To illustrate the process of estimating crack growth behavior in a
structure with reinforcement in order to set inspection limits.
Overview of Problem Description
This problem focuses on the
crack growth life of a rear wing spar accounting for the reinforcing effect of the wing
skins. The rear spar is C shaped. The dimensions of the vertical web, top and bottom
flanges, as well as the thickness change along the span of the spar are accounted for. The bottom flange is connected to the wing skin
with two staggered rows of rivets. It is
assumed that a crack will initiate from one of these attachments, since the specified
loading spectrum causes this flange to be under the highest tension stress.
Figure SIE4.1. Rear
Wing Spar
Figure SIE4.2. Wing Lower Skin at Critical Location on Rear Wing
Spar.
Figure SIE4.3. View Looking Up at the Cracked Rear Spar.
Structural Model
The wing skins provide reinforcement to the assembly affecting the
overall load transfer as the crack propagates. Where
the overall load transfer is greatly affected by the crack, an established finite element
approach is available. It is important to
note that in these crack configuration cases, the crack will always be a through the
thickness crack and thus a two dimensional model of the crack tip vicinity is appropriate.
The finite element approach that is used in this example employs a
special cracked finite element that contains the crack tip singularity within it. A cracked finite element can be found in
MSC/NASTRAN or other similar programs. The
stress intensity factor is automatically generated and is uniquely defined. The overall mesh requirements are not demanding;
relatively crude meshes give excellent results utilizing this element.
The proper singular behavior of stress and strain distributions
from the near field solutions of linear fracture mechanics are embedded into the shape
functions of the cracked element. The
interelement compatibility conditions of displacement and tractions are maintained
through the use of Langrangian multiplier techniques.
Consequently, the stress intensity factors are solved directly as unknowns in the
final algebraic system of equations along with the nodal displacements. It has been shown that by using only 20 to 50
degrees of freedom, the stress intensity factor can be computed using the cracked element
with accuracy of one percent or better.
Although this example uses a specific cracked element, the intent
of this problem is to show how to incorporate these types of elements into a damage
tolerance analysis. This is why the
derivations of the specific element are not presented here.
Figure
SIE4.4. Example of a Finite Element Mesh with a Cracked Element.
Model Geometry Description
For an initial crack at the rear spar to skin attachment, the
fatigue crack propagation life is divided into two phases.
The first phase, Phase 1, consists of a through the thickness crack starting from a
rivet hole in the rear spar bottom flange in the row farthest from the web, and growing
towards the edge of the flange. This is
simulated as a crack emanating from an eccentric hole in a plate, as wide as the lower
flange, subject to constant uniform stresses (W = 2.443 in., t = 0.25 in., D = 0.312 in.). In this first phase, the wing skin has negligible
effects, and the crack growth analysis is performed using standard stress intensity
solutions.
The second phase, Phase 2, occurs after the crack in Phase 1
reaches the edge of the flange. In this
second phase, the crack becomes an edge crack growing towards the web of the spar. Phase 2 is analyzed as an edge crack in a
semiinfinite plate subject to uniform stresses that change with crack length. This change in stress with crack length is due to
the reinforcing effect of the wing skins.
This phase of crack growth in the rear spar was studied using the
aforementioned finite element analysis with a cracked element to determine the stress
intensity factor versus crack length.
Figure
SIE4.5. Initial, Phase 1, Crack Growth
Model.
Figure SIE4.6. Phase
2 Crack Growth Model.
In Phase 2, ten different finite element models representing the
rear spar at BL35.8, each with a different crack length, were generated. The following figures show respectively the mesh
for the case where the crack is partially along the flange and where the flange is fully
cracked with the crack tip on the beam web.
These models utilize symmetry through the crack place and model the
structure on one side of the crack with the appropriate boundary conditions along the
crack plane.

Figure SIE4.7.
Partially Cracked Flange. 

Figure SIE4.8.
Fully Cracked Flange. 
The upper and lower skins were modeled as intact. It is appropriate to model these skins as intact
as long as fatigue cracks will not be generated in the skins as the crack grows in the
rear spar. The effect of the crackstopper
attached to the web was conservatively ignored. The
applied load for the stress intensity factor analysis consisted of a unit moment applied
at a distance of 24 inches from the crack.
Table SIE4.1 contains the obtained stress intensity factors for
various crack lengths.
Table SIE4.1.
Stress Intensity Factors for Skin Intact.
Figures SIE4.9 and SIE4.10 show the variation of b
and K/s with length. Note the reduction in K/s as the crack progresses through the spar flange and
then increases after it gets near the spar web.
Figure SIE4.9. Crack
Length vs b.
Figure SIE4.10.
Crack Length vs K/s.
Inspection Capabilities and Crack Limits
The crack growth analysis was driven by the simple criteria that
any fatigue damage is required to be discoverable prior to the failure of a critical
element that could lead to the loss of the airplane.
The goal of this analysis was to show that the rear wing spar and skin would have a
crack growth life, with an initial through the thickness crack, greater than the design
life of 20,000 hours and that the threshold inspection would be based on past experience
and practice.
Structural Loading and Stress History Description
The design lifetime for the aircraft is 20,000 hours. Based on the Airplane flight profile combined with
gust maneuver exceedance curves, and the stress profiles on the assembly, a stress
spectrum is developed. This spectrum is
defined in blocks for the analysis.
The spectrum block that was used for crack growth analysis is 500
hours long and repeated 40 times in a lifetime, so that the frequency of the highest load
is 40 per lifetime. This is a conservative
clipping level for the high loads, since less frequently applied loads would cause more
crack growth retardation.
Based on a truncation study described below, a stress range
truncation level of 1.6 ksi was used for the low loads.
Due to the large number of cycles occurring in this spectrum block, first the
cycles representing 100 flight hours were generated and then the 100 hour block was
repeated five times with proper intermediate overloads to form the 500 spectrum block.
A truncation study was conducted for two locations located on the
rear spar. Table
SIE4.2 shows the growth lives as a function of the spectrum truncation level.
Table SIE4.2.
Truncation Level.
Location 
Truncation
Level (ksi) 
Life
in Hours 
1 
0.8 
28,000 
1 
0.85 
29,000 
1 
0.9 
30,000 
1 
1.0 
34,000 
2 
0.7 
746,000 
2 
0.8 
782,000 
2 
0.9 
953,500 
For both locations, the growth lives stabilized at an amplitude
stress level truncation value of 0.8 ksi. This
corresponds to a stress range truncation value of 1.6 ksi.
This truncation level was used for spectrum generation of the critical location.
There are four flight types in the mission mix. The following table shows the airplane life
profile. For each flight, the duration and
sequence of segments are indicated. The last
column represents the percentages of design life for each segment of a flight. There are twenty different segment types.
Table SIE4.3.
Airplane Flight Profile.
Detailed exceedance curves for the takeoff segment were not
generated because it was simpler to use a slightly conservative ground load for all
flights. Therefore, a total of 19 gust stress
exceedance curves, 19 maneuver exceedance curves, and 19 1G stresses were supplied for the
critical location.
The steps in generation of the 100 hour spectrum block are
described below. In steps a and b, peak and
valley stresses are generated without consideration of their sequence in the spectrum
block. Step h describes the inclusion of
fuselage pressure.
a. For
each segment type (i.e. climb, cruise, descent, and approach and landing) the total number
of excursions of positive and negative gusts and maneuvers in a spectrum block of 100
hours is determined. The total number of
excursions is the product of the exceedances per flight hour at the stress truncation
level and the number of blocks.
b. There
are separate exceedance curves for maneuver peaks and valleys. Gust peaks and valleys are related in that their
exceedances have the same absolute values but opposite signs. For each segment type, all gust peaks and maneuver
peaks and valleys are picked from respective exceedance curves on a random basis and
without replacement. The total number of
excursions to be picked for each exceedance curve was generated in step a. Three files of stress values are generated for
each of the nineteen segments.
c. The
number of flights of each type (there are a total of four types) in the spectrum block is
determined using their percentages of occurrence; their sequence is determined on a random
basis.
d. For
each segment of a flight, the numbers of cycles of gusts and maneuvers are determined from
the duration of the segment within a flight (as compared to the total time in that segment
in the design life).
e. For
each segment of a flight, gust peaks are picked from the information generated in step b
and in the order appearing there. The cycles
for gust are formed by coupling each peak with its opposite value as a valley. The segment 1G stress is then added to the peaks
and valleys. The stress cycles within the
same segment for two flights of the same type are in general different.
f. For
each segment of a flight, maneuver peaks and valleys are picked separately from the
respective information generated in step b and in the order appearing there. The segment 1G stress is then added to the peaks
and valleys. The gust cycles for a segment
are placed before those of maneuver.
g. A
ground stress (compression for the lower wing) is placed between each flight.
h. For
reasons of simplicity, the stresses due to fuselage pressure for various flight conditions
are taken to be equal to that of the most critical pressure condition (9.0 psi). It is added to all the spectrum peaks and valleys
except to the ground stresses. In general,
the net effect is to increase the mean stress by this stress value.
i.
The highest peak in the block is then identified and the portion of the spectrum
above it is moved to the end of the spectrum so that in the resequenced spectrum, the
highest peak is at the beginning. In
addition, the values in the spectrum not defining a peak or valley are eliminated. The resequencing is done to enable cycle pairing
to be performed using the Rainflow method (this was accomplished with an inhouse
program). The cycle pairing is important to
define the representative stress cycles for the crack growth analysis.
As discussed previously, a conservative clipping level is defined
for the highest load. To calculate this, the
combined exceedance curve of all 19 segments of gust and maneuver was generated and used
to assemble a 500 hour block from the 100 hour block.
The clipping level at a frequency of 40 per lifetime corresponds to the highest
stress S_{500} occurring in a 500 hour spectrum; S_{500} is determined
from the composite exceedance curve.
Composite exceedance curves were obtained by combining the 19 gust
exceedance data, maneuver exceedance data, and 1G stress values for the 19 segment types
using their associated percentage of occurrence. The
composite exceedance curve was used to determine the stress level occurring 40 times in a
lifetime. This was then used as the clipping
level. The composite exceedance curve for the
wing rear spar is shown below.
Figure SIE4.11.
Exceedance Curve for Wing Rear Spar.
S_{100} is the highest stress occurring in a 100 hour
block; it can be determined either from the composite exceedance curve or from the 100
hour spectrum block. Four overloads, all
larger than S_{100}, are used to extend the 100 hour block to a 500 hour block. The largest of these overloads is S_{500}. The other three overloads are determined by
picking three equally distant values on logarithmic scale between S_{500} and S_{100}. The 500 hour block is formed by the following
sequence:
S_{1}+S_{5}+S_{3}+S_{5}+S_{2}+S_{5}+S_{4}+S_{5}
Where S_{1} through S_{4} are the four overloads in
decreasing order and S_{5} is the 100 hour spectrum block.
Material Property Description
The material properties for the rear spar, 7075T7351LS (long/short
transverse), are given below. The inhouse
crack growth program used the modified Willenborg retardation model and the Chang
acceleration option. For the alloys used in
the wing, these models have been shown to be a reasonable representation of the crack
growth retardation and acceleration that occurs due to the interaction of high and low
loads in the loading spectrum. The bislope
representation of the crack growth rate versus DK
was opted for the aluminum alloys. The crack
growth program uses a Walker stress ratio effect model.
Table SIE4.4.
Material Properties and Growth Rate Data.
Parameter 
DK<12.1 
DK>12.1 
C 
5.044
x 10^{9} 
2.584
x 10^{7} 
n 
3.189 
1.608 
m 
0.4 
0.4 
q 
1.0 
1.0 
da/dn 
1.42
x 10^{5} 
1.42
x 10^{5} 
K_{C} 
61.5 
61.5 
K_{IC} 
29.4 
29.4 
F_{ty} 
58 
58 
DK_{th} 
0.01 
0.01 
Solution Technique
Although the crack growth analysis for this problem was solved
using an inhouse computer program, NASGRO3.0 could be also be used. Using standard stress intensity factors for Phase
1 and the stress intensity factor versus crack length chart developed with the cracked
finite element for Phase 2, the crack growth analysis for this type of problem is
conveniently solved using NASGRO3.0. This
would require a separate crack growth run for each phase.
Note that for Phase 2, the stress intensity factors versus crack length can be
input into NASGRO3.0 as a table. In this
example, the cycles for crack growth life will be converted into hours with the assumption
of 0.67 hours per groundairground (GAG) cycle.
Results
Life:
Using the relationships for b, the crack growth rate data for
7075T7351LS, and the appropriate stress spectrum, it was found that the total crack
growth duration of parts 1 and 2 up to the crack stopper on the flange is 60,000 hours
corresponding to three design lifetimes. Based upon this, the initial inspection should be
based upon past experience and practice and is not impacted by the damage tolerance
analysis.
Figure SIE4.12. Crack Growth Life for Problem SIE4.