198.9 2.5 397.8 125.3 0.512 0.0 473.4 211.2 2.2 422.5 133.1 0.427 0.0 447.2 223.6 2.0 447.2 140.9 0.360 0.0 316.2 316.2 1.0 632.5 199.2 0.127 0.0 A [mm'] 100,000 t[mm] 4.7625 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm^] buckling ratio 1000.0 100.0 10.0 200.0 42.0 13.582 0.0 890.0 112.4 7.9 . 224.7 47.2 9.574 0.0 801.8 124.7 6.4 249.4 52.4 7.000 0.0 729.5 137.1 5.3 274.2 57.6 5.272 0.0 669.2 149.4 4.5 298.9 62.8 4.069 0.0 618.0 161.8 3.8 323.6 67.9 3.206 0.0 574.2 174.2 3.3 348.3 73.1 2.571 0.0 536.1 186.5 2.9 373.0 ,78.3 2.093 0.0 502.8 198.9 2.5 397.8 83.5 1.726 0.0 . 473.4 211.2 2.2 422.5 88.7 1.441 0.0 447.2 223.6 2.0 447.2 93.9 1.215 0.0 316.2 316.2 1.0 632.5 132.8 0.429 0.0 157 APPENDIX A: TABULATED DATA A [mm2] 100,000 t [mm] 1.5875 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 1000.0 100.0 10.0 50.0 31.5 1.820 0.0 890.0 112.4 7.9 56.2 35.4 1.283 0.0 801.8 124.7 6.4 62.4 39.3 0.938 0.0 729.5 137.1 5.3 68.5 43.2 0.706 0.0 669.2 149.4 4.5 74.7 47.1 0.545 0.0 618.0 161.8 3.8 80.9 51.0 0.430 0.0 574.2 174.2 3.3 87.1 54.9 0.344 0.0 536.1 186.5 2.9 93.3 58.7 0.280 0.0 502.8 198.9 2.5 99.4 62.6 0.231 0.0 473.4 211.2 2.2 105.6 66.5 0.193 0.0 447.2 223.6 2.0 111.8 70.4 0.163 0.0 316.2 316.2 1.0 158.1 99.6 0.058 0.1 A [mm2] 100,000 t [mm] 3.1750 b/R [rad] 2.0 L b AR Radius R/t buckling press [Nmm"2] buckling ratio [mm] [mm] [mm] 1000.0 100.0 10.0 50.0 15.7 14.558 0.0 890.0 112.4 7.9 56.2 17.7 10.263 0.0 801.8 124.7 6.4 62.4 19.6 7.504 0.0 729.5 137.1 5.3 68.5 21.6 5.651 0.0 669.2 149.4 4.5 74.7 23.5 4.362 0.0 618.0 161.8 3.8 80.9 25.5 3.437 0.0 574.2 174.2 3.3 87.1 27.4 2.756 0.0 536.1 186.5 2.9 93.3 29.4 2.243 0.0 502.8 198.9 2.5 99.4 31.3 1.851 0.0 473.4 211.2 2.2 105.6 33.3 1.544 0.0 447.2 223.6 2.0 111.8 35.2 1.302 0.0 316.2 316.2 1.0 158.1 49.8 0.460 0.0 A [mm2] 100,000 t [mm] 4.7625 b/R [rad] 2.0 L b AR Radius R/t buckling press [Nmm"2] buckling ratio [mm] [mm] [mm] 1000.0 100.0 10.0 50.0 10.5 49.133 0.0 890.0 112.4 7.9 56.2 11.8 34.636 0.0 801.8 124.7 6.4 62.4 13.1 25.325 0.0 729.5 137.1 5.3 68.5 14.4 19.074 0.0 669.2 149.4 4.5 74.7 15.7 14.721 0.0 618.0 161.8 3.8 80.9 17.0 11.599 0.0 574.2 174.2 3.3 87.1 i8.3 9.300 0.0 536.1 186.5 2.9 93.3 19.6 7.571 0.0 502.8 198.9 2.5 99.4 20.9 6.245 0.0 473.4 211.2 2.2 105.6 22.2 5.212 0.0 447.2 223.6 2.0 111.8 23.5 4.395 0.0 316.2 316.2 1.0 158.1 33.2 1.554 0.0 158 APPENDIX A: TABULATED DATA A [mm2] 100,000 t [mm] 1.5875 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1000.0 100.0 10.0 25.0 15.7 2.408 0.0 890.0 112.4 7.9 28.1 17.7 1.698 0.0 801.8 124.7 6.4 31.2 19.6 1.241 0.0 729.5 137.1 5.3 34.3 21.6 0.935 0.0 669.2 149.4 4.5 37.4 23.5 0.722 0.0 618.0 161.8 3.8 40.5 25.5 0.569 0.0 574.2 174.2 3.3 43.5 27.4 0.456 0.0 536.1 186.5 2.9 46.6 29.4 0.371 0.0 502.8 198.9 2.5 49.7 31.3 0.306 0.0 473.4 211.2 • 2.2 52.8 33.3 0.255 0.0 447.2 223.6 2.0 55.9 35.2 0.215 0.0 316.2 316.2 1.0 79.1 49.8 0.076 0.1 A [mm2] 100,000 t [mm] 3.1750 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press ' [Nmm'2] buckling ratio 1000.0 100.0 10.0 25.0 7.9 — — 890.0 112.4 7.9 28.1 8.8 — — 801.8 124.7 . 6.4 31.2 9.8 — — 729.5 137.1 5.3 34.3 10.8 7.480 0.0 669.2 149.4 4.5 37.4 11.8 5.773 0.0 618.0 161.8 3.8 40.5 12.7 4.549 0.0 574.2 174.2 3.3 43.5 13.7 3.647 0.0 536.1 186.5 2.9 46.6 14.7 2.969 0.0 502.8 198.9 2.5 49.7 15.7 2.449 0.0 473.4 211.2 2.2 52.8 16.6 2.044 0.0 447.2 223.6 2.0 55.9 17.6 1.723 0.0 316.2 316.2 1.0 79.1 24.9 0.609 0.0 A [mm2] 100,000 t[mm] 4.7625 • b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 1000.0 100.0 10.0 25.0 5.2 — — 890.0 112.4 7.9 28.1 5.9 — — 801.8 124.7 6.4 31.2 6.5 — — 729.5 137.1 5.3 34.3 7.2 — — 669.2 149.4 4.5 37.4 7.8 — — 618.0 161.8 3.8 40.5 8.5 — — 574.2 174.2 3.3 43.5 9.1 — — 536.1 186.5 2.9 46.6 9.8 . . . . — 502.8 198.9 2.5 49.7 10.4 8.266 0.0 473.4 211.2 2.2 52.8 11.1 6.898 0.0 447.2 223.6 2.0 55.9 11.7 5.816 0.0 316.2 316.2 1.0 79.1 16.6 2.056 0.0 159 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 200,000 t [mm] 1.5875 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] • buckling ratio 1414.2 141.4 10.0 1414.2 890.8 0.036 0.1 1258.6 158.9 7.9 1589.0 1001.0 0.025 0.2 1133.9 176.4 6.4 1763.8 1111.1 0.018 0.2 1031.7 193.9 5.3 1938.6 1221.2 0.014 0.3 946.3 211.3 4.5 2113.4 1331.3 0.011 0.4 874.0 228.8 3.8 2288.2 1441.4 0.008 0.5 812.0 246.3 3.3 2463.1 1551.5 0.007 0.6 758.2 263.8 2.9 2637.9 1661.6 0.006 0.8 711.1 281.3 2.5 2812.7 1771.8 0.005 0.9 669.5 298.7 2.2 2987.5 1881.9 0.004 1.1 632.5 316.2 2.0 3162.3 1992.0 0.003 1.3 447.2 447.2 1.0 4472.1 2817.1 0.001 3.7 A [mm2] 200,000 t [mm] 3.1750 b/R [rad] 0.1 L b AR Radius R/t buckling press [Nmm'2] buckling ratio [mm] [mm] [mm] 1414.2 141.4 10.0 1414.2 445.4 0.286 0.0 1258.6 158.9 7.9 1589.0 500.5 0.202 0.0 1133.9 176.4 6.4 1763.8 555.5 0.148 0.0 1031.7 193.9 5.3 1938.6 610.6 0.111 0.0 946.3 211.3 4.5 2113.4 665.7 0.086 0.0 874.0 228.8 3.8 2288.2 720.7 0.068 0.1 812.0 246.3 3.3 2463.1 775.8 0.054 0.1 758.2 263.8 2.9 2637.9 830.8 0.044 0.1 711.1 281.3 2.5 2812.7 885.9 0.036 0.1 669.5 298.7 2.2 2987.5 940.9 0.030 0.1 632.5 316.2 2.0 3162.3 996.0 0.026 0.2 447.2 447.2 1.0 4472.1 1408.5 0.009 0.5 A [mm2] 200,000 t [mm] 4.7625 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 1414.2 141.4 10.0 1414.2 296.9 0.966 0.0 1258.6 158.9 7.9 1589.0 333.7 0.681 0.0 1133.9 176.4 6.4 1763.8 370.4 0.498 0.0 1031.7 193.9 5.3 1938.6 407.1 0.375 0.0 946.3 211.3 4.5 2113.4 443.8 0.290 0.0 874.0 228.8 3.8 2288.2 480.5 0.228 0.0 812.0 246.3 3.3 2463.1 517.2 0.183 0.0 758.2 263,8 2.9 2637.9 553.9 0.149 0.0 711.1 281.3 2.5 2812.7 590.6 0.123 0.0 669.5 298.7 2.2 2987.5 627.3 0.102 0.0 632.5 316.2 2.0 3162.3 664.0 0.086 0.0 447.2 447.2 1.0 4472.1 939.0 0.031 0.1 160 APPENDIX A : T A B U L A T E D D A T A A [mm2] 200,000 t [mm] 1.5875 b/R [rad] 0.5 L. b AR Radius R/t . buckling press [Nmm 2] buckling ratio [mm] [mm] [mm] 1414.2 141.4 10.0 282.8 178.2 0.178 0.0 1258.6 158.9 7.9 317.8 200.2 0.125 0.0 1133.9 176.4 6.4 352.8 222.2 0.092 0.0 1031.7 193.9 5.3 387.7 244.2 0.069 0.1 946.3 211.3 4.5 422.7 266.3 0.053 0.1 874.0 228.8 3.8 457.6 288.3 0.042 0.1 812.0 246.3 3.3 492.6 310.3 0.034 0.1 758.2 263.8 2.9 527.6 332.3 0.027 0.2 711.1 281.3 2.5 562.5 354.4 0.023 0.2 669.5 298.7 2.2 597.5 376.4 0.019 0.2 632.5 316.2 2.0 632.5 ' 398.4 0.016 0.3 447.2 447.2 1.0 894.4 563.4 0.006 0.7 A [mm2] 200,000 t [mm] 3.1750 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1414.2 141.4 10.0 282.8 89.1 1.423 0.0 1258.6 158.9 7.9 317.8 100.1 1.003 0.0 1133.9 176.4 6.4 352.8 111.1 0.733 0.0 1031.7 193.9 5.3 387.7 122.1 0.552 0.0 946.3 211.3 4.5 422.7 133.1 0.426 0.0 874.0 228.8 3.8 457.6 144.1 0.336 0.0 812.0 246.3 3.3 492.6 155.2 0.269 0.0 758.2 263.8 2.9 527.6 166.2 0.219 0.0 711.1 281.3 2.5 562.5 177.2 0.181 0.0 669.5 298.7 2.2 597.5 188.2 0.151 0.0 632.5 316.2 2.0 632.5 199.2 0.127 0.0 447.2 447.2 1.0 894.4 281.7 0.045 0.1 A [mm2] 200,000 t[mm] 4.7625 b/R [rad] • 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 1414.2 141.4 10.0 282.8 59.4 4.802 0.0 1258.6 158.9 7.9 317.8 66.7 3.385 0.0 1133.9 176.4 6.4 352.8 74.1 2.475 0.0 1031.7 193.9 5.3 387.7 81.4 1.864 0.0 946.3 211.3 4.5 422.7 88.8 1.439 0.0 874.0 228.8 3.8 457.6 96.1 1.134 0.0 812.0 246.3 3.3 492.6 103.4 0.909 0.0 758.2 263.8 2.9 527.6 110.8 0.740 0.0 711.1 281.3 2.5 562.5 118.1 0.610 0.0 669.5 298.7 2.2 597.5 125.5 0.509 0.0 632.5 316.2 2.0 632.5 132.8 0.429 0.0 447.2 447.2 1.0 894.4 187.8 0.152 0.0 161 APPENDIX A : T A B U L A T E D D A T A A [mm2] 200,000 t [mm] 1.5875 b/R [rad] 2.0 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm'2] 1414.2 141.4 10.0 70.7 44.5 0.643 0.0 1258.6 158.9 7.9 79.5 50.0 0.454 0.0 1133.9 176.4 6.4 88.2 55.6 0.332 0.0 1031.7 193.9 5.3 96.9 61.1 0.250 0.0 946.3 211.3 4.5 105.7 66.6 0.193 0.0 . 874.0 228.8 3.8 114.4 72.1 0.152 0.0 812.0 246.3 3.3 123.2 77.6 0.122 0.0 758.2 263.8 2.9 131.9 83.1 0.099 0.0 711.1 281.3 2.5 140.6 88.6 0.082 0.1 669.5 298.7 2.2 149.4 94.1 0.068 0.1 632.5 316.2 2.0 158.1 99.6 0.058 0.1 447.2 447.2 1.0 223.6 140.9 0.020 0.2 A [mm2] 200,000 t [mm] 3.1750 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1414.2 141.4 10.0 70.7 22.3 5.147 0.0 1258.6 158.9 7.9 79.5 25.0 3.628 0.0 1133.9 176.4 6.4 88.2 27.8 2.653 0.0 1031.7 193.9 5.3 96.9 30.5 1.998 0.0 946.3 211.3 4.5 105.7 33.3 1.542 0.0 874.0 228.8 3.8 114.4 36.0 1.215 0.0 812.0 246.3 3.3 123.2 38.8 0.974 0.0 758.2 263.8 • 2.9 131.9 41.5 0.793 0.0 711.1 281.3 2.5 140.6 44.3 0.654 0.0 669.5 298.7 . 2.2 149.4 47.0 0.546 0.0 632.5 316.2 2.0 158.1 49.8 0.460 0.0 447.2 447.2 1.0 223.6 70.4 0.163 0.0 A [mm2] 200,000 t [mm] 4.7625 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1414.2 141.4 10.0 70.7 14.8 17.371 0.0 1258.6 158.9 7.9 79.5 16.7 12.246 0.0 1133.9 176.4 6.4 88.2 18.5 8.954 0.0 1031.7 193.9 5.3 96.9 20.4 6.744 0.0 946.3 211.3 4.5 105.7 22.2 5.205 0.0 874.0 228.8 3.8 114.4 24.0 4.101 0.0 812.0 246.3 3.3 123.2 25.9 3.288 0.0 758.2 263.8 2.9 131.9 27.7 2.677 0.0 711.1 281.3 2.5 140.6 29.5 2.208 0.0 669.5 298.7 2.2 149.4 31.4 1.843 0.0 632.5 316.2 2.0 158.1 33.2 1.554 0.0 447.2 447.2 1.0 223.6 47.0 0.549 0.0 162 APPENDIX A: T A B U L A T E D D A T A A [mm2] 200,000 t [mm] 1.5875 b/R [rad] 4.0 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm 2] 1414.2 141.4 10.0 35.4 22.3 0.852 0.0 1258.6 158.9 7.9 39.7 25.0 0.600 0.0 1133.9 176.4 6.4 44.1 27.8 0.439 0.0 1031.7 193.9 5.3 48.5 30.5 0.331 0.0 946.3 211.3 4.5 52.8 33.3 0.255 0.0 874.0 228.8 3.8 57.2 36.0 0.201 0.0 812.0 246.3 3 3 • 61.6 38.8 0.161 0.0 758.2 263.8 2.9 65.9 41.5 0.131 0.0 711.1 281.3 2.5 70.3 44.3 0.108 0.0 669.5 298.7 2.2 74.7 47.0 0.090 0.0 632.5 316.2 2.0 79.1 49.8 0.076 0.1 447.2 447.2 1.0 111.8 70.4 0.027 0.2 A [mm2] 200,000 t [mm] 3.1750 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1414.2 141.4 10.0 35.4 11.1 6.812 0.0 1258.6 158.9 7.9 39.7 12.5 4.802 0.0 1133.9 176.4 6.4 44.1 13.9 3.511 0.0 1031.7 193.9 5.3 48.5 15.3 2.645 0.0 946.3 211.3 4.5 52.8 16.6 2.041 0.0 874.0 228.8 3.8 57.2 18.0 1.608 0.0 812.0 246.3 3.3 61.6 19.4 1.289 0.0 758.2 263.8 2.9 65.9 20.8 1.050 0.0 711.1 281.3 2.5 70.3 22.1 0.866 0.0 669.5 298.7 2.2 74.7 23.5 0.723 0.0 632.5 316.2 2.0 79.1 24.9 0.609 0.0 447.2 447.2 1.0 111.8 35.2 0.215 0.0 A [mm2] 200,000 t [mm] 4.7625 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1414.2 141.4 10.0 35.4 7.4 — — 1258.6 158.9 7.9 39.7 8.3 — — 1133.9 176.4 6.4 44.1 9.3 . . . i — 1031.7 193.9 5.3 48.5 10.2 8.925 0.0 946.3 211.3 4.5 52.8 11.1 6.889 0.0 874.0 228.8 3.8 57.2 12.0 5.428 0.0 812.0 246.3 3.3 61.6 12.9 4.352 0.0 758.2 263.8 2.9 65.9 13.8 3.543 0.0 711.1 281.3 2.5 70.3 14.8 2.923 0.0 669.5 298.7 2.2 74.7 15.7 2.439 0.0 632.5 316.2 2.0 79.1 16.6 2.056 0.0 447.2 447.2 1.0 111.8 23.5 0.727 0.0 163 APPENDIX A: T A B U L A T E D D A T A A [mm2] 300,000 t [mm] 1.5875 b/R [rad] 0.1 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm 2] -1732.1 173.2 10.0 1732.1 1091.1 0.019 0.2 1541.5 194.6 7.9 1946.1 1225.9 0.014 0.3 1388.7 216.0 6.4 2160.2 1360.8 0.010 0.4 1263.5 237.4 5.3 2374.3 1495.6 0.008 0.6 1159.0 258.8 4.5 2588.4 1630.5 0.006 0.7 1070.5 280.3 3.8 2802.5 1765.4 0.005 0.9 994.5 . 301.7 3.3 3016.6 1900.2 0.004 1.1 928.6 323.1 2.9 3230.7 2035.1 0.003 1.4 870.9 344.5 2.5 3444.8 2170.0 0.002 1.7 819.9 365.9 2.2 3658.9 2304.8 0.002 2.0 774.6 387.3 2.0 3873.0 2439.7 0.002 2.4 547.7 547.7 1.0 5477.2 3450.2 0.001 6.8 A [mm2] 300,000 t [mm] 3.1750 b/R [rad] 0.1 L b AR Radius R/t buckling press [Nmm 2] buckling ratio [mm] [mm] [mm] 1732.1 173.2 10.0 1732.1 545.5 0.156 0.0 1541.5 194.6 7.9 1946.1 613.0 0.110 0.0 1388.7 216.0 6.4 2160.2 680.4 0.080 0.1 1263.5 237.4 5.3 2374.3 747.8 0.060 0.1 1159.0 258.8 4.5 2588.4 815.3 0.047 0.1 1070.5 280.3 3.8 2802.5 882.7 0.037 0.1 994.5 301.7 3.3 3016.6 950.1 0.029 0.1 928.6 323.1 2.9 3230.7 1017.5 0.024 0.2 870.9 344.5 2.5 3444.8 1085.0 0.020 0.2 819.9 365.9 2.2 3658.9 1152.4 0.017 0.3 774.6 387.3 2.0 3873.0 1219.8 0.014 0.3 547.7 547.7 1.0 5477.2 1725.1 0.005 0.9 A [mm2] 300,000 t [mm] 4.7625 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 1732.1 173.2 10.0 1732.1 363.7 0.526 0.0 1541.5 194.6 7.9 1946.1 408.6 0.371 0.0 1388.7 216.0 6.4 2160.2 453.6 0.271 0.0 1263.5 237.4 5.3 2374.3 498.5 0.204 0.0 1159.0 258.8 4.5 2588.4 543.5 0.158 0.0 1070.5 280.3 3.8 2802.5 588.5 0.124 0.0 994.5 301.7 3.3 3016.6 633.4 0.100 0.0 928.6 323.1 2.9 3230.7 678.4 0.081 0.1 870.9 344.5 2.5 3444.8 723.3 0.067 0.1 819.9 365.9 2.2 3658.9 768.3 0.056 0.1 774.6 387.3 2.0 3873.0 813.2 . 0.047 0.1 547.7 547.7 1.0 5477.2 1150.1 0.017 0.3 164 APPENDIX A: T A B U L A T E D D A T A A [mm2] 300,000 t [mm] 1.5875 b/R [rad] 0.5 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm 2] 1732.1 173.2 10.0 346.4 218.2 0.097 0.0 1541.5 194.6 7.9 389.2 245.2 0.068 0.1 1388.7 216.0 6.4 432.0 272.2 0.050 0.1 1263.5 237.4 5.3 474.9 299.1 0.038 0.1 1159.0 258.8 4.5 517.7 326.1 0.029 0.1 1070.5 280.3 3.8 560.5 353.1 0.023 0.2 994.5 301.7 3.3 603.3 380.0 0.018 0.2 928.6 323.1 2.9 646.1 407.0 0.015 0.3 870.9 344.5 2.5 689.0 434.0 0.012 0.3 819.9 365.9 2.2 731.8 461.0 0.010 0.4 774.6 387.3 2.0 774.6 487.9 0.009 0.5 547.7 547.7 1.0 1095.4 690.0 0.003 1.4 A [mm2] 300,000 t [mm] 3.1750 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmnr2] buckling ratio 1732.1 173.2 10.0 346.4 109.1 0.774 0.0 1541.5 194.6 7.9 389.2 122.6 0.546 0.0 1388.7 216.0 . 6.4 432.0 136.1 0.399 0.0 1263.5 237.4 5.3 474.9 149.6 0.301 0.0 1159.0 258.8 4.5 517.7 163.1 0.232 0.0 1070.5 280.3 3.8 560.5 176.5 0.183 0.0 994.5 301.7 3.3 603.3 190.0 0.147 0.0 928.6 323.1 2.9 646.1 203.5 0.119 0.0 870.9 344.5 2.5 689.0 217.0 0.098 0.0 819.9 365.9 2.2 - 731.8 230.5 0.082 0.1 774.6 387.3 2.0 774.6 244.0 0.069 0.1 547.7 547.7 1.0 1095.4 345.0 0.024 0.2 A [mm2] 300,000 t [mm] 4.7625 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 1732.1 173.2 10.0 346.4 72.7 2.614 0.0 1541.5 194.6 7.9 389.2 81.7 1.843 0.0 1388.7 216.0 6.4 432.0 90.7 . 1.347 0.0 1263.5 237.4 5.3 474.9 99.7 1.015 0.0 1159.0 258.8 4.5 517.7 108.7 0.783 0.0 1070.5 280.3 3.8 560.5 117.7 0.617 0.0 994.5 301.7 3.3 603.3 126.7 0.495 0.0 928.6 323.1 • 2.9 646.1 135.7 0.403 0.0 870.9 344.5 2.5 689.0 144.7 0.332 0.0 819.9 365.9 2.2 731.8 153.7 0.277 0.0 774.6 387.3 2.0 774.6 162.6 0.234 0.0 547.7 547.7 1.0 1095.4 230.0 0.083 0.1 165 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 300,000 t [mm] 1.5875 b/R [rad] 2.0 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm'] 1732.1 173.2 10.0 86.6 54.6 0.350 0.0 1541.5 194.6 7.9 97.3 61.3 0.247 0.0 1388.7 216.0 6.4 108.0 68.0 0.181 0.0 1263.5 237.4 5.3 118.7 74.8 0.136 0.0 1159.0 258.8 4.5 129.4 81.5 0.105 0.0 1070.5 280.3 3.8 140.1 88.3 0.083 0.1 994.5 301.7 3.3 150.8 95.0 0.066 0.1 928.6 323.1 2.9 161.5 101.8 0.054 0.1 870.9 344.5 2.5 172.2 108.5 0.045 0.1 819.9 365.9 2.2 182.9 115.2 0.037 0.1 774.6 387.3 2.0 193.6 122.0 0.031 0.1 547.7 547.7 1.0 273.9 172.5 0.011 0.4 A [mm2] 300,000 t [mm] 3.1750 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmrrf2] buckling ratio 1732.1 173.2 10.0 86.6 27.3 2.802 0.0 1541.5 194.6 7.9 97.3 30.6 1.975 0.0 1388.7 216.0 6.4 108.0 34.0 1.444 0.0 1263.5 237.4 5.3 118.7 37.4 1.088 0.0 1159.0 258.8 4.5 129.4 40.8 0.839 0.0 1070.5 280.3 • 3.8 140.1 44.1 0.661 0.0 994.5 301.7 3.3 150.8 47.5 0.530 0.0 928.6 323.1 2.9 161.5 50.9 0.432 0.0 870.9 344.5 2.5 172.2 54.2 0.356 0.0 819.9 365.9 2.2 182.9 57.6 0.297 0.0 774.6 387.3 2.0 193.6 61.0 0.251 0.0 547.7 547.7 1.0 273.9 86.3 0.089 0.0 A [mm2] 300,000 t [mm] 4.7625 b/R [rad] 2.0 • L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm -2] buckling ratio 1732.1 173.2 10.0 86.6 18.2 9.456 0.0 1541.5 194.6 7.9 97.3 20.4 6.666 0.0 1388.7 216 :0 6.4 108.0 22.7 4.874 0.0 1263.5 237.4 5.3 118.7 24.9 3.671 0.0 1159.0 258.8 4.5 129.4 27.2 2.833 0.0 1070.5 280.3 3.8 140.1 29.4 2.232 0.0 994.5 301.7 3.3 150.8 31.7 1.790 0.0 928.6 323.1 2.9 161.5 33.9 1.457 0.0 870.9 344.5 2.5 172.2 36.2 1.202 0.0 819.9 365.9 2.2 182.9 38.4 . 1.003 0.0 774.6 387.3 2.0 193.6 40.7 0.846 0.0 547.7 547.7 1.0 273.9 57.5 0.299 0.0 166 APPENDIX A: T A B U L A T E D D A T A A [mm2] 300,000 t [mm] 1.5875 b/R [rad] 4.0 L b AR Radius R/t buckling press [Nmm 2] buckling ratio [mm] [mm] [mm] 1732.1 173.2 10.0 43.3 27.3 0.464 0.0 1541.5 1.94.6 7.9 ' 48.7 30.6 0.327 0.0 1388.7 216.0 6.4 54.0 34.0 0.239 0.0 1263.5 237.4 5.3 59.4 37.4 0.180 0.0 1159.0 258.8 4.5 64.7 40.8 0.139 0.0 1070.5 280.3 3.8 70.1 44.1 0.109 0.0 994.5 301.7 3.3 75.4 47.5 0.088 0.0 928.6 323.1 2.9 80.8 50.9 0.071 0.1 870.9 344.5 2.5 86.1 54.2 0.059 0.1 819.9 365.9 2.2 91.5 57.6 0.049 0.1 774.6 387.3 2.0 96.8 61.0 0.041 0.1 547.7 547.7 1.0 136.9 86.3 0.015 0.3 A [mm2] 300,000 t [mm] 3.1750 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 1732.1 173.2 10.0 43.3 13.6 3.708 0.0 1541.5 194.6 7.9 48.7 15.3 2.614 0.0 1388.7 216.0 6.4 54.0 17.0 1.911 0.0 1263.5 237.4 5.3 59.4 18.7 1.440 0.0 1159.0 258.8 4.5 64.7 20.4 1.111 0.0 1070.5 280.3 3.8 70.1 22.1 0.875 0.0 994.5 301.7 3.3 75.4 23.8 0.702 0.0 928.6 323.1 2.9 80.8 25.4 0.571 0.0 870.9 344.5 2.5 86.1 27.1 0.471 0.0 819.9 365.9 2.2 91.5 28.8 0.393 0.0 774.6 387.3 2.0 96.8 30.5 0.332 0.0 547.7 547.7 1.0 136.9 43.1 0.117 0.0 A [mm2] 300,000 t [mm] 4.7625 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 1732.1 173.2 10.0 43.3 9.1 — — 1541.5 194.6 7.9 48.7 10.2 8.822 0.0 1388.7 216.0 6.4 54.0 11.3 6.451 0.0 1263.5 237.4 5.3 59.4 12.5 4.858 0.0 1159.0 258.8 4.5 64.7 13.6 3.750 0.0 1070.5 280.3 3.8 70.1 14.7 2.954 0.0 994.5 301.7 3.3 75.4 15.8 2.369 0.0 928.6 323.1 2.9 80.8 17.0 1.928 0.0 870.9 344.5 2.5 86.1 18.1 1.591 0.0 819.9 365.9 2.2 91.5 19.2 1.328 0.0 774.6 387.3 2.0 96.8 20.3 1.119 0.0 547.7 547.7 1.0 136.9 28.8 0.396 0.0 167 APPENDIX A: TABULATED DATA A [mm2] 500,000 t [mm] 1.5875 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 2236.1 223.6 10.0 2236.1 1408.5 0.009 0.5 1990.1 251.2 7.9 2512.5 1582.7 0.006 0.7 1792.9 278.9 6.4 2788.9 1756.8 0.005 0.9 1631.2 306.5 5.3 3065.2 1930.9 0.004 1.2 1496.3 334.2 4.5 3341.6 2105.0 0.003 1.5 1382.0 361.8 3.8 3618.0 2279.1 0.002 2.0 1283.9 389.4 3.3 3894.4 2453.2 0.002 2.5 1198.8 417.1 2.9 4170.8 2627.3 0.001 3.0 1124.3 444.7 2.5 4447.2 2801.4 0.001 3.6 1058.5 472.4 2.2 4723.6 2975.5 0.001 4.4 1000.0 500.0 2.0 5000.0 3149.6 0.001 5.2 707.1 707.1 1.0 7071.1 4454.2 0.000 14.7 A [mm2] 500,000 t [mm] 3.1750 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 2236.1 223.6 - 10.0 2236.1 704.3 0.072 0.1 1990.1 251.2 7.9 2512.5 791.3 0.051 0.1 1792.9 278.9 6.4 2788.9 878.4 0.037 0.1 1631.2 306.5 5.3 3065.2 965.4 0.028 0.1 1496.3 334.2 4.5 3341.6 1052.5 0.022 0.2 1382.0 361.8 3.8 3618.0 1139.5 0.017 0.2 1283.9 389.4 3.3 3894.4 1226.6 0.014 0.3 1198.8 417.1 2.9 4170.8 1313.6 0.011 0.4 1124.3 444.7 2.5 4447.2 1400.7 0.009 0.5 1058.5 472.4 2.2 4723.6 1487.8 0.008 0.5 1000.0 500.0 2.0 5000.0 1574.8 0.006 0.6 707.1 707.1 1.0 7071.1 2227.1 0.002 1.8 A [mm2] 500,000 t [mm] 4.7625 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 2236.1 223.6 10.0 2236.1 469.5 0.244 0.0 1990.1 251.2 7.9 2512.5 527.6 0.172 0.0 1792.9 278.9 6.4 2788.9 585.6 0.126 0.0 1631.2 306.5 5.3 3065.2 643.6 0.095 0.0 1496.3 334.2 4.5 3341.6 701.7 0.073 0.1 1382.0 361.8 3.8 3618.0 759.7 0.058 0.1 1283.9 389.4 3.3 3894.4 817.7 0.046 0.1 1198.8 417.1 2.9 4170.8 875.8 0.038 0.1 1124.3 444.7 2.5 4447.2 933.8 0.031 0.1 1058.5 472.4 2.2 4723.6 991.8 0.026 0.2 1000.0 500.0 2.0 5000.0 1049.9 0.022 0.2 707.1 707.1 1.0 7071.1 1484.7 0.008 0.5 168 APPENDIX A : T A B U L A T E D D A T A A [mm2] 500,000 t [mm] 1.5875 , b/R [rad] 0.5 L b AR Radius R/t buckling press [Nmm 2] buckling ratio [mm] [mm] [mm] 2236.1 223.6 10.0 447.2 281.7 0.045 0.1 1990.1 251.2 7.9 502.5 316.5 0.032 0.1 1792.9 278.9 6.4 557.8 351.4 0.023 0.2 1631.2 306.5 5.3 613.0 386.2 0.017 0.2 1496.3 334.2 4.5 668.3 421.0 0.013 0.3 1382.0 361.8 .3.8 723.6 455.8 0.011 0.4 1283.9 389.4 3.3 778.9 490.6 0.009 0.5 1198.8 417.1 2.9 834.2 525.5 0.007 0.6 1124.3 444.7 2.5 889.4 560.3 0.006 0.7 1058.5 472.4 2.2 944.7 595.1 0.005 0.9 1000.0 500.0 2.0 1000.0 629.9 0.004 1.0 707.1 707.1 1.0 1414.2 890.8 0.001 3.0 A [mm2] 500,000 t [mm] 3.1750 b/R [rad] 0.5 L b AR Radius R/t buckling press [Nmm"2] buckling ratio [mm] [mm] [mm] 2236.1 223.6 10.0 447.2 140.9 0.360 0.0 1990.1 251.2 7.9 502.5 158.3 0.254 0.0 1792.9 278.9 6.4 557.8 175.7 0.186 0.0 1631.2 306.5 5.3 613.0 193.1 0.140 0.0 1496.3 334.2 4.5 668.3 210.5 0.108 0.0 1382.0 361.8 3.8 723.6 227.9 0.085 0.0 1283.9 389.4 3.3 778.9 245.3 0.068 0.1 1198.8 417.1 2.9 834.2 262.7 0.055 0.1 1124.3 444.7 . 2.5 889.4 280.1 0.046 0.1 1058.5 472.4 2.2 944.7 297.6 0.038 0.1 1000.0 500.0 2.0 1000.0 315.0 0.032 0.1 707.1 707.1 •1.0 1414.2 445.4 0.011 0.4 A [mm2] 500,000 t [mm] 4.7625 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 2236.1 223.6 10.0 447.2 93.9 1.215 0.0 1990.1 251.2 7.9 502.5 105.5 0.856 0.0 1792.9 278.9 6.4 557.8 11.7.1 0.626 0.0 1631.2 306.5 5.3 613.0 128.7 0.472 0.0 1496.3 334.2 4.5 668.3 140.3 0.364 0.0 1382.0 361.8 3.8 723.6 151.9 0.287 0.0 1283.9 389.4 3.3 778.9 163.5 0.230 0.0 1198.8 417.1 2.9 834.2 175.2 0.187 0.0 1124.3 444.7 2.5 889.4 186.8 0.154 0.0 1058.5 472.4 2.2 944.7 198.4 0.129 0.0 1000.0 500.0 2.0 1000.0 210.0 0.109 0.0 707.1 707.1 1.0 1414.2 296.9 0.038 0.1 169 APPENDIX A : T A B U L A T E D D A T A A [mm2] 500,000 t [mm] 1.5875 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 2236.1 223.6 10.0 111.8 70.4 0.163 0.0 1990.1 251.2 7.9 125.6 79.1 0.115 0.0 1792.9 278.9 6.4 139.4 87.8 0.084 0.1 1631.2 306.5 5.3 153.3 96.5 0.063 0.1 1496.3 334.2 4.5 167.1 105.2 0.049 0.1 1382.0 361.8 3.8 180.9 114.0 0.038 0.1 1283.9 389.4 3.3 194.7 122.7 0.031 0.1 1198.8 417.1 2.9 208.5 131.4 0.025 0.2 1124.3 444.7 2.5 222.4 140.1 0.021 0.2 1058.5 472.4 2.2 .236.2 148.8 0.017 0.2 1000.0 500.0 2.0 250.0 157.5 0.015 0.3 707.1 707.1 1.0 353.6 222.7 0.005 0.8 A [mm2] 500,000 t [mm] 3.1750 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm-2] buckling ratio 2236.1 223.6 10.0 111.8 35.2 1.302 0.0 1990.1 251.2 7.9 125.6 39.6 0.918 0.0 1792.9 278.9 6.4 139.4 43.9 0.671 0.0 1631.2 306.5 5.3 153.3 48.3 0.505 0.0 1496.3 334.2 4.5 167.1 52.6 0.390 0.0 1382.0 361.8 3.8 180.9 57.0 0.307 0.0 1283.9 389.4 3.3 194.7 61.3 0.246 0.0 1198.8 417.1 2.9 208.5 65.7 0.201 0.0 1124.3 444.7 2.5 222.4 70.0 0.166 0.0 1058.5 472.4 2.2 236.2 74.4 0.138 0.0 1000.0 500.0 2.0 250.0 78.7 0.116 0.0 707.1 707.1 1.0 353.6 111.4 0.041 0.1 A [mm2] 500,000 t [mm] 4.7625 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 2236.1 223.6 10.0 111.8 23.5 4.395 0.0 1990.1 251.2 7.9 125.6 26.4 3.098 0.0 1792.9 278.9 6.4 139.4 29.3 2.265 0.0 1631.2 306.5 5.3 153.3 32.2 1.706 0.0 1496.3 334.2 4.5 167.1 35.1 1.317 0.0 1382.0 361.8 3.8 180.9 38.0 1.037 0.0 1283.9 389.4 3.3 194.7 40.9 0.832 0.0 1198.8 417.1 2.9 208.5 43.8 0.677 0.0 1124.3 . 444.7 2.5 222.4 46.7 0.559 0.0 1058.5 472.4 2.2 236.2 49.6 0.466 0.0 1000.0 500.0 2.0 250.0 52.5 0.393 0.0 707.1 707.1 1.0 353.6 74.2 0.139 0.0 170 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 500,000 t [mm] 1.5875 b/R [rad] 4.0 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm 2] 2236.1 223.6 10.0 55.9 35.2 0.215 0.0 1990.1 251.2 7.9 62.8 39.6 0.152 0.0 1792.9 278.9 6.4 69.7 43.9 0.111 0.0 1631.2 306.5 5.3 76.6 48.3 0.084 0.1 1496.3 334.2 4.5 83.5 52.6 0.065 0.1 1382.0 361.8 3.8 90.5 57.0 0.051 0.1 1283.9 389.4 3.3 97.4 61.3 0.041 0.1 1198.8 417.1 2.9 104.3 65.7 0.033 0.1 1124.3 444.7 2.5 111.2 70.0 0.027 0.2 1058.5 472.4 2.2 118.1 74.4 0.023 0.2 1000.0 500.0 2.0 125.0 78.7 0.019 0.2 707.1 707.1 1.0 176.8 111.4 0.007 0.6 A [mm2] 500,000 t [mm] 3.1750 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 2236.1 223.6 10.0 55.9 17.6 1.723 0.0 1990.1 251.2 7.9 62.8 19.8 1.215 0.0 1792.9 278.9 6.4 69.7 ' 22.0 0.888 0.0 1631.2 306.5 5.3 76.6 24.1 0.669 0.0 1496.3 334.2 4.5 83.5 26.3 • 0.516 0.0 1382.0 361.8 3.8 90.5 28.5 0.407 0.0 1283.9 389.4 3.3 97.4 30.7 0.326 0.0 1198.8 417.1 2.9 104.3 32.8 0.266 0.0 1124.3 444.7 2.5 111.2 35.0 0.219 0.0 1058.5 472.4 2.2 118.1 37.2 0.183 0.0 1000.0 500.0 2.0 125.0 39.4 0.154, 0.0 707.1 707.1 1.0 176.8 55.7 0.054 0.1 A [mm2] 500,000 t [mm] 4.7625 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 2236.1 223.6 10.0 55.9 11.7 5.816 0.0 1990.1 251.2 7.9 62.8 13.2 4.100 0.0 1792.9 278.9 6.4 69.7 14.6 2.998 0.0 1631.2 306.5 5.3 76.6 16.1 2.258 0.0 1496.3 334.2 4.5 83.5 17.5 1.743 0.0 1382.0 361.8 3.8 90.5 19.0 1.373 0.0 1283.9 389.4 3.3 97.4 20.4 1.101 0.0 1198.8 417.1 2.9 104.3 21.9 0.896 0.0 1124.3 444.7 2.5 111.2 23.3 0.739 0.0 1058.5 472.4 2.2 118.1 24.8 0.617 0.0 1000.0 500.0 2.0 125.0 26.2 0.520 0.0 707.1 707.1 1.0 176.8 37.1 0.184 0.0 171 APPENDIX A: T A B U L A T E D D A T A A [mm2] 1,000,000 t [mm] 1.5875 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 3162.3 316.2 10.0 3162.3 1992.0 0.003 1.3 2814.4 355.3 7.9 3553.2 2238.2 0.002 1.9 2535.5 394.4 6.4 3944.0 2484.4 0.002 2.5 2306.9 433.5 5.3 4334.9 2730.7 0.001 3.4 2116.0 472.6 4.5 4725.8 2976.9 0.001 4.4 1954.4 511.7 3.8 5116.7 3223.1 0.001 5.6 1815.7 550.8 3.3 5507.6 3469.3 0.001 6.9 1695.4 589.8 2.9 5898.4 3715.5 0.000 8.5 1590.0 628.9 2.5 6289.3 3961.8 0.000 10.3 1497.0 668.0 2.2 6680.2 4208.0 0.000 12.4 1414.2 707.1 2.0 7071.1 4454.2 0.000 14.7 1000.0 1000.0 1.0 10000.0 6299.2 0.000 41.5 A [mm2] 1,000,000 t [mm] 3.1750 b/R [rad] 0.1-L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 3162.3 316.2 10.0 3162.3 996.0 0.026 0.2 2814.4 355.3 7.9 3553.2 1119.1 0.018 0.2 2535.5 394.4 6.4 3944.0 1242.2 0.013 0.3 2306.9 433.5 5.3 4334.9 1365.3 0.010 0.4 2116.0 472.6 4.5 4725.8 1488.4 0.008 0.5 1954.4 511.7 3.8 5116.7 1611.6 0.006 0.7 1815.7 550.8 3.3 5507.6 1734.7 0.005 0.9 1695.4 589.8 2.9 5898.4 1857.8 0.004 1.1 1590.0 628.9 2.5 6289.3 1980.9 0.003 1.3 1497.0 668.0 2.2 6680.2 2104.0 0.003 1.5 1414.2 707.1 2.0 7071.1 2227.1 0.002 1.8 1000.0 1000.0 1.0 10000.0 3149.6 0.001 5.2 A [mm2] 1,000,000 t [mm] 4.7625 b/R [rad] 0.1 L [mm] b [mm] AR Radius, [mm] R/t buckling press [Nmm 2] buckling ratio 3162.3 316.2 10.0 3162.3 664.0 0.086 0.0 2814.4 355.3 7.9 3553.2 746,1 0.061 0.1 2535.5 394.4 6.4 3944.0 828.1 0.045 0.1 2306.9 433.5 5.3 4334.9 910.2 0.034 0.1 2116.0 472.6 4.5 4725.8 992.3 0.026 0.2 1954.4 511.7 3.8 5116.7 1074.4 0.020 0.2 1815.7 550.8 3.3 5507.6 1156.4 0.016 0.3 1695.4 589.8 2.9 5898.4 1238.5 0.013 0.3 1590.0 628.9 2.5 6289.3 1320.6 0.011 0.4 1497.0 668.0 2.2 6680.2 1402.7 0.009 0.5 1414.2 707.1 2.0 7071.1 1484.7 0.008 0.5 1000.0 1000.0 1.0 10000.0 2099.7 0.003 1.5 172 APPENDIX A: T A B U L A T E D D A T A A [mm2] 1,000,000 t [mm] 1.5875 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 3162.3 316.2 10.0 632.5 398.4 0.016 0.3 2814.4 355.3 7.9 710.6 447.6 0.011 0.4 2535.5 394.4 6.4 788.8 496.9 0.008 0.5 2306.9 433.5 5.3 867.0 546.1 0.006 0.7 2116.0 472.6 4.5 945.2 595.4 0.005 0.9 1954.4 511.7 3.8 1023.3 644.6 0.004 1.1 1815.7 550.8 3.3 1101.5 693.9 0.003 1.4 1695.4 589.8 2.9 1179.7 743.1 0.002 1.7 1590.0 628.9 2.5 1257.9 792.4 0.002 2.1 1497.0 668.0 2.2 1336.0 841.6 0.002 2.5 1414.2 707.1 2.0 1414.2 890.8 0.001 3.0 1000.0 1000.0 1.0 2000.0 1259.8 0.001 8.3 A [mm2] 1,000,000 t [mm] 3.1750 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 3162.3 316.2 10.0 632.5 199.2 0.127 0.0 2814.4 355.3 7.9 710.6 223.8 0.090 0.0 2535.5 - 394.4 6.4 788.8 248.4 0.066 0.1 2306.9 433.5 5.3 867.0 273.1 0.049 0.1 2116.0 472.6 4.5 945.2 297.7 0.038 0.1 1954.4 511.7 3.8 1023.3 322.3 0.030 0.1 1815.7 550.8 3.3 1101.5 346.9 0.024 0.2 1695.4 589.8 2.9 1179.7 371.6 0.020 0.2 1590.0 628.9 2.5 1257.9 396.2 0.016 0.3 .1497.0 668.0 2.2 1336.0 420.8 0.013 0.3 1414.2 707.1 2.0 1414.2 445.4 0.011 0.4 1000.0 1000.0 1.0 2000.0 629.9 0.004 1.0 A [mm2] 1,000,000 t [mm] 4.7625 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 3162.3 316.2 10.0 632.5 132.8 0.429 0.0 2814.4 355.3 7.9 ' 710.6 149.2 0.303 0.0 2535.5 394.4 6.4 ' 788.8 165.6 0.221 0.0 2306.9 433.5 5.3 867.0 182.0 0.167 0.0 2116.0 472.6 4.5 945.2 198.5 0.129 0.0 1954.4 511.7 3.8 1023.3 214.9 0.101 0.0 1815.7 550.8 3.3 1101.5 231.3 0.081 0.1 1695.4 589.8 2.9 1179.7 247.7 0.066 0.1 1590.0 628.9 2.5 1257.9 264.1 0.055 0.1 1497.0 668.0 2.2 1336.0 280.5 0.046 0.1 1414.2 707.1 2.0 1414.2 296.9 0.038 0.1 1000.0 1000.0 1.0 2000.0 419.9 0.014 0.3 173 APPENDIX A: T A B U L A T E D D A T A A [mm2] 1,000,000 t [mm] 1.5875 b/R [rad] 2.0 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm 2] 3162.3 316.2 10.0 158.1 99.6 0.058 0.1 2814.4 355.3 7.9 177.7 111.9 0.041 0.1 2535.5 394.4 6.4 197.2 124.2 0.030 0.1 2306.9 433.5 5.3 216.7 136.5 0.022 0.2 2116.0 472.6 4.5 236.3 148.8 0.017 0.2 1954.4 511.7 3.8 255.8 161.2 0.014 0.3 1815.7 550.8 3.3 275.4 173.5 0.011 0.4 1695.4 589.8 2.9 294.9 185.8 0.009 0.5 1590.0 628.9 2.5 314.5 198.1 0.007 0.6 1497.0 668.0 2.2 334.0 210.4 0.006 0.7 1414.2 707.1 2.0 353.6 222.7 0.005 0.8 1000.0 1000.0 1.0 500.0 315.0 0.002 2.3 A [mm2] 1,000,000 t [mm] 3.1750 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 3162.3 316.2 10.0 158.1 49.8 0.460 0.0 2814.4 355.3 7.9 177.7 56.0 0.325 0.0 2535.5 394.4 6.4 197.2 62.1 0.237 0.0 2306.9 433.5 5.3 216.7 68.3 0.179 0.0 2116.0 472.6 4.5 236.3 74.4 0.138 0.0 1954.4 511.7 3.8 255.8 80.6 0.109 0.0 1815.7 550.8 3.3 275.4 86.7 0.087 0.0 1695.4 589.8 2.9 294.9 92.9 0.071 0.1 1590.0 628.9 2.5 314.5 99.0 0.059 0.1 1497.0 668.0 2.2 334.0 105.2 0.049 0.1 1414.2 707.1 2.0 353.6 111.4 0.041 0.1 1000.0 1000.0 1.0 500.0 157.5 0.015 0.3 A [mm2] 1,000,000 t [mm] 4.7625 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 3162.3 316.2 10.0 158.1 33.2 1.554 0.0 2814.4 355.3 7.9 177.7 37.3 1.095 0.0 2535.5 394.4 6.4 197.2 41.4 0.801 0.0 2306.9 433.5 5.3 216.7 45.5 0.603 0.0 2116.0 472.6 4.5 236.3 49.6 0.466 0.0 1954.4 511.7 3.8 255.8 53.7 0.367 0.0 1815.7 550.8 3.3 275.4 57.8 0.294 0.0 1695.4 589.8 2.9 294.9 61.9 0.239 0.0 1590.0 628.9 2.5 314.5 66.0 0.197 0.0 1497.0 668.0 2.2 334.0 70.1 0.165 0.0 1414.2 707.1 2.0 353.6 74.2 0.139 0.0 1000.0 1000.0 1.0 500.0 105.0 0.049 0.1 174 APPENDIX A: T A B U L A T E D D A T A A [mm2] 1,000,000 t [mm] 1.5875 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 3162.3 316.2 10.0 79.1 49.8 0.076 0.1 2814.4 355.3 7.9 88.8 56.0 0.054 0.1 2535.5 394.4 6.4 98.6 62.1 0.039 0.1 2306.9 433.5 5.3 108.4 68.3 0.030 0.1 2116.0 472.6 4.5 118.1 74.4 0.023 0.2 1954.4 511.7 3.8 127.9 80.6 0.018 0.2 1815.7 550.8 3.3 137.7 86.7 0.014 0.3 1695.4 589.8 2.9 147.5 92.9 0.012 0.4 1590.0 628.9 2.5 157.2 99.0 0.010 0.4 1497.0 668.0 2.2 167.0 105.2 0.008 0.5 1414.2 707.1 2.0 176.8 111.4 0.007 0.6 1000.0 1000.0 1.0 250.0 157.5 0.002 1.7 A [mm2] 1,000,000 t [mm] 3.1750 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 3162.3 316.2 10.0 79.1 24.9 0.609 .0.0 2814.4 355.3 7.9 88.8 28.0 0.430 0.0 2535.5 394.4 6.4 98.6 31.1 0.314 0.0 2306.9 433.5 5.3 108.4 34.1 0.237 0.0 2116.0 472.6 4.5 118.1 37.2 0.183 0.0 1954.4 511.7 3.8 127.9 40.3 0.144 0.0 1815.7 550.8 3.3 137.7 43.4 0.115 0.0 1695.4 589.8 2.9 147.5 46.4 0.094 0.0 1590.0 628.9 2.5 157.2 49.5 0.077 0.1 1497.0 668.0 2.2 167.0 52.6 0.065 0.1 1414.2 707.1 2.0 176.8 55.7 0.054 0.1 1000.0 1000.0 1.0 250.0 78.7 0.019 0.2 A [mm2] 1,000,000 t [mm] 4.7625 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 3162.3 316.2 10.0 79.1 16.6 2.056 0.0 2814.4 355.3 7.9 88.8 18.7 1.450 0.0 2535.5 394.4 6.4 98.6 20.7 1.060 0.0 2306.9 433.5 5.3 108.4 22.8 0.798 0.0 2116.0 472.6 4.5 118.1 24.8 , 0.616 0.0 1954.4 511.7 3.8 127.9 26.9 0.485 0.0 1815.7 550.8 3.3 137.7 28.9 0.389 0.0 1695.4 589.8 2.9 147.5 31.0 0.317 0.0 1590.0 628.9 2.5 157.2 33.0 0.261 0.0 1497.0 668.0 2.2 167.0 35.1 0.218 0.0 1414.2 707.1 2.0 176.8 37.1 0.184 0.0 1000.0 1000.0 1.0 250.0 52.5 0.065 0.1 175 APPENDIX A : T A B U L A T E D D A T A A [mm2] 2,000,000 t [mm] 1.5875 b/R [rad] 0.1 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm 2] 4472.1 447.2 10.0 4472.1 2817.1 0.001 3.7 3980.2 502.5 7.9 5024.9 3165.3 0.001 5.3 3585.7 557.8 6.4 5577.7 3513.5 0.001 7.2 3262.4 613.0 5.3 6130.5 3861.7 0.000 9.6 2992.5 668.3 4.5 6683.3 4209.9 0.000 12.4 2763.9 723.6 3.8 7236.1 4558.2 0.000 15.7 2567.8 778.9 3.3 7788.9 4906.4 0.000 19.6 2397.6 834.2 2.9 8341.6 5254.6 0.000 24.1 2248.6 889.4 2.5 8894.4 5602.8 0.000 29.2 2117.0 944.7 2.2 9447.2 5951.0 0.000 35.0 2000.0 1000.0 2.0 10000.0 6299.2 0.000 . 41.5 1414.2 1414.2 1.0 14142.1 8908.4 0.000 117.4 A [mm2] 2,000,000 t [mm] . 3.1750 b/R [rad] 0.1 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm 2] 4472.1 447.2 10.0 . 4472.1 1408.5 0.009 0.5 3980.2 502.5 7.9 5024.9 1582.7 0.006 0.7 3585.7 557.8 6.4 5577.7 1756.8 0.005 0.9 3262.4 613.0 5.3 6130.5 1930.9 0.004 1.2 2992.5 668.3 4.5 6683.3 2105.0 0.003 1.5 2763.9 723.6 3.8 7236.1 2279.1 0.002 2.0 2567.8 778.9 3.3 7788.9 • 2453.2 0.002 2.5 2397.6 834.2 2.9 8341.6 2627.3 0.001 3.0 2248.6 889.4 2.5 8894.4 2801.4 0.001 3.6 2117.0 944.7 2.2 9447.2 2975.5 0.001 4.4 2000.0 1000.0 2.0 10000.0 3149.6 0.001 5.2 1414.2 1414.2 1.0 14142.1 4454.2 0.000 14.7 A [mm2] 2,000,000 t [mm] 4.7625 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 4472.1 447.2 10.0 4472.1 939.0 0.031 0.1 3980.2 502.5 7.9 5024.9 1055.1 0.022 0.2 3585.7 557.8 6.4 5577.7 1171.2 0.016 0.3 3262.4 613.0 5.3 6130.5 1287.2 0.012 0.4 2992.5 668.3 4.5 6683.3 1403.3 0.009 0.5 2763.9 723.6 3.8 7236.1 1519.4 0.007 0.6 2567.8 778.9 3.3 7788.9 1635.5 0.006 0.7 2397.6 834.2 2.9 8341.6 1751.5 0.005 0.9 2248.6 889.4 2.5 8894,4 1867.6 0.004 1.1 2117.0 944.7 2.2 9447.2 1983.7 0.003 1.3 2000.0 1000.0 2.0 10000.0 2099.7 0.003 1.5 1414.2 1414.2 1.0 14142.1 2969.5 0.001 4.3 176 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 2,000,000 t [mm] 1.5875 b/R [rad] 0.5 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm'2] 4472.1 447.2 10.0 894.4 563.4 0.006 0.7 3980.2 502.5 7.9 1005.0 633.1 0.004 1.1 3585.7 557.8 6.4 1115.5 702.7 0.003 1.4 3262.4 613.0 5.3 1226.1 772.3 0.002 1.9 2992.5 668.3 4.5 1336.7 842.0 0.002 2.5 2763.9 723.6 3.8 1447.2 911.6 0.001 3.2 2567.8 778.9 3.3 1557.8 981.3 0.001 3.9 2397.6 834.2 2.9 1668.3 1050.9 0.001 4.8 2248.6 889.4 2.5 1778.9 1120.6 0.001 5.9 2117.0 944.7 2.2 1889.4 1190.2 0.001 7.0 2000.0 1000.0 2.0 2000.0 1259.8 0.001 8.3 1414.2 1414.2 1.0 2828.4 1781.7 0.000 23.6 A [mm2] 2,000,000 t [mm] 3.1750 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 4472.1 447.2 10.0 894.4 281.7 0.045 0.1 3980.2 502.5 7.9 1005.0 316.5 0.032 0.1 3585.7 557.8 6.4 1115.5 351.4 0.023- 0.2 3262.4 613.0 5.3 1226.1 386.2 0.017 0.2 2992.5 668.3 4.5 1336.7 421.0 0.013 0.3 2763.9 723.6 3.8 1447.2 455.8 0.011 0.4 2567.8 778.9 3.3 1557.8 490.6 0.009 0.5 2397.6 834.2 2.9 1668.3 525.5 0.007 0.6 ' 2248.6 889.4 2.5 1778.9 560.3 0.006 0.7 2117.0 944.7 2.2 1889.4 595.1 0.005 0.9 2000.0 1000.0 2.0 2000.0 629.9 0.004 1.0 1414.2 1414.2 1.0 2828.4 890.8 0.001 3.0 A [mm2] 2,000,000 t [mm] 4.7625 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 4472.1 447.2 10.0 894.4 187.8 0.152 0.0 3980.2 502.5 . 7.9 1005.0 211.0 0.107 0.0 3585.7 557.8 6.4 1115.5 234.2 0:078 • 0.1 3262.4 613.0 5.3 1226.1 257.4 0.059 0.1 2992.5 668.3 4.5 1336.7 280.7 0.045 0.1 2763.9 723.6 3.8 1447.2 303.9 0.036 0.1 2567.8 778.9 3.3 1557.8 327.1 0.029 0.1 2397.6 834.2 2.9 1668.3 350.3 0.023 0.2 2248.6 889.4 2.5 1778.9 373.5 0.019 0.2 2117.0 944.7 2.2 1889.4 396.7 0.016 0.3 2000.0 1000.0 2.0 2000.0 419.9 0.014 0.3 1414.2 1414.2 1.0 2828.4 593.9 0.005 0.9 177 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 2,000,000 t [mm] 1.5875 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 4472.1 447.2 10.0 223.6 140.9 0.020 0.2 3980.2 502.5 7.9 251.2 158.3 0.014 0.3 3585.7 557.8 6.4 • 278.9 175.7 0.010 0.4 3262.4 613.0 5.3 306.5 193.1 0.008 0.5 2992.5 668.3 4.5 334.2 210.5 0.006 0.7 2763.9 723.6 3.8 361.8 227.9 0.005 0.9 2567.8 778.9 3.3 389.4 245.3 0.004 1.1 2397.6 ' 834.2 2.9 417.1 262.7 0.003 1.3 2248.6 889.4 2.5 444.7 280.1 0.003 1.6 2117.0 944.7 2.2 472.4 297.6 0.002 1.9 2000.0 1000.0 2.0 500.0 315.0 0.002 2.3 1414.2 1414.2 1.0 707.1 445.4 0.001 6.5 A [mm2] 2,000,000 t [mm] 3.1750 b/R [rad] 2.0 L b AR Radius R/t buckling press [Nmm'2] buckling ratio [mm] [mm] [mm] 4472.1 447.2 10.0 223.6 70.4 0.163 0.0 3980.2 502.5 7.9 251.2 79.1 0.115 0.0 3585.7 557.8 6.4 278.9 87.8 0.084 0.1 3262.4 613.0 5.3 306.5 96.5 0.063 0.1 2992.5 668.3 4.5 334.2 105.2 0.049 0.1 2763.9 723.6 3.8 361.8 114.0 0.038 0.1 2567.8 778.9 3.3 389.4 122.7 0.031 0.1 2397.6 834.2 2.9 417.1 131.4 0.025 0.2 2248.6 889.4 2.5 444.7 140.1 0.021 0.2 2117.0 944.7 2.2 472.4 148.8 0.017 0.2 2000.0 1000.0 2.0 500.0 157.5 0.015 0.3 1414.2 1414.2 1.0 707.1 222.7 0.005 0.8 A [mm2] 2,000,000 t [mm] 4.7625 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 4472.1 447.2 10.0 223.6 47.0 0.549 0.0 3980.2 502.5 7.9 251.2 52.8 0.387 0.0 3585.7 557.8 6.4 278.9 58.6 0.283 0.0 3262.4 613.0 5.3 306.5 64.4 0.213 0.0 2992.5 668.3 4.5 334.2 70.2 0.165 0.0 2763.9 723.6 3.8 - 361.8 76.0 0.130 0.0 2567.8 778.9 3.3 389.4 81.8 0.104 0.0 2397.6 834.2 2.9 417.1 87.6 0.085 0.0 2248.6 889.4 2.5 444.7 93.4 0.070 0.1 2117.0 944.7 2.2 472.4 99.2 0.058 0.1 2000.0 1000.0 2.0 500.0 105.0 0.049 0.1 1414.2 1414.2 1.0 707.1 148.5 0.017 0.2 178 A P P E N D I X A : T A B U L A T E D D A T A A [mm2) 2,000,000 t [mm] 1.5875 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 4472.1 447.2 10.0 111.8 70.4 0.027 0.2 3980.2 502.5 7.9 125.6 79.1 0.019 0.2 3585.7 557.8 6.4 139.4 87.8 0.014 0.3 3262.4 613.0 5.3 153.3 96.5 0.010 0.4 2992.5 668.3 4.5 167.1 105.2 0.008 0.5 2763.9 723.6 3.8 180.9 114.0 0.006 0.7 2567.8 778.9 3.3 194.7 122.7 0.005 0.8 2397.6 834.2 2.9 208.5 131.4 0.004 1.0 2248.6 889.4 2.5 222.4 140.1 0.003 1.2 2117.0 944.7 2.2 236.2 148.8 0.003 1.5 2000.0 1000.0 2.0 250.0 157.5 0.002 1.7 1414.2 1414.2 1.0 353.6 222.7 0.001 4.9 A [mm2] 2,000,000 t [mm] 3.1750 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 4472.1 447.2 10.0 111.8 35.2 0.215 0.0 3980.2 502.5 7.9 125.6 39.6 0.152 0.0 3585.7 557.8 6.4 139.4 43.9 0.111 0.0 3262.4 613.0 ' 5.3 153.3 48.3 0.084 0.1 2992.5 668.3 4.5 167.1 52.6 0.065 0.1 2763.9 723.6 3.8 180.9 57.0 0.051 0.1 2567.8 778.9 3.3 194.7 61.3 0.041 0.1 2397.6 .834.2 2.9 208.5 65.7 0.033 0.1 2248.6 889.4 2.5 222.4 70.0 0.027 0.2 2117.0 944.7 2.2 236.2 74.4 0.023 0.2 2000.0 1000.0 2.0 250.0 78.7 0.019 0.2 1414.2 1414.2 1.0 353.6 111.4 0.007 0.6 A [mm2] 2,000,000 t [mm] 4.7625 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 4472.1 447.2 10.0 111.8 23.5 0.727 0.0 3980.2 502.5 7.9 125.6 26.4 0.513 0.0 3585.7 557.8 6.4 139.4 29.3 0.375 0.0 3262.4 613.0 5.3 153.3 32.2 0.282 0.0 2992.5 668.3 4.5 167.1 35.1 0.218 0.0 2763.9 723.6 3.8 180.9 38.0 0.172 0.0 2567.8 778.9 3.3 194.7 40.9 0.138 0.0 2397.6 834.2 2.9 208.5 43.8 0.112 0.0 2248.6 889.4 2.5 222.4 46.7 0.092 0.0 2117.0 944.7 2.2 236.2 49.6 0.077 0.1 2000.0 1000.0 2.0 250.0 52.5 0.065 0.1 1414.2 1414.2 1.0 353.6 74.2 0.023 0.2 179 APPENDIX A: T A B U L A T E D D A T A A [mm2] 4,000,000 t [mm] 1.5875 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 6324.6 632.5 10.0 6324.6 3984.0 0.000 10.5 5628.8 710.6 7.9 7106.3 4476.4 0.000 14.9 5070.9 788.8 6.4 7888.1 4968.9 0.000 20.4 4613.7 867.0 5.3 8669.8 5461.3 0.000 27.0 4232.1 945.2 4.5 9451.6 5953.8 0.000 35.0 3908.8 1023.3 3.8 10233.3 6446.2 0.000 44.5 3631.4 1101.5 3.3 11015.1 6938.6 0.000 55.5 3390.7 1179.7 2.9 11796.9 7431.1 0.000 68.1 3180.0 1257.9 2.5 12578.6 7923.5 0.000 82.6 2993.9 1336.0 2.2 13360.4 8416.0 0.000 99.0 2828.4 1414.2 2.0 14142.1 8908.4 0.000 117.4 2000.0 2000.0 1.0 20000.0 12598.4 0.000 332.0 A [mm2] 4,000,000 t [mm] 3.1750 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 6324.6 632.5 10.0 6324.6 1992.0 0.003 1.3 5628.8 710.6 7.9 7106.3 2238.2 0.002 1.9 5070.9 788.8 6.4 7888.1 2484.4 0.002 2.5 4613.7 867.0 5.3 8669.8 2730.7 0.001 3.4 4232.1 945.2 4.5 9451.6 2976.9 0.001 4.4 3908.8 1023.3 3.8 10233.3 3223.1 0.001 ' 5.6 3631.4 1101.5 3.3 11015.1 3469.3 0.001 6.9 3390.7 1179.7 2.9 11796.9 3715.5 0.000 8.5 3180.0 1257.9 2.5 12578.6 3961.8 0.000 10.3 2993.9 1336.0 2.2 13360.4 4208.0 0.000 12.4 2828.4 1414.2 2.0 14142.1 4454.2 0.000 14.7 2000.0 2000.0 1.0 20000.0 6299.2 0.000 41.5 A [mm2] 4,000,000 t [mm] 4.7625 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 6324.6 632.5 10.0 6324.6 1328.0 0.011 0.4 5628.8 710.6 7.9 7106.3 1492.1 0.008 0.6 5070.9 788.8 6.4 7888.1 1656.3 0.006 0.8 4613.7 867.0 5.3 8669.8 1820.4 0.004 1.0 4232.1 945.2 4.5 9451.6 1984.6 0.003 1.3 3908.8 1023:3 3.8 10233.3 2148.7 0.003 1.6 3631.4 1101.5 3.3 11015.1 2312.9 0.002 2.1 3390.7 1179.7 2.9 11796.9 2477.0 0.002 2.5 3180.0 1257.9 2.5 12578.6 2641.2 0.001 3.1 2993.9 1336.0 2.2 13360.4 2805.3 0.001 3.7 2828.4 1414.2 2.0 14142.1 2969.5 0.001 4.3 2000.0 2000.0 1.0 20000.0 4199.5 0.000 12.3 180 APPENDIX A : T A B U L A T E D D A T A A [mm2] 4,000,000 t [mm] 1.5875 b/R [rad] 0.5 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm 2] 6324.6 632.5 10.0 1264.9 796.8 0.002 2.1 5628.8 710.6 7.9 1421.3 895.3 0.001 3.0 5070.9 788.8 6.4 1577.6 993.8 0.001 4.1 4613.7 867.0 5.3 1734.0 1092.3 0.001 5.4 4232.1 945.2 4.5 1890.3 1190.8 0.001 7.0 3908.8 1023.3 3.8 2046.7 1289.2 0.000 8.9 3631.4 1101.5 3.3 2203.0 1387.7 0.000 11.2 3390.7 1179.7 2.9 2359.4 1486.2 0.000 13.7 3180.0 1257.9 2.5 2515.7 1584.7 0.000 16.6 2993.9 1336.0 2.2 2672.1 1683.2 0.000 19.9 2828.4 1414.2 2.0 2828.4 1781.7 0.000 23.6 2000.0 2000.0 1.0 4000.0 2519.7 0.000 66.8 A [mm2] 4,000,000 t [mm] 3.1750 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 6324.6 632.5 10.0 1264.9 398.4 0.016 0.3 5628.8 710.6 7.9 1421.3 447.6 0.011 0.4 5070.9 788.8 6.4 1577.6 496.9 0.008 0.5 4613.7 867.0 5.3 1734.0 546.1 0.006 0.7 4232.1 945.2 4.5 1890.3 595.4 0.005 0.9 3908.8 1023.3 3.8 2046.7 644.6 0.004 1.1 3631.4 1101.5 3.3 2203.0 693.9 0.003 1.4 3390.7 1179.7 2.9 2359.4 743.1 0.002 1.7 3180.0 1257.9 2.5 2515.7 792.4 0.002 2.1 2993.9 1336.0 2.2 2672.1 841.6 0.002 2.5 2828.4 1414.2 2.0 2828.4 890.8 0.001 3.0 2000.0 2000.0 1.0 4000.0 1259.8 0.001 8.3 A [mm2] 4,000,000 t [mm] 4.7625 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 6324.6 632.5 10.0 1264.9 265.6 0.054 0.1 5628.8 710.6 7.9 1421.3 298.4 0.038 0.1 5070.9 788.8 , 6.4 1577.6 331.3 0.028 0.2 4613.7 867.0 5.3 1734.0 364.1 0.021 0.2 4232.1 945.2 4.5 1890.3 396.9 0.016 0.3 3908.8 1023.3 3.8 2046.7 429.7 0.013 0.3 3631.4 1101.5 3.3 2203.0 462.6 0.010 0.4 3390.7 1179.7 2.9 2359.4 495.4 0.008 0.5 3180.0 1257.9 2.5 2515.7 528.2 0.007 0.6 2993.9 1336.0 2.2 2672.1 561.1 0.006 0.7 2828.4 1414.2 2.0 2828.4 593.9 0.005 0.9 2000.0 2000.0 1.0 4000.0 839.9 0.002 2.5 181 APPENDIX A : T A B U L A T E D D A T A A [mm2] 4,000,000 t [mm] 1.5875 b/R [rad] 2.0 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm -2] 6324.6 632.5 10.0 316.2 199.2 0.007 0.6 5628.8 710.6 7.9 355.3 223.8 0.005 0.8 5070.9 788.8 6.4 394.4 248.4 0.004 1.1 4613.7 867.0 5.3 433.5 273.1 0.003 1.5 4232.1 945.2 4.5 472.6 297.7 0.002 1.9 3908.8 1023.3 3.8 511.7 322.3 0.002 2.5 3631.4 1101.5 3.3 550.8 346.9 0.001 3.1 3390.7 1179.7 2.9 589.8 371.6 0.001 3.8 3180.0 1257.9 2.5 628.9 396.2 0.001 4.6 2993.9 1336.0 2.2 668.0 420.8 0.001 5.5 2828.4 1414.2 2.0 707.1 445.4 0.001 6.5 2000.0 2000.0 1.0 1000.0 629.9 0.000 18.5 A [mm2] 4,000,000 t [mm] 3.1750 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 6324.6 632.5 10.0 316.2 99.6 0.058 0.1 5628.8 710.6 7.9 355.3 111.9 0.041 0.1 5070.9 788.8 6.4 394.4 124.2 0.030 0.1 4613.7 867.0 5.3 433.5 136.5 0.022 0.2 4232.1 945.2 4.5 472.6 148.8 0.017 0.2 3908.8 1023.3 3.8 511.7 161.2 0.014 0.3 3631.4 1101.5 3.3 550.8 173.5 0.011 0.4 3390.7 1179.7 2.9 589.8 185.8 0.009 0.5 3180.0 1257.9 2.5 628.9 198.1 0.007 0.6 2993.9 1336.0 2.2 668.0 210.4 0.006 0.7 2828.4 1414.2 2.0 707.1 222.7 0.005 0.8 2000.0 2000.0 1.0 1000.0 315.0 0.002 2.3 A [mm2] 4,000,000 t [mm] 4.7625 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 6324.6 632.5 10.0 316.2 66.4 0.194 0.0 5628.8 710.6 7.9 355.3 74.6 0.137 0.0 5070.9 788.8 6.4 394.4 82.8 0.100 0.0 4613.7 867.0 5.3 433.5 91.0 0.075 0.1 4232.1 945.2 4.5 472.6 99.2 0.058 0.1 3908.8 1023.3 3.8 511.7 107.4 0.046 0.1 3631.4 1101.5 3.3 550.8 115.6 0.037 0.1 3390.7 1179.7 2.9 589.8 123.9 0.030 0.1 3180.0 1257.9 2.5 628.9 132.1 0.025 0.2 2993.9 1336.0 2.2 668.0 140.3 0.021 0.2 2828.4 1414.2 2.0 707.1 148.5 0.017 0.2 2000.0 2000.0 1.0 1000.0 210.0 0.006 0.7 182 APPENDIX A: T A B U L A T E D D A T A A [mm2] 4,000,000 t [mm] 1.5875 b/R [rad] 4.0 L b AR Radius R/t buckling press [Nmm"2] buckling ratio [mm] [mm] [mm] 6324.6 632.5 10.0 158.1 99.6 0.010 0.4 5628.8 710.6 7.9 177.7 111.9 0.007 0.6 5070.9 788.8 6.4 197.2 124.2 0.005 0.9 4613.7 867.0 5.3 216.7 136.5 0.004 1.1 4232.1 945.2 4.5. 236.3 148.8 0.003 1.5 3908.8 1023.3 3.8 255.8 161.2 0.002 1.9 3631.4 1101.5 3.3 275.4 173.5 0.002 2.3 3390.7 1179.7 2.9 294.9 185.8 0.001 2.9 3180.0 1257.9 2.5 314.5 198.1 0.001 3.5 2993.9 1336.0 2.2 334.0 210.4 0.001 4.2 2828.4 1414.2 2.0 353.6 222.7 0.001 4.9 2000.0 2000.0 1.0 500.0 315.0 0.000 14.0 A [mm2] 4,000,000 t [mm] 3.1750 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 6324.6 632.5 10.0 158.1 49.8 0.076 0.1 5628.8 710.6 7.9 177.7 56.0 0.054 0.1 5070.9 788.8 6.4 197.2 62.1 0.039 0.1 4613.7 867.0 5.3 216.7 68.3 0.030 0.1 4232.1 945.2 4.5 236.3 74.4 0.023 0.2 3908.8 1023.3 3.8 255.8 80.6 0.018 0.2 3631.4 1101.5 3.3 275.4 86.7 0.014 0.3 3390.7 1179.7 2.9 294.9 92.9 0.012 0.4 3180.0 1257.9 2.5 314.5 99.0 0.010 0.4 2993.9 1336.0 2.2 334.0 105.2 0.008 0.5 2828.4 1414.2 2.0 353.6 111.4 0.007 0.6 2000.0 2000.0 1.0 500.0 157.5 0.002 1.7 A [mm2] 4,000,000 t [mm] 4.7625 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 6324.6 632.5 10.0 158.1 33.2 0.257 0.0 5628.8 710.6 7.9 177.7 37.3 0.181 0.0 5070.9 788.8 6.4 197.2 41.4 0.132 0.0 4613.7 867.0 5.3 216.7 45.5 0.100 0.0 4232.1 945.2 4.5 236.3 49.6 0.077 0.1 3908.8 1023.3 3.8 255.8 53.7 0.061 0.1 3631.4 1101.5 3.3 275.4 57.8 0.049 0.1 3390.7 1179.7 2.9 294.9 61.9 0.040 0.1 3180.0 1257.9 2.5 314.5 66.0 0.033 0.1 2993.9 1336.0 2.2 334.0 70.1 0.027 0.2 2828.4 1414.2 2.0 353.6 74.2 0.023 0.2 2000.0 2000.0 1.0 500.0 105.0 0.008 0.5 183 APPENDIX A: T A B U L A T E D D A T A A [mm2] 10,000,000 t [mm] 1.5875 b/R [rad] 0.1 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm 2] 10000.0 1000.0 10.0 10000.0 6299.2 0.000 41.5 8899.9 1123.6 7.9 11236.1 7077.8 0.000 58.9 8017.9 1247.2 6.4 12472.1 7856.5 0.000 80.5 7294.9 1370.8 5.3 13708.2 8635.1 0.000 106.9 6691.5 1494.4 4.5 14944.3 9413.7 0.000 138.5 6180.3 1618.0 3.8 16180.3 10192.3 0.000 175.8 5741.7 1741.6 3.3 17416.4 10971.0 0.000 219.2 5361.2 1865.2 2.9 18652.5 11749.6 0.000 269.3 5028.0 1988.9 2.5 19888.5 12528.2 0.000 326.4 4733.8 2112.5 2.2 21124.6 13306.8 0.000 391.2 4472.1 2236.1 2.0 22360.7 14085.5 0.000 463.9 3162.3 3162.3 1.0 31622.8 19919.9 0.000 1312.1 A [mm2] 10,000,000 t [mm] 3.1750 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 10000.0 1000.0 10.0 10000.0 3149.6 .• 0.001 5.2 8899.9 1123.6 7.9 11236.1 - 3538.9 0.001 7.4 8017.9 1247.2 6.4 12472:1 3928.2 0.000 10.1 7294.9 1370.8 5.3 13708.2 4317.5 0.000 13.4 6691.5 1494.4 4.5 14944.3 4706.9 0.000 17.3 6180.3 1618.0 3.8 16180.3 5096.2 0.000 22.0 5741.7 •1741.6 3.3 17416.4 5485.5 0.000 27.4 5361.2 1865.2 2.9 18652.5 5874.8 0.000 33.7 5028.0 1988.9 2.5 19888.5 6264.1 0.000 40.8 4733.8 2112.5 2.2 21124.6 6653.4 0.000 48.9 4472.1 2236.1 2.0 22360.7 7042.7 0.000 58.0 3162.3 3162.3 1.0 31622.8 9959.9 0.000 164.0 A [mm2] 10,000,000 t [mm] 4.7625 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 10000.0 1000.0 10.0 10000.0 2099.7 0.003 1.5 8899.9 1123.6 7.9 11236.1 2359.3 0.002 2.2 8017.9 1247.2 6.4 ' 12472.1 2618.8 0.001 3.0 7294.9 1370.8 5.3 13708.2 2878.4 0.001 4.0 6691.5 1494.4 4.5 14944.3 3137.9 0.001 5.1 6180.3 1618.0 3.8 16180.3 3397.4 0.001 6.5 5741.7 1741.6 3.3 17416.4 3657.0 0.001 8.1 5361.2 1865.2 2.9 • 18652.5 3916.5 0.000 10.0 5028.0 1988.9 2.5 19888.5 ' 4176.1 0.000 12.1 4733.8 2112.5 2.2 21124.6 4435.6 0.000 14.5 4472.1 2236.1 2.0 22360.7 4695.2 0.000 17.2 3162.3 3162.3 1.0 31622.8 6640.0 0.000 48.6 184 APPENDIX A : T A B U L A T E D D A T A A [mm2] 10,000,000 t [mm] 1.5875 b/R [rad] 0.5 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm"2] 10000.0 . 1000.0 10.0 2000.0 1259.8 0.001 8.3 8899.9 1123.6 7.9 2247.2 1415.6 0.000 11.8 8017.9 1247.2 6.4 2494.4 1571.3 0.000 16.2 7294.9 1370.8 5.3 2741.6 1727.0 0.000 21.5 6691.5 1494.4 4.5 2988.9 1882.7 0.000 27.9 6180.3 1618.0 3.8 3236.1 2038.5 0.000 35.4 5741.7 1741.6 3.3 3483.3 2194.2 0.000 44.1 5361.2 1865.2 2.9 3730.5 2349.9 0.000 54.2 5028.0 1988.9 2.5 3977.7 2505.6 0.000 65.7 4733.8 2112.5 2.2 4224.9 2661.4 0.000 78.7 4472.1 2236.1 2.0 4472.1 2817.1 0.000 93.4 3162.3 3162.3 1.0 6324.6 3984.0 0.000 264.0 A [mm2] 10,000,000 t [mm] 3.1750 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 10000.0 1000.0 10.0 2000.0 629.9 0.004- 1.0 8899.9 1123.6 7.9 2247.2 707.8 0.003 1.5 8017.9 1247.2 6.4 2494.4 785.6 0.002 2.0 7294.9 1370.8 5.3 2741.6 863.5 0.002 2.7 6691.5 1494.4 4.5 2988.9 941.4 0.001 3.5 6180.3 1618.0 3.8 3236.1 1019.2 0.001 4.4 5741.7 1741.6 3.3 3483.3 1097.1 0.001 5.5 5361.2 1865.2 2.9 3730.5 1175.0 0.001 6.8 5028.0 1988.9 2.5 3977.7 1252.8 0.001 8.2 4733.8 2112.5 2.2 4224.9 1330.7 0.000 9.8 4472.1 2236.1 2.0 4472.1 1408.5 0.000 11.7 3162.3 3162.3 1.0 6324.6 1992.0 0.000 33.0 A [mm2] 10,000,000 t [mm] 4.7625 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 10000.0 1000.0 10.0 2000.0 419.9 0.014 0.3 8899.9 1123.6 7.9 2247.2 471.9 0.010 0.4 8017.9' 1247.2 6.4 2494.4 523.8 . 0.007 0.6 7294.9 1370.8 5.3 2741.6 575.7 0.005 0 8 6691.5 1494.4 4.5 2988.9 627.6 0.004 1.0 6180.3 1618.0 3.8 3236.1 679.5 0.003 1.3 5741.7 1741.6 3.3 3483.3 731.4 0.003 1.6 5361.2 1865.2 2.9 3730.5 783.3 0.002 2.0 5028.0 1988.9 2.5 3977.7 835.2 0.002 2.4 4733.8 2112.5 2.2 4224.9 887.1 0.001 2.9 4472.1 2236.1 2.0 4472.1 939.0 0.001 3.5 3162.3 3162.3 1.0 6324.6 1328.0 0.000 9.8 185 APPENDIX A: T A B U L A T E D D A T A A [mm2] 10,000,000 t [mm] 1.5875 b/R [rad] 2.0 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm 2] 10000.0 1000.0 10.0 500.0 315.0 0.002 2.3 8899.9 1123.6 7.9 561.8 353.9 0.001 3.3 8017.9 1247.2 6.4 , 623.6 392.8 0.001 4.5 7294.9 1370.8 5.3 . 685.4 431.8 0.001 5.9 6691.5 1494.4 4.5 747.2 470.7 0.001 7.7 6180.3 1618.0 3.8 809.0 509.6 0.000 9.8 5741.7 1741.6' 3.3 870.8 548.5 0.000 12.2 5361.2 1865.2 2.9 932.6 587.5 0.000 15.0 5028.0 1988.9 2.5 994.4 626.4 0.000 18.2 4733.8 2112.5 2.2 1056.2 665.3 0.000 21.8 4472.1 2236.1 2.0 1118.0 704.3 0.000 25.8 3162.3 3162.3 1.0 1581.1 996.0 0.000 73.0 A [mm2] 10,000,000 t [mm] 3.1750 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 10000.0 1000.0 10.0 500.0 157.5 0.015 0.3 8899.9 1123.6 7.9 561.8 176.9 0.010 0.4 8017.9 1247.2 6.4 623.6 196.4 0.008 0.6 7294.9 1370.8 5.3 685.4 215.9 0.006 0.7 6691.5 1494.4 4.5 747.2 235.3 0.004 1.0 6180.3 1618.0 3.8 809.0 254.8 0.003 1.2 5741.7 1741.6 3.3 870.8 274.3 0.003 1.5 5361.2 1865.2 2.9 932.6 293.7 0.002 1.9 5028.0 1988.9 2.5 994.4 313.2 0.002 2.3 4733.8 2112.5 2.2 1056.2 332.7 0.002 2.7 4472.1 2236.1 2.0 1118.0 352.1 0.001 3.2 3162.3 3162.3 1.0 1581.1 . 498.0 0.000 9.1 A [mm2] 10,000,000 t[mm] 4.7625 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 10000.0 1000.0 10.0 500.0 105.0 0.049 0.1 8899.9 1123.6 7.9 561.8 118.0 0.035 0.1 8017.9 1247.2 6.4 623.6 130.9 0.025 0.2 7294.9 1370.8 5.3 685.4 143.9 0.019 0.2 6691.5 1494.4 4.5 747.2 156.9 0.015 ' 0.3 6180.3 1618.0 3.8 809.0 169.9 0.012 0.4 5741.7 1741.6 3.3 870.8 182.8 0.009 0.5 5361.2 1865.2 2.9 932.6 195.8 0.008 0.6 5028.0 1988.9 2.5 994.4 208.8 0.006 0.7 4733.8 2112.5 2.2 1056.2 221.8 0.005 0.8 4472.1 2236.1 2.0 1118.0 234.8 0.004 1.0 3162.3 3162.3 1.0 1581.1 332.0 0.002 2.7 186 APPENDIX A : T A B U L A T E D D A T A A [mm2] 10,000,000 t [mm] 1.5875 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 10000.0 1000.0 10.0 250.0 157.5 0.002 1.7 8899.9 1123.6 7.9 280.9 176.9 0.002 2.5 8017.9 1247.2 6.4 311.8 196.4 0.001 3.4 7294.9 1370.8. 5.3 342.7 215.9 0.001 4.5 6691.5 1494.4 4.5 373.6 235.3 0.001 5.8 6180.3 1618.0 3.8 404.5 254.8 0.001 7.4 5741.7 1741.6 3.3 435.4 274.3 0.000 9.2 5361.2 1865.2 2.9 466.3 293.7 0.000 11.3 5028.0 1988.9 2.5 497.2 313.2 0.000 13.7 4733.8 2112.5 2.2 528.1 332.7 0.000 16.4 4472.1 2236.1 2.0 559.0 352.1 0.000 19.5 3162.3 3162.3 1.0 790.6 498.0 0.000 55.1 A [mm2] 10,000,000 t [mm] 3.1750 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 10000.0 1000.0 10.0 250.0 78.7 0.019 0.2 8899.9 1123.6 7.9 280.9 88.5 0.014 0.3 8017.9 1247.2 6.4 311.8 98.2 0.010 0.4 7294.9 1370.8 5.3 342.7 107.9 0.007 0.6 6691.5 1494.4 4.5 373.6 117.7 0.006 0.7 6180.3 1618.0 3.8 404.5 127.4 0.005 0.9 5741.7 1741.6 3.3 435.4 137.1 0.004 1.2 5361.2 1865.2 2.9 466.3 146.9 0.003 1.4 5028.0 1988.9 2.5 497.2 156.6 0.002 1.7 4733.8 2112.5 2.2 528.1 166.3 0.002 2.1 4472.1 2236.1 2.0 559.0 176.1 0.002 2.4 3162.3 3162.3 1.0 790.6 249.0 0.001 6.9 A [mm2] 10,000,000 t [mm] 4.7625 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 10000.0 1000.0 10.0 . 250.0 52.5 0.065 0.1 8899.9 1123.6 7.9 280.9 59.0 0.046 0.1 8017.9 1247.2 6.4 311.8 65.5 0.034 0.1 7294.9 1370.8 5.3 342.7 72.0 0.025 0.2 6691.5 1494.4 4.5 373.6 78.4 0.019 0.2 6180.3 1618.0 3.8 404.5 84.9 0.015 0.3 5741.7 1741.6 3.3 435.4 91.4 0.012 0.3 5361.2 1865.2 2.9 466.3 97.9 0.010 0.4 5028.0 1988.9 2.5 497.2 104.4 0.008 0.5 4733.8 2112.5 2.2 528.1 110.9 0.007 0.6 4472.1 2236.1 2.0 559.0 117.4 0.006 0.7 3162.3 3162.3 1.0 790.6 166.0 0.002 2.0 187 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 20,000,000 t [mm] 1.5875 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 14142.1 1414.2 10.0 14142.1 8908.4 0.000 117.4 12586.4 1589.0 7.9 15890.2 10009.6 0.000 166.5 11339.0 1763.8 6.4 17638.3 11110.7 0.000 227.7 10316.5 1938.6 5.3 19386.3 12211.9 0.000 302.3 9463.2 2113.4 4.5 21134.4 13313.0 0.000 391.7 8740.3 2288.2 3.8 22882.5 14414.1 0.000 497.2 8120.0 2463.1 3.3 24630.5 15515.3 0.000 620.0 7581.9 2637.9 2.9 26378.6 16616.4 0.000 761.6 7110.7 2812.7 2.5 28126.6 17717.6 0.000 923.3 6694.6 2987.5 2.2 29874.7 18818.7 0.000 1106.4 6324.6 3162.3 2.0 31622.8 19919.9 0.000 1312.1 4472.1 4472.1 1.0 44721.4 28170.9 0.000 3711.3 A [mm2] 20,000,000 t [mm] 3.1750 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 14142.1 1414.2 10.0 14142.1 4454.2 0.000 14.7 12586.4 1589.0 7.9 15890.2 5004.8 0.000 20.8 11339.0 1763.8 6.4 17638.3 5555.4 0.000 28.5 10316.5 1938.6 5.3 19386.3 6105.9 0.000 37.8 9463.2 2113.4 4.5 21134.4 6656.5 0.000 49.0 8740.3 2288.2 3.8 22882.5 7207.1 0.000 62.1 8120.0 2463.1 3.3 24630.5 7757.6 0.000 77.5 7581.9 2637.9 2.9 26378.6 8308.2 0.000 95.2 7110.7 2812.7 2.5 28126.6 8858.8 0.000 115.4 6694.6 2987.5 2.2 • 29874.7 9409.4 0.000 138.3 6324.6 3162.3 2.0 31622.8 9959.9 0.000 164.0 4472.1 4472.1 1.0 44721.4 14085.5 0.000 463.9 A [mm2] 20,000,000 t [mm] 4.7625 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 14142.1 1414.2 10.0 14142.1 2969.5 0.001 . 4.3 12586.4 1589.0 7.9 15890.2 3336.5 0.001 6.2 11339.0 1763.8 6.4 17638.3 3703.6 0.000 8.4 10316.5 1938.6 5.3 19386.3 4070.6 0.000 11.2 9463.2 2113.4 4.5 21134.4 4437.7 0.000 14.5 8740.3 2288.2 3.8 22882.5 4804.7 0.000 18.4 8120.0 2463.1 3.3 24630.5 5171.8 0.000 23.0 7581.9 2637.9 2.9 26378.6 5538.8 0.000 28.2 7110.7 2812.7 2.5 28126.6 5905.9 0.000 34.2 6694.6 2987.5 2.2 29874.7 6272.9 0.000 41.0 6324.6 3162.3 2.0 31622.8 6640.0 0.000 48.6 4472.1 4472.1 1.0 44721.4 9390.3 0.000 137.5 188 APPENDIX A : T A B U L A T E D D A T A A [mm2] 20,000,000 t [mm] 1.5875 b/R [rad] 0.5 L [mm] • b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 14142.1 1414.2 10.0 2828.4 1781.7 0.000 23.6 12586.4 1589.0 7.9 3178.0 2001.9 0.000 33.5 11339.0 1763.8 6.4 3527.7 2222.1 0.000 45.8 10316.5 1938.6 5.3 3877.3 2442.4 0.000 60.8 9463.2 2113.4 4.5 4226.9 2662.6 0.000 78.8 8740.3 2288.2 3.8 4576.5 2882.8 0.000 100.0 8120.0 2463.1 3.3 4926.1 3103.1 0.000 124.8 7581.9 2637.9 2.9 5275.7 3323.3 0.000 153.3 7110.7 2812.7 2.5 5625.3 3543.5 0.000 185.8 6694.6 2987.5 2.2 5974.9 3763.7 0.000 222.6 6324.6 3162.3 2.0 6324.6 3984.0 0.000 264.0 4472.1 4472.1 1.0 8944.3 5634.2 0.000 746.8 A [mm2] 20,000,000 t [mm] 3.1750 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 14142.1 1414.2 10.0 2828.4 890.8 0.001 3.0 12586.4 1589.0 7.9 3178.0 1001.0 0.001 4.2 11339.0 1763.8 6.4 3527.7 1111.1 0.001 5.7 10316.5 1938.6 5.3 3877.3 1221.2 0.001 7.6 9463.2 2113.4 4.5 4226.9 1331.3 0.000 9.9 8740.3 2288.2 3.8 4576.5 1441.4 0.000 12.5 8120.0 2463.1 3.3 4926.1 1551.5 0.000 15.6 7581.9 2637.9 2.9 5275.7 1661.6 0.000 19.2 7110.7 2812.7 2.5 5625.3 1771.8 0.000 23.2 6694.6 2987.5 2.2 5974.9 1881.9 0.000 27.8 6324.6 3162.3 2.0 6324.6 1992.0 0.000 33.0 4472.1 4472.1 1.0 8944.3 2817.1 0.000 93.4 A [mm2] 20,000,000 t [mm] 4.7625 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 14142.1 1414.2 .10.0 2828.4 593.9 0.005 0.9 12586.4 1589.0 7.9 3178.0 667.3 0.003 1.2 11339.0 1763.8 6.4 3527.7 740.7 0.002 1.7 10316.5 1938.6 5.3 3877.3 814.1 0.002 2.3 9463.2 2113.4 4.5 4226.9 887.5 0.001 2.9 8740.3 2288.2 3.8 4576.5 960.9 0.001 3.7 8120.0 2463.1 3.3 4926.1 1034.4 0.001 4.6 7581.9 2637.9 2.9 5275.7 1107.8 0.001 5.7 7110.7 2812.7 2.5 5625.3 1181.2 0.001 6.9 6694.6 2987.5 2.2 5974.9 1254.6 0.001 8.2 6324.6 3162.3 2.0 6324.6 1328.0 0.000 9.8 4472.1 4472.1 1.0 8944.3 1878.1 0.000 27.7 189 APPENDIX A : T A B U L A T E D D A T A A [mm2] 20,000,000 t [mm] 1.5875 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 14142.1 1414.2 10.0 707.1 445.4 0.001 6.5 12586.4 1589.0 7.9 794.5 500.5 0.000 9.3 11339.0 1763.8 6.4 881.9 555.5 0.000 12.7 10316.5 1938.6 5.3 969.3 610.6 0:000 16.8 9463.2 2113.4 4.5 1056.7 665.7 0.000 21.8 8740.3 2288.2 3.8 1144.1 720.7 0.000 27.7 8120.0 2463.1 3.3 1231.5 775.8 0.000 34.5 7581.9 2637.9 2.9 1318.9 830.8 0.000 42.4 7110.7 2812.7 2.5 1406.3 885.9 0.000 51.4 6694.6 2987.5 2.2 1493.7 940.9 0.000 61.5 6324.6 3162.3 2.0 1581.1 996.0 0.000 73.0 4472.1 4472.1 1.0 2236.1 1408.5 0.000 206.4 A [mm2] 20,000,000 t [mm] 3.1750 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 14142.1 1414.2 10.0 707.1 222.7 0.005 0.8 12586.4 1589.0 7.9 794.5 250.2 0.004 1.2 11339.0 1763.8 6.4 881.9 277.8 0.003 1.6 10316.5 1938.6 5.3 969.3 305.3 0.002 2.1 9463.2 2113.4 4.5 1056.7 332.8 0.002 2.7 8740.3 2288.2 3.8 1144.1 360.4 0.001 3.5 8120.0 2463.1 3.3 1231.5 387.9 0.001 4.3 7581.9 2637.9 2.9 1318.9 415.4 0.001 5.3 7110.7 2812.7 2.5 1406.3 442.9 0.001 6.4 6694.6 2987.5 2.2 1493.7 470.5 0.001 7.7 6324.6 3162.3 2.0 1581.1 498.0 . 0.000 9.1 4472.1- 4472.1 1.0 2236.1 •704.3 0.000 25.8 A [mm2] 20,000,000 t [mm] 4.7625 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 14142.1 1414.2 10.0 707.1 148.5 0.017 0.2 12586.4 1589.0 7.9 794.5 166.8 0.012 0.3 11339.0 1763.8 6.4 881.9 185.2 0.009 0.5 10316.5 1938.6 ' 5.3 969.3 203.5 0.007 0.6 9463.2 2113.4 4.5 1056.7 221.9 0.005 0.8 8740.3 2288.2 3.8 1144.1 240.2 0.004 1.0 8120.0 2463.1 3.3 1231.5 258.6 0.003 1.3 7581.9 2637.9 2:9 1318.9 276.9 0.003 1.6 7110.7 2812.7 2.5 1406.3 295.3 0.002 1.9 6694.6 2987.5 2.2 1493.7 313.6 0.002 2.3 6324.6 3162.3 2.0 1581.1 332.0 0.002 2.7 4472.1 4472.1 1.0 2236.1 469.5 0.001 7.6 190 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 20,000,000 t[mm] 1.5875 b/R [rad] 4.0 L b AR Radius R/t buckling press [Nmm"2] buckling ratio [mm] [mm] [mm] 14142.1 1414.2 10.0 353.6 222.7 0.001 4.9 12586.4 1589.0 7.9 397.3 250.2 0.001 7.0 11339.0 1763.8 6.4 441.0 277.8 0.000 9.6 10316.5 1938.6 5.3 484.7 305.3 0.000 • 12.7 9463.2 2113.4 4.5 528.4 332.8 0.000 16.5 8740.3 2288.2 3.8 572.1 360.4 0.000 20.9 8120.0 2463.1 3.3 615.8 387.9 0.000 26.1 7581.9 2637.9 2.9 659.5 415.4 0.000 32.0 7110.7 2812.7 2.5 703.2 442.9 0.000 38.8 6694.6 2987.5 2.2 746.9 470.5 0.000 46.5 6324.6 3162.3 2.0 790.6 498.0 0.000 55.1 4472.1 4472.1 1.0 1118.0 704.3 0.000 156.0 A [mm2] 20,000,000 t [mm] 3.1750 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 14142.1 1414.2 10.0 353.6 111.4 0.007 0.6 12586.4 1589.0 7.9 397.3 125.1 0.005 0.9 11339.0 1763.8 6.4 441.0 138.9 0.004 1.2 10316.5 1938.6 5.3 484.7 152.6 0.003 1.6 9463.2 2113.4 4.5 528.4 166.4 0.002 2.1 8740.3 2288.2 3.8 572.1 180.2 0.002 2.6 8120.0 2463.1 3.3 615.8 193.9 0.001 3.3 7581.9 2637.9 2.9 659.5 207.7 0.001 4.0 7110.7 2812.7 2.5 703.2 221.5 0.001 4.9 6694.6 2987.5 2.2 746.9 235.2 0.001 5.8 6324.6 3162.3 2.0 790.6 249.0 0.001 6.9 4472.1 4472.1 1.0 1118.0 352.1 0.000 19.5 A [mm2] 20,000,000 t [mm] 4.7625 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 14142.1 1414.2 10.0 353.6 74.2 0.023 0.2 12586.4 1589.0 7.9 397.3 83.4 0.016 0.3 11339.0 1763.8 6.4 441.0 92.6 0.012 0.4 10316.5 1938.6 5.3 484.7 101.8 0.009 0.5 9463.2 2113.4 4.5 528.4 110.9 0.007 0.6 8740.3 2288 2 3.8 572.1 120.1 0.005 0.8 8120.0 2463.1 3.3 615.8 129.3 0.004 1.0 7581.9 2637.9 2.9 659.5 138.5 0.004 1.2 7110.7 2812.7 2.5 703.2 147.6 0.003 1.4 6694.6 2987.5 2.2 746.9 156.8 0.002 1.7 6324.6 3162.3 2.0 790.6 166.0 0.002 2.0 4472.1 4472.1 1.0 1118.0 234.8 0.001 5.8 191 APPENDIX A: T A B U L A T E D D A T A A [mm2] 40,000,000 t [mm] 1.5875 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 20000.0 2000.0 10.0 20000.0 12598.4 0.000 332.0 17799.8 2247.2 7.9 22472.1 14155.7 0.000 470.9 16035.7 2494.4 6.4 24944.3 15712.9 0.000 644.0 14589.8 2741.6 5.3 27416.4 17270.2 0.000 855.1 13383.1 2988.9 4.5 29888.5 18827.4 0.000 1107.9 12360.7 3236.1 3.8 32360.7 20384.7 0.000 1406.2 11483.4 3483.3 3.3 34832.8 21941.9 0.000 1753.7 10722.4 3730.5 2.9 37305.0 23499.2 0.000 2154.2 10056.0 3977.7 2.5 39777.1 25056.4 0.000 ' 2611.5 9467.6 4224.9 2.2 42249.2 26613.7 0.000 3129.2 8944.3 4472.1 '2.0 44721.4 28170.9 0.000 3711.3 6324.6 6324.6 1.0 63245.6 39839.7 0.000 10497.2 A [mm2] 40,000,000 t [mm] 3.1750 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 20000.0 2000.0 10.0 20000.0 6299.2 0.000 41.5 17799.8 2247.2 7.9 22472.1 7077.8 0.000 58.9 16035.7 2494.4 6.4 24944.3 7856.5 0.000 80.5 14589.8 2741.6 5.3 27416.4 8635.1 0.000 106.9 13383.1 2988.9 4.5 29888.5 9413.7 0.000 138.5 12360.7 3236.1 3.8 32360.7 10192.3 0.000 175.8 11483.4 3483.3 3.3 34832.8 10971.0 0.000 219.2 10722.4 3730.5 2.9 37305.0 11749.6 0.000 269.3 10056.0 3977.7 2.5 39777.1 12528.2 0.000 326.4 9467.6 4224.9 2.2 42249.2 13306.8 0.000 391.2 8944.3 4472.1 2.0 44721.4 14085.5 0.000 463.9 6324.6 6324.6 1.0 63245.6 19919.9 0.000 1312.1 A [mm2] 40,000,000 t [mm] 4.7625 b/R [rad] 0.1 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm'2] 20000.0 2000.0 10.0 20000.0 4199.5 0.000 12.3 17799.8 2247.2 7.9 22472.1 4718.6 0.000 17.4 16035.7 2494.4 6.4 24944.3 5237.6 0.000 23.9 14589.8 2741.6 5.3 27416.4 5756.7 0.000 31.7 13383.1 2988.9 4.5 29888.5 6275.8 0.000 41.0 12360.7 3236.1 3.8 32360.7 6794.9 0.000 52.1 11483.4 3483.3 3.3 34832.8 7314.0 0.000 65.0 10722.4 3730.5 2.9 37305.0 7833.1 0.000 79.8 10056.0 3977.7 2.5 39777.1 8352.1 0.000 96.7 9467.6 4224.9 2.2 42249.2 8871.2 0.000 115.9 8944.3 4472.1 2.0 44721.4 9390.3 0.000 137.5 6324.6 6324.6 1.0 63245.6 13279.9 0.000 388.8 192 APPENDIX A: T A B U L A T E D D A T A A [mm2] 40,000,000 t [mm] 1.5875 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 20000.0 2000.0 10.0 4000.0 2519.7 0.000 66.8 17799.8 2247.2 7.9 4494.4 2831.1 0.000 94.8 16035.7 2494.4 6.4 4988.9 3142.6 0.000 129.6 14589.8 2741.6 5.3 5483.3 3454.0 0.000 172.1 13383.1 2988.9 4.5 5977.7 3765.5 0.000 222.9 12360.7 3236.1 3.8 6472.1 4076.9 0.000 283.0 11483.4 3483.3 3.3 6966.6 4388.4 0.000 352.9 10722.4 3730.5 2.9 7461.0 4699.8 0.000 433.5 10056.0 3977.7 2.5 7955.4 5011.3 0.000 525.5 9467.6 4224.9 2.2 8449.8 5322.7 0.000 629.7 8944.3 4472.1 2.0 8944.3 5634.2 .0.000 746.8 6324.6 • 6324.6 1.0 12649.1 7967.9 0.000 2112.3 A [mm2] 40,000,000 t [mm] 3.1750 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 20000.0 2000.0 10.0 4000.0 1259.8 0.001 8.3 17799.8 2247.2 7.9 4494.4 1415.6 0.000 11.8 16035.7 2494.4 6.4 4988.9 1571.3 0.000 16.2 14589.8 2741.6 5.3 5483.3 1727.0 0.000 21.5 13383.1 2988.9 4.5 5977.7 1882.7 0.000 27.9 12360.7 3236.1 3.8 6472.1 2038.5 0.000 35.4 11483.4 3483.3 3.3 6966.6 2194.2 0.000 44.1 10722.4 3730.5 2.9 7461.0 2349.9 0.000 54.2 10056.0 3977.7 2.5 7955.4 2505.6 0.000 65.7 9467.6 4224.9 2.2 8449.8 2661.4 0.000 78.7 8944.3 4472.1 2,0 8944.3 2817.1 0.000 93.4 6324.6 6324.6 1.0 12649.1 3984.0 0.000 264.0 A [mm2] 40,000,000 t [mm] 4.7625 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 20000.0 2000.0 10.0 4000.0 839.9 0.002 2.5 17799.8 2247.2 7.9 4494.4 943.7 0.001 3.5 16035.7 2494.4 6.4 4988.9 1047.5 0.001 4.8 14589.8 2741.6 5.3 5483.3 1151.3 0.001 6.4 13383.1 2988.9 4.5 5977.7 1255.2 0.001 8.3 12360.7 3236.1 3.8 6472.1 1359.0 0.000 10.5 11483.4 3483.3 3.3 6966.6 1462.8 0.000 13.1 10722.4 3730.5 2.9 7461.0 1566.6 0.000 16.1 10056.0 3977.7 2.5 7955.4 1670.4 0.000 19.5 9467.6 4224.9 2.2 8449.8 1774.2 0.000 23.3 8944.3 4472.1 2.0 • 8944.3 1878.1 0.000 27.7 6324.6 6324.6 1.0 12649.1 2656.0 0.000 78.2 193 APPENDIX A: T A B U L A T E D D A T A A [mm2] 40,000,000 t [mm] 1.5875 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 20000.0 2000.0 10.0 1000.0 629.9 0.000 18.5 17799.8 2247.2 7.9 1123.6 707.8 0.000 26.2 16035.7 2494.4 6.4 1247.2 785.6 0.000 35.8 14589.8 2741.6 5.3 1370.8 863.5 0.000 47.6 13383.1 2988.9 4.5 1494.4 941.4 0.000 61.6 12360.7 3236.1 3.8 1618.0 1019.2 0.000 78.2 11483.4 3483.3 3.3 1741.6 1097.1 0.000 97.5 10722.4 3730.5 2.9 1865.2 1175.0 0.000 119.8 10056.0 3977.7 2.5 1988.9 1252.8 0.000 145.3 9467.6 4224.9 2.2 2112.5 1330.7 0.000 174.1 8944.3 4472.1 2.0 2236.1 1408.5 0.000 206.4 6324.6 6324.6 1.0 3162.3 1992.0 0.000 583.9 A [mm2] 40,000,000 t [mm] 3.1750 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 20000.0 2000.0 10.0 1000.0 315.0 0.002 2.3 17799.8 2247.2 7.9 1123.6 353.9 0.001 3.3 16035.7 2494.4 6.4 1247.2 392.8 0.001 4.5 14589.8 2741.6 5.3 1370.8 431.8 0.001 5.9 13383.1 2988.9 4.5 1494.4 470.7 o.ooi 7.7 12360.7 3236.1 3.8 1618.0 509.6 0.000 9.8 11483.4 3483.3 3.3 1741.6 548.5 0.000 12.2 10722.4 3730.5 2.9 1865.2 587.5 0.000 15.0 10056.0 3977.7 2.5 1988.9 626.4 0.000 18.2 9467.6 4224.9 2.2 2112.5 665.3 0.000 21.8 8944.3 4472.1 2.0 2236.1 704.3 0.000 25.8 6324.6 6324.6 1.0 3162.3 996.0 0.000 73.0 A [mm2] 40,000,000 t [mm] 4.7625 b/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 20000.0 2000.0 10.0 1000.0 210.0 0.006 0.7 17799.8 2247.2 7.9 1123.6 235.9 0.004 1.0 16035.7 2494.4 6.4 1247.2 261.9 0.003 1.3 14589.8 2741.6 5.3 1370.8 287.8 0.002 1.8 13383.1 2988.9 4.5 1494.4 313.8 0.002 2.3 12360.7 3236.1 3.8 1618.0 339.7 0.001 2.9 11483.4 3483.3 3.3 1741.6 365.7 0.001 3.6 10722.4 3730.5 2.9 1865.2 391.7 0.001 4.4 10056.0 3977.7 2.5 1988.9 417.6 0.001 5.4 9467.6 4224.9 2.2 2112.5 443.6 0.001 6.4 8944.3 4472.1 2.0 2236.1 469.5 0.001 7.6 6324.6 6324.6 1.0 3162.3 664.0 0.000 21.6 194 APPENDIX A : T A B U L A T E D D A T A A [mm2] 40,000,000 t [mm] 1.5875 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 20000.0 2000.0 10.0 500.0 315.0 0.000 14.0 17799.8 2247.2 7.9 561.8 353.9 0.000 19.8 16035.7 2494.4 6.4 623.6 392.8 0.000 27.1 14589.8 2741.6 5.3 685.4 431.8 0.000 35.9 13383.1 2988.9 4.5 747.2 470.7 0.000 46.6 12360.7 3236.1 3.8 809.0 509.6 0.000 59.1 11483.4 3483.3 3.3 870.8 548.5 0.000 73.7 10722.4 3730.5 2.9 932.6 587.5 0.000 90.5 10056.0 3977.7 2.5 994.4 626.4 0.000 109.7 9467.6 4224.9 2.2 1056.2 665.3 0.000 131.5 8944.3 4472.1 2.0 1118.0 704.3 0.000 156.0 6324.6 6324.6 1.0 1581.1 996.0 0.000 441.2 A [mm2] 40,000,000 t [mm] 3.1750 b/R [rad] 4.0 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm"2] 20000.0 2000.0 10.0 500.0 157.5 0.002 1.7 17799.8 2247.2 7.9 561.8 176.9 0.002 2.5 16035.7 2494.4 6.4 623.6 196.4 0.001 3.4 14589.8 2741.6 5.3 685.4 215.9 0.001 4.5 13383.1 2988.9 4.5 747.2 235.3 0.001 5.8 12360.7 3236.1 3 8 809.0 254.8 0.001 7.4 11483.4 3483.3 3.3 870.8 274.3 0.000 9.2 10722.4 3730.5 2.9 932.6 293.7 0.000 11.3 10056.0 3977.7 2.5 994.4 313.2 o:ooo 13.7 9467.6 4224.9 2.2 1056.2 332.7 0.000 16.4 8944.3 4472.1 2.0 1118.0 352.1 0.000 19.5 6324.6 6324.6 1.0 1581.1 498.0 0.000 55.1 A [mm2] 40,000,000 t [mm] 4.7625 b/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 20000.0 2000.0 10.0 500.0 105.0 0.008 0.5 17799.8 2247.2 7.9 561.8 118.0 0.006 0.7 16035.7 2494.4 6.4 623.6 130.9 0.004 1.0 14589.8 2741.6 5.3 685.4 143.9 0.003 1.3 13383.1 2988.9 4.5 747.2 156.9 0.002 1.7 12360.7 3236.1 3.8 809.0 169.9 0.002 2.2 11483.4 3483.3 3.3 870.8 182.8 0.002 2.7 -10722.4 3730.5 2.9 932.6 195.8 0.001 3.4 10056.0 3977.7 2.5 994.4 208.8 0.001 4.1 9467.6 4224.9 2.2 1056.2 221.8 0.001 4.9 8944.3 4472.1 2.0 1118.0 234.8 0.001 5.8 6324.6 6324.6 1.0 1581.1 332.0 0.000 16.3 195 APPENDIX A . T A B U L A T E D D A T A t = 3.175mm, b/R = 0.1 0.00E+00 5.00E+05 1 00E+06 1.50E+06 2.00E+06 2.50E+06 3.00E+06 3.50E+06 4.00E+06 Area /mm 2 196 APPENDIX A : T A B U L A T E D D A T A t = 4.7625mm, b/R = 0.1 1.000 o. 0,800 i 0.600 0.400 0.200 0.000 / / / / / / X }S\ p = 4.2 k N m m 2 1 I / /// 1 / / / / ' / J X 0.00E+00 1 00E+06 2 .00E+06 3 .00E+06 4 .00E+06 Area /mm2 - 1 0 0 -7.9 6.4 5 3 - 4 . 5 - 3 8 - 2 . 9 - 2 0 - 1 . 0 5.00E+06 6.00E+06 7.00E+06 8.00E+06 t = 1.5875mm, b/R = 0.5 1.20 1.00 0.80 0-0.60 3 m 0.20 0.00 O.OOE+00 5.00E+05 2 .50E+06 3 .00E+06 197 APPENDIX A : T A B U L A T E D D A T A t = 3.175mm, b/R = 0.6 t = 4.7625mm, b/R = 0.5 198 A P P E N D I X A : T A B U L A T E D D A T A t = 1.5875mm, b/R = 2.0 t = 3.175mm, b/R = 2.0 199 APPENDIX A: T A B U L A T E D D A T A t = 4.7625mm, b/R = 2.0 200 Maximum allowable plate area acording to analytical buckling analysis t = 1.5875mm t = 3.175mm t= 4.7625mm AR b/R = 0.1 b/R = 0.5 b/R = 2.0 b/R = 0.1 b/R = 0.5 b/R = 2.0 b/R = 0.1 b/R = 0.5 b/R = 2.0 1 7.5E+04 2.8E+05 5.8E+05 3.2E+05 9.9E+05 2.3E+06 7.5E+05 2.2E+06 5.0E+06 2 1.6E+05 4.9E+05 1.1 E+06 6.8E+05 2.0E+06 4.5E+06 1.5E+06 4.4E+06 1.0E+07 2.9 2.4E+05 7.0E+05 1.7 E+06 9.6E+05 2.8E+06 6.5E+06 2.2E+06 6.2E+06 1.5E+07 3.8 3.2E+05 9.3E+05 2.2E+06 1.3E+06 3.8E+06 8.8E+06 2.9E+06 8.4E+06 2.0E+07 4.5 3.8E+05 1.1 E+06 2.6E+06 1.5E+06 4.3E+06 1.0E+07 3.4E+06 9.6E+06 2.3E+07 5.3 4.5E+05 1.3E+06 3.1 E+06 1.8E+06 5.1 E+06 1.2E+07 4.0E+06 1.2E+07 2.7E+07 6.4 5.3E+05 1.6E+06 3.7E+06 2.2E+06 6.2E+06 1.5E+07 4.8E+06 1.4E+07 3.3E+07 7.9 6.6E+05 1.9E+06 4.5E+06 2.7E+06 7.8E+06 1.8E+07 5.9E+06 1.8E+07 4.1E+07 10 8.4E+05 2.5E+06 5.7E+06 3.5E+06 9.8E+06 2.3E+07 7.6E+06 2.2E+07 5.4E+07 A P P E N D I X A : T A B U L A T E D D A T A t = 1/16" [1.5875 mm] 6.0E+06 5.0E+06 4.0E+06 E E n 3.0E+06 s < 2.0E+06 1.0E+06 O.OE+00 - » - b / R = 0.1 -*~b /R = 0.5 b/R = 2.0 p = 4.2 kNmm 4 5 6 Aspect Ratio (L/b) 9 10 t = 2/16" [3.175 mm] 2.5E+07 2.0E+07 1.5E+07 < 1 .OE+07 5.0E+06 0.0E-00 -b/R = 0.1 -b/R = 0.5 b/R = 2.0 p = 4 . 2 k N m m ' : 4 5 6 Aspect Ratio (L/b) 202 APPENDIX A: TABULATED DATA i .OE+07 t = 3/16" [4.7625 mm] 3.0E+07 • 3) < - • - b / R = 0.1 - • - b / R = 0.5 b/R = 2 0 p = 4.2 kNmm'2 \ 2.0E+07 1.0E+07 0.0E+00 4 5 6 Aspect Ratio (L/b) 10 203 APPENDIX A: T A B U L A T E D D A T A p = 4.2 kNmrrr2 APPENDIX A: T A B U L A T E D D A T A A [mm2] 100,000 t [mm] 1.5875 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1414.2 70.7 20.0 707.1 445.42159 0.00 117.4 1162.8 86.0 13.5 860.0 541.73411 0.00 65.2 987.3 101.3 9.7 1012.9 638.04663 0.00 39.9 857.8 116.6 7.4 1165.8 734.35914 0.00 26.2 758.3 131.9 5.8 1318.7 830.67166 0.00 18.1 679.5 147.2 4.6 1471.6 926.98418 0.00 13.0 615.6 162.4 3.8 1624.5 1023.29669 0.00 9.7 562.6 177.7 3.2 1777.4 1119.60921 0.00 7.4 518.1 193.0 2.7 1930.3 1215.92172 0.00 5.8 480.0 208.3 2.3 2083.2 1312.23424 0.00 4.6 447.2 223.6 2.0 2236.1 1408.54676 0.00 3.7 316.2 316.2 1.0 3162.3 1991.98593 0.00 1.3 A [mm2] 100,000 t [mm] 3.1750 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1414.2 70.7 20.0 707.1 222.71080 0.00 14.7 1162.8 86.0 13.5 860.0 270.86706 0.00 8.2 987.3 101.3 9.7 1012.9 319.02331 0.00 5.0 857.8 116.6 7.4 1165.8 367.17957 0.00 3.3 758.3 131.9 5.8 1318.7 415.33583 0.00 2.3 679.5 147.2 4.6 1471.6 463.49209 0.00 1.6 615.6 162.4 3.8 1624.5 511.64835 0.00 1.2 562.6 177.7 3.2 1777.4 559.80460 0.00 0.9 518.1 193.0 2.7 1930.3 607.96086 0.01 0.7 480.0 208.3 2.3 2083.2 656.11712 0.01 0.6 447.2 223.6 2.0 2236.1 704.27338 0.01 0.5 316.2 316.2 1.0 3162.3 995.99296 0.03 0.2 A [mm2] 100,000 t [mm] 4.7625 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1414.2 70.7 20.0 707.1 148.47386 0.00 4.3 1162.8 86.0 13.5 860.0 180.57804 0.00 2.4 987.3 101.3 9.7 1012.9 212.68221 0.00 1.5 857.8 116.6 7.4 1165.8 244.78638 0.00 1.0 758.3 131.9 5.8 1318.7 276.89055 0.01 0.7 679.5 147.2 4.6 1471.6 308.99473 0.01 0.5 615.6 '162.4 3.8 1624.5 341.09890 0.01 0.4 562.6 177.7 3.2 1777.4 373.20307 0.02 0.3 518.1 193.0 2.7 1930.3 405.30724 0.02 0.2 480.0 208.3 2.3 2083.2 437.41141 0.02 0.2 447.2 223.6 2.0 2236.1 469.51559 0.03 0.1 316.2 316.2 1.0 3162.3 663.99531 0.09 0.0 205 APPENDIX A : T A B U L A T E D D A T A A [mm2] 100,000 t [mm] 1.5875 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1414.2 70.7 20.0 141.4 89.08432 0.00 23.6 1162.8 86.0 13.5 172.0 108.34682 0.00 13.1 987.3 101.3 9.7 202.6 127.60933 0.00 8.0 857.8 116.6 7.4 233.2 146.87183 0.00 5.3 758.3 131.9 5.8 263.7 166.13433 0.00 3.6 679.5 147.2 4.6 294.3 185.39684 0.00 2.6 615.6 162.4 3.8 324.9 204.65934 0.00 1.9 562.6 177.7 3.2 355.5 223.92184 0.00 1.5 518.1 193.0 2.7 386.1 243.18434 0.00 1.2 480.0 208.3 2.3 416.6 262.44685 0.00 0.9 447.2 223.6 2.0 447.2 281.70935 0.01 0.7 316.2 316.2 1.0 632.5 398.39719 0.02 0.3 A [mm2] 100,000 . . t [mm] 3.1750 b/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 1414.2 70.7 20.0 141.4 44.54216 0.00 3.0 1162.8 86.0 13.5 172.0 54.17341 0.00 1.6 987.3 101.3 9.7 202.6 63.80466 0.00 1.0 857.8 116.6 7.4 233.2 73.43591 0.01 0.7 758.3 131.9 5.8 263.7 83.06717 0.01 0.5 679.5 147.2 4.6 294.3 92.69842 0.01 0.3 615.6 162.4 3.8 324.9 102.32967 0.02 0.2 562.6 177.7 3.2 355.5 111.96092 0.02 0.2 518.1 193.0 2.7 386.1 121.59217 0.03 0.1 480.0 208.3 2.3 416.6 131.22342 0.04 0.1 447.2 223.6 2.0 447.2 140.85468 0.04 0.1 316.2 316.2 1.0 632.5 199.19859 ' 0.13 0.0 A [mm2] 100,000 t [mm] 4.7625 b/R [rad] 0.5 L [mm] . b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1414.2 70.7 20.0 141.4 29.69477 0.00 0.9 1162.8 86.0 13.5 172.0 36.11561 0.01 0.5 987.3 101.3 9.7 202.6 42.53644 0.01 0.3 857.8 116.6 7.4 233.2 48.95728 0.02 0.2 758.3 131.9 5.8 263.7 55.37811 0.03 0.1 679.5 147.2 4.6 294.3 61.79895 0.04 0.1 615.6 162.4 3.8 324.9 68.21978 0.06 0.1 562.6 177.7 3.2 355.5 74.64061 0.08 0.1 518.1 193.0 2.7 386.1 81.06145 0.10 0.0 480.0 208.3 2.3 416.6 87.48228 0.12 0.0 447.2 223.6 2.0 447.2 93.90312 0.15 0.0 316.2 316.2 1.0 632.5 132.79906 0.43 0.0 206 APPENDIX A : T A B U L A T E D D A T A A [mm2] 100,000 t [mm] 1.5875 L/R [rad] 2.0 L b AR Radius R/t buckling press [Nmm"2] buckling ratio [mm] [mm] [mm] 1414.2 70.7 20.0 35.4 22.27108 0.00 6.5 1162.8 86.0 13.5 43.0 27.08671 0.00 3.6 987.3 101.3 9.7 50.6 31.90233 0.00 2.2 857.8 116.6 7.4 58.3 36.71796 0.00 1.5 758.3 131.9 5.8 65.9 41.53358 0.00 1.0 679.5 147.2 4.6 73.6 46.34921 0.01 0.7 615.6 162.4 3.8 81.2 51.16483 0.01 0.5 562.6 177.7 3.2 88.9 55.98046 0.01 0.4 518.1 193.0 2.7 96.5 60.79609 0.01 0.3 480.0 208.3 2.3 104.2 65.61171 0.02 0.3 ' 447.2 223.6 2.0 111.8 70.42734 0.02 0.2 316.2 316.2 1.0 158.1 99.59930 0.06 0.1 A [mm2] 100,000 t [mm] 3.1750 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1414.2 70.7 20.0 35.4 11.13554 0.01 0.8 1162.8 86.0 13.5 43.0 13.54335 0.01 0.5 987.3 101.3 9.7 50.6 15.95117 0.02 0.3 857.8 116.6 7.4 58.3 18.35898 0.02 0.2 758.3 131.9 5.8 65.9 20.76679 0.03 0.1 679.5 147.2 4.6 73.6 23.17460 0.05 0.1 615.6 162.4 3.8 81.2 25.58242 0.06 0.1 562.6 177.7 3.2 88.9 27.99023 0.08 0.1 518.1 193.0 2.7 96.5 30.39804 0.10 0.0 480.0 208.3 2.3 104.2 32.80586 0.13 0.0 447.2 223.6 2.0 111.8 35.21367 0.16 0.0 316.2 316.2 1.0 158.1 49.79965 0.46 0.0 A [mm2] 100,000 t [mm] 4.7625 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1414.2 70.7 20.0 35.4 7.42369 — — 1162.8 86.0 13.5 43.0 9.02890 — — 987.3 101.3 9.7 50.6 10.63411 0.05 0.1 857.8 116.6 7.4 58.3 12.23932 0.08 0.1 758.3 131.9 5.8 65.9 13.84453 0.11 0.0 679.5 147.2 4.6 73.6 15.44974 0.16 0.0 615.6 162.4 3.8 81.2 17^05494 0.21 0.0 562.6 177.7 3.2 88.9 18.66015 0.28 0.0 518.1 193.0 2.7 96.5 20.26536 0.35 0.0 480.0 208.3 2.3 104.2 21.87057 0.44 0.0 447.2 223.6 2.0 111.8 . 23.47578 0.55 0.0 316.2 316.2 1.0 158.1 33.19977 1.55 0.0 207 APPENDIX A: T A B U L A T E D D A T A A [mm2] 100,000 t [mm] 1.5875 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 1414.2 70.7 20.0 17.7 11.13554 0.00 4.9 1162.8 86.0 13.5 21.5 .13.54335 0.00 2.7 987.3 101.3 9.7 25.3 15.95117 0.00 1.7 857.8 116.6 7.4 29.1 18.35898 0.00 1.1 758.3 131.9 5.8 33.0 20.76679 0.01 0.8 679.5 147.2 4.6 36.8 23.17460 0.01 0.5 615.6 162.4 3.8 40.6 25.58242 0.01 0.4 562.6 177.7 3.2 44.4 27.99023 0.01 0.3 518.1 193.0 2.7 48.3 30.39804 0.02 0.2 480.0 208.3 2.3 52.1 32.80586 0.02 0.2 447.2 223.6 2.0 55.9 35.21367 0.03 0.2 316.2 316.2 1.0 79.1 49.79965 0.08 0.1 A [mm2] 100,000 t [mm] 3.1750 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 1414.2 70.7 20.0 17.7 5.56777 — — 1162.8 86.0 13.5 21.5 6.77168 — — 987.3 101.3 9.7 25.3 7.97558 — — 857.8 116.6 7.4 29.1 9.17949 — — 758.3 131.9 5.8 33.0 10.38340 0.04 0.1 679.5 147.2 4.6 36.8 11.58730 0.06 0.1 615.6 162.4 3.8 40.6 12.79121 0.08 0.1 562.6 177.7 3.2 44.4 13.99512 0.11 0.0 518.1 193.0 2.7 48:3 15.19902 0.14 0.0 480.0 208.3 2.3 52.1 16.40293 0.17 0.0 447.2 223.6 2.0 55.9 17.60683 0.22 0.0 316.2 316.2 1.0 79.1 24.89982 0.61 0.0 A [mm2] 100,000 t [mm] 4.7625 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 1414.2 70.7 20.0 17.7 3.71185 — — 1162.8 86.0 13.5 21.5 • 4.51445 — — 987.3 101.3 9.7 25.3 5.31706 — — 857.8 116.6 7.4 29.1 6.11966 — — 758.3 131.9 5.8 33.0 6.92226 — — 679.5 147.2 4.6 36.8 7.72487 — — 615.6 162.4 3.8 40.6 8.52747 — — 562.6 177.7 3.2 44.4 9.33008 — — 518.1 193.0 2.7 48.3 10.13268 0.47 0.0 480.0 208.3 2.3 52.1 10.93529 0.59 0.0 447.2 223.6 2.0 55.9 11.73789 0.73 0.0 316.2 316.2 1.0 79.1 16.59988 2.06 0.0 208 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 200,000 t [mm] 1.5875 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 2000.0 100.0 20.0 1000.00000 629.92 0.0 332.0 1644.4 121.6 135 1216.22777 766.13 0.0 184.5 1396.2 143.2 9.7 1432.45553 .902.33 0.0 112.9 1213.1 164.9 7.4 1648.68330 1038.54 0.0 74.1 1072.4 186.5 5.8 1864.91106 1174.75 0.0 51.2 961.0 208.1 4.6 2081.13883 1310.95 0.0 36.8 870.6 229.7 3.8 2297.36660 1447.16 0.0 27.4 795.7 251.4 3.2 2513.59436 1583.37 0.0 20.9 732.6 273.0 2.7 2729.82213 1719.57 0.0 16.3 678.9 294.6 2.3 2946.04989 1855.78 0.0 13.0 632.5 316.2 2.0 3162.27766 1991.99 0.0 10.5 447.2 447.2 1.0 4472.13595 2817.09 0.0 3.7 A [mm2] 200,000 t [mm] 3.1750 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 2000.0 100.0 20.0 1000.00000 314.96 0.0 41.5 1644.4 121.6 13.5 1216.22777 383.06 0.0 23.1 1396.2 143.2 9.7 1432.45553 451.17 0.0 14.1 1213.1 164.9 7.4 1648.68330 519.27 0.0 9.3 1072.4 186.5 5.8 1864.91106 587.37 0.0 6.4 961.0 208.1 4.6 2081.13883 655.48 0.0' 4.6 870.6 229.7 3.8 2297.36660 723.58 0.0 3.4 795.7 251.4 3.2 2513.59436 791.68 0.0 2.6 732.6 273.0 2.7 2729.82213 859.79 0.0 2.0 678.9 294.6 2.3 2946.04989 927.89 0.0 1.6 632.5 316.2 2.0 3162.27766 995.99 0.0 1.3 447.2 447.2 1.0 4472.13595 1408.55 0.0 0.5 A [mm2] 200,000 t [mm] 4.7625 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 2000.0 100.0 20.0 1000.00000 209.97 0.0 12.3 1644.4 121.6 13.5 1216.22777 255.38 0.0 6.8 1396.2 143.2 9.7 1432.45553 300.78 0.0 4.2 1213.1 164.9 7.4 1648.68330 346.18 0.0 2.7 1072.4 186.5 5.8 1864.91106 391.58 0.0 1.9 961.0 208.1 4.6 2081.13883 436.98 0.0 1.4 870.6 229.7 3.8 2297.36660 482.39 0.0 1.0 795.7 251.4 3.2 2513.59436 527.79 0.0 0.8 732.6 273.0 2.7 2729.82213 573.19 0.0 0.6 678.9 - 294.6 2.3 2946.04989 618.59 0.0 0.5 632.5 316.2 2.0 3162.27766 664.00 0.0 0.4 447.2 447.2 1.0 4472.13595 939.03 0.0 0.1 209 APPENDIX A: T A B U L A T E D D A T A A [mm2] 200,000 t [mm] 1.5875 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 2000.0 100.0 20.0 200.00000 125.98 0.0 66.8 1644.4 121.6 13.5 243.24555 153.23 0.0 37.1 1396.2 143.2 9.7 286.49111 180.47 0.0 22.7 1213.1 164.9 7.4 329.73666 207.71 0.0 14.9 1072.4 186.5 5 8 372.98221 234.95 0.0 10.3 961.0 208.1 4.6 416.22777 262.19 0.0 7.4 870.6 229.7 3.8 459.47332 289.43 0.0 5.5 795.7 251.4 3.2 502.71887 316.67 0.0 4.2 732.6 273.0 2.7 545.96443 343.91 0.0 3.3 678.9 294.6 2.3 589.20998 371.16 0.0 2.6 632.5 316.2 2.0 632.45553 398.40 0.0 2.1 447.2 447.2 1.0 894.42719 563.42 0.0 0.7 A [mm2] 200,000 t [mm] 3.1750 L/R [rad] 0.5 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm"2] 2000.0 100.0 20.0 200.00000 62.99 0.0 8.3 1644.4 121.6 13.5 243.24555 76.61 0.0 4.6 1396.2 143.2 . 9.7 286.49111 90.23 0.0 2.8 1213.1 164.9 7.4 329.73666 103.85 0.0 1.9 1072.4 186.5 5.8 372.98221 117.47 0.0 1.3 961.0 208.1 4.6 416.22777 131.10 0.0 0.9 870.6 229.7 3.8 459.47332 144.72 0.0 0.7 795.7 251.4 3.2 502.71887 158.34 0.0 0.5 732.6 273.0 2.7 545.96443 171.96 0.0 0.4 678.9 294.6 2.3 589.20998 185.58 0.0 0.3 632.5 316.2 2.0 632.45553 199.20 0.0 0.3 447.2 447.2 1.0 894.42719 281.71 0.0 0.1 A [mm2] 200,000 t [mm] 4.7625 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 2000.0 100.0 20.0 200.00000 41.99 0.0 2.5 1644.4 121.6 13.5 243.24555 51.08 0.0 1.4 1396.2 143.2 9.7 286.49111 60.16 0.0 0.8 1213.1 164.9 7.4 329.73666 69.24 0.0 0.6 1072.4 186.5 5.8 372.98221 78.32 0.0 0.4 961.0 208.1 4.6 416.22777 87.40 0.0 0.3 870.6 229.7 3.8 459.47332 96.48 0.0 0.2 795.7 251.4 3.2 502.71887 105.56 0.0 0.2 732.6 273.0 2.7 545.96443 114.64 0.0 0.1 678.9 294.6 2.3 589.20998 123.72 0.0 0.1 632.5 316.2 2.0 632.45553 132.80 0.1 0.1 447.2 447.2 1.0 894.42719 187.81 0.2 0.0 210 APPENDIX A: T A B U L A T E D D A T A A [mm2] 200,000 t [mm] 1.5875 L/R [rad] 2.0 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm 2] 2000.0 100.0 20.0 50.00000 31.50 0.0 18.5 1644.4 121.6 13.5 60.81139 38.31 0.0 10.3 1396.2 143.2 9.7 71.62278 45.12 0.0 . 6.3 1213.1 164.9 7.4 82.43416 51.93 0.0 4.1 1072.4 186.5 5.8 93.24555 58.74 . 0.0 2.8 961.0 208.1 4.6 104.05694 65.55 0.0 2.0 870.6 229.7 3.8 114.86833 72.36 0.0 1.5 795.7 251.4 3.2 125.67972 79.17 0.0 1.2 732.6 273.0 2.7 136.49111 85.98 0.0 0.9 678.9 294.6 2.3 147.30249 92.79 0.0 0.7 632.5 316.2 2.0 158.11388 99.60 0.0 0.6 447.2 447.2 1.0 223.60680 140.85 0.0 0.2 A [mm2] 200,000 t [mm] 3.1750 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 2000.0 100.0 20.0 50.00000 15.75 0.0 2.3 1644.4 121.6 13.5 60.81139 19.15 0.0 1.3 1396.2 143.2 9.7 71.62278 22.56 0.0 0.8 1213.1 164.9 7.4 82.43416 25.96 0.0 0.5 1072.4 186.5 5.8 93.24555 29.37 0.0 0.4 961.0 208.1 4.6 104.05694 32.77 0.0 0.3 870.6 229.7 3.8 114.86833 36.18 0.0 0.2 795.7 251.4 3.2 125.67972 39.58 0.0 0.1 732.6 273.0 2.7 136.49111 42.99 0.0 0.1 678.9 294.6 2.3 147.30249 46.39 0.0 0.1 632.5 316.2 2.0 158.11388 49.80 0.1 0.1 447.2 447.2 1.0 223.60680 70.43 0.2 0.0 A [mm2] 200,000 t [mm] 4.7625 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 2000.0 100.0 20.0 50.00000 10.50 0.0 0.7 1644.4 121.6 13.5 60.81139 12.77 0.0 0.4 1396.2 143.2 9.7 71.62278 15.04 0.0 0.2 1213.1 164.9 7.4 82.43416 17.31 0.0 0.2 1072.4 186.5 5.8 93.24555 19.58 0.0 0.1 961.0 208.1 4.6 104.05694 21.85 0.1 0.1 870.6 229.7 3.8 114.86833 24.12 0.1 0.1 795.7 251.4 3.2 125.67972 26.39 0.1 0.0 732.6 273.0 2.7 136.49111 28.66 0.1 0.0 678.9 294.6 2.3 147.30249 30.93 0.2 0.0 632.5 316.2 2.0 158.11388 33.20 0.2 0.0 447.2 447.2 1.0 223.60680 46.95 0.5 0.0 211 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 200,000 t[mm] 1.5875 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 2000.0 100.0 20.0 25.00000 15.75 0.0 14.0 1644.4 121.6 13.5 30.40569 19.15 0.0 7.8 1396.2 143.2 9.7 35.81139 22.56 0.0 4.7 1213.1 164.9 7.4 41.21708 25.96 0.0 3.1 1072.4 186.5 5.8 46.62278 29.37 0.0 2.2 961.0 208.1 4.6 52.02847 32.77 0.0 1.5 870.6 229.7 3.8 57.43416 36.18 0.0 1.2 795.7 251.4 3.2 62.83986 39.58 0.0 0.9 732.6 273.0 2.7 68.24555 42.99 0.0 0.7 678.9 294.6 2.3 73.65125 46.39 0.0 0.5 632.5 316.2 2.0 79.05694 49.80 0.0 0.4 447.2 447.2 1.0 111.80340 70.43 0.0 0.2 A [mm2] 200,000 t [mm] 3.1750 L/R [rad] ' 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 2000.0 100.0 20.0 25.00000 7.87 — — 1644.4 121.6 13.5 30.40569 9.58 — — 1396.2 143.2 9.7 35.81139 11.28 0.0 0.6 1213.1 164.9 7.4 41.21708 12.98 0.0 0.4 1072.4 186.5 5.8 46.62278 14.68 0.0 0.3 961.0 208.1 4.6 52.02847 16.39 0.0 0.2 870.6 229.7 3.8 57.43416 18.09 0.0 0.1 795.7 251.4 3.2 62.83986 19.79 0.0 0.1 732.6 273.0 2.7 68.24555 21.49 0.0 0.1 678.9 294.6 2.3 73.65125 23.20 0.1 0.1 632.5 316.2 2.0 79.05694 24.90 0.1 0.1 447.2 447.2 1.0 111.80340 35.21 0.2 0.0 A [mm2] 200,000 t [mm] 4.7625 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 2000.0 100.0 20.0 25.00000 5.25 — — 1644.4 121.6 13.5 30.40569 6.38 — — 1396.2 143.2 9.7 35.81139 7.52 — — 1213.1 164.9 7.4 41.21708 8.65 — — 1072.4 186.5 5.8 46.62278 9.79 — — 961.0 208.1 4.6 52.02847 10.92 0.1 0.1 870.6 229.7 3.8 57.43416 12.06 0.1 0.0 795.7 251.4 3.2 62.83986 13.19 0.1 0.0 732.6 273.0 2.7 68.24555 14.33 0.2 0.0 678.9 294.6 2.3 73.65125 15.46 0.2 0.0 632.5 316.2 2.0 79.05694 16.60 0.3 0.0 447.2 447.2 1.0 111.80340 23.48 0.7 0.0 212 APPENDIX A: T A B U L A T E D D A T A A [mm2] 300,000 t [mm] 1.5875 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 2449.5 122.5 20.0 1224.7 771.5 0.0 609.8 2014.0 149.0 13.5 1489.6 938.3 0.0 339.0 1710.0 175.4 9.7 1754.4 1105.1 0.0 207.5 1485.7 201.9 7.4 2019.2 1271.9 0.0 136.1 1313.5 228.4 5.8 2284.0 1438.8 0.0 94.0 1177.0 254.9 4.6 2548.9 1605.6 0.0 67.7 1066.2 281.4 3.8 2813.7 1772.4 0.0 50.3 974.5 307.9 3.2 3078.5 1939.2 0.0 38.4 897.3 334.3 2.7 3343.3 2106.0 0.0 30.0 831.4 360.8 2.3 . 3608.2 2272.9 0.0 23.9 774.6 387.3 2:0 3873.0 2439.7 0.0 19.3 547.7 547.7 1.0 5477.2 3450.2 0.0 6.8 A [mm2] 300,000 t [mm] 3.1750 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 2449.5 122.5 20.0 1224.7 385.7 0.0 76.2 2014.0 149.0 13.5 1489.6 469.2 0.0 42.4 1710.0 175.4 9.7 1754.4 552.6 0.0 25.9 1485.7 201.9 7.4 2019.2 636.0 0.0 17.0 1313.5 228.4 5.8 2284.0 719.4 0.0 11.8 1177.0 254.9 4.6 2548.9 802.8 0.0 8.5 1066.2 281.4 3.8 2813.7 886.2 0.0 6.3 974.5 307.9 3.2 3078.5 969.6 0.0 4.8 897.3 334.3 2.7 3343.3 1053.0 0.0 3.7 831.4 360.8 2.3 3608.2 1136.4 0.0 3.0 774.6 387.3 2.0 3873.0 1219.8 0.0 2.4 547.7 547.7 1.0 5477.2 1725.1 0.0 0.9 A [mm2] 300,000 • t [mm] 4.7625 b/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 2449.5 122.5 20.0 1224.7 257.2 0.0 22.6 2014.0 149.0 13.5 1489.6 312.8 0.0 12.6 1710.0 175.4 9.7 1754.4 368.4 0.0 7.7 1485.7 201.9 7.4 2019.2 424.0 0.0 5.0 1313.5 228.4 5.8 2284.0 479.6 0.0 3.5 1177.0 254.9 4.6 2548.9 535.2 0.0 2.5 1066.2 281.4 3.8 2813.7 590.8 0.0 1.9 974.5 307.9 3.2 3078.5 646.4 0.0 1.4 897.3 334.3 2.7 3343.3 702.0 0.0 1.1 831.4 360.8 2.3 3608.2 757.6 0.0 0.9 774.6 387.3 2.0 3873.0 813.2 0.0 0.7 547.7 547.7 1.0 5477.2 1150.1 0.0 0.3 213 APPENDIX A: TABULATED DATA A [mm2] 300,000 t [mm] 1.5875 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 2449.5 122.5 20.0 244.9 154.3 0.0 122.7 2014.0 149.0 13.5 297.9 187.7 0.0 68.2 1710.0 175.4 9.7 350.9 221.0 0.0 41.7 1485.7 201.9 7.4 403.8 254.4 0.0 27.4 1313.5 228.4 5.8 456.8 287.8 0.0 18.9 1177.0 254.9 4.6 509.8 321.1 0.0 13.6 1066.2 281.4 3.8 562.7 354.5 0.0 10.1 974.5 307.9 3.2 615.7 387.8 0.0 7.7 897.3 334.3 2.7 668.7 421.2 0.0 6.0 831.4 360.8 2.3 721.6 454.6 0.0 4.8 774.6 387.3 2.0 774.6 487.9 0.0 3.9 547.7 547.7 1.0 1095.4 690.0 0.0 1.4 A [mm2] 300,000 t [mm] 3.1750 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 2449.5 122.5 20.0 244.9 77.1 0.0 15.3 2014.0 149.0 13.5 297.9 93.8 0.0 8.5 1710.0 175.4 9.7 350.9 110.5 0.0 5.2 1485.7 201.9 7.4 403.8 127.2 0.0 3.4 1313.5 228.4 5.8 456.8 143.9 0.0 2.4 1177.0 254.9 4.6 509.8 160.6 0.0 1.7 1066.2 281.4 3.8 562.7 177.2 0.0 1.3 974.5 307.9 3.2 615.7 193.9 0.0 1.0 897.3 334.3 2.7 668.7 210.6 0.0 0.8 831.4 360.8 2.3 721.6 227.3 0.0 0.6 774.6 387.3 2.0 774.6 244.0 0.0 0.5 547.7 547.7 1.0 1095.4 345.0 0.0 0.2 A [mm2] 300,000 t [mm] 4.7625 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 2449.5 122.5 20.0 244.9 51.4 0.0 4.5 2014.0 149.0 13.5 297.9 62.6 0.0 2.5 1710.0 175.4 9.7 350.9 73.7 0.0 1.5 1485.7 201.9 7.4 403.8 84.8 0.0 1.0 1313.5 228.4 5.8 456.8 95.9 0.0 0.7 1177.0 254.9 4.6 509.8 107.0 0.0 0.5 1066.2 281.4 3.8 562.7 118.2 0.0 0.4 974.5 307.9 3.2 615.7 129.3 0.0 0.3 897.3 334.3 2.7 668.7 140.4 0.0 0.2 831.4 360.8 2.3 721.6 151.5 0.0 0.2 774.6 387.3 2.0 774.6 162.6 0.0 0.1 547.7 547.7 1.0 1095.4 230.0 0.1 0.1 214 APPENDIX A : T A B U L A T E D D A T A A [mm2] 300,000 t [mm] 1.5875 L/R [rad] 2.0 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm 2] 2449.5 122.5 20.0 61.2 38.6 0.0 33.9 2014.0 149.0 13.5 74.5 46.9 0.0 18.9 1710.0 175.4 9.7 87.7 55.3 0.0 11.5 1485.7 201.9 7.4 101.0 63.6 0.0 7.6 1313.5 228.4 5.8 114.2 71.9 0.0 5.2 1177.0 254.9 4.6 127.4 80.3 0.0 3.8 1066.2 281.4 3.8 140.7 88.6 0.0 2.8 974.5 307.9 3.2 153.9 97.0 0.0 2.1 897.3 334.3 2.7 167.2 105.3 0.0 1.7 831.4 360.8 2.3 180.4 113.6 0.0 1.3 774.6 387.3 2.0 193.6 122.0 0.0 1.1 547.7 547.7 1.0 273.9 172.5 0.0 0.4 A [mm2] . 300,000 t [mm] 3.1750 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 2449.5 122.5 20.0 61.2 19.3 0.0 4.2 2014.0 149.0 13.5 74.5 23.5 0.0 2.4 1710.0 175.4 9.7 87.7 27.6 0.0 1.4 1485.7 201.9 7.4 101.0 31.8 0.0 0.9 1313.5 228.4 5.8 114.2 36.0 0.0 0.7 1177.0 254.9 4.6 127.4 40.1 0.0 0.5 1066.2 281.4 3.8 140.7 44.3 0.0 0.3 974.5 307.9 3.2 153.9 48.5 0.0 0.3 897.3 334.3 2.7 167.2 52.7 0.0 0.2 831.4 360.8 2.3 180.4 56.8 0.0 0.2 774.6 387.3 2.0 193.6 61.0 0.0 0.1 547.7 547.7 1.0 273.9 86.3 0.1 0.0 A [mm2] 300,000 t[mm] 4.7625 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 2449.5 122.5 20.0 61.2 12.9 0.0 1.3 2014.0 149.0 13.5 74.5 15.6 0.0 0.7 1710.0 175.4 9.7 87.7 18.4 0.0 0.4 1485.7 201.9 7.4 101.0 21.2 0.0 0.3 1313.5 228.4 5.8 114.2 24.0 0.0 0.2 1177.0 254.9 4.6 127.4 26.8 0.0 0.1 1066.2 281.4 3.8 140.7 29.5 0:0 0.1 974.5 307.9 3.2 153.9 32.3 0.1 0.1 897.3 334.3 2.7 167.2 35.1 0.1 0.1 831.4 360.8 2.3 180.4 37.9 0.1 0.0 774.6 387.3 2.0 193.6 40.7 0.1 0.0 547.7 547.7 1.0 273.9 57.5 0.3 0.0 215 APPENDIX A : T A B U L A T E D D A T A A [mm2] 300,000 t [mm] 1.5875 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 2449.5 122.5 20.0 30.6 19.3 0.0 25.6 2014.0 149.0 13.5 37.2 23.5 0.0 14.2 1710.0 175.4 9.7 43.9 27.6 0.0 8.7 1485.7 201.9 7.4 50.5 31.8 0.0 5:7 1313.5 228.4 5.8 57.1 36.0 0.0 4.0 1177.0 254.9 4.6 63.7 40.1 0.0 2.8 1066.2 281.4 3.8 70.3 44.3 0.0 2.1 974.5 307.9 3.2 •77.0 48.5 0.0 1.6 897.3 334.3 2.7 83.6 52.7 0.0 1.3 831.4 360.8 2.3 90.2 56.8 0.0 1.0 774.6 387.3 2.0 96.8 61.0 0.0 0.8 547.7 547.7 1.0 136.9 86.3 0.0 0.3 A [mm2] 300,000 t [mm] 3.1750 UR [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 2449.5 122.5 20.0 30.6 9.6 — — 2014.0 149.0 13.5 37.2 11.7 0.0 1.8 1710.0 175.4 9.7 43.9 13.8 0.0 1.1 1485.7 201.9 7.4 50.5 15.9 0.0 0.7 1313.5 228.4 5.8 57.1 18.0 0.0 0.5 1177.0 254.9 4.6 63.7 20.1 0.0 0.4 1066.2 281.4 3.8 70.3 22.2 0.0 0.3 974.5 307.9 3.2 77.0 24.2 0.0 0.2 897.3 334.3 2.7 83.6 26.3 0.0 0.2 831.4 360.8 2.3 90.2 28.4 0.0 0.1 774.6 387.3 2.0 96.8 30.5 0.0 0.1 547.7 547.7 1.0 136.9 43.1 0.1 0.0 A [mm2] 300,000 t [mm] 4.7625 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 2449.5 122.5 20.0 30.6 6.4 — — 2014.0 149.0 13.5 37.2 7.8 — — 1710.0 175.4 9.7 43.9 9.2 — — 1485.7 201.9 7.4 50.5 10.6 0.0 0.2 1313.5 228.4 5.8 57.1 12.0 0.0 0.1 1177.0 254.9 4.6 63.7 13.4 0.0 0.1 1066.2 281.4 3.8 70.3 14.8 0.1 0.1 974.5 307.9 3.2 77.0 16.2 0.1 0.1 897.3 334.3 2.7 83.6 17.6 0.1 0.0 831.4 360.8 2.3 90.2 18.9 0.1 0.0 774.6 387.3 2.0 96.8 20.3 0.1 0.0 547.7 547.7 1.0 136.9 28.8 0.4 0.0 216 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 500,000 t [mm] 1.5875 UR [rad] 0.1 L b AR Radius R/t buckling press [Nmm"2] buckling ratio [mm] [mm] [mm] 3162.3 158.1 20.0 1581.1 996.0 0.0 1312.1 2600.1 192.3 13.5 1923.0 1211.4 0.0 729.4 2207.6 226.5 9.7 2264.9 1426.7 0.0 446.4 1918.1 260.7 7.4 2606.8 1642.1 0.0 292.8 1695.7 294.9 5.8 2948.7 1857.4 0.0 202.3 1519.5 329.1 4.6 3290.6 • 2072.8 0.0 145.6 1376.5 363.2 3.8 3632.5 2288.2 0.0 : 108.2 1258.1 397.4 3.2 3974.3 2503.5 0.0 82.6 1158.4 431.6 2.7 4316.2 2718.9 0.0 64.5 1073.4 465.8 2.3 4658.1 2934.2 0.0 51.3 1000.0 500.0 2.0 5000.0 3149.6 0.0 41.5 707.1 707.1 1.0 7071.1 4454.2 0.0 14.7 A [mm2] 500,000 • t [mm] 3.1750 L/R [rad] 0.1 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm"2] 3162.3 158.1 20.0 1581.1 498.0 0.0 164.0 2600.1 . 192.3 13.5 1923.0 605.7 0.0 91.2 2207.6 226.5 9.7 2264.9 713.4 0.0 55.8 1918.1 260.7 7.4 2606.8 821.0 0.0 36.6 1695,7 294.9 5.8 2948.7 928.7 0.0 25.3 1519.5 329.1 4.6 3290.6 1036.4 0.0 18.2 1376.5 363.2 3.8 3632.5 1144.1 0.0 13.5 1258.1 397.4 3.2 3974.3 1251.8 0.0 10.3 1158.4 431.6 2.7 4316.2 1359.4 0.0 8.1 1073.4 465.8 2.3 4658.1 1467.1 0.0 6.4 1000.0 500.0 2.0 5000.0 1574.8 0.0 5.2 707.1 707.1 1.0 7071.1 2227.1 0.0 1.8 A [mm2] 500,000 t [mm] 4.7625 L/R [rad] 0.1 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm 2] 3162.3 158.1 20.0 1581.1 332.0 0.0 48.6 2600.1 192.3 13.5 1923.0 403.8 0.0 27.0 2207.6 226.5 9.7 2264.9 475.6 0.0 16.5 1918.1 260.7 7.4 2606.8 547.4 0.0 10.8 1695.7 294.9 5.8 2948.7 619.1 0.0 7.5 1519.5 329.1 4.6 3290.6 690.9 0.0 5.4 1376.5 363.2 3.8 3632.5 762.7 0.0 4.0 1258.1 397.4 3.2 3974.3 834.5 0.0 3.1 1158.4 431.6 2.7 4316.2 906.3 0.0 2.4 1073.4 465.8 2.3 4658.1 978.1 0.0 1.9 1000.0 500.0 2.0 5000.0 1049.9 0.0 1.5 707.1 707.1 1.0 7071.1 1484.7 0.0 0.5 217 APPENDIX A : T A B U L A T E D D A T A A [mm2] 500,000 t [mm] 1.5875 L/R [rad] 0.5 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm 2] 3162.3 158.1 20.0 316.2 199.2 0.0 264.0 2600.1 192.3 13.5 384.6 242.3 0.0 146.8 2207.6 226.5 9.7 453.0 285.3 0.0 89.8 1918.1 260.7 7.4 521.4 328.4 0.0 58.9 1695.7 294.9 5.8 589.7 371.5 0.0 40.7 1519.5 329.1 4.6 658.1 414.6 0.0 29.3 1376.5 363.2 3.8 726.5 457.6 0.0 21.8 1258.1 397.4 3.2 794.9 500.7 0.0 16.6 1158.4 431.6 2.7 863.2 543.8 0.0 13.0 1073.4 465.8 2.3 931.6 586.8 0.0 10.3 1000.0 500.0 2.0 1000.0 629.9 0.0 8.3 707.1 707.1 1.0 1414.2 890.8 0.0 3.0 A [mm2] 500,000 t [mm] 3.1750 L/R [rad] 0.5 L b AR Radius R/t buckling press [Nmm'2] buckling ratio [mm] [mm] [mm] 3162.3 158.1 20.0 316.2 99.6 0.0 33.0 2600.1 192.3 13.5 384.6 121.1 0.0 18.3 2207.6 226.5 9.7 453.0 142.7 0.0 11.2 1918.1 260.7 7.4 521.4 164.2 0.0 7.4 1695.7 294.9 5.8 589.7 185.7 0.0 5.1 1519.5 329.1 4.6 658.1 207.3 0.0 3.7 1376.5 363.2 3.8 726.5 228.8 0.0 2.7 1258.1 397.4 3.2 794.9 250.4 0.0 2.1 1158.4 431.6 2.7 863.2 271.9 0.0 1.6 1073.4 465.8 2.3 931.6 293.4 0.0 1.3 1000.0 500.0 2.0 1000.0 315.0 0.0 1.0 707.1 707.1 1.0 1414.2 445.4 0.0 0.4 A [mm2] 500,000 t[mm] 4.7625 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 3162.3 158.1 20.0 316.2 66.4 0.0 9.8 2600.1 192.3 13.5 384.6 80.8 0.0 5.4 2207.6 226.5 9.7 453.0 95.1 0.0 3.3 1918.1 260.7 7.4 521.4 109.5 0.0 2.2 1695.7 294.9 5.8 589.7 123.8 0.0 1.5 1519.5 329.1 4.6 658.1 138.2 0.0 1.1 1376.5 363.2 3.8 726.5 152.5 0.0 0.8 1258.1 397.4 3.2 794.9 166.9 0.0 0.6 1158.4 431.6 2.7 863.2 181.3 0.0 0.5 1073.4 465.8 2.3 931.6 195.6 0.0 0.4 1000.0 500.0 2.0 1000.0 210.0 0.0 0.3 707.1 707.1 1.0 1414.2 296.9 0.0 0.1 218 APPENDIX A : T A B U L A T E D D A T A A [mm2] 500,000 t[mm] 1.5875 UR [rad] 2.0 L b AR Radius R/t buckling press [Nmm 2] buckling ratio [mm] [mm] [mm] 3162.3 158.1 20.0 79.1 49.8 0.0 73.0 2600.1 192.3 13.5 96.2 60.6 0.0 40.6 2207.6 226.5 9.7 113.2 71.3 0.0 24.8 1918.1 260.7 7.4 130.3 82.1 0.0 16.3 1695.7 294.9 5.8 147.4 92.9 0.0 11.3 1519.5 329.1 4.6 164.5 103.6 0.0 8.1 1376.5 363.2 3.8 181.6 114.4 0.0 6.0 1258.1 397.4 3.2 198.7 125.2 0.0 4.6 1158.4 431.6 2.7 215.8 135.9 0.0 3.6 1073.4 465.8 2.3 232.9 146.7 0.0 2.9 1000.0 500.0 2.0 250.0 157.5 0.0 2.3 707.1 707.1 1.0 353.6 222.7 0.0 0.8 A [mm2] 500,000 t [mm] 3.1750 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 3162.3 158.1 20.0 79.1 24.9 0.0 9.1 2600.1 192.3 13.5 96.2 30.3 0.0 5.1 2207.6 226.5 9.7 113.2 35.7 0.0 3.1 1918.1 260.7 7.4 130.3 41.1 0.0 2.0 1695.7 294.9 5.8 147.4 46.4 0.0 1.4 1519.5 329.1 4.6 164.5 51.8 0.0 1.0 1376.5 363.2 3.8 181.6 57.2 0.0 0.8 1258.1 397.4 3.2 198.7 62.6 0.0 0.6 1158.4 431.6 2.7 215.8 68.0 0.0 0.4 1073.4 465.8 2.3 232.9 73.4 0.0 0.4 1000.0 500.0 2.0 250.0 78.7 0.0 0.3 707.1 707.1 1.0 353.6 111.4 0.0 0.1 A [mm2] 500,000 t [mm] 4.7625 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 3162.3 158.1 20.0 79.1 16.6 0.0 2.7 2600.1 192.3 13.5 96.2 20.2 0.0 1.5 2207.6 226.5 9.7 113.2 23.8 0.0 0.9 1918.1 260.7 7.4 130.3 27.4 0.0 0.6 1695.7 294.9 5.8 147.4 31.0 0.0 0.4 1519.5 329.1 4.6 164.5 34.5 0.0 0.3 1376.5 363.2 3.8 181.6 38.1 0.0 0.2 1258.1 397.4 3.2 198.7 41.7 0.0 0.2 1158.4 431.6 2.7 215.8 45.3 0.0 0.1 1073.4 465.8 2.3 232.9 48.9 0.0 0.1 1000.0 500.0 2.0 250.0 52.5 0.0 0.1 707.1 707.1 1.0 353.6 74.2 0.1 0.0 219 A P P E N D I X A : T A B U L A T E D D A T A A [mm2] 500,000 t [mm] 1.5875 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 3162.3 158.1 20.0 39.5 24.9 0.0 55.1 2600.1 192.3 13.5 48.1 30.3 0.0 30.7 2207.6 226.5 9.7 56.6 35.7 0.0 18.8 1918.1 260.7 7.4 65.2 41.1 0.0 12.3 1695.7 294.9 5.8 73.7 46.4 0.0 8.5 1519.5 329.1 4.6 82.3 51.8 0.0 6.1 1376.5 363.2 3.8 90.8 57.2 0.0 4.5 1258.1 397.4 3.2 99.4 62.6 0.0- 3.5 1158.4 431.6 2.7 107.9 68.0 0.0 2.7 1073.4 465.8 2.3 116.5 73.4 0.0 2.2 1000.0 500.0 2.0 125.0 78.7 0.0 1.7 707.1 707.1 1.0 176.8 111.4 0.0 0.6 A [mm2] 500,000 t [mm] 3.1750 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 3162.3 . 158.1 20.0 39.5 12.4 0.0 6.9 2600.1 192.3 13.5 48.1 15.1 0.0 3.8 2207.6 226.5 9.7 56.6 17.8 0.0 2.3 1918.1 260.7 7.4 65.2 20.5 0.0 1.5 1695.7 294.9 5.8 73.7 23.2 0.0 1.1 1519.5 329.1 4.6 82.3 25.9 0.0 0.8 1376.5 363.2 3.8 90.8 28.6 0.0 0.6 1258.1 397.4 3.2 99.4 31.3 0.0 0.4 1158.4 431.6 2.7 107.9 34.0 0.0 0.3 1073.4 465.8 2.3 116.5 36.7 0.0 0.3 1000.0 500.0 2.0 125.0 39.4 0.0 0.2 707.1 707.1 1.0 176.8 55.7 0.1 0.1 A [mm2] 500,000 t [mm] 4.7625 L/R [rad] 4.0 L b AR Radius R/t buckling press [Nmm 2] buckling ratio [mm] [mm] [mm] 3162.3 158.1 20.0 39.5 8.3 — — 2600.1 192.3 13.5 48.1 10.1 0.0 1.1 2207.6 226.5 9.7 56.6 11.9 0.0 0.7 1918.1 260.7 7.4 65.2 13.7 0.0 0.5 1695.7 294.9 5.8 73.7 15.5 0.0 0.3 1519.5 329.1 4.6 82.3 17.3 0.0 0.2 1376.5 363.2 3.8 90.8 19.1 0.0 0.2 1258.1 397.4 3.2 99.4 20.9 0.0 0.1 1158.4 431.6 2.7 107.9 22.7 0.0 0.1 1073.4 465.8 2.3 116.5 24.5 0.1 0.1 1000.0 500.0 2.0 125.0 26.2 0.1 0.1 707.1 707.1 1.0 ' 176.8 37.1 0.2 0.0 220 APPENDIX A: T A B U L A T E D D A T A A [mm2] 1,000,000 t [mm] 1.5875 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 4472.1 223.6 20.0 2236.1 1408.5 0.0 3711.3 3677.1 272.0 13.5 2719.6 1713.1 0.0 2062.9 3122.0 320.3 9.7 3203.1 2017.7 0.0 1262.7 2712.6 368.7 7.4 3686.6 2322.2 0.0 828.2 2398.0 417.0 5.8 4170.1 2626.8 0.0 572.2 2148.9 465.4 4.6 4653.6 2931.4 0.0 411.7 1946.6 513.7 3.8 5137.1 3235.9 0.0 306.1 1779.2 562.1 3.2 5620.6 3540.5 0.0 233.7 1638.3 610.4 2.7 6104.1 3845.1 0.0 182.4 1518.0 658.8 2.3 6587.6 4149.6 0.0 145.1 1414.2 707.1 2.0 7071.1 4454.2 0.0 117.4 1000.0 1000.0 1.0 10000.0 6299.2 0.0 4T.5 A [mm2] 1,000,000 t [mm] 3.1750 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 4472.1 223.6 20.0 2236.1 704.3 0.0 463.9 3677.1 272.0 13.5 2719.6 856.6 0.0 257.9 3122.0 320.3 9.7 3203.1 1008.8 0.0 157.8 2712.6 368.7 7.4 3686.6 1161.1 0.0 103.5 2398.0 417.0 5.8 4170.1 1313.4 0.0 71.5 2148.9 465.4 4.6 4653.6 1465.7 0.0 51.5 1946.6 513.7 3.8 5137.1 1618.0 0.0 38.3 1779.2 562.1 3.2 5620.6 1770.3 0.0 29.2 1638.3 610.4 2.7 6104.1 1922.5 0.0 22.8 1518.0 658.8 2.3 6587.6 2074.8 0.0 18.1 1414.2 707.1 2.0 7071.1 2227.1 0.0 14.7 1000.0 1000.0 1.0 10000.0 3149.6 0.0 5.2 A [mm2] 1,000,000 t [mm] 4.7625 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 4472.1 223.6 20.0 2236.1 469.5 0.0 137.5 3677.1 272.0 13.5 2719.6 571.0 0.0 76.4 3122.0 320.3 9.7 3203.1 672.6 0.0 46.8 2712.6 368.7 7.4 3686.6 774.1 0.0 30.7 2398.0 417.0 5.8 4170.1 875.6 0.0 21.2 2148.9 465.4 4.6 4653.6 977.1 0.0 15.2 1946.6 513.7 3.8 5137.1 1078.6 0.0 11.3 1779.2 562.1 3.2 5620.6 1180.2 0.0 8.7 1638.3 610.4 2.7 6104.1 1281.7 0.0 6.8 1518.0 658.8 2.3 6587.6 1383.2 0.0 5.4 1414.2 707.1 2.0 7071.1 1484.7 0.0 4.3 1000.0 1000.0 1.0 10000.0 2099.7 0.0 1.5 221 APPENDIX A: T A B U L A T E D D A T A A [mm2] 1,000,000 t[mm] 1.5875 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 4472.1 223.6 20.0 447.2 281.7 0.0 746.8 3677.1 272.0 13.5 543.9 342.6 0.0 415.1 3122.0 320.3 9.7 640.6 403.5 0.0 254.1 2712.6 368.7 7.4 737.3 464.4 0.0 166.6 2398.0 417.0 5.8 834.0 525.4 0.0 115.1 2148.9 4654 4.6 930.7 586.3 0.0 82.9 1946.6 513.7 3.8 1027.4 647.2 0.0 61.6 1779.2 562.1 3.2 1124.1 708.1 0.0 47.0 1638.3 610.4 2.7 1220.8 769.0 0.0 36.7 1518.0 658.8 2.3 1317.5 829.9 0.0 29.2 1414.2 707.1 2.0 1414.2 890.8 0.0 23.6 1000.0 1000.0 1.0 2000.0 1259.8 0.0 8.3 A [mm2] 1,000,000 t [mm] 3.1750 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 4472.1 223.6 20.0 447.2 140.9 0.0 93.4 3677.1 272.0 13.5 543.9 171.3 0.0 51.9 3122.0 320.3 9.7 640.6 201.8 0.0 31.8 2712.6 368.7 7.4 737.3 232.2 0.0 20.8 2398.0 417.0 5.8 834.0 262.7 0.0 14.4 2148.9 465.4 4.6 930.7 293.1 0.0 10.4 1946.6 513.7 3.8 1027.4 323.6 0.0 7.7 1779.2 562.1 3.2 1124.1 354.1 0.0 5.9 1638.3 610.4 2.7 1220.8 384.5 0.0 4.6 1518.0 658.8 2.3 1317.5 415.0 0.0 3.7 1414.2 707.1 2.0 1414.2 445.4 0.0 3.0 1000.0 1000.0 1.0 2000.0 629.9 0.0 1.0 A [mm2] 1,000,000 t [mm] 4.7625 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 4472.1 223.6 20.0 447.2 93.9 0.0 27.7 3677.1 272.0 13.5 543.9 114.2 0.0 15.4 3122.0 320.3 9.7 640.6 134.5 0.0 9.4 2712.6 368.7 7.4 737.3 154.8 0.0 6.2 2398.0 417.0 5.8 834.0 175.1 0.0 4.3 2148.9 465.4 4.6 930.7 195.4 0.0 3.1 1946.6 513.7 3.8 1027.4 215.7 0.0 2.3 1779.2 562.1 3.2 1124.1 236.0 0.0 1.7 1638.3 610.4 2.7 1220.8 256.3 0.0 1.4 1518.0 ' 658.8 2.3 1317.5 276.6 0.0 1.1 1414.2 707.1 2.0 1414.2 296.9 0.0 0.9 1000.0 1000.0 1.0 2000.0 419.9 0.0 0.3 222 APPENDIX A: T A B U L A T E D D A T A A [mm2] 1,000,000 t [mm] 1.5875 L/R [rad] 2.0 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm'2] 4472.1 223.6 20.0 111.8 70.4 0.0 206.4 3677.1 272.0 13.5 136.0 85.7 0.0 114.7 3122.0 320.3 9.7 160.2 100.9 0.0 70.2 2712.6 368.7 7.4 184.3 116.1 0.0 46.1 2398.0 417.0 5.8 208.5 131.3 0.0 31.8 2148.9 465.4 4.6 232.7 146.6 0.0 22.9 1946.6 513.7 3.8 256.9 161.8 0.0 17.0 1779.2 562.1 3.2 281.0 177.0 0.0 13.0 1638.3 610.4 2.7 305.2 192.3 0.0 10.1 1518.0 658.8 2.3 329.4 207.5 0.0 8.1 1414.2 707.1 2.0 353.6 222.7 0.0 6.5 1000.0 1000.0 1.0 500.0 315.0 0.0 2.3 A [mm2] 1,000,000 t [mm] 3.1750 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 4472.1 223.6 20.0 111.8 35.2 0.0 25.8 3677.1 272.0 13.5 136.0 42.8 0.0 14.3 3122.0 320.3 9.7 160.2 50.4 0.0 8.8 2712.6 368.7 7.4 184.3 58.1 0.0 5.8 2398.0 417.0 5.8 208.5 65.7 0.0 4.0 2148.9 465.4 4.6 232.7 73.3 0.0 2.9 1946.6 513.7 3.8 256.9 80.9 0.0 2.1 1779.2 562.1 3.2 281.0 88.5 0.0 1.6 1638.3 610.4 2.7 305.2 96.1 . 0.0 1.3 1518.0 658.8 2.3 329.4 103.7 0.0 1.0 1414.2 707.1 2.0 353.6 111.4 0.0 0.8 1000.0 1000.0 1.0 500.0 157.5 0.0 0.3 A [mm2] 1,000,000 t [mm] 4.7625 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 4472.1 223.6 20.0 111.8 23.5 0.0 7.6 3677.1 272.0 13.5 136.0 28.6 0.0 4.2 3122.0 320.3 9.7 160.2 33.6 0.0 2.6 2712.6 368.7 7.4 184.3 38.7 0.0 1.7 2398.0 417.0 5.8 -208.5 43.8 0.0 1.2 2148.9 465.4 4.6 232.7 48.9 0.0 0.8 1946.6 513.7 3.8 256.9 53.9 0.0 0.6 1779.2 562.1 3.2 281.0 59.0 0.0 0.5 1638.3 610.4 2.7 305.2 64.1 0.0 0.4 1518.0 658.8 2.3 329.4 69.2 0.0 0.3 1414.2 707.1 2.0 353.6 74.2 0.0 0.2 1000.0 1000.0 1.0 500.0 105.0 0.0 0.1 223 APPENDIX A: T A B U L A T E D D A T A A [mm2] 1,000,000 t [mm] 1.5875 L/R [rad] 4.0 L b AR Radius R/t buckling press [Nmm"2] buckling ratio [mm] [mm] [mm] 4472.1 223.6 20.0 55.9 35.2 0.0 156.0 3677.1 272.0 13.5 68.0 42.8 0.0 86.7 3122.0 320.3 9.7 80.1 50.4 0.0 53.1 2712.6 368.7 7.4 92.2 58.1 0.0 34.8 2398.0 417.0 5.8 104.3 65.7 0.0 24.0 2148.9 465.4 4.6 116.3 73.3 0.0 17.3 1946.6 513.7 3.8 128.4 80.9 0.0 12.9 1779.2 562.1 3.2 140.5 88.5 0.0 9.8 1638.3 610.4 2.7 152.6 96.1 0.0 7.7 1518.0 658.8 2.3 164.7 103.7 0.0 6.1 1414.2 707.1 2.0 176.8 111.4 0.0 4.9 1000.0 1000.0 1.0 250.0 157.5 0.0 1.7 A [mm2] 1,000,000 t[mm] 3.1750 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 4472.1 223.6 20.0 55.9 17.6 0.0 19.5 3677.1 272.0 13.5 68.0 21.4 0.0 10.8 3122.0 320.3 9.7 80.1 25.2 0.0 6.6 2712.6 368.7 7.4 92.2 29.0 0.0 4.4 2398.0 417.0 5.8 104.3 32.8 0.0 3.0 2148.9 465.4 4.6 116.3 36.6 0.0 2.2 1946.6 513.7 3.8 128.4 40.4 0.0 1.6 1779.2 562.1 3.2 140.5 44.3 0.0 1.2 1638.3 610.4 2.7 152.6 48.1 0.0 1.0 1518.0 658.8 2.3 164.7 51.9 0.0 0.8 1414.2 707.1 2.0 176.8 55.7 0.0 0.6 1000.0 1000.0 1.0 250.0 78.7 0.0 0.2 A [mm2] 1,000,000 t [mm] 4.7625-L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 4472.1 223.6 20.0 55.9 11.7 0.0 5.8 3677.1 272.0 13.5 68.0 14.3 0.0 3.2 3122.0 320.3 9.7 80.1 16.8 0.0 2.0 2712.6 368.7 7.4 92.2 19.4 0.0 1 3 2398.0 417.0 5.8 104.3 21.9 0.0 0.9 2148.9 465.4 4.6 116.3 24.4 0.0 0.6 1946.6 513.7 . 3.8 128.4 27.0 0.0 0.5 1779.2 562.1 3.2 140.5 29.5 0.0 0.4 1638.3 610.4 2.7 152.6 32.0 0.0 0.3 1518.0 658.8 2.3 164.7 34.6 0.0 0.2 1414.2 707.1 2.0 176.8 37.1 0.0 0.2 1000.0 1000.0 1.0 250.0 52.5 0.1 0.1 224 APPENDIX A: T A B U L A T E D D A T A A [mm2] 2,000,000 t [mm] 1.5875 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 6324.6 316.2 20.0 3162.3 1992.0 0.0 10497.2 5200.1 384.6 13.5 3846.0 2422.7 0.0 5834.8 4415.2 453.0 9.7 4529.8 2853.4 0.0 3571.3 3836.1 521.4 7.4 5213.6 3284.2 0.0 2342.4 3391.3 589.7 5.8 5897.4 3714.9 0.0 1618.4 3039.0 658.1 4.6 6581.1 4145.6 0.0 1164.6 2753.0 726.5 3.8 7264.9 4576.3 0.0 865.7 2516.1 794.9 3.2 7948.7 5007.0 0.0 661.0 2316.8 863.2 2.7' 8632.5 5437.8 0.0 516.0 2146.8 931.6 2.3 9316.2 5868.5 0.0 410.5 2000.0 1000.0 2.0 10000.0 6299.2 0.0 332.0 1414.2 1414.2 1.0 14142.1 8908.4 0.0 117.4 A [mm2] 2,000,000 t [mm] 3.1750 L/R [rad] 0.1 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm"2] 6324.6 316.2 20.0 3162.3 996.0 0.0 1312.1 5200.1 384.6 13.5 3846.0 1211.4 0.0 729.4 4415.2 453.0 9.7 4529.8 1426.7 0.0 446.4 3836.1 521.4 7.4 5213.6 1642.1 0.0 292.8 3391.3 589.7 5.8 5897.4 1857.4 0.0 202.3 3039.0 658.1 4.6 6581.1 2072.8 0.0 145.6 2753.0 726.5 3.8 7264.9 2288.2 0.0 108.2 2516.1 794.9 3.2 7948.7 2503.5 0.0 82.6 2316.8 863.2 2.7 8632.5 2718.9 0.0 64.5 2146.8 931.6 2.3 9316.2 2934.2 0.0 51.3 2000.0 1000.0 2.0 10000.0 3149.6 0.0 41.5 1414.2 1414.2 1.0 14142.1 4454.2 0.0 14.7 A [mm2] 2,000,000 t [mm] 4.7625 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 6324.6 316.2 20.0 3162.3 664.0 0.0 388.8 5200.1 384.6 13.5 3846.0 807.6 0.0 216.1 4415.2 453.0 9.7 4529.8 951.1 0.0 132.3 3836.1 521.4 7.4 5213.6 1094.7 0.0 86.8 3391.3 589.7 5.8 5897.4 1238.3 0.0 59.9 3039.0 658.1 4.6 6581.1 1381.9 0.0 43.1 2753.0 726.5 3.8 7264.9 1525.4 0.0 32.1 2516.1 794.9 3.2 7948.7 1669.0 0.0 24.5 2316.8 863.2 2.7 8632.5 1812.6 0.0 19.1 2146.8 931.6 2.3 9316.2 1956.2 0.0 15.2 2000.0 1000.0 2.0 10000.0 2099.7 0.0 12.3 1414.2 1414.2 1.0 14142.1 2969.5 0.0 4.3 225 APPENDIX A: T A B U L A T E D D A T A A [mm2] 2,000,000 t [mm] 1.5875 UR [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 6324.6 316.2 20.0 632.5 398.4 0.0 2112.3 5200.1 384.6 13.5 769.2 484.5 0.0 1174.1 4415.2 453.0 9.7 906.0 570.7 0.0 718.6 3836.1 521.4 7.4 1042.7 656.8 0.0 471.3 3391.3 589.7 5.8 1179.5 743.0 0.0 325.7 3039.0 658.1 4.6 1316.2 829.1 0.0 234.3 2753.0 726.5 3.8 1453.0 915.3 0.0 174.2 2516.1 794.9 3.2 1589.7 1001.4 0.0 133.0 2316.8 863.2 2.7 1726.5 1087.6 0.0 103.8 2146.8 931.6 2.3 1863.2 •' 1173.7 0.0 82.6 2000.0 1000.0 2.0 2000.0 1259.8 0.0 66.8 1414.2 1414.2 1.0 2828.4 1781.7 0.0 23.6 A [mm2] 2,000,000 t [mm] 3.1750 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 6324.6 316.2 20.0 632.5 199.2 0.0- 264.0 5200.1 384.6 13.5 769.2 242.3 0.0 146.8 4415.2 453.0 9.7 906.0 285.3 0.0 89.8 3836.1 521.4 7.4 1042.7 328.4 0.0 58.9 3391.3 589.7 5.8 1179.5 371.5 0.0 40.7 3039.0 658.1 4.6 1316.2 414.6 0.0 29.3 2753.0 726.5 3.8 1453.0 457.6 0.0 21.8 2516.1 794.9 3.2 1589.7 500.7 0.0 16.6 2316.8 863.2 2.7 1726.5 543.8 0.0 13.0 2146.8 931.6 2.3 1863.2 586.8 0.0 10.3 2000.0 1000.0 2.0 2000.0 629.9 0.0 8.3 1414.2 1414.2 1.0 2828.4 890.8 0.0 3.0 A [mm2] 2,000,000 t [mm] 4.7625 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 6324.6 316.2 20.0 632.5 132.8 0.0 78.2 5200.1 384.6 13.5 769.2 161.5 0.0 43.5 4415.2' 453.0 9.7 906.0 190.2 0.0 26.6 3836.1 521.4 7.4 1042.7 218.9 0.0 17.5 3391.3 589.7 5.8 1179.5 247.7 0.0 .12.1 3039.0 658.1 4.6 1316.2 276.4 0.0 8.7 2753.0 726.5 3.8 1453.0 305.1 0.0 6.5 2516.1 794.9 3.2 1589.7 333.8 0.0 4.9 2316.8 863.2 2.7 1726.5 362.5 0.0 3.8 2146.8 931.6 2.3 1863.2 391.2 0.0 3.1 2000.0 1000.0 2.0 2000.0 419.9 0.0 2.5 1414.2 1414.2 1.0 2828.4 593.9 0.0 0.9 226 APPENDIX A : T A B U L A T E D D A T A A [mm2] 2,000,000 t [mm] 1.5875 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 6324.6 316.2 20.0 158.1 99.6 0.0 583.9 5200.1 384.6 13.5 192.3 121.1 0.0 324.6 4415.2 453.0 9.7 . 226.5 142.7 0.0 198.6 3836.1 521.4 7.4 260.7 164.2 0.0 130.3 3391.3 589.7 5.8 294.9 185.7 0.0 90.0 3039.0 658.1 4.6 329.1 207.3 0.0 64.8 2753.0 726.5 3.8 363.2 228.8 0.0 48.2 2516.1 794.9 3.2 397.4 250.4 0.0 36.8 2316.8 863.2 2.7 431.6 271.9 0.0 28.7 2146.8 931.6 2.3 465.8 293.4 0.0 22.8 2000.0 1000.0 2.0 500.0 315.0 0.0 18.5 1414.2 1414.2 1.0 707.1 445.4 0.0 6.5 A [mm2] 2,000,000 t [mm] 3.1750 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 6324.6 316.2 20.0 158.1 49.8 0.0 73.0 5200.1 384.6 13.5 192.3 60.6 0.0 40.6 4415.2 453.0 9.7 226.5 71.3 0.0 24.8 3836.1 521.4 7.4 260.7 82.1 0.0 16.3 3391.3 589.7 5.8 294.9 92.9 0.0 11.3 3039.0 658.1 4.6 329.1 103.6 0.0 8.1 2753.0 726.5 3.8 363.2 114.4 0.0 6.0 2516.1 794.9 3.2 397.4 • 125.2 0.0 4.6 2316.8 863.2 2.7 431.6 135.9 0.0 3.6 2146.8 931.6 2.3 465.8 146.7 0.0 2.9 2000.0 1000.0 2.0 500.0 157.5 0.0 2.3 1414.2 1414.2 1.0 707.1 222.7 0.0 0.8 A [mm2] 2,000,000 t [mm] 4.7625 L/R [rad] 2.0 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm 2] 6324.6 316.2 20.0 158.1 33.2 0.0 21.6 . 5200.1 384.6 13.5 192.3 40.4 0.0 12.0 4415.2 453.0 9.7 226.5 47.6 0.0 7.4 3836.1 521.4 7.4 260.7 54.7 0.0 4.8 3391.3 589.7 5.8 294.9 61.9 0.0 3.3 3039.0 658.1 4.6 329.1 69.1 0.0 2.4 2753.0 726.5 3.8 363.2 76.3 0.0 1.8 2516.1 794.9 3.2 397.4 83.5 0.0 1.4 2316.8 863.2 2.7 431.6 •• 90.6 0.0 1.1 2146.8 931.6 2.3 465.8 97.8 0.0 0.8 2000.0 1000.0 2.0 500.0 105.0 0.0 0.7 1414.2 1414.2 1.0 707.1 148.5 0.0 0.2 227 APPENDIX A: T A B U L A T E D D A T A A [mm2] 2,000,000 t [mm] 1.5875 L/R [rad] 4.0 L b AR Radius R/t buckling press [Nmm 2] buckling ratio [mm] [mm] [mm] 6324.6 316.2 20.0 79.1 49.8 0.0 441.2 5200.1 384.6 13.5 96.2 60.6 0.0 245.2 4415.2 453.0 9.7 113.2 71.3 0.0 150.1 3836.1 521.4 7.4 130.3 82.1 0.0 98.4 3391.3 589.7 5.8 147.4 92.9 0.0 68.0 3039.0 658.1 4.6 164.5 103.6 0.0 48.9 2753.0 726.5 3.8 181.6 114.4 0.0 36.4 2516.1 794.9 3.2 198.7 125.2 0.0 27.8 2316.8 863.2 2.7 215.8 135.9 0.0 21.7 2146.8 931.6 2.3 232.9 146.7 0.0 17.3 2000.0 1000.0 2.0 250.0 157.5 0.0 14.0 1414.2 1414.2 1.0 353.6 222.7 0.0 4.9 A [mm2] 2,000,000 t [mm] 3.1750 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 6324.6 316.2 20.0 79.1 24.9 0.0 55.1 5200.1 384.6 13.5 96.2 30.3 0.0 30.7 4415.2 453.0 9.7 113.2 35.7 0.0 18.8 3836.1 521.4 7.4 130.3 41.1 0.0 12.3 3391.3 589.7 5.8 147.4 46.4 0.0 8.5 3039.0 658.1 4.6 164.5 51.8 0.0 6.1 2753.0 726.5 3.8 181.6 57.2 0.0 4.5 2516.1 794.9 3.2 198.7 62.6 0.0 3.5 2316.8 863.2 2.7 215.8 68.0 0.0 2.7 2146.8 931.6 2.3 232.9 73.4 0.0 2.2 2000.0 1000.0 2.0 250.0 78.7 0.0 1.7 1414.2 1414.2 1.0 353.6 111.4 0.0 0.6 A [mm2] 2,000,000 t [mm] 4.7625 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 6324.6 316.2 20.0 79.1 16.6 0.0 16.3 5200.1 384.6 13.5 96.2 20.2 0.0 9.1 4415.2 453.0 9.7 113.2 23.8 0.0 5.6 3836.1 521.4 7.4 130.3 27.4 0.0 3.6 3391.3 589.7 5.8 147.4 31.0 0.0 2.5 3039.0 658.1 4.6 164.5 34.5 0.0 1.8 2753.0 726.5 3.8 181.6 38.1 0.0 1.3 2516.1 794.9 3.2 198.7 41.7 0.0 1.0 2316.8 863.2 2.7 215.8 45.3 0.0 0.8 2146.8 931.6 2.3 232.9 48.9 0.0 0.6 2000.0 1000.0 2.0 250.0 52.5 0.0 0.5 1414.2 1414.2 1.0 353.6 74.2 0.0 0.2 228 APPENDIX A: T A B U L A T E D D A T A A [mm2] 4,000,000 t [mm] 1.5875 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 8944.3 447.2 20.0 4472.1 2817.1 0.0 29690.5 7354.1 543.9 13.5 5439.1 3426.2 0.0 16503.4 6244.0 640.6 9.7 6406.1 4035.4 0.0 10101.2 5425.1 737.3 7.4 7373.1 4644.5 0.0 6625.3 4796.1 834.0 5.8 8340.1 5253.6 0.0 4577.7 4297.8 930.7 4.6 9307.1 5862.8 0.0 3293.9 3893.3 1027.4 3.8 10274.1 6471.9 0.0 2448.7 3558.4 1124.1 3.2 11241.1 7081.0 0.0 1869.5 3276.5 1220.8 2.7 12208.1 7690.2 0.0 1459.5 ' 3036.0 1317.5 2.3 13175.1 8299.3 0.0 1161.2 2828.4 1414.2 2.0 14142.1 8908.4 0.0 938.9 2000.0 2000.0 1.0 20000.0 12598.4 0.0 332.0 A [mm2] 4,000,000 t [mm] 3.1750 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm2] buckling ratio 8944.3 447.2 20.0 4472.1 1408.5 0.0 3711.3 7354.1 543.9 13.5 5439.1 1713.1 0.0 2062.9 6244.0 640.6 9.7 6406.1 2017.7 0.0 1262.7 5425.1 737.3 7.4 7373.1 2322.2 0.0 828.2 4796.1 834.0 5.8 8340.1 2626.8 0.0 572.2 4297.8 930.7 4.6 . 9307.1 2931.4 0.0 411.7 3893.3 1027.4 3.8 10274.1 3235.9 0.0 306.1 3558.4 1124.1 3.2 11241.1 3540.5 0.0 233.7 3276.5 1220.8 2.7 12208.1 3845.1 0.0 182.4 3036.0 1317.5 2.3 13175.1 4149.6 0.0 145.1 2828.4 1414.2 2.0 14142.1 4454.2 0.0 117.4 2000.0 2000.0 1.0 20000.0 6299.2 0.0 41.5 A [mm2] 4,000,000 t [mm] 4.7625 L/R [rad] 0.1 L b AR Radius R/t buckling press [Nmm'2] buckling ratio [mm] [mm] [mm] 8944.3 447.2 20.0 4472.1 939.0 0.0 1099.6 7354.1 543.9 • 13.5 5439.1 1142.1 0.0 611.2 6244.0 640.6 9.7 6406.1 1345.1 0.0 374.1 5425.1 737.3 7.4 7373.1 1548.2 0.0 245.4 4796.1 834.0 5.8 8340.1 1751.2 0.0 169.5 4297.8 930.7 4.6 9307.1 1954.3 0.0 122.0 3893.3 1027.4 3.8 10274.1 2157.3 0.0 90.7 3558.4 1124.1 3.2 11241.1 2360.3 0.0 69.2 3276.5 1220.8 2.7 12208.1 2563.4 0.0 54.1 3036.0 1317.5 2.3 13175.1 2766.4 0.0 43.0 2828.4 1414.2 2.0 14142.1 2969.5 0.0 34.8 2000.0 2000.0 1.0 20000.0 4199.5 0.0 12.3 229 APPENDIX A: T A B U L A T E D D A T A A [mm2] 4,000,000 t [mm] 1.5875 L/R [rad] 0.5 L b AR Radius R/t buckling press [Nmm"2] buckling ratio [mm] [mm] [mm] 8944.3 447.2 20.0 894.4 563.4 0.0 5974.4 7354.1 543.9 13.5 1087.8 685.2 0.0 3320.9 6244.0 640.6 9.7 1281.2 807.1 0.0 2032.6 5425.1 737.3 7.4 1474.6 928.9 0.0 1333.2 4796.1 834.0 5.8 1668.0 1050.7 . 0.0 921.1 4297.8 930.7 4.6 1861.4 1172.6 0.0 662.8 3893.3 1027.4 3.8 2054.8 • 1294.4 0.0 492.7 3558.4 1124.1 3.2 2248.2 1416.2 0.0 376.2 3276.5 1220.8 2.7 2441.6 1538.0 0.0 293.7 3036.0 1317.5 2.3 2635.0 1659.9 0.0 233.7 2828.4 1414.2 2.0 2828.4 1781.7 0.0 188.9 2000.0 2000.0 1.0 4000.0 2519.7 0.0 66.8 A [mm2] 4,000,000 t [mm] 3.1750 L/R [rad] 0.5 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm"2] 8944.3 447.2 20.0 894.4 281.7 0.0 746.8 7354.1 543.9 13.5 1087.8 342.6 0.0 415.1 6244.0 640.6 9.7 1281.2 403.5 0.0 254.1 5425.1 737.3 7.4 1474.6 464.4 0.0 166.6 4796.1 834.0 5.8 1668.0 525.4 0.0 115.1 4297.8 930.7 4.6 1861.4 586.3 0.0 82.9 3893.3 1027.4 3.8 2054.8 647.2 0.0 61.6 3558.4 1124.1 3.2 2248.2 708.1 0.0 47.0 3276.5 1220.8 2.7 2441.6 769.0 0.0 36.7 3036.0 1317.5 2.3 2635.0 829.9 0.0 29.2 2828.4 1414.2 2.0 2828.4 890.8 0.0 23.6 2000.0 2000.0 1.0 4000.0 1259.8 0.0 8.3 A [mm2] 4,000,000 t [mm] 4.7625 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t ' buckling press [Nmm 2] buckling ratio 8944.3 447.2 20.0 894.4 187.8 0.0 221.3 7354.1 543.9 13.5 1087.8 228.4 0.0 123.0 6244.0 640.6 9.7 1281.2 269.0 0.0 75.3 5425.1 737.3 7.4 1474.6 309.6 0.0 49.4 4796.1 834.0 5.8 1668.0 350.2 0.0 34.1 4297.8 930.7 4.6 1861.4 390.9 0.0 24.5 3893.3 1027.4 3.8 2054.8 431.5 0.0 18.2 3558.4 1124.1 3.2 2248.2 472.1 0.0 13.9 3276.5 1220.8 2.7 2441.6 512.7 0.0 10.9 3036.0 1317.5 2.3 2635.0 553.3 0.0 8.7 2828.4 1414.2 2.0 2828.4 593.9 0.0 7.0 2000.0 2000.0 1.0 4000.0 839.9 0.0 2.5 230 APPENDIX A : T A B U L A T E D D A T A A [mm2] 4,000,000 t [mm] 1.5875 L/R [rad] 2.0 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm2] 8944.3 447.2 20.0 223.6 140.9 0.0 1651.5 7354.1 543.9 13.5 272.0 171.3 0.0 918.0 6244.0 640.6 9.7 320.3 201.8 0.0 561.9 5425.1 737.3 7.4 368.7 232.2 0.0 368.5 4796.1 834.0 5.8 417.0 262.7 0.0 254.6 4297.8 930.7 4.6 465.4 293.1 0.0 183.2 3893.3 1027.4 3.8 513.7 323.6 0.0 136.2 3558.4 1124.1 3.2 562.1 354.1 0.0 104.0 3276.5 1220.8 2.7 610.4 384.5 0.0 81.2 3036.0 1317.5 2.3 658.8 415.0 0.0 64.6 2828.4 1414.2 2.0 707.1 445.4 0.0 52.2 2000.0 2000.0 1.0 1000.0 629.9 0.0 18.5 A [mm2] 4,000,000 t [mm] 3.1750 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm2] buckling ratio 8944.3 447.2 20.0 223.6 70.4 0.0 206.4 7354.1 543.9 13.5 272.0 85.7 0.0 114.7 6244.0 640.6 9.7 320.3 100.9 0.0 70.2 5425.1 737.3 7.4 368.7 116.1 0.0 46.1 4796.1 834.0 5.8 417.0 131.3 0.0 31.8 4297.8 930.7 4.6 465.4 146.6 0.0 22.9 3893.3 1027.4 3.8 513.7 161.8 0.0 17.0 3558.4 1124.1 3.2 562.1 177.0 0.0 13.0 3276.5 1220.8 2.7 610.4 192.3 0.0 10.1 3036.0 1317.5 2.3 658.8 207.5 0.0 8.1 2828.4 1414.2 2.0 707.1 222.7 0.0 6.5 2000.0 2000.0 1.0 1000.0 315.0 0.0 2.3 A [mm2] 4,000,000 t [mm] 4.7625 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm2] buckling ratio 8944.3 447.2 20.0 223.6 47.0 0.0 61.2 7354.1 543.9 13.5 272.0 57.1 0.0 34.0 6244.0 640.6 9.7 320.3 67.3 0.0 20.8 5425.1 737.3 7.4 368.7 77.4 0.0 13.6 4796.1 834.0 5.8 417.0 87.6 0.0 9.4 4297.8 930.7 4.6 465.4 97.7 0.0 6.8 3893.3 1027.4 3.8 513.7 107.9 0.0 5.0 3558.4 1124.1 3.2 562.1 118.0 0.0 3.9 3276.5 1220.8 2.7 610.4 128.2 0.0 3.0 3036.0 1317.5 2.3 658.8 138.3 0.0 2.4 2828.4 1414.2 2.0 707.1 148.5 0.0 1.9 2000.0 2000.0 1.0 1000.0 210.0 0.0 0.7 231 APPENDIX A : T A B U L A T E D D A T A A [mm2] 4,000,000 t [mm] 1.5875 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 8944.3 447.2 20.0 111.8 70.4 0.0 1247.8 7354.1 543.9 13.5 136.0 85.7 0.0 693.6 6244.0 640.6 9.7 160.2 100.9 0.0 424.5 5425.1 737.3 7.4 184.3 116.1 0.0 278.4 4796.1 834.0 5.8 208.5 131.3 0.0 192.4 4297.8 930.7 4.6 232.7 146.6 0.0 138.4 3893.3 1027.4 3.8 256.9 161.8 0.0 '102.9 3558.4 1124.1 3.2 281.0 177.0 0.0 78.6 3276.5 1220.8 2.7 305.2 192.3 0.0 61.3 3036.0 1317.5 2.3 329.4 207.5 0.0 48.8 2828.4 1414.2 2.0 353.6 222.7 0.0 39.5 2000.0 2000.0 1.0 500.0 315.0 0.0 14.0 A [mm2] 4,000,000 t[mm] 3.1750 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 8944.3 447.2 20.0 111.8 35.2 0.0 156.0 7354.1 543.9 13.5 136.0 42.8 0.0 86.7 6244.0 640.6 9.7 160.2 50.4 0.0 53.1 5425.1 737.3 7.4 184.3 58.1 0.0 34.8 4796.1 834.0 5.8 208.5 65.7 0.0 24.0 4297.8 930.7 4.6 232.7 73.3 0.0 17.3 3893.3 1027.4 3.8 256.9 80.9 0.0 12.9 3558.4 1124.1 3.2 281.0 88.5 0.0 9.8 3276.5 1220.8 2.7 305.2 96.1 0.0 7.7 3036.0 1317.5 2.3 329.4 103.7 0.0 6.1 2828.4 1414.2 2.0 353.6 111.4 0.0 4.9 2000.0 2000.0 1.0 500.0 157.5 0.0 1.7 A [mm2] 4,000,000 t [mm] 4.7625 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 8944.3 447.2 20.0 111.8 23.5 0.0 46.2 7354.1 543.9 13.5 136.0 28.6 0.0 25.7 6244.0 640.6 9.7 160.2 33.6 0.0 15.7 5425.1 737.3 7.4 184.3 38.7 0.0 10.3 4796.1 834.0 5.8 208.5 43.8 0.0 7.1 . 4297.8 930.7 4.6 232.7 48.9 o.o' 5.1 3893.3 1027.4 3.8 256.9 ' 53.9 0.0 3.8 3558.4 1124.1 3.2 281.0 59.0 0.0 2.9 3276.5 1220.8 2.7 305.2 64.1 0.0 2.3 3036.0, 1317.5 2.3 329.4 69.2 0.0 1.8 2828.4 1414.2 2.0 353.6 74.2 0.0 1.5 2000.0 2000.0 1.0 500.0 105.0 0.0 0.5 232 APPENDIX A : T A B U L A T E D D A T A A [mm2] 10,000,000 t [mm] 1.5875 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 14142.1 707.1 20.0 7071.1 4454.2 0.0 117362.1 11627.9 860.0 13.5 8600.0 5417.3 0.0 65235.4 9872.7 1012.9 9.7 10129.0 ; 6380.5 0.0 39928.6 8577.8 1165.8 7.4 11658.0 7343.6 0.0 26188.8 7583.3 1318.7 5.8 13186.9 8306.7 0.0 18094.8 6795.4 1471.6 4.6 14715.9 9269.8 0.0 13020.4 6155.8 1624.5 3.8 16244.8 10233.0 0.0 9679.1 5626.3 1777.4 3.2 17773.8 11196.1 0.0 7390.0 5180.6 1930.3 2.7 19302.8 12159.2 0.0 5769.3 4800.4 2083.2 2.3 20831.7 13122.3 0.0 4589.9 4472.1 2236.1 2.0 22360.7 14085.5 • 0.0 3711.3 3162.3 3162.3 1.0 31622.8 19919.9 0.0 1312.1 A [mm2] 10,000,000 t [mm] 3.1750 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm2] buckling ratio 14142.1 707.1 20.0 7071.1 2227.1 0.0 14670.3 11627.9 860.0 13.5 8600.0 2708.7 0.0 8154.4 9872.7 1012.9 9.7 10129.0 3190.2 0.0 4991.1 8577.8 1165.8 7.4 11658.0 3671.8 0.0 3273.6 7583.3 1318.7 5.8 13186.9 4153.4 0.0 2261.8 6795.4 1471.6 4.6 • 14715.9 4634.9 0.0 1627.6 6155.8 1624.5 3.8 16244.8 5116.5 0.0 1209.9 5626.3 1777.4 3.2 17773.8 5598.0 0.0 923.7 5180.6 1930.3 2.7 19302.8 6079.6 0.0 721.2 4800.4 2083.2 2.3 20831.7 6561.2 0.0 573.7 4472.1 2236.1 2.0 22360.7 7042.7 0.0 463.9 3162.3 3162.3 1.0 31622.8 9959.9 0.0 164.0 A [mm2] 10,000,000 t [mm] 4.7625 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 14142.1 707.1 20.0 7071.1 1484.7 0.0 4346.7 11627.9 860.0 13.5 8600.0 1805.8 0.0 2416.1 9872.7 1012.9 9.7 10129.0 2126.8 0.0 1478.8 8577.8 1165.8 7.4 11658.0 2447.9 0.0 970.0 7583.3 1318.7 5.8 13186.9 2768.9 0.0 670.2 6795.4 1471.6 4.6 14715.9 3089.9 0.0 482.2 6155.8 1624.5 3.8 16244.8 3411.0 0.0 358.5 5626.3 1777.4 3.2 17773.8 3732.0 0.0 273.7 5180.6 1930.3 2.7 19302.8 4053.1 0.0 213.7 4800.4 2083.2 2.3 20831.7 4374.1 0.0 170.0 4472.1 2236.1 2.0 22360.7 4695.2 0.0 137.5 3162.3 3162.3 1.0 31622.8 6640.0 0.0 48.6 233 APPENDIX A : T A B U L A T E D D A T A A [mm2] 10,000,000 t [mm] 1.5875 L/R [rad] 0.5 L b AR Radius R/t buckling press [Nmm"2] buckling ratio [mm] [mm] [mm] 14142.1 707.1 20.0 1414.2 890.8 0.0 23616.0 11627.9 860.0 13.5 1720.0 1083.5 0.0 13126.9 9872.7 1012.9 9.7 2025.8 1276.1 0.0 8034.6 8577.8 1165.8 7.4 2331.6 1468.7 0.0 5269.8 7583.3 1318.7 5.8 2637.4 1661.3 0.0 3641.1 6795.4 1471.6 4.6 2943.2 1854.0 0.0 2620.0 6155.8 1624.5 3.8 3249.0 2046.6 0.0 1947.7 5626.3 1777.4 3.2 3554.8 2239.2 0.0 1487.0 5180.6 1930.3 2.7 3860.6 2431.8 0.0 1160.9 4800.4 2083.2 2.3 4166.3 2624.5 0.0 923.6 4472.1 2236.1 2.0 4472.1 2817.1 0.0 746.8 3162.3 3162.3 1.0 6324.6 3984.0 0.0 264.0 A [mm2] 10,000,000 t [mm] 3.1750 L/R [rad] 0.5 L b AR Radius R/t buckling press [Nmm"2] buckling ratio [mm] [mm] [mm] 14142.1 707.1 20.0 1414.2 445.4 0.0 2952.0 11627.9 860.0 13.5 1720.0 541.7 0.0 1640.9 9872.7 1012.9 9.7 2025.8 638.0 0.0 1004.3 8577.8 1165.8 7.4 2331.6 734.4 0.0 658.7 7583.3 1318.7 5.8 2637.4 830.7 0.0 455.1 6795.4 1471.6 4.6 2943.2 927.0 0.0 327.5 6155.8 1624.5 3.8 3249.0 1023.3 0.0 243.5 5626.3 1777.4 3.2 3554.8 1119.6 0.0 185.9 5180.6 1930.3 2.7 3860.6 1215.9 0.0 145.1 4800.4 2083.2 2.3 4166.3 1312.2 0.0 115.5 4472.1 2236.1 2.0 4472.1 1408.5 0.0 93.4 3162.3 3162.3 1.0 6324.6 1992.0 0.0 33.0 A [mm2] 10,000,000 t [mm] 4.7625 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 14142.1 . 707.1 20.0 1414.2 296.9 0.0 874.7 11627.9 860.0 13.5 1720.0 361.2 0.0 486.2 9872.7 1012.9 9.7 2025.8 425.4 0.0 297.6 8577.8 1165.8 7.4 2331.6 489.6 0.0 195.2 7583.3 1318.7 5.8 2637.4 553.8 0.0 134.9 6795.4 1471.6 4.6 2943.2 618.0 0.0 97.0 6155.8 1624.5 3.8 3249.0 682.2 0.0 72.1 5626.3 1777.4 3.2 3554.8 746.4 0.0 55.1 5180.6 1930.3 2.7 3860.6 810.6 0.0 43.0 4800.4 2083.2 2.3 4166.3 874.8 0.0 34.2 4472.1 2236.1 2.0 4472.1 939.0 0.0 27.7 3162.3 3162.3 1.0 6324.6 1328.0 0.0 9.8 234 APPENDIX A: T A B U L A T E D D A T A A [mm2] 10,000,000 t [mm] 1.5875 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm -2] buckling ratio 14142.1 707.1 20.0 353.6 222.7 0.0 6528.0 11627.9 860.0 13.5 430.0 270.9 0.0 3628.6 9872.7 1012.9 9.7 506.4 319.0 0.0 • 2221.0 8577.8 1165.8 7.4 582.9 367.2 0.0 1456.7 7583.3 1318.7 5.8 659.3 415.3 0.0 1006.5 6795.4 1471.6 4.6 735.8 463.5 0.0 724.2 6155.8 1624.5 3.8 812.2 511.6 0.0 538.4 5626.3 1777.4 3.2 888.7 559.8 0.0 411.1 5180.6 1930.3 2.7 965.1 608.0 0.0 320.9 4800.4 2083.2 2.3 1041.6 656.1 0.0 255.3 4472.1 2236.1 2.0 1118.0 704.3 0.0 206.4 3162.3 3162.3 1.0 1581.1 996.0 0.0 73.0 A [mm2] 10,000,000 t [mm] 3.1750 L/R [rad] 2.0 L b AR Radius R/t buckling press [Nmm -2] buckling ratio [mm] [mm] [mm] 14142.1 707.1 20.0 353.6 111.4 0.0 816.0 11627.9 860.0 13.5 430.0 135.4 0.0 453.6 9872.7 1012.9 9.7 506.4 159.5 0.0 277.6 8577.8 1165.8 7.4 582.9 183.6 0.0 182.1 7583.3 1318.7 5.8 659.3 207.7 0.0 125.8 6795.4 1471.6 4.6 735.8 231.7 0.0 90.5 6155.8 1624.5 3.8 812.2 255.8 0.0 67.3 5626.3 1777.4 3.2 888.7 279.9 0.0 51.4 5180.6 1930.3 2.7 965.1 304.0 0.0 40.1 4800.4 2083.2 2.3 1041.6 328.1 0.0 31.9 4472.1 2236.1 2.0 1118.0 352.1 0.0 25.8 3162.3 3162.3 1.0 1581.1 498.0 0.0 9.1 A [mm2] 10,000,000 t [mm] 4.7625 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 14142.1 707.1 20.0 353.6 74.2 0.0 241.8 11627.9 860.0 13.5 430.0 90.3 0.0 134.4 9872.7 1012.9 9.7 506.4 106.3 0.0 82.3 8577.8 1165.8 7.4 582.9 122.4 0.0 54.0 7583.3 1318.7 5.8 659.3 138.4 0.0 37.3 6795.4 1471.6 4.6 735.8 154.5 0.0 26.8 6155.8 1624.5 3.8 812.2 170.5 0.0 19.9 5626.3 1777.4 3.2 888.7 • 186.6 0.0 15.2 5180.6 1930.3 2.7 965.1 202.7 0.0 11.9 4800.4 2083.2 2.3 1041.6 218.7 0.0 9.5 4472.1 2236.1 2.0 1118.0 234.8 0.0 7.6 3162.3 3162.3 1.0 1581.1 332.0 0.0 2.7 235 A P P E N D I X A: T A B U L A T E D D A T A A [mm2] 10,000,000 t [mm] 1.5875 L/R [rad] 4.0 L b AR Radius R/t buckling press [Nmm'2] buckling ratio [mm] [mm] [mm] 14142.1 707.1 20.0 176.8 111.4 0.0 4932.3 11627.9 860.0 13.5 215.0 135.4 0.0 2741.6 9872.7 1012.9 9.7 253.2 159.5 0.0 1678.1 8577.8 1165.8 7.4 291.4 183.6 0.0 1100.6 7583.3 1318.7 5.8 329.7 207.7 0.0 760.5 6795.4 1471.6 4.6 367.9 231.7 0.0 547.2 6155.8 1624.5 3.8 406.1 255.8 0.0 406.8 5626.3 1777.4 3.2 444.3 279.9 0.0 310.6 5180.6 1930.3 2.7 482.6 304.0 0.0 242.5 4800.4 2083.2 2.3 520.8 328.1 0.0 192.9 4472.1 2236.1 2.0 559.0 352.1 0.0 156.0 3162.3 3162.3 1.0 790.6 498.0 0.0 55.1 A [mm2] 10,000,000 t [mm] 3.1750 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 14142.1 707.1 20.0 176.8 55.7 0.0 616.5 11627.9 860.0 13.5 215.0 67.7 0.0 342.7 9872.7 1012.9 9.7 253.2 79.8 0.0 209.8 8577.8 1165.8 7.4 291.4 91.8 0.0 137.6 7583.3 1318.7 5.8 329.7 103.8 0.0 95.1 6795.4 1471.6 4.6 367.9 115.9 0.0 68.4 6155.8 1624.5 3.8 406.1 127.9 0.0 50.8 5626.3 1777.4 3.2 444.3 140.0 0.0 38.8 5180.6 1930.3 2.7 482.6 152.0 0.0 30.3 4800.4 2083.2 2.3 520.8 164.0 0.0 24.1 4472.1 2236.1 2.0 559.0 176.1 0.0 19.5 3162.3 3162.3 1.0 790.6 249.0 0.0 6.9 A [mm2] 10,000,000 t [mm] 4.7625 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 14142.1 707.1 20.0 176.8 37.1 0.0 182.7 11627.9 860.0 13.5 215.0 45.1 0.0 101.5 9872.7 1012.9 9.7 253.2 53.2 0.0 62.2 8577.8 1165.8 7.4 291.4 61.2 0.0 40.8 7583.3 1318.7 5.8 329.7 69.2 0.0 28.2 6795.4 1471.6 4.6 367.9 77.2 0.0 20.3 6155.8 1624.5 3.8 406.1 85.3 0.0 15.1 5626.3 1777.4 3.2 444.3 93.3 0.0 11.5 5180.6 1930.3 2.7 482.6 101.3 0.0 9.0 4800.4 2083.2 2.3 520.8 109.4 0.0 7.1 4472.1 2236.1 2.0 559.0 117.4 0.0 5.8 3162.3 3162.3 1.0 790.6 166.0 0.0 2.0 236 APPENDIX A : T A B U L A T E D D A T A A [mm2] 20,000,000 t [mm] 1.5875 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] ' R/t buckling press [Nmm"2] buckling ratio 20000.0 1000.0 20.0 10000.0 6299.2 0.0 331950.2 16444.3 1216.2 13.5 12162.3 7661.3 0.0 184513.5 13962.0 1432.5 9.7 14324.6 9023.3 0.0 112935.2 12130.9 1648.7 7.4 16486.8 10385.4 0.0 74073.2 10724.4 1864.9 5.8 18649.1 11747.5 0.0 51179.8 9610.1 2081.1 4.6 20811.4 13109.5 0.0 36827.3 8705.6 2297.4 3.8 22973.7 14471.6 0.0 27376.8 7956.7 2513.6 3.2 25135.9 15833.7 0.0 20902.0 7326.5 2729.8 2.7 27298.2 17195.7 0.0 16318.1 6788.8 2946.0 2.3 29460.5 18557.8 0.0 12982.3 6324.6 3162.3 2.0 31622.8 19919.9 0.0 10497.2 4472.1 4472.1 1.0 44721.4 28170.9 0.0 3711.3 A [mm2] 20,000,000 t [mm] 3.1750 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 20000.0 1000.0 20.0 10000.0 3149.6 0.0 41493.8 16444.3 1216.2 13.5 12162.3 3830.6 0.0 23064.2 13962.0 1432.5 9.7 14324.6 4511.7 0.0 14116.9 12130.9 1648.7 7.4 16486.8 5192.7 0.0 9259.2 10724.4 1864.9 5.8 18649.1 5873.7 0.0 6397.5 9610.1 2081.1 4.6 20811.4 6554.8 0.0 4603.4 8705.6 2297.4 3.8 22973.7 7235.8 0.0 3422.1 7956.7 2513.6 3.2 25135.9 7916.8 0.0 2612.7 7326.5 2729.8 2.7 27298.2 8597.9 0.0 2039.8 6788.8 2946.0 2.3 29460.5 9278.9 0.0 1622.8 6324.6 3162.3 2.0 31622.8 9959.9 0.0 1312.1 4472.1 4472.1 1.0 44721.4 14085.5 0.0 463.9 A [mm2] 20,000,000 t [mm] 4.7625 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 20000.0 1000.0 20.0 10000.0 2099.7 0.0 12294.5 16444.3 1216.2 13.5 12162.3 2553.8 0.0 6833.8 13962.0 1432.5 9.7 14324.6 3007.8 0.0 4182.8 12130.9 1648.7 7.4 16486.8 3461.8 0.0 2743.5 10724.4 1864.9 5.8 18649.1 3915.8 0.0 1895.5 9610.1 2081.1 4.6 20811.4 4369.8 0.0 1364.0 8705.6 2297.4 3.8 22973.7 4823.9 0.0 1014.0 7956.7 2513.6 3.2 25135.9 5277.9 0.0 774.1 7326.5 2729.8 2.7 27298.2 5731.9 0.0 604.4 6788.8 2946.0 2.3 29460.5 6185.9 0.0 480.8 6324.6 3162.3 2.0 31622.8 6640.0 0.0 388.8 4472.1 4472.1 1.0 44721.4 9390.3 0.0 137.5 237 APPENDIX A: T A B U L A T E D D A T A A [mm2] 20,000,000 t [mm] 1.5875 L/R [rad] 0.5 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm"2] 20000.0 1000.0 20.0 2000.0 1259.8 0.0 66796.2 16444.3 1216.2 13.5 2432.5 1532.3 0.0 37128.5 13962.0 1432.5 9.7 2864.9 1804.7 0.0 22725.2 12130.9 1648.7 7.4 3297.4 2077.1 0.0 14905.3 10724.4 1864.9 5.8 3729.8 2349.5 0.0 10298.6 9610.1 2081.1 4.6 4162.3 2621.9 0.0 7410.5 8705.6 2297.4 3.8 4594.7 2894.3 0.0 5508.9 7956.7 2513.6 3.2 5027.2 3166.7 0.0 4206.0 7326.5 2729.8 2.7 5459.6 3439.1 0.0 3283.6 6788.8 2946.0 2.3 5892.1 3711.6 0.0 2612.4 6324.6 3162.3 2.0 6324.6 3984.0 0.0 2112.3 4472.1 4472.1 1.0 ' 8944.3 5634.2 0.0 746.8 A [mm2] 20,000,000 t [mm] 3.1750 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press • [Nmm"2] buckling ratio 20000.0 1000.0 20.0 2000.0 629.9 0.0 8349.5 16444.3 1216.2 13.5 2432.5 766.1 0.0 4641.1 13962.0 1432.5 9.7 2864.9 902.3 0.0 2840.7 12130.9 1648.7 7.4 3297.4 1038.5 0.0 1863.2 10724.4 1864.9 5.8 3729.8 1174.7 0.0 1287.3 9610.1 2081.1 4.6 4162.3 1311.0 0.0 926.3 8705.6 2297.4 3.8 4594.7 1447.2 o:o 688.6 7956.7 2513.6 3.2 5027.2 1583.4 0.0 525.7 7326.5 2729.8 2.7 5459.6 1719.6 0.0 410.4 6788.8 2946.0 2.3 5892.1 1855.8 0.0 326.5 6324.6 3162.3 2.0 6324.6 1992.0 0.0 264.0 • 4472.1 4472.1 1.0 8944.3 2817.1 0.0 93.4 A [mm2] 20,000,000 t [mm] 4.7625 L/R [rad] 0.5 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm"2] 20000.0 1000.0 20.0 2000.0 419.9 0.0 2473.9 16444.3 1216.2 13.5 2432.5 510.8 0.0 1375.1 13962.0 1432.5 9.7 2864.9 601.6 0.0 841.7 12130.9 1648.7 7.4 3297.4 692.4 0.0 552.0 10724.4 1864.9 5.8 3729.8 783.2 0.0 381.4 9610.1 2081.1 4.6 4162.3 874.0 0.0 274.5 8705.6 2297.4 3.8 4594.7 964.8 0.0 204.0 7956.7 2513.6 3.2 5027.2 1055.6 0.0 155.8 7326.5 2729.8 2.7 5459.6 1146.4 0.0 121.6 6788.8 2946.0 2.3 5892.1 1237.2 0.0 96.8 6324.6 3162.3 2.0 6324.6 1328.0 0.0 78.2 4472.1 4472.1 1.0 8944.3 1878.1 0.0 27.7 238 APPENDIX A : T A B U L A T E D D A T A A [mm2] 20,000,000 t[mm] 1.5875 . L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 20000.0 1000.0 20.0 500.0 315.0 0.000000 18464.1 16444.3 1216.2 13.5 608.1 383.1 0.000000 10263.2 13962.0 1432.5 9.7 716.2 451.2 0.000001 6281.8 12130.9 1648.7 7.4 824.3 519.3 0.000001 4120.2 10724.4 1864.9 5.8 932.5 587.4 0.000001 2846.8 9610.1 2081.1 4.6 1040.6 655.5 0.000002 2048.4 8705.6 2297.4 3.8 1148.7 723.6 0.000003 1522.8 7956.7 2513.6 3.2 1256.8 791.7 0.000004 1162.6 7326.5 2729.8 2.7 1364.9 859.8 0.000005 907.7 6788.8 2946.0 2.3 1473.0 927.9 0.000006 722.1 6324.6 3162.3 2.0 1581.1 996.0 0.000007 583.9 4472.1 4472.1 1.0 2236.1 1408.5 0.000020 206.4 A [mm2] 20,000,000 t [mm] 3.1750 L/R [rad] 2.0 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm'2] 20000.0 1000.0 20.0 500.0 157.5 0.000002 2308.0 16444.3 1216.2 13.5 608.1 191.5 0.000003 1282.9 13962.0 1432.5 9.7 716.2 225.6 0.000005 785.2 12130.9 1648.7 7.4 824.3 259.6 0.000008 515.0 10724.4 1864.9 5.8 932.5 293.7 0.000012 355.8 9610.1 2081.1 4.6 1040.6 327.7 0.000016 256.1 8705.6 2297.4 3.8 1148.7 361.8 0.000022 190.3 7956.7 2513.6 3.2 1256.8 395.8 0.000029 145.3 7326.5 2729.8 2.7 1364.9 429.9 0.000037 113.5 6788.8 2946.0 2.3 1473.0 463.9 0.000047 90.3 6324.6 3162.3 2.0 1581.1 498.0 0.000058 73.0 4472.1 4472.1 1.0 2236.1 704.3 0.000163 25.8 A [mm2] 20,000,000 t[mm] 4.7625 L/R [rad] 2.0 L b AR Radius R/t buckling press [Nmm 2] buckling ratio [mm] [mm] [mm] 20000.0 1000.0 20.0 500.0 105.0 0.000006 683.9 16444.3 1216.2 13.5 608.1 127.7 0.000011 380.1 13962.0 1432.5 9.7 716.2 150.4 0.000018 232.7 12130.9 1648.7 7.4 824.3 173.1 0.000028 152.6 10724.4 1864.9 5.8 932.5 195.8 0.000040 105.4 9610.1 2081.1 4.6 1040.6 218.5 0.000055 75.9 8705.6 2297.4 3.8 1148.7 241.2 0.000074 56.4 7956.7 2513.6 3.2 1256.8 263.9 0.000098 43.1 7326.5 2729.8 2.7 1364.9 286.6 0.000125 33.6 6788.8 2946.0 2.3 1473.0 309.3 0.000157 26.7 6324.6 3162.3 2.0 1581.1 332.0 0.000194 21.6 4472.1 4472.1 1.0 2236.1 469.5 0.000549 7.6 239 APPENDIX A: T A B U L A T E D D A T A A [mm2] 20,000,000 t [mm] 1.5875 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 20000.0 1000.0 20.0 250.0 157.5 0.0 13950.6 16444.3 1216.2 13.5 304.1 191.5 0.0 7754.4 13962.0 1432.5 9.7 358.1 225.6 0.0 4746.2 12130.9 1648.7 7.4 412.2 259.6 0.0 3113.0 10724.4 1864.9 5.8 466.2 293.7 0.0 2150.9 9610.1 2081.1 4.6 520.3 327.7 0.0 1547.7 8705.6 2297.4 3.8 574.3 361.8 0.0 1150.5 7956.7 2513.6 3.2 628.4 395.8 0.0 878.4 7326.5 2729.8 2.7 682.5 429.9 0.0 685.8 6788.8 2946.0 2.3 736.5 463.9 0.0 545.6 6324.6 3162.3 2.0 790.6 498.0 0.0 441.2 4472.1 4472.1 1.0 1118.0 704.3 0.0 156.0 A [mm2] 20,000,000 t [mm] 3.1750 L/R [rad] 4.0 L b AR Radius R/t buckling press [Nmm"2] buckling ratio [mm] [mm] [mm] 20000.0 1000.0 20.0 250.0 78.7 0.0 1743.8 16444.3 1216.2 13.5 304.1 95.8 0.0 969.3 13962.0 1432.5 9.7 358.1 112.8 0.0 593.3 12130.9 1648.7 7.4 412.2 129.8 0.0 389.1 10724.4 1864.9 5.8 466.2 146.8 0.0 268.9 9610.1 2081.1 4.6 520.3 163.9 0.0 193.5 8705.6 2297.4 3.8 574.3 180.9 0.0 143.8 7956.7 2513.6 3.2 628.4 197.9 0.0 109.8 7326.5 2729.8 2.7 682.5 214.9 0.0 85.7 6788.8 2946.0 2.3 736.5 232.0 0.0 68.2 6324.6 3162.3 2.0 790.6 249.0 0.0 55.1 4472.1 4472.1 1.0 1118.0 352.1 0.0 19.5 A [mm2] 20,000,000 t [mm] 4.7625 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 20000.0 1000.0 20.0 250.0 52.5 0.0 516.7 16444.3 1216.2 13.5 304.1 63.8 0.0 287.2 13962.0 1432.5 9.7 358.1 75.2 0.0 175.8 12130.9 1648.7 7.4 412.2 86.5. 0.0 115.3 10724.4 1864.9 5.8 466.2 97.9 0.0 79.7 9610.1 2081.1 4.6 520.3 109.2 0.0 57.3 8705.6 2297.4 3.8 574.3 120.6 0.0 42.6 7956.7 2513.6 3.2 628.4 131.9 0.0 32.5 7326.5 2729.8 2.7 682.5 143.3 0.0 25.4 6788.8 2946.0 2.3 736.5 154.6 0.0 20.2 6324.6 3162.3 2.0 790.6 166.0 0.0 16.3 4472.1 4472.1 1.0 1118.0 234.8 0.0 5.8 240 APPENDIX A: T A B U L A T E D D A T A A [mm2] 40,000,000 . t [mm] 1.5875 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 28284.3 1414.2 20.0 14142.1 8908.4 0.0 938897.0 23255.7 1720.0 13.5 17200.1 10834.7 0.0 521883.1 19745.3 2025.8 • 9.7 20258.0 12760.9 0.0 319429.1 17155.7 2331.6 7.4 23315.9 14687.2 0.0 209510.7 15166.6 2637.4 5.8 26373.8 16613.4 0.0 144758.3 13590.8 2943.2 4.6 29431.7 18539.7 0.0 104163.3 12311.6 3249.0 3.8 32489.7 20465.9 0.0 77433.2 11252.5 3554.8 3.2 35547.6 22392.2 0.0 59119.7 10361.2 3860.6 2.7 38605.5 24318.4 0.0 46154.6 9600.7 4166.3 2.3 41663.4 26244.7 0.0 36719.6 8944.3 4472.1 2.0 44721.4 28170.9 0.0 29690.5 6324.6 6324.6 1.0 63245.6 39839.7 0.0 10497.2 A [mm2] 40,000,000 t [mm] ' 3.1750 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 28284.3 1414.2 20.0 14142.1 4454.2 0.0 117362.1 23255.7 1720.0 13.5 17200.1 5417.3 0.0 65235.4 19745.3 2025.8 9.7 20258.0 6380.5 0.0 39928.6 17155.7 2331.6 7.4 23315.9 7343.6 0.0 26188.8 15166.6 2637.4 5.8 26373.8 8306.7 0.0 18094.8 13590.8 2943.2 4.6 29431.7 9269.8 0.0 13020.4 12311.6 3249.0 3.8 32489.7 10233.0 0.0 9679.1 11252.5 3554.8 3.2 35547.6 11196.1 0.0 7390.0 10361.2 3860.6 2.7 38605.5 12159.2 0.0 5769.3 9600.7 4166.3 2.3 41663.4 13122.3 0.0 4589.9 8944.3 4472.1 2.0 44721.4 14085.5 0.0 3711.3 6324.6 6324.6 1.0 63245.6 19919.9 0.0 1312.1 A [mm2] 40,000,000 t [mm] 4.7625 L/R [rad] 0.1 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 28284.3 1414.2 20.0 14142.1 2969.5 0.0 34774.0 23255.7 1720.0 13.5 17200.1 3611.6 0.0 19329.0 19745.3 2025.8 9.7 20258.0 4253.6 0.0 11830.7 17155.7 2331.6 7.4 23315.9 4895.7 0.0 7759.7 15166.6 2637.4 5.8 26373.8 5537.8 0.0 5361.4 13590.8 2943.2 4.6 29431.7 6179.9 0.0 3857.9 12311.6 3249.0 3.8 32489.7 6822.0 0.0 2867.9 11252.5 . 3554.8 3.2 35547.6 7464.1 0.0 2189.6 10361.2 3860.6 2.7 38605.5 8106.1 0.0 1709.4 9600.7 4166.3 2.3 41663.4 8748.2 0.0 1360.0 8944.3 4472.1 2.0 44721.4 9390.3 0.0 1099.6 6324.6 6324.6 . 1.0 63245.6 13279.9 0.0 388.8 241 APPENDIX A : T A B U L A T E D D A T A A [mm2] 40,000,000 t [mm] 1.5875 L/R [rad] 0.5 L b AR Radius R/t buckling press buckling ratio [mm] [mm] [mm] [Nmm2] 28284.3 1414.2 20.0 2828.4 1781.7 0.0 188928.2 23255.7 1720.0 13.5 3440.0 2166.9 0.0 105015.2 19745.3 2025.8 • 9.7 4051.6 2552.2 0.0 64276.7 17155.7 2331.6 7.4 4663.2 2937.4 0.0 42158.5 15166.6 2637.4 5.8 5274.8 3322.7 0.0 29128.8 13590.8 2943.2 4.6 5886.3 3707.9 0.0 20960.1 12311.6 3249.0 3.8 6497.9 4093.2 0.0 15581.4 11252.5 3554.8 3.2 7109.5 4478.4 0.0 11896.3 10361.2 3860.6 2.7 7721.1 4863.7 0.0 9287.4 9600.7 4166.3 2.3 8332.7 5248.9 0.0 7388.8 8944.3 4472.1 2.0 8944.3 5634.2 0.0 5974.4 6324.6 6324.6 1.0 12649.1 7967.9 0.0 2112.3 A [mm2] 40,000,000 t [mm] 3.1750 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm2] buckling ratio 28284.3 1414.2 20.0 2828.4 890.8 0.0 23616.0 23255.7 1720.0 13.5 3440.0 1083.5 0.0 13126.9 19745.3 2025.8 9.7 4051.6 1276.1 0.0 8034.6 17155.7 2331.6 7.4 4663.2 1468.7 0.0 5269.8 15166.6 2637.4 5.8 5274.8 1661.3 0.0 3641.1 13590.8 2943.2 4.6 5886.3 1854.0 0.0 2620.0 12311.6 3249.0 3.8 6497.9 2046.6 0.0 1947.7 11252.5 3554.8 3.2 7109.5 2239.2 0.0 1487.0 10361.2 3860.6 2.7 7721.1 2431.8 0.0 1160.9 9600.7 4166.3 2.3 8332.7 2624.5 0.0 923.6 8944.3 4472.1 2.0 8944.3 2817.1 0.0 746.8 6324.6 6324.6 1.0 12649.1 3984.0 0.0 264.0 A [mm2] 40,000,000 t [mm] 4.7625 L/R [rad] 0.5 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 28284.3 1414.2 20.0 2828.4 593.9 0.0 6997.3 23255.7 1720.0 13.5 3440.0 722.3 0.0 3889.5 19745.3 2025.8 9.7 4051.6 850.7 0.0 2380.6 17155.7 2331.6 7.4 4663.2 979.1 0.0 1561.4 15166.6 2637.4 5.8 5274.8 1107.6 0.0 1078.8 13590.8 2943.2 4.6 5886.3 1236.0 0.0 776.3 12311.6 3249.0 3.8 6497.9 1364.4 0.0 577.1 11252.5 3554.8 3.2 7109.5 1492.8 0.0 440.6 10361.2 3860.6 2.7 7721.1 1621.2 0.0 344.0 9600.7 4166.3 2.3 8332.7 1749.6 0.0 273.7 8944.3 4472.1 2.0 8944.3 1878.1 0.0 221.3 6324.6 . 6324.6 1.0 12649.1 2656.0 0.0 78.2 242 APPENDIX A : T A B U L A T E D D A T A A [mm2] 40,000,000 t [mm] 1.5875 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm2] buckling ratio 28284.3 1414.2 20.0 707.1 445.4 0.0 52224.4 23255.7 1720.0 13.5 860.0 541.7 0.0 29028.8 19745.3 2025.8 9.7 1012.9 638.0 0.0 17767.6 17155.7 2331.6 7.4 1165.8 734.4 0.0 11653.6 15166.6 2637.4 5.8 1318.7 830.7 0.0 8051.9 13590.8 2943.2 4.6 1471.6 927.0 0.0 5793.9 12311.6 3249.0 3.8 1624.5 1023.3 0.0 4307.1 11252.5 3554.8 3.2 1777.4 1119.6 0.0 3288.4 10361.2 3860.6 2.7 1930.3 1215.9 0.0 2567.3 9600.7 4166.3 2.3 2083.2 1312.2 0.0 2042.5 8944.3 4472.1 2.0 2236.1 1408.5 0.0 1651.5 6324.6 6324.6 1.0 3162.3 1992.0 0.0 583.9 A [mm2] 40,000,000 t [mm] 3.1750 L>R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm"2] buckling ratio 28284.3 1414.2 20.0 707.1 222.7 0.0 6528.0 23255.7 1720.0 13.5 860.0 270.9 0.0 3628.6 19745.3 2025.8 9.7 1012.9 319.0 0.0 2221.0 17155.7 2331.6 7.4 1165.8 367.2 0.0 1456.7 15166.6 2637.4 5.8 1318.7 415.3 0.0 1006.5 13590.8 2943.2 4.6 1471.6 463.5 0.0 724.2 12311.6 3249.0 3.8 1624.5 511.6 0.0 538.4 11252.5 3554.8 3.2 1777.4 559.8 0.0 411.1 10361.2 3860.6 2.7 1930.3 608.0 0.0 320.9 9600.7 4166.3 2.3 2083.2 656.1 0.0 255.3 8944.3 4472.1 2.0 2236.1 704.3 0.0 206.4 6324.6 6324.6 1.0 3162.3 996.0 0.0 73.0 A [mm2] 40,000,000 t [mm] 4.7625 L/R [rad] 2.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm2] buckling ratio 28284.3 1414.2 20.0 707.1 148.5 0.0 ' 1934.2 23255.7 1720.0 13.5 860.0 180.6 0.0 1075.1 19745.3 2025.8 9.7 1012.9 212.7 0.0 658.1 17155.7 2331.6 7.4 1165.8 244.8 0.0 431.6 15166.6 2637.4 5.8 1318.7 276.9 0.0 298.2 13590.8 2943.2 4.6 1471.6 309.0 0.0 214.6 12311.6 3249.0 3.8 1624.5 341.1 0.0 159.5 11252.5 3554.8 3.2 1777.4 373.2 0.0 121.8 • 10361.2 3860.6 2.7 1930.3 405.3 0.0 95.1 9600.7 4166.3 2.3 2083.2 437.4 0.0 75.6 8944.3 4472.1 2.0 2236.1 469.5 0.0 61.2 6324.6 6324.6 1.0 3162.3 664.0 0.0 21.6 243 APPENDIX A : T A B U L A T E D D A T A A [mm2] 40,000,000 t [mm] 1.5875 L/R [rad] 4.0 L b AR Radius R/t buckling press [Nmm'2] buckling ratio [mm] [mm] [mm] 28284.3 1414.2 20.0 353.6 222.7 0.0 39458.3 23255.7 1720.0 13.5 430.0 270.9 0.0 21932.8 19745.3 2025.8 9.7 506.4 319.0 0.0 13424.4 17155.7 2331.6 7.4 582.9 367.2 0.0 8805.0 15166.6 2637.4 5.8 659.3 415.3 0.0 6083.7 13590.8 2943.2 4.6 735.8 463.5 0.0 4377.6 12311.6 3249.0 3.8 812.2 511.6 0.0 3254.2 11252.5 3554.8 3.2 888.7 559.8 0.0 2484.6 10361.2 3860.6 2.7 965.1 608.0 0.0 1939.7 9600.7 4166.3 • 2.3 1041.6 656.1 0.0 1543.2 8944.3 4472.1 2.0 1118.0 704.3 0.0 1247.8 6324.6 6324.6 1.0 1581.1 996.0 0.0 441.2 A [mm2] 40,000,000 t [mm] 3.1750 L/R [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm'2] buckling ratio 28284.3 1414.2 20.0 353.6 111.4 0.0 4932.3 23255.7 1720.0 13.5 430.0 135.4 0.0 2741.6 19745.3 2025.8 9.7 506.4 159.5 0.0 1678.1 17155.7 2331.6 7.4 582.9 183.6 0.0 1100.6 15166.6 2637.4 5.8 659.3 207.7 0.0 760.5 13590.8 2943.2 4.6 735.8 231.7 0.0 547.2 12311.6 3249.0 3.8 812.2 255.8 0.0 406.8 11252.5 3554.8 3.2 888.7 279.9 0.0 310.6 10361.2 3860.6 2.7 965.1 304.0 0.0 242.5 9600.7 4166.3 2.3 1041.6 328.1 0.0 192.9 8944.3 4472.1 2.0 1118.0 352.1 0.0 156.0 6324.6 6324.6 1.0 1581.1 498.0 0.0 55.1 A [mm2] 40,000,000 t [mm] 4.7625 UR [rad] 4.0 L [mm] b [mm] AR Radius [mm] R/t buckling press [Nmm 2] buckling ratio 28284.3 1414.2 20.0 353.6 74.2 0.0 1461.4 23255.7 1720.0 13.5 430.0 90.3 0.0 812.3 19745.3 2025.8 9.7 506.4 106.3 0.0 497.2 17155.7 2331.6 7.4 582.9 122.4 0.0 326.1 15166.6 2637.4 5.8 659.3 138.4 0.0 225.3 13590.8 2943.2 4.6 735.8 154.5 0.0 162.1 12311.6 3249.0 3.8 812.2 170.5 0.0 120.5 11252.5 3554.8 3.2 888.7 186.6 0.0 92.0 10361.2 3860.6 2.7 965.1 202.7 0.0 71.8 9600.7 4166.3 2.3 1041.6 218.7 0.0 57.2 8944.3 4472.1 2.0 1118.0 234.8 0.0 46.2 6324.6 6324.6 1.0 1581.1 332.0 0.0 16.3 244 APPENDIX A: T A B U L A T E D D A T A t= 1.5875mm, L/R = 0.1 245 APPENDIX A : T A B U L A T E D D A T A t = 4.7625mm, L/R = 0.1 0.00 -I , , , , , , , , , i 1 O.OOE+OO 1.00E+05 2.00E+05 3.00E+05 4.00E+05 5.00E+05 6.00E+05 7.00E+05 8.00E+05 9.00E+05 1.00E+06 Area /mm 2 t= 1.5875mm, L/R = 0.5 246 APPENDIX A : T A B U L A T E D D A T A t = 4.7625mm, b/R = 0.5 O.OOE+00 5.00E+05 1.00E+06 1.50E+06 2.00E+06 2.50E+06 Area /mm 2 247 APPENDIX A : T A B U L A T E D D A T A t = 1.5875mm, L/R = 2.0 0.0 \ 1 1 1 , , 1 O.OOE+OO 1.00E+05 2.00E+05 3.00E+05 4.00E+05 5.00E+05 6.00E+05 Area /mm 2 t = 3.175mm, L/R = 2.0 0.00E+00 5.00E+05 1.00E+06 1.50E+06 2.00E+06 2.50E+06 Area /mm 2 248 APPENDIX A : T A B U L A T E D D A T A 249 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS APPENDIX B: VISUAL BASIC PROGRAM CODES AND FORMATTED SPREADSHEETS Bl Flat plates: 250 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS PROJECT MegaWheel Cladding Panels DATE 8/3/2006 FILE flatplate.xls TIME 10:00 AM INPUT 1 Plate Width b = 2138 2138 [mm] Plate length L = 4677 4677 [mm] Plate thickness t = 3.175 3.18 [mm] Plate boundary conditions be = "F" F [SS,F] Normal pressure P = 0.0042 0.0042 [Nmm'2] Young's Modulus E = 70000 70000 [MPa} Simply Supported (SS) Fixed (F) COMPUTATIONS 1 Geometric Properties Aspect ratio AR = L/b 2.19 Area A = L*b 9999426 [mm2] S = tA2/6 1.68 [mm2] Representative length bav = (L+b)/2 3407.50 [mm] High aspect ratio (>2) Solution of cubic governing load distribution Total load W = P*A 41998 [N] Deflection coefficient k_b_ss = 0.14 0.14 1 Deflection coefficient k_b_f = 0.028 0.028 1 Deflection coefficient k_b = IF(bc="SS",k_b_ss,k_b_f) 0.028 Deflection coefficient k_m = 0.34 0.34 1 Moment coefficient k2 b ss = 0.125 0.125 1 Moment coefficient k2_b_f = 0.08 0.080 1 Moment coefficient k2_b = IF(bc="SS",k2_b_ss,k2_b_f) 0.080 Cubic coefficient A' = k_b*bA3/(E*L*tA3) 0.0261 Cubic coefficient B' = k_m*(bA3/(E*L*t))A(1/3) 0.7176 Cubic coefficient C" = (A7B')A3 4.82E-05 Cubic coefficient m' = 1/C 2.07E+04 Cubic coefficient n' = W / C 8.71 E+08 Cubic coefficient g' = n72+(SQRT(n'A2+4/27*m'A3))/2 8.71 E+08 2 Cubic coefficient g" = n'/2-(SQRT(n'A2+4/27*m'A3))/2 -3.80E+02 • 2 Cubic coefficient g = IF(g'>=0.0,g,,g") 8.71 E+08 2 Cubic coefficient a" = gA(1/3) 9.55E+02 2 Cubic coefficient b" = m'/(3*a") 7.24E+00 2 Load taken by bending W_b = a"-b" 947.91 [N] 2 Load taken by membrane W_m = W-W_b 41049.68 [N] 251 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS Bending action Moment/unit width Stress Deflection Membrane action Tension stress Deflection Max stress high AR Low aspect ratio (<2) M_b s_b d b s_m d m k2_b*W_b*b/L M_b/S k_b*W_b*bA3/(E*L*tA3) = 0.37*((W_m/(L*t))A(2/3))*EA(1/3) = k_m*(W_m*bA3/(E*L*t))A(1/3) s max har = s m+s b 34.67 [Nmm1] 20:63 [MPa] 24.75 [mm] 30.04 [MPa] 24.75 [mm] 50.67 [MPa] Solution of cubic governing load distribution Deflection coefficient k'_b_ss = Deflection coefficient k'_b_f = Deflection coefficient k'_b = Deflection coefficient k'_m = Moment coefficient k2'_b_ss = Moment coefficient . k2'_t_f = Moment coefficient k2' b = Cubic coefficient Cubic coefficient Cubic coefficient Cubic coefficient Cubic coefficient Cubic coefficient Cubic coefficient Cubic coefficient Cubic coefficient Cubic coefficient Load taken by bending Load taken by membrane Bending action Moment/unit width Stress Deflection Membrane action Tension stress Deflection Max stress low AR Deflection Max stress AA' BB' C C mm' nn' gg' gg" gg aa" bb" W_b' W m' M_b' s_b' d b' s_m d m' s_max_lar defl s max 0.043 0.014 IF(bc= 0.28 0.05 0.052 IF(bc= ,SS",k'_b_ss,k'_b_f) SS",k2,_b_ss,k2'_b_f) k'_b*bavA2/(E*tA3) k'_m*(bavA2/(E*t))A(1/3) (AA'/BB')A3 1/CC W/CC nn72+(SQRT(nn,A2+4/27*mm'A3))/2 nn72-(SQRT(nn,A2+4/27*mm'A3))/2 IF(gg'>=0.0,gg',gg") ggAd/3) mm'/(3*aa") aa"-bb" W-W b' k2'_b*W_b' M_b'/S k'_b*W_b'*bavA2/(E*tA3) 0.28*((W_m7(bav*t))A(2/3))*EA(1/3) k'_m*(W_m*bavA2/(E*t))A(1 /3) s_m'+s_b' IF(AR<1.999,(d_b'+d_m')/2,d_b) IF(AR<1.999,s_max_lar,s_max_har) 0.04 0.01 0.01 0.28 0.050 0.052 0.052 0.0726 1.0467 3.33E-04 3.00E+03 1 26E+08 1.26E+08 -7.95E+00 1.26E+08 5.01 E+02 2.00E+00 499.47 [N] 41498.12 [N] 25.97 [Nmm1] 15.46 [MPa] 36.24 [mm] 28.28 [MPa] 36.11 [mm] 43.74 [MPa] 24.75 [mm] 50.67 [MPa] 1 Strength of Aluminum, Cedric Marsh, Alcan, 1983 2 http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Quadratic_etc_equations.html 252 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS Run 1 Visual Basic code: Private Sub Calculate l _ C l i c k ( ) FormatSheet End Sub FormatSheet Visual Basic code: Sub FormatSheet() D i m varName( 1 To maxNumRows) A s String D i m va rRow( l To maxNumRows) A s Integer D i m startRow&, sheetNameS, varCount&, rowEmpty&, s R o w & , fCharS, vName$ D i m ErrorNumber&, Er rorLine& D i m f lnt&, dupFlag&, modFlag&, tNameS, txtForm$, rPos&, varLen&, flag& D i m i & , j & , t R o w & , formCount&, newForm$ ' Sheet information startRow = 6 sheetName = Application.ActiveSheet.Name ' Counters varCount = 0 rowEmpty = 0 O n Error G o T o ErrorHandler ' Create array of variables. First check that the variable name does not ' already exist and the name is legal. A legal variable name starts with ' an alphabetic character and is less than 16 characters in length sRow = startRow Whi le ((rowEmpty < 40) A n d varCount = 0) Or (rowEmpty < maxNumRowEmpty) vName = Worksheets(sheetName).Cells(sRow, 3).Value If vName o "" Then rowEmpty = 0 ' Check if the variable name is legal fChar = Left(vName, 1) Er rorN umber = 0 tint = Clnt(fChar) If (ErrorNumber o 0) A n d Len(vName) < maxVarLen Then dupFlag = 0 For i = 1 To varCount If vName = varName(i) Then dupFlag =1 Exi t For End If Next i If dupFlag = 0 Then varCount = varCount + 1 varName( varCount) = vName 253 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS varRow(varCount) = sRow Worksheets(sheetName).Cells(sRow, 4).Value = "=" Else Worksheets(sheetName).Activate Worksheets(sheetName).Cells(sRow, 3).Activate M s g B o x "Duplicate variable name "' & vName & "' on row " & Format(sRow, "0"), _ v b O K , "Format Error" Exi t Sub End If Else Worksheets(sheetName). Activate Worksheets(sheetName).Cells(sRow, 3).Activate If Len(vName) >= maxVarLen Then M s g B o x "Improper variable name length on row " & Format(sRow, "0"), _ v b O K , "Format Error" Else M s g B o x "Improper variable name format on row " & Format(sRow, "0"), _ v b O K , "Format Error" End If Exi t Sub End If Else rowEmpty = rowEmpty + 1 End If sRow = sRow + 1 Wend ' Sort the list of variable names to ensure the search/replace algorithm ' works correctly. If the name of one variable is contained in another, the ' longer variable name should be replaced first. modFlag = True While modFlag = True modFlag = False For i = 1 To varCount For j = i + 1 To varCount If InStr( l , varName(j), varName(i)) > 0 Then tName = varName(i) tRow = varRow(i) varName(i) = varName(j) varRow(i) = varRow(j) varNametj) = tName varRow(j) = tRow modFlag = True Exi t For End If N e x t j I f modFlag = True Then Exi t For N e x t i Wend ' Copy text formulas from Column E and create cell formula in Column G sRow = startRow formCount = 0 rowEmpty = 0 While ((rowEmpty < 60) A n d formCount = 0) Or (rowEmpty < maxNumRowEmpty) txfForm = Worksheets(sheefName).Cells(sRow, 5).Value If txtForm o "" Then 254 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS rowEmpty = 0 formCount = formCount + 1 ' Check, eacli variable name to see i f it is in the formula For i = 1 To varCount rPos = InStr( l , txtForm, varName(i)) varLen = Len(varName(i)) While rPos > 0 flag = 0 If rPos = 1 Then flag = ifNameChar(Mid(txtForm, rPos + varLen, 1)) E l s e l f rPos = Len(txtForm) Then flag = ifNameChar(Mid(txtForm, rPos - 1 , 1 ) ) Else flag = ifNameChar(Mid(txtForm, rPos - 1, 1)) flag = flag + ifNameChar(Mid(txtForm, rPos + varLen, 1)) End If rff lag = OThen newForm = Left(txtForm, rPos - 1) newForm = newForm & " G " & Format(varRow(i), "0") newForm = newForm & Mid(txtForm, rPos + Len(varName(i))) txtForm = newForm End If rPos = InStr(rPos + varLen, txtForm, varName(i)) Wend Next i Worksheets(sheefName).Cells(sRow, 6).Value = "=" Worksheets(sheetName).Cells(sRow, 7).Value = "=" & txtForm Else rowEmpty = rowEmpty + 1 End If sRow = sRow + 1 Wend Exi t Sub ErrorHandler: ErrorN umber = Err ErrorLine = E r l Resume Next End Sub 255 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS PROJECT Mega Wheel Cladding Panels DATE 8/3/2006 FILE flatplate.xls TIME 10:00 AM REF INPUT 2 Area [mm2] 4000000 Thickness [mm] 2 Pressure [Nmm2] 0.0042 Boundary [SS,F] "SS" No. data points 20 Min AR 2 Max AR 10 Run2 Run2: Calculates deflection and stresses based on the input values for INPUT2 for geometry, and pressure, and material properties from INPUT1, for several different aspect ratios. Tabulated below. Simply Supported (SS) Fixed (F) L b AR A a [mm] [mm] [mm] [Mpa] 6324.6 632.5 10.0 0.0 20.7 5938.2 673.6 8.8 0.0 21.5 5596.4 714.7 7.8 0.0 22.1 5291.8 755.9 7.0 0.0 22.8 5018.6 797.0 6.3 0.0 23.4 4772.2 838.2 5.7 0.0 23.9 4548.9 879.3 5.2 0.0 24.5 4345.6 920.5 4.7 0.0 25.0 4159.7 961.6 4.3 0.0 25.4 3989.0 1002.8 4.0 0.0 25.9 3831.8 1043.9 3.7 0.0 26.3 3686.5 1085.1 3.4 0.0 26.6 3551.8 1126.2 3.2 0.0 27.0 3426.6 1167.3 2.9 0.0 27.3 3309.9 1208.5 2.7 0.0 27.6 3200.9 1249.6 2.6 0.0 27.9 3098.9 1290.8 2.4 0.0 28.2 3003.2 1331.9 2.3 0.0 28.4 2913.2 1373.1 2.1 0.0 28.6 2828.4 1414.2 2.0 0.0 28.8 2000.0 2000.0 1.0 0.0 30.0 Run 2 Visual Basic Code: Private Sub Run2_Click() D i m Area, Thick, Pres, B M A X , B M I N A s Double D i m M i n A R , M a x A R , S tep l , B M I N 2 A s Double D i m Boundary A s String D i m Nodatap, i , j , k A s Integer D i m b() A s Double D i m L() A s Double D i m d() A s Double D i m S() A s Double D i m A R ( ) A s Double 256 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS 'Read input form J N P U T 2 section Area = Cells(10, 12) Thick = C e l l s ( l l , 12) Pres = Cells(12, 12) Boundary = Cells( 13, 12) Nodatap = Cells(14, 12) M i n A R = Cells(15, 12) M a x A R = C e l l s ( 1 6 , 12) 'Dimension arrays R e D i m b(Nodatap + 1) R e D i m L(Nodatap+ 1) R e D i m d(Nodatap + I) R e D i m S(Nodatap + 1) R e D i m AR(Nodatap + 1) 'Calculate the maximum and minimum plate width for 'Given plate area and desired maximum/minimum aspect ratios B M A X = (Area / M i n A R ) A 0.5 B M 1 N = (Area / M a x A R ) A 0.5 Stepl = ( B M A X - B M 1 N ) / (Nodatap - 1) 'Determine plate geometry for each iteration For i = 1 To Nodatap b(i) = B M 1 N + (i - 1 )* Stepl L( i ) = Area / b(i) A R ( i ) = L ( i ) / b ( i ) Next i For j = 1 To Nodatap 'Populate I N P U T 1 section with plate geometry Cel ls (10,5) = b(j) C e l l s ( l l , 5 ) = L(j) Cells(12, 5) = Thick Cells(13, 5) = Boundary Cells(14, 5) = Pres 'Cal l FormatSheet subroutine to calculate buckling pressures 'and ratios. Extract the values. FormatSheet dG) = Cells(100, 7) SG) = Cel ls(101,7) N e x t j 'Pertbrm similar calculations for an aspect ratio of 1.0 B M I N 2 = Area A 0.5 257 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS b(Nodatap + 1) = B M I N 2 L(Nodatap+ 1) = B M I N 2 A R ( N o d a t a p + 1)= 1 Cells(10, 5) = b(Nodatap + 1) C e l l s ( l l , 5 ) = L(Nodatap+ 1) FormatSheet d(Nodatap + 1) = Cells(100, 7) S(Nodatap + 1) = Cells(101, 7) 'Tabulate data For k = 1 To Nodatap + 1 Cells(23 + k, 10) = L(k) Cells(23 + k, l l ) = b(k) Cells(23 + k, 12) = A R ( k ) Cells(23 + k, 13) = d(k) Cells(23 + k, 14) = S(k) Next k Visual basic program code for tabulating data: Sub Flat() D i m t() A s Double D i m A() A s Double D i m m, n, d u m l , dum2, dum3 A s Integer D i m Area, Thick, Pres, B M A X , B M I N A s Double D i m M i n A R , M a x A R , S tep l , B M I N 2 A s Double D i m Boundary A s String D i m Nodatap, i , j , k, x, y A s Integer D i m b() A s Double D i m L() A s Double D i m d() A s Double D i m SO A s Double D i m A R ( ) A s Double D i m Lab 10 A s String D i m Lab2() A s String D i m InThick, InArea, InNodatap A s Integer InThick = 3 InArea = 10 InNodatap =11 R e D i m t(InThick) R e D i m L a b i (InThick) t ( l ) = 25.4 * 1 / 16 t(2) = 25.4 * 2 / 1 6 t(3) = 25.4 * 3 / 16 258 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS L a b l ( l ) = "t= 1.5875" Lab l (2 ) = "t = 3.175" Lab l (3 ) = "t = 4.7625" R e D i m A(InArea) R e D i m Lab2(InArea) A ( l ) = 100000 - A(2) = 200000 A(3) = 300000 A(4) = 500000 A ( 5 ) = 1000000 A(6) = 2000000 A(7) = 4000000 A ( 8 ) = 10000000 A(9) = 20000000 A(10) = 40000000 Lab2( l ) = " A = 100,000" Lab2(2) = " A = 200,000" Lab2(3) = " A = 300,000" Lab2(4) = " A = 500,000" Lab2(5) = " A = 1,000,000" Lab2(6) = " A = 2,000,000" Lab2(7)= " A = 4,000,000" Lab2(8) = " A = 10,000,000" Lab2(9) = " A = 20,000,000" Lab2(10) = " A = 40,000,000 d u m l = 1 For m = 1 To InThick dum2 = 1 For n = 1 To InArea Worksheets("Flat").Activate Area = A(n) Thick = t(m) Dummy 1 InNodatap, d u m l , dum2, Area, Thick, L a b i , Lab2, n, m dum2 = dum2 + 6 Next n duml = d u m l + InNodatap + 6 duml = duml + 1 Next m Worksheets("Flat-out"). Activate Cells.Select Selection.Column Width = 14# 259 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS With Selection .HorizontalAlignment = xlCenter .VerticalAlignment = xlBot tom .WrapText = False .Orientation = 0 .Addlndent = False .IndentLevel = 0 .ShrinkToFit = False .ReadingOrder = xlContext .MergeCells = False End With End Sub Sub Dummyl(InNodatap, d u m l , dum2, Area, Thick, L a b i , Lab2, n, m) D i m Pres, B M A X , B M I N A s Double D i m M i n A R , M a x A R , S tep l , B M I N 2 A s Double D i m Boundary A s String D i m Nodatap, i , j , k, x, y A s Integer D i m b() A s Double D i m L() A s Double D i m d() A s Double D i m S() A s Double D i m A R ( ) A s Double Nodatap = InNodatap Pres = Cells(12, 12) Boundary = Cells( 13, 12) M i n A R = Cells(15, 12) M a x A R = C e l l s ( l 6 , 12) R e D i m b(Nodatap + 1) R e D i m LfNodatap + 1) R e D i m d(Nodatap + 1) R e D i m S(Nodatap + 1) R e D i m AR(Nodatap +1 ) B M A X = (Area / M i n A R ) A 0.5 B M I N = (Area / M a x A R ) A 0.5 Stepl = ( B M A X - B M I N ) / (Nodatap - 1) For i = 1 To Nodatap b(i) = B M I N + (i - 1 ) * Stepl L( i ) = Area / b(i) A R ( i ) = L( i ) / b(i) Next i For j = 1 To Nodatap Cel lsQO, 5) = bG) C e l l s ( l l , 5 ) = L(j) Cells(12, 5) = Thick 260 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS Cells(13, 5) = Boundary Cells(14, 5) = Pres FormatSheet d(j) = Cells(100, 7) S(j) = Cel ls(101,7) Next j B M r N 2 = Area A 0.5 b(Nodatap+ 1) = B M I N 2 " L(Nodatap+ 1) = B M I N 2 AR(Noda tap+ 1)= 1 Cells(10, 5) = b(Nodatap + 1) C e l l s ( l l , 5) = L(Nodatap + 1) FormatSheet d(Nodatap + 1) = Cells(100, 7) S(Nodatap + 1) = Cells( 101, 7) Worksheets("Flat-out").Activate Ce l l s (duml ,dum2) = " A " Cel ls (duml + l , d u m 2 ) = "t" Ce l l s (duml , dum2 + 2) = Lab2(n) Cel ls (duml + 1, dum2 + 2) = L a b l ( m ) Ce l l s (duml , dum2 + 1) = Area Cel ls (duml + 1, dum2 + 1) = Thick Cel l s (duml + 3 , dum2) = " L " Cel l s (duml + 3, dum2 + 1) = "b" Cel l s (duml + 3, dum2 + 2) = " A R " Cel l s (duml + 3, dum2 + 3) = " D " Cel l s (duml + 3, dum2 + 4) = "s" Cel ls (duml + 4, dum2) = "[mm]" Cel l s (duml + 4, dum2 + 1) = "[mm]" Cel l s (duml + 4, dum2 + 3) = "[mm]" Cel l s (duml + 4, dum2 + 4) = " [MPa]" For k = 1 To Nodatap + 1 Cel l s (duml + k + 4, dum2) = L(k) Cel l s (duml + k + 4, dum2 + 1) = b(k) Cel ls (duml + k + 4, dum2 + 2) = A R ( k ) Cel l s (duml + k + 4, dum2 + 3) = d(k) Cel l s (duml + k + 4, dum2 + 4) = S(k) Next k Dummy2 d u m l , dum2, Nodatap End Sub 2 6 1 APPENDIX B: V I S U A L BASIC P R O G R A M C O D E S A N D F O R M A T T E D SPREADSHEETS Sub Dummy2(duml , dum2, Nodatap) D i m C e l l l , Ce l l2 , Ce l l3 , Cel l4 A s Range D i m Cel l5 , Ce l l6 , Cel l7 , Cel l8 A s Range D i m Ce l l9 , C e l l 10 A s Range Set C e l l l = Ce l l s (duml , dum2) Set Cel l2 = Cel l s (duml + 1, dum2 + 2) Range(Cel l l ,Cel l2) .Selec t Selection.Borders(xlDiagonalDown).LineStyle = xlNone Selection.Borders(xlDiagonalUp).LineStyle = xlNone With Selection.Borders(xlEdgeLeft) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic End Wi th Wi th Selection.Borders(xlEdgeTop) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic End With With Selection.Borders(xlEdgeBottom) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic End Wi th With Selection.Borders(xlEdgeRight) .LineStyle = xlContinuous .Weight = x l M e d i u m .Colorlndex = xlAutomatic End Wi th Selection.Borders(xlInsideVertical).LineStyle = xlNone Selection.Borders(xllnsideHorizontal).LineStyle = x lNone Set Cel l5 = Cel ls (duml + 3, dum2 + 1) Set Cel l6 = Cel ls (duml + Nodatap + 5, dum2 +1) Range(Cell5, Cell6).Select Selection.Borders(xlDiagonalDown).LineStyle = xlNone Selection.Borders(xlDiagonalUp).LineStyle = x lNone With Selection.Borders(xlEdgeLeft) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End Wi th With Se lection. Borders(xlEdgeTop) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End Wi th With Selection.Borders(xlEdgeBottom) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End With With Selection.Borders(xlEdgeRight) 262 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS .LineStyle = xlContinuous .Weight = x lThin .Colorlndex = xlAutomatic End Wi th Selection.Borders(xlInside Vertical).LineStyle = xlNone Selection.Borders(xlInsideHorizontal).LineStyle = xlNone Set Cel l7 = Cells(dum 1 + 3 , dum2 + 3) Set Cel l8 = Cells(dum 1 + Nodatap + 5, dum2 + 3) Range(Cell7, Cell8).Select Selection.Borders(xlDiagonalDown).LineStyle = xlNone Selection.Borders(xlDiagonalUp).LineStyle = xlNone With Selection.Borders(xlEdgeLeft) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End Wi th Wi th Selection. Borders(xlEdgeTop) .LineStyle = xlContinuous .Weight = x lThin .Colorlndex = xlAutomatic End Wi th Wi th Selection.Borders(xlEdgeBottom) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End Wi th With Selection.Borders(xlEdgeRight) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic E n d Wi th Selection.Borders(xlInsideVertical).LineStyle = xlNone Selection.Borders(xlInsideHorizontal).LineStyle = xlNone Set Cel l3 = Cel l s (duml + 3, dum2) Set Cel l4 = Cel ls (duml + Nodatap + 5, dum2 + 4) Range(Cell3, Cell4).Select Selection. Borders(xlDiagonalDown).LineStyle = xlNone Selection.Borders(xlDiagonalUp).LineStyle = xlNone With Selection.Borders(xlEdgeLeft) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic End With With Selection.Borders(xlEdgeTop) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic End Wi th Wi th Selection.Borders(xlEdgeBottom) .LineStyle = xlContinuous .Weight = x l M e d i u m .Colorlndex = xlAutomatic 263 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS End Wi th With Selection.Borders(xlEdgeRight) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic End Wi th Set Cel l9 = Cel l s (duml + 3, dum2) Set Ce l l 10 = Cel ls(duml + 4, dum2 + 4) Range(Cell9, Cell lO).Select Selection.Borders(xlDiagonalDown).LineStyle = xlNone Selection.Borders(xlDiagonalUp).LineStyle = xlNone With Selection.Borders(xlEdgeLeft) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic End Wi th Wi th Selection. Borders(xlEdgeTop) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic End Wi th Wi th Selection.Borders(xlEdgeBottom) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic End With Wi th Selection.Borders(xlEdgeRight) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic End Wi th End Sub 264 APPENDIX B: V ISUAL BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS B2 Curved plates: 265 APPENDIX B: V ISUAL BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS PROJECT PANASONIC MEGAWHEEL AL CLADDING DATE 7/25/2006 FILE Short edge curved.xls TIME 1:37 PM REF INPUT 1 Plate Width b = 6324.55532 = 6325 [mm] Plate length L = 6324.55532 = 6325 [mm] Plate thickness t = 4.7625 = 4.76 [mm] Angle (b/R) ang = 4 = 4.00 [rad] Normal pressure P = 0.0042 = 0.0042 [Nmm2] Young's Modulus E = 70000 = 70000 [MPa] Shear Modulus G = 26000 = 26000 [MPa] Poisson's ratio nu = 0.3 = 0.3 Run1 COMPUTATIONS 1 Geometric Properties Aspect ratio AR Plan area A Sectional area Asec Second moment of area I Torsional stiffness J Torsional Rigidity C Representative length bav Radius R alpha alpha Arch rise arise Projected horizontal width bho L/b L*b L*t L*tA3/12 L*tA3/3 G*J (L+b)/2 b/ang ang/2 R*(1-COS(alpha)) 2*R*SIN(alpha) 1.00 40000000 30121 5.69E+04 2.28E+05 5.92E+09 6324.56 1581 2.00 2239.12 2875.45 [mm2] [mm2] [mm4] [mm4] [mm] [mm] [rad] [mm] [mm] Buckling with inflection point duml duml Buckling pressure q' Buckling ratio bu_rat 12*RA3*(1-nuA2) E*tA3*(PI()A2/alphaA2-1)/dum1 P/q' 4.32E+10 2.57E-04 [Nmm'2] 16.34 Buckling without inflection (snap-through) m dum22 dum2 Buckling pressure 2 Buckling ratio2 m dum22 dum2 q'2 bu rat2 4*l/(Asec*ariseA2) (4/27)*(((1-m)A3)/mA2) IF(dum22<0,1 E99,dum22) (E*l*arise/bhoA4)*(1+(dum2)A0.5)*384/5 P/q'2 1.51E-06 6.51E+10 6.51E+10 2.56E+06 [Nmm2] 0.000000002 Lateral buckling dum3 dum4 Buckling pressure 3 Buckling ratio3 dum3 = PI()A2+angA2*(E*l/C) dum4 = (PI()A2-angA2)A2 q'3 = ETdum4/(RA3*angA2' bu_rat3 = P/q'3 dum3) 20.64 37.58 1.15E-01 0.036605 266 APPENDIX B: V I S U A L BASIC P R O G R A M C O D E S A N D F O R M A T T E D SPREADSHEETS Buckling pressure q = MIN(q',q'2,q'3) = 0.00025705 [Nmm'2] Buckling ratio bu_rat_m = P/q = 16.34 Buckling check bu_check = IF(bu_rat_m<1.0,"Ok !!","Buckling!!") = Buckling I! 1 Theory of Elastic Stability, Timoshenko and Gere, 2nd edition, 1961 Runl VBA program code: Private Sub Calculate l _ C l i c k ( ) FormatSheet End Sub Formatsheet VBA program code: See section B1 267 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS PROJECT PANASONIC MEGAWHEEL AL CLADDING DATE 7/25/2006 FILE Short edge curved.xls TIME 1:37 PM REF INPUT 2 Area [mm2] 4,000,000 Thickness [mm] 1.5875 Pressure [Nmm"2] 0.0042 Angle (b/R) [rad] 0.1 No. data points 20 Min AR 2 Max AR 20 Run2: Calculates buckling ratios and pressures based on the input values for INPUT2 for geometry, and external pressure, and material properties from INPUT1, for several different aspect ratios. Tabulated below. Radial: Calculates buckling ratios and pressures based on the input values for INPUT2 for aspect ratio and pressure, and INPUT 1 for material properties, for] several different plate thicknesses, areas, included angles (curvature) and aspect ratios. The plate thicknesses, areas and included angles are selected in the program code. Tabulated in Rad-out Run2 Radial L b A R Radius Buckling press. bu_rat1 bu_rat2 bu_rat3 Buckling ratio [mm] [mm] [mm] [Nmm2] 8944.3 447.2 20.0 4472 0.00113167 3.71 0.00 0.00 3 7 8030.4 498.1 16.1 4981 0.00081902 5.13 0.00 0.00 5.1 7285.9 549.0 13.3 5490 0.00061170 6.87 0.00 0.00 6.9 6667.8 599.9 11.1 5999 0.00046885 8.96 0.00 0.01 9.0 6146.4 650.8 9.4 6508 0.00036723 11.44 0.00 0.01 11.4 5700.5 701.7 8.1 7017 0.00029298 14.34 0.00 0.01 14.3 5315.0 752.6 7.1 7526 0.00023747 17.69 0.00 0.01 17.7 4978.4 803.5 6.2 8035 0.00019514 21.52 0.00 0.02 21.5 4681.8 854.4 5.5 8544 0.00016230 25.88 0.00 0.02 25.9 4418.6 905.3 4.9 9053 0.00013644 30.78 0.00 0.03 30.8 4183.4 956.2 4.4 9562 0.00011579 36.27 0.00 0.04 36.3 3972.0 1007.1 3.9 10071 0.00009911 42.38 0.00 0.05 42.4 3780.9 1058.0 3.6 10580 0.00008548 49.13 0.00 0.06 49.1 3607.4 1108.8 3.3 11088 0.00007424 56.57 0.00 0.07 566 3449.0 1159.7 3.0 11597 0.00006489 64.72 0.00 0.08 64.7 3304.1 1210.6 2.7 12106 0.00005705 73.62 0.00 0.10 73.6 3170.8 1261.5 2.5 12615 0.00005042 83.31 0.00 0.12 83.3 3047.8 1312.4 2.3 13124 0.00004478 93.80 0.00 0.14 93.8 2934.0 1363.3 2.2 13633 0.00003995 105.14 0.00 0.16 105.1 2828.4 1414.2 2.0 14142 0.00003579 117.36 0.00 0.18 117.4 2000.0 2000.0 1.0 20000 0.00001265 331.95 0.00 0.73 332.0 Run2 V B A program code: Private Sub Run2_Click() D i m Area, Thick, Pres, B M A X , B M I N , A n g A s Double D i m M i n A R , M a x A R , S tep l , B M I N 2 A s Double D i m Boundary A s String D i m Nodatap, i , j , k A s Integer D i m b() A s Double D i m L() A s Double D i m bupress() A s Double D i m buratio() A s Double 268 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS D i m buratA() A s Double D i m buratB() A s Double D i m buratC() A s Double D i m A R ( ) A s Double D i m Rad() A s Double 'Read in data form INPUT 2 section Area = Cells(10, 12) Thick = C e l l s ( l l , 12) Pres = Cells(12, 12) . A n g = Cells(13, 12) Nodatap = Cells(14, 12) M i n A R = Cells( 15, 12) M a x A R = Cells( 16, 12) 'Dimension the arrays to hold data R e D i m b(Nodatap + 1) R e D i m L(Nodatap +1) R e D i m Rad(Nodatap + 1) R e D i m bupress(Nodatap + 1) R e D i m buratio(Nodatap +1 ) R e D i m buratA(Nodatap + 1) R e D i m buratB(Nodatap +1) R e D i m buratC(Nodatap + 1) R e D i m AR(Nodatap + 1) 'Calculate the maximum and minimum plate width from the 'Inputed plate area and maximum/minimum aspect ratios B M A X = (Area / M i n A R ) A 0.5 B M I N = (Area / M a x A R ) A 0.5 Stepl = ( B M A X - B M I N ) / (Nodatap - 1) 'Determine plate geometry for each iteration For i = 1 To Nodatap b(i) = B M I N + (i - 1 )* Stepl L( i ) = Area / b(i) A R ( i ) = L ( i ) / b ( i ) Rad(i) = b(i) / A n g Next i For j = 1 To Nodatap 'Place plate geometry in the INPUT! section Cells(10, 5) = b(j) C e l l s ( l l , 5) = L(j) Cells(12, 5) = Thick Cells(13, 5) = A n g Cells(14, 5) = Pres 'Call the FormatSheet subroutine and extract the buckling ratios 269 APPENDIX B: V I S U A L BASIC P R O G R A M C O D E S A N D F O R M A T T E D SPREADSHEETS FormatSheet buratAtj) = Cells(49, 7) buratB(j) = Cells(56, 7) buratCO) = Cells(62, 7) bupress(j) = Cells(64, 7) buratioO) = Cells(65, 7) Next j 'Perform similar calculations for an aspect ratio of 1.0 B M 1 N 2 = Area A 0.5 b(Nodatap+ 1) = B M I N 2 L(Nodatap+ 1) = B M I N 2 AR(Noda tap+ 1)= 1 Rad(Nodatap + 1) = L(Nodatap + 1) / A n g Cells(10, 5) = b(Nodatap + 1) C e l l s ( l l , 5 ) = L(Nodatap+ 1) FormatSheet bupress(Nodatap + 1) = Cells(64, 7) buratio(Nodatap + 1) = Cells(65, 7) buratA(Nodatap + 1) = Cells(49, 7) buratBfNodatap + 1) = Cells(56, 7) buratC(Nodatap + 1) = Cells(62, 7) For k = 1 To Nodatap + 1 Cells(32 + k, 10) = L(k) Cells(32 + k, l l ) = b(k) Cells(32 + k, 12) = A R ( k ) Cells(32 + k, 13) = Rad(k) Cells(32 + k, 14) = bupress(k) Cells(32 + k, 15) = buratA(k) Cells(32 + k, 16) = buratB(k) Cells(32 + k, 17) = buratC(k) Cells(32 + k, 18) = buratio(k) N e x t k E n d Sub Radial VBA program code: Option Expl ic i t Sub RadialO D i m t() A s Double D i m A() A s Double 270 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS D i m A n g O A s Double D i m m, n, d u m l , dum2, dum.3, counter A s Integer D i m Area, Thick, Pres, B M A X , B M I N A s Double D i m M i n A R , M a x A R , S tep l , B M I N 2 , Angle A s Double D i m Boundary A s String D i m Nodatap, i , j , k, x, y, z, dum4 A s Integer D i m b() A s Double D i m L() A s Double D i m bupress() A s Double D i m buratio() A s Double D i m A R ( ) A s Double D i m L a b l ( ) A s String D i m Lab2() A s String D i m Lab3() A s String D i m Lab4() A s String D i m InThick, InArea, InNodatap, InAng A s Integer 'Hardwired input parameters for the number of different thickness's 'area's, aspect ratios and included angles (curvature) InThick = 3 InArea = 10 InNodatap = 11 InAng = 4 counter = (InArea - 1) * 8 R e D i m t(InThick) R e D i m L a b i (InThick) 'The thicknesses are multiples of sixteenths of an inch t ( l ) = 25.4 * 1 / 16 t(2) = 2 5 . 4 * 2 / 16 t(3) = 25.4 * 3 / 1 6 R e D i m A(InArea) R e D i m Lab2(InArea) 'The plate areas to be analyzed are inputed A ( l ) = 100000 A(2) = 200000 A(3) = 300000 A(4) = 500000 A ( 5 ) = 1000000 A(6) = 2000000 A(7) = 4000000 A ( 8 ) = 10000000 A(9) = 20000000 A(10) = 40000000 Lab2( l ) = " A = 100,000" Lab2(2) = " A = 200,000" Lab2(3) = " A = 300,000" 271 APPENDIX B: V I S U A L BASIC P R O G R A M C O D E S A N D F O R M A T T E D SPREADSHEETS Lab2(4) = " A = 500,000" Lab2(5) = " A = 1,000,000" Lab2(6)= " A = 2,000,000" Lab2(7)= " A = 4,000,000" Lab2(8) = " A = 10,000,000" Lab2(9) = " A = 20,000,000" Lab2(10) = " A = 40,000,000 R e D i m Ang(InAng) R e D i m Lab3(lnAng) 'The included angles (curvatures) are inputed A n g ( l ) = 0.1 Ang(2) = 0.5 Ang(3) = 2 Ang(4) = 4 dum3 = 0 dum4 = 1 For z = 1 To InAng duml = 1 For m = 1 To InThick dum2 = 1 Sheets("Rad").Select 'For each specific plate area, thickness and included angle (curvature) 'call the subroutine 'RadDummy I' to perform the calculations For n = 1 To InArea Worksheets("Rad").Activate Area = A(n) Thick = t(m) Angle = Ang(z) R a d D u m m y l InNodatap, d u m l , dum2, Area, Thick, L a b i , Lab2, n, m, z, Angle , dum3, Lab3, Lab4, dum4 dum2 = dum2 + 8 Next n d u m l = d u m l + InNodatap + 7 d u m l = duml + 1 dum4 = dum4 + 1 Next m dum3 = dum3 + dum 1 + 1 272 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS Range(Cells(dum3 - 1 , 1 ) , Cells(dum3 - 1, counter + 7)).Select Wi th Selection.Interior .Colorlndex =15 .Pattern = x lSo l id End With Next z 'Format the data in the tables Worksheets("Rad-out").Activate Cells.Select Selection.ColumnWidth = 14# With Selection .HorizontalAlignment = xlCenter .VerticalAlignment = xlBottom .WrapText = False .Orientation = 0 .Addlndent = False .IndentLevel = 0 .ShrinkToFit = False .ReadingOrder = xlContext .MergeCells = False End With Wi th Selection.Font .Name = " A r i a l " .FontStyle = "Regular" Size = 9 .Strikethrough = False .Superscript = False .Subscript = False .OutlineFont = False .Shadow = False .Underline = xlUnderlineStyleNone .Colorlndex = xlAutomatic End With End Sub Sub RadDummyl(InNodatap, d u m l , dum2, Area, Thick, L a b i , Lab2, n, m, z, Angle , dum3, Lab3, Lab4, dum4) 'Sub routine to perform calculations D i m Pres, B M A X , B M I N , A n g A s Double D i m M i n A R , M a x A R , S tep l , B M I N 2 A s Double D i m Boundary A s String D i m Nodatap, i , j , k, x, y, r l , r2, c l , c2 A s Integer D i m b() A s Double D i m L() A s Double D i m bupress() A s Double D i m buratio() A s Double 273 A P P E N D I X B : V I S U A L B A S I C P R O G R A M C O D E S A N D F O R M A T T E D S P R E A D S H E E T S D i m A R ( ) A s Double D i m Thickrat() A s Double D i m C e l l l , Cel l2 , Ce l l3 , Cel l4 A s Range D i m aa, bb, cc, dd A s Range Nodatap = InNodatap 'Read in plare pressure, minimum and maximum aspect ratio from 'INPUT 2 section Pres = Cells(12, 12) M i n A R = C e l l s ( 1 5 , 12) M a x A R = Cells(16, 12) x = m * n 'Dimension arrays to hold the data, each array represents a specific 'plate area, thickness and included angle, and various aspect ratios R e D i m b(Nodatap + 1) R e D i m L(Nodatap+ 1) R e D i m Rad(Nodatap +1 ) R e D i m bupress(Nodatap + 1) R e D i m buratio(Nodatap + 1) R e D i m AR(Nodatap +1) R e D i m Thickrat(Nodatap + 1) 'Determine the minimum and maximum plate width from the inputed aspect 'ratios and plate area B M A X = (Area / M i n A R ) A 0.5 B M I N = (Area / M a x A R ) A 0.5 Stepl = ( B M A X - B M I N ) / (Nodatap - 1) A n g = Angle 'Determine plate geometry For i = 1 T o Nodatap b(i) = B M I N + (i - 1 )* Stepl L(i ) = A r e a / b ( i ) A R ( i ) = L ( i ) / b ( i ) Rad(i) = b(i) / A n g Thickrat(i) = Rad(i) / Thick Next i For j = 1 To Nodatap 'Populate the INPUT 1 section with plate geometry Cells(10, 5) = b(j) • C e l l s ( l l , 5 ) = L(j) Cells(12, 5) = Thick Cells(13, 5) = A n g 274 APPENDIX B: V ISUAL BASIC PROGRAM CODES A N D FORMATTED SPREADSHEETS Cells(14, 5) = Pres ' C a l l the formatsheet subroutine to calculate b u c k l i n g ratios 'and extract the b u c k l i n g ratios and pressure FormatSheet bupresstj) = Cells(64, 7) buratioG) = Cells(65, 7) Next j 'Per form same calculat ions for an aspect ratio o f 1.0 B M I N 2 = Area A 0.5 b(Nodatap+ 1) = B M I N 2 L(Nodatap+ 1) = B M I N 2 AR(Noda tap+ 1)= 1 Rad(Nodatap + 1) = L(Nodatap + 1) / A n g ThickratfNodatap + 1) = RadfNodatap + 1) / Thick Cells(10, 5) = b(Nodatap + 1) C e l l s ( l l , 5 ) = L(Nodatap+ 1) FormatSheet bupress(Nodatap + 1) = Cells(64, 7) buratio(Nodatap + 1) = Cells(65, 7) ' F o r m the tables in the output sheet Worksheets("Rad-out").Activate Cells(dum3 + d u m l , dum2) = " A [mm2]" Cells(dum3 + d u m l + 1, dum2) = "t [mm]" Cells(dum3 + duml + 2, dum2) = " L / R " Cells(dum3 + d u m l , dum2 + 1) = Area Cells(dum3 + duml + 1, dum2 + 1) = Thick Cells(dum3 + duml + 2, dum2 + 1) = Angle Cells(dum3 + duml + 4, dum2) = " L " Cells(dum3 + d u m l + 4, dum2+ 1) = "b" Cells(dum3 + duml + 4, dum2 + 2) = " A R " Cells(dum3 + duml + 4, dum2 + 3) = "Radius" Cells(dum3 + d u m l + 4, dum2 + 4) = "R/t" Cells(dum3 + duml + 4, dum2 + 5) = "buckling press" Cells(dum3 + duml + 4, dum2 + 6) = "buckling ratio" Cells(dum3 + d u m l + 5, dum2) = "[mm]" Cells(dum3 + d u m l + 5, dum2 + 1) = "[mm]" Cells(dum3 + d u m l + 5, dum2 + 3) = "[mm]" Cells(dum3 + duml + 5, dum2 + 5) = "[Nmm-2]" ' F i l l the tables with data 275 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS For k = 1 To Nodatap + 1 Cells(dum3 + duml + k + 5, dum2) = L(k) Cells(dum3 + dum 1 + k + 5, dum2 + 1) = b(k) Cells(dum3 + duml + k + 5, dum2 + 2) = A R ( k ) Cells(dum3 + d u m l + k + 5, dum2 + 3) = Rad(k) Cells(dum3 + d u m l + k + 5, dum2 + 4) = Thickrat(k) Cells(dum3 + dum 1 + k + 5, dum2 + 5) = bupress(k) Cells(dum3 + d u m l + k + 5, dum2 + 6) = buratio(k) 'The theory is not applicable for R/10 less than 10 I fThickra t (k)< 10 Then Cells(dum3 + d u m l + k + 5, dum2 + 5) = " — " Cells(dum3 + dum 1 + k + 5, dum2 + 6) = " — " End If Next k r l = duml + dum3 + 6 r2 = dum3 + dum 1 + Nodatap + 6 c 1 = dum2 + 2 c2 = dum2 + 6 'Call a subroutine to format the tables RadDummy2 d u m l , dum2, Nodatap, dum3 End Sub Sub RadDummy2(dum 1, dum2, Nodatap, dum3) D i m C e l l l , Ce l l2 , Ce l l3 , Cel l4 A s Range D i m Cel lS , Ce l l6 , Cel l7 , Cel l8 A s Range D i m Cel l9 , C e l l 10, C e l l l 1, C e l l 12 A s Range D i m C e l l 13, C e l l 14 A s Range Set C e l l l = Cells(dum3 + d u m l , dum2) Set Cel l2 = Cells(dum3 + d u m l + 2, dum2 + 1) RangefCe l l l , Cell2).Select Selection.Borders(xlDiagonalDown).LineStyle = xlNone Selection.Borders(xlDiagonalUp).LineStyle = xlNone With Selection.Borders(xlEdgeLeft) .LineStyle = xlContinuous .Weight = x l M e d i u m .Colorlndex = xlAutomatic End With Wi th Selection.Borders(xlEdgeTop) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic End Wi th Wi th Selection.Borders(xlEdgeBottom) .LineStyle = xlContinuous .Weight = x l M e d i u m .Colorlndex = xlAutomatic 276 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS End Wi th With Selection.Borders(xlEdgeRight) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic End Wi th Selection.Borders(xllnsideVertical).LineStyle = xlNone Selection.Borders(xlInsideHorizontal).LineStyle = xlNone Set Cel l5 = Cells(dum3 + d u m l + 4, dum2 + 1) Set Cel l6 = Cells(dum3 + duml + Nodatap + 6, dum2 + 1) Range(Cell5, Cell6).Select Selection.Borders(xlDiagonalDown).LineStyle = xlNone Selection.Borders(xlDiagonalUp).LineStyle = xlNone With Selection.Borders(xlEdgeLeft) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End Wi th With Selection.Borders(xlEdgeTop) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End Wi th Wi th Selection.Borders(xlEdgeBottom) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End Wi th With Selection.Borders(xlEdgeRight) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End With Selection.Borders(xllnsideVertical).LineStyle = xlNone Selection.Borders(xlInsideHorizontal).LineStyle = xlNone Set Cel l7 = Cells(dum3 + dum 1 + 4, dum2 + 3) Set Cel l8 = Cells(dum3 + d u m l + Nodatap + 6, dum2 + 3) Range(Cell7, Cell8).Select Selection.Borders(xlDiagonalDown).LineStyle = xlNone Selection.Borders(xlDiagonalUp).LineStyle = xlNone With Selection.Borders(xlEdgeLeft) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End Wi th With Selection. Borders(xlEdgeTop) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End Wi th With Selection. Borders(xlEdgeBottom) 277 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End With Wi th Selection.Borders(xlEdgeRight) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End With Selection.Borders(xlInsideVertical).LineStyle = xlNone Selection.Borders(xlInsideHorizontal).LineStyle = xlNone Set C e l l l 1 = Cells(dum3 + duml + 4, dum2 + 5) Set C e l l 12 = Cells(dum3 + duml + Nodatap + 6, dum2 + 5) RangefCel l l 1, Cel l l2) .Select Selection.Borders(xlDiagonalDown).LineStyle = xlNone Selection.Borders(xlDiagonalUp).LineStyle = xlNone With Selection.Borders(xlEdgeLeft) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End With Wi th Selection.Borders(xlEdgeTop) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End Wi th Wi th Selection. Borders(xlEdgeBottom) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End With Wi th Selection.Borders(xlEdgeRight) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic E n d With Selection.Borders(xlInsideVertical).LineStyle = x lNone Selection.Borders(xlInsideHorizontal).LineStyle = xlNone Set C e l l 13 = Cells(dum3 + d u m l + 4, dum2 + 6) Set C e l l 14 = Cells(dum3 + d u m l + Nodatap + 6, dum2 + 6) Range(Cel l l3 , Cel l l4) .Select Selection.Borders(xlDiagonalDown).LineStyle = xlNone Selection.Borders(xlDiagonalUp).LineStyle = xlNone With Selection.Borders(xlEdgeLeft) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End Wi th With Selection.Borders(xlEdgeTop) .LineStyle = xlContinuous .Weight = x lTh in . 278 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS .Colorlndex = xlAutomatic End Wi th Wi th Selection.Borders(xlEdgeBottom) .LineStyle = xlContinuous .Weight = x lThin .Colorlndex = xlAutomatic End Wi th With Selection.Borders(xlEdgeRight) .LineStyle = xlContinuous .Weight = x lTh in .Colorlndex = xlAutomatic End Wi th Selection.Borders(xlInsideVertical).LineStyle = xlNone Selection.Borders(xlInsideHorizontal).LineStyle = xlNone Set Cel l3 = Cells(dum3 + d u m l + 4, dum2) Set Cel l4 = Cells(dum3 + d u m l + Nodatap + 6, dum2 + 6) Range(Cell3, Cell4).Select Selection.Borders(xlDiagonalDown).LineStyle = xlNone Selection.Borders(xlDiagonalUp).LineStyle = xlNone With Selection.Borders(xlEdgeLeft) .LineStyle = xlContinuous .Weight = x l M e d i u m .Colorlndex = xlAutomatic End Wi th With Selection.Borders(xlEdgeTop) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic End With Wi th Selection.Borders(xlEdgeBottom) .LineStyle = xlContinuous .Weight = x l M e d i u m .Colorlndex = xlAutomatic End Wi th Wi th Selection.Borders(xlEdgeRight) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic End Wi th Set Cel l9 = Cells(dum3 + duml + 4, dum2) Set C e l l 10 = Cells(dum3 + duml + 5, dum2 + 6) Range(Cell9, Cell lO).Select Selection.Borders(xlDiagonalDown).LineStyle = xlNone Selection.Borders(xlDiagonalUp).LineStyle = xlNone With Selection.Borders(xlEdgeLeft) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic End Wi th With Selection.Borders(xlEdgeTop) .LineStyle = xlContinuous 279 APPENDIX B: VISUAL BASIC PROGRAM CODES AND FORMATTED SPREADSHEETS .Weight = x lMed ium .Colorlndex = xlAutomatic E n d Wi th With Selection.Borders(xlEdgeBottom) .LineStyle = xlContinuous .Weight = x lMed ium .Colorlndex = xlAutomatic E n d Wi th With Selection.Borders(xlEdgeRight) .LineStyle = xlContinuous .Weight = x l M e d i u m .Colorlndex = xlAutomatic End Wi th End Sub 280 APPENDIX B: VISUAL BASIC PROGRAM CODES AND FORMATTED SPREADSHEETS 2 8 1 APPENDIX B: VISUAL BASIC PROGRAM CODES AND FORMATTED SPREADSHEETS PROJECT PANASONIC MEGAWHEEL AL CLADDING PANELS DATE 7/25/2006 FILE Long edge curved.xls TIME 12:44 PM REF INPUT 1 Plate Width b = 800 = 800 [mm] Plate length L = 2000 = 2000 [mm] Plate thickness t = 1.5875 = 1.59 [mm] Angle (L/R) ang = 0.1 = 0.10 [rad] Normal pressure P = 0.0042 = 0.0042 [Nmm2] Young's Modulus E = 70000 = 70000 [MPa] Shear Modulus G = 26000 = 26000 [MPa] Poisson's ratio nu = 0.3 = 0.3 Run1 COMPUTATIONS 1 Geometric Properties Aspect ratio AR = L/b 2.50 Plan area A = L*b 1600000 [mm2] Sectional area Asec = b*t 1270 [mm2] Second moment of area I = b*tA3/12 2.67E+02 [mm4] Torsional stiffness J = b*tA3/3 1 07E+03 [mm4] Torsional Rigidity C = G*J 2.77E+07 Representative length bav = (L+b)/2 1400.00 [mm] Radius R = L/ang = 20000 [mm] alpha alpha = ang/2 = 0.05 [rad] Arch rise arise = R*(1-COS(alpha)) 24.99 [mm] Projected horizontal length Lho 2*R*SIN(alpha) 1999.17 [mm] Buckling with inflection point duml duml = 12*RA3*(1-nuA2) 8.74E+13 Buckling pressure q' = E*tA3*(PI()A2/alphaA2-1)/dum1 1.27E-05 [Nmm Buckling ratio bu_rat = P/q' 331.95 Buckling without inflection (snap-through) m m = 4*l/(Asec*ariseA2) = 1.34E-03 dum22 dum22 = (4/27)*(((1-m)A3)/mA2) 8.16E+04 dum2 dum2 = IF(dum22<0,1E99,dum22) 8.16E+04 Buckling pressure 2 q'2 = (E*l*arise/LhoA4)*(1 +(dum2)A0.5)*384/5 6.43E-01 [Nmm Buckling ratio2 bu_rat2 = P/q'2 0.0065 Lateral buckling dum3 dum3 = PI()A2+angA2*(E*l/C) 9.88 dum4 dum4 = (PI()A2-angA2)A2 97.21 Buckling pressure 3 q'3 = E*l*dum4/(RA3*angA2*dum3) 2.30E-03 Buckling ratio3 bu_rat3 = P/q'3 1.828387 282 A P P E N D I X B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS Buckling pressure Buckling ratio Buckling check q = MIN(q',q,2,q'3) bu_rat_m = P/q bu_check = IF(bu_rat_m<1.0,"Ok !!","Buckling !!") 0.00001 331.9502 Buckling !! [Nmm'"1] 1 Roark's Formulas for Stress and Strain, W Young, 6th edition 2 Theory of Elastic Stability, Timoshenko and Gere, 2nd edition, 1961 The Visual basic program code for Runl, Run2 and Radial are almost identical to the codes for the short edge curved plate case, presented previously. PROJECT PANASONIC MEGAWHEEL AL CLADDING PANELS DATE 7/25/2006 FILE Long edge curved.xls TIME 12.44 PM I REF INPUT 2 Area [mm2] 4,000,000 Thickness [mm] 1.5875 Pressure [Nmm2] 0.0042 Angle (L/R) [rad] 0.1 No. data points 20 Min AR 2 Max AR 20 Run2 Run2: Calcula tes buckling ratios and pressures based on the input values for I N P U T 2 for geometry, and external pressure, and material properties from I N P U T ! , for several different aspect ratios. Tabulated below. Radial: Calcula tes buckling ratios and pressures based on the input values for I N P U T 2 for aspect ratio and pressure, and I N P U T 1 for material properties, for] several different plate th icknesses , areas, included angles {curvature) and aspect ratios. The plate th icknesses , areas and included angles are selected in the program code. Tabulated in Rad-out Radial L b AR Radius Buckling press. bu_rat1 bu_rat2 bu_rat3 Buckling ratio [mm] [mm] [mm] [Nmm2] 8944 447 20 4472 0.00 29691 0 293 29691 8030 498 16 4981 0.00 21488 0 190 21488 7286 549 13 5490 0.00 16049 0 129 16049 6668 600 11 5999 0.00 12301 0 90 12301 6146 651 9 6508 0.00 9635 0 65 9635 5701 702 8 7017 0.00 7687 0 48 7687 5315 753 7 7526 0.00 6230 0 36 6230 4978 803 6 8035 0.00 5120 0 28 5120 4682 854 5 8544 0.00 4258 0 22 4258 4419 905 5 9053 0.00 3580 0 17 3580 4183 956 4 9562 0.00 3038 0 14 3038 3972 1007 4 10071 0.00 2600 0 11 2600 3781 1058 4 10580 0.00 2243 0 9 2243 3607 1109 3 11088 0.00 1948 0 8 1948 3449 1160 3 11597 0.00 1702 0 6 1702 3304 1211 3 12106 0.00 1497 0 5 1497 3171 1262 3 12615 0.00 1323 0 5 1323 3048 1312 2 13124 0.00 1175 0 4 1175 2934 1363 2 13633 0.00 1048 0 3 1048 2828 1414 2 14142 0.00 939 0 3 939 2000 2000 1 20000 0.00 332 0 1 332 283 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS 284 APPENDIX B: VISUAL BASIC PROGRAM CODES AND FORMATTED SPREADSHEETS PROJECT MegaWhee l Cladding Panels DATE 8/3/2006 FILE 1D heat model.xls TIME 10:13 AM REF INPUT 1 Outside surface cladding temp Tel 83 83 [°C] Inside surface cloadding temp Te2 = Te1 83 [ °C] Outside air temp Ta1 40 40 [ °C] Inside air temp Ta2 = 50 = 50 [°C] Incident solar flux qA = 1000 1000 [W/m2] Plate thickness t = 0.0015875 = 1.5875E-03 [m] Plate surface area Ax = 1 = 1.00 [m2] thermal conductivity of cladding k = 202 3 202 [W/m°C] absorptivity for solar radiation alpsol = (0.96+0.16)/2 = 0.56 absorptivity for low temp radiation alplt = (0.04+0.95)/2 = 0.50 Stefan boltzman constant sigma = 5.669E-08 = 5.669E-08 [W/m2K] gravitational acceleration 3 9.81 9.81 [m/s2] Run Solve Te1 Solve Ta2 External air conditions Interior air conditions 9 •§»• %mM q n ( d ! ,4 q t . , n v j -4 , ^ t|j(Wii1 COMPUTATIONS 1 Miscelaneous initial calcs Temperatures in kelvin Properties of air at temp Te 1 Viscosity of air Conductive resistance of air Prandtl number of air Properties of air at temp Te2 Viscosity of air conductive resistance of air Prandtl number of air Incident Solar radiation Solar energy transfer Te1k Te2k Talk Ta2k nu1 kairl Pr1 nu2 kair2 Pr2 qsolar Te 1+273 Te2+273 Ta1+273 Ta2+273 VLOOKUP(Te1k,air!A3:D353,2) VLOOKUP(Te1k,air!A3:D353,3) VLOOKUP(Te1 k,air!A3:D353,4) VLOOKUP(Te2k,air!A3 VLOOKUP(Te2k,air!A3 VLOOKUP(Te2k,air!A3 qA*Ax D353.2) D353.3) D353.4) 356 [K] 356 [K] 313 [K] 323 [K] 2.14E-05 [m2/s] 0.03046 [W/(m°C] 0.69604 2.14E-05 [m/s] 0.03046 [W/(m°C] 0.69604 1000 [W] 285 A P P E N D I X B: V I S U A L B A S I C P R O G R A M C O D E S A N D F O R M A T T E D S P R E A D S H E E T S Conduction Conductive resistance Rcond conductive energy transfer qcond Convection Cladding outer surface beta of air at temp Te betal Grashof number GM Convection heat transfer coef hcondl Convective resistance Rconvl Convective energy transfer qconvl Cladding inner surface beta of air at temp Te beta2 Grashof number Gr2 Convection heat transfer coef hcond2 Convective resistance Rconv2 Convective energy transfer qconv2 Total convective energy transfer qconv Radiation Cladding outer surface resistance factor hr1 radiative resistance Rradl radiative energy transfer qradl Cladding inner surface resistance factor hr2 radiative resistance Rrad2 radiative energy transfer qrad2 Total radiation energy transfer qrad Total energy transfer qtotal Energy discrepancy qdiff t/(k*Ax) (Te1-Te2)/Rcond 1/Te1k g*beta 1 *qA*(AxA0.5)A4/(kair1 *nu 1A2) 0.17*(Gr1 *Pr1 )A0.25*kair1 /(AxA0.5) 1/(hcond1*Ax) (Te1-Ta1)/Rconv1 1/Te2k g*beta2*qA*(AxA0.5)A4/(kair2*nu2A2) 0.17*(Gr2*Pr2)A0.25*kair2/(AxA0.5) 1/(hcond2*Ax) (Te2-Ta2)/Rconv2 qconvl+qconv2 sigma*(Te1 kA2+Ta1 kA2)*(Te1 k+Ta1 k)*alplt/alpsol 1/(hr1*Ax) (Te1-Ta1)/Rrad1 sigma*(Te2kA2+Ta2kA2)*(Te2k+Ta2k)*alplt/alpsol 1/(hr2*Ax) (Te2-Ta2)/Rrad2 qrad1+qrad2 qconv+qrad (qsolar-qtotal)/(qsolar)*100 7.859E-06 [ C/W] 0.00 [W] 0.003 [1/K] 1.98.E+12 5.611 [W/(m 2°C]| 0.178 [°C/W] 241 [W] 0.003 [1/K] 1.98.E+12 5.611 [W/(m 2 oC]| 0.178 [°C/W] 185 [W] 426 [W] 7.5329 [W/(nri K] 0.1328 [°C/W] 324 [W] 7.8619 [W7(nrfK] 0.1272 [°C/W] 259 [W] 583 [W] 1010 [W] -0.98 [%] 1 Heat transfer, J.P. Holman, Fifth edition,1981 Run VBA program code: Private Sub Calcu la te l_Cl ick( ) FormatSheet End Sub 286 APPENDIX B: V I S U A L BASIC P R O G R A M C O D E S A N D F O R M A T T E D SPREADSHEETS FormatSheet VBA program code: See section Bl Solve Tel VBA program code: Private Sub CommandBut tonl_Cl ick( ) D i m curtemp, i , j , startemp, optemp A s Integer D i m curdif, duml A s Single 'Start at Tel = -20 and iterate until get a minimum in error 'when evaluating the equilibrium equations, at each iteration 'increase temp by 10 degrees (Celsius) 'Populate the input 1 section with the Tel trial temp and 'use formatsheet subroutine to calculate error startemp = -20 i = 1 j = 0 d u m l = 100000000000# D o While i < 2 curtemp = startemp + j Cells(10, 5) = curtemp FormatSheet curdif = ((Cells(104, 7)) A 2) A 0.5 If curdif < duml Then duml = curdif optemp = curtemp Else l f curdif > duml Then i = 10 End If j = j + 10 Loop 'Perform same iteration around the previous optimum temperature 'to get temperature with smallest error, to the nearest degree startemp = optemp - 10 i = 1 j = 0 duml = 100000000000# 287 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS Do While i < 2 curtemp = startemp + j Cells(10, 5) = curtemp FormatSheet curdif = ((Cells(104, 7)) A 2) A 0.5 If curdif < d u m l Then dum 1 = curdif optemp = curtemp Else l f curdif > duml Then i = 10 End If j = j + l Loop Cells(10, 5) = optemp FormatSheet End Sub Solve Ta2 VBA program code: Private Sub CommandButton2_Click() D i m curtemp, i , j , startemp, optemp A s Integer D i m curdif, d u m l A s Single 'Start at Ta2 = -20 and iterate until get a minimum in error 'when evaluating the equilibrium equations, at each iteration 'increase temp by 10 degrees (Celsius) 'Populate the input! section with the Ta2 trial temp and 'use formatsheet subroutine to calculate error startemp = -20 i = 1 j = 0 d u m l = 100000000000# Do While i < 2 curtemp = startemp + j Cells(13, 5) = curtemp FormatSheet curdif = ((Ce!ls(104, 7)) A 2) A 0.5 288 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS If curdif < d u m l Then dum 1 = curdif optemp = curtemp Else l f curdif > d u m l Then i = 10 End I f j = j + 1 0 Loop 'Perform same iteration around the previous optimum temperature 'to get temperature with smallest error, to the nearest degree startemp = optemp - 10 i = 1 j = 0 duml = 100000000000# Do While i < 2 curtemp = startemp + j Cells(13, 5) = curtemp FormatSheet curdif = ((Cells(104, 7)) A 2) A 0.5 If curdif < d u m l Then dum 1 = curdif optemp = curtemp Else l f curdif > d u m l Then i = 10 End If j = j + l Loop Cells(13, 5) = optemp FormatSheet End Sub APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS P R O J E C T M e g a W h e e l Cladding Pane ls D A T E 8/3/2006 F I L E 1D heat model xls T I M E 10:13 AM I N P U T Start outside temp Ta1 20 Outside temp step 5 Number of steps 5 Start inside temp Ta2 20 Inside temp step 5 Number of steps 5 External uir conditions tf«Mvl Interior air conditions Form temp table T a 1 T a 2 Te1 [ ° C ] C C ] l°C] 2 0 20 62 2 0 25 6 4 2 0 30 6 6 2 0 35 68 2 0 4 0 7 0 2 0 4 5 72 2 5 20 6 4 2 5 2 5 6 6 2 5 30 6 8 2 5 3 5 70 2 5 4 0 72 2 5 4 5 74 3 0 20 6 6 3 0 25 6 8 3 0 30 70 3 0 35 72 3 0 4 0 74 3 0 4 5 7 6 3 5 20 68 3 5 2 5 7 0 3 5 3 0 72 3 5 3 5 74 3 5 4 0 7 6 3 5 4 5 78 4 0 20 70 4 0 25 72 4 0 30 74 4 0 35 7 6 4 0 4 0 78 4 0 4 5 81 4 5 2 0 72 4 5 25 74 4 5 30 76 4 5 35 78 4 5 4 0 81 4 5 4 5 8 3 290 APPENDIX B: V I S U A L BASIC P R O G R A M CODES A N D F O R M A T T E D SPREADSHEETS Form temp table VBA program code: Private Sub CommandButton3_Click() D i m curtemp, i , j , k, 1, m, n, startemp, optemp A s Integer D i m startoutempl, startoutemp2, outempstepl, outempstep2 A s Integer D i m numoutstepl, numoutstep2, counter, outempl , outemp2 A s Integer D i m curdif, duml A s Single 'Read input parameters startoutempl = Cells(10, 11) outempstep 1 = Cells( 11, 11) numoutstepl =Cel ls(12, 11) startoutemp2 = Cells(13, 11) outempstep2 = Cells(14, 11) numoutstep2 = Cells(15, 11) k = 0 counter = 0 'From the inputed Tal and Ta2 temperatures, iterate to find the equilibrium Tel temperature to the nearest degree by populating the Input I section 'and using the FormatSheet subroutine Do While k < numoutstepl + 1 outempl = startoutempl + k * outempstepl Cells(12, 5) = outempl 1 = 0 Do While 1 < numoutstep2 + 1 outemp2 = startoutemp2 + 1 * outempstep2 Cells(13, 5) = outemp2 startemp = 0 i = 1 j = 0 d u m l = 100000000000# Do Whi le i < 2 curtemp = startemp + j Cells(10, 5) = curtemp FormatSheet curdif = ((Cells(104, 7)) A 2) A 0.5 If curdif < duml Then duml = curdif optemp = curtemp APPENDIX B: V I S U A L BASIC P R O G R A M C O D E S A N D F O R M A T T E D SPREADSHEETS Else l f curdif > d u m l Then i = 10 End If j = j + 10 Loop startemp = optemp - 10 i = 1 j = 0 duml = 100000000000# D o While i < 2 curtemp = startemp + j Cells(10, 5) = curtemp FormatSheet curdif = ((Cells(104, 7)) A 2) A 0.5 If curdif < d u m l Then d u m l = curdif optemp = curtemp , E l se l f curdif > duml Then i = 10 End If j = j + l Loop Cells(45 + counter, 11) = outempl Cells(45 + counter, 13) = optemp Cells(45 + counter, 12) = outemp2 1 = 1+1 counter = counter + 1 Loop k = k + 1 Loop 292 APPENDIX C: ANSYS INPUT FILES APPENDIX C: ANSYS INPUT FILES CI Un-stiffened plates: The Ansys Program Design Language (APDL) file; plate.mac, is used when performing analysis of flat or curved un-stiffened plates. The analysis type is specified when 'calling' the file (plate.mac), a second separate file is then automatically accessed which contains commands used to build the model geometry and specify the loading and boundary conditions. There is a separate file for the flat plate (flatplate.dat) and curved plate (curvedplate.dat) models. Plate.mac APDL file: ! M A C R O TO B A T C H R U N M L G A W H E E L A L C L A D D I N G F L A T P L A T E M O D E L ! PANASONIC M E G A WHEEL ! Created by Devari Fitch June 2005 ! Usage: plate.runMode _runMode=argl IrunMode determines which type of analysis is to be performed ! 1.1 - linear static • ! 1.2 - stress stiffening effects included ! 1.3 - eigenvalue buckling ! 1.4 - nonlinear buckling finish parsav,all /clear !The input file containing model geometry and loads is •declared here as 'theFile' parres,new ItheFile = 'flatplate' theFile = 'curvedplate' !The model file is accessed and subroutines called to [complete the model creation *ulib,%thefile%,dat 293 APPENDIX C: ANSYS INPUT FILES /prep7 *USE,create *use,modgeom *use,loadbc ! Display sellings for the model are declared /pbc,u„ 1 /psf ,pres„2 /vup„z *ulib allsel,all ! Different analysis types are performed, depending on the input ! 1.1 - Linear static analysis * if,_runMode,eq, 1.1 ,then finish /solu *ulib,%thefile%,dat *ulib /solu allsel,all solve finish /postl ! 1.2 - Stress stiffening effects are included *elseif,_runMode,eq, 1.2,then finish /solu *ulib,%thefile%,dat *ulib /solu allseLall S O L C O N T R O L , on N L G E O M . o n solve finish /postl ! 1.3 Eigenvalue buckling analysis *elseif,_runMode,eq, 1.3,then finish /solu A N T Y P E , s t a t i c P S T R E S , o n solve finish /solu A N T Y P E , b u c k l e 294 APPENDIX C: ANSYS INPUT FILES B U C 0 P T , S U B S P , 4 M X P A N D S O L V E F I N I S H /POST1 SETJ i s t SET.first P L D l S P , l F I N I S H ! 1.4 - Nonlinear buckling analysis *elseif,_runMode,eq, 1.4,then INonlinear buckling analysis finish / S O L U *ulib,%thefile%,dat *ulib allsel,all A N T Y P E , S T A T I C S O L C O N T R O L , on N L G E O M , o n O U T R E S „ l N S U B S T , 1 0 0 „ , ! A U T O T S , o n A R C L E N , o n ! N C N V , 2 , 5 S O L V E F I N I S H IDetermine nodes of maximum and minimum displacement in ' !z-direction /POST1 N S E L , a l l N S O R T , u , z , l , *GET,duml,sort ,0 , imax *GET,dum2,sort ,0,imin F I N I S H ! Formatting of non linear stress deflection plot / P O S T 2 6 N S O L , 2 , d u m l , u , z NSOL,3 ,dum2,u ,z P R O D , 4 , l , „ L o a d „ , 4 . 2 PROD,5 ,2 , „pos def l„ , l PROD,6 ,3 , „neg defl ,„- l / A X L A B , Y , D e f l e c t i o n (mm) / A X L A B , X , N o r m a l pressure Nmm-2 / G R I D . l / X R A N G E , 0 , 4 . 2 X V A R , 4 P L V A R , 6 *endif 295 APPENDIX C: ANSYS INPUT FILES Flatplate.dat A P D L file: create Panasonic Mega Wheel A L Cladding Flat Plate Finite Element Model F E M Generation Script Created by Devan Fitch June 2005 / t i t l e , M E G A W H E E L 05 / P R E P 7 ! Default units are kN,mm SECTION 0: Definition of basic parameters PI = 3.14159265 ! Input relevant plate geometry pb = 1228.0 p L = 8146.0 pth = 25.4*2/16 !plate width mm plate length mm plate thickness mm SECTION 1: Definition of elements & materials Define element type et,15,shell63 ! Define material properties of steel mp,ex, 1,200 mp,ey, 1,200 mp,ez, 1,200 mp,dens,l ,7.85E-12 mp,nuxy, 1,0.3 mp,nuyz, 1,0.3 mp,nuxz, 1,0.3 mp,gxy, 1,79.29 mp ,a lpx , l , l 1.7E-6 ! Young's modulus in kN/mm2 Young's modulus in kN/mm2 Young's modulus in k.N/mni2 density in kN-secA2/mmA4 (w/ a=9810 mm/seeA2) Poisson's ratio Poisson's ratio Poisson's ratio shear ratio in kN/mm2 thermal expansion coeff. / deg. C ! Define material properties of aluminum 296 A P P E N D I X C : A N S Y S I N P U T F I L E S mp,ex,2,70 mp,ey,2,70 mp,ez,2,70 Young's modulus in kN/inm2 Young's modulus in kN/mm2 Young's modulus in kN/mm2 density in kJN-secA2/mmA4 (w/ a=9810 mm/secA2) Poisson's ratio Poisson's ratio Poisson's ratio shear ratio in kN/mm2 shear ratio in kN/mm2-shear ratio in kN/min2 thermal expansion coeff. / deg. C mp,dens,2,2.710E-12 mp,nuxy,2,0.34 mp,nuyz,2,0.34 mp,nuxz,2,0.34 mp,gxy,2,26.96 mp,gyz,2,26.96 mp,gxz,2,26.96 mp,alpx,2,23.6E-6 Def. of Real Data (for each element type) /eof modgeom SECTION 2: Model geometry / P R E P 7 ICreate four corner keypoints in the x-y plane k, 11,0,0,0 k,12,pb,0,0 k,13,pb,pL,0 k,14,0,pL,0 [create the plate area numstr,area, 100 a,l 1,12,13,14 'Define the material, real data, and element type type, 15 mat,2 real, 15 !Define a name for the plate area asel ,s ,area„ 100,110 cm,plateArea,area !Define element size and mesh the plate r,15,pth !Plate thickness csys,0 ICartesian co-ordinates 297 APPENDIX C: ANSYS INPUT FILES esize,50 amesh,all . allsel,all /eof loadbc ! SECTION 3: Boundary conditions and loading ! Define surface pressure applied to plate cmsel,s,plateArea sfa,all,l,pres,4.2e-6 !kN/mm2 allsel,all •Restrain edge degrees of freedom to create simply supported or fixed !boundary conditions nsel.all nsel,s,loc,x,0.0 nsel,a,loc,x,pb d,al l ,ux,0„„uy,uz d,all,roty,0,„„ nsel,s,loc,y,0.0 nsel,a,loc,y,pL d,al l ,ux,0„„uy,uz d,all,rotx,0„,„ allsel,all /eof post SECTION 4: Postprocessing /format,,, 14,4,5000,239 /page„ ,9999 ,240 /output,rst,rst prrsol ese l , s , type„20,29 etable,fxi,smisc,l etable,fyi,smisc,2 298 APPENDIX C: ANSYS INPUT FILES etable,fzi,smisc,3 etable,mxi,smisc,4 etable,myi,smisc,5 etable,mzi,smisc,6 etable,fxj,smisc,7 etable,fyj,smisc,8 etable,fzj,smisc,9 etable,mxj,smisc,10 etable,myj,smisc, 11 etable,mzj,smisc, 12 etable,sdiri ,LS,l etable,sbyti,LS,2 etable,sbybi,LS,3 etable,sbzti,LS,4 etable,sbzbi,LS,5 etable,sdirj,LS,6 etable,sbytj,LS,7 etable,sbybj,LS,8 etable,sbztj,LS,9 etable,sbzbj,LS,10 etable,smaxi,nmisc, 1 etable,smini,nmisc,2 etable,smaxj,nmisc,3 etable,srninj,nmisc,4 etable,strni ,LEPEL, 1 etable,strnj,LEPEL,6 e table ,s tmpre ,LEPEL,l 1 pretab,fxi,fyi,fzi,mxi,myi,mzi pretab,rxj,ryj,fzj,mxj,myj,mzj pretab,smaxi,smini,sdiri,sbyti,sbybi,sbzti,sbzbi pretab,smaxj ,sminj ,sdirj,sbytj,sbybj ,sbztj ,sbzbj esel , s , type„10,12 etable,fxi,smisc,l etable,sdiri ,LS,l pretab,fxi,sdiri /output esel,all .! Prepare information for a timestamp *get , t ime,act ive„t ime,wal l *get ,date ,act ive„dbase, ldate year=nint(date/l 0000-.5) month=date-year* 10000 month=nint(month/100-.5) day=date-year* 10000-month* 100 hour=nint(time-.5) minute=60*(time-hour) minute=nint(minute-.5) second=60*(time-hour)-minute second=60 * second 299 APPENDIX C: ANSYS INPUT FILES /eof Curvedplate.dat APDL file: create ! Panasonic Mega Wheel A L Cladding ! Curved Plate Finite Element Model ! FEM Generation Script ! Created by Devan Fitch July 2005 /title, M E G A W H E E L 05 / P R E P 7 SECTION 0: Definition of basic parameters PI = 3.14159265 ! Input section for geometric properties of the plate pb = 2000 ! Curved edge plate dimension p L = 20000 ! Straight edge plate dimension Theta = 2.0 ! Plate curvature (included angle) Thick = 3 ! Plate thickness in sixteenths of an inch ! Further plate geometry values calculated from input values rad=pb/Theta ! Radius of curvature of the plate X X = sin(Theta/2)*rad*2 ! Projected straight dimension of curved edge pth = 25.4*Thick/16 ! Platethicknessin mm SECTION I: Definition of elements & materials ! Define element type et,15,she!163 ! Shell element 300 APPENDIX C: ANSYS INPUT FILES mp,ex, 1,200 mp,ey, 1,200 mp,ez, 1,200 mp,dens,l,7.85E-12 mp,nuxy, 1,0.3 mp,nuyz, 1,0.3 mp.nuxz, 1,0.3 mp,gxy, 1,79.29 mp,alpx,l ,11.7E-6 Young's modulus in kN/mm2 Young's modulus in k.N/mm2 Young's modulus in kN/mm2 density in kN-sec A 2 /mm A 4 (w/ a=98l0 mm/sec A 2) Poisson's ratio Poisson's ratio Poisson's ratio shear ratio in kN/min2 thermal expansion coeff. / deg. C ! Define material properties of aluminum mp,ex,2,70 mp,ey,2,70 mp,ez,2,70 mp,dens,2,2.710E-I mp,nuxy,2,0.34 mp,nuyz,2,0.34 mp,nuxz,2,0.34 mp,gxy ,2,26.96 mp,gyz,2,26.96 mp,gxz,2,26.96 mp,alpx,2,23.6E-6 Young's modulus in kN/inin2 Young's modulus in kN/inni2 Young's modulus in kN/mm2 density in kN-secA2/mmA4 (w/ a=9810 mm/secA2) Poisson's ratio Poisson's ratio Poisson's ratio shear ratio in k'N/mm2 shear ratio in klSI/mm2 shear ratio in kN/mm2 thermal expansion coeff. / deg. C Def. of Real Data for each element type r, 11, pth csys,0 /eof ! Plate thickness ! Cartesian co-ordinate system modgeom S E C T I O N 2: Model geometry / P R E P 7 ! Define key-points at four plate corners k, 11,0,0,0 k ,12 ,XX,0 ,0 k , 1 3 , X X , p L , 0 k,14,0,pL,0 ! Define keypoints below midpoint of each curved edge ! for defining curved lines later 301 APPENDIX C: ANSYS INPUT FILES k,15 ,XX/2 ,0 , -2 k , 1 6 , X X / 2 , p L , - 2 numstr,line,100 ! Define lines from keypoints L , l 1,14 L,12,13 L A R C , l l , 1 2 , 1 5 , r a d LARC,14 ,13 ,16 , rad ! Left straight edge ! Right straight edge ! Front curved edge ! Back curved edge numstr,area,100 ! Define plate area AL,100,101,102,103 ! Define material, real data and element type type, 15 mat,2 real, 11 ! Mesh the plate area asel,s,area„100 ! Plate area esize, 100 ! Element size in mm amesh,all ! Assign a name to the plate area asel , s ,area„100 cm,PlateArea,area SECTION 3: Boundary conditions and loading ! Define normal pressure 4.2 kN/m2 applied to plate cmsel,s,PlateArea sfa,all,l,pres,4.2e-6 allsel,all i Simply supported along straight edges /eof loadbc 302 APPENDIX C: ANSYS INPUT FILES nsel,all nsel,s,loc,z,0 d ,a l l ,ux ,0„„uy,uz allsel,all /eof post S E C T I O N 4: Post processing /format,,, 14,4,5000,239 /page„ ,9999 ,240 /output,rst,rst prrsol ese l , s , type„20,29 etable,fxi,srnisc, 1 etable,fyi,smisc,2 etable,fzi,smisc,3 etable,mxi,smisc,4 etable,myi,smisc,5 etable,mzi,smisc,6 etable,fxj,smisc,7 etable,fyj,smisc,8 etable,fzj,smisc,9 etable,mxj,smisc,10 etabie,myj,smisc,l 1 etable,mzj,smisc, 12 etable,sdiri ,LS,l etable,sbyti,LS,2 etable,sbybi,LS,3 etable,sbzti,LS,4 etable,sbzbi,LS,5 etable,sdirj,LS,6 etable,sbytj,LS,7 etable,sbybj,LS,8 etable,sbztj,LS,9 etable,sbzbj,LS,10 etable,smaxi,nmisc, 1 etable,smini,nmisc,2 etable,smaxj,nmisc,3 etable,sminj,nmisc,4 etable,strni ,LEPEL, 1 etable,strnj,LEPEL,6 etable,strnpre,LEPEL,l 1 303 APPENDIX C: ANSYS INPUT FILES pretab,fxi,tyi,fzi,mxi,myi,mzi pretab,frj,fyj,fzj,mxj,myj,mzj pretab,smaxi,srnini,sdiri,sbyti,sbybi,sbzti,sbzbi pretab,smaxj,sminj,sdirj,sbytj,sbybj,sbztj,sbzbj esel , s , type„10,12 etable,fxi,smisc, 1 etable,sdiri ,LS,l pretab,fxi,sdiri /output esel,all ! Prepare information for a time-stamp *get , t ime,act ive„t ime,wal l *get ,date ,act ive„dbase, ldate year=nint(date/l 0000-.5) month=date-year* 10000 month=nint(month/l 00-.5) day=date-year* 10000-month* 100 hour=nint(time-.5) minute=60*(time-hour) minute=nint(minute-.5) second=60*(time-hour)-minute second=60*second /eof 304 APPENDIX C: ANSYS INPUT FILES C 2 Longitudinally stiffened plates The file longstiff.mac is used to analyze singly curved plates, with angle stiffeners along the un-curved edges. These stiffeners are simply supported along a set width at a number of discrete points. The file longstiffopt.mac is used to perform design optimization, this file references the longstiff.mac file. Longstiff.mac APDL file: /title, M E G A W H E E L 05 / P R E P 7 Panasonic Mega Wheel A L Cladding Longitudinally Stiffened Curved Plate Finite Element Model F E M Generation Script Created by Devan Fitch July 2007 SECTION 0: Definition of basic parameters PI = 3.14159265 Geometric properties of the plate pb = 723.71 pL = 2171.4 theta= 0.1 pth = 2.5751 support = 6 swidth= 100 Plate curved edge length Plate straight edge length Plate curvature (included angle) Plate thickness in mm number of discrete support points per longitudinal stiffener Support width at each support point (SS) ! From plate geometry, calculate a mesh size such that the shortest plate ! dimension has at least 20 elements, therefore 400 elements minimum for panel dumspll=pb/20 dumspl2=pL/20 dummy=0 *If,dumspll,gt,dumspl2,THEN dumspl=dumspl2 *ELSE 305 APPENDIX C: ANSYS INPUT FILES dumspl=dumspl 1 * E N D I F ! Calculate remainig plate geometric items based on input values plarea=pb*pL ! Plate area rad=pb/theta ! Radius of curvature o f the plate X X = sin(Theta/2)*rad*2 ! Projected straight length of curved edge plvol=plarea*pth ! Plate volume ! Geometric properties o f the longitudinal stiffeners (assumed to be angles) lonth = 4.35 ! Thickness of the angle stiffeners lonw = 26.22 ! Width o f the angle stiffeners ! Calculate remaining stiffener geometric items based on input values lonvol==lonth*lonw*2*pL*2 !Total stiffener volume Atheta=lonw/rad !Stiffener included angle ! Calculate projected straight x and y dimensions of curved stiffener leg AXX=2*rad*SIN(Atheta/2)*COS(Theta/2-Atheta/2) AYY=2*rad*SIN(Atheta/2)*SIN(Theta/2-Atheta/2) ! Calculate stiffener slenderness ratio (according to C A N / C S A S16.1) lslen=((300**0.5)/200)*(lonw/lonth) ! S E C T I O N I: Definition of elements & materials ! Define element type et,15,shell63 mpDensl =7.85E-12 mp,ex, 1,200 mp,ey, 1,200 mp,ez, 1,200 mp,dens,l , mpDensl mp,nuxy, 1,0.3 mp,nuyz,l ,0.3 mp,nuxz, 1,0.3 mp,gxy, 1,79.29 mp,alpx,l ,11.7E-6 ! Young's modulus in k N mm2 ! Young's modulus in kN/mm2 ! Young's modulus in kN/mm2 ! density in kN-sec A 2 /mm A 4 (w/ a=9810 mm/sec A2) ! Poisson's ratio ! Poisson's ratio ! Poisson's ratio ! shear ratio in kN/mm2 ! thermal expansion coeff / deg. C ! Define material properties o f aluminum 306 APPENDIX C: ANSYS INPUT FILES mpDens2 = 2.710E-12 mp,ex,2,70 mp,ey,2,70 mp,ez,2,70 mp,dens,2,mpDens2 mp,nuxy,2,0.34 mp,nuyz,2,0.34 mp,nuxz,2,0.34 mp,gxy,2,26.96 mp,gyz,2,26.96 mp,gxz,2,26.96 mp,alpx,2,23.6E-6 Young's modulus in kN/mm2 Young's modulus in kN/mm2 Young's modulus in kN/mm2 density in kN-sec A 2 /mm A 4 (w/ a=98I0 mm/secA2) Poisson's ratio Poisson's ratio Poisson's ratio shear ratio in kN/mm2 shear ratio in kN/mm2 shear ratio in kN/mm2 thermal expansion coeff. / deg. C Def. of Real Data for each element type r , l l , p t h r,12,Ionth csys,0 ! Plate ! Longitudinal stiffener !• Cartesian co-ordinates SECTION 2: Model geometry-Plate keypoints k, 11,0,0,0 k ,12 ,XX,0 ,0 k , 1 3 , X X , p L , 0 k,14,0,pL,0 k ,15 ,XX/2 ,0 , -2 k , 1 6 , X X / 2 , p L , - 2 ! Plate corners ! Keypoint below the midpoint of each curved edge ! Stiffener keypoints k , 1 7 , A X X , 0 , A y y k , 1 8 , A X X , p L , A y y k , 1 9 , X X - A X X , 0 , A y y k , 2 0 , X X - A X X , p L , A y y k , 2 1 , A X X , 0 , A y y k , 2 2 , A X X , p L , A y y k , 2 3 , X X - A X X , 0 , A y y k , 2 4 , X X - A X X , p L , A y y k,31,0,0,-lonw k,32,0,pL,-lonw k ,33 ,XX,0 , - lonw k , 3 4 , X X , p L , - l o n w ! Left longitudinal stiffener keypoints ! Right longitudinal stiffener keypoints ! Additional set of keypoints for left longitudinal stiffener ! Additional set of keypoints for right longitudinal stiffener ! Left longitudinal stiffener leg keypoints ! Right longitudinal stiffener leg keypoints 307 APPENDIX C: ANSYS INPUT FILES ! Create plate lines from keypoints numstr,line,100 L , l l , 1 4 L,12,13 L A R C , l l , 1 7 , 1 5 , r a d LARC,17,19 ,15 , rad LARC,19 ,12 ,15 , rad LARC,14,18 ,16 , rad LARC,18 ,20 ,16 , rad LARC,20 ,13 ,16 , rad L,17,18 L , 19,20 Left straight edge Right straight edge Front edge arc of plate above left longitudinal stiffener Front edge arc of plate between longitudinal stiffeners Front edge arc of plate above right longitudinal stiffener Back edge arc of plate above left longitudinal stiffener Back edge arc of plate between longitudinal stiffeners Back edge arc of plate above right longitudinal stiffener Left longitudinal division of plate (stiffener edge) Right longitudinal division of plate (stiffener edge) ! Create plate areas from lines numstr.area, 100 A L , 102,108,100,105 AL,104,109,101,107 AL,103,109,108,106 ! Define material, real data and element type type,15 mat,2 real, 11 ! Assign area names and mesh areas appropriately asel,s,area„ 100,101 ! Plate area bonded to longitudinal stiffeners cm,platebondArea,area esize,dumspl amesh,all asel ,s ,area„ 102,102 cm,MidplateArea,area esize,dumspl*2 amesh,all ase l , s ,area„100,102 cm,PlateArea,area Plate area between longitudinal stiffeners ! Total.plate area ! Create stiffener lines from keypoints numstr,line,200 L A R C , 11,21,15,rad ! Front edge arc of left longitudinal sti ffener L A R C , 14,22,16,rad ! Front edge arc of right longitudinal stiffener L A R C , 12,23,15,rad ! Back edge arc of left longitudinal stiffener 308 APPENDIX C: ANSYS INPUT FILES L A R C , 13,24,16,rad L,21,22 L,23,24 ! Back edge arc o f right longitudinal stiffener ! Right edge of left longitudinal stiffener ! Left edge of right longitudinal stiffener numstr,line,300 L , l l , 3 1 L , 14,32 L,31,32 L,12,33 L , 13,34 L,33,34 ! create stiffener areas from keypoints numstr,area,200 A L , 100,200,201,204 AL,101,202,203,205 numstr,area,300 A L , 100,300,301,302 A L , 101,303,304,305 ! Define material, real data and element type type, 15 mat, l real, 12 ! Assign area names and mesh areas appropriately asel , s ,area„200,201 ! Longitudinal stiffener area bonded to plate cm,AnglebondArea,area esize,dumspl amesh,all ase l , s ,area„300,301 ! Longitudinal stiffener leg area cm,AngleLegArea,area esize,dumspl amesh,all The plate and longitudinal stiffeners are both modeled with shell elements, these shell elements occupy the same space in the bond areas, the connectivity between nodes must be defined to ensure the plate and angles act monolothically, for this purpose it is assumed that the bond stiffness is the same as the material sliffnessj.e. no slip between plate and longitudinal stiffeners ! Select nodes associated with plate area above left longitudinal stiffener 309 APPENDIX C: ANSYS INPUT FILES asel ,s ,area„ 100 N S L A , s , l NSEL,u, loc ,x ,0 .0 ,0 .0 ! Determine maximum and minimum node numbers and the number of nodes in the ! selected set * G E T , D m a x , N O D E , 0 , N U M , M A X * G E T , D U M m i n , N O D E , 0 , N U M , M l N * G E T , D U M c o u n t , N O D E , 0 , C O L T N T ! Dimension arrays appropriately * D I M , X N , A r r a y , D U M c o u n t * D I M , Y N , A r r a y , D U M c o u n t * D I M , Z N , A r r a y , D U M c o u n t * D I M , N O D , A r r a y , D U M c o u n t * D I M , N O D 2 , A r r a y , D U M c o u n t ! Perform a loop to populate the arrays with the x,y,z co-ordinates and node ! numbers D U M n o w = D U M m i n p=l * G E T , x n o w , N O D E , D U M n o w , l o c , X * G E T , y n o w , N O D E , D U M n o w , l o c , Y * G E T , z n o w , N O D E , D U M n o w , l o c , z * S E T , X N ( l ) , x n o w * S E T , Y N ( l ) , y n o w * S E T , Z N ( l ) , z n o w * S E T , N O D ( l ) , D U M n o w ind=100 * D O , p , 2 , D U M c o u n t , l D U M t h e n = D U M n o w * G E T , D U M n o w , N O D E , D U M t h e n , N X T H ! Get the next highest node number in set * G E T , x n o w , N O D E , D U M n o w , l o c , X * G E T , y n o w , N O D E , D U M n o w , l o c , Y * G E T , z n o w , N O D E , D U M n o w , l o c , z *SET,XN(p) ,xnow *SET,YN(p) ,ynow *SET,ZN(p) ,znow * S E T , N O D ( p ) , D U M n o w * E N D D O ! Select nodes associated with left longitudinal stiffener area below plate asel ,s ,area„200 N S L A , s , l NSEL,u, loc ,x ,0 .0 ,0 .0 ! Get the node number of each stiffener area node with the same co-ordinates ! as the equivalent node in the plate area * D O , p , l , D U M c o u n t , l N O D 2 ( p ) = N O D E ( X N ( p ) , Y N ( p ) , Z N ( p ) ) 310 APPENDIX C: ANSYS INPUT FILES * E N D D O asel.none nsel,none ! Select nodes associated with plate area above right longitudinal stiffener asel ,s ,area„101 N S L A , s , l NSEL,u , loc ,z ,0 ,0 ! Determine maximum and minimum node numbers and the number of nodes in the ! selected set * G E T , D U M m a x 2 , N O D E , 0 , N U M , M A X * G E T , D U M m i n 2 , N O D E , 0 , N U M , M I N * G E T , D U M c o u n t 2 , N O D E , 0 , C O L T N T ! Dimension arrays appropriately * D I M , X N 2 , A r r a y , D U M c o u n t 2 * D l M , Y N 2 , A r r a y , D U M c o u n t 2 * D I M , Z N 2 , Ar ray ,DUMcoun t2 * D I M , N O D 3 , A r r a y , D U M c o u n t 2 * D l M , N O D 4 , A r r a y , D U M c o u n t 2 ! Perform a loop to populate the arrays with the x.y.z co-ordinates and node ! numbers D U M n o w 2 = D U M m i n 2 j = l * G E T , x n o w 2 , N O D E , D U M n o w 2 , l o c , X * G E T , y n o w 2 , N O D E , D U M n o w 2 , l o c , Y * G E T , z n o w 2 , N O D E , D U M n o w 2 , l o c , z * S E T , X N 2 ( l ) , x n o w 2 * S E T , Y N 2 ( l ) , y n o w 2 * S E T , Z N 2 ( l ) , z n o w 2 * S E T , N O D 3 ( 1 ) , D U M n o w 2 * D O , j , 2 , D U M C O U N T 2 , l D U M t h e n 2 = D U M n o w 2 * G E T , D U M n o w 2 , N O D E , D U M t h e n 2 , N X T H * G E T , x n o w 2 , N O D E , D U M n o w 2 , l o c , X * G E T , y n o w 2 , N O D E , D U M n o w 2 , l o c , Y * G E T , z n o w 2 , N O D E , D U M n o w 2 , l o c , z *SET,XN2( j ) ,xnow2 *SET,YN2( j ) ,ynow2 *SET,ZN20) , znow2 * S E T , N O D 3 G ) , D U M n o w 2 * E N D D O 311 APPENDIX C: ANSYS INPUT FILES ! Select nodes associated with right longitudinal stiffener area below plate asel ,s ,area„201 nsla,s,l NSEL,u , loc ,z ,0 ,0 ! Get the node number o f each stiffener area node with the same co-ordinates ! as the equivalent node in the plate area * D O j , l , D U M c o u n t 2 , l NOD4( j )=NODE(XN20) ,YN2( j ) ,ZN2( j ) ) * E N D D O allsel,all nsel,all ! Couple the degrees o f freedom in the bond area o f the left longitudinal ! stiffener * D O , p , l , D U M c o u n t , l C P „ a l l , N O D ( p ) , N O D 2 ( p ) * E N D D O ! Couple the degrees o f freedom in the bond area of the right longitudinal ! stiffener * D O , p , l , D U M c o u n t 2 , l C P „ a l l , N O D 3 ( p ) , N O D 4 ( p ) • E N D D O S E C T I O N 3: Boundary conditions and loading ! Apply normal pressure of 4.2 k'N/m.2 to the plate cmsel,s, Plate Area sfa,all,2,pres,4.2e-6 allsel.all ! Apply simply supported boundary condition to nodes along bottom edge o f ! longitudinal stiffener legs within the support length swidth ! Either end of each stiffener nsel,all nsel,s,loc,z,-lonw nsel,r,loc,y,0,swidth d,all ,ux,0„„uy>uz nsel,all 312 APPENDIX C: ANSYS INPUT FILES nsel,s,loc,z,-lonw nsel,r , loc,y,pL,pL-swidth d ,a l l ,ux ,0„„uy,uz ! A l l intermediate support points (between the two stiffener ends) delta=pL/(support-1) dumdelta 1 =delta-swidth dumdelta2=dumdelta 1 +swidth *DO,i , l , suppor t -2 , l nsel,all nsel,s,loc,z,-lonw nsel,r,loc,y,dumdeltal ,dumdelta2 d ,a l l ,ux ,0„„uy,uz dumdeltal=dumdeltal+delta dumdelta2=dumdelta2+delta * E N D D O allsel,all SECTION 4: Analysis and solution ! Perform linear static analysis as a pre-requisite for eigenvalue buckling ! and to determine Von Mises stress /solu A N T Y P E , s t a t i c P S T R E S , o n solve F I N I S H /postl N S O R T , S , E Q V * G E T , m a x s e q v , S O R T , 0 , M A X ! Calculate efficiency function of total weight divided by plate area weightA=(plvol*mpDens2+ lonvol*mpDensl)/plarea finish ! Perform eigen value buckling analysis /solu A N T Y P E , b u c k l e B U C O P T , S U B S P , 4 M X P A N D S U B O P T , „ , 1 0 0 0 313 APPENDIX C: ANSYS INPUT FILES S O L V E * G E T , m o d e f , M O D E , 1 , F R E Q F I N I S H :here A penalty in terms of efficiency and stress can be applied here during optimisation runs for any models that are automatically generated which are not feasible but satisfy the design variable limits *IF,dummy,eq, 1 , T H E N weightA= 1000000000000 maxseqv=l *EndIF F I N I S H Longstiffopt.mac APDL file: /opt Panasonic Mega Wheel A L Cladding Longitudinally Stiffened Curved Plate Optimization File Created by Devan Fitch July 2007 SECTION 0: Definition of optimisation variables and model generation file ! Identify the model generation file OPANLj longs t i f fmac ! Identify' design variables defined in the model generation file and ! input desired value range OPVAR,pb,DV,400,2500 OPVAR,pL,DV,400,8500 OPVAR,pth ,DV, l ,3 OPVAR,lonth,DV,2,25 OPVAR,lonw,DV,25,250 ! Identify state variables defined in the model generation file and ! input allowable limits OPVAR,maxseqv ,SV,0 .0 ,0 .150 APPENDIX C: ANSYS INPUT FILES O P V A R , m o d e f , S V , 1.3,2.1 O P V A R , l s l e n , S V , 0 . 0 , 1 . 0 ! Identify the optimisation function O P V A R , w e i g h t A , O B J SECTION 1: Specify optimisation type and controls !Save the best-set results and database file O P K E E P , O N ! Perforin random iterations in design space O P T Y P E , R A N D O P R A N D , 4 0 0 , 2 0 O P E X E ! Keep all feasible design sets O P S E L , - l ! Perform iterations using subproblem approximation method O P T Y P E , S U B P OPSUBP,100 ,15 O P E X E ! Perform a sweep of the design variables from the best set O P T Y P E , S W E E P O P S W E E P , B E S T , 1 0 F I N I S H 315 APPENDIX C: ANSYS INPUT FILES C3 Radially stiffened plates: The file radstiff.mac is used to analyze singly curved plates, with angle stiffeners along the un-curved edges and two or more radial stiffeners. The longitudinal stiffeners are simply supported along a set width at a number of discrete points. The file stopt.mac is used to perform design optimization, this file references the radstiff.mac file. Radstiff.mac APDL file: / P R E P 7 /title, M E G A W H E E L 05 Panasonic Mega Wheel A L Cladding Finite Element Model F E M Generation Script Created bv Devan Fitch August 2005 SECTION 0: Definition of basic parameters PI = 3.14159265 ! Plate geometry input Theta = 0.1 ! Plate curvature pb = 2918.3 ! Plate curved edge length p L = 1395.6 ! Plate straight edge length Thick2 = 2.0 ! Plate thickness in 16ths of an inch swid th=100 ! Calculate further geometric items from input Thick=NINT(Thick2) ! Plate thickness is rounded to nearest sixteenth of an inch plarea = pb*pL ! Plate area pth = 25.4*Thick/16 ! Plate thickness in mm plvol = plarea*pth rad=pb/Theta ! Radius of curvature of the plate X X = sin(Theta/2)*rad*2 ! Projected straight length of curved edge ! From plate geometry, calculate a mesh size such that the shortest plate 316 APPENDIX C: ANSYS INPUT FILES ! dimension has at least 20 elements, therefore 400 elements minimum for panel dumspll=pb/20 dumspl2=pL/20 dummy=0 *If ,dumspll ,gt ,dumspl2,THEN dumspl=dumspl2 * E L S E dumspl=dumspll * E N D I F ! Stiffener geometry input noradst2 = 6.0 noradst=NLNT(noradst2) radth = 5.0725 lonth = 8.6663 radw = 35.931 lonw = 32.975 ! Number of radial stiffeners ! The number of radial stiffeners is rounded to nearest integer ! Radial stiffener thickness (flat bar) ! Longitudinal stiffener thickness (angle) ! Radial stiffener leg length (angle stiffener) ! Longitudinal stiffener leg length (angle stiffener) ! Calculate further stiffener geometric items from input radvol = radth*radw*2*noradst*(pb-2*lonw) lonvol = lonth*lonw*2*2*pL Atheta = lonw/rad A X X = 2*rad*SIN(Atheta/2)*COS(Theta/2-Atheta/2) A Y Y = 2*rad*SlN(Atheta/2)*SIN(Theta/2-Atheta/2) ! Calculate total width o f longitudinal stiffeners and radial stiffeners rstiffw=noradst*radw lstiffw=2*lonw ! Check i f stiffener width exceeds plate dimensions, i f they do, then model ! generation and analysis is not performed, as the system w i l l 'crash' *!F,rstiffw,gt,pL-100,Then dummy=l *GO,:here *ELSEIF,lstiffw,gt,pb-100,Then dummy=l *GO,:here * E N D 1 F ! From plate geometry, calculate a mesh size such that is one fifth of the ! shortest plate dimension dumsstl=pb/5 dumsst2=pL/5 317 APPENDIX C: ANSYS INPUT FILES *If,dumsstl ,gt ,dumsst2,THEN dumsst=dumsst2 * E L S E dumsst=dumsstl * E N D I F ! SECTION I: Definition of elements & materials i I = = = = = = = = — = = = = ^ = — = = = = = = = = = — = = = ! Define element type et,15,shell63 ! Define material properties of steel mpDensl=7.85E-12 Es=200 mp,ex, l ,Es mp,ey, l ,Es mp,ez, l ,Es mp,dens, l ,mpDensl mp,nuxy, 1,0.3 mp,nuyz, 1,0.3 mp,nuxz,l ,0.3 mp,gxy, 1,79.29 mp ,a lpx , l , l 1.7E-6 ! Young's modulus in kN/mm2 ! Young's modulus in kN/mm2 ! Young's modulus in kN/mm2 ! density in kN-sec A 2 /mm A 4 (w/ a=9810 mm/sec A 2) ! Poisson's ratio ! Poisson's ratio ! Poisson's ratio ! shear ratio in kN/mm2 ! thermal expansion coeff. / deg. C ! Define material properties of aluminum mpDens2=2.710E-12 Ea = 70 mp,ex,2,Ea mp,ey,2,Ea mp,ez,2,Ea mp,dens,2,mpDens2 mp,nuxy ,2,0.34 mp,nuyz,2,0.34 mp,nuxz,2,0.34 mp,gxy,2,26.96 mp,gyz,2,26.96 mp,gxz,2,26.96 mp,alpx,2,23.6E-6 Young's modulus in kN/mm2 Young's modulus in kN/nim2 Young's modulus in kN/nim2 density in kN-secA2/mmA4 (w/ a=9810 mm/secA2) Poisson's ratio Poisson's ratio Poisson's ratio shear ratio in kN/mm2 shear ratio in kN/mm2 shear ratio in kN/mm2 thermal expansion coeff. / deg! C Def. of Real Data • ! The radial stiffeners and leg of the longitudinal stiffener (angle) bonded to ! the plate are smeared into the plate, creating a locally thicker plate the ! additional thickness is factored to account for the difference in Young's ! Modulus ! Smear radial stiffeners 318 APPENDIX C: ANSYS INPUT FILES Iradvleg = radth*(radw**3)/12 Iradhleg = (radw-radth)*(radth**3)/12 Ipl = radw*(pth**3)/12 Idummy 1 = Iradvleg+Iradhleg ypl = pth/2+radw yradvleg = radw/2 yradhleg = radw-radth/2 A p l e f f = radw*pth Aradvleg = radw*radth Aradhleg = (radw-radth)*radth Atotal = Apleff+Aradvleg+Aradhleg A y p l = yp l*Ap le f f Ayradvleg =yradvleg*Aradvleg Ayradhleg = yradhleg*Aradhleg Aytotal = Aypl+Ayradvleg+Ayradhleg Ybar = Aytotal/Atotal Idummy2 = Aradvleg *(yradvleg-ybar)**2 Idummy3 = Aradhleg *(yradhleg-ybar)**2 Idummy4 = Apleff*(ypl-ybar)**2 El to tduml = Es*(Idummyl+Idummy2+Idummy3) + Ea*(Idummy4+Ipl) radstth = (12*EItotduml/(Ea*radw))**(l/3) Ilonhleg = (lonw-lonth)*(lonth**3)/12 Ipl2 = lonw*(pth**3)/12 ypl2 = pth/2+lonth ylonhleg = lonth/2 Apleff2 = lonw*pth Alonhleg = (lonw-lonth)*lonth Atotal2 = Apleff2+Alonhleg A y p l 2 = ypl2*Apleff2 Aylonhleg = ylonhleg*Alonhleg Aytotal2 = Aypl2+Aylonhleg Ybar2 = Aytotal2/Atotal2 Idummy5 = Alonhleg *(ylonhleg-ybar2)**2 Idummy6 = Apleff2*(ypl2-ybar2)**2 EItotdum2 = Es*(Idummy5+Ilonhleg) + Ea*(Idummy6+Ipl2) lonstth = (12*EItotdum2/(Ea*lonw))**(l/3) 319 APPENDIX C: ANSYS INPUT FILES ! The angle radial stiffeners are smeared into the plate to create an effective ! plate thickness, this is done by equating the bending stiffness (EI) ! The horizontal leg of each angle longitudinal stiffener is also smeared into the ! plate thickness, done by factoring the young's modulus E ! For simplicity, determine equivalent aluminum longitudinal stiffener vertical ! leg thickness, based on E ratio of steel to aluminum lonth2=(Es/Ea)*lonth r, 11 ,pth ! Normal plate thickness r,12,lonth2 ! Effective longitudinal stiffener leg thickness r,13,lonstth ! Effective plate thickness in region bonded to longitudinal stiffener r,14,radstth ! Effective plate thickness in region bonded to radial stiffener csys,0 ! Cartesion co-ordinate system j=l/(noradst-l) SECTION 2 : Model geometry ! Define keypoints ! Bottom edge of left longitudinal stringer k,101,0,0,-lonw k,102,0,pL,-lonw ! Left edge of plate k,201,0,0,0 k,202,0,pL,0 ! Right edge of left longitudinal stiffener', left edge of radial stiffeners k,301 , A X X , 0 , A Y Y ! First rib - front curved plate edge k , 3 0 2 , A X X , r a d w , A Y Y ! Iterative loop to determine keypoint locations for left side of intermediate ! radial stiffeners stgap = (pL-(noradst*radw))*j dumloop1=301 ydum 1 = radw+stgap 320 APPENDIX C: ANSYS INPUT FILES *DO,integ, 1 ,noradst-2,1 dumloop 1 =dumloop 1 +2 k,dumloop 1, A X X , y d u m 1, A Y Y y d u m l = y d u m l + radw k,dumloop 1 +1 , A X X , y d u m 1, A Y Y y d u m l = y d u m l + stgap * E N D D O dumlastl = dumloop 1+2 k , d u m l a s t l , A X X , p L - r a d w , A Y Y ! Last rib - back curved plate edge k,dumlast 1+1, A X X , p L , A Y Y ! Left edge of right longitudinal stiffener, right edge of radial stiffeners k,401 , X X - A X X , 0 , A y y ! First rib k , 4 0 2 , X X - A X X , r a d w , A y y ! Iterative loop to determine keypoint locations for right side of intermediate ! radial stiffeners dumloop2=401 ydum2 = radw+stgap *DO,integ,l,noradst-2,l dumloop2=dumloop2+2 k , d u m l o o p 2 , X X - A X X , y d u m 2 , A y y ydum2=ydum2+radw k,dumloop2+1 , X X - A X X , y d u m 2 , A y y ydum2=ydum2+stgap * E N D D O dumlast2 = dumloop2+2 k , d u m l a s t 2 , X X - A X X , p L - r a d w , A y y ! Last rib k,dumlast2+1 , X X - A X X , p L , A y y ! Right edge of plate k ,501 ,XX,0 ,0 k , 5 0 2 , X X , p L , 0 ! Bottom edge of right longitudinal stiffener k ,601 ,XX,0 , - lonw k , 6 0 2 , X X , p L , - l o n w ! Dummy keypoints for defining curvature of plate and radial stiffeners k, 1 , A X X / 2 , 0 , - 2 ! Front edge k ,2 ,XX/2 ,0 , -2 k , 3 , X X - A X X / 2 , 0 , - 2 321 APPENDIX C: ANSYS INPUT FILES k,4,XX/2,radw,-2 ! inner edge of first radial stiffener ! Iterative loop for intermediate radial stiffeners dumloop3=3 ydum3 = radw + stgap *DO,integ, 1 ,noradst-2,1 dumloop3=dumloop3+2 k,dumloop3 ,XX/2 ,ydum3 ,-2 ydum3 =ydum3 +radw k,dumloop3+l ,XX/2 ,ydum3,-2 ydum3 =ydum3+stgap * E N D D O dumlast3 = dumloop3+2 k,dumlast3,XX/2,pL-radw,-2 ! inner edge of last radial stiffener k ,dumlas t3+ l ,AXX/2 ,pL , -2 ! Back edge k,dumlast3+2,XX/2,pL,-2 k ,dumlas t3+3 ,XX-AXX/2 ,pL , -2 ! Create lines from keypoints ! Left stiffener leg numstr,line,100 L,101,201 L,201,202 L,202,102 L,102,101 ! Right edge of left longitudinal stiffmer numstr,line,300 * DO.integ, 1,2 *noradst-1,1 duml=300+integ dum2=301+integ L,duml ,dum2 * E N D D O ! Left edge of right longitudinal stiffener numstr,line,400 *DO,integ, l ,2*noradst- l , l duml=400+integ dum2=401+integ L,duml ,dum2 * E N D D O 322 APPENDIX C: ANSYS INPUT FILES ! Right stiffener leg numstr,line,200 L,501,601 L,601,602 L,602,502 L,502,501 ! Radial stiffener curved edges and front and back curved edge of plate numstr,line,500 LARC,201,301,1 , rad ! Front edge LARC,301,401,2 , rad LARC,401,501,3 , rad dum 1 =3 +2 *(noradst-1)+1 dum3 =301+2* (noradst-1)+1 dum4=401 +2 * (noradst-1)+1 LARC,202 ,dum3,duml , rad ! Back edge L A R C , d u m 3 ,dum4,dum 1 +1 ,rad LARC,dum4,502,duml+2, rad *Do,integ, 1,2 *(noradst-1), 1 duml=3+integ dum3=301+integ dum4=401+integ LARC,dum3,dum4,duml , rad * E N D D O numstr,line,600 allsel.all ! Create areas by draging groups of lines along paths numstr,area,300 numstr,line,3000 l s e l , s , l o c , x , A X X lsel , r , l ine„300,399 A D R A G , a l l „ „ „ 5 0 0 numstr,area,400 numstr,line,4000 l s e l , s , l o c , x , X X - A X X lsel , r , l ine„400,499 A D R A G , a l I , „ „ „ 5 0 2 allsel,all numstr,area,500 323 APPENDIX C: ANSYS INPUT FILES AL,300,501,400,506 *DO,integ, 1 ,noradst-2,1 dum3=300+integ*2 dum4=400+integ*2 dum5a=506+integ*2 dum5b=505+integ*2 AL,dum3,dum5b,dum4,dum5a * E N D D O dum3=298+noradst*2 dum4=398+noradst*2 dum5=505+2*(noradst-1) AL,dum3,dum5,dum4,504 numstr,area,600 * D O , integ, 1 ,noradst-1,1 dum3=299+integ*2 dum4=399+integ*2 dum5a=504+integ*2 dum5b=505+integ*2 AL,dum3,dum5b,dum4,dum5a * E N D D O numstr,area, 100 lsel ,s , l ine„3000,3999 lsel,r,loc,x,0 lsel,r,loc,z,0 A D R A G , a I l „ „ „ 1 0 0 numstr,area,200 lse l , s , l ine„4000,4999 l se l , r , loc ,x ,XX lsel,r,loc,z,0 A D R A G , a l l „ „ „ 2 0 0 ! Assign area names and mesh areas appropriately ASEL,s , a rea„100 ,199 ASEL,a , a rea„200 ,299 cm, LonstlegArea,area ASEL,s , a rea„300 ,399 ASEL,a , a r ea„400 ,499 cm, Lonstplate Area,area ASEL,s , a rea„500 ,599 cm,RadstplateArea,area ASEL,s , a rea„600 ,699 cm,NostplateArea,area 324 APPENDIX C: ANSYS INPUT FILES Asel , s ,a rea„400,499 Ase l ,a ,a rea„500 ,599 Asel ,a ,a rea„600,699 A R E V E R S E , a l l cmsel,s,LonstplateArea cmsel,a,RadstplateArea cmsel,a,NostplateArea cm,PlateArea,area allsel,all type, 15 mat,2 real, 11 esize,dumspl cmsel,s,NostplateArea amesh,all !esize,dumsst mat,2 real, 12 cmsel,s,LonstlegArea amesh,all mat,2 real, 13 cmsel,s, Lonstplate Area amesh,all mat,2 real, 14 cmsel,s,RadstplateArea amesh,all S E C T I O N 3: Boundary conditions and loading ! Apply normal pressure of 4,2 kN/m2 to the plate cmsel,s,PlateArea sfa,all,2,pres,4.2e-6 !kN/mm2 allsel,all ! App ly simply supported boundary condition to nodes along bottom edge o f ! longitudinal stiffener legs within the support length swidth nsel,all nsel,s,loc,z,-lonw nsel,r,loc,y,0,swidth 325 APPENDIX C: ANSYS INPUT FILES d,a l l ,ux ,0„„uy,uz nsel,all nsel,s,loc,z,-lonw nsel,r , loc,y,pL,pL-swidth d ,a l l ,ux ,0„„uy,uz allseLall i S E C T I O N 4: Analysis and solution i ! Perform linear static analysis as a pre-requisite for eigenvalue buckling ! and to determine Von Mises stress /solu A N T Y P E , s t a t i c P S T R E S , o n !IRLF,-1 solve F I N I S H /postl N S O R T , S , E Q V * G E T , m a x s e q v , S O R T , 0 , M A X ! * G E T , m a s s x , E L E M , 0 , M T O T , X ! Calculate efficiency function of total weight divided by plate area weightA=(plvol*mpDens2+(radvol+lonvol)*mpDensl)/plarea finish ! Perform eigen value buckling analysis /solu A N T Y P E , b u c k l e BUCOPT,SUBSP,4 M X P A N D S U B O P T „ „ 1 0 0 0 S O L V E * G E T , m o d e f , M O D E , 1 , F R E Q F I N I S H ! A penalty in terms of efficiency and stress can be applied here during ! optimisation runs for any models that are automatically generated which are ! not feasible (wi l l crash the analysis) but satisfy the design variable limits ! the analysis and model generation are skipped - see section 0 326 APPENDIX C: ANSYS INPUT FILES :here * IF ,dummy,eq , l ,THEN weight= 1000000000000 maxseqv=T *End lF FINISH Stoptmac APDL file: /opt Panasonic Mega Wheel A L Cladding Longitudinally and Radially Stiffened Curved Plate Optimization File Created by Devan Fitch July 2007 SECTION 0: Definition of optimisation variables and model generation file ! Identify the model generation file OPANL,radst i f f ,mac ! Identify design variables defined in the model generation file and ! input desired value range O P V A R , p b , D V , 1200,6000 O P V A R , p L , D V , 1200,6000 ! O P V A R , T h i c k 2 , D V , 1,3,1 !OPVAR,noradst2 ,DV,2,10,1 O P V A R , r a d t h , D V , 5 , 2 5 O P V A R , l o n t h , D V , 5 , 2 5 O P V A R , r a d w , D V , 2 5 , 2 5 0 O P V A R , l o n w , D V , 2 5 , 2 5 0 ! Identify state variables defined in the model generation file and i input allowable limits OPVAR,maxseqv ,SV,0 .0 ,0 .150 O P V A R , m o d e f , S V , 1.3,2.1 ! Identify the optimisation function O P V A R , w e i g h t A , O B J SECTION I: Specify optimisation type and controls 327 APPENDIX C: ANSYS INPUT FILES •Save the best-set results and database file O P K E E P , O N ! Perform random iterations in design space O P T Y P E , R A N D O P R A N D , 2 0 0 0 , 5 0 O P E X E ! Keep all feasible design sets O P S E L , - l ! Perform iterations using subproblem approximation method O P T Y P E , S U B P OPSUBP,200 ,20 O P E X E ! Perform a sweep of the design variables from the best set O P T Y P E , S W E E P O P S W E E P , B E S T , 1 0 F I N I S H 328 

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