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- 1. Wing and Fuselage Structural Optimization Considering Alternative Material Systems By Jonathan Lusk B.S.A.E., University of Kansas, 2006Submitted to the Department of Aerospace Engineering and the Faculty of theGraduate School of the University of Kansas in partial fulfillment of the requirementsfor the degree of Masters of Engineering. ______________________________ Professor in Charge-Dr. Mark Ewing ______________________________ Dr. Richard Hale ______________________________ Dr. Stan Rolfe ___________________ Date Project Accepted
- 2. The Thesis Committee for Jonathan Lusk Certifies that this is the approved Version of the following thesis:Wing and Fuselage Optimization Considering Alternative Material Systems Committee: ______________________________ Professor in Charge-Dr. Mark Ewing ______________________________ Dr. Richard Hale ______________________________ Dr. Stan Rolfe ___________________ Date defended i
- 3. Abstract This study is a sensitivity analysis to compare weight benefits for a transportaircraft airframe from potential mechanical property enhancements of CFRP (CarbonFiber Reinforced Plastic) Laminate and Aluminum Alloy. The computationalframework is based on a simplified skin-stringer-frame/rib configuration to model thefuselage and the wings of a generic narrow and wide body jet transport. Simple(Strength of Materials) mechanics were used to predict the stresses in the skin andstringers. Strength allowables and panel buckling equations are used in conjunctionwith an iterative optimizer to calculate the structural airframe weight. The baselinematerials include 7075-T6 Aluminum Alloy and a fictitious intermediate moduluscarbon epoxy. For the CFRP material, the optimized weight results show Open Holecompression enhancement produces the most weight benefit. The Fatigue strength isthe most sensitive material property for the baseline Aluminum Alloy structure. Theresults also indicate that the current CFRP laminate minimum gauge limits weightreduction from potential material property enhancements especially on the small jettransport. ii
- 4. Acknowledgements First of all I would like to thank my academic advisor Dr. Mark Ewing for hisguidance, assistance, advice, and support throughout my project. I would also like tothank Dr. Mark Ewing for teaching me the fundamentals of structural optimizationand introducing me to computational resources to conduct my analysis. I’d like tothank Dr. Abdel Abusafieh and Cytec Engineering Materials for providing fundingfor my research and also providing me their time, information, and resources for meto carryout my analysis. I would also like to acknowledge Dr. Abdel Abusafieh fortaking the time to review my documentation and results and discussing as well asenlightening me on the interest and concerns of the industry. I’d like to thank Dr. Richard Hale for teaching me the mechanics and designof composite materials and making me aware of details that should be considered inthis type of analysis. I’d like to thank Dr. Stan Rolfe for teaching me the factors thatpertain to a complete fatigue analysis and for kindly participating in my examinationboard committee. iii
- 5. Table of ContentsAbstract........................................................................................................................ iiAcknowledgements...................................................................................................... iiiList of Figure ............................................................................................................. viiList of Tables ............................................................................................................. xiiNomenclature ........................................................................................................... xiv1 Introduction......................................................................................................... 1 1.1 Objective ....................................................................................................... 1 1.2 Prior Work .................................................................................................... 22 Methodology ........................................................................................................ 4 2.1 Geometry....................................................................................................... 4 2.1.1 Wing ..................................................................................................... 4 2.1.2 Fuselage.............................................................................................. 10 2.2 Stiffened Panel Geometry ........................................................................... 11 2.2.1 Wing ................................................................................................... 12 2.2.2 Fuselage.............................................................................................. 15 2.3 Loads........................................................................................................... 18 2.3.1 Wing ................................................................................................... 18 2.3.2 Fuselage.............................................................................................. 27 2.4 Stress Analysis ............................................................................................ 33 2.4.1 Thin Walled Idealized Beam Theory (Ref 2).................................. 33 2.4.2 Hoop and Longitudinal Stress (Ref 5)............................................. 35 iv
- 6. 2.4.3 Buckling Analysis.............................................................................. 36 2.5 Materials ..................................................................................................... 51 2.5.1 CFRP.................................................................................................. 51 2.5.2 CFRP Lamination Scheme of Aircraft Structure .......................... 52 2.5.3 Aluminum .......................................................................................... 54 2.5.4 Material Enhancement Analysis Cases........................................... 55 2.6 Failure Modes and Criteria ......................................................................... 57 2.6.1 CFRP.................................................................................................. 57 2.6.2 Aluminum Alloy ................................................................................ 60 2.7 Optimization ............................................................................................... 61 2.7.1 Optimized Variables ......................................................................... 61 2.7.2 Fixed Variables.................................................................................. 61 2.7.3 Concept of Minimum Gauge............................................................ 63 2.7.4 Discretization..................................................................................... 63 2.7.5 Optimization...................................................................................... 703 Verification ......................................................................................................... 78 3.1 Regression Analysis.................................................................................... 78 3.1.1 Wing ................................................................................................... 80 3.1.2 Fuselage.............................................................................................. 83 3.2 Finite Element............................................................................................. 86 3.2.1 Wing ................................................................................................... 86 3.2.2 Fuselage.............................................................................................. 89 v
- 7. 4 Results and Discussion...................................................................................... 925 Conclusions...................................................................................................... 1096 Recommendations ........................................................................................... 110References................................................................................................................ 112A Appendix........................................................................................................... A.1B Appendix............................................................................................................B.1C Appendix............................................................................................................C.1D Appendix........................................................................................................... D.1 vi
- 8. List of FigureFigure 2.1: Wing Plan form with Kick ......................................................................... 4Figure 2.2: Wing Chord Distribution............................................................................ 6Figure 2.3 Wingbox and Wing Planform...................................................................... 7Figure 2.4: Illustration of Wingbox Thickness Taper (this in not a configurationconcept illustration and there is no rib geometry displayed in this figure)................... 8Figure 2.5: Fuselage Geometry................................................................................... 10Figure 2.6: "I" beam Section Stiffened Panel ............................................................. 12Figure 2.7: Geometry of an "I"-Beam Section Stiffener............................................. 13Figure 2.8: "Hat"-Section Stiffened Panel .................................................................. 15Figure 2.9:Geometry of a Hat Stiffener ...................................................................... 16Figure 2.10: Generic Load Profile for 2.5g Positive Maneuver ................................. 18Figure 2.11: Generic Load Profile for Negative 1.0g Gust......................................... 19Figure 2.12: Trapezoidal Lift Distribution (1g Steady Level Flight) ......................... 20Figure 2.13: Elliptical Lift Distribution (1g Steady Level Flight).............................. 20Figure 2.14: Plot of Trapezoid and Elliptic Chord Distribution ................................ 21Figure 2.15: Lift Distribution...................................................................................... 21Figure 2.16: Wing Structural Weight Distribution ..................................................... 23Figure 2.17: Fuel in Wing Weight Distribution.......................................................... 24Figure 2.18: Distributed Load Summary Profile of Positive 2.5g maneuver (does notinclude engine point load)........................................................................................... 25 vii
- 9. Figure 2.19: Shear and Moment Diagram (Positive 2.5g maneuver) ......................... 26Figure 2.20: Correction for Wing Sweep Illustration ................................................. 27Figure 2.21: Fuselage Structure Loading Modeled as Cantilever Beam .................... 29Figure 2.22: Shear and Moment Distribution for Fuselage Load Case 1 ( Positive 2.5gManeuver) ................................................................................................................... 31Figure 2.23: Shear and Moment Distribution for Fuselage Load Case 2 (Negative 2.0gHard Landing)............................................................................................................. 32Figure 2.24: Stiffened Panels on the Wing Box Skin ................................................. 37Figure 2.25: Principal Stress Components of Pure Shear Buckling (Ref 2)............... 39Figure 2.26: Bending Buckling Illustration (Ref 7).................................................... 40Figure 2.27: Compression Shear Buckling Interaction (Ref 7) .................................. 42Figure 2.28: One Edge Free Crippling (logarithmic scale) (Ref 4) ........................... 45Figure 2.29: No Edge Free Crippling (logarithmic scale) (Ref 4).............................. 45Figure 2.30: Crippling Curves .................................................................................... 46Figure 2.31: Skin Buckled Stiffened Panel and Effective Width Illustration............. 49Figure 2.32: Lumping Areas on the Wing Box........................................................... 65Figure 2.33:Generic Axial Stress and Skin Thickness Relationship .......................... 73Figure 2.34: Generic Buckling Stability and Axial Stress vs Skin ThicknessRelationship ................................................................................................................ 75Figure 2.35: Optimization Flow Chart (example for fuselage)................................... 77Figure 3.1: Linear Regression Correlation between PDCYL and Calculated Results 81 viii
- 10. Figure 3.2: :Linear Regression Correlation between Actual Load Bearing Weight andCalculated Results....................................................................................................... 81Figure 3.3: Linear Regression Correlation between Actual Primary Weight andCalculated Results....................................................................................................... 82Figure 3.4: : Linear Regression Correlation between Actual Total Structural Weightand Calculated Results................................................................................................ 82Figure 3.5: Linear Regression Correlation between PDCYL and Calculated ModelResults......................................................................................................................... 83Figure 3.6: Linear Regression Correlation between Actual Load Bearing Weight andCalculated Model Results ........................................................................................... 84Figure 3.7: Linear Regression Correlation between Actual Primary Weight andCalculated Model Results ........................................................................................... 84Figure 3.8: Linear Regression Correlation between Actual Total Structural Weightand Calculated Model Results .................................................................................... 85Figure 3.9: Wing Finite Element Geometry ............................................................... 86Figure 3.10: Axial Stress in Skin ................................................................................ 87Figure 3.11: Top Skin Element ID Numbers .............................................................. 87Figure 3.12: Axial and Shear Stresses in Elements 4213 and 4214 (“F06 file”)........ 88Figure 3.13: Axial and Shear Stress from Model ....................................................... 88Figure 3.14: Axial Stress............................................................................................. 89Figure 3.15: Axial Stress from Model ........................................................................ 90Figure 3.16: Shear Stress ............................................................................................ 91 ix
- 11. Figure 3.17: Shear Stress Calculated in Model........................................................... 91Figure 4.1: Percent Load Bearing Structural Weight Benifit of CFRP over Aluminumfor a Range of Potential Fatigue Performance........................................................... 96Figure 4.2: Skin Thickness Design Stress Relationship ............................................. 98Figure 4.3: Failure Mode Distribution on Wing Structure for Wide and Narrow body(CFRP Baseline and all cases except for narrow body Improved Composite 1)........ 99Figure 4.4: Failure Mode Distribution for Narrow Body Wing (CFRP ImprovedComposite 1)............................................................................................................. 100Figure 4.5: Failure Mode Distribution on Wide Body Fuselage (CFRP) ................. 101Figure 4.6: Failure Mode Distribution on Narrow Body Fuselage (CFRP) ............. 102Figure 4.7: Failure Mode Distribution for Wide Body and Narrow Body Wing(Aluminum)............................................................................................................... 105Figure 4.8: Failure Mode Distribution for Wide Body Fuselage (Aluminum)......... 107Figure 4.9: : Failure Mode Distribution for Narrow Body Fuselage (Aluminum) ... 108Figure A.1: Medium Body Jet Transport.................................................................. A.1Figure A.2: V-N Diagram Specifications for Military Airplanes (Ref 6)................. A.2Figure B.1:WB Wing Running Loads .......................................................................B.5Figure B.2: NB Wing Running Loads .......................................................................B.5Figure B.3: Crippling of CFRP Laminate "I"-Section Beam With Ex=14.5Msi.......B.6Figure B.4: Euler Column Buckling of CFRP Laminate Ex=14.5Msi .....................B.7Figure B.5: Radius of Gyration for Euler Buckling Calculation with Contribution ofEffective Width..........................................................................................................B.8 x
- 12. Figure C.1: 7075 Aluminum Alloy Mechanical Properties (Ref 4)...........................C.1Figure C.2: "K" Values for Compression and Shear Panel Buckling (Ref 7) ...........C.2Figure C.3: Stiffness Correction ................................................................................C.3Figure D.1: Illustration of Fuselage Modeled as Idealized Tube.............................. D.1Figure D.2: Optimized Thicknesses for Fuselage Section 46................................... D.1Figure D.3: Lateral Cut of Fuselage as a Idealized Tube ......................................... D.3Figure D.4: Center of Mass (in)................................................................................ D.4Figure D.5: Second Moment of Area (in4)............................................................... D.5Figure D.6: Internal Loads (shear(lbs), moment(lbs*in))......................................... D.6Figure D.7: Maximum and Minimum Principle Stress in Crown (psi) .................... D.7Figure D.8: Minimum Principle Stress in Belly (psi) ............................................... D.8Figure D.9: Max In-plane Shear Stress (psi) .......................................................... D.10Figure D.10: Idealized Stringer Section.................................................................. D.11Figure D.11: Crippling Stress of Belly Stringer ..................................................... D.13Figure D.12: Geometry of Hat Stiffener ................................................................. D.13Figure D.13: Hat Stiffener Including Effective Width ........................................... D.16Figure D.14: Belly Stringer Buckling Strength ...................................................... D.17Figure D.15: Output for WB CFRP Baseline ......................................................... D.19 xi
- 13. List of TablesTable 2.1: Wing Model Geometry Values.................................................................... 9Table 2.2: Fuselage Model Geometry Values............................................................. 11Table 2.3: Wing Stiffened Panel Geometry Values (CFRP) ...................................... 12Table 2.4: Wing Striffened Panel Geometry Values (Aluminum) ............................. 12Table 2.5: Wing Stringer Geometry (CFRP) .............................................................. 14Table 2.6: Wing Stringer Geometry (Aluminum)....................................................... 14Table 2.7: Fuselage Stiffened Panel Geometry Values (CFRP) ................................. 15Table 2.8: Fuselage Stiffened Panel Geometry Values (Aluminum) ......................... 15Table 2.9:Fuselage Stringer Geometry (CFRP).......................................................... 17Table 2.10: Fuselage Stringer Geometry (Aluminum) ............................................... 17Table 2.11: Fuel Weight Breakdown .......................................................................... 24Table 2.12: CFRP Lamina Properties ......................................................................... 51Table 2.13: CFRP Laminate Properties ...................................................................... 52Table 2.14: 7075-T6 Aluminum Alloy Properties...................................................... 55Table 3.1: Wing Regression Analysis Data ................................................................ 80Table 3.2: Fuselage Regression Analysis Data........................................................... 83Table 4.1: CFRP Baseline and Enhanced Material Weights (% Structural WeightReduction)................................................................................................................... 93Table 4.2: Aluminum Baseline and Enhanced Material Weights (% Structural WeightReduction)................................................................................................................... 94 xii
- 14. Table 4.3: Stiffened and Un-stiffened Structural Weights.......................................... 95 xiii
- 15. Nomenclature English Symbol Description Units [A] Linear Inequality Constraint Matrix ~ [C] Non-Linear Constraint Matrix ~ [D] Laminate Flexure Stiffness Matrix in-lb [LB] Optimization Lower Bound ~ [UB] Optimization Upper Bound ~ {b} Non-linear Inequality Limit ~ a Stiffened Panel Length in. A Cross Section Area in2 ai Discrete Lumped Area in2 b Wing Span ft bc Stiffener Cap Panel Width in bf Stiffener Flange Width in bw Stiffener Web Width in c Wing Chord ft D Fuselage Diameter ft Dmn Flexural Stiffness Matrix Term in-lb E Elastic Modulus psi F Stress (Military Handbook Notation) psi xiv
- 16. f Optimization Cost Function in2g Inequality Constraint ~I Second Moment of Area in4 Distance to Wing Kick (in percent% wing halfk span) ~l Fuselage Length ftL Length of Longitudinal Stiffener in.L Distributed Lift lb/inM Internal Moment Load lb-inn Load Factor ~p Net Distributed Load lb/inq Shear Flow lb/inQ First Moment of Area in3R Critical Buckling Stress Ratio ~S Wing Planform Surface Area ft2T Gross Engine Thrust lbt Thickness inV Internal Shear Load lbW Weight lbw Thin Plate Deflection in.x Structure Axial Coordinate fty Structure Lateral Coordinate in. xv
- 17. Acronyms Description AR Wing Aspect Ratio ~ M.S. Margin of Safety ~ OHC Open Hole Compression Strength psi OHT Open Hole Tension Strength psi S.F. Safety Factor ~ W.S. Wing Station ft PDCYL NASAs Structural Weight EstimateGreek Symbol Description Δσ Change in Stress Allowable psi Δt Change in Thickness in. λ Taper Ratio ~ Λ Sweep Angle Degrees ν Poissons Ratio ~ θ Stiffener Web Inclination Angle Degrees ρ Radius of Gyration in. σ Axial or Transverse Stress psi τ Shear Stress psi xvi
- 18. Subscript Description 0 Initial Value 1 Maximum Principal Direction/Fiber Direction 2 Minimum Principal Direction/Transverse Direction Laminate Flexural Stiffness Response to In-Plane Bending 11 Strain Laminate Flexural Stiffness Response to Transverse 12 Bending Strain Laminate Transverse Flexural Response to Transverse 22 Bending Strain Laminate Torsional Flexure Response to In-Plane Torsional 66 Strain 1,2i Principal Stress of a Discrete Lumped Area 12su Lamina Ultimate Shear Strength 1cu Lamina Ultimate Compression Strength in Fiber Direction 1tu Lamina Ultimate Tensile Strength in Fiber Direction Lamina Ultimate Compression Strength Transverse to Fiber 2cu Direction Lamina Ultimate Tensile Strength Transverse to Fiber 2tu Direction All Allowable Stress b Bending Stress xvii
- 19. c Compression Stress cc Crippling Strength CL Wing Center Line cr Critical Buckling Strength cr_stiff Critical Bucklng Strength of Stiffener cu Ultimate Compression Strength Design Design Stress e Effective Length engine/pylon Engine and Pylon Weight exp Exposed Wing Area Fracture Margin of Safety/Flange Width/ Forward Fuselage f Location/Fiber Volume fi Final Thickness of Discrete Structural Area Fract_sk_Bottom Critical Fracture of Bottom Skin Fract_sk_Top Critical Fracture of Top Skin Fract_st_Bottom Critical Fracture of Bottom Stiffener Fract_st_Top Critical Fracture of Top StiffenerFracture_Allowable Fracture Strength fuselage Fuselage Cross Secitonal Area G Center of Area GTO Gross Takeoff Weight h High Relative Stress Range xviii
- 20. i Discrete Area, Thickness or Stress Notation xix
- 21. Subscript Description kick Wing Thickness/Depth at Wing Kick l Low Relative Stress Range L Longitudinal Stress Ratio LE Leading Edge Sweep/ Swept Span Limit Stress at Limit Load Linear Linear Inequality Constraint MGTO Maximum Gross Takeoff WeightNon-Linear Non-Linear Inequality Constraint OHC Open Hole Compression Strength OHT Open Hole Tension Strength oi Initial Thickness of Discrete Lumped Structure p Panel Geometry r Root Chord/Thickness r_wb Root Chord of Wing Box ref Reference s Shear Stress Ratio skin Fuselage Cross Section Skin AreaSpar_Caps Wing Cross Section Spar Caps AreaSpar_Webs Wing Cross Section Spar Web Area stiff Stress in Stiffener xx
- 22. stringers Wing/Fuselage Cross Section Stringer Area su Shear Ultimate t Tip Chord/Thickness t_wb Tip Chord of Wing Box tu Tensile Ultimate wing Wing Cross Section Area x In-Plane Axial Mechanical Properties/Stress In-Plane Axial Mechanical Properties/Stress of Discrete xi Structure xy In-Plane Shear Mechanical Properties/Stress In-Plane Shear Mechanical Properties/Stress of Discrete xyi Structure y Transverse Axial Mechanical Properties/Stress Lateral First/Second Moment of Area (Referenced Text z Notation)z_LC1 Vertical Load Factor for Load Case 1z_LC2 Vertical Load Factor for Load Case 2 Lateral First/Second Moment of Area (Referenced Text zz Notation) xxi
- 23. 1 Introduction1.1 Objective The main objective of this study is to determine if potential mechanicalproperty enhancements of CFRP material would lend themselves to the application onsmaller transport aircraft, referred to as the narrow body aircraft. This is determinedby identifying and comparing critical failure modes of CFRP and Aluminum Alloyson medium and small commercial jet transport. It is currently known that CFRPmaterials show benefits over Aluminum Alloys for current medium jet transportaircraft, referred to as the wide body aircraft. This study is a weight sensitivityanalysis only and does not take into account factors such as Acquisition and LifeCycle cost and is not intended to compare Aluminum Alloy to CFRP Laminatematerials. The general questions that this analysis addresses are:How does the weight performance benefits from the application of CFRP andAluminum Alloy on a medium transport aircraft compare with that on a smalltransport aircraft?What CFRP material enhancements offer the most weight benefit on the wideand narrow body aircraft and what are the critical failure modes? 1
- 24. What Aluminum Alloy material enhancements offer the most weight benefit onthe wide and narrow body aircraft and what are the critical failure modes? This study presents a method of estimating the wing box and fuselagegeometry from fundamental preliminary design parameters. Also presented is ananalytical method using optimization to evaluate the weight benefits of altering thematerial properties of the structure. This optimization model will be useful instudying the effects and limitations of enhancing aspect of a unidirectional materialon the wing and fuselage structure. This method determines the primary load bearingstructural mass by sizing the skin and the stringers, but not the fuselage frames orwing ribs. The only input parameters in this analysis are mechanical properties of thematerial. Therefore there is not a configuration trade study included and theconfiguration concept is fixed, though the dimensions of the structural concept areoptimized for the specific baseline material and transport aircraft size.1.2 Prior Work A parallel study was found in a NASA Technical Memorandum Titled:Analytical Fuselage and Wing Weight Estimation of Transport Aircraft. Thedocument presents an methodical procedure in defining the structure geometry, loads,and failure criteria in order to estimate the structural load, similar to the procedurepresented in this document. Unlike this study, the procedure does not include 2
- 25. compression and shear buckling interaction, sub-structure of different in-planestiffness, and did not study “I”-beam and “Hat”-section stiffened concepts. It doesinclude, unlike this study, curvature of fuselage geometry, deflection analysis, thesizing and weight of fuselage frames and wing ribs, and the study of “Z” –sectionand Sandwich Honeycomb stiffened configurations. The technical memorandum tabulates the actual structural weights of eightAluminum Alloy transport aircrafts and compares the weights with the calculatedstructural weight of their model. This study uses those published weights to verify theoptimized weight calculations for the eight transport aircraft using the optimizedmodel documented in this study. The results are presented in the verification section. 3
- 26. 2 Methodology2.1 Geometry2.1.1 Wing The wing planform geometry for a transport aircraft is estimated using FigureA.1 in Appendix A. Figure A.1 shows that the wing does not have a straight taper, buta “kick” from the edge of the fuselage to the position where the engine attaches to thewing. The wing planform with the “kick” is illustrated in Figure 2.1. The kick k ,which is given as a fraction of the half span, is located where there is a geometrytransition and the wing taper is discontinuous. D Fuselage Wall (Root) 2 Λ LEc CL cr k ct b 2 Figure 2.1: Wing Plan form with Kick 4
- 27. Figure 2.1 illustrates the geometry for the wing for the wide body jettransport. The known wing geometry includes the Aspect ratio AR , span b , andleading edge sweep Λ LE . The wing reference surface area S ref is determined from theaspect ratio and span in Equation 1. The span, reference surface area, and the chorddistribution for the wide body, presented in Figure 2.2, is used to calculate the meangeometric chord c . The formula used to calculate mean geometric chord is shown inEquation 2. This chord distribution is determined from Figure A.1 in Appendix A.Since the chord distribution is known the root chord c r and the tip chord ct areknown. Figure 2.1 shows that the leading edge has a constant sweep. The quarterchord sweep Λ c 4 is determined by Equation 3. The exposed wing reference areaS exp is equal to the reference wing area minus the referenced wing area within thefuselage diameter (or fuselage wall) shown in Figure 2.2. ( 1) b2 AR = S ref ( 2) b 2 2 c= ∫c 2 dy S ref 0 5
- 28. ( 3) 3 tan Λ c = tan Λ LE 4 4 c( y ) cCL cr ct y D k 2 b 2 Figure 2.2: Wing Chord Distribution The wing box structure extends the span of the wing and the width of the wingbox is a fraction of the chord for a specific wing station. The wing box is shownwithin the geometry of the wing planform in Figure 2.3. It is assumed for simplicityof analysis that the center of the wing box is located at the quarter chord of the wingplanform which is also assumed to be the location of the aerodynamic center andcenter of pressure. This implied that the aerodynamic resultant force is located at thecenter of the wing box and that there is no pitching moment on the wing. Theconfigurations for the wide and narrow body transport structures are both two sparconcepts with rib spacing indicated by the stiffened panel geometry length. 6
- 29. c r _ wbcr c t _ wb ct Figure 2.3 Wingbox and Wing Planform The wing-box thickness (not to be confused with skin thickness) and wing boxwith taper transition at the “kick” is illustrated in Figure 2.4. This is an importantdetail to recognize because there is a large thickness taper from the root to the “kick”and a subtle thickness taper from the “kick” to the tip, which has a large effect on therunning load profile distribution over the wing span. There is a beam that runs behindthe rear landing gear, from the reference wing planform centerline to the “kick” thisbeam is neglected in this analysis because the details needed to size this beam werenot readily available. The geometry parameters of the wing are presented in Table2.1. 7
- 30. Wingbox to fuselage Intersection Front Spar tr t kick Rear Spar ttFigure 2.4: Illustration of Wingbox Thickness Taper (this in not a configuration concept illustration and there is no rib geometry displayed in this figure) 8
- 31. Table 2.1: Wing Model Geometry ValuesParameter Parameter Name and Narrrow Body (NB) Wide Body (WB)Designation Description Units Value Value Maximum Gross Takeoff WGTO Weight lbs 123,675 453,000 b Wing Span ft 112 170 D Fuselage Diameter ft 13 19 2 AR Aspect Ratio, AR=b /Sref ~ 9.55 9.55 ΛLE Leading Edge Sweep degrees 40 40 Reference Chord Taper λref Ratio, λref=ct/cCL ~ NA NA Exposed Chord Taper λexp Ratio, λexp=ct/cr ~ 0.1875 0.1875 ⎛t⎞ Thickness to Chord Ratio ⎜ ⎟ ⎝ c ⎠r at Wing Root, (t/c)r= tr/cr ~ 0.10 0.10 Exposed Thickness Ratio, τexp τexp= tt/tr ~ 0.1883 0.1883 Wingbox to Wing Planform Chord Ratio at rr Root, rr=cr_wb/cr ~ 0.41 0.41 Wingbox to Wing Planform Chord Ratio at rt Tip, rt=ct_wb/ct ~ 0.31 0.31 Distance to Kick from Wing Root (as fraction of wing half span), kick kick=k/(b/2) ~ 0.254 0.254 ⎛t⎞ Thickness to Chord Ratio ⎜ ⎟ ⎝ c ⎠k kick, (t/c)k=tkick/ckick ~ 0.10 0.10 9
- 32. 2.1.2 Fuselage The Fuselage is modeled as an un-tapered cylinder neglecting wing andempennage attachment structure. The maximum diameter of the actual fuselagestructure is used far the diameter of the cylinder. The length of the cylinder extendsfrom the nose to the tail cone. WBD 19ft l Figure 2.5: Fuselage Geometry ( 4)l = 0.67WGTO 0.43 (Ref 2) The fuselage geometry is defined in terms of the fineness ratio l D , and themaximum gross takeoff weight. The length and diameter of the fuselage areillustrated in Figure 2.5. The Gross Takeoff Weight and fuselage slenderness ratio forthe wide and narrow body aircraft is given in Table 2.2. Equation 4 estimates thefuselage length from the gross takeoff weight. 10
- 33. Table 2.2: Fuselage Model Geometry Values Parameter Parameter Name and Narrow Body (NB) Wide Body (WB) Designation Description Units Value Value WGTO Gross Takeoff Weight lb 123,675 453,000 l D Fuselage Length to Diameter Ratio (~) 7.8 9.62.2 Stiffened Panel Geometry In order to compare the stability of the stiffened panels at the proper stress levelthe stiffener spacing and geometry is found using an iterative process presented inAppendix B. The “I”-beam section is used to stiffen the wing skin panels and the“Hat”-section stiffener is used on the fuselage. Figure 2.6 shows an “I” beamstiffened panel using a finite element modeling software. Figure 2.8 shows a solidmodel of a hat stiffened panel with the panel width b and panel length a defined. Tables 2.3 and 2.4 give the wing stiffened panel geometry values for CFRP andAluminum Alloy respectively. Tables 2.7 and 2.8 give the stiffened panel geometryfor the CFRP and Aluminum Alloy fuselage. The stiffened panel geometry iscompared for CFRP and Aluminum Alloy for the specific wide or narrow body wingor fuselage structural with the un-stiffened structural weight in the results. The b tstiffener spacing to panel thickness ratio is a function of material stiffness E , stress inthe panel σ x , Poisson’s ratio ν , and boundary conditions. The panel width to skin 11
- 34. thickness ratio b t should be similar for every like material geometry combination(i.e. The Aluminum Alloy narrow and wide body wing structure).2.2.1 Wing Figure 2.6: "I" beam Section Stiffened Panel Table 2.3: Wing Stiffened Panel Geometry Values (CFRP) Parameter Parameter Name and Narrow Body (NB) Wide Body (WB) Designation Description Units Value Value b Panel Width in 7.0 10.6 17.0 a Panel Length in 14.0 24 34.0 Table 2.4: Wing Striffened Panel Geometry Values (Aluminum) Parameter Parameter Name and Narrow Body (NB) Wide Body (WB) Designation Description Units Value Value b Panel Width in 2.3 10.6 5.7 a Panel Length in 4.6 24 11.4 12
- 35. Figure 2.7 shows the geometry of the “I”-section stiffener. They i displacements are used as inputs in determining the radius of gyration and effectivewidth in the model, there values are not tabulated in this section. Table 2.5 and 2.6give the geometry values of the CFRP and Aluminum Alloy stiffeners for the wingrespectively. The stiffener is assumed to be bonded to the wing skin. The “I” beamstiffener configuration is chosen because it is what is currently being used in industryon a medium transport aircraft with CFRP as its primary load bearing material. bc y ct bw tw yw y tc y cb ts Figure 2.7: Geometry of an "I"-Beam Section Stiffener 13
- 36. Table 2.5: Wing Stringer Geometry (CFRP) Parameter Parameter Name Narrow Body (NB) Wide Body (WB)Designation and Description Units Value Value Stiffener I-Section bw Web Width in. 1.00 2.00 Stiffener I-Section bf Flange Width in. 0.55 1.30 Table 2.6: Wing Stringer Geometry (Aluminum) Parameter Parameter Name Narrow Body (NB) Wide Body (WB)Designation and Description Units Value Value Stiffener I-Section bw Web Width in. 0.40 1.00 Stiffener I-Section bf Flange Width in. 0.25 0.65 14
- 37. 2.2.2 Fuselage a b Figure 2.8: "Hat"-Section Stiffened Panel Table 2.7: Fuselage Stiffened Panel Geometry Values (CFRP) Parameter Parameter Name and Narrow Body (NB) Wide Body (WB) Designation Description Units Value Value b Panel Width in 6.3 10.6 11.0 a Panel Length in 9.4 24 16.4 Table 2.8: Fuselage Stiffened Panel Geometry Values (Aluminum) Parameter Parameter Name and Narrow Body (NB) Wide Body (WB) Designation Description Units Value Value b Panel Width in 2.5 10.6 4.4 a Panel Length in 3.8 24 6.5 15
- 38. Figure 2.8 shows the geometry of the “Hat”-section stiffened panel. The websof the stiffener are actually at an angle like what is shown in Figure 2.9. The angle ofthe hat stiffener web is used when determining the radius of gyration of the stiffener.The flange of the stiffener, with width b f in Figure 2.9, is attached to the skin of thefuselage. All flanges are assumed to be bonded to the fuselage skin. Like the “I”-section stiffener the y i lateral locations of the legs of the stiffener are inputs in themodel to compute the radius of gyration and are functions of the stiffener dimensionsand the wing or fuselage skin thickness (i.e. f (bi , t ) ). The geometry of the “Hat”-section stiffener is given in Table 2.9 and 2.10 for the CFRP and Aluminum Alloymaterial respectively. The “Hat”-section stiffener is chosen because it is currentlybeing used in industry on a medium transport aircraft with CFRP as its primary loadbearing material. The stiffened panel geometry given in this section is a fixed input inthe model and is not allowed to vary since the scope of this study is reduced tobenefits of the material mechanical properties only. The stiffened panel geometry isoptimized prior to the sensitivity analysis. bc θ bw y yc yw yf bf *yi is lateral location of bi Figure 2.9:Geometry of a Hat Stiffener 16
- 39. Table 2.9:Fuselage Stringer Geometry (CFRP) Parameter Parameter Name and Narrow Body (NB) Wide Body (WB)Designation Description Units Value Value Stiffener Hat-Section bw Web Width in. 0.66 1.00 Stiffener Hat-Section bf Flange Width in. 0.50 0.50 Stiffener Hat-Section bc Cap Width in. 1.00 1.50 Stiffener Hat-Section θ Web Angle Degrees 43 43 Table 2.10: Fuselage Stringer Geometry (Aluminum) Parameter Parameter Name and Narrow Body (NB) Wide Body (WB)Designation Description Units Value Value Stiffener Hat-Section bw Web Width in. 0.35 0.70 Stiffener Hat-Section bf Flange Width in. 0.30 0.45 Stiffener Hat-Section bc Cap Width in. 0.70 1.40 Stiffener Hat-Section θ Web Angle Degrees 43 43 17
- 40. 2.3 Loads2.3.1 Wing The critical load cases being analyzed is a positive 2.5g maneuver and anegative 1.0g gust at limit load that is a standard specification for Jet TransportAircraft (Ref 7). Figure 2.10 and 2.11 illustrates the difference between the two loadcases. Figure A.2, in Appendix A, shows the V-N diagram for transport militaryaircraft. The load case for the medium transport military aircraft is the same for thecommercial medium transport. The medium transport loading capability is marked inFigure A.2. All load cases are symmetric about the center axis of the fuselage. Thereis no torsion, drag, or dynamic load cases applied to the wing because the detailsrequired for these load cases is out of the scope of this sensitivity analysis. b 2 ∫ b L ( x ) dx = 2 . 5 W MGT O − 2 L(x) 2.5WMGTO b Figure 2.10: Generic Load Profile for 2.5g Positive Maneuver 18
- 41. b 2 ∫ L ( x)dx = −W b MGTO − 2 b W TO Figure 2.11: Generic Load Profile for Negative 1.0g Gust2.3.1.1 Lift Profile Using Shrenk’s Approximation (Ref 3) it is required to find the trapezoidalchord distribution of the wing, shown in Figure 2.12, and the theoretical ellipticalchord distribution of the wing illustrated in Figure 2.13. According to Shrenk’sapproximation the lift distribution is proportional to the average of the Trapezoidaland the elliptic chord distribution. Figure 2.14 plots the comparison between the twodistributions. Figure 2.15 shows the resultant lift distribution of the two geometricallyaveraged. Figure 2.15 gives a linear approximation of the lift distribution which canbe used as a short hand method to assure that the distribution sums to the grosstakeoff weight; this calculation is shown in Equation 5 (453,000 lb is the grosstakeoff weight of the wide body aircraft). Half the gross takeoff weight would beexperienced by the half span of the wing in steady level 1.0g flight. 19
- 42. Lift per unit length (lb/ft) S=Area Chord(y) W GTO W 37.77 S1 167.16 2..12 GTO 1 2.12 b 88 b S2 557.43 S3 424.82 S4 363.91 S1 Swing 1513.32 W GT ) W 1.1.33 GTO 17 GTO 32 23.61 bb S3 S2 W W 0.40 GTO 0..39 GTO 0 35 GTO 7.08 b b S4 0.055b 23.61 b 0.127 b 182 00.5b y Wing Stationb Wing Station (ft) .500 75.5 ft Figure 2.12: Trapezoidal Lift Distribution (1g Steady Level Flight) Lift per unit length W W 1127 GTO . .13 GTO bb Wing Station (ft) 0.055b 0.500b Figure 2.13: Elliptical Lift Distribution (1g Steady Level Flight) 20
- 43. 40.0 35.0 30.0Chord Distribution (ft) 25.0 Trapezoid 20.0 Elliptic 15.0 10.0 5.0 0.0 0.0 20.0 40.0 60.0 80.0 Wing Station (ft) Figure 2.14: Plot of Trapezoid and Elliptic Chord Distribution Lift Distribution y = -39.513x + 4406.9 2 R = 0.9966 5000.0 4500.0 4000.0 Lift Distribution (lb/ft) 3500.0 3000.0 Lift 2500.0 Linear (Lift) 2000.0 1500.0 1000.0 500.0 0.0 0.0 20.0 40.0 60.0 80.0 W.S. (ft) Figure 2.15: Lift Distribution 21
- 44. ( 5) ⎡⎛ lb ⎞ ⎤ ⎢ ⎜ 4500 − 1500 ⎟(75 ft ) lb ⎜ ⎟ ⎥ ⎢⎝ ft ft ⎠ ⎛ lb ⎞Total _ Load = 2 + ⎜1500 ⎟(75 ft )⎥ = 450,000 ≈ 453,000lb ⎜ ⎢ 2 ⎝ ft ⎟ ⎠ ⎥ ⎢ ⎥ ⎢ ⎣ ⎥ ⎦2.3.1.2 Inertial Relief2.3.1.2.1 Wing Weight The wing total structural weight is modeled with wing trapezoidal chorddistribution, shown in Figure 2.16, which is similar to the trapezoidal approximationof the lift distribution. The weight of the wing is subtracted from the lift profile. Thisis consistent with the load case because when the wing is experiencing a positive 2.5gmaneuver the wing weight is resisting the motion with the same load factor. The wingtotal structural weight is estimated to be 12% of the Gross Takeoff Weight. The wingstructural weight distribution is a fixed function of the aircraft gross takeoff weight ofthe aircraft regardless of the material applied to the structure this is done becausethere is a large amount of the structural mass in the wing that cannot be accounted forin this sensitivity analysis. 22
- 45. 0.055b 0.182b 0.500b Wing Station (ft) W W −00042 GTO . .05 GTO b b W W − 0..14 GTO 016 GTO W wing=50% W structural W structural 108600 lb b b W wing 54300 lb W W − 0..25 GTO 0 23 GTO b Wing Weight Distribution (lb/ft) Figure 2.16: Wing Structural Weight Distribution2.3.1.2.2 Engine and Pylon Weight The engine weight load is applied at the location of the wing kick (25.4% ofthe wing half-span outboard of the wing-fuselage intersection). The weight of theengine and the pylon is applied as a discrete point load on the wing. There are twoengines, one on each wing, for both the wide and narrow body aircraft. Equation 6estimates the engine pylon weight using the gross takeoff weight and the maximumthrust to weight ratio. ( 6) 1.1 ⎛ 1 T ⎞ WEngine / Pylon = 0.064⎜ ⎜ # _ of _ Engines W WGTO ⎟ ⎟ ⎝ ⎠ (Ref 3) 23
- 46. 2.3.1.2.3 Fuel Weight It is assumed that the portion of the total fuel in the wing, specified in Table2.12, is 80%. The fuel distribution, shown in Figure 2.17, is assumed to taper linearlyto zero within the span of one wing. The fuel weight, along with the engine and wingstructural weight, is subtracted from the positive lift distribution for the resultantloading on the wing structure. Table 2.11: Fuel Weight Breakdown Fuel % Weightfuel 35%W TO %Weight fuel in wing 80%W fuel %bspan occupied by fuel 100%bspan 0.055b 0.500b Wing Station (ft) W - 0.315 WGTO −0.63 GTO b b Fuel Weight Distribution (lb/ft) Figure 2.17: Fuel in Wing Weight Distribution 24
- 47. 2.3.1.3 Load Summary The lift distribution for the positive 2.5g load case is given in Figure 2.18.The negative 1.0g load case would take the same profile but in the negative liftdirection with 40% of the magnitude. Both load cases on the wing structure are quasi-static and symmetric. 1.000 0.900 0.800 0.700 (lb/ft) 0.600 0.500 Load Distribution Location of Engine 0.400 0.300 0.200 0.100 0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 Wing Station× b (ft) Figure 2.18: Distributed Load Summary Profile of Positive 2.5g maneuver (does not include engine point load) 25
- 48. 2.3.1.4 Internal Loading Engine Point Load Distributed Load 0.3 0.25 (lb) 0.2 Shear × WGTO Shear 0.15 0.1 0.05 0 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 Wing Station 6.0 5.0 (lbs-ft) 4.0 Moment × WGTO b 3.0 2.0 1.0 0.0 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 Wing Station × b (ft) Figure 2.19: Shear and Moment Diagram (Positive 2.5g maneuver) 26
- 49. 2.3.1.5 Load Distribution Correction for Sweep The load distribution p(x ) is reduced to its component perpendicular to thequarter chord swept span, to account for the effects of a wing sweep.. This will stretchthe shear distribution over the swept span while maintaining the same shear loading atthe root. The moment at the root will increase. The calculation for quarter chordsweep is given in Equation 3. Λ LE Quarter Chord Span b c 4 Λ c 4 C ⎛ ⎞ 4 p((x ) cos⎜ Λ c ⎟ C x) ⎜ ⎟ ⎝ 4 ⎠ c 4 b Figure 2.20: Correction for Wing Sweep Illustration2.3.2 Fuselage The fuselage structure is sized using two load cases: a positive 2.5g maneuverand a negative 2.0g hard landing. Static load cases are used, for this sizing analysis, 27
- 50. which exclude external aerodynamic pressure forces from drag or bending momentsfrom thrust lines of the engines. The hard landing load case takes into account thenose down pitch rate of the fuselage. This model only takes into account symmetrical load conditions, thereforethere is no torsion induced on the fuselage structure. Two load cases are appliedindividually. The worst case scenario for the crown and belly of any fuselage sectionis taken for a given failure mode. The model does account for pressurization, butneglects the effects of pressurization on fuselage sections in compression and sectionsthat are failure critical from load case 2. This is because it is assumed that thefuselage is not fully pressurized in the negative 2.0g hard landing load case.2.3.2.1 Loading Profile The loading profile for the fuselage structure includes the fuselage structuralweight, the airplane payload weight, the front landing gear weight, and theempennage structural weight. It is assumed that the rear landing gear is mounted inthe wing fuselage attachment and therefore the load is taken by structural loadingpoints that are not sized in this model. The loading profile is illustrated in Figure 2.21. 28
- 51. Fixed at Airplane Front Landing Gear Wing Load Empennage Load x1 x2 xf xt Fuselage Structure Distributed Load Passenger/Cargo Distributed Load Figure 2.21: Fuselage Structure Loading Modeled as Cantilever Beam The fuselage is modeled as two cantilever beams facing outwards, as shown inFigure 2.21, from the wing box quarter chord point. Modeling the fuselage as twocantilever beam is convenient because modeling it as one beam makes it anindeterminate problem. Each cantilever beam has its own coordinate referencesystem. x f is the fuselage station coordinate starting at the nose of the fuselage andending at the fuselage quarter chord. xt is the coordinate originating from the tail ofthe airplane and ending at the wing box quarter chord point.2.3.2.2 Internal Loading For the loading given in Figure 2.21 the bottom of the fuselage is incompression which corresponds to a negative bending moment. With the signconvention in Equation 7 and 8, the coordinate system using x f and x t , a maximumpositive shear, and minimum negative bending moment takes place at the wing box 29
- 52. quarter chord. The internal load distribution for load case 1 and 2 are given in Figures2.22 and 2.23 respectively. ( 7) V y ( x) = ∫ − p( x)dx ( 8) ∂M x = −Vy ∂x 30
- 53. Tension Compression 41 43 44 46 47 48 541in. 276in. 336in 396in 273in 379in. WB 0.2 0.18 0.16 Shear × GTO (lb) Shear × WWGTO (lb) 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fuselage Station × b (ft) Fuselage Station ×b (ft) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 (lbs-ft) × 10 × WGTO WGTO b (lbs-ft) -1 -2 × 10 −2 × b -3 Moment −2oment -4 M -5 -6 Fuselage × b × b (ft) (ft) Fuselage Station StationFigure 2.22: Shear and Moment Distribution for Fuselage Load Case 1 ( Positive 2.5g Maneuver) 31
- 54. Tension Compression 41 43 44 46 48 47 541in. 276in. 336in 396in 273in 379in. WB 0.7 0.6 0.5 0.4 (lb) 0.3 × WGTO 0.2 Shea 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.1 -0.2 -0.3 Fuselage Station ×Station Fuselage b (ft) × b (ft) 6 4 2 Moment × 10 −2 × WGTO b (lb-ft) 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -2 -4 -6 -8 -10 -12 -14 Fuselage Station × b (ft) Fuselage Station ×b(ft)Figure 2.23: Shear and Moment Distribution for Fuselage Load Case 2 (Negative 2.0g Hard Landing) 32
- 55. The shear and moment distribution profiles are the same for the narrow andwide body, though they have different magnitudes and length of distributions. Themaneuver load case has a maximum negative moment at the intersection of the rearfuselage and the wing box. The hard landing condition has a maximum negativemoment at the rear landing gear and a maximum positive moment on the forwardfuselage at the mid point between the front and rear landing gear.2.3.2.3 PressurizationThe fuselage skin has a tensile stress from an ultimate pressurization load of 18 psi,applied to all fuselage cabin structure. The pressurization load is the same betweenthe narrow and wide body aircraft, and also the same for the analysis of the CFRP andAluminum Alloy materials.2.4 Stress Analysis2.4.1 Thin Walled Idealized Beam Theory (Ref 2)The axial stress formula is given in Equation 9. M z is the moment about the lateralaxis. y i is the discrete location of stress. 33
- 56. ( 9) − M z ( yi − yG ) σx = I ZZThe Shear Flow and Shear Stress formula is given in Equation 10 and 11. ( 10) V y Qz Δq = I zz , ( 11) qi τ xy = tiQ z and I z are the first and second moment of area respectively, and are given inEquation 12. The first and second moment of inertia using discrete areas is given inEquation 13. ( 12) Q z = ∫∫ ydA A I z = ∫∫ y 2 dA A ( 13) n Q z = ∑ Ai ( y i − y G ) i =1 n I z = ∑ Ai ( y i − y G ) 2 i =1 34
- 57. y G is the lateral neutral axis (or y-component of the mass center) and its formula isgiven in Equation 14. The discretized calculation of the y-component of the centroidis also in Equation 14. ( 14) n 1 ∑Ay i i yG = ∫∫ ydA = i =1 n ∑A A A i i =12.4.2 Hoop and Longitudinal Stress (Ref 5) The hoop stress is the circumferential stress in a pressure vessel wall. Thelongitudinal stress is parallel to the axis of symmetry in the pressure vessel wall. p isthe pressure in psi. r is the radius of the pressure vessel. t is the thickness of thepressure vessel wall. ( 15) pr σ Hoop = t pr σ Longitudinal = 2t 35
- 58. 2.4.3 Buckling Analysis2.4.3.1 Compression Buckling The buckling strength of a panel depends on thickness, width, stiffness, aspectratio, Poisson’s effects and the lay-up or bending stiffness matrix of the laminate.Figure 2.24 shows the geometry and loads on individual panels of width b and lengtha that are used to determine buckling stability. Since the buckling strength forlaminates depends on thickness and stacking sequence the flexural stiffnesscoefficients are used, referred to as the [D ] matrix. The [D ] matrix terms are used tocalculate the buckling strength of a composite laminate in Equation 16. N x is the critical running load in the laminate panel expressed in (lb/in.). m isthe number of half sine waves formed by the buckled material. The value of m ischosen that gives the minimum buckling strength for a given aspect ratio. ( 16) π2 ⎡ ⎤ 2 2 2 ⎛b⎞ N x = 2 ⎢ D11 m ⎜ ⎟ + 2 2 (D12 + 2 D66 ) + D22 ⎛ a ⎞ ⎛ 1 ⎞ ⎥ ⎜ ⎟ ⎜ ⎟ b ⎢⎣ ⎝a⎠ ⎝b⎠ ⎝m⎠ ⎥ ⎦ 36
- 59. Panel for Buckling Analysis Aspect Ratio = a/b a b a b Compression Stringers Shear ribs Figure 2.24: Stiffened Panels on the Wing Box Skin Buckling of aluminum depends on thickness, width, stiffness, poisons ratioand aspect ratio. The equations to calculate buckling strength are for isotropicmaterials. The isotropic buckling equations are used to calculate the buckling strengthof an Aluminum Alloy panel as shown in Equation 17, where E c is the ElasticModulus of the material in compression. The plot illustrating the K values for thecompression and shear buckling are in Appendix C, Figure C.2. ( 17) π 2 Ec ⎛ t ⎞ 2 2 ⎛t⎞ ⎛a⎞ σ cr = C c 2 ⎜ ⎟ = K c E c ⎜ ⎟ where K c = f ⎜ ⎟ 12(1 − ν ) ⎝ b ⎠ ⎝b⎠ ⎝b⎠ 37
- 60. 2.4.3.2 Shear Buckling The shear buckling strength formula in Equation 18 is found in Reference 8for a composite laminate. It is also a function of the flexure coefficients. Thisequation is derived assuming the long plate assumption which makes it a conservativeshear buckling strength calculation for the geometry of the panels in this structure. Kis a parameter that accounts for bend-twist coupling in the stability of a laminate. ( 18) D12 + 2 D66 K= D11 D22 D11 D22 (8.125 + 5.045K ) when K ≤ 1 4 N xy = 4 3 b2 ⎛ 1.46 ⎞ D22 (D12 + 2 D66 )⎜11.71 + 2 ⎟ when K < 1 4 N xy = 4 b2 ⎝ K ⎠ In Equation 19 the shear buckling strength τ cr , for the isotropic AluminumAlloy, is a function of axial Elastic Modulus E . Equation 19 uses the ElasticModulus rather than the shear modulus, because pure shear produces equalcompressive and tensile principal stresses on the diagonal plane with respect to theedge of the plate. The concept of pure shear as diagonal principal stresses is displayedin Figure 2.25 (Ref 2). 38
- 61. ( 19) π 2 Ec ⎛ t ⎞ 2 2 ⎛t⎞ ⎛a⎞ τ cr = Cs 2 ⎜ ⎟ = K s E c ⎜ ⎟ where K s = f ⎜ ⎟ 12(1 − ν ) ⎝ b ⎠ ⎝b⎠ ⎝b⎠ τ τ σ 45 o τ σ τ τ τ σ τ σ τ Figure 2.25: Principal Stress Components of Pure Shear Buckling (Ref 2)2.4.3.3 Bending Buckling The bending buckling strength formula, Equation 20, for a composite laminateis found in Reference 8. Like the shear buckling equation for composite laminate,Equation 20 assumes the long plate assumption. ( 20) Nx = π2 b2 [13.9 ] D11 D22 + 11.1(D12 + 2 D66 ) 39
- 62. The bending buckling allowable, used for the Aluminum Alloy, is found usingEquation 21 (Ref 7). The buckling coefficient k b is found from Figure C5.15 fromReference 7. The Elastic Modulus for Compression E = E c is used for AluminumAlloy in Equation 21. Bending buckling is illustrated, in Figure 2.26, where w is thedeflection of the web. Buckling, in Figure 2.26, is positive for “out of the page” andnegative for “in the page”. ( 21) π 2 Ec ⎛ t ⎞ 2 σ b = kb ⎜ ⎟ 12(1 − ν 2 ) ⎝ b ⎠ w=deflection 2 b 3 σb w+ w- w+ w- w+ w- w+ w- b w=0 0 0 0 a Figure 2.26: Bending Buckling Illustration (Ref 7) 40

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