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 K_R 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 [1973], Swift and Wang [1969], Swift [1971], Creager and Liu [1971], and Wilhem and Ratwani [1974].

Application of the stress intensity factor parameter, b, and the stringer load concentration factor, L, were proposed by Vlieger [1973] and Swift and Wang [1969].

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 . . . F_n.  This case can be considered as superposition of three others, shown in the second line of Figure 4.5.6.  Namely:

a.       A uniformly loaded cracked sheet.

b.      A sheet without a crack, loaded with forces F₁ . . . F_n.

c.       A 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 [1944].  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 = sÖpa.  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 . . . F_n.  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 [1974] 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 [1974].  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 deflection.

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 . . . F_n.  The adhesive was considered by dividing it into a series of discrete segments.  The forces F₁, F₂ . . . F_n 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