Stress Analysis of Doors and WindowsStress Analysis of Doors and WindowsStress Analysis of Doors and WindowsStress Analysis of Doors and Windows

of BOEINGof BOEINGof BOEINGof BOEING����787 under Uniform Shear at Infinity787 under Uniform Shear at Infinity787 under Uniform Shear at Infinity787 under Uniform Shear at Infinity

Rajesh Kumar08310031

M.Tech. (Design)

Guide: Prof. V.G. UkadgaonkerGuide: Prof. V.G. Ukadgaonker

Department of Mechanical Engineering

Indian Institute of Technology, Bombay

May, 2010

Outline� Introduction

� Boeing-787

� Forces and Their Effect

� ProblemDefinition

� Literature Review

� Complex Variable Method

� Schwarz Alternating Technique

� Mapping of Door andWindow

� Mathematical Formulation

� Finite Element Analysis

� Results

� Conclusions and Future Prospects

� References

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• 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-787

A mid-sized, wide-body jet airliner currently under development by Boeing Commercial Airplanes

Composite materials to construct fuselage - 15% Al, 50% composites and 12% titanium

Allows high cabin pressure during flight Ref [1]

Openings – passenger door, emergency door, cargo door and windows

Main passenger door and the Window nearest to this door - Dimensions

*All Dims in inchesMaterial Properties : E1=139.3 Gpa, E2=11.3 Gpa, G12=6 Gpa, ν21=0.3,ν23=0.4

Forces and Their Effect� During the steady flight, main forces acting on

aircraft fuselage are

1. Body forces - Differential internal pressure-Hoop and longitudinal stresses (Biaxialtensile state)

2. Engine thrust and wing drag - Engine thrustacts in forward direction, wind drag acts inthe opposite direction of the motion of theaircraft – Longitudinal bending moment(out of plane load)(out of plane load)

3. Due to manoeuvering of aircraft -Differential pressure acts on the wings whiletaking turn in air+inertia of the aircraft -torsional forces - Shear stresses in theaircraft skin

Problem DefinitionTo 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 of

curvature of fuselage iscurvature of fuselage is

large compared to the

dimensions of the doors

and window, the fuselage

is modelled as an infinite

plate with single and

multiple openings.

Literature ReviewSingle Hole Problem

• Krisch and Muskhelishvili --- the problem of infinite plate with singlecircular hole subjected to uniaxial stress at infinity

• Krisch --- Airy's stress function, Muskhelishvili --- complex variablemethod

• Muskhelishvili --- various boundary value problem --- complex variablemethod and conformal mapping technique

• Lekhnitskii --- the problem of anisotropic plates --- both in-plane andout of plane loading --- stress functions by series method

• Savin --- isotropic and anisotropic plates --- conformal mapping ---• Savin --- isotropic and anisotropic plates --- conformal mapping ---circular, triangular, rectangular and elliptical single hole

• Ukadgaonker and Awasare --- principle of superposition andMuskhelishvili’s complex variable approach --- solution for infinite platecontaining, circular, elliptical, triangular, rectangular holes --- ellipticalhole in anisotropic medium

• Ukadgaonker and Rao --- solution for stress field around various holegeometries in an anisotropic medium --- subjected to biaxial and shearstress at infinity, uniform internal pressure at hole boundary, uniformshear stress at hole boundary in detail

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

Literature Review (continued…)

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

Door and Windows of Passenger Aircraft

� Gandhi --- door and windows of Boeing-747 --- analytical formulation fora single rectangular hole for tensile loading --- problem of multipleopening done by FEM

� Upadhyay, Sharma --- door and windows of Boeing-777 aircraft --- stressfunctions for single rectangular hole under tensile load and bendingmoment (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

Literature Review (continued…)

vicinity using ANSYS

� Vasnik --- door and windows of Boeing-777 with crack --- stress intensityfactor were obtained using FEM as well as complex variable approach

Gaps Identified in Literature

• Very few analytical solutions are available considering rectangular hole inan infinite plate of anisotropic material.

• The interaction effect of two rectangular holes has not been yet studiedusing Schwarz’s alternating method.

Schwarz’s Alternating Technique

• The problem of multiply connected regions is solved as simply connectedregion and successively relaxing the boundary conditions on the holes.

• First complex solution in terms of stress functions is obtained for platewithout hole by mapping the physical Z-plane into ζ-plane.

• Boundary condition at the fictitious circular hole is determined usingthese stress functions.

• The second approximate solution is obtained by the application of thenegative value of the boundary condition on the circular boundary.

• Addition of these two solutions gives the solution valid near the circular• Addition of these two solutions gives the solution valid near the circularhole.

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• Generalized form of mapping function for Door and Window

• Mapping Constants

▫ Door

▫ Window

m1

m3

m5

m7

R

-0.2570 -0.1555 0.0240 0.0111 34.3980

m1

m3

m5

R

-0.2460 -0.1565 .0231 8.6500

• Door and window generated by using Matlab

Mapping of Door and Window (continued…)

Window

Door

In mapped plane

Complex Variable ApproachGeneralised Hooke’s law for plane stress

Stresses in terms of Airy’s stress function

Biharmonic equation asCompatibility equation for2D- elasticity problem

Its roots are,

Hence,

Introducing the stress functions φ(z1), ψ(z2) and their conjugate

Stresses in terms of stress functions are

Complex Variable Approach (continued…)

We can obtain the solution using the following steps

• First stage solution

• Second stage solution

• First Approximation

• Second Approximation

Boundary Conditions

Stress Function of Single Hole Problem under Remote Loading

First Stage – Stress functions for hole free plate

Second Stage – Plate having single rectangular hole

Mathematical Formulation

Second Stage – Plate having single rectangular hole

where

From Schwarz’s technique,

where

Mathematical Formulation (continued…)

Final Solution – Obtained by superposition of the stress functions of the first and the second stage

These stress functions give the stresses around rectangular hole.

First Approximation� Stress functions for the door

� Stress functions for the window

(continued…)

� These Stress functions do not consider the interaction effect of door and window.

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.

(continued…)

ζ

The boundary conditions for anisotropic plate is given by

Corrected stress functions around the window can be given by,

,

Using Cauchy’s integral formulae,

where, a =

(continued…)

and

b

(continued…)

We get the corrected stress functions as

By superposition of transformed and corrected stress functions we get the stress function for window considering the interaction effect of door

(continued…)

Second Approximation (Door)

This gives

Using these stress functions we can find the stresses around door andwindow with interaction effect.

Finite Element Analysis

A numerical technique to find approximate solution of PDE

ANSYS – A software to solve structural, static, transient, etc. problems

Anisotropic thin infinite-plate with plain stress condition

E1=139.3 GPa, E2=11.3 GPa, G12=6 GPa, ν21=0.3, ν23=0.4

Steps involved are- Preprocessing, Solution, Post processingSteps involved are- Preprocessing, Solution, Post processing

PLANE82

� 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 plasticity Ref: ANSYS Help

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.

Meshing

Meshing

Meshing

Results (Single Hole)Type of opening Max. Stress Concentration

Factor (SCF)

Error (%) Angular Position

Analytical Numerical

Passenger door 3.44 3.39 1.4 640

Window 2.27 2.16 4.8 680

MATLAB plot of SCF ANSYS plot of SCF

Results (Single hole)

Results (Single hole)

Results (Single hole)

Results (Single hole)

Results (Two Hole)Type of opening Max. Stress Concentration

Factor (SCF)

Error (%) Angular Position

Analytical Numerical

Passenger door 3.44 3.40 1.2 1190

Window 2.24 2.27 1.4 1220

MATLAB plot of SCF ANSYS plot of SCF

Results (Two Hole)

Results (Two Hole)

Results (Two Hole)

Results

Type of opening Max. Stress Concentration Factor (SCF)

Analytical

Difference(%)

Without Interaction With Interaction

Passenger door 3.44 3.44 00

Window 2.27 2.24 1.3

Type of opening Max. Stress Concentration Factor (SCF)

Numerical

Difference(%)

Without Interaction With Interaction

Passenger 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.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.

�The variation of SCF for other geometries and with different

parameters like length, width and thickness can be analyzed.

�The problem has been solved for the case of shear loading. The

other loadings can be considered for the analysis like in-plane and out

of plane bending loads.

Future Prospects

of plane bending loads.

�The curvature of aircraft fuselage can be taken into consideration

to solve a problem of three dimensional curved plate subjected to

different 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.

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.

(continued…)

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.

Thank You

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