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Section 6.3. Slow Crack Growth Structure
The purpose of this example is to demonstrate the lowest level
of damage tolerance analysis that can be undertaken. This example problem will be set up to use only a hand-held
calculator for all calculations. Some
simplifying assumptions to obtain engineering estimates will also be
demonstrated.
EXAMPLE
6.3.1 Wing Attachment
Fitting
Problem Definition
A training aircraft has been discovered to have cracks in the wing
attachment fitting. A redesign
and retrofit will be necessary. Cracks
have been found in two aircraft that have been grounded. The problem is to determine inspection
intervals for the remainder of the force until the modifications can be
performed.
Wing Attachment Fitting
Material Property Data
The material for the
attachment fitting is 7079 aluminum forging with the following properties:
KIc
= 22.5 ksiÖin
and a Forman equation
describes the crack growth rate behavior:
Structural Loads and
Stress History
Each aircraft is equipped
with a counting accelerometer. The data
has been collected and published in the form of nz counts per 500 hours, as shown in the table.
The stress analysis for the
aircraft gives the l-g stress as 7.0 ksi., and using this, the nz values are converted to
stresses. Assuming the 1-g stress is
the minimum stress, the stress ratios R
can be calculated. These values are
also shown in the table.
Stress History for 500 Hours
|
nz
|
Counts/500
Hours
|
Smax
(ksi)
|
R
|
|
5.1
|
80
|
35.7
|
0.20
|
|
4.5
|
1200
|
31.5
|
0.22
|
|
3.5
|
2500
|
24.5
|
0.29
|
|
3
|
12500
|
21.0
|
0.33
|
|
2
|
22000
|
14.0
|
0.50
|
Initial Flaw Sizes
The structure is assumed to be slow crack growth
structure. A special inspection program
has demonstrated an initial flaw size inspection capability of 0.02 inches.
Geometry Model
The critical configuration is determined to be a radial through
flaw at the edge of the hole. The
stress-intensity factor for this geometry, while well known, is not amenable to
closed form solutions. However,
applying the approximation techniques discussed in Section 11 leads to an
approximate expression for K as
follows:
This equation represents a K
solution for a through crack in a plate multiplied by the stress concentration
factor, Kt, for a
hole. Using this expression the initial
K’s for each load level are
determined, as shown in the table.
Residual Strength Diagram
The residual strength diagram for this configuration is
obtained simply by setting K in the
above equation equal to KIc
and solving for a, which gives:
Plotting this function gives the residual
strength diagram, as shown.
Residual
Strength Diagram
Fatigue Crack Growth Analysis
The basic purpose of this analysis is simply to determine the
life under the given stress history.
Since the shape of the crack growth curve is not of prime importance
because of the imminent retrofit, a damage index approach can be used to
estimate the life. The Forman Equation
may be integrated to give the life from an initial crack size to critical crack
size for nz level.
Performing this integration gives:
This function is evaluated to give Nallow for each stress level in the history. The results are shown in the next table.
Using a fatigue damage analogy, a damage index (DI) is
calculated for each stress level by dividing the number of counts in 500 hours
by Nallow. For nz
= 5.1, the damage index is:
The life is then obtained by dividing 500 hours by the sum of
the damage indices:
Crack Propagation
Analysis Using Linear Damage Indices
|
nz
|
Count/500 Hours
|
Smax
(ksi)
|
R
|
Ko
(ksi√in)
|
Nallow
|
Damage Index
|
|
5.1
|
80
|
35.7
|
0.20
|
8.49
|
2320.92
|
0.034
|
|
4.5
|
1200
|
31.5
|
0.22
|
7.49
|
4260.63
|
0.282
|
|
3.5
|
2500
|
24.5
|
0.29
|
5.83
|
13957.88
|
0.179
|
|
3
|
12500
|
21.0
|
0.33
|
4.99
|
28875.60
|
0.433
|
|
2
|
22000
|
14.0
|
0.50
|
3.33
|
222173.42
|
0.100
|
|
|
|
|
|
|
|
Sum = 1.027
|