This section illustrates parametric analyses available to an
engineer for evaluating the sensitivity of residual strength to geometric and
material parameters. As discussed in
Section 4, the residual strength relates load carrying capability to material
toughness and crack size in a unique way for each structure. Two methods are generally available for
describing the residual strength of a structure: the first is with a relationship between
residual strength and crack length, and the second is with a
relationship between residual strength and time. These relationships are summarized in Figure
9.7.1. The first relationship is
best used to describe the effects of toughness or of global geometry. The second relationship is best used to
describe the effects of crack growth resistance and of global geometry.

** **

**Figure
9.7.1.** Types of Residual Strength
Relationships

Constructing residual strength-crack length relationships are
relatively straight forward. To do so
requires both fracture toughness data for the material and a stress-intensity
factor analysis for the cracked structure.
Fracture toughness data can be found in the Damage Tolerant Design
(Data) Handbook [Skinn, et al., 1994].
The stress-intensity factor for a given cracked structure can be
obtained through the methods discussed in Section 11.

Constructing residual strength-life relationships requires the
same information as above plus a description of the crack growth life behavior
under the service loading. This
additional information can be generated by integrating wide area crack growth
rate equations or by summing incremental damage on a cycle-by-cycle basis. Cycle-by-cycle damage summation presumes (a)
that a stress history is available for the cracked structure and (b) that
constant amplitude fatigue crack growth rate data are available for the
material. The additional complexity
associated with generating residual strength-life relationships is one reason
why residual strength data are normally only presented as a function of crack
length.

A series of examples have been prepared to describe the effects
of material properties, spectrum stress level, and structural geometry on the
residual strength of relatively simple structures. The approach taken could be duplicated for other more complicated
situations related to specific structural repairs.

For each example, the Irwin abrupt fracture criterion is
employed to obtain the relationship between residual strength and crack
size. Simply stated, failure is
presumed to occur when the applied stress-intensity factor (*K*) is greater than or equal to the
fracture toughness (*K*_{cr})
of the material, i.e.

then failure occurs.
Because the stress-intensity factor is a function of stress and crack
size, Equation 9.7.1 provides the relationship between residual strength and
critical crack length.

To facilitate a general overview of the residual strength-life
relationship, the wide area crack growth rate equation methods developed in
Section 9.3 are utilized. In this
section, the wide area equation is expressed as

where the crack growth rate (*da/dN*) is given appropriately as a function of cycles, flights, or
flight hours depending on the given structural situation. Also, the characteristic stress-intensity
factor (_{}) in Equation 9.7.2 is related to the characteristic stress (_{}) through

where (*K/**s*) is the stress-intensity factor coefficient
(dependent only on geometric parameters, such as crack length and edge
distances). The residual strength-life
relationship is obtained by cross
correlating the residual strength calculated from Equation 9.7.1 with the life
calculated from

The cross correlation is accomplished using the same value of
critical crack length (*a*_{cr})
for both the residual strength and life calculations.

Example 9.7.1 considers the effect
of fracture toughness on both the residual strength-crack length and residual
strength-life relationships. These
relationships are established for an open hole with a through-the-thickness
radial crack, and for a wide area crack growth rate equation defined as

Examples 9.7.2, 9.7.3, and 9.7.4 consider
how the characteristic stress level (_{}), the constant *C,*
and the exponential constant *n*,
respectively, affect the residual strength-life relationship. Subsequently, Examples
9.7.5 and 9.7.6 present the effect of geometrical
and loading changes on the residual strength-crack length and residual
strength-life relationships.

EXAMPLE
9.7.1 __Effect of Fracture
Toughness__

Evaluated
the residual strength relationships be as a function of fracture toughness for
the structural geometry and loading described in the figure shown here.

To determine the residual
strength-crack length relationship, Equation 9.7.1 is utilized in conjunction
with the stress-intensity factor coefficient obtained from Section 11. The stress-intensity factor coefficient for
the tension-loaded, open-hole with a radial-through-the-thickness crack in a
wide plate is given by:

_{}

where _{}

Solving Equation 9.7.1 in
conjunction with this equation leads to the relationship between residual
stress and critical crack size, thus

_{}

where _{}

Defining a series of
critical crack sizes for a given value of *K*_{cr}
is the easiest method for evaluating the relationship. The following plot describes the
relationship between residual strength and crack length, evaluated in this
manner, for several given values of *K*_{cr}. As the plot illustrates, a substantial
difference exists between the residual strength curves at any crack

When
the crack growth life calculation is based on the integral formulation of
Equation 9.7.2 (a power law), i.e. on Equation 9.7.4, the shape of the crack
growth-life curve is as shown here.
This specific curve was obtained for a characteristic stress level (_{}) of 20 ksi and employed

*C* = 1310^{-8}
and *n* = 3.0 as the constants in the
growth rate equation. The initial crack
length (*a*_{o}) was chosen as
0.050 inches.

Crack
Growth-Life Relationship for the Baseline Geometry

The
curve has been marked to indicate the stress-intensity factor at various crack
length levels. These levels correspond
to the lower fracture toughness levels shown in the previous plot. One consequence of using a power law equation
to describe crack growth rate behavior is that the crack growth life curve does
not indicate a rapid increase as the stress-intensity factor approaches the
fracture toughness level. From a
practical standpoint, only a slight error in the life calculation occurs due to
inaccurately modeling the crack growth rate in the fracture toughness regime.

When
the crack length-life data are cross correlated with the residual
strength-crack length data, one obtains the relationships between residual
strength and life shown below. Each
residual strength-life data point is associated with a common crack length that
relates the data in the previous figures.
The figure shows that the highest values of fracture toughness are again
associated with the highest values of residual strength. The figure also shows that a material with
high fracture toughness will maintain a high residual strength capability
longer than one with low fracture toughness, all other conditions being equal. Interestingly, for the conditions given for
this example, the residual strength capability decays in a linear fashion for
most of the life. The only
non-linearity occurs in the earliest part of life where the crack is in a
severe stress-intensity factor gradient.
Other factors which affect the extent of the nonlinear region will be
discussed later.

Effect of Fracture
Toughness on the Residual Strength-Life Relationship

EXAMPLE
9.7.2 __Effect of
Characteristic Stress Level__

Because the operational
stress level significantly affects the crack length life of a structure, an
engineer might wish to consider its effect on residual strength. For this evaluation, assume that the
material is known to have a fracture toughness (*K*_{cr}) of 30 ksi √in and a crack growth rate
behavior given by Equation 9.7.5 as:

_{}

The residual strength-crack
length relationship will not be affected by the operational stress level; thus,
the *K*_{cr} = 30 ksi √in
curve in Example 9.7.1 describes the relationship
for this example.

The corresponding crack
growth-life curves for characteristic stress levels (_{}) of 15, 20, and 25 ksi are presented. As anticipated, the highest stress produces
the fastest crack growth-life behavior.
Based on Equation 9.7.4, the curves are related to each other by a life
factor given by

_{}

where the lives *L*_{1} and *L*_{2} are calculated at the same crack length (any choice
of *a*_{cr} applies) for
characteristic stress levels _{}and _{}.

EXAMPLE
9.7.3 __Effect of
Pre-Exponential Constants__

This example and Example 9.7.4 collectively consider the effect of
modifying the material’s crack growth rate response on the residual strength
capability. In both examples, the
baseline conditions stated in Example 9.7.1 are
used unless otherwise specified.

It is noted that the crack
growth rate resistance can be changed independent of the fracture toughness
(fracture resistance), so that the residual strength-crack length relationship
is again given by the *K*_{cr}
= 30 ksi √in curve in Example 9.7.1.

In a somewhat decoupling
fashion, the effect of varying the coefficients in the growth rate equation,
(Equation 9.7.2)

_{}

are considered separately. In this example, only the constant *C* is varied to reflect decreasing the
crack growth rate response in a systematic manner from the baseline condition
where *C* = 1310^{-8}. In Example 9.7.4,
the effect of varying the exponent *n*
is considered.

A change in the constant *C* is equivalent to shifting the crack
growth rate curve to a new position but with the same slope. If the constant *C* is reduced by a factor of 2, i.e. *C*=0.5310^{-8},
then the growth rate *da/dN* decreases
by a factor of 2.

Variation of Crack Growth
Behavior Resulting from

a Shift in the Power Law Curve

Based on an analysis of
Equation 9.7.4, it is seen that the life difference that results from a change
in *C* can be expressed as a life ratio

_{}

Thus, if a baseline crack
growth-life curve and a baseline residual strength-life curve exist, new curves
can be generated by factoring the lives from the baseline condition to the new
material conditions using this equation.

The figure below describes
the residual strength-life curves for the baseline and two lower values of the
constant *C*. From the figure, it is seen that the increased crack growth
resistance, i.e. lower *C* values,
results in slower rates of residual strength decay. The new curves are exactly a factor of two and of four removed
from the baseline curve.

Effect of Constant *C* on Residual Strength-Life Relationship

Increasing the material’s
crack growth resistance has an immediate effect of increasing the number of
flights (amount of flight hours) until the residual strength capability decays
to the residual strength requirement.

EXAMPLE
9.7.4 __Effect of Exponential
Constant__

In this example, the
exponential constant (*n*) in Equation
9.7.2 is varied along with the constant *C*
to reflect a defined rate of crack growth (_{} = 10310^{-6}
in/cyclic unit) for a given characteristic stress-intensity factor level of _{} = 10 ksi Öin. The baseline constants of *n* = 3.0 and *C* = 1310^{-8}
yield an equation which passes through the point (10, 10310^{-6}). This figure illustrates the three choices of
*n* considered in this example.

Crack Growth Curves Shown
Passing Through Common Point

For this baseline geometry,
fracture toughness level, and stress level, the characteristic stress-intensity
factor varies between about 15 and 30 ksi Öin as a result of the
crack growth change. When the common
point for the power law equations is located at a stress-intensity factor level
that corresponds to a crack length within the crack length interval associated
with the life calculation, one can not immediately interpret the effect of the
crack growth rate behavior. However,
based on the crack growth rate behavior defined in the figure, the curve with *n* = 3.5 will yield crack growth rates
faster than the baseline throughout the crack length interval of interest. Thus, for the conditions stated, an engineer
would expect a more accelerated crack growth behavior and a more rapidly
decaying residual strength behavior for the *n*
= 3.5 material than for the baseline.
The following figures bear out this expectation.

Effect
of Exponent *n* on the Crack Growth
Life Relationship

Effect
of Exponent *n* on the Residual
Strength-Life Relationship

One observation made in
studying the residual strength-life behavior presented in the figure is that
the decay in residual strength is slightly nonlinear in the long life region
for the two non-baseline crack growth rate behaviors. For the *n* = 2.5
material, the residual strength-life curve is slightly concave up while the *n* = 3.5 material produces a slightly
concave down shape. Thus, a second
factor that produces nonlinear decay effects is the exponent *n*.
Generally speaking, nonlinear decay effects would be expected when the
crack growth rate behavior can not be described by a power law equation with *n* = 3.
While the nonlinear behavior is evident, it is important to note that it
is slight. As a result, local regions
of the residual strength life curve can be easily described by linear line
segments, and the procedures presented in Examples
9.7.1, 9.7.2, and 9.7.3
can be utilized to extrapolate from a segmented baseline curve.

The rate of decay in residual strength as a function of service
loading has been shown by the above examples to be an important function of
material behavior and of load level.
The residual strength decay rate can also be significantly affected by
geometric parameters and loading conditions.
In Example 9.7.5, the effect of global and
crack geometry is considered; and then in Example 9.7.6,
the effect of localized fastener loading is evaluated.

EXAMPLE
9.7.5 __Effect of Geometrical
Parameters__

Using the
through-the-thickness, radially-cracked, open hole geometry (shown in Example 9.7.1) as the baseline geometry, two other
geometrical configurations are considered: (1) the through-the-thickness,
center-crack and (2) an open hole with a radial crack which transitions from a
one-quarter-circular, corner-crack shape to a through-the-thickness-crack
shape. In all cases, the width of the
structure is considered sufficient so that it does not influence the
results. Baseline material properties (*K*_{cr}, *C*, and *n*), initial crack
length (*a*_{o}), and
characteristic stress level (_{}) are as defined in Example 9.7.1
and apply to all three geometries. The
center crack geometry does not have a central (starter) hole; its total initial
length is 2*a*_{o}. The radius (*r*) of the hole with the transitioning crack is 0.125 inch, the same
as the baseline geometry.

The information presented at
the introduction of this section described how the residual stress
relationships could be developed using Equations 9.7.1 and 9.7.4 and the
stress-intensity factor coefficient for the geometry. The only factor that changes as a function of geometrical
parameters is the stress-intensity factor coefficient; Example
9.7.1 provides this coefficient for the baseline case. For the through-the-thickness, center crack
configuration in an infinite plate, the stress-intensity factor coefficient is
given by

_{}

The
case of the transitioning corner crack requires that the crack growth shape be
known throughout the interval of crack growth.
The stress intensity factor solution for this geometry is given in
Section 11.

When
the stress-intensity factor coefficients for the given geometries are utilized
in conjunction with Equation 9.7.1, the residual strength-crack length
relationships are determined (*K*_{cr}=30
ksi Öin). As expected, the transitioning corner crack
geometry exhibits residual strength that is greater than that of the
through-the-thickness crack geometry (baseline) for shorter cracks. For crack lengths greater than 0.250 inch,
the transitioning radial crack and baseline configurations exhibit the same
residual strength (since the stress-intensity factor coefficients are the same
here). One interesting feature of this
plot is that the residual strength of the center crack configuration is higher
than the radially cracked holes for short crack lengths but rapidly decreases
with crack length and eventually falls below the residual strength exhibited by
the cracked hole. One might puzzle
through this observation by noting that the center crack has a total length of
2*a*, whereas the radially cracked hole
has an equivalent length of (*a*+*2r*).

Effect of Geometry on the Residual
Strength-Crack Length Relationship

Equation
9.7.4 was utilized to calculate the crack growth life relationships for the
three geometries and these are shown below.
Because the stress-intensity factor for the through-the-thickness radial
crack is initially higher than those of the other two configurations, the
baseline configuration exhibits the fastest crack growth behavior. The transitioning radial-corner-crack configuration
initially exhibits slower crack growth behavior than the baseline but
eventually these two crack growth curves become parallel (when the
stress-intensity factor is the same, i.e. when *a *> 0.250 inch). The
center crack configuration exhibits the slowest initial growth behavior, and
this is primarily because the stress-intensity factor for small crack lengths
is substantially below that of the other two configurations.

By cross-correlating the information presented in
these figures, one is able to construct the residual strength-life
relationships shown in the next figure.
As anticipated, the baseline configuration has the lowest residual
strength capability and the center crack configuration exhibits the highest
residual strength capacity. Both the
baseline and center crack configurations are also shown to exhibit an extensive
region of linear residual strength decay as a function of time-in-service. The nonlinear residual strength decay
exhibited by the transitioning radial corner crack is attributed to the
gradient in the stress-intensity factor coefficient for relatively short cracks
and the transition to a through crack.
Based on observations in this and other examples in Section
9.7, it would appear that one of the most important factors contributing to
nonlinear behavior is the severity of stress-intensity factor gradient (as a
function of crack length).

Effect
of Geometry on the Crack Growth-Life Relationship

Effect of Geometry on the
Residual Strength-Life Relationship

EXAMPLE
9.7.6 __Effect of Hole
Loading__

As
a means of evaluating the effect of fastener loading on the residual strength,
this example combines the baseline remote loading configuration described in Example 9.7.1 with localized pin loading shown
here. To calculate residual strength,
the baseline material properties are utilized in Equations 9.7.1 and 9.7.4
along with the stress-intensity factor associated with the combined loading.

Because the structural response
is linear elastic, stress-intensity factor solutions for the remote and
localized loadings can be added to obtain the stress-intensity factor for the
combined loading; thus,

*K*_{Total }= *K*_{remote }+*K*_{local}

where *K*_{remote} is obtained from the product of the remote
stress (_{}) and the stress-intensity factor coefficient for a wide
plate given in Example 9.7.1, so that:

_{}

and _{}

where _{}

*K*_{local} is the stress-intensity factor associated with
the pin loading. From Section 11, this
stress intensity factor is given by

_{}

where

_{}

and

_{}

As a method of comparing
the effect of pin loading in conjunction with remote stress loading, the
bearing to bypass ratio was used. The
bearing to bypass ratio is the ratio between the bearing stress (*P/2rt*) and the remote stress _{}, i.e.

_{}

Various
combinations of this ratio were chosen and the residual strength relationships
were then generated using Equations 9.7.1 and 9.7.4. The residual strength-crack length relationships are shown for *K*_{cr}=30 ksi √in; and,
the crack length-life are shown for *C*=1310^{-8} and *n*=3.0. By cross-correlating
this information, one can generate the residual strength-life relationships, as
shown.

Effect
of Pin Loading on the Residual Strength-Crack Growth Relationship

Effect
of Pin Loading on the Crack Growth-Life Relationship

Effect
of Pin Loading on the Residual Strength-Life Relationship

Based on the results
presented in these figures, it would appear that bearing to bypass ratios less
than 0.4 cause a relatively small change in the residual strength/crack
length/life relationships. As the
bearing to bypass ratio increases from 0 to 2, (a) the residual strength decays
very rapidly in the short crack region, (b) a significant reduction occurs in
the crack growth-life curves, and (c) the residual strength-life curves are
progressively lower. The collective sum
of these observations indicate that when substantial hole loading is present,
it is necessary to account for the hole loading when assessing the residual
strength and crack growth life behavior.