When the cracked thin sheet structure of high fracture
toughness material is considered, the solutions based on linear elastic
behavior for the calculation of residual strength are no longer valid due to
the large scale yielding at the crack tip.
For fail-safe structures with crack arrest capabilities, the residual
strength analysis becomes complicated.
However, using the R-curve based on ÖJR concept as
the failure criterion Ratwani and Wilhem  developed a step-by-step
procedure for predicting the residual strength of built-up skin stringer
structure composed of tough material exhibiting tearing type fractures.
The residual strength prediction procedure is briefly outlined
here to show step-by-step, the required data and analysis. It should not be assumed that by reading
this step-by-step procedure that the uninitiated can perform a residual
strength prediction. It is strongly
recommended that the details of the preceding subsections and Ratwani and
Wilhem  be examined prior to attempting a structural residual strength
analysis based on the following ten procedural steps:
Model the structure for finite-element analysis or use an existing
finite-element modeling remembering –
That structural idealizations are typically two-dimensional,
That no out-of-plane bending is permitted,
To use a proper fastener model (a flexible fastener model for
riveted or bolted structure, or a shear spring model for bonded structure).
To use material property data from skin and substructure of
interest (i.e., E, Ety and
To select the most critical crack location (normally highest
To take advantage of structural symmetry.
Step 2. Select one crack length (2a
or a) of interest (based on
inspection capability or detailed damage tolerance requirement). Based on this “standard” crack length, five
other crack lengths are selected for a Dugdale type elastic plastic
analysis. These crack lengths should be
selected such that crack length to stiffener spacing (2a) ratios vary between 0.15 to 1.1 remembering –
That the greatest variation in J values will take place near reinforcements, and
To select at least one crack size shorter than “standard”.
Step 3. With the finite-element model (from Step 1) and assumed crack lengths
(from Step 2), perform an analysis assuming Dugdale type plastic zones for each
crack size remembering –
To select the first increment of plastic zone length at 0.2
inches and sufficient successive increments (normally 6) to reach
Bueckner-Hayes calculated stresses up to 85 percent to Fty.
To make judicious selection of plastic zone increments so as
to take advantage of overlapping ae
(effective crack length) (e.g., 3.2, 3.5, 4.2, 5.0 inches for a 3 inch physical
crack and 4.2, 4.5, 5.0 inches, etc., for a 4 inch physical crack). If overlapping is done, those cases where
the crack surfaces are loaded throughout the crack length will be common for
two or more physical crack sizes hence the computer programs need be run only
once (e.g. 4.2 and 5.0 inches) thus reducing computer run times.
Step 4. From Step 3, obtain stresses in stiffeners for Dugdale analysis
and elastic analysis. Plot stiffener
stresses as function of applied stress.
Step 5. From the crack surface displacement data of Step 3, plot ÖJ
(obtained by Bueckner-Hayes approach) versus applied stress to Fty ratio for each crack
Step 6. From Step 5, cross plot the data in the form of ÖJ
versus crack size (a) at specific
values of applied stress to Fty
Step 7. Employing the data of Step 4 and the “standard” crack size
determine, gross panel stress to yield strength ratio, s/Fty
at ultimate strength (Ftu)
for the stiffener material - assuming zero slow crack growth. This information will be used subsequently
to determine if a skin or stiffener critical case is operative.
Step 8. Obtain crack growth resistance data for skin material (see Volume
II of reference 26) remembering --
To use thickness of interest (i.e., if the skin material is
chemically milled, use the experimentally obtained R-curve for the same
chemically milled material)
Use proper crack orientation (LT, TL, or off angle)
corresponding to anticipated direction structural cracking.
Step 9. Plot ÖJ
versus DaPHY curve
as shown in Figure 4.5.24 from the data obtained in
Figure 4.5.24. Square Root of Jr Resistance
Step 10. Determine structural residual strength. On the ÖJ versus crack size (a)
plots obtained in Step 6 for the structure, overlay the ÖJR versus DaPHY material plot of Step 9 at the
initial crack length of interest as shown in Figure
4.5.25. Determine if –
Figure 4.5.25. Failure Analysis Based on J critical Curve
At the gross panel stress obtained from Step 7, significant
slow tear (> 0.25 inch) will occur as indicated from the intersection
of the ÖJR
versus DaPHY curve
with the constant s/Fty curve
at a stringer ultimate strength (see Step 7).
Interpolation will probably be necessary between values of constant s/Fty.
Then proceed as follows:
If significant slow tear occurs (> 0.25 inch) the structure
can be considered to be skin critical
(at that particular crack length).
Tangency of ÖJR
versus DaPHY and ÖJ
versus aPHY at constant
applied stress can be used to determine extent of slow tear and residual
strength at failure as a percentage of Fty
If significant slow tear does not occur (DaPHY <
0.25 inch) the structure will normally be stiffener critical. To determine a conservative value of residual
strength (for that crack length) use the Dugdale curve of Step 4 and stiffener