Because the linear elastic fracture mechanics approach is based
on elasticity, one can determine the effects of more than one type of loading
on the crack tip stress field by linearly adding the stressintensity factor due to each type of loading. The process of adding stressintensity
factor solutions for the same geometry is sometimes referred to as the
principle of superposition. The only constraint
on the summation process is that the stressintensity factors must be associated
with the same structural geometry, including crack geometry. Thus, stressintensity factors associated
with edge crack problems cannot be added to that of a crack growing radially
from a hole. An example will illustrate
the conditions under which one might linearly add stressintensity factors.
EXAMPLE 11.2.1 Axial and Bending Loads Combined
An edge crack of length a
is subjected to a combination of axial and bending loads as shown. The stressintensity factor for the edge
crack geometry subjected to the tensile load (P) is given by
_{}
while that due to the
bending moment (M) is given by
_{}
with _{}
The stress intensity
factor resulting from the combination of tensile and bending loads is given by
the sum of K_{P} and K_{M}, so that
K_{TOTAL} = K_{P} + K_{M}
Edge Crack Geometry
Loaded With Axial and Bending Loads
As shown by Example 11.2.1, if the
geometry of the structure is described, the effect of each loading condition
can be separately determined and the effect of all the loading conditions can
be obtained by summing the individual conditions, i.e.,
K_{TOTAL} = K_{1}
+ K_{2} + K_{3} + . . .

(11.2.3)

This particular property is quite useful in the analysis of
complex structures. Example 11.2.2 (Wilhelm, 1970) further illustrates the principle of superposition.
EXAMPLE
11.2.2 Remote Loading and
Concentrated Forces Combined
Many times in a particular
aircraft design a part may develop cracks at rivet holes where the skin is
attached to the frame or stringer. This
situation is depicted in the figure below.
It will be analyzed as a simple case in which the sheet is in uniaxial
tension and the rivets above and below the crack are influential in keeping the
crack closed. (Tests of panels with
concentrated forces superimposed on the uniform tension loading simulate crack
growth behavior in the presence of rivets.)
The insert of the figure shows
the local parameters necessary for determining the stressintensity factors.
Crack at Rivet In a
Riveted SkinStringer Panel (No Crack Buckling)
Assuming that a crack grows from the rivet hole, the total
stressintensity factor for this geometry is obtained using the linear
superposition of stressintensity factors.
Closer examination of the figure
indicates that the loading can be decomposed as shown in the next
figure. The total stressintensity
factor is the sum of the remote loading and concentrated load induced stressintensity
factors.
Note: The concentrated
force induced stressintensity factor solution presented is only applicable if
the concentrated forces are applied along the centerline of the sheet and at a
distance greater than 3 or 4 times the hole diameter. Inasmuch as the concentrated forces are in an opposite direction
to the uniform stress, and tend to close the crack, this stressintensity is
subtracted from the uniform extensional stressintensity factor.
With knowledge of the stressintensity solution for this
geometry, it is possible to determine what effect the rivet closure forces have
on the local stress field for similar problems.
Combined

Uniform Tension

Concentrated Force




(a)

(b)

(c)



_{}

=

_{}


_{}












Superposition of Stress intensities for Uniform
Tension and Concentrated Force
In some cases, the additive property of the stressintensity
factor can be used to derive solutions for loading conditions that are not
readily available. The process of
deriving the stressintensity factor for a center crack geometry, which is
uniformly loaded with a pressure (p), shown in Figure
11.2.1, illustrates this feature. Figure 11.2.2 describes the process whereby the
remotely loaded center crack geometry is decomposed into a set of two center
crack geometries which have loading conditions, that when added, result in the
canceling of the crack line loadings.
The stressintensity factor (K_{1}) for the plate loaded
with the remote stress condition (s) and the crack
closing stresses (also equal to s) is zero, i.e. K_{1} = 0, because
the crack is clamped closed under such conditions. Thus, the equation for addition of stressintensity factors
reduces to
so that the stressintensity factor for a pressurized center
crack with pressure (p) equal to s is the same as that
associated with remote loading, i.e.
Figure 11.2.1. Internally Pressurized Center Crack

Remotely Loaded

=

Remotely
Loaded With Crack Line Closing Stresses

+

Crack Line Opening Stresses

K_{Total}

=

K_{1}

+

K_{2}

Figure 11.2.2. Principle of Superposition Illustrated for
Center Cracked Geometry
Sometimes, it is difficult to visualize how one arrives at the
values of the crack closing stresses.
Consider the uncracked body with the uniformly applied remote loading as
shown in Figure 11.2.3a. Determination of the stresses along the
dotted line lead to the observation that the stresses here are equal to the
remote stress (s). To obtain a stressfree condition along the
dotted line, and thus simulate a cracked structural configuration, one must
apply opposing stresses of magnitude s
along the length of the dotted line as shown in Figure
11.2.3b. The stresses along the
dotted line generated by the applied remote stresses are the opening stresses (Figure 11.2.3a). The equal but opposite stresses are the crack closing
stresses. The reader should note that
the stresses on the dotted line that are generated by the remote loading lead
to the crack opening condition; these opening stresses lead to nonzero values
for the stressintensity factor (see Figure 11.2.2).


Figure 11.2.3a. Uniform
Stresses Along Dotted Line Generated by Remote Loading

Figure 11.2.3b Opposing
Stresses Applied Along the Dotted Line

Figure 11.2.4 presents the concept
of linear superposition of elastic solutions in a slightly different way so
that the reader has a full appreciation of the procedure. The structural element B is noted to be
exactly the same as element A; the crack closing stresses exactly balance the
effect of the remote stresses along the line so the structural element B still
experiences uniform tension throughout.
Structural element B is further decomposed into elements D and E. Note that the crack loading stresses shown
on the structural element E are crack closing stresses and, therefore, result
in a stressintensity factor which is the negative of the remotely applied
loading case, i.e. K_{E} =
K_{D}.
Remote Loading W/O Crack


Remote Loading With Crack

Crack Loading Stresses On Crack


K_{A }= 0

K_{B }= 0

K_{B }= K_{D}+K_{E}=0
K_{E }= K_{D}

Figure 11.2.4. Illustration of Superposition Principle
Since K_{D} is known,
it follows that
As we noted before, if the direction of stress in element E is
reversed (becomes crack opening) then the stressintensity factor is
The loading on structural element A in Figure
11.2.5 can be decomposed into the series of loadings shown. The stressintensity factor for element A is
obtained from the superposition of the three other loadings:
K_{A} = K_{B} + K_{D}  K_{E}

(11.2.7)

Since it is obvious that the loadings in elements A and E will
result in the same stressintensity factor, i.e. K_{A} = K_{E},
the stressintensity factor for element A becomes
The stressintensity factors for elements B and D are known,
i.e.
and, therefore
Figure 11.2.5. Application of Superposition Principle
Now a more complex example is presented using the principle of
superposition applied in a twostep process.
Shown in Figure
11.2.6 is a structural element (F) in which intermediate values of
load transfer occur through a pin loaded hole.
As shown, Step 1 consists of decomposing element F into two parts, such
that in one part the pin reacts its entire load and the other part is remotely
loaded. The stressintensity factor for
element F is the sum of those generated by the decomposed elements, i.e.,
where the superscript denotes the loading.
Step 2 involves the determination of K_{A}. The pin reactive
loading on element A is decomposed into the loading shown in Figure 11.2.6. Using the logic previously
illustrated in Figure 11.2.5, K_{A} is
determined as
The stressintensity factor for the loading on element F is
Note that while the stressintensity factor solution formula
for element B is the same in Steps 1 and 2, the stresses used in each
calculation are different (as indicated by the superscripts).
Figure 11.2.6. Stress Intensity Factor for PinLoaded Hole
(Bearing Bypass Problem)
Obtained by
Superposition