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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
        • 0. Residual Strength
        • 1. Introduction
        • 2. Failure Criteria
        • 3. Residual Strength Capability
          • 0. Residual Strength Capability
          • 1. Single Load Path Residual Strength Diagrams
          • 2. Built-Up Structure Residual Strength Diagrams
        • 4. Single Load Path Structure
        • 5. Built-Up Structures
        • 6. References
      • 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
    • Examples

Section 4.3.1. Single Load Path Residual Strength Diagrams

For a single load path structure, such as an unstiffened panel, the residual strength diagram under plane strain conditions, consists of a single curve as shown in Figure 4.3.1.  The procedure for developing the residual strength diagram involves the calculation of the critical stress sf, for the critical crack length ac, using the relationship


where Kcr is the known value of fracture toughness of the material.  (Kcr may be equal to KIc or Kc depending on the problem.)  The plot of sf vs. ac then provides the necessary residual strength diagram required in design analysis for the simple configuration.

Figure 4.3.1.  Residual Strength Diagram for Abrupt Failure of a Single Load Path Structure

The available fracture mechanics solution techniques, as given in Section 11, can be employed in the calculation of the crack-tip stress-intensity factor K to construct the residual strength diagram.  Depending on the complexity of the structure, K can be calculated either numerically or through closed form solutions.  These techniques, in conjunction with an appropriate failure criterion, can then be used to determine the residual strength capabilities of a given structure.

In general, the construction of a residual strength diagram involves three steps:

(a)    The development of the relationship between the applied stress s, the crack length parameter a, and the applied stress-intensity factor K for the given structural configuration (see Section 11).

(b)   The selection of an appropriate failure criterion based for the expected material behavior at the crack tip (see Section 4.2.1).

(c)    The fracture strength (sf) values for critical crack sizes (ac) are obtained utilizing the results of the first two steps and residual strength diagram (sf vs. ac) for the given structural configuration is plotted.

To understand these three steps for constructing a residual strength diagram, the following example is considered.  The example considers a wide thin panel with a central crack that has a simple relationship for the stress intensity factor.  This example illustrates the importance of the stress-intensity factor for constructing the residual strength diagram.


Construct the residual strength diagram for the wide unstiffened panel shown here, assuming that the structure is made from 7075-T6 aluminum sheet material, with a fracture toughness of 40 ksiÖin.



Step 1.  Define the stress-intensity factor relationship.  From Section 11, the stress intensity factor for a wide unstiffened, center crack panel is given by


Step 2.  Define the failure criterion.  For this problem, it is assumed that an abrupt fracture occurs and the condition that defines the fracture is


Step 3.  Utilize the results of the first two steps to derive a relationship between fracture strength (sf) and critical crack size (ac), the sf vs. a relationship is given by


For a half crack size (ac) of 2.0 inch, the fracture strength (sf) is about 16 ksi.  Other (sf  vs. ac) values can be similarly obtained.  Once a sufficient number of values are available, the residual strength diagram can be developed, or one could also attack the problem in the graphic manner that is explained using the following:

Step 1.  Construct a plot of K vs a by using the equations in Step 1 for various values of stress and crack lengths.

Step 3.  Complete the residual strength diagram.  Utilize the intersection points of the horizontal line with curves where the failure criterion is satisfied, i.e. where Kcr = sfÖpac.  The values of the respective stresses and the crack sizes at these points are termed to be the failure stresses and the critical crack sizes for the given structure, i.e., the unstiffened panel.  The residual strength diagram is finally constructed by plotting the sf  vs. ac curve.