To illustrate the process of estimating crack growth behavior to
set inspection limits.
Overview of Problem Description
This problem focuses on the main
cargo surround doubler attachment to the existing fuselage skin at stringers 2R and 26L. The skin is considered to be a single load path
structure under the total hoop stress before the doubler attachment. The critical location is in the skin at the first
row of fasteners because the skin sees both bypass and bearing stresses at this row, where
as, at the other fastener rows the load is in both the doubler and the skin with each row
having lower load transfer.
The fuselage skin was fabricated from 2024-T3 aluminum. The fasteners are 0.188 in diameter, and join the
skin and surround doubler.
The specific area is shown in View A of Figure
SIE-1.1, with the specific details and MSD crack path shown in Figure
SIE-1.3. Note that the skin at this first
row of fasteners is a single load path as shown in Figure SIE-1.2.
Figure SIE-1.1. Main Cargo Door Doubler Installation
Figure SIE-1.2. Structural Detail for Critical Area
Figure SIE-1.3. Detail Geometry of Critical Location, View A.
Model Geometry Description
The crack growth analysis is based on the Fatigue Crack Growth
Computer Program NASGRO3.0. This computer
program calculates crack growth for a single crack for several standard crack cases. Crack growth rate calculations use the
“NASGRO” equation with elements developed by Forman, Newman, de Koning, and
Henriksen (see NASGRO reference manual). This
is a modified Paris equation to account for fatigue crack closure, stress ratio effects,
and upper and lower fatigue crack growth rate asymptotes for threshold and critical crack
growth.
The analysis uses the NASGRO3.0 material libraries for the crack
growth rate equation constants. Non-interaction
of loads and constants for the Forman crack growth rate equation are used.
Since the standard crack models in NASGRO3.0 are for crack growth
of single cracks, no influence of one crack upon another is calculated in NASGRO3.0 for
these standard cases. MSD scenarios involve
fatigue damage at multiple locations. This
causes the potential of crack interactions. The
analysis presented here includes these crack interaction effects by iterating though a
series of NASGRO3.0 computer runs tracking the growth of multiple cracks and modifying the
stress intensity factors appropriately. The
increased stress intensity factors are based on the crack sizes of the interacting cracks
from the previous iteration and correction factors based on the compounding of analytical
stress intensity solutions.
This iteration procedure is accomplished in an Excelâ Spreadsheet
utilizing Visual Basic Programming to submit a NASGRO3.0 computer run for each crack at
each iteration. The spreadsheet reads the
NASGRO3.0 output files for cycles and current crack lengths. Based on these crack lengths, correction factors
are calculated and input into the NASGRO3.0 input file for the next iteration, which is
automatically submitted by the spreadsheet.
The correction factors are accounted for by increasing the stress
scaling factors input into NASGRO3.0. These
correction factors for crack interaction account for the condition of interactions of
cracks in parts that are analyzed for multiple site damage.
These increased stress scaling factors can be input based on the following:
_{}
The correction factors for crack interaction are based on
compounding of analytical stress intensity solutions.
Two correction factors are used in this analysis.
The first correction factor is termed “Bowie” and is for equal length
cracks growing from opposite sides of a hole. The
second correction factor is termed “Periodic” and is used for equal length
cracks emanating from holes approaching one another.
Compounding of the first and second correction factors is done, and
is termed “Bo + Bp’. This product
of the “Bowie” and the “Periodic” correction factors is what is used for typical MSD situations where there are
multiple fastener holes in a row and assumed imperfection flaws equal in size growing from opposite sides of each hole towards one
another.
Figure SIE-1.4. Bowie
Correction Factor
Figure SIE-1.5. Periodic
Correction Factor
Figure SIE-1.6. Bowie +
Periodic Correction Factor
These correction factors are based on through the thickness cracks. They are used for part through cracks when defined
with an equivalent crack length. The
equivalent crack length is based on equating the area of a part through crack as a quarter
ellipse to that of an equivalent through crack as a rectangular area with thickness, t:
_{}
Note the correction factors are used in conjunction with standard
crack models in NASGRO3.0. Therefore, to
obtain a correction factor, the ratio of an analytical solution for a specific crack
problem to a solution approximating the standard crack model is calculated. The analytical
solutions used in deriving these correction factors are taken from H. Tada, P Paris, and
G. Irwin, “The Stress Analysis of Cracks Handbook”, Third Edition.
The “Bowie” correction factor is derived by comparing
Bowie’s solution for equal length cracks from both sides of a hole to the solution of
a single crack from one side of a hole. The
ratio of these solutions, from pages 19.1 and 19.2 of the
Handbook, respectively yield the “Bowie” correction factor:
_{}
The “Periodic” correction factor is derived from the
ratio of the beta factor for a periodic array of cracks from page 7.1 of the Handbook to the solution of a crack in an infinite width plate. Note the solution for a periodic array of cracks
is an exact solution. Including the diameter
of the hole in the crack length and defining the pitch of the hole spacing to be the pitch
of the crack spacing yields the “Periodic”
correction factor:
_{}
Note the periodic correction factor is based on the solution of an
infinite width plate, while the standard crack model in NASGRO3.0 is for a finite width
plate. The use of this correction factor is
conservative based on the following calculations.
A comparison of the stress intensity factor calculated for a finite
width plate with that of a periodic array of cracks in an infinite plate is made to
estimate the effects on crack growth as cracks approach one another. Note that the cracks are analyzed with a finite
width plate using the width equal to the pitch of the fastener spacing. Solutions for both problems are taken from H.
Tada, P Paris, and G. Irwin, “The Stress Analysis of Cracks Handbook”, Third
Edition.
Koiter found the exact limit for the beta factor for the finite
width plate at a/b=1 as:
_{}
For the periodic array of cracks in an infinite plate, the beta
factor is:
_{}
Using the following expansion on the tan(x) , taking the limit of
a/b ®1 and
representing the second and higher terms of the series as the constant C_{1:}
_{}
Comparing the beta factors for both solutions as a/b ®1 and noting
that the constant term is multiplied by zero:
_{}
Based on this comparison, it is conservative to approximate the
beta factor of a periodic array of cracks in an infinite plate with that of a finite width
plate of the same dimension. Note that this
comparison is at the limit as a/b ®1. This ratio increases to 1.0 as a/b ®0.
Therefore, the crack growth model for the main cargo door surround
doubler attachment to the fuselage skin at stringer 2R employs the NASGRO3.0 corner crack
from a hole centered in a plate, CC02, with the correction factors for both the influence
of cracks growing from opposite sides of a hole , “Bowie”, and the influence of
a periodic array of cracks approaching one another, “Periodic”.
These correction factors are used along with the NASGRO3.0 standard crack growth model CC02. The
following dimensional values are used for the CC02 crack growth model.
_{}
_{}
.
Figure SIE-1.7. NASGRO3.0 Crack Model, CC02.
Two identical NASGRO files are created for the multiple site damage
cracks and submitted to the Excel interaction spreadsheet.
The spreadsheet accesses NASGRO and grows both cracks for 100 flights. The b
correction factors are calculated for the crack lengths at that time and the resulting
increased stress scaling factors are plugged back into the NASGRO files. The interaction spreadsheet grows the two cracks
until there is a 10% change in crack length (this could also be done in increments of
flights), recalculates the b correction and stress
scaling factors, and continues to grow the cracks until they become critical or the edge
of the plate is reached, whichever occurs first.
Inspection Capabilities and Crack Limits
The holes in the fuselage skin at the attachment of the first row
of the surround doubler attachment at 2R (and 26L) are directly accessible from the
inside. Therefore, these areas are inspected
by HFEC surface probe. With a HFEC
inspection, the minimum detectable crack size in the field is assumed to be a 0.0625 inch
crack past the fastener head.
Structural Loading and Stress History Description
The stress spectrum is considered to have a remote stress due to
cabin pressurization. Cabin pressurization
primarily causes hoop tension in the fuselage. The
GAG pressurization load is based on FAR25.571. The pressure condition is comprised of a
7.8 psi normal operating differential pressure and an additional 0.5 psi external
aerodynamic pressure. A factor of 1.1 is
only applied to the normal operating pressure for residual strength.
_{}
The bypass and bearing load at the critical fastener row is
calculated using a displacement compatibility analysis as described by Swift
(“Repairs to Damage Tolerant Aircraft,” presented to the International Symposium
on Structural Integrity of Aging Airplanes, Atlanta, Georgia, USA, 1990). Layer “a” is the fuselage skin and an
existing bonded doubler. Layer “b”
is the main cargo door surround doubler. The
surround doubler becomes fully effective after the first three rows. This analysis shows the most critical fastener
location is the first row of fasteners.
Table SIE-1.1.
Fastener Transfer Calculations.
Based on these results, 30% of the load is taken through bearing in
the first row of fasteners. This first row of
fasteners therefore has 30% as a bearing load and the remaining 70% as a bypass load.
The axial stress and bearing stress acting on this section are:
_{}
The limit stress used for residual strength purposes in this
scenario is calculated as stated earlier according to FAR25.571.
_{}
The residual strength axial stress and bearing stress acting on
this section are:
_{}
Material Property Description
The outer skin and doubler are made from 2024-T3 IAW QQ-A-250/5. The material properties from the NASGRO3.0
libraries are used for the fracture toughness and the crack growth rate properties. The material properties used are for 2024-T3;
Clad, Plate and Sheet; T-L; LA & HHA NASGRO material code M2EA12AB1.
Table SIE-1.2. Material
Properties and Growth Rate Data.
MATL
1: 2024-T3
Clad Plt & Sht; L-T; LA & HHA
Material
Properties:
:Matl: UTS : YS : K1e
: K1c :
Ak : Bk : Thk : Kc :
Keac :
:
No.: : : : : :
: : : :
:----:------:------:------:------:-----:-----:-------:------:------:
: 1 : 66.0: 53.0: 46.0: 33.0: 1.00: 1.00:
0.036: 66.0: :
:Matl:---------------
Crack Growth Eqn Constants -------------------:
:
No.: C :
n :
p : q :
DKo : Cth+ :Cth- : Rcl:Alpha:Smax/:
: :
: : :
: : : : :
:SIGo :
:----:---------:-----:----:----:------:------:-----:----:-----:-----:
: 1 :0.829D-08:3.284:0.50:1.00: 2.90: 1.50:
0.10:0.70: 1.50: 0.30:
Solution Technique
This type of problem is conveniently solved using NASGRO3.0 with
the crack growth interactions previously discussed. The
input files for the equal length cracks growing from opposites sides of a hole are
identical for the NASGRO3.0 analysis shown in Table SIE-1.3. The spectrum is included as a constant amplitude
GAG cycle with 100 flights per block, with a single block applied per schedule.
Table SIE-1.3.
NASGRO Input File for Problem SIE-1.
Data |
Description |
71fc1-2cout |
Output file
name |
1 |
1=US units |
D |
D=direct |
71fc1-2 skin at
upper and lower doubler edges |
Problem name |
CC |
Crack model
type |
2 |
Crack model no. |
0.036 |
Thk,
t |
5 |
W |
0.188 |
D |
2.5 |
Hole center to
edge |
0.33 |
Poisson's Ratio |
U |
U=User defined
crack |
0.005 |
Initial a |
1 |
Initial a/c |
1 |
Number of
materials |
N |
Non
Interaction |
1 |
Matl input
choice |
w |
File input
choice |
M |
Material
Category |
2 |
Material type |
EA |
Material alloy |
1 |
Material heat
treat information |
Stress on skin
at upper/lower edged |
Spectrum name |
N |
Flag for
identifying steps |
100000 |
No. times to
apply schedule |
1 |
No. distinct
blocks |
N |
Don't display
spec blocks |
1 |
Num steps/block |
3 |
Schedule option |
1 |
Load step
number |
100 |
Number of
cycles |
0 |
FMIN(1) t1 S_{0} |
11.857 |
FMAX(1) t2 S_{0} |
0 |
FMIN(2) t1 S_{1} |
0 |
FMAX(2) t2 S_{1} |
0 |
FMIN(3) t1 S_{3} |
26.018 |
FMAX(3) t2 S_{3} |
0 |
End manual
input |
1 |
Scaling Factor
S_{0} |
1 |
Scaling Factor
S_{1} |
1 |
Scaling Factor
S_{3} |
Y |
Reference
stress input |
13 |
REFACT(1,1,1) S_{0} |
2 |
Ref Stress at
t2 |
0 |
REFACT(2,1,1) S_{1} |
2 |
Ref Stress at
t2 |
28.527 |
REFACT(4,1,1) S_{3} |
2 |
Ref Stress at
t2 |
N |
Do not enter
schedule from file |
1 |
Sblock case |
1 |
Number of times
to apply |
0 |
End Spectrum
input |
Results
Critical crack
size/Residual Strength
This case is considered to address an MSD case where 0.005 inch
cracks are grown from both sides of a fastener hole over an effective width to represent
multiple fastener holes with cracks growing from both sides. There is potential for these crack to link prior
to failure. The following calculations
estimate this potential.
The half distance between the edges of adjacent holes is:
_{}
The plastic zone size of the MSD type cracks is estimated to be:
_{}
If fast fracture failure does not occur first, the two MSD type
cracks approaching one another will link at a crack length of:
_{}
Life:
Based on the calculations for growing the crack in NASGRO and the
MSD crack growth interactions, the life from initial crack size to failure is determined
to be 52,161 flights. The results of crack
length and crack depth versus life are shown in Figure SIE-1.8. The life is given in numbers of flights.
Figure SIE-1.8. Crack Growth Life for Problem SIE-1.
Inspection Intervals
The threshold and repeat intervals are calculated using the life
reduction factors shown below.
Life Reduction
Factors:
K_{1} = 2.0
K_{2} = 3.0, MSD crack scenario
Detectable crack
length (HFEC around fastener head):
_{}
Number of flights @ detectable crack length, N_{det} =
44,085 flights
Critical crack
length: c = 0.327 in.
Number of flights @ critical crack length, N_{crit} =
52,161 flights
_{}