The residual strength diagram of a simple panel with two stringers and a central crack can be constructed as follows.  Consider first a crack in plane stress, which starts propagating slowly at s_o = K_onset/Öpa_o and becomes unstable at s_c = K_c/Öpa_c in a sheet without stringers as shown in Figure 4.5.1a.

When the panel is stiffened with stringers, the stress-intensity factor is reduced to K = bsÖpa where b < 1.  As a result, both the stress for slow stable crack growth, s_o, and the stress for unstable crack growth, s_f, are altered to give s_o = K_onset/Öpa_o  and s_cf = K_c/Öpa_c, respectively.

Hence, these events take place at higher stresses in the stiffened panel than in the unstiffened panel.  This means that the lines in Figure 4.5.1a are raised by a factor 1/b for the case of the stiffened panel, as depicted in Figure 4.5.1b.  Since b decreases as the crack approaches the stringer, the curves in Figure 4.5.1b turn upward for crack sizes on the order of the stringer spacing.

Figure 4.5.1.  Elements of Residual Strength Diagram

The possibility of stringer failure should be considered also.  The stringer will fail when its stress reaches the ultimate tensile stress (s_UTS).  As the stringer stress is Ls, where s is the nominal stress in the panel away from the crack, failure will occur at s_sf, given by Ls_sf = s_UTS.  Using L, a measure of the load transferred to the stringer, the panel stress at which stringer failure occurs is shown in Figure 4.5.1c.  The stringer may yield before it fails.  This means that its capability to take overload from the cracked skin decreases.  As a result, b will be higher and L will be lower.  The stress-intensity analysis should account for this effect.

Figure 4.5.2 shows the residual strength diagram of the stiffened panel.  It is a composite of the critical conditions shown in Figure 4.5.1.  In the case when the crack is still small at the onset of instability (2a <<2s, where 2s is stringer spacing), the stress condition at the crack tip will hardly be influenced by the stringers and the stress at unstable crack growth initiation will be the same as that of an unstiffened sheet of the same size (Point B in Figure 4.5.2).  When the unstably growing crack approaches the stiffener, the load concentration in the stiffener will be so high that the stiffener fails (Point C) without stopping the unstable crack growth (line BC).

Figure 4.5.2.  Residual Strength Diagram for a Stiffened Panel

When the panel contains a crack extending almost from one stiffener to the other (2a  2s), the stringer will be extremely effective in reducing the peak stress at the crack tips (b small), resulting in a higher value of the stress at crack growth initiation.  With increasing load, the crack will grow stably to the stiffener (line LMIF) and due to the inherent increase of stiffener effectiveness, the crack growth will remain stable.  Fracture of the panel will occur at the same stress level corresponding to the point F due to the fact that the stiffener has reached its failure stress and the stress reduction in the skin is no longer effective after stringer failure.

For cracks of intermediate size (2a = 2a₁), there will be unstable crack growth at a stress slightly above the fracture strength of the unstiffened sheet (point H), but this will be stopped under the stiffeners at I.  After crack arrest, the panel load can be further increased at the cost of some additional stable crack growth until F, where the ultimate stringer load is reached.

Since b and L depend upon stiffening ratio, the residual strength diagram of Figure 4.5.2 is not unique.  Figure 4.5.2 shows the case where stringer failure is the critical event.  For other stiffening ratios, skin failure may be the critical event as depicted in Figure 4.5.3.  Due to a low stringer load connection, the curve e and g do not intersect.  A crack of size 2a₁ will show stable growth at point B and become unstable at point C.  Crack arrest occurs at D from where further slow growth can occur if the load is raised.  Finally, at point E, the crack will again become unstable, resulting in panel fracture.  It is, therefore, obvious then that a criterion for crack arrest has to involve the two alternatives of stringer failure and skin failure, and these depend upon the relative stiffness of sheet and stringer.

Figure 4.5.3.  Panel Configuration with Heavy Stringers; Skin-Critical Case

The foregoing clearly shows that for crack arrest it is not essential that the crack run into a fastener hole.  Crack arrest basically results from the reduction of stress-intensity factor due to load transmittal to the stringer.

For the particular case depicted in Figure 4.5.4, the residual strength is not determined by stringer failure solely but also by fastener failure (point K).  A crack of length 2a₁ will show slow growth from E to F and instability from F to G.  After crack arrest at G, further slow growth occurs until at point K the fasteners fail.  The latter could cause panel failure, but this cannot be directly determined from the diagram.

Figure 4.5.4.  Criterion for Fastener Failure

In fact, a new residual strength diagram must now be calculated with omission of the first row of rivets at either side of the crack.  Fastener failure will affect load transmittal from the skin to the stringer: line e will be lowered, line g will be railed.  The intersection point H¢ of the new lines g¢ and e¢ may still be above K and hence, the residual strength will still be determined by stringer failure at H¢.

In reality, the behavior will be more complicated due to plastic deformation.  Shear deformation of the fasteners, hole deformation, and plastic deformation of the stringers will occur before fracture takes place.  Plastic deformation always reduces the ability of the stringer to take load from the skin that implies that line g in actuality will be raised and line e will be lowered.  The intersection of the two lines (failure point) will not be affected a great deal, however, (compare points H and H¢ in Figure 4.5.4).  For this reason the residual strength of a stiffened panel can still be predicted reasonably well, even if plasticity effects are ignored.  Nevertheless, a proper treatment of the problem requires that plasticity effects be taken into account.