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material is considered, the solutions based on linear elastic behavior for the calculation of residual strength are no longer valid due to the large scale yielding at the crack tip.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 ÖJ_R concept as the failure criterion Ratwani and Wilhem [1974] 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 [1974] be examined prior to attempting a structural residual strength analysis based on the following ten procedural steps:

Step 1. Model the structure for finite-element analysis or use an existing finite-element modeling remembering –

a.                   That structural idealizations are typically two-dimensional,

b.                  That no out-of-plane bending is permitted,

c.                   To use a proper fastener model (a flexible fastener model for riveted or bolted structure, or a shear spring model for bonded structure).

d.                  To use material property data from skin and substructure of interest (i.e., E, E_ty and F_tu),

e.                   To select the most critical crack location (normally highest stressed area),

f.                    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 –

a.                   That the greatest variation in J values will take place near reinforcements, and

b.                  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 –

a.                   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 F_ty.

b.                  To make judicious selection of plastic zone increments so as to take advantage of overlapping a_e (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 F_ty ratio for each crack size.

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 F_ty ratio.

Step 7.  Employing the data of Step 4 and the “standard” crack size determine, gross panel stress to yield strength ratio, s/F_ty at ultimate strength (F_tu) 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 --

a.                   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)

b.                  Use proper crack orientation (LT, TL, or off angle) corresponding to anticipated direction structural cracking.

Step 9.  Plot ÖJ versus Da_PHY curve as shown in Figure 4.5.24 from the data obtained in Step 8.

Figure 4.5.24.  Square Root of J_r Resistance Curve

Step 10.  Determine structural residual strength.  On the ÖJ versus crack size (a) plots obtained in Step 6 for the structure, overlay the ÖJ_R versus Da_PHY 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 ÖJ_R versus Da_PHY curve with the constant s/F_ty curve at a stringer ultimate strength (see Step 7).  Interpolation will probably be necessary between values of constant s/F_ty.  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 ÖJ_R versus Da_PHY and ÖJ versus a_PHY at constant applied stress can be used to determine extent of slow tear and residual strength at failure as a percentage of F_ty

If significant slow tear does not occur (Da_PHY < 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 ultimate strength.