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 so
and becomes unstable at sc
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, so,
and the stress for unstable crack growth, sf,
are altered to give so =
Konset/Öpao and scf
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 (sUTS).
As the stringer stress is Ls, where s is the nominal stress in the panel away from the crack,
failure will occur at ssf, given by Lssf = sUTS.
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.
and L depend upon stiffening ratio,
the residual strength diagram of Figure 4.5.2 is not
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¢
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.