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