Probabilistic durability analysis methods for metallic airframes

Probabilistic durability analysis methods for metallic airframes J. N. Yang

School of Engineering and Applied Science, The George Washington University, Washington DC 20052, USA S. D. Manning

General Dynamics, Fort Worth Division, PO Box 748, Forth Worth, T X 76101, USA J. L. Rudd and M. E. Arfley

Air Force Wright Aeronautical Laboratories, Flight Dynamics Laboratory, AFWAL/FIBEC, Wright-Patterson Air Force Base, OH 45433, USA Probabilistic-based deterministic and stochastic crack growth approaches are compared and evaluated for durability analysis applications dealing with fatigue cracking in metallic fastener holes, Crack exceedance predictions for the deterministic crack growth approach correlated well with experimental results. Correlations for the stochastic crack growth approach were more conservative and less accurate than those for the deterministic crack growth approach.

1. INTRODUCTION Fatigue cracks commonly originate in fastener holes in metallic aircraft structure 1. The fatigue crack growth accumulation affects the damage tolerance and durability of the airframe. The U.S. Air Force has damage tolerance design requirements 2'3 for ensuring structural safety and durability design requirements 2,3 for minimizing functional impairment problems such as excessive cracking (e.g., crack length < 2.54 ram), fuel leakage, and ligament breakage (e.g., crack length 12.7 mm19.05 mm). This paper is concerned with the structural durability problem which affects structural maintenance requirements, total life-cycle costs, aircraft performance, and operational readiness. A typical fighter airframe contains numerous fastener holes (e.g., > 50000). Each hole is a potential site for crack initiation, and durability is concerned with the subsequent fatigue crack growth accumulation in the entire fastener hole population at any service life. For this reason, a statistical approach is best suited for analytically assuring that the durability design requirements are met. Probabilistic-based durability analysis tools have been developed for analytically ensuring aircraft structural durability in the small crack size region 4-s. These tools can be used to analytically predict the statistical distribution of crack size as a function of service life, taking into account the statistical variabilities of the equivalent initial flaw size (EIFS). The crack growth rate can be either deterministic or stochastic. The deterministic crack growth approach (DCGA) reflects the variablity of the crack growth rate in the 0266-8920/87/010009-0752.00 © 1987 Computational MechanicsPublications

definition of the initial fatigue quality. The entire population of crack sizes is grown from one time to another using a deterministic service crack growth master curve. Whereas, the stochastic crack growth approach (SCGA) accounts for the variability of the crack growth rates directly using a stochastic crack growth law. The DCGA has been verified for full-scale aircraft structure for relatively small cracks (e.g., < 2.54 mm) in fastener holes where excessive cracking is the issue. Advanced durability analysis methods are currently being developed for predicting the crack exceedance for large through-the-thickness cracks (e.g., 12.7 mm-19.05 mm) associated with fuel leakage and ligament breakage. Both the DCGA and the SCGA are being considered in the advanced durability analysis development. The DCGA is less complicated than the SCGA but the SCGA may be more accurate for durability analysis than the DCGA. This paper will: (1) review durability analysis methods and concepts, (2) describe the DCGA and the SCGA, (3) present methods for defining the initial fatigue quality, and compare and evaluate the effectiveness of the DCGA and the SCGA for the durability analysis of fastener holes. 2. DURABILITY ANALYSIS APPROACHES Both the DCGA and the SCGA approaches use a probabilistic format and an equivalent initial flaw size distribution (EIFSD) to represent the initial fatigue quality of the structural details. The EIFSD for the DCGA and the SCGA is based on a deterministic and a stochastic crack growth model, respectively. Once the

Probabilistic Engineering Mechanics, 1987, Vol. 2, No. 1 9

Probabitistic durability: d. N. gang et al. EIFSD has been established for each approach, predictions for crack exceedance at any service time and the cumulative distribution of time to crack initiation (TTCI) at any given crack size can be made using the applicable crack growth models for each approach. The crack growth rate variance for the DCGA is reflected in the resulting EIFSD. Whereas, the SCGA directly accounts for the variance in the crack growth rate using the lognormal random variable modeff - ~ . Essential features of the D C G A and the SCGA are conceptually described in Fig. 1 (a) and l(b), respectively.

in this paper the special case b = 1 is used, resuhing ir~ the crack size-time relationship as follows,

re,t) = a,O,~exp(Qt)

The cumulative distribution of crack size a(t) at ae.y time t, denoted by F,m(x)= P[a(t)~< x], can be obtained from the Weibull compatible distribution of a(O) given by equation (t) through the transfbrmation of equation (3) as follows <

dx)=exp

- I,

-

L L_

2.1 Deterministic crack growth approach (DCGA) The Weibull compatible distribution function has been found to be reasonable for representing the EIFS cumulative distribution 4~8

F={m(x)=exp{-[ln(~dx) f }

~

.j )

The probability that the crack size a(t) at any service time z will exceed any given crack size x,, d e n o t ~ b y p(i, r) = P[a(r) > xi], is referred to as the crack exceedance probability. It is obtained from equation (4) as

O<<.x<<.x= p(~,z)= 1 - r ~ ; t x ~ =

~ -exio~ -(

x> x u

= 1.0

f~

no

h

" t ;" .~) (5)

in which F,{o)(X ) = P[a(0) ~
da(t )/dt = Q[a(t )] b

(2)

where da(t)/dt = crack growth rate, a(t) = crack size at any time t, and Q and b are emprical crack growth rate parameters.

If Fr(t) denotes the cumulative distribution of time to initiate a given crack size xx, referred to as time to crack initiation (T or TTCt), then it follows that FT(t)= P[T<~t] = 1-F,m(xl)=p(i, t). Hence, the prediction of the cumulative distribution of TTCI can be made using equation (5).

2.2 Stochastic crack growth approach (SCGA) A randomized version of the crack growth rate equation, equation (2), has been shown to be reasonable for fastener holes under spectrum loading 9-~4

da(t )/dt = X Q[a(t )] b

xI

CRACK SIZE

!

I

u

I

TINE

l

(b)

expi i/log ,--5[-~

fx{u)= /2--~ua=

U~2) ) f

> 0

!7}

m which a= is the standard deviation of the normal random variable Z = log X. Taking the logarithm of both sides of equauon (6i for 5 = 1 yields

"!I:

Y=U ÷q+Z

TiME

7"

(a) Fig. J. Probabilistic approaches, (a) deterministic crack growth approach (DCGA), (b) stochastic crack growth approach (SCGA) 10

in which X is a lognormal random variable with a me&an of 1.0. This model, referred to as the °Iognormal random variable model " 4 is the simplest stochastic crack growth model for structural applications. Such a model with b = 1 will be used in this paper. The lognormal random variable. X0 in equation (6) accounts for the crack growth rate variability, such as the variabilities due to materiat cracking resistance° crack geometry, crack modelling, spectrum loading, etc. 9-~4 The probability density function of the tognormmi random variable X with a median !.0 is given by log e

W

CRACK SIZE

(6)

Probabilistic Engineering Mechanics, 1987, VoI. 2, No. ]

~8}

where° g = l o g da(t l/dr. U = l o g a(t), q = log Q and Z = log X. Since X is a Iognormal random variable with a median of 1.0. Z = t o g X is a normal random variable with zero mean and standard deviation %. The crack growth rate parameter Q and the standard deviation, G, of Z can be estimated from the log crack growth rate, log da(t)'dt = K versus log crack size, tog a(t)= U, data, denoted by i Yi, U~) for i = 1, 2 . . . N , using equatior, (g) and a linear regression analysis.

Probabilistic durability: J. N. Yan9 et al. The randomized version of the crack size-time relationship is obtained from equation (6) as follows

a(t) = a(O)exp(XQt)

(9)

The cumulative distribution of the crack size at any time t can be obtained from the EIFS cumulative disFibution, F,(m(x), using the randomized crack sizetime relationship of equation (9) and the theorem of total probability. The resulting expression is given in equation (10).

C°~

exp( -[Vln(x./x)+uQt~ g .j ~) . . . . (10)

in which fx(u) is given by equation (7). The probability of crack exceedance, p(i, z) = P[a(z) > x~], at any time r can be obtained from equation (10) as follows.

p(i, z) = 1 -- Fa~)(xl)

= 1-?

C~

( [ln(xJxO+uQz] ~] . . . . exp,L- ~ -j ;j~tu)au

(11)

Due to the statistical compatibility of F,(,)(x,) and Fr(t), i.e., Fr(t)= 1-F,,)(x,), the right hand side of equation (11) with z being replaced by t is equal to the cumulative distribution of TTCI, Fr(t). Equations (10) and (11) are not amenable to analytical integrations. However, these equations can easily be solved by numerical integration.

transfer). Pooling EIFS data sets effectively increases the sample size and confidence in the EIFSD parameters. Likewise, it is a reasonable approach to justify using an EIFSD for more general applications. This is an important perspective for practical durability analysis. Essential steps of data pooling are: (1) acquire suitable fractographic data sets for a given material, type spectrum (e.g., fighter, bomber, transport), and applicable fastener hole type/fit, (2) determine the EIFSs for each fractographic data set using either the DCGA or the SCGA, (3) a least square sum is determined for each EIFS data set using the ranked EIFSs and the selected EIFS cumulative distribution function, F,(o)(X), and (4) the desired EIFSD parameters (e.g., xu, c~and q~ in equation (1)) are obtained by minimizing the combined least square sum for all the EIFS data sets. The last two steps are referred to as the combined least square sums approach (CLSSA). In other words, EIFSs for each data set are treated separately rather than combining them into a single EIFS population. The candidate EIFSD is justified for durability analysis by showing that reasonable crack exceedance predictions, p(i,z), can be obtained for a fractographic data set (or sets) that is not used for the determination of the EIFSD.

3.1 SCGA EIFSs With the SCGA, an EIFS value for each specimen can be determined using equation (9), fractographic results in the crack size range AL-AU, and a least squares fit procedure. Equation (9) is transformed into a linear least square fit form by taking the natural log of both sides. In a(t) = XQt + In a(0)

3. INITIAL FATIGUE QUALITY AND EQUIVALENT INITIAL FLAW SIZE COMBINED LEAST-SQUARE SUMS APPROACH (CLSSA) Initial fatigue quality (IFQ) defines the initial manufactured state of a structural detail or details with respect to initial flaws in a part, component, or airframe prior to service. The IFQ for a group of replicate details (e.g., fastener holes) is represented by an equivalent initial flaw size (EIFS) distribution. An equivalent initial flaw is an artificial initial crack which results in an actual crack size at an actual point in time when the initial flaw is grown forward. An EIFS can be determined by backextrapolating fractographic results using a reasonable crack growth law, such as equation (2). A simple crack growth law (e.g., equation (2)) bridges the small crack size region where the effects of microstructure, persistent slip bands, plasticity, crack branching, etc. exist. Once the small crack size region has been 'bridged', linear elastic fracture mechanics methods or a suitable crack growth model can be used to make crack exceedance predictions needed to analytically ensure compliance with durability design requirements. After the EIFSD parameters have been defined, the EIFSD is justified by demonstrating that reasonable p(i, z) and/or Fr(t) predictions can be made for different fractographic data sets. Correlations for p(i, z) and Fr(t) predictions are emphasized rather than the goodness-offit of the cumulative distribution to the artificial EIFSs. Data pooling is a practical way to define the EIFSD parameters for different data sets (e.g., same type of load spectrum but different stress levels and percent bolt load

(12)

Let Xq=tj, Yo=lna(t) and Ni=total number of (Xij, Y~j) pairs in the AL-AU range of the ith specimen. Then, the EIFS value, denoted by ai(0), and the crack growth rate parameter Q~= X~Q for the ith specimen can be determined from equation (12) using the least square procedure as follows ai(0) = exp[(~ Yq)(~ X~)-(~, Xij)(~ XqYq) 1

N,Z X~j_(ZX,j)2

Qi=XiQ = [Ni Z

~

(13)

XijYii-(ZXq)(~,Vo)]/ [NiZ X ~ - ( Z X i ; ) 2] (14)

in which all summations are for j = 1 to Ni.

3.2 DCGA EIFSs In the DCGA, the crack growth rate, equation (2), is considered to be deterministic, i.e., the crack growth rate parameter Q is identical for each specimen. Such a parameter will be evaluated by pooled the Qi values for a given fractographic data set. The expression for the pooled Q value using a least-squares fit procedure has been developed as follows Q = e x p { l i ~ l In Qi}

(15)

in which N = total number of specimens in the replicate data set. The EIFS value for the ith specimen is obtained by back extrapolation of the corresponding fractographic

Probabilistic Engineerin9 Mechanics, 1987, Vol. 2, No. 1 11

Probabilistic durability: J. N. Yan9 et al. data in AL-AU range using the common Q value and a least squares fit procedure. The result is given in the following (i6) where the summation is for j = 1 to N~.

Suppose a total of M EIFS data sets are pooled together for the determination of e, and ,& Let xt~ ,%:~ i = t, 2 . . . . . M a n d j = 1, 2 . . . . . N,: be the jlb EIFS value in the ith EIFS data set, where N.: is the total number of EIFS data in the ith data set. Further, let I~ be the total number of fastener holes per specimen for t~e ith E!FS data set. Then, equations (i 9) and (20) car., be modified by virtue of equation (17) for the determination of ~ and 4~as follows

3.3 Scaling EIFS data The IFQ or EIFSD for fastener holes is defined for a single hole population. Test specimens for acquiring fatigue crack growth data may have one or more fastener holes per specimen. Some specimens may not be fatigue tested to failure. Also, every fastener hole in each replicate test specimen may not contain a measurable fatigue crack or else the crack is too small or complex (e.g., multiple crack origins and branching) for fractographic analysis. Hence, a statistical scaling technique should be established such that the equivalent initial flaw size distribution for a single hole population can be obtained using the largest fatigue crack per specimen. This minimizes the fractographic reading requirements, permits a maximum utilization of the available fractographic data and allows for mixing and matching of fractographic data for specimens with a different number of holes. Let the cumulative distribution of EIFS for a single hole population be denoted by F=~o}(x) and that of the maximum EIFS, based on the largest fatigue crack per specimen with I fastener holes, be denoted by F,,(m(x). With the assumption that fatigue cracking in each fastener hole of a specimen is statistically independent of that of the other holes, F,,(o~(X) is related to F=(m(x) through the following

F.iol(X ) = F.~o{X) ~

~ ,~ ,

where l = n u m b e r of fastener holes per specimen. if all the EIFS values obtained previously are from specimens with single holes, then the scaling procedure is not necessary. In this case, the EIFSD parameters can be estimated by transforming equation (i) into a linear least° squares fit format and using the least-squares fit procedure. The transformed equation for each EIFS sample is given by A~= :~D; + B

(I 8)

in which A~=ln[-lnF~101(x;) ], D;=lnln(x,/xj), and B = - c ~ i n q~, where the EIFS values are ranked in an ascending order with xj being the jth sample and F~(o.(xj) =j/(N + 1) for j = 1, 2 . . . . . N. The parameters ~ and ~b i~ equation (18) are estimated for a given x, using the following least-squares procedure;

O~= N Z DjAj- f Dj i A \

j=l

j=~

q5= exp

(19) (20)

where N = total number of (Ds, Aj) pairs in the EIFS data set.

12

@= exp~

h', "~r~M N~'N, A -: , ES= , D,;aL~,= , z~.,: ~.,%.;

m

t

0{ S i =

1 ~q'-"

"

(22)

[

"

where A~j = in{ -ln[f~,,~o~(xu)] la,} = l n

g ; - In ~ [~

T-/a5 " ( {23 }

D/j = tn[in(x./x0) ~

{24)

Note that equations (i9) and (20) are special cases of equation (21) and (22) in which M = i and l~--I. The parameter x~ in equations (21}-(22) defines the EIFS upper bound limit.The following limitsare p~aced on x,;: iargest EIFS in data sets < x~< 1.27 ram. An upper bound limit of 1.27 mm was set based on the inifiaI flaw size associated with the US Air Force damage tolerance design requirements 2"3. The expression for the standard e r r o r denoted by SE, is given by

M

The SE value is an indication of the goodnessoofofit, it can reasonably be assumed that the best combination of~ and 6 for a given x, results in a minimum standard erro> Thus. the following kerative procedure can be used to obtain a reasonable set of x.,, ~ and ~ values: (t) assume an x, value within allowable limits; (2) compute ~ and for selected EIFS data sets using equations (2! }and (22); (3) determine the standard error using equation (25): and (4) repeat steps 1-3 until a minimum standard error has been obtained, 4. C O R R E L A T I O N STUDY

4A General description

3=~

(NjZ=ID2-(j~=iDJ)2)

&i=V'~ i N r~'viLz~i=I ~F=V~'i DiyAij]-EEf=-

Probabitistic Engineering Mechanics, t987, Vol. 2, No. i

Three fractograph data sets d.e. WPF, XWPF,, and WWPF} used in the correlation study are described in Table I and spemmen details are shown in Fig. 2. The W W P F specimen geometry was the same as that of the WPF specimen except the W W P F specimen was wider. The material for all spemmens was 7475-T735! aluminum. All specimens were dog-bone specimens reflecting 6.35 mm diameter straight-bore fastener holes with protruding head steel bolts 0'qAS 6204) and a clearance fit. Replicate WPF. XWPF0 and W W P F

Probabilistic durability: J. N. Yang et al. ~= ,=------152.4 m m - - ~ ~

704.8 mm

estimated using equations (15) and (26),

-

a2

38.1 mm

(a)

4.2 Results and discussion Using the pooled EIFS distribution and the pooled Q and a~ values shown in Table 1, theoretical crack exceedance predictions (solid curves) for both the DCGA and the SCGA for the WPF, XWPF and W W P F data sets were computed using equations (5) and (11), respectively, and the results are presented as solid curves in Figs 3-8. Also shown in these figures as solid circles are the corresponding test results. For comparison purposes, crack exceedance predictions for both DCGA and SCGA approaches are made for the same flight hours for a given data set. For example, z = 14 800, 12400 and 18 400 flight hours are used for the WPF, XWPF and W W P F data sets, respectively. For structural durability analysis, the large cracks in the lower tail of the crack exceedance curve may cause functional impairment problems, such as fuel leakage and ligament breakage. In Figs 3, 5, and 7, the theoretical

370.2 mm

I

--]--

I

'

(26)

In equation (26) N = the total number of specimens in the data set, Qi=the crack growth rate parameter for specimen i in a given data set (see equation (14)), and Q = the pooled crack growth rate parameter for the data set (see equation (15)). The resulting pooled Q and a z values for each data set are shown in Table 1. These values were used to grow the pooled EIFS distribution forward for the correlation study.

I I 9.525 mm

d~

~i~1 [ln(Qi/Q)] 2 N

',

38.1 mm

I

4.775 mm

-I

I "-'--7-(b)

Fig. 2. Test specimen geometries, (a) W P F specimen, (b) X W P F specimen

Table 1. Description of fractographic data sets used in correlation study

specimens were fatigue tested in a laboratory air environment using a fighter spectrum with a maximum gross stress of 234.4 MPa. Specimens for the W P F and XWPF data sets were fatigue tested to 16 000 flight hours or failure, which ever came first. Whereas, all specimens for the W W P F data set were fatigue tested to failure. Specimens were fatigue tested without intentional flaws in the fastener holes and natural fatigue cracks were allowed to occur. Following the fatigue test, the largest crack in each specimen was evaluated fractographically. Fractographic results (i.e., a(t) verus t records) for the W P F and X W P F data sets are given in Ref. 15 and those for the W W P F data set are given in Ref. 16. Two baseline fractographic data sets, i.e., WPF and XWPF, in the crack size range of 0.254mm-l.27mm were used to represent the IFQ of straight-bore fastener holes in 7475-T7351 aluminum. Equations (13) and (16) were employed to estimate the EIFS values of the W P F and XWPF specimens for the SCGA and the DCGA, respectively. Then, these EIFS data sets were pooled together with the aid of the statistical scaling procedures described in the previous section. The pooled EIFS distribution parameters e and ~b were determined using an upper bound of x, = 0.762 mm. The parametric values were found to be e=4.552, ~b=4.357 for the DCGA, and ~=3.771, ~b=4.554 for the SCGA. In order to predict the crack exceedance probability in the large crack size regions, pooled value of Q and a= for the WPF, XWPF, and W W P F individual data sets were

Data set

l (3)

No. of specimens

% LT

Specimen width mm

Q x 104 1/hr

(7 z

WPF (1) XWPF (1) WWPF (2)

1 2 1

33 32 13

0 15 0

38A 38.1 76.2

2.364 3.266 2.924

0.162 0.271 0.153

Notes: (1) Baseline data set (Ref. 15) (2) Same geometry as WPF specimen except for width (Ref. 16) (3) No. of fastener holes per specimen

1.0

WPF

,x.,

0,8

0.6

< u tr

0.4

>-

0.2 n-. n

0.0

i

0.0

3.30

6.60

9.90

CRACK SIZE. m m

Fig. 3. Analytical~experimental correlations for W P F specimens - DCGA

Probabilistic Engineering Mechanics, 1987, Vol. 2, No. 1

13

Probabitistic durability: J. N. Yang et aL Theoretically, the statistical dispersion of~ the E).FS values based on the SCGA should be smaller than ',:hat of the EiFS based on the DCGA, This is because in the SCGA, the crack growth variability has been filtered out in determining the EIFS values. As a result, the SCG/~, should provide more accurate crack exceedan~c¢ predictions than those for the DCGA. Un~tortuaateiy, ti~e numerical results indicate the opposite ":rend~ T!~s problem is currently being investigated. The DCOA is welt-suited for predicting crack exceedance probabilities in the small crack size regioe

Z

< WPF u

0.8

5<

0.8

O

0.4

<

0.2

× g~

@

8

m

O.O

3.30

0.0

~.60

9,90

1.0

CRACK SIZE, m m

,;
Fig. 4. Ana!yticaI/experimemat correlations for W P F specimens - SCGA e, x..

e

X i/1

,~

XWPF

0.8 0.2. L

,5 <

%

0 0.0 .

B

.

0.0

>.,

.

m

.

1.78

'

3.56

5,24

CRACKS~ZE. mm Fig. 6. Analytical~experimental correlations for X , , P.F specimens SCGA

0 O: 00

~ 0.0

~ 1.78

~

~ 3.56

~

1.0~ ~

5°34

CRACK SIZE, m m

WWPF

Fig. 5. Analytical~experimental correlations for X W P F specimens - DCGA

0.8. o <

0.6[

5 crack exceedance predictions for the DCGA correlate reasonably weli with the actual test results for data sets WPF, XWPF and WWPF, respectively. However, the theoretical predictions in the lower tail portion of the crack size cumulative distribution are generally conservative using the SCGA as illustrated from Figs 4, and 8. The statistical disperson of EIFS for the DCGA was much smaller than that for the SCGA. Such a trend was not expected. The conservative nature of the SCGA is attributed to the large statistical dispersions in the EWSs and the crack growth rates.

o\

o\ o\ \ ee "~

[

0.2 0

oE 0,.

0.01 0.0

8.35 CSAC~ SIZE.

i2~70

~,O5

mm

Fig. 7. Analytical~experimental correZations for W W P F specimens - DCGA

WWF~F 5.

Durability analysis methods and concepts for metallic airframes have been reviewed. The DCGA and the SCGA for durability analysis have been described. Methods for determining the IFQ of fastener holes have been presented. The deterministic and stochastic crack growth approaches have been compared and evaluated. The combined least-square sums approach (CLSSA) is reasonable for estimating the EIFSD parameters for the Weibult compatible distribution. Crack exceedance predictions for the DCGA correlated well with the actual tests results. Generally more conservative and less accurate correlations were obtained for the SCGA than for the DCGA.

14

!

CONCLUSIONS <

[5

Probabilistic Engineering Mechanics, t987, VoL 2, No. !

o\

02

r |

0.0 L 0.0

~,

6,3,5

12o70

"~9.05

c R A C K SIZE. r n m

Fig. 8. Analytical, experimental correlations for W W P F specimens - SCGA

Probabilistic durability: J. N . Yang et al. (e.g., 0.254 m m - l . 2 7 mm). T h e S C G A , however, s h o u l d be m o r e a c c u r a t e for describing the c r a c k g r o w t h b e h a v i o u r a n d hence the c r a c k exceedance p r o b a b i l i t y in the large c r a c k size region. I n this c o n n e c t i o n , a t w o segment a p p r o a c h is c u r r e n t l y being investigated. W i t h the t w o - s e g m e n t a p p r o a c h , t h e E I F S D is defined using the D C G A a n d the f r a c t o g r a p h i c d a t a in the small c r a c k region (e.g., 0.254 m m - l . 2 7 mm). T h e resulting E I F S D is g r o w n f o r w a r d in the small c r a c k size region (0.254 m m 1.27 m m ) using the D C G A . O n c e the initial flaws have been g r o w n to 1.27 m m using the D C G A , the S C G A is used to g r o w the flaws further into the large c r a c k size region. Such a t w o - s e g m e n t a p p r o a c h is expected to p r o v i d e m o r e a c c u r a t e c r a c k exceedanee predictions. The results of the t w o - s e g m e n t a p p r o a c h will be r e p o r t e d in the n e a r future.

REFERENCES 1

2 3 4

5

Tiffany, C. F. Durability and damage tolerance assessments of United States air force aircraft, Proc., the AIAA Structural Durability and Damage Tolerance Workshop, Washington DC, April 6-7, 1978, 25-47 Aircraft Structural Integrity Program, Airplane Requirements, Military Standard MIL-STD-1530A, Air Force Aeronaitical Systems Division, WPAFB, OH, December 1975 General Specification for Aircraft Structures, Military Specification MIL-A-87221, Air Force Aeronautical System Division, WPAFB, OH, February 1985 Manning, S. D. and Yang, J. N. USAF Durability design handbook: Guidelines for the analysis and design of durable aircraft structures, AFWAL-TR-83-3027, Air Force Wright Aeronautical Laboratories, WPAFB, OH, January 1984 Rudd, J. L., Yang, J. N., Manning, S. D. and Garver, W. R. Durability design requirements and analysis for metallic airframes, Design of Fatigue and Fracture Resistant Structures, ASTM STP 761, (eds P. R. Abelkis and C. M. Hudson), American Society for Testing and Materials, 1982, 133-151

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7

8

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Rudd, J. L., Yang, J. N., Manning, S. D. and Yee, B. G. W. Damage assessment of mechanically fastened joints in the small crack size range, Proc., Ninth US National Congress of Applied Mechanics, Symposium on Structural Reliability and Damage Assessment, Cornell University, Ithaca, NY, June 21-25, 1985, 329-338 Rudd, J. L., Yang, J. N., Manning, S. D. and Yee, B. G. W. Probabilistic fracture mechanics analysis methods for structural durability, Proc, Conference on the Behavior of Short Cracks in Airframe Components, AGARD-CP-328, Toronto, Canada, September 1982, 10-1 through 10-23 Yang, J. N., Manning, S. D. and Rudd, J. L. Evaluation of a stochastic initial fatigue quality model for fastener holes, to appear in Fatigue in Mechanically Fastened Composite and Metallic Joints, ASTM STP 927, (ed. J. M. Potter), American Society for Testing and Materials, Philadelphia, 1986 Yang, J. N., Salivar, G. C. and Annis, C. G. Statistical modeling of fatigue crack growth in a nickel-based superalloy, Journal of Engineering Fracture Mechanics, June 1983, 18(2), 257-270 Yang, J. N. and Donath, R. C. Statistical fatigue crack propagation in fastener holes under spectrum loading, Journal of Aircraft, AIAA, December 1983, 20(12), 1028-1032 Yang, J. N., Manning, S. D., Rudd, J. L. and Hsi, W. H. Stochastic crack propagation in fastener holes, Journal of Aircraft, AIAA, September 1985, 22(9), 810--817 Yang, J. N. and Chen, S. Fatigue reliability of gas turbine engine components under scheduled inspection maintenance, Journal of Aircraft, AIAA, May 1985, 22(5), 415-422 Yang, J. N. and Chen, S. An exploratory study of retirement-forcause for gas turbine engine components, Journal of Propulsion and Power, AIAA, January 1986, 2(1), 38-49 Yang, J. N., Hsi, W. H., Manning, S. D. and Rudd, J. L. Stochastic crack growth models for application to aircraft structures, to appear in Probabilistic Fracture Mechanics and Reliability, Martinus Nijhoff publishers, The Netherlands, 1986, Ch. IV, 6, 171-211 Noronha, P. J. et al. Fastener hole quality, AFFDL-TR-78-206, Air Force Flight Dynamics Laboratory, WPAFB, OH, December 1978, II Gordon, D. E. et al. Advanced durability analysis, Fractographic Test Data, AFWAL-TR-86-3017, Air Force Wright Aeronautical Laboratories, WPAFB, OH, 1986, III

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