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 s_f, for the critical crack length a_c, using the relationship

where K_cr is the known value of fracture toughness of the material.  (K_cr may be equal to K_Ic or K_c depending on the problem.)  The plot of s_f vs. a_c 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 (s_f) values for critical crack sizes (a_c) are obtained utilizing the results of the first two steps and residual strength diagram (s_f vs. a_c) 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.

SOLUTION:

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 (s_f) and critical crack size (a_c), the s_f vs. a relationship is given by

For a half crack size (a_c) of 2.0 inch, the fracture strength (s_f) is about 16 ksi.  Other (s_f  vs. a_c) 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 K_cr = s_fÖpa_c.  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 s_f  vs. a_c curve.