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DTD Handbook

Handbook for Damage Tolerant Design

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    • Sections
      • 1. Introduction
      • 2. Fundamentals of Damage Tolerance
      • 3. Damage Size Characterizations
      • 4. Residual Strength
      • 5. Analysis Of Damage Growth
      • 6. Examples of Damage Tolerant Analyses
      • 7. Damage Tolerance Testing
      • 8. Force Management and Sustainment Engineering
      • 9. Structural Repairs
      • 10. Guidelines for Damage Tolerance Design and Fracture Control Planning
      • 11. Summary of Stress Intensity Factor Information
        • 0. Summary of Stress Intensity Factor Information
        • 1. Background of Stress Intensity Factors
        • 2. Methodology For Determining Stress Intensity Factors
          • 0. Methodology For Determining Stress Intensity Factors
          • 1. Principle of Superposition
          • 3. Finite Element Methods
        • 3. Selected Stress Intensity Factor Cases
        • 5. Computer Codes
        • 6. References
    • Examples

Section 11.2.1. Principle of Superposition

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 stress-intensity factor due to each type of loading.  The process of adding stress-intensity 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 stress-intensity factors must be associated with the same structural geometry, including crack geometry.  Thus, stress-intensity 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 stress-intensity 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 stress-intensity 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 KP and KM, so that

            KTOTAL = KP + KM

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.,

KTOTAL = K1 + K2 + K3 + . . .

(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 uni-axial 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 stress-intensity factors.

Crack at Rivet In a Riveted Skininger Panel (No Crack Buckling)

Assuming that a crack grows from the rivet hole, the total stress-intensity factor for this geometry is obtained using the linear superposition of stress-intensity factors.  Closer examination of the figure indicates that the loading can be decomposed as shown in the next figure.  The total stress-intensity factor is the sum of the remote loading and concentrated load induced stress-intensity factors. 

Note:  The concentrated force induced stress-intensity 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 stress-intensity is subtracted from the uniform extensional stress-intensity factor.

With knowledge of the stress-intensity 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 stress-intensity factor can be used to derive solutions for loading conditions that are not readily available.  The process of deriving the stress-intensity 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 stress-intensity factor (K1) for the plate loaded with the remote stress condition (s) and the crack closing stresses (also equal to s) is zero, i.e. K1 = 0, because the crack is clamped closed under such conditions.  Thus, the equation for addition of stress-intensity factors

(11.2.4)

reduces to

(11.2.5)

so that the stress-intensity factor for a pressurized center crack with pressure (p) equal to s is the same as that associated with remote loading, i.e.

(11.2.6)

Figure 11.2.1.  Internally Pressurized Center Crack

Remotely Loaded

=

Remotely Loaded With Crack Line Closing Stresses

+

Crack Line Opening Stresses

KTotal

=

K1

+

K2

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 stress-free 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 non-zero values for the stress-intensity 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 stress-intensity factor which is the negative of the remotely applied loading case, i.e.  KE = -KD.



Remote Loading W/O Crack

 

Remote Loading With Crack

Crack Loading Stresses On Crack

KA = 0

KB = 0

KB = KD+KE=0

KE = -KD

Figure 11.2.4.  Illustration of Superposition Principle

Since KD is known,

 

it follows that

 

As we noted before, if the direction of stress in element E is reversed (becomes crack opening) then the stress-intensity factor is

 

The loading on structural element A in Figure 11.2.5 can be decomposed into the series of loadings shown.  The stress-intensity factor for element A is obtained from the superposition of the three other loadings:

KA = KB + KD - KE

(11.2.7)

Since it is obvious that the loadings in elements A and E will result in the same stress-intensity factor, i.e. KA = KE, the stress-intensity factor for element A becomes

(11.2.8)

The stress-intensity factors for elements B and D are known, i.e.

and

and, therefore

(11.2.9)

Figure 11.2.5.  Application of Superposition Principle

Now a more complex example is presented using the principle of superposition applied in a two-step 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 stress-intensity 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 KA.   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, KA is determined as

 

The stress-intensity factor for the loading on element F is

(11.2.10)

Note that while the stress-intensity 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 Pin-Loaded Hole (Bearing By-pass Problem)

Obtained by Superposition