This document summarizes a thesis analyzing stress concentrations around doors and windows on the Boeing 787 aircraft under uniform shear loading. It presents analytical solutions using complex variable and Schwarz alternating techniques to model openings as rectangular holes in an infinite plate. Finite element analysis is also conducted and results show good agreement with analytical solutions. Stress concentrations are highest at corners and depend on geometry. Door and window interaction increases window stresses up to 4.8% but negligibly impacts door stresses.

1. Stress Analysis of Doors and Windows
of 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

2. Outline
Introduction
Boeing-787
Forces and Their Effect
Problem Definition
Literature Review
Complex Variable Method
Schwarz Alternating Technique
Mapping of Door and Window
Mathematical Formulation
Finite Element Analysis
Results
Conclusions and Future Prospects
References

3. 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

4. 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
Ref [1]
Allows high cabin pressure during flight
Openings – passenger door, emergency
door, cargo door and windows
Main passenger door and the Window
nearest to this door - Dimensions
Material Properties : E1=139.3 Gpa, *All Dims in inches
E2=11.3 Gpa, G12=6 Gpa,
ν21=0.3,ν23=0.4

5. 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 (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

6. 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 of
curvature 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.

7. Literature Review
Single Hole Problem
• Krisch and Muskhelishvili --- the problem of infinite plate with single
circular hole subjected to uniaxial stress at infinity
• Krisch --- Airy's 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

8. 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

9. 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 approach
Gaps 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.

10. 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

11. Mapping of Door and Window
Conformal Mapping
• A conformal map is a function which preserves angles.
• Any conformal mapping of a complex variable which has continuous
partial derivatives is analytic. An analytic function is conformal at any
point where it has a nonzero derivative.
• Conformal mapping helps in transforming very complicated shapes into
much simpler ones.
•It allow the basic complex variable formulations to extend to the
transformed problem.
• Generalized form of mapping function for Door and Window

12. 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

13. Complex Variable Approach
Generalised Hooke’s law for plane stress
Stresses in terms of Airy’s stress function
Compatibility equation for Biharmonic equation as
2D- elasticity problem
Its roots are,
Hence,
Introducing the stress functions φ(z1), ψ(z2) and their conjugate

14. 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

15. Mathematical Formulation
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
where

16. Mathematical Formulation (continued…)
From Schwarz’s technique,
where
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.

17. (continued…)
First Approximation
Stress functions for the door
Stress functions for the window
These Stress functions do not consider the interaction effect of door and window.

18. (continued…)
Second Approximation (Window)
In order to account for the interaction effect of door on the stress
functions of the window, the stress functions of the door is transformed to
the centre of the window by translation through a distance C0, given by Z0 =
ω(C0 ) such that |C0|>1.
ζ
The boundary conditions for anisotropic plate is given by
Corrected stress functions around the window can be given by,
,

19. (continued…)
Using Cauchy’s integral formulae,
where, a =

20. (continued…)
and
b

21. (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
Second Approximation (Door)
This gives
Using these stress functions we can find the stresses around door and
window with interaction effect.

22. 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 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
Ref: ANSYS Help
plasticity

23. 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.

24. Meshing

25. Meshing

26. Meshing

27. Results (Single Hole)
Type of opening Max. Stress Concentration Error (%) Angular Position
Factor (SCF)
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

28. Results (Single hole)

29. Results (Single hole)

30. Results (Single hole)

31. Results (Single hole)

32. Results (Two Hole)
Type of opening Max. Stress Concentration Error (%) Angular Position
Factor (SCF)
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

33. Results (Two Hole)

34. Results (Two Hole)

35. Results (Two Hole)

36. Results
Type of opening Max. Stress Concentration Factor (SCF) Difference(%)
Analytical
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) Difference(%)
Numerical
Without Interaction With Interaction
Passenger door 3.39 3.40 0.3
Window 2.16 2.27 4.8

37. 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.

38. Future Prospects
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.
The curvature of aircraft fuselage can be taken into consideration
to solve a problem of three dimensional curved plate subjected to
different loads.

39. References
1. 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.

40. (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.

41. Thank You
Questions