The fatigue crack growth life of structural components is
significantly affected by the level of applied (repeating) stress and the
initial crack size. This section
addresses the effect of applied stress level on any structural component and
provides examples whereby relative life estimates can be utilized to facilitate
the damage tolerant analysis of structural repairs. Section 9.5 discusses the effect that initial crack size has on
the crack growth life of a repaired hole.
The simple method for evaluating the effect of stress level on
the fatigue crack growth life is based on the general form of Equation 9.3.14
and an available crack growth life curve for the structural geometry of
interest. The general form of Equation
9.3.14 related the life (Ls) at the current
stress level (s) to the life (Lxs)
at the new stress level (x.s) through the equation
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(9.4.1)
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As explained in Subsection 9.3.3, Equation 9.4.1 will estimate
life in a relative sense for any structural detail if (1) the crack growth life
is known for a defined stress history and (2) the flight-by-flight crack growth
rate behavior is described by the power law equation
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(9.4.2)
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Equation 9.4.1 does not allow one to calculate relative life
for changes in crack interval, in crack geometry, or in mission mix (unless a
master crack growth curve is available for different mission mixes). The above restrictions do not minimize the
extensive usefulness of Equation 9.4.1.
Rewriting Equation 9.4.1 so that it relates the unknown crack
growth life (Lxs) to the known
life results in
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(9.4.3)
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Equation 9.4.3, in essence, provides a scaling factor that
would be applied to the complete crack growth life curve for any structural
detail; Figure 9.4.1 illustrates this concept
schematically. Note that the life
scaling factor (x-p) is
independent of the shape of the crack growth life curve.
Figure 9.4.1. Schematic Describing the Use of Equation 9.4.3 to Scale the Crack
Growth Life Curve Based on a Stress Level Change from s
to x.s
where x < 1
Due to the generality of the life scaling factor for
constructing life estimates, it is instructive to evaluate this factor as a
function of the stress scaling factor.
The relationship is described in Table 9.4.1
for four different values of the crack growth rate exponent p.
Table 9.4.1 shows that the smallest life scaling factors for x < 1 are
associated with the lowest exponential value (p = 2.2). For x < 1, the new stress level is lower than the current level and as
one would guess (see Table 9.4.1 and Figure 9.4.2), the greater the reduction in stress the
longer the life (the higher the life scaling factors).
Table 9.4.1. Relationship Between Stress Scaling Factor x and Life
Scaling Factor Lxs
Defined for Values of the Crack Growth Exponent p
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Stress Scaling Factor 
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Life Scaling Factor
(for x-p)
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p = 2.2
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p = 2.5
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p = 3.0
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p = 3.0
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0.50
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4.60
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5.66
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8.00
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11.31
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0.60
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3.08
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3.59
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4.63
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5.98
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0.70
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2.19
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2.44
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2.92
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3.48
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0.80
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1.63
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1.75
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1.95
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2.18
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0.85
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1.43
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1.50
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1.63
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1.77
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0.90
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1.26
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1.30
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1.37
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1.46
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0.92
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1.20
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1.23
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1.28
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1.34
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0.94
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1.15
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1.17
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1.20
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1.24
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0.96
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1.09
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1.11
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1.13
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1.15
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0.98
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1.04
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1.05
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1.06
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1.07
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1.00
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1.00
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1.00
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1.00
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1.00
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1.02
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0.96
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0.95
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0.94
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0.93
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1.04
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0.92
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0.91
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0.89
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0.87
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1.06
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0.88
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0.86
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0.84
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0.81
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1.08
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0.84
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0.83
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0.79
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0.76
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1.10
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0.81
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0.79
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0.75
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0.72
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1.15
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0.73
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0.71
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0.66
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0.61
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1.20
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0.67
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0.63
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0.58
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0.53
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1.30
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0.56
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0.52
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0.46
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0.40
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1.40
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0.48
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0.43
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0.36
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0.31
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1.50
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0.41
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0.36
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0.30
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0.24
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Figure 9.4.2. Life Scaling Factor (New Life / Current Life) as a
Function of the Stress Scaling Factor (x
= New Stress / Current Stress)
The life benefit achieved by reducing the general level of
stress in a structural detail that has experienced
crack problems can be estimated from Equation 9.4.3. If the power law exponent p
is not available for this particular structural detail, it is recommended that
a conservative estimate of p be made,
i.e. for a stress reduction chose p =
2.2, and evaluate the increase in life on this basis.
EXAMPLE
9.4.1 Modify to Achieve
Lower Stress Levels
The doubler shown below has
been modified to reduce the general level of stress at the cracking site
identified by ten (10) percent. The
original doubler on a 6000 hour aircraft had a mean service life of 3400 flight
hours to a crack size which would functionally impare the use of this aircraft. How much life will the replacement doubler
have? No crack growth life curve exists
for the doubler nor for the general area of the wing where it is located. A wide area master curve for the wing is
described by a power law equation with exponent p = 2.89.
SOLUTION:
The aircraft is presumed to
fly the same type of missions with the same frequency after the repair
modification as before. Since a master
crack growth rate curve is available for the wing, the analyst would evaluate
the life of the repair using Equation 9.4.3 with a power law exponent of
2.89. The modification is expected to
result in a new life (Lnew)
for a ten (10) percent reduction in stress level (the new stress level is 0.9
times the current stress level). The
new life is given by
when Lcurrent is equal to 3400 flight hours, this reduces to
= 1.356 .
3400
= 4610 flight hours
Thus, a first order estimate
indicates the life of the replacement doubler will be 35 percent greater than
the original doubler. If the original
doubler are removed at 2500 hours and replaced with the doubler with the lower
stress, it is anticipated that the replacement doubler will not fail during the
remaining life of the aircraft (2500 + 4610 7110 hours > 6000 hour life requirement).
If
no information on the crack growth rate behavior existed for this region where
the doubler was located, then it is suggested that the equation be evaluated
with p = 2.2. The result of this evaluation is 4285 hours,
which still indicates that the replacement doubler will outlast the aircraft
(2500 + 4285 = 6785 hours > 6000 hour life requirement).
As a cautionary note, it is important to recognize that the
best estimate of the exponent p will
result in the best life estimate. The
exponent p is expected to vary as a
function (due to material and stress event effects on damage) so if values of
the exponent p are available for a
given location in a component, it is more accurate to utilize the exponent p for that location.
Another direct application of Equation 9.4.4 comes from moving
from a stress analysis control point where a complete crack growth life
analysis is available to a new location where the cracking behavior is expected
to be similar due to geometrical material conditions, but where only a strength
of materials analysis is available. An
example illustrates the approach here.
EXAMPLE 9.4.2 Local
Stress Scaling
The figure describes a local
area (Location A) of an aircraft structure that has been experiencing
distress. Only the most critical hole
(Location B) in the region was analyzed during a damage tolerance analysis;
this analysis is summarized in the figure.
The exponent p associated with
the aircraft’s standard operational missions is 3.2 for location B.
A strength of materials
analysis was conducted to evaluate the difference in stress levels at the two
location (A & B) for a given external loading; these stress levels are
defined in the geometry. Provide an estimate
of the life for the hole identified at Location A.
Description
of Structural Geometry and Definition of
Analysis Location and of Crack Site
SOLUTION:
The
crack at Location A is presumed to grow in the same manner illustrated for
Location B. The stress history at
Location A is identical to that at Location B except that the stresses are
scaled to a lower level x given by
So
that the life (LA) at
Location A is found using Equation 9.4.3 and the crack growth life curve in
Figure 9.4.5, which describes the life (LB)
to any given crack size for location B:
LA = (0.951)-3.2(LB)
From the Location B crack
life curve, the flights required to break the ligament and to fracture the
component are 7300 and 12100 flights, respectively. From the equation, the corresponding lives at Location A are 8570
and 14210 flights, respectively, a 17 percent over that of location B.
If cracks are observed with
a greater frequency at location A than at Location B, and if the crack sizes at
location A are longer than that anticipated at location B for the same
operational conditions, then the analyst might reverse the analysis, i.e. use
the life ratios for specific crack sizes to obtain a better indication of the
stresses at the distressed location.
EXAMPLE
9.4.3 Stress Estimated from
Crack Behavior
Cracks have been noted
during PDM in a number of aircraft at Location A show for Example 9.4.2.
From the available inspection data, it appears that the cracks reach a
length of 0.150 inches after about 3600 flights. The DTA established crack growth life curve indicated that 0.150
inch long cracks should not appear until 5800 flights. Estimate the stress level difference between
location A and B. Also estimate the
number of flights required to fail the ligament and the component.
Details
of Cracking Process at Location B and Life Curve
SOLUTION:
The
method suggested for determining the stress level difference is with Equation
9.4.3, i.e.
LA = x-p.LB
where
it’s known that LA = 3600,
LB = 5800, and p = 3.2. Solving for x, the
stress ratio between Location A and B yields
and
the stress ratio is
So the stresses at the
cracking site (Location A) are expected to be 16 percent greater than that at
the DTA location (Location B).
Equation 9.4.6 can be now
used to estimate the lives to grow the crack (at Location A) to fail the
ligament and the component with x known, the lives are given by
LA
= (1.16)-3.2LB
And with LB = 7300 and 12100 flights
for the Location B critical conditions, LA
= 4540 and 7525 flights, respectively, to fail the ligament and the component
at Location A.