[American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures,...

American Institute of Aeronautics and Astronautics

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Reliability Based Design and Inspection of Stiffened Panels against Fatigue

Amit A. Kale * ([email protected]) and Raphael T. Haftka †([email protected]) Bhavani V. Sankar‡([email protected])

Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL-32608

This paper addresses the problem of developing optimum structural design and inspection strategy for fatigue crack growth in stiffened panels subjected to uncertainty in material properties and loading. The approach is based on the application of methods of structural reliability analysis. An optimization problem is formulated to minimize expected lifetime cost while maintaining a minimum acceptable reliability level. The effect of structural design and inspection schedule on operational cost and reliability is explored and tradeoff of structural weight and cost between load carrying structural members (skin and stiffeners) and inspections is conducted. The panel is assumed to be under plane stress condition, and stiffeners are modeled as rectangular bars discretely attached to the panel by fasteners. The stress intensity factor is first determined by enforcing displacement compatibility at fastener locations, fatigue crack growth rate is then obtained by numerical integration. Optimization revealed that improving the structure by designing it for multiple load path capability combined with the use of inspections can be very cost effective compared to unstiffened structure.

Nomenclature a = crack size ai = initial crack size ac = critical crack size a50 = crack size at which probability of detection is 50% D = paris model parameter Fc = fuel cost Ic = inspection cost K = stress intensity factor KIC = fracture toughness m = paris model exponent Mc = material manufacturing cost for aluminum Nf = fatigue life W = structural weight Pfth = required probability of failure Pd = probability of detection r = fuselage radius p = fuselage pressure differential h = panel width t = panel thickness Ns = number of stiffeners As = area of a stiffener acY = critical crack length causing yield of net section of panel acH = critical crack length due to hoop stress acL = critical crack length for transverse stress * Graduate Student, Department of Mechanical and Aerospace Engineering, Student Member AIAA † Distinguished Professor, Department of Mechanical and Aerospace Engineering, Fellow AIAA. ‡ Professor, Department of Mechanical and Aerospace Engineering, Associate Fellow AIAA.

46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference18 - 21 April 2005, Austin, Texas

AIAA 2005-2145

Copyright © 2005 by Amit Kale. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


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Nub = number of intact stiffeners Y = yield stress s = stress ψ = geometric factor

I. Introduction he structural integrity of aircraft components is affected by damage, such as fatigue cracks in metal structures or delamination in composite structures. Damage may reduce the residual strength of the structure below what is

needed to carry the flight loads. In a fail-safe design, structural safety can be maintained by inspecting the components and repairing the detected damage. Alternatively the structure can be strengthened by increasing structural thicknesses so that damage never grows to a critical length in the service life —so called safe-life design.

Stiffened panels are popular in aerospace applications. Stiffeners improve the load carrying capacity of

structures subjected to fatigue by providing alternate load path so that load gets redistributed to stiffeners as cracks progress. Typical stiffening members include stringers in longitudinal directions, frame, fail-safe-straps and doublers in circumferential direction of fuselage. Fracture analysis of stiffened panels has been performed by Swift (1984) and Yu (1988). They used a displacement compatibility method to obtain the stress intensity factor due to stiffening. Swift (1984) performed a parameteric study to demonstrate the effect of stiffener area, skin thickness and stiffener spacing on the stress intensity factor of panels. He also discussed failure due to fastener unzipping and effect of stiffening on residual strength of panel.

It is easier to perform reliability based structural optimization of safe-life structures than of fail-safe structures,

because the optimization of the former involves only structural sizes, while for the latter the inspection regime also needs to be optimized. Nees and Canfield (1998) and Arrieta et al. (2000) performed safe-life structural optimization of F-16 wing panels to obtain minimum structural weight for fatigue crack growth under a service load spectrum.

For fail-safe design, reliability-based optimization has been applied to the design of the inspection schedules. Harkness et al. (1994), Provan et al. (1994), Fujimoto et al. (1998), Toyoda-Makino (1999), Enright et al. (2000), Wu et al. (2000), Garbatov et al. (2001), and Wu et al. (2003) developed optimum inspection schedules for a given structural design to maintain a specified probability of failure. Brot (1994) demonstrated that using multiple inspection types could minimize cost.

Backman (2001) also used multiple inspection types to develop an optimum inspection schedule. However, he

also considered the tradeoff between the cost of inspection and the cost of additional structural weight for maintaining the same probability of failure. Using approximate relationship between structural weight and damage propagation, he concluded that increasing structural weight is more cost effective than increasing inspection frequency.

Our previous work (Kale et al. 2002, 2003, 2004) demonstrated the combined structural design and optimization

of inspection schedule of an unstiffened panel. The objective of this paper is to perform reliability based design and inspection scheduling of stiffened structures and demonstrate that if structures are designed together with the inspection schedule, then cost of additional structural weight can be traded against the cost of inspections. In addition we also seek to demonstrate the tradeoff of weight between the stiffeners and the skin and show that multiple load path structure can be safer and cheaper than having single load path. We use the method described by Swift (1984) to calculate the stress intensity factor of centrally cracked stiffened panel containing through the thickness crack.

Section II outlines the damage growth in structure subjected to cyclic pressure loading. Section III illustrates the

use of probabilistic method to incorporate uncertainty in material data and damage growth and the effect of inspections on cracks sizes and fatigue life. Section IV outlines a general approach for calculating inspection schedule for fixed level of reliability. Section V describes the combined optimization of structural design and inspection schedule for reducing cost and Section VI describes the results and conclusions.

T


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II. Damage growth Model A. Fatigue crack growth

We consider the case of fatigue crack growth in a stiffened panel with initial crack size of ai subjected to load cycles with constant amplitude. We focus on fuselage structure where the main fatigue loading is due to pressurization, with stress varying between a maximum value of s to a minimum value of zero in one flight. Like many other researchers(e.g., Tisseyre et al., 1994, Harkness et al. (1994), Yang et al., (1987), Lin et al., (2000), Soboyejo et al. 2001), we assume that damage growth follows the Paris law (Equation 1).

( )mKDdNda ∆= (1)

where a is the crack size, N is the number of cycles (flights), dNda

is the crack growth rate, D and m are material

parameters related by Equation 2 ( )47.122.3 −−= meD (2)

relation between D and m was obtained by fitting exponential function to values from Pierie et al. (1981) for 7075-T651 aluminum alloy (e.g., Harkness et al.1994).The stress intensity factor range K∆ for cracked stiffened panel is calculated as a function of stress s and crack length a following Swift. (1984) is explained in Appendix A.

aK πψσ=∆ (3.1) The effect of stiffening on the stress intensity is characterized by the geometric factor ψ which is the ratio of stress intensity factor with stiffeners to that of stress intensity of an unstiffened panel. The crack size after N cycles of fatigue loading (aN) can be obtained by solving Equation 3.2 for aN .

( )∫=N

i

a

am

aD

daNπψσ

(3.2)

Here we focus on designing fuselage for fatigue failure caused by hoop stressess. We consider the effect of fail-safe stiffening members in circumferential direction such as frame, fail-safe straps and doublers by modeling them as one dimensional rods attached to the panel by fasteners. The hoop stress is given by

ss ANth

rph+

=σ (3.3)

where r is the fuselage radius , p is the pressure differential inside fuselage, h is the panel width, t is panel thickness, Ns is the number of stiffeners and As is the area of single stiffener. The stiffened panel geometry is shown in Figure 1. The panel size is assumed to be small compared to the fuselage radius so it is modeled as a flat panel. The critical crack length ac at which failure will occur is dictated by considerations of residual strength or crack stability. Here we assume that failure occurs at a crack length that fails one of the two criteria.

Figure 1. Fuselage stiffened panel geometry and applied loading in hoop direction (crack grows Perpendicular to the direction of hoop stress)


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−−=t

ANYtrphha sub

cY 5.0 (4.1)

2

=

πψσIC

cHK

a (4.2)

2

2

=π

σIC

cLK

a (4.3)

( )cLcHcYc aaaa ,,min= (5) the critical crack length for determining failure of the structure is obtained from Equation 5. Equation 4.1 gives the crack length acY at which the residual strength of the panel will be less than yield stress. Y is yield stress and Nub is the number of intact stiffeners. Equation 4.2 determines the critical crack length for failure due to hoop stress s and Equation 4.3 determines the critical crack length for failure due to transverse stress which is approximately half of the hoop stress. This is required to prevent fatigue failure in longitudinal direction where skin is the only load carrying member. The critical length is taken as the minimum of Equation 4 and failure occurs if the crack size aN after N flights is greater than the critical length (ac). Typical material properties for 7075-T651 aluminum-alloy most commonly used in aerospace application are presented in Table 1. The applied load due to fuselage pressurization is assumed to be 0.06 MPa. B. Response surface approximation for ψ For fatigue life calculation it is assumed that only three stiffeners adjacent to crack centerline are effective in reducing the stress intensity factor. So, we model the aircraft fuselage structure by a periodic array of through the thickness center crack with three stiffeners on either sides of centerline. For reliability based optimization calculations it will be sufficient to consider the geometry of Figure 1. Typical values of design variables are listed in Table 2. As the crack propagates in a stiffened panel, load is transferred out of the cracked sheet into the intact stiffeners through the fasteners. The resultant stress intensity factor at the crack tip is much less than that of an unstiffened panel for same cracks length. Stiffeners can break during crack growth if stiffener strength is exceeded causing it to break thereby increasing the stress intensity factor. To estimate the fatigue life we first determine the crack tip stress intensity factor using the method described by Swift, 1984. The stress intensity factor in center cracked stiffened panel is given by Equation 3.1and calculated in Appendix A. Computational cost of calculating the geometric factor is reduced by using response surface approximation described in appendix B.

Table 1: Fatigue properties of 7075-T651 Aluminum alloy, Pierie et al. 1990, (Harkness et al. 1994)

Property Y (yield stress)MPa

ai (meters) mean,

iaµ

standard

deviationiaχ

m mean,

iaµ

standard

deviationiaχ

Load (MPa)

Distribution Type , mean,

standard deviation

500.0 Lognormal

iaµ =0.0002,

iaχ =0.00007

Lognormal

mµ =2.97

mχ = 1.05

Lognormal µs = 0.06

COV = 5%


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The geometric factorψ accounts for the effect of stiffening and depends on crack length a, panel dimensions, stiffener dimensions and stiffener spacing. Due to symmetry for a given crack length and stress, one of these cases might exist, (a) all stiffeners intact (b) two inner stiffener broken (c) four inner stiffener broken (d) all stiffeners broken. The response curves for geometric factor ψ for typical aircraft panel dimensions are shown in Figure 2. Since the applied stress is a random variable itself, computation of exact geometric factor from the displacement compatibility method can be extremely time consuming for calculating crack growth during the reliability based optimization. To reduce the computational cost we use response surface approximation to estimate ψ at crack tip and maximum force on stiffener as a function of plate thickness, stiffener area and crack length (Appendix B).

To take into account the state of stiffener (broken or intact) in computing fatigue life, we first obtain the maximum number of stiffeners that could have been broken at the given state of structure (structural design, crack length) and use the appropriateψ . To do this we construct four RSAs for ψ and three RSAs for maximum force on stiffeners for each of the cases above. The maximum stress on each stiffener is then calculated using the three RSAs for

maximum stress on each stiffenermax

enerFirstStiffF , max

fenerSecondStifF , max

enerThirdStiffF . If this stress exceeds yield stress of

material than the stiffener is broken. Depending on state of stiffener the appropriate RSA for SIF is used, e.g. if none of the stiffeners are broken than ψ0_broken , if all stiffeners are broken than ψ = ψ3_broken

Effect of stiffeners on the geometric factor

0

0.5

1

1.5

2

2.5

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

half crack length (m)

Geo

met

ric

fact

or All stiffeners

intact

2 innerstiffenersbroken4 innerstiffenersbrokenall stiffenersbroken

Figure 2. Response curves for effect of stiffening on geometric factor ψ for a stiffener area of 1.5 mm2 and skin thickness of 1.5 mm


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III. Computation of fatigue life and crack sizes C. Inspection Model When the structure is subjected to periodic inspections, cracks are detected and repaired or the structural part is replaced. Following Harkness et al. 1994 we assume that the probability of detecting a crack of length a is given by Palmberg equation. (Palmberg et al., 1987)

Pd(a) = β

β

+

50

50

1aa

aa

(6)

Where a50 = 1.0 mm is the crack length with 50% probability of detection and β = 3 are inspection parameters obtained from Harkness et al. (1994). The probability of detection for different crack length is shown in Figure 3.

Table 2: Structural design for fuselage (Source: Swift 1984, Jane’s All the World’s Aircraft

1995-1996 Airframe Structural Design by Niu, M.C.Y.) Fuselage Radius ,r 3.25 m

Length ,l 68.3 m Panel width ,h 1.72 m

Frame Spacing, b 0.6 m Number of stiffeners, Ns 6 Number of fasteners per stiffener 10 Fastener spacing , s 3.1 cm Rivet diameter, d 4.8 mm Stiffener thickness ,ts 5 mm Number of panels, Np 1350

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

normalized crack length

prob

abili

ty o

f de

tect

ion

Figure 3. Probability of detection curve for inspection

a/a50


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D. Estimating crack sizes using response surface approximation The uncertainty in material properties (ai, m) and applied stress (s) can be incorporated into the structural design

and optimization by using probabilistic methods. The probability of failure at any time is the probability that the damage is greater than a critical size ac, and that it is not detected in all the previous inspections. Our previous work (Kale et al. 2002, 2003, 2004) used the adaptation of first order reliability method (FORM) proposed by Harkness et al. (1994). The key idea of their approach is that instead of updating the distribution of crack sizes after each inspection, FORM is adapted to take into account the effect of inspections directly. The underlying assumption is that components repaired after inspection will not fail again, thus greatly reducing the computational cost of the reliability analysis. The results from this approximation are accurate if the first inspection time is after 50% of the service life because the replaced components will not fail before the end of service. If the first inspection time is before half of the service life, there is a substantial probability that the replaced components will fail before the end of service life.

Estimating crack size distribution after inspection

When inspection and replacement of structural components are scheduled the damage size distribution changes because defective parts are replaced with new parts having smaller value of damage size. Updating the damage distribution after an inspection can be easily done by using Monte Carlo simulation (MCS) with large sample size. The crack size aN after N cycle of fatigue loading is given by solving Equation 3.2. To obtain the crack size mean and standard deviation after an inspection conducted after N cycles of fatigue loading (flights), we produce 50000 random numbers for each of random variable in Equation 3.2 (ai, m, s) and obtain the final crack size aN. To simulate the effect of inspections in updating the probability distribution of cracks we produce uniformly distributed random numbers Pdtrue between 0 and 1 for each of the 500000 final cracks ,aN. If the probability of detection of the inspection obtained using Equation 6 is greater than Pdtrue then crack is detected and replaced by another new component with same initial crack distribution (ai , Table 1) . After all cracks are analyzed for detection, the updated crack sizes are used to fit a lognormal distribution and obtain the mean and standard deviation. This serves as initial crack distribution for next inspection. Here we assume that inspections do not change the type of distribution and that damaged components are replaced by new components with original assumed distribution. For an unstiffened panel the computational cost for calculating crack size distribution after inspection is very low and is calculated by Monte Carlo simulation during the optimization. However, for stiffened panel we use response surface approximation (RSA) constructed by fitting data obtained from MCS at some locations in design space (Appendix C).

IV. Calculating an inspection schedule Calculating failure probability The probability of failure after N cycles of fatigue loading following the nth inspection is

( ) ( )( )cnif aaNaPnNP ≥= ,,, (7)

where ai,n is the crack size distribution after nth inspection and ac is the critical crack given by Equation 5. This probability is calculated by the first order reliability method (FORM) and the crack size distribution parameter after the inspection is obtained by fitting a lognormal distribution using Monte Carlo simulation. For a given structural thickness optimum inspection times are obtained such that the probability of failure before the inspection is just equal to the maximum allowed value (Pfth, reliability constraint). The probability of failure decreases after the inspection because cracks are detected and repaired. With number of cycles of loading the failure probability increases until it hits the threshold value again defining the next inspection. To illustrate this method, inspections schedule developed for a 2.00 mm thick unstiffened panel for a required reliability level of 10-7 is shown in Table 3. The inspection times are determined that Pf (S1) = Pfth, Pf (S2) = Pfth, etc. Using Equation 7, the first inspection time S1 is obtained by solving Equation 8.

( )( ) 01 =−= thc PfaSaP (8)

After S1 is obtained, the crack size mean and standard deviation ai(S1) after the inspection is determined by Monte Carlo simulation (Appendix A). This becomes the updated initial crack size distribution for the second inspection. The second inspection time S2 is determined by solving Equation 9 with S2-S1 as effective crack growth time.


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( )( )( )2 1 1, 0i c thP a S S a S a Pf− = − = (9)

The crack size distribution is again updated and third inspection time S3 is determined by solving Equation 10

( )( )( )3 2 2, 0i c thP a S S a S a Pf− = − = (10)

where ac is the critical crack size. Other inspection times are obtained similarly. The probability term ( )( )ci aSaP = is obtained using the FORM method .Equations 8 through 10 can be solved for

times Si by using the bisection method. To find if the number of inspections is adequate; the probability of failure at the end of service (40,000 flights) is calculated using Equation 7. If it is not, additional inspections must be added.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

10-50

10-40

10-30

10-20

10-10

100

number of cycles

failu

re p

roba

bilit

y Pf without inspectionPf with inspections

Figure 4. Variation of failure probability with number of cycles for a 2.00 mm thick unstiffened panel with inspections scheduled for Pfth = 10-7

Table 3: Inspection schedule and crack size distribution after inspection for an unstiffened plate thickness of 2.00 mm and a threshold probability of 10-7

Number of Inspections,

n Inspection Time

(flights) Inspection Interval (flights) 1−− nn SS

Crack size distribution after inspection (Mean,

COV)

0 -- -- Initial crack distribution (2.00 mm, 35%)

1 9,288 9,288 (0.300mm, 86% ) 2 15,540 6,252 (0.326mm, 90%) 3 20,741 5,201 (0.335mm, 87%) 4 26,223 5,482 (0.342mm, 87%) 5 31,649 5,426 (0.345mm, 86%) 6 37,100 5,451 (0.347mm, 86%)


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Figure 4 illustrates the variation of probability of failure with and without inspection for a given structural design and threshold reliability level. Table 3 presents the inspection times and the crack size distribution parameters after the inspections. It can be seen that inspections are very helpful in maintaining reliability of structure. From Table 3 it can be seen that first inspection interval is largest. After the first inspection the repaired components are replaced with same initial crack distribution (mean = 2.0 mm and COV = 35%), however some cracks could have escaped detection. This leads to smaller intervals with time until the inspection intervals become constant. From the crack size distribution parameter shown in last column of Table 3 we can conclude that the crack size distribution after each inspection essentially remains unchanged after a certain number of inspections are done, leading to uniform inspection intervals. We can infer that towards the end of service the rate at which unsafe cracks are introduced in the structure due to replacement is same as the rate at which cracks are detected by the inspections.

V. Combined optimization of structural design and inspection schedule

We seek to demonstrate the trade offs between increasing the structural thickness and performing additional inspections to maintain reliability. We assume a single inspection type costing $1,000,000. Cost associated with change in structural weight for aluminum is obtained from Venter (1998) for metals are shown in Table 4. The service life is assumed to be 40,000 flights. The application of proposed methodology is demonstrated for fuselage structure. The structural weight is assumed directly proportional to panel thickness. It is desired to optimize the structure and inspection schedule for minimum life cycle cost.

The life cycle cost is calculated as

Cost = Mc W + Fc W Ns + N i Ic (11) and

( )ρthbbANNW ssp += (12)

Where W = Fatigue controlled structural weight Mc = material and manufacturing cost Fc = Fuel cost t = Skin thickness Ns = Service Life (in number of flights) Ic = Inspection Cost Ni = Number of Inspections Np = Number of panels During the optimization the structural thickness t and the stiffener area As are changed which changes the structural weight according to the Equations 12. The optimum inspection schedule is determined for this structural design and the total cost is obtained from Equation 13. Reliability based design optimization requires the computation of failure probability for scheduling inspections during fitness evaluation of structural design. Computational cost is very high because of large number of iterations on design variables to find the best possible design. Optimum combination of

Table 4: Cost factors Material and manufacturing cost per pound (Mc)

$ 150.0 for Aluminum

Fuel cost per pound per flight (Fc)

$ 0.015

Density of Aluminum (?) 2670

3mkg

Inspection Cost (Ic) $ 1,000,000 Inspection parameters

(a50 , ß ) (1mm, 3)


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skin thickness, stiffener area, inspection times need to be determined for minimizing cost. To reduce computational cost for scheduling inspections and evaluating reliability, we uses response surface approximation (RSA) to predict the crack size mean and standard deviation after an inspection as a function of skin thickness, stiffener area, standard deviation in applied stress, initial crack mean, initial crack standard deviation and crack growth time. The reliability index is also approximated by RSA in same 6 variables (Appendix C). The optimal design can be obtained by finding minimum total cost of structural design plus cost of inspection schedule required to maintain the required safety level for that design. Matlab’s fmincon function was used to do the optimization.

VI. Results Structural design can have large effect on operational cost and weight of the structure. When inspections and maintenance are not feasible safety can be maintained by having conservative (thicker) structural design. To demonstrate this we first obtain safe-life design required to maintain desired level of reliability throughout the service life for unstiffened and stiffened structures. Table 5 shows the safe life design of and unstiffened panel and Table 6 shows the safe-life design of a stiffened panel. We study the tradeoff between skin and stiffener weights by calculating the ratio of total s tiffener area to that of total area of skin plus stiffeners. As = area of six stiffeners on the panel ATotal = cross sectional area of skin panel + AS

Unstiffened panel design is a typical example of a single load path structure. This design does not have multiple load transfer capability so that failure of a single component will lead to failure of entire structure. From Table 5 it can be seen that such a design will be very heavy because thicker design is required to maintain required safety levels. Comparing this to Table 6 we can see that if structure is designed with multiple load transfer capability then the cost can be reduced by about 10 – 17 % and the weight can be reduced by same amount. Next, we demonstrate the effect of inspections on structural safety and operational cost. Inspections improve the reliability by detecting and removing cracks. If this effect is taken to optimize the structural design together with inspection schedule, then structural weight could be traded against inspection cost to reduce overall operational cost. To demonstrate the effectiveness of inspections optimum structural design and inspection schedule was first obtained for an unstiffened panel design as shown in Table 7. It can be seen that inspection and repair can play a

Table 5: Safe–Life design of an unstiffened panel Required

probability of failure

Minimum required skin

thickness (mm)

Life cycle cost $ × 106

Structural weight, lb

% increase in cost of

improving reliability by a factor of 10

10-6 3.95 24.62 32829 -- 10-7 4.08 25.42 33902 3.25 10-8 4.20 26.16 34880 2.91 10-9 4.24 26.34 35129 0.68

Table 6: Safe–Life design of a stiffened panel Required probability of failure

Total Stiffener Area 10-3 m2

Skin thickness (mm)

100Total

s

AA

% Life cycle cost $ × 106

Structural weight, lb

% increase in cost of improving reliability by a factor of 10

10-7 2.10 2.168 36.04 21.09 28122 -- 10-8 2.10 2.192 35.68 21.20 28277 0.52 10-9 2.44 2.286 38.23 23.03 30710 8.63


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vital role in maintaining structural safety at lower cost. Typically, about 25% saving in operational cost compared to safe-life unstiffened panel design and 20% cost saving compared to stiffened panel safe-life design can be achieved if inspections are scheduled at optimum times to enhance their effectiveness in damage detection. When stiffeners are added (Table 8), it is found that their effect is smaller than that without inspections. The reasons is that when the primary load carrying member is too thin (compared to safe life design in Table 5) then most of the load will be transferred to stiffeners causing them to break during crack growth thereby decreasing safety. Even then, design of structure with multiple load paths with inspection schedule can lead to substantial savings compared to both the single load path design with inspection or safe-life multiple load path design without any inspection.

It can be seen that the cost of improving the reliability by a factor of 10 decreases with increase in reliability level as shown in last column of each table. This is because structures which are already too safe will require little extra improvement to make them safer, but unsafe structure will require more improvement to make them safer (e.g. cdf of normal distribution). Table 9 presents the exact evaluation of failure probability without any RSA for the optimum obtained from Table 8. It can be seen that RSAs can be used to predict the optimum with sufficient accuracy at very low computational cost.

Table 7: Optimum structural design and inspection schedule of an unstiffened panel

Required probability of failure

skin thickness (mm)

required number of inspections

Optimum Inspection Times (flights)


Structural weight


10-7 2.30 3 12346,22881, 31365 17.28 19109 lb -- 10-8 2.43 3 13158,23496,31496 18.15 20199 lb 5.03 10-9 2.56 3 13927,24016,31682 18.97 21295 lb 4.51

Table 8: Optimum structural design and inspection schedule for stiffened panel

Required probability of fa ilure

Total Stiffener Area (AS) × 10-4 m2

required skin thickness (mm)

100TotalAAs

% Optimal Inspection Times (flights)


Structural weight


10-7 6.82 1.87 17.43 13774,22496, 31263

17.13 18803 lb --

10-8 6.90 1.98 16.78 14405,23018,31574 17.82 19763 lb 4.03 10-9 8.24 1.98 19.41 14511,23088,31593 18.31 20412 lb 2.75


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Table 9 and 10 show the exact calculation of failure probability for the optimums obtained from the RSA for reliability index. From Table 9 it can be seen that RSA leads to a more conservative design leading to heavier structure. Cost can be reduced by scaling down the design variables. The last column on table 9 shows the percentage of structural weight that can be reduced by scaling the design variables to maintain the required reliability level. For the optimum design with inspections shown in Table 10 a different RSA for reliability index was used and it can be seen that it gives an unconservative design and structural weight will have to be increased by to achieve the desired reliability level. This discrepancy is caused because of inaccuracies in the RSA’s. Here, response surfaces are used at three stages (1) Geometric factor ? (2) Crack sizes after inspection (3) reliability index. The geometric factor ? follows a pattern shown in figure 3 and can be improved by using separate RSA for each section of spacing between stiffeners. The error in failure probability can be improved by fitting PSF (probabilistic sufficiency factor) instead of reliability index.

Conclusion Combined optimization of structural design and inspection schedule was conducted using response surface approximation to reduce computational cost. Optimum combination of structural design and inspection schedule was obtained that will maintain the desired reliability level during service at minimum cost. The important outcome of this research is

Table 10: Exact evaluation of structural reliability for optimum obtained from RSA for stiffened panel with inspection

Total Stiffener Area (AS) × 10-4 m2

Minimum required skin thickness (mm)

100Total

s

AA

% Inspection Times and Pf before Inspection

Actual Pf at end of service life = 40000 flights

% change in structural design variables from RSA required for accuracy

6.82 1.87 17.43 Pf13774 = 4.02 × 10-

7, Pf22496 = 5.3 × 10-

8 , Pf31263 = 5.8 ×

10-7

5.91 ×10-7

15.5

6.90 1.98 16.78 Pf14405 = 1.04 × 10-

7, Pf23018 = 8.79 × 10-9

, Pf31674 = 5.11 × 10-8

6.12 ×10-8 10.2

8.24 1.98 19.41 Pf14511 = 4.25 × 10-

9, Pf23088 = 1.95 × 10-9

, Pf31593 = 2.43 × 10-9

2.27 ×10-9

8.1

Table 9: Exact evaluation of structural reliability for optimum obtained from RSA for stiffened panel without any inspection

probability of failure from RSA

Total Stiffener Area 10-3 m2

Skin thickness (mm) Actual probability of failure

Structural weight

% change in structural design variables from RSA required for accuracy

10-7 2.10 2.168 7.66 ×10-9

19109 lb 8.2

10-8 2.10 2.192 3.01×10-9 20199 lb 4.7

10-9 2.44 2.286 5.57 ×10-11 21295 lb --


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(1) Designing structures for multiple load transfer capability can be much more cost effective and failure resistant than single load path structures

(2) Inspections are most effective in maintaining reliability levels through damage detection and replacement. There is a 25 % in cost saving due to inspection over a no-inspection design

(3) Inspection interval becomes closer with time initially due to deterioration in structural safety with service usage until stabilizing to constant intervals towards the end of life.

(4) For higher reliability levels it will be more useful to use multiple load transfer structure with inspections.

Appendix A. Stress intensity factor for stiffened panel

The displacement compatibility method developed by Swift, 1984 and Yu 1988 is used to obtain the stress intensity factor for stiffened panel. The fastener forces are first determined by enforcing the displacement at fastener locations and their affect on reducing ψ is then evaluated. The effect of stiffening is measured by the geometric factor ψ which is the ratio of stress intensity factor with stiffening and that without stiffening. We consider a stiffened panel applied to a uniform remote tension. The panel is assumed to be in a state of plane stress and the stiffeners are assumed to be one dimensional rods placed symmetrically across the crack. It is also assumed that stiffeners are discretely attached to the panel by fasteners which are located symmetrically about the crack so that only a quarter space is needed for the analysis. The displacements in the panel at fastener locations are obtained by superposition of four cases

(1) V1, the displacement anywhere in the cracked sheet caused by the applied gross stress. (2) V2, the displacement in the uncracked sheet resulting from fastener loads, F. (3) V3, the displacement in the cracked sheet resulting from stress applied to the crack face equal and opposite

to the stresses caused by rivet loads.

(4) Stiffener displacement at location yi resulting from direct fastener load, Ey i

Gi

σδ =

Figure A1: Half section of stiffened panel


14

Figure A2: Description of displacements on stiffened panel due to Displacement V1 The displacement in the cracked sheet resulting from overall gross stress can be determined using Westregaard’s stress function as follows.

( )

+

−−+−

+

= yrr

yrvrr

EV ν

θθθ

θθσ

21

21

21211

22cos1

2sin2 (A1)

Displacement V2

The stress distribution anywhere in and infinite plate resulting from concentrated force F can be determined from work of Love, 1944. Superposing the forces due to rivet load, the resulting displacement can be obtained as follows


15

( ) ( )( )

( ) ( )( )

( ) ( )( )

( ) ( )( )

( ) ( )( )

−+

−

−+

−

−+

+

−+

−−

+

+−+−

−−

++++

+

+

+−+−

−−

++++

+

−+=

−−

−−

12

tan1

2tan

12

tan1

2tan

31

4

11

log111

log1

11

log111

log1

1631

,,,

221

221

221

221

22

22

22

22

22

22

22

22

2

AB

BB

BB

BB

BA

AA

AA

AA

BB

ABB

BB

ABB

BA

AAA

BA

AAA

jjii

XYY

YXYY

Y

XYY

YXYY

Y

YXYX

XYXYX

X

YXYX

XYXYX

X

EBF

yxyxV

νν

πνν

(A2) Where,

( ) ( ) ( ) ( )jiBjiAjiBjiA yyD

YyyD

YxxD

XxxD

X +

=−

=−

=−

=

2,

2,

2,

2

the ith term is the point at which displacement is required and the jth term reflects coordinates of the forces. Displaceme nt V3

( ) ( ) ( )

+−= ∫

a

iijjjj dbbyxbyxFEB

yV

023 ,,,,

21 εα

πν (A3)

Where

( ) ( ) ( )( )

( )[ ]( )

( )[ ]222

2

222

2

2222

221113

,,jj

j

jj

j

jjjjjj

yxb

xb

yxb

xb

yxbyxbbyx

++

+−

+−

−−

+++

+−

++

=νν

α

and ( ) ( ) ( ) ( ) 5.02

5.02

5.05.0 ,,, axrDaxrCaxrBaxrA iiiiii −−=++=+−=−+=

Stiffener displacement resulting from direct fastener load

+= ∑∑

=

+=

=

+=

nj

ijji

ij

njjjD FyyF

AEi

2

11

1δ (A4)

Stiffener displacement due to far field stress

Ey i

Gi

σδ = (A5)

For displacement compatibility


16

iDδ - ( ) i

n

jj GVVVF δσ −=−∑

=1

132 (A6)

The displacement of fasteners and the stiffeners at fastener locations are also calculated. Compatibility condition is enforced by requiring that the skin displacement is equal to stiffener plus fastener displacement at each fastener location. The set of linear equations can be solved to obtain fastener forces. The effect of these point forces at fastener location on the crack tip stress intensity factor can be calculated by Love’s solution. The geometric factor ψ is the ratio of this stress intensity factor with stiffeners to that of an unstiffened panel. A complete description of procedure can be found in Swift, (1984) and Yu, (1988).

B. RSA for geometric factor ψ due to stiffening The RS for geometric factorψ due to stiffening can be obtained by first computing it for some selected design points in the domain using the displacement compatibility analysis. The design of experiments for fitting the RS to ψ is obtained by dividing the domain into equispaced grid with 8 levels of 3 design variable, plate thickness, stiffener area, and crack length. A cubic response surface is fitted to these points by minimizing the square error between the predicted and the true value at the selected locations. The polynomial used for fitting a cubic response surface is shown in Equation B1. Table B1 shows the bounds on design variables used to construct the design of experiment, the bounds on stiffener area and skin thickness were obtained by successive refinement and bound on crack length is based on maximum possible crack length for the unstiffened panel safe life design shown in Table 5. Table B2 show the error estimate for the RS used to approximateψ .

( )

3211922318

123173

22161

22153

21142

2113

3312

3211

3110329

318217236

225

2143322110321 ,,

xxxbxxb

xxbxxbxxbxxbxxbxbxbxbxxb

xxbxxbxbxbxbxbxbxbbxxx

++

+++++++++

++++++++=ψ

(B1)

Table B1: Bounds on design variables used to evaluate response surface approximation for safe life design

Design Variable

Upper bound Lower bound

Skin thickness ,

mm ,x1

2.73 2.11

Stiffener area, m2 , x2

2.1 × 10-3 9.0 × 10-4

Crack length, x3

0.01 mm 0.40 m

Table B2: Error estimate of analysis response surfaces used to obtain safe -life stiffened panel design.

Error estimates ?

Normalized values ?

typical value eav erms R2 R2adj

ψ0_broken 0.5 – 1.0 0.026 0.032 0.986 0.980 ψ1_broken >1 0.062 0.085 0.949 0.927 ψ2_broken >1 0.065 0.084 0.986 0.980 ψ3_broken >1 0.065 0.084 0.990 0.987

max

enerFirstStiffF >1 0.033 0.040 0.996 0.995

max

fenerSecondStifF >1 0.0024 0.0028 0.956 0.955

max

enerThirdStiffF >1 10-4 1.29 ×10-4 0.999 0.998


17

The bounds on design variable used to construct the design of experiment for stiffened panel optimization with inspection is shown in Table B3. The upper bound on thickness is based on maximum possible plate thickness for unstiffened panel optimum with inspection (2.61 mm + thickness equivalent to cost of 3 inspections), bounds on stiffener area were reduced based on successive refinement. Table B4 shows the error estimate of the RSA.

Table B3: Bounds on design variables used to evaluate response surface approximation for inspection based

design.

Design Variable

Upper bound Lower bound

Skin thickness

3.20 mm 1.0 mm

Stiffener area

1.5 × 10-3 m2 3.0 × 10-4 m2

Crack length 0.01 mm 0.40 m

Table B4: Error estimate of analysis response surfaces used to obtain inspection based stiffened panel design.

Error estimates ?

Normalized values ?

Nominal value eav erms R2 R2adj

ψ0_broken 0.5 – 1.0 0.030 0.036 0.916 0.913 ψ1_broken >1 0.056 0.078 0.908 0.902 ψ2_broken >1 0.065 0.084 0.906 0.901 ψ3_broken >1 0.055 0.081 0.914 0.913

max

enerFirstStiffF >1 0.12 0.16 0.983 0.932

max

fenerSecondStifF >1 0.0030 0.0036 0.995 0.995

max

enerThirdStiffF >1 1.1 × 10-4 1.5 ×10-4 0.999 0.998

C. Effect of inspections on crack size distribution and probability of failure

Computational cost of reliability analysis with inspection scheduling is reduced by using a cubic response surface approximation to predict the crack size mean and standard deviation after an inspection as a function of skin thickness, stiffener area, standard deviation in applied stress, initial crack mean, initial crack standard deviation and crack growth time. The reliability index is also approximated by a cubic RSA in same 6 variables (Appendix C).A minimax LHS design of experiment with 400 level of each design variable is used. The range pf each variable is shown in Table C1 and the error estimates are shown in Table C2 and C3.


18

Table C1: Bounds on design variables used to evaluate response surface for crack sizes and reliability index

Design Variable Upper bound Lower bound

Skin thickness 3.20 mm 1.0 mm

Stiffener Area 1.5 × 10-3 m2 3.0 × 10-4 m2

Initial crack mean µai

0.01 mm 0.40 m

Initial crack standard deviation sai

2.0 mm 0.2 mm

Crack growth time , Nf

40000 flights 1000 flights

Standard deviation in Stress

10 Mpa 2 Mpa

Table C2: Error estimate of crack size response surfaces used to estimate the distribution after the inspection

Error Estimates Nominal target value ,

mm

eav,, mm erm, mm R2 R2adj

Mean ai after inspection

0.2 – 2.0 0.00628 0.00786 0.999 0.998

Standard deviation of ai after inspection

0.2 – 2.0 0.0286 0.0480, 0.985 0.983

Table C3: Error estimate of reliability index response surfaces

Error Estimates Nominal target value

eav erms R2 R2adj

Beta 5.0 0.29 0.52 0.951 0.943

Acknowledgments . The present effort has been supported by the Institute for Future Space Transport. The authors thank Thomas

Swift for providing with valuable inputs in implementing the displacement compatibility method for calculating effect of stiffening on stress intensity


19

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20

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