In this subsection analytical procedures are presented for the
residual strength capability analyses.
Methods for evaluating the unknown fastener force and the
stress-intensity factors for the stiffened panel are presented. Since the equations for the solution
procedures have been based on linear elastic fracture mechanics, the failure
criterion used in these analyses are also based on fracture toughness values
for abrupt fracture conditions and KR
resistance curve data for tearing fracture conditions.
Analysis methods for stiffened panels have been developed
independently by Romualdi, et al 1957], Poe [1970, 1971], Vlieger , Swift
and Wang , Swift , Creager and Liu , and Wilhem and Ratwani
Application of the stress intensity factor parameter, b, and the stringer load concentration factor, L, were proposed by Vlieger  and
Swift and Wang .
From the residual strength capability analysis as discussed in
the preceding subsections, it is evident that the construction of residual
strength diagrams for built-up structures also requires the estimation of the
stress-intensity factor K. A number of approaches for determining K have been developed. Solutions for complicated structural
geometries can sometimes be obtained from the basic stress field solutions
combined with displacement compatibility requirements for all the structural
members involved. This approach has
been shown by several investigators to be useful in the analysis of built-up
sheet structure. While the analysis is
based on closed form solutions, the actual analyses are computerized for
efficient solutions. The essentials of
this technique are described below.
In calculating b
and L, two methods can be used. There are the finite-element method and an
analytical method based on closed-form solutions. The analytical method has advantages over the finite-element
method in that the effect of different panel parameters on the residual
strength of a certain panel configuration can be easily assessed, so that the
stiffened panel can be optimized with respect to fail-safe strength. It allows direct determination of the
residual-strength diagram. In the case of
the finite-element method, a new analysis has to be carried out when the dimensions
of certain elements are changed because a new idealization has to be made. An advantage of the finite-element analysis,
on the other hand, is that such effects as stringer eccentricity, hole
deformation, and stringer yielding can be incorporated with relative ease. Details of the calculations can be found in
the referenced papers.
The procedure for analytical calculation is outlined in Figure 4.5.6.
The stiffened panel is split up into its composite parts, the skin and
the stringer. Load transmission from
the skin to the stringer takes place through the fasteners. As a result, the skin will exert forces F1, F2, etc., on the stringer, and the stringer will exert
reaction forces F1, F2, etc. on the skin. This is depicted in the upper line of Figure 4.5.6.
Figure 4.5.6. Analysis of Stiffened Panel
The problem is now reduced to that of an unstiffened plate
loaded by a uniaxial stress, s,
and fastener forces F1 . . . Fn. This case can be considered as superposition
of three others, shown in the second line of Figure
uniformly loaded cracked sheet.
sheet without a crack, loaded with forces F1
. . . Fn.
cracked sheet with forces on the crack edges given by the function p(x).
The forces p(x) represent the
load distribution given by Love .
When the slit CD is cut, these forces have to be exerted on the edges of
the slit to provide the necessary crack-free edges.
The three cases have to be analyzed individually. For case a, the stress-intensity factor is K
For case b, K = 0. The stress intensity for case c is a
complicated expression that has to be solved numerically. However, once the K value for case c is determined, the stress-intensity factor for
the whole stiffened panel can be obtained by adding the K values for cases a and b.
The determination of K
requires calculations of fastener forces F1,
F2 . . . Fn.
To calculate these forces, the displacement compatibility conditions
which require equal displacements in sheet and stringer at the corresponding
fastener locations, can be used. These
compatibility requirements deliver a set of n
(n = number of fasteners) independent
algebraic equations from which the fastener forces can be obtained. These equations can be solved numerically
using Gauss-Seidal or Gauss-Jordan iterative methods.
The number of fasteners to be included in the calculation
depends somewhat upon geometry and crack size.
According to Swift  and shown in Figure
4.5.7, 15 fasteners at either side of the crack seems to be sufficient to
get a consistent result. Similar
results were obtained by Sanga .
Swift’s analysis provides a detailed description of how to incorporate
nonelastic behavior in this kind of analysis.
The method can account for (1) stiffener flexibility and stiffener
bending, (2) fastener flexibility, and (3) biaxiality. Stringer yielding, fastener flexibility, and
hole flexibility are lumped together in an empirical equation for fastener
Figure 4.5.7. Effect of Number of Fasteners Included in
Analysis on Calculated Stress-Intensity Factor
The effect of fastener flexibility and stiffener bending on b and L is shown
in Figure 4.5.8.
Although the effects are quite large, the vertical position of the
crossover of critical stress-intensity factor curve and stringer stress curve
is not affected too much (compare points A and B in Figure
4.5.8). The level of the crossover
determines the residual strength, as pointed out in the previous
subsections. This explains why the
residual strength can be reasonably well predicted if the flexibility of the
fasteners is neglected.
Figure 4.5.8. Skin-Stress-Reduction b and Stringer-Load-Concentration L as Affected by Fastener Flexibility
and Stiffener Bending
In the case of adhesively bonded stiffeners, the displacement
compatibility approach was used to calculate the fastener loads F1, F2 . . . Fn. The adhesive was considered by dividing it
into a series of discrete segments. The
forces F1, F2 . . .
Fn correspond to the segments shown in Figure
4.5.9. Using an appropriate computational
method as explained for riveted fastener, the unknown fastener forces can be
evaluated. The method of superposition
results in an expression in terms of a complex integral for the
stress-intensity factor. A typical
residual strength diagram for a bonded structure as compared to the riveted
structure is shown in Figure 4.5.10. The required expressions and the solution
techniques are discussed in the example problem for a riveted skin-stringer
combination with a central crack in the skin.
Figure 4.5.9. Bonded Fastener Divided into Discrete Segments
Figure 4.5.10. Residual Strength Diagram Comparing Riveted
and Bonded Structures