Figure 4.5.16.  Riveted Panel with a Central Crack Between Two Stringers

In this subsection, the specific details are covered which are associated with conducting the stress-intensity factor analysis as well as the analysis to determine the stresses in the stringers and fastener loads.  To simplify the detailed calculations, it is assumed that only one fastener (rivet) on either side of the crack is active, as shown in Figure 4.5.17 and that this rivet is assumed to be rigid.  Thus, there is only one unknown fastener force F transferred between the stringers and the skin by this rivet.

Figure 4.5.17.  Stiffened Structure Broken into Components

Typically, the analysis proceeds by splitting up the structure shown in Figure 4.5.16 into its component parts as shown in Figure 4.5.17.  The unknown force F can be calculated from the displacement compatibility condition between the skin and the stringer.  The complicated expressions which correspond to the displacements V_s, V_F, and V_P due to the applied stress, s, the fastener force F and the distributed pressure P(x), respectively, can be obtained using a procedure suggested by Westergaard [1939] and by Love [1944].  The detailed discussions on the methods of obtaining the required relationships are presented by Broek [1974].  The necessary relationships for V_s, V_F, V_p and V_st (displacement in the stringer) are given as:

	(4.5.6)
	(4.5.7)
	(4.5.8)
	(4.5.9)

where

	(4.5.10)
	(4.5.11)
	(4.5.12)

where

(4.5.13)

 

and

(4.5.14)

 

The geometric variables r, r₁, r₂, q₁, q₂ and q are shown in Figure 4.5.18.  The displacement compatibility condition requires equal displacements in corresponding points of sheet and stringer; it yields the following equation to calculate the unknown fastener force F.

Figure 4.5.18.  Geometrical and Displacement Parameters Relative to the Crack Tip

 

Vs + VF + Vp = Vst

(4.5.15)

substituting the expressions 4.5.6 - 4.5.9 for V_s, V_F, V_p, and V_st in the above relationship, and reassembling, we get

(4.5.16)

The next step is to obtain an expression for the stress-intensity factor for the entire stiffened panel configuration.  Using superposition, the stress-intensity factor is obtained as the sum of the stress-intensity factors for the three cases shown in Figure 4.5.17.  It can easily be seen that for Case I: K = sÖpa and for Case II: K = 0.  The stress-intensity factor (K) for Case III is a fairly complicated expression and it is given by,

(4.5.17)

where

(4.5.18a)

and

(4.5.18b)

where  are normalized with respect to the rivet pitch.  The estimation of K_III requires solution of the above integrals by numerical methods.  Replacing the fastener force F by the expression and rearranging the expression for K_III, the stress-intensity factor K for the stiffened panel then becomes

(4.5.19)

where

(4.5.20a)

and

(4.5.20b)

The stress-intensity factor K can be finally expressed in the following form,

 

(4.5.21)

where

(4.5.22)

 

To calculate K for a given stiffened panel the values of F_s, f_F, f_p, f_st, and l₁ have to be obtained.  These variables are numerically calculated and plotted as shown in Figures 4.5.19 to 4.5.23 for various values of .  For the given example data, we can now construct the residual strength diagram using the values obtained from these plots.

Figure 4.5.19.  Normalized Panel Displacement Function (f_s /p) Due to Applied Stress vs. Normalized Crack Length (a/p) for Various Stringer Spacing (s=S/p)

Figure 4.5.20.  Panel Displacement Function Due to Fastener Force vs. Normalized RivetDiameter (d/p) for All Stiffener Spacings

Figure 4.5.21.  Normalized Panel Displacement Function (F_p /p) Due to Crack Distributed Pressure Along Crack vs. Normalized Crack Length (a/p) for Various Stringer Spacings (s=s/p)

Figure 4.5.22.  Stringer Displacement Function vs. Normalized Rivet Diameter (d/p) for Various Half-Stringer Widths

 

Figure 4.5.23.  Parameter l₁ Vs. Normalized Crack Length (a/p) for Various Normalized Stringer Spacings (s/p)

For a critical crack size (2a) of 4.0 inch, what is the fracture strength and for an operating stress of 20 ksi, what is the critical crack size?

 

Structural Geometry and Material Properties for Example 4.5.1

SOLUTION:

The first step is to obtain the stress-intensity factor by means of Equation 4.5.21 that involves the parameters l₁ and l₂.  For various crack lengths, these two variables can be calculated using Equation 4.5.20.  The calculations involve the values of f_s, f_F, f_p, f_st and l which are obtained from the plots for various values of  for the given  = 20 and d/p = 3/16.  Knowing the values of l₁ and l₂, the geometric parameter b can be estimated from Equation 4.5.22.  It is then straightforward to obtain the K vs. a plot by substituting the sets of values of a and b in the stress-intensity factor equation (K = sbÖpa) for a particular value of the applied stress s.  The

corresponding K vs. a plot is shown for s = 5, 10, and 15 ksi.  This figure shows that the stress-intensity factor decreases rapidly when the crack approaches the stringer.  The figure also shows the effect of stringer to panel thickness ratio on the stress-intensity factor.

Stress Intensity Factor Diagram for Panel and Riveted Stringers

The next step is to apply a failure criterion to evaluate the fracture stresses, s_cr, for various crack sizes.  Assuming that the material exhibits negligible subcritical crack growth, the fracture toughness failure criterion (K = K_cr) based on the plane stress condition can then be applied.  For K = K_c in Equation 4.5.12, s_f can be evaluated for a particular crack size and the corresponding b which was obtained through Equation 4.5.22.  The residual strength diagram, i.e., the plot of s_f vs. a_c for the given data (K_c = 65 ksiÖin), is shown in the following figure.

Residual Strength Diagram for Panel and Riveted Stringers (Light Stringers)

The residual strength curves of the fastener and stiffeners are obtained by combining the equations for fastener failure and the equations stringer failure.  The corresponding equations are given by:

             (Fastener)

and

             (Stringer)

where l is a function of a, and the values of l for various crack lengths can be obtained using the Equation 4.5.16.  To obtain this Equation 4.5.24, note that the maximum stringer load (P_max) is the source of the fastener force (F = sl) and the remote stringer force (sA_s).  The composite residual strength diagram as shown in the figure above contains the three failure curves corresponding to panel, stringer, and fastener.  The stringer failure curve corresponds to a = 1 (light stringer).

For the crack length given (2a = 4 inches), the corresponding residual strength is found from the figure for a half crack length (a) of 2 inches.  Point A in this figure identifies the skin failure condition which occurs at a stress level of 25.9 ksi.  For the operating stress level of 20 ksi, the panel can be effective without catastrophic failure for cracks with length less than the critical crack (a_cr) of 3.4 inch (note 2a_cr = 6.8 inch).  If the panel develops a crack less than a_cr, it will not fail by unstable crack growth.  However, for any other crack size which is equal or

Assume that the panel develops a crack of size a_cr.  At point B in the figure, the crack extends rapidly.  When the rapidly extending crack becomes 15 inches, the stress level in the stiffener (point C) reaches its critical value and the stiffener fails.  Due to the stiffener failure, the stiffener becomes ineffective, leading to the total failure of the panel without any crack arrest possibilities.

In the next figure, the stiffener failure curve is plotted for a strong stiffener with a = 4 (the stiffener thickness if “assumed” four times the panel thickness).  If the panel develops a crack size a_cr, the crack will extend rapidly from point D to point E as shown in the next figure.  At point E, the fastener fails, leading to an ineffective stringer (loads are no longer transferred to the stringer).  Thus, the failure of the panel is unavoidable and the unstable crack growth without effective crack arrest leads to the total failure of the structure.

Residual Strength Diagram for Panel and Riveted Stringers

(Heavy Stringers)