Stress Analysis of Bullet-Holes
on the
Boeing 737 Fuselage
Submitted To: Prof Vlassak, Nanshu Lu
Submitted By: Jane Yoon, Sun Min Jung
Date: January 11th, 2008
ES 240 Project
Overview
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Introduction / Objectives
Variable / Parameter Definitions
Case 1: Single Bullet Hole
Case 2: Two Bullet Holes
Case 3: Single Bullet Hole Near Window
Case 4: Two Bullet Holes Near Window
Case 5: Two Holes Approaching Each Other
Extra: Fracture due to Crack-Propagation
Introduction / Objectives

Introduction:

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
How we came about
doing this project
Why this analysis is
important
How is this applicable
to real-life engineering

Objectives



Go through various case
examples
Use ABAQUS CAE to
model and view stress,
analyze more complex
scenarios
Compare results with
theoretical calculations
Variable / Parameter Definitions

Material Constants:
Aluminum 7079-T6
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Young’s Modulus:
E = 71.7 GPa
Poisson’s Ratio:
V = 0.33
Density: 2740 kg/m^3
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Physical Dimensions

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Dimension of Fuselage:
Radius = 5 m
Thickness = 0.2 m
Dimension of Bullet Hole
Diameter = 1
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Dimension of Window
20 x 10
General Assumptions and Simple Calculations:
Uniaxial Load in Compression: 0.19 MPa
Case 1: Single Bullet Hole
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S11
Case 1: ABAQUS CAE Analysis
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S22
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S12
Case 2: Two Bullet Holes
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S11
Case 2: ABAQUS CAE Analysis
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S22
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S12
Case 3: Single Bullet Hole
Near Window
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S11
Case 3: ABAQUS CAE Analysis
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S22
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S12
Case 4: Two Bullet Holes Near
Window
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S11
Case 4: ABAQUS CAE Analysis
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S22
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S12
Case 5: Two Holes Approaching
Each Other
Case 5: ABAQUS CAE Analysis
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S11
Case 5: ABAQUS CAE Analysis
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S11
Case 5: ABAQUS CAE Analysis
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S11
Case 5: ABAQUS CAE Analysis

S11
Case 5: ABAQUS CAE Analysis

S22
Case 5: ABAQUS CAE Analysis

S22
Case 5: ABAQUS CAE Analysis

S22
Case 5: ABAQUS CAE Analysis

S22
Case 5: ABAQUS CAE Analysis

S12
Case 5: ABAQUS CAE Analysis
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S12
Case 5: ABAQUS CAE Analysis
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S12
Case 5: ABAQUS CAE Analysis
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S12
Orthotropic Stress Equations
(Ellipse)
Stress-Strain Relations:
1
"
# XX =
! XX $ YX ! YY
EX
EY
# YY =
" XY
Differential Equation 1:
) XY
1 !4F ' 1
%
+
(
2
EY !x 4 %& µ XY
EX
"
1
! YY $ XY ! XX
EY
EX
$ !4F
1 !4F
"" 2 2 +
=0
4
E X !y
# !x !y
Differential Equation Simplified:
1
=
! XY
µ XY
!4F
!4F
!4F
+ 2# 2 2 + 4 = 0
4
!x
!x !"
!"
Compatibility Equation:
where:
! 2" XX ! 2" YY ! 2" XY
+
=
2
2
!y
!x
!x!y
)=
E X EY
2
& 1
2( XY
$$
'
µ
EX
% XY
Airy Stress Function:
" XX
!2F
= 2
!y
" YY
!2F
= 2
!x
# XY
!2F
="
!x!y
" = !y
and
! =4
#
!!
"
EX
EY
Orthotropic Stress Equations
(Ellipse)
Solve for Roots of Equation:
r1 = ie
!
2
r3 = "ie
r2 = ie
!
2
"!
2
r4 = "ie
"!
2
Biharmonic Equation Form:
Specific Solution Form:
) A /1
, &
2
2
(
)
z
0
w
+
y
log(
z
+
w
)
+#
1
1
1
1
1
# 2 -2
*
+ # P" 2
# 2 y1 .
F = Re (
%+
2
B
1
/
, # 2!
2
2
#
(z 0 w2 ) + y2 log( z 2 + w2 )* #
#' 2 y 2 2 -. 2 2
+$
2
2
& '2
#& ' 2
#
2 '
2 '
$$ 2 + *
!
$
!F = 0
+
)
2 !$
2
2 !
'
x
'
(
'
x
'
(
%
"%
"
where:
where:
z1 = x + i!"
z 2 = x + i!"
" 2 = e!
w1 = z1 ! y1
" 2 = e #!
w2 = z 2 ! y 2
2
2
2
2
2
General Solution Form:
F = Re[F1 (x + i#! )+ F2 (x + i"! )]+ FP (x, ! )
y1 = a 2 # " 2! 2b 2
2
y 2 = a 2 # " 2! 2b 2
Orthotropic Stress Equations
(Ellipse)
x = a cos !
# = "y = "b sin !
P = pressure
Use Boundary Conditions to
solve for constants of integration
and unknown variables…
Orthotropic Stress Equations
(Ellipse)
& Pb , (a + 10b)
.2F
(a + /0b) ) #
= 2 YY = Re %
*
'"
2
.x
0
(
1
/
)
w
(
z
+
w
)
w
(
z
+
w
)
1
2
2
2 (!
+ 1 1
$
& P0b , - 1 2 (a + 10b) / 2 (a + /0b) ) #
.2F
= 2 XX = Re %
+
*
'" + P
2
.y
w2 ( z 2 + w2 ) ( !
$ (1 - / ) + w1 ( z1 + w1 )
& Pb , - 1i (a + 10b) /i (a + /0b) ) #
.2F
= 2 XY = Re %
+
*
'"
.x.3
w2 ( z 2 + w2 ) ( !
$ (1 - / ) + w1 ( z1 + w1 )
To obtain the stress equations for a circle,
set a=b in the stress equations for an ellipse.
Isotropic Stress Equations (Ellipse)
For an isotropic material:
# = " =! =1
Therefore, specific solution (from previous) becomes:
*
$
$$
Pb
F = R)
#
1#
$ 2! 0. e 2 1 e 2
$ .
$( /
0
1
01
- -'
2
2
(
)
(
)
.1
z
1
w
+
y
log
z
+
w
.
1
1
1
1
1 + + +$
#
2
/
, +$
.
2
2
a
1
e
!
b
.
+$& + P"
2
-.
1
01
-+$ 2!
2
2
+.
. (z 2 1 w2 ) + y 2 log(z 2 + w2 )++
#
+.
1
,+$
, / a 1 e 2 !b / 2
,$%
Isotropic Stress Equations (Ellipse)
) 0
' *
' *
' *
' *
' *
' *
' *
' *
' *
' *
' *
' *
' *
' *
' *
Pb
F = lim ' R /
#
7#
# 80
' *! 64 e 2 + e 2
' * 4
' * 5
' *
' *
' *
' *
' *
' *
' *
' *
' *
' *
' *
' *
( .
#
6
3 -&
6
3
6
3
# 2 2
2
4
1 *$
4 (z1 7 w1 )4 #
1
1
z1ie " + e ! b
4
1 *$
4
1
4 ie 2" 7
1
2
w1
4
1 *$
4
1
4
1
5
2
4
1 *$
1
4
1
#
47
1 *$
4
1
#
#
# 2 2
2
4 a 7 e 2 !b 4
1 *$
1
z
ie
"
+
e
!
b
1
2
4
1 *$
2 ie " +
4
1
w1
# 2 2
4
1 *$
4 + y1
1
7 e ! b log(z1 + w1 )1
4 2
4
1 *$
z1 + w1
5
2
4
1 *$
#
4
1 *$
6 (z1 7 w1 )2
3
e 2 !b
2
47
1 *$
4
+ y1 log(z1 + w1 )11
2 4
#
4
1 *$
2
6
3 5
2
2
4
1 *$
244 a 7 e !b 11
4
1*$ P" 2
5
2
4
1 ,$ +
2
7#
34
6
3
1*$ 2!
6
3
7
#
7
#
2
2
2
14
4
11
(z 2 7 w2 )4 2 z 2 ie " + e ! b 1
1
4
1 1 *$
4 7 ie " +
1
24
2
w2
4
1 1 *$
4
1
4
5
2
1
4
1 1 *$
4+
7#
4
1 1 *$
7#
4
7#
1 1 *$
z 2 ie 2 " + e 7# ! 2 b 2
2
4 a 7 e 2 !b 4
7
ie
"
7
2
4
1 1 *$
4
y
w
*$
7
#
2
2
2
2
4+
+ e ! b log(z 2 + w2 )11 1*$
4
4 2
z 2 + w2
5
2 1 *$
4
7#
4
1 *$
2
4
1 *$
e !b
61
3
2
2
4 (z 2 7 w2 ) + y 2 log(z 2 + w2 )1
47
1 *$
2
7#
52
2
6
3
4
1 *$
244 a 7 e 2 !b 11
4
1 *$
5
2
5
2 +%
Isotropic Stress Equations (Ellipse)
(
)
' $
1 iyw . iyz . b 2 *
b
2*
+
(
+
(
(
)
(
)
.
z
.
w
.
z
.
w
!
!
+
( 2(a . b )2
w
(!
! Pb + a . b ,
)
F = R& +
(#
2
2
2
*
a
+
b
b
y
! +.
(!
++ iy +
(( . b log(z + w)+
(!
! +
w ,
z + w)
b
)"
% ,
For simplification, set a=b to obtain stress equations for a circle:
& Pa , a 3 a 3 2ay a 3
y 2 )#
''"
F = R % ** 2 - 2 - 2 - a log 2 z +
2
z
a
z
2
z
z
(!
$ +
In Polar Coordinates:
#
P & a4
F = $$ ' 2 cos 2( ' 4a 2 sin 2 ( + 2r 2 sin 2 ( ' 2a 2 log r !!
4% r
"
(
P &$ 2
r 2 ' a2
2
=
r ' 2a log r '
4 $%
r2
2
) cos 2( #!
!
"
where:
w = z 2 ! a2 + b2
Extra: Fracture due to Crack-Propagation


Fast Fracture of aircraft fuselage due to Crack Propagation
Linear-Elastic Fracture Mechanics (LEFM)
Condition for Fast Fracture
Plane
Strain:
Plane
Stress:
References
ES 240 Lecture Notes provided by
Professor Joost Vlassak
 “Theory of Plates and Shells” by
Timoshenko and Woinowsky-Krieger
 “Elementary Engineering Fracture
Mechanics” by David Broek
 “Engineering Materials 3: Materials Failure
Analysis” by D.R.H. Jones
