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Stress Analysis of Doors and Windowsof BOEING 787 under Uniform Shear at Infinity Rajesh Kumar 08310031 M.Tech. (Design) Guide: Prof. V.G. Ukadgaonker Department of Mechanical Engineering Indian Institute of Technology, Bombay May, 2010
OutlineIntroductionBoeing-787Forces and Their EffectProblem DefinitionLiterature ReviewComplex Variable MethodSchwarz Alternating TechniqueMapping of Door and WindowMathematical FormulationFinite Element AnalysisResultsConclusions and Future ProspectsReferences
Introduction• Aircraft design • Optimum material utilization • High fatigue strength with minimum weight• Non-uniform stress distribution in components • Irregularities • Intrinsic defect /Flaws • Functional features like door, window, hole for fasteners, keyways etc. • Manufacturing defect• Non-uniform stress distribution causes localization of stress in the vicinity of any discontinuity (Stress Concentration)• Stress analysis is a tool to know stresses and its direction at various points• Major failures occurs due to crack initiation at points of maximum stress concentration (Critical points)• Stress Concentration Factor
Boeing-787A mid-sized, wide-body jet airlinercurrently under development by BoeingCommercial AirplanesComposite materials to constructfuselage - 15% Al, 50% composites and12% titanium Ref Allows high cabin pressure during flightOpenings – passenger door, emergencydoor, cargo door and windowsMain passenger door and the Windownearest to this door - DimensionsMaterial Properties : E1=139.3 Gpa, *All Dims in inchesE2=11.3 Gpa, G12=6 Gpa,ν21=0.3,ν23=0.4
Forces and Their EffectDuring the steady flight, main forces acting onaircraft fuselage are 1. Body forces - Differential internal pressure- Hoop and longitudinal stresses (Biaxial tensile state) 2. Engine thrust and wing drag - Engine thrust acts in forward direction, wind drag acts in the opposite direction of the motion of the aircraft – Longitudinal bending moment (out of plane load) 3. Due to manoeuvering of aircraft - Differential pressure acts on the wings while taking turn in air+inertia of the aircraft - torsional forces - Shear stresses in the aircraft skin
Problem Definition To obtain stress concentration factor around the rectangular door and window of the Boeing-787 aircraft subjected to uniform shear at infinity. Also, to obtain the stress concentration factor around the door due to the interaction effect of the presence of a nearby window and vice versa.As the radius ofcurvature of fuselage islarge compared to thedimensions of the doorsand window, the fuselageis modelled as an infiniteplate with single andmultiple openings.
Literature ReviewSingle Hole Problem• Krisch and Muskhelishvili --- the problem of infinite plate with single circular hole subjected to uniaxial stress at infinity• Krisch --- Airys stress function, Muskhelishvili --- complex variable method• Muskhelishvili --- various boundary value problem --- complex variable method and conformal mapping technique• Lekhnitskii --- the problem of anisotropic plates --- both in-plane and out of plane loading --- stress functions by series method• Savin --- isotropic and anisotropic plates --- conformal mapping --- circular, triangular, rectangular and elliptical single hole• Ukadgaonker and Awasare --- principle of superposition and Muskhelishvili’s complex variable approach --- solution for infinite plate containing, circular, elliptical, triangular, rectangular holes --- elliptical hole in anisotropic medium• Ukadgaonker and Rao --- solution for stress field around various hole geometries in an anisotropic medium --- subjected to biaxial and shear stress at infinity, uniform internal pressure at hole boundary, uniform shear stress at hole boundary in detail
Literature Review (continued…)Two Hole Problem• Ukadgoanker and Avarigarimath --- infinite plate having two unequal collinear elliptical holes subjected to uniaxial tension and uniform shear --- complex variable approach as well as FEM• Ukadgaonker and Koranne --- infinite plate containing two unequal arbitrary oriented elliptical holes and cracks subjected to uniaxial tensile and shear loading --- complex variable approach, method of photoelasticity, FEM• Ukadgaonker and Awasare --- interaction effect of rectangular and arbitrarily oriented elliptical hole in infinite plate subjected to uniform tensile loading at infinity• Ukadgaonker and Sharma --- infinite plate containing two unequal arbitrarily oriented circular holes --- biaxial tensile, uniform shear, biaxial moment and torsion --- complex variable approach and FEM
Literature Review (continued…)Door and Windows of Passenger Aircraft Gandhi --- door and windows of Boeing-747 --- analytical formulation for a single rectangular hole for tensile loading --- problem of multiple opening done by FEM Upadhyay, Sharma --- door and windows of Boeing-777 aircraft --- stress functions for single rectangular hole under tensile load and bending moment (Upadhyay) and under biaxial bending (Sharma) Shrivastava --- door and windows of Boeing-777 aircraft with FEM --- effect on stress field of one hole due to the presence of another hole in its vicinity using ANSYS Vasnik --- door and windows of Boeing-777 with crack --- stress intensity factor were obtained using FEM as well as complex variable approachGaps Identified in Literature• Very few analytical solutions are available considering rectangular hole in an infinite plate of anisotropic material.• The interaction effect of two rectangular holes has not been yet studied using Schwarz’s alternating method.
Schwarz’s Alternating Technique• The problem of multiply connected regions is solved as simply connected region and successively relaxing the boundary conditions on the holes.• First complex solution in terms of stress functions is obtained for plate without hole by mapping the physical Z-plane into ζ-plane.• Boundary condition at the fictitious circular hole is determined using these stress functions.• The second approximate solution is obtained by the application of the negative value of the boundary condition on the circular boundary.• Addition of these two solutions gives the solution valid near the circular hole. Solution of single hole problem
Mapping of Door and WindowConformal Mapping• A conformal map is a function which preserves angles.• Any conformal mapping of a complex variable which has continuouspartial derivatives is analytic. An analytic function is conformal at anypoint where it has a nonzero derivative.• Conformal mapping helps in transforming very complicated shapes intomuch simpler ones.•It allow the basic complex variable formulations to extend to thetransformed problem.• Generalized form of mapping function for Door and Window
Mapping of Door and Window (continued…)• Mapping Constants ▫ Door m1 m3 m5 m7 R -0.2570 -0.1555 0.0240 0.0111 34.3980 ▫ Window m1 m3 m5 R -0.2460 -0.1565 .0231 8.6500• Door and window generated by using Matlab Window In mapped plane Door
Complex Variable ApproachGeneralised Hooke’s law for plane stress Stresses in terms of Airy’s stress functionCompatibility equation for Biharmonic equation as2D- elasticity problem Its roots are, Hence, Introducing the stress functions φ(z1), ψ(z2) and their conjugate
Complex Variable Approach (continued…)Stresses in terms of stress functions are We can obtain the solution using the following steps • First stage solution • Second stage solution • First Approximation • Second Approximation
Mathematical FormulationBoundary ConditionsStress Function of Single Hole Problem under Remote LoadingFirst Stage – Stress functions for hole free plateSecond Stage – Plate having single rectangular hole where
Mathematical Formulation (continued…)From Schwarz’s technique, whereFinal Solution – Obtained by superposition of the stress functions of the firstand the second stageThese stress functions give the stresses around rectangular hole.
(continued…)First ApproximationStress functions for the doorStress functions for the windowThese Stress functions do not consider the interaction effect of door and window.
(continued…)Second Approximation (Window) In order to account for the interaction effect of door on the stressfunctions of the window, the stress functions of the door is transformed tothe centre of the window by translation through a distance C0, given by Z0 =ω(C0 ) such that |C0|>1. ζThe boundary conditions for anisotropic plate is given byCorrected stress functions around the window can be given by, ,
(continued…)Using Cauchy’s integral formulae,where, a =
(continued…)We get the corrected stress functions asBy superposition of transformed and corrected stress functions we getthe stress function for window considering the interaction effect of doorSecond Approximation (Door)This givesUsing these stress functions we can find the stresses around door andwindow with interaction effect.
Finite Element AnalysisA numerical technique to find approximate solution of PDEANSYS – A software to solve structural, static, transient, etc. problemsAnisotropic thin infinite-plate with plain stress conditionE1=139.3 GPa, E2=11.3 GPa, G12=6 GPa, ν21=0.3, ν23=0.4Steps involved are- Preprocessing, Solution, Post processingPLANE82 eight nodes having two translational degrees of freedom at each node more accurate results for mixed quadrilateral and triangular elements well suited to model curved boundaries and have compatible displacement shapes has large deflection, large strain capabilities and Ref: ANSYS Help plasticity
Models Plate: Length=1000 in., Width= 1000 in. Door: Length= 42 in., Width= 74 in., Corner Radius= 7 in. Window: Length= 10.74 in., Width= 18.44 in., Corner Radius= 5 in. Distance between door and window= 58.95 in.
ResultsType of opening Max. Stress Concentration Factor (SCF) Difference(%) Analytical Without Interaction With InteractionPassenger door 3.44 3.44 00 Window 2.27 2.24 1.3Type of opening Max. Stress Concentration Factor (SCF) Difference(%) Numerical Without Interaction With InteractionPassenger door 3.39 3.40 0.3 Window 2.16 2.27 4.8
Conclusions• Higher stress concentrations occur near the corner locations.• The SCF depends on the side ratio and corner radius.• Less is the side ratio higher is stress concentration factor.• Due to interaction, there is negligible change in stress field around door but the stress field around window gets affected significantly.• Door has higher maximum SCF compared to window both with and without interaction effect.• Analytical and numerical results are in good agreement.
Future Prospects The variation of SCF for other geometries and with differentparameters like length, width and thickness can be analyzed. The problem has been solved for the case of shear loading. Theother loadings can be considered for the analysis like in-plane and outof plane bending loads. The curvature of aircraft fuselage can be taken into considerationto solve a problem of three dimensional curved plate subjected todifferent loads.
References1. Boeing official website: www.boeing.com.2. Muskhelishvili, N.I., Some Basic Problems of Mathematical Theory of Elasticity, P. Noordhoff Ltd., Groningen, The Netherlands, 1963.3. Lekhnitskii, S.G, Anisotropic Plates, Gordon and Breach Science Publishers, New York 1968.4. Savin, G.N., Stress Concentration around Holes, Pergamom Press New York, 1961.5. Ukadgaonker,V.G, Awasare, P.J, A Novel Method of Stress Analysis of Infinite Plate with Elliptical Hole with Uniform Tensile Stress, Journal of the Institution of Engineers (India), MC, 73, 1993 pp.309-311.6. Ukadgaonker,V.G, Awasare, P.J, A Novel Method of Stress Analysis of Infinite Plate with Circular Hole with Uniform Loading at Infinity, Indian Journal of Technology, 31, 1993, pp.539-541.7. Ukadgaonker,V.G, Awasare, P.J, A Novel Method of Stress Analysis of Infinite Plate with Small Radius Equilateral Triangular hole with Uniform Tensile Stress, Journal of the Institution of Engineers(India), MC, 73, 1993, pp.312-317.8. Ukadgaonker,V.G, Awasare, P.J, A Novel Method of Stress Analysis of Infinite Plate with Rounded Corners of a Rectangular Hole under Uniform edge Loading, Indian Journal of Engineering and Material Sciences (India), 1994, pp.17-25.9. Rao, D.K.N., Some General Solutions for Stresses around Holes in Anisotropic Plates, Ph.D. thesis, IIT Bombay, 2000.10. Ukadgaonker, V.G., A Novel Method of Stress Analysis of Infinite Plate with rounded corners of a rectangular Hole, Indian Journal Technology, 26 (1988) 549-559.
(continued…)11. Ukadgaonker, V.G. and Avarigarimath, R.R., Stress Analysis Of An infinite Plate Containing Two Unequal Elliptical Holes under In-Plane Stresses at Infinity, Presented at 12th Canadian Congress of Applied Mechanics, Carleton University, Ottawa, Canada, May-June 1989.12. Ukadgaonker, V.G., Stress Analysis Of A Plate With Two Unequal Circular Holes Subjected To Tangential Stresses, AIAA Journal, pp. 125-128, January 1980.13. Ukadgaonker, V.G. and Koranne, S.D., Interaction Effect On Stresses In An Infinite Plate With Two Unequal Arbitrary Oriented Elliptical Holes Or Cracks, Proceedings Of International Conference On Advances In Structural Testing, Analysis And Design, Bangalore, pp 996-1001, Aug. 1990.14. Ukadgaonker, V. G. and Awasare, P. J., Interaction effect of rectangular hole and arbitrarily oriented elliptical hole or crack in infinite plate subjected to uniform tensile loading at infinity, Indian Journal of Engineering & Material Sciences, Vol.6, pp.125-134, June 1999.15. Sharma, D.S, “Stress analysis of cracks emanating from two unequal circular holes in an anisotropic plate”, Ph. D. Thesis, IIT. Bombay, 2008.16. Gandhi, B.S., Stress Analysis of Stiffened Doors and Windows of Boeing-747, M.Tech. Dissertation 2000.17. Upadhyay, A., Stress Analysis of Boeing-777 Aircraft with Reinforced Doors and Windows, M.Tech. Dissertation 2005.18. Shrivastava, D., Stress Analysis of Boeing-777 Aircraft Using FEM, M.Tech. Dissertation 2005.19. Sharma, V., Stresses near the Door and Windows of a Passenger Aircraft Subjected to Biaxial Bending with FEM, M.Tech. Dissertation 2005.20. Vasnik, T., Stress Analysis of Boeing-777 Aircraft with crack at the Door and Window, M.Tech. Dessertation 2005.21. Huo, H., Bobet, A., Fernandez, A., Ramirez, J., Analytical Solution for Deep Rectangular Structures Subjected to Far-field Stress, Elsevier, pp. 613 -625, 2005.