In single load path structures, the residual strength analysis
involved only one failure criterion for a given structural geometry. In built-up structures, due to the complex
geometrical configuration, one or more failure criterion may have to be
considered in the determination of residual strength for the whole
structure. The following paragraphs
examine these aspects of the residual strength analysis of built-up structures.

It was explained earlier that safety can be achieved by
designing aircraft structure either as slow crack growth or as fail-safe. The latter case can further be classified
into two cases: Multiple Load Path and Crack Arrest. Typically, both Multiple Load Path and Crack Arrest structures
are built-up structures. In Section
1.3, the definitions and requirements for these two types of built-up structure
are discussed. For completeness, the
structure shown in Figure 4.3.2 is analyzed to
further explain the features inherent in multiple load path, built-up
structure.

As long as the central member is not failed, all three elements
carry a share of the total load *P*. In the event of failure of the center
member, the total load *P* (actually
1.15*P*) must be transmitted by the
other two members at the instant of failure, if the structure is to stay
intact.

The residual
strength capability for the multiple load path structure shown in Figure 4.3.2 can be explained with Figure 4.3.3.
When one element fails, Figure 4.3.3 shows
that the remaining parallel members are able to carry the required load without
failure. The residual capability is
shown to degrade as the crack in the central member extends and as the cracks
in the remaining elements fail. Figure 4.3.3 shows the discontinuous change in the
strength capability as a result of element failures. Since the load levels in other members dramatically increase, if
the load *P* must be maintained, the
remaining members will have short lives.
Thus, the second member may fail after the time (*t*_{2}). The
residual strength capability is shown to drop below the safe level somewhere in
time between *t*_{1} and *t*_{2}. The duration of the time interval between the failure of the first element and the failure of the structure may
be short or long depending on the “type of failure” of the first member
and the load requirements subsequent to this failure. This time interval is available for the detection of the failure
of the first member and the repair of the structure.

**Figure 4.3.2.** Multiple Load Path (Built-up) Structure with a Crack in the
Central Member

**Figure 4.3.3.** Reduction of Residual Strength During
Successive Failure of Members in the Structure Shown on Figure 4.3.2

The failure stress or the critical flaw size level of the
central member (any one of the parallel members) can be estimated by treating
the problem in a manner similar to the single load path structure. Using a fatigue crack growth analysis, the
crack propagation curve is obtained from the minimum detectable crack size to
the critical crack length as illustrated in Figure 4.3.4. In multiple load path structure, partial
failure of the structure can occur during its operating period. But this failure must be detected at an
inspection before catastrophic failure of the entire structure occurs. A suitable inspection schedule must include
analysis of structural characteristics along with the operational requirements
for the intervals between inspections.

**Figure 4.3.4.** Crack Growth for Multiple Load Path Structure Shown in Figure 4.3.2

To illustrate the analysis involved in the estimation of
residual strength of complex structures, consider an axially loaded
skin-stringer combination with longitudinal stiffening as shown in Figure 4.3.5.
Assuming that the fasteners are rigid, the displacements of adjacent
points in skin and stringers will be equal.
(If skin and stringers are made from the same material, the stresses in
the two will also be equal for the case of no crack.) Let a transverse crack develop in the skin. This will cause larger displacement in the
skin, and the stringers must follow this larger displacement. As a result, they take load from the skin,
thus decreasing the skin stress at the expense of higher stringer stress. Consequently, the displacements in the
cracked skin will be smaller than in an unstiffened plate with the same size of
crack. This implies that the skin
stresses are lower and that the stress-intensity factor is lower. The closer the stringers are to the crack,
the more effective is the load transfer.

**Figure 4.3.5.** Skin-Structure Built-Up Structure

If the stress-intensity factor for a small crack in an
unstiffened panel is approximated by *K = **sÖpa*,
the stress-intensity factor for the stiffened plate will be *K = **bsÖpa*. The reduction factor, *b = K/**sÖpa*,
will decrease when the crack tip approaches a stringer. Since the stringers take load from the skin,
the stringer stress will increase from *s*
to *L**s*,
where *L* increases as the crack tip
approaches the stringer. Obviously, 0
< *b* __<__ 1, and *L* __>__ 1. These values depend upon stiffening ratios,
the stiffness of the attachment, and the ratio of crack size to stringer
spacing. As will be shown subsequently,
*b* and *L* can be readily calculated; at this
point it is sufficient to note that *b*
and *L* vary with crack length as shown
in Figure 4.3.6.

**Figure 4.3.6.** Variation of *B* and *L* with Crack Length
in Stiffened Panel with a Crack Between the Stiffeners

Due to the complexity of stiffened skin structure, the
construction of a residual strength diagram is considerably more
difficult. Consider first the condition
where an abrupt failure in the skin occurs.
When the crack is small as compared to the stiffener spacing, the
residual strength of the skin is not influenced by the stiffeners and the
initial portion of the diagram follows the plot for an unstiffened panel (see
point A in Figure 4.3.7). Once the crack size is long enough that the skin cannot sustain
the applied load any further, the stringer will take some of the load from the
skin, thus decreasing the skin stress.
Consequently, the crack-tip stress-intensity factor will be lower due to
the reduced stress and so the residual strength of the skin structure will
increase with crack length as shown in Figure 4.3.6. As the crack size increases further toward
the stiffener location, the load transferred from the skin to the stiffener
also increases significantly, thus reducing the stress-intensity factor. The residual strength of the stiffened panel
continues to increase as shown in the figure for longer cracks. It can also be noted from the figure that
the residual strength diagram for an unstiffened panel would have followed the
dotted line, i.e., the continuous decay in the residual strength as the crack
size increases. This is because there
is no inherent feature present in the single load path structure to decrease
the crack tip stress-intensity factor.

**Figure 4.3.7.** Residual Strength of the Cracked Panel as a
Function of Crack Length for Built-Up Skin-Stiffened Structure Compared with
Unstiffened Panel. Abrupt Failure
Criterion Used to Determine Residual Strength

The residual strength diagram for the skin-stiffened structure
is repeated in Figure 4.3.8 where several additional
points of interest are defined for the analyst. For a structure with a crack of length *a*_{A}, the residual strength is identified as point A. Since point A is associated with a failure
stress that is above the peak stress *(**s*_{peak}), the crack extends abruptly and
completely fails the panel. If the
structure contains a crack of length *a*_{C},
in the range between *a*_{B}
and *a*_{D}, the crack extends
abruptly but then arrests at crack length *a*_{E},
where the residual strength available is greater than the applied (failure)
stress. This crack extension and arrest
feature of skin-stringer construction greatly facilitates meeting inspection
requirements for fail-safe structures.

**Figure 4.3.8.** Residual Strength of the Cracked Panel as a
Function of Crack Length for Built-Up Skin Stiffened Structure. Only Skin Failure Mode Considered. Abrupt Failure Criterion Used to Determine
Residual Strength

Before the panel fails completely, the failure stress level at
point C/E must be increased to the level associated with point F, i.e. to *s*_{peak}.
As the stress is increased above the level of point E, the crack extends
from *a*_{E} to maintain an
equilibrium between the input stress and the residual strength. When the stress reaches *s*_{peak},
the crack has extended to *a*_{F},
at which point the crack abruptly extends causing failure of the panel. A schematic illustrating the load crack length
diagram observed during an abrupt crack extension/arrest situation in a
skin-stringer structure is presented in Figure 4.3.9. Thus, it is seen that the residual strength
curve ABCDEF shown in Figure 4.3.8 can be replaced
for all practical purposes with a curve that connects points ABF.

**Figure 4.3.9.** Load-Crack Length Behavior Observed in Skin-Stiffened
Construction with Arrest Features

In the design of fail-safe structure, a frequent objective is
to design the structure for limiting or arresting unstable crack growth so that
catastrophic failure can be prevented.
A number of arrest techniques are described in Bluhm [1969], Romauldi
& Sanders [1959-1960] and Broek [1974].
The fundamental concept in crack arrest design is to provide within the
structure a means to reduce the crack tip stress intensity factor. This concept requires the use of additional
stiffening members such as stiffeners, reinforcing rings, etc., to produce a
decrease in the stress. These are
inherently present in built-up structures, such as aircraft wings, fuselages,
etc..

In general, the residual strength analysis of a structure with
crack arrest capabilities may involve more than one failure criterion. For instance, in a stiffened skin structure
or an aircraft wing, the analysis should consider stringer failure, fastener
failure, and skin crack failure criteria.
Built-up panels loaded to fail-safe levels tend to exhibit substantial
local deformations of critical elements.
Failure criteria are thus dependent also on elastic-plastic deflection
allowables for both fastener and skin/stringer elements. Gunther and Wozumi [1982] provide additional
details on the residual strength analysis of complex panels based on the
ultimate stringer strain.

The residual strength
diagram for the structure that exhibits slow crack growth behavior will contain
two curves as shown in Figure 4.3.10. The lower curve corresponds to the critical
level of stress at which slow crack extension starts. The onset of slow tearing is then described by this lower curve.
The upper curve provides the critical stress level at which the unstable
rapid crack extension occurs.
When the crack approaches the stiffener, as explained earlier, the
residual strength levels, corresponding to the onset of slow cracking and the
rapid extension, start increasing.

**Figure 4.3.10.** Residual Strength of Cracked Panel as a
Function of Crack Length for Built-up Skin-Stringer Structure. Tearing Failure Criterion Used to Determine
Residual Stress

For a crack length *a*_{i},
as shown in Figure 4.3.10, the slow crack extension
begins at point B. This stable
extension continues up to point B¢ where the rapid failure is supposed to occur. However, due to the continuous rise in the
residual strength of the stiffened panel, the stable crack extension continues
to occur beyond point B¢ and up to point C.
Since the residual strength of the panel starts reducing at this point,
any further increase in the applied load will lead to the rapid unstable crack
extension.

The construction of the residual strength diagram follows the
three steps presented in Section 4.3.1.
Due to the complexity of the structural geometry, however, estimating
requires the calculation of the loads that are transferred to the stiffening or
secondary members from the main load carrying member of the structure. Depending upon the complexity, the *K* vs. *a* curves can be obtained either
through an appropriate numerical method or through the method of superposition.
The methods for constructing
residual strength diagrams and for the residual strength capability analyses
are further discussed in the following sections with various example problems.