@prefix vivo: .
@prefix edm: .
@prefix ns0: .
@prefix dcterms: .
@prefix skos: .
vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Civil Engineering, Department of"@en ;
edm:dataProvider "DSpace"@en ;
ns0:degreeCampus "UBCV"@en ;
dcterms:creator "Fitch, Devan Carless"@en ;
dcterms:issued "2011-03-15T20:03:43Z"@en, "2006"@en ;
vivo:relatedDegree "Master of Applied Science - MASc"@en ;
ns0:degreeGrantor "University of British Columbia"@en ;
dcterms:description """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, if 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 of 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 will 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 decisions and changes can be made more readily and with fewer consequences in terms of
budget and schedule over-runs."""@en ;
edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/32467?expand=metadata"@en ;
skos:note "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 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 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 ( 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\",-> 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: Anad 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 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 "@en ;
edm:hasType "Thesis/Dissertation"@en ;
edm:isShownAt "10.14288/1.0063262"@en ;
dcterms:language "eng"@en ;
ns0:degreeDiscipline "Civil Engineering"@en ;
edm:provider "Vancouver : University of British Columbia Library"@en ;
dcterms:publisher "University of British Columbia"@en ;
dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ;
ns0:scholarLevel "Graduate"@en ;
dcterms:title "A case study in the conceptual design of an anodized aluminium cladding system for a steel torus structure"@en ;
dcterms:type "Text"@en ;
ns0:identifierURI "http://hdl.handle.net/2429/32467"@en .
**