• DTDHandbook
• Contact
• Contributors
• Sections
• 1. Introduction
• 2. Fundamentals of Damage Tolerance
• 3. Damage Size Characterizations
• 4. Residual Strength
• 5. Analysis Of Damage Growth
• 6. Examples of Damage Tolerant Analyses
• 7. Damage Tolerance Testing
• 8. Force Management and Sustainment Engineering
• 0. Force Management and Sustainment Engineering
• 1. Force Structural Management
• 2. Sustainment Engineering
• 0. Sustainment Engineering
• 2. Corrosion
• 3. Structural Risk Analysis
• 3. References
• 9. Structural Repairs
• 10. Guidelines for Damage Tolerance Design and Fracture Control Planning
• 11. Summary of Stress Intensity Factor Information
• Examples

# Section 8.2.3. Structural Risk Analysis

The complex combinations of potential cracking within and between structural elements and the unknown state of the fatigue damage in aging aircraft essentially preclude the use of deterministic crack growth calculations for estimating the onset of WFD.  Accordingly, structural risk analyses are being used to quantify structural capability.  Current practice is to express structural risk in terms of the single-flight probability of failure as a function of experienced flights or flight hours from a reference age.  According to Lincoln [2000], in the USAF, the acceptable upper bound on the single-flight failure probability is 10-7.  This degree of risk implies that less than one failure would be expected in any given fleet.

When an airframe enters service, estimates-of-failure probability would be based on the growth of monolithic cracks at the most severe, known critical locations and would be extremely small.  Such estimates would be made on the basis of a probabilistic characterization of initial quality.  Currently, the equivalent initial flaw size distribution is used to model the crack sizes at the critical locations.  In the probabilistic approach to maintenance scheduling, inspections would be planned at intervals that keep the failure probabilities of the monolithic structures below 10-7.  In the aging aircraft scenarios, crack size distributions are obtained for the critical locations in the complete load path as the basis for the estimates of failure probability.  Structural failure probability is then calculated as the conditional probability of inadequate strength given the condition of the elements in the load path.  For example, assume there is a 10-3 probability of a discrete source damage event, such as a sudden fatigue crack linkup across two bays in a fuselage lap joint.  To maintain an overall catastrophic failure probability less than 10-7, the probability of failure in this damaged state must be less than 10-4.  In this example, loss of fail safety can be said to occur at the number of flight hours when the WFD reaches the state at which the probability of surviving the discrete event exceeds 10-4.  The number of flight hours to reach such a state of fatigue cracking has been suggested as the definition of the onset of WFD [Lincoln, 1997].

There are several approaches that can be used to calculate single flight failure probability, but the USAF has available a computer program named PRobability Of Fracture (PROF) for risk analysis in aging aircraft.  PROF is a computer program that runs in the Windows environment on a personal computer and was specifically written to interface with the data that is available as a result of ASIP.  See Berens, et al. [1991] and Hovey, et al. [1998] for complete descriptions of the development of the program and its update to the Windows environment.  Figure 8.2.1 is a schematic of the program for calculating probability of failure as a function of flight hours for a monolithic crack.  The figure illustrates the types of data required to perform an analysis and the probability of failure (POF) output that is calculated as a function of flight hours.  Another calculation module in PROF calculates probability of failure due to a discrete source damage event.

Figure 8.2.1   Schematic of the PROF Computer Program

Under ASIP, crack life predictions (a versus T) are available for every known critical location.  This implies the availability of:

a)    the flight by flight stress spectrum, from which the distribution of maximum stress per flight can be obtained;

b)    stress intensity factors as a function of crack size, a versus K/s; and,

c)    fracture toughness, Kcr, from which a distribution of fracture toughness can be inferred.

The initiating crack size distribution can be obtained from inspection feedback, tear-down inspections, or equivalent initial flaw sizes.  Probability of detection as a function of crack size, POD(a), is from a characterization of the capability of the non-destructive inspection system used during the safety inspections.

The starting point of a PROF analysis can be representative of any arbitrary number of hours in the life of the fleet.  PROF uses the deterministic a versus T curve to project the percentiles of the initiating crack size distribution as a function of flight hours.  At defined flight hour increments, the single-flight probability of fracture is calculated from the distributions of crack size, maximum stress per flight, and fracture toughness.  That is, the single-flight fracture probability is the probability that the maximum stress intensity factor (combination of the distributions of maximum stress per flight and crack sizes) during the flight exceeds the critical stress intensity factor.

At a maintenance cycle, the distribution of crack sizes is changed in accordance with the POD(a) function and the equivalent repair crack size distribution.  It is assumed that all detected cracks are repaired and the equivalent repair crack size distribution accounts for the repaired cracks.  PROF produces files of both the pre- and post-inspection crack size distributions.  The availability of these distributions allows changing the analysis conditions at inspection times set by the analyst.

The a versus T, a versus K/s, and crack size distributions are input to PROF in tabular form.  Fracture toughness is modeled by a normal distribution and requires values for the mean and standard deviation.  Maximum stress per flight is modeled by the Gumbel extreme value distribution and the parameters of the distribution can be obtained from a fit of either a flight- by-flight stress spectrum, or an exceedance curve of all of the stresses in the spectrum.  The POD(a) function is modeled by a cumulative lognormal distribution with parameters m and s.  Fifty percent of the cracks of size m would be detected.  The parameter, s, determines the flatness of the POD(a) function with smaller values implying steeper POD(a) functions.

The module for the calculation of failure probability given discrete source damage also requires an evaluation of the residual strength in the presence of partial structural failure.  Procedures for determining residual strength in the presence of discrete source damage for a number of representative aircraft skin structures can be found in Swift [1993].

Sensitivity studies have been performed on the application of PROF in representative problems [Berens, et al., 1991].  These studies have indicated that, although the absolute magnitudes of the fracture probabilities are strongly dependent on the input, relative magnitudes tend to remain consistent when factors are varied one at a time.  Because of the indefinite nature of some of the input data, particularly the crack size information, absolute magnitudes of the fracture probabilities are suspect.  However, it is believed that relative differences resulting from consistent variations in the better-defined input factors are meaningful.

A single run of PROF analyzes the growth of a crack for a single geometry, including crack type and shape.  The analysis would apply to the population of structural details that both have this geometry and are subject to an equivalent stress spectrum.  The output includes fracture probabilities for a single structural detail, for a single aircraft when there are multiple equivalent details, and for the entire fleet.  The inspection intervals are set by the analyst, including the possibility for an immediate inspection at time zero.

More complex problems can be analyzed by combining the results of multiple runs.  First, intermediate output can be used to initiate new runs for changed conditions.  Examples of such analyses would include the introduction of corrosive thinning of the material, the effect of over-sizing holes during repairs, and the effects of changing usage.  The results from multiple runs for different details can also be combined to model more complex scenarios.  Examples of such scenarios include the analyses of multi-element and multi-site damage.

There are four examples of the application of risk analysis in sustainment scenarios in the Sample Problem Section of the Handbook.  The sample problems addressed are:

a)      Problem No. UDRI-2 – Structural Risk Assessment for a Discrete Source Damage Threat to the Fail Safety Capability of Stringer 7 in a Boeing 707 JSTARS Airframe.

b)      Problem NO. UDRI-3 – Structural Risk Assessment for the Multiple-Element Damage Scenario at WS 405 of a C-141 Airframe.

c)      Problem No. UDRI-4Comparative Risk Assessment of the Thinning Effect of Corrosion on a Representative Lap Joint.

d)      Problem No. NRC-3 – Effect of Discontinuity States on the Risk Assessment of Corroded Fuselage Lap Joints.