An analytic model is developed to calculate the reliability of a structure after it is inspected for cracks. The model accounts for the growth of undiscovered cracks between inspections and their effect upon the reliability after subsequent inspections. The model is based upon a differential form of Bayes' Theorem for reliability, and upon fracture mechanics for crack growth

Davidson, J. R.

English

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NASA TECHNICAL
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(NASA-TM-X-71934) RELIABILITY AND
STRUCTURAL INTEGRITY (NASA) 14 p HC
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RELIABILITYANDSTRUCTURALINTEGRITY
by John R. Davidson
Langley Research Center
Hampton, Virginia
Presented at the lOth Anniversary Meeting
of the Society of Engineering Science,
Raleigh, North Carolina, Nov. 5-7, 1973
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_sLIABIL.[TY AND _)IRUC_U_AL INTEGRITY
J. R. Davidson
NASA T.ang_ey Research Center, 1{ampton, Va., 23665
ABSTRACT
An analytic model is developed to calculate the reliability
of a structure after it is inspected for cracks. The model
accounts for the growth of undiscovered cracks between inspec-
tions and their effect upon the reliability after subsequent
inspections. The model is based upon a differential form of
Bayes' Theorem for reliability, and upon fracture mechanics for
crack growth.
SYMBOLS
a,b Half-ier4_th of crack
a
c
Critical haiti-length of crack
a
o
Threshold half-length of crack below which no cracks
are detected
a
s
a
Half-length of crack whose probability of detection is
specified to be 0.9
Half-length of crack which grows to infinity
between inspections
B Event "a crack is indicated"
B ! Event "a crack is not indicated"
f[ lal
Constants
Probability that a crack is indicated, given that it
is present
ina_ half-length of crack
Re2 Probability of surviving one period with no initial
inspecoion
RI2 Probability of surviving one period with an initial
inspection
L-9124
P:robab:il.ity of surviving one period, given that the
:rtruc'tnre p1_ssecl inspection at ts, survived one
|n:_pec:t:i.o_period, then passed a second inspection
Half-length of crack which grows to a between
:i.nzpect; i.onz c
INTRODUCTION
Cracks occasionally appear in structures. They grow larger
under repeated loads. If a structure is to remain strong, the
cracks must be detected and repaired. The reliability of the
structure will depend upon how many cracks are present, how long_
they are, and how well they can be detected so that they can be
repaired.
Recent (unpublished) studies of crack detectability have
established some probabilities of crack detection, given that a
crack exists. The purpose of the present paper is to develop the
methodology by which the re].iability after inspection can be
calculated from the reliability before inspection and the proba-
bility of crack detection. The method takes into account the
variability of detection probability with crack size and the
growth of cracks between inspections. The probability of multi-
ple cracks in a structure is taken to be small compared with the
probabilities of one crack or no crack; the multiple crack situa-
tions are beyond the scope of this paper.
CRACK SIZES AND DETECTION
In any structure, the crack population can be divided into
two categories: potential fatigue cracks which have not initi-
ated yet and which will not become detectably large for one or
two orders of magnitude more flights than the second category of
cracks; and cracks which were there initially, or which have
already been initiated by cyclic or repeated stresses. A proba-
bility density function for cracks which represents both cate-
gories is
-82 a
I C 5(a - o) + C2 _2 e for a _ ac
g(a) : (1)
[ 0 for a > a c
where 5(a - o) is the Dir_c delta function and CO is the frac-
tion of cracks which have not yet initiated. (Note: Mathemati-
cally, CO is a normalizing constant.) It was assumed that no
structures will apDear for routine inspection if they already
have cracks longer than the critical crack length, a c. Under
these conditions C2 and _2 are related.
C2=
i - C0 (2)
Consequently, the parameters CO and 62 are independently
adjustable to fit Equation (1) to data from actual experiences.
A typical curve is sketched in Figure 1.
g(a)
SPIKE ACCOUNTS FOR NON-INITIATED CRACKS
Figure I. An illustration of the probability density function
for flaw sizes. Most values of parameters gave functions which
decreased much more rapidly _han the curve shown.
Crack detectability varies with crack size. In general,
during nondestructive inspection, large cracks are more easily
found than small cracks. A detection function which represents
the detectability, given that a crack is present, is
I°(f(B/a)= -61cI i - e (a-a°)#,
for a < a
-- O
for a > a
-- O
(3)
where ao is the _hreshold o? de;ection, and CI < I is the
asymptote for the probability of de_ection (see Fig. 2). The
inequality, CI < l_ represents t_e fact that occasionally quite
large cracks are overlooked.
1.0
.9
f(Bla) .5
0
r_ o = 0.5
as=2
I I I
I0 20 30
HALF-LENGTH OF CRACK, a, mm
a
c
Figalre 2. Probability a crack is discovered if it exists.
For this paper_ the numerical values used for the probabil-
ity of detection function were C I = 0.98; the threshold ao
was O. 5 ram} the crack length which was detectable with probabil-
ity 0.9 was as = 2.0 ram; the critical crack length was
ae = 50 ram.
CRACK GROWT]I AND INSPECTION PERIOD
Unrepaired cracks grow. From fracture mechanicsj a simple
crack growth expression is
da n (h)
d-_,_= C_ a
where F is the number of l'light:_ and C_ and n are material
constants. For aluminum_ titanium_ and aircraft steels 1 < n < 2
[I]. One way to dete_.ine C 5 might be by fllght-by-flight t_e
laboratory tests. Equation (_) can be integrated,
Ib _ da fO F C3-- = dyII
from which
7nwYY =b-(%-zD-]1 - (n - l)c3 F
(5)
where F is the number of flights between inspections, a is the
crack length just after the jth inspection, and % is the length
Just before the (J +l) th inspection. Define "m" as the number
of intervals of length (tc - ts)/m needed for a crack to grow from
as at time ts to ac at time t e (see Fi_. 3)_ then
F = mC3(n_l) a = C3(n_l ) (6)
where
c_= _ n-I
S C
TIME OF EVENT
ts tc._
tc- ts
m
I I
1 2 3 (m+l)
INSPECTION NUMBER
Figure 3. Relations among times, inspection periods, and
inspection numbers.
From Equations (5) and (6)
1 1
Define z as the size of the cr_ck which will _row to ac during one
inspe_,tion period.
,m __
ii
Also, ] eL
side of Equ_tJon (7) becomes ze]7o told
1
a =
Figure 1 shows how a*, and
shows how z
between t s
a* 1)e the v_lue o#' "b" n,i,which 'the rIRht,-hn.nd
"7." becomes inl'_nito.
aas,_ z, ac are ordered. Figure Ifand vary with the number of inspections_ m,
and tc .
m
(_)
(9)
40_ w//a_
/
10 -
1 I
0 10 20
NIIHflI!ROF TN_PI!CTTON PI_RIOI]S, m
Figure 4. Crack, z, which reaches ac between _nspectionsj and
crack, a*, which reaches infinity between inspections.
3O
IIAI,F-I,ENGTII OF
CRACK, mm
2O
ANALYSIS FOR RELIABILITY
The unro]I:i_aIbi]_ity, or probability that a structure will not
last from one inspection to the next, equals the probability that
an undiscovered crack of length "bl" (_lere z < bI <_ac)
remains in the structure at the start of the period between
inspcct_ions; such a crack wi]l grow to excced a¢ before the
next .inspnction.
The probability that the structure will last through one
inspection interval without a previous inspection is
a
ROI = 1 -fz c
g(a)da
=i-c 2 (e"p2z- e-"_2%) (10)
However, the reliability if inspected will depend upon the prob-
ability density function (p.d.f.) for undiscovered cracks. This
p.d.f, can be found from Bayes' Theorem [2]. Tne p.d.f, for
undiscovered cracks is
h[a/B']= ..... (n)
a
fo c c[B'l_]_(a)a_
where the prime indicates "not." The reliability after the first
inspection is
(12)
where
_i = C2(i- Cl).(e"_2z - e"B2ac)
-(Pl+_2)z+61ao . e'(Pl+P2)ac+8lao]
and
( )_:':L = CO + C2 1 - e FI2a° + 02(1-C])(e "_2a° - e'132ac)
CIC2_2 [e'_Aao "(_].+_2)ac+_iL_o]
+ (1_1 +I_2) - e
Not all of the cracks which were undiscovered during the
first inspection will cause failurej most cracks were shorter
than z. But now, during the period 'between the first and second
inspection, these overlooked or undetectable cracks will grow,
and some may become longer th_1 z; if these are overlooked during
the second inspection, they will grow to exceed ac before the
third inspection. Consequently, the unreliability (probability
of not surviving until the third inspection) is the integral from
z to a c of a new d±_ribution function which represents grown
cracks. This function can be obtained from h[alB'] [3]:
h2[ZIB'] = h2(7_) =
dQf(]_, I_)h(QIB') aT
de,f[_, IZ]dZh[Q,I]3'] aT
(13)
_ £
_2
where
Q = + (14)
Specifically,
C _ "_32QldQo8(_-o)+c_p2e ]_- for O<l <a 0
C2132e ].- C I+C I e o _dQ for ao_<%<_ -
f
-_l (Q'%)/(1.t "_l (_-do) /
"I32Qkl. Cl+Cle C l+Cle
dQ
C2J32e d'-'_-
8
_'i n-I
-oi<_
for _ > a
C
This function does not account for structures in which cracks
grew longer than ac before the second inspection) these struc-
tures were regarded _s having removed themselves from further
consideration. The reliability of those structures which survive
until - and then pass - the second inspection is
a
(15)
where
2 (e'B2Q(z) "B2ac)_2 = C2(I - CI) - e
+ CIC2_2(I- CI) e_lao{e-(_l+J32)Q(z)\
(_l + _2)
_ e-(_l+_2)z)
+ Cl(l - Cl)C 2 e_laO(e "_32Q(z)'_31z _e'(_l+J32)ac)
+
C12C282 (e-_31z-(_l+_2)Q(z)
(_l + _2)
-e [_3!ac'(_31+_32)z) e2f31a°
a
. Cl(l - CI)C2f32 e_31ao _ c
-_2Q-_I _
e d%
a
C12C2_I_2 281a o,_ c
(_i + _2) e _z
e d%
9
.......... r
_'2 -- CO + c2 i - + c2(1 - c.i. ) e
(-r_2_,(ao) - _lY(aol_(t_l-_2)ao)
+ ClC2\c - - e
+ C2(]. - C1)2(e "82a°- e-[_2 z)
4- C2_2C1(1 " Cl) l_lao(e'(_l+_2)a 0 "(_1+82 )z)
(_i + _2') e - e
+ C2C1(1 - C1)(e-82ao "pl(y(ao)'a°)
c2c12_2 ( -_lY(a°)'(_l+_2)a°
+ (_i + _2 ) e
. e'_2z-Pl(ac "ao))
. e'Plac'(61+82)Z)e281ao
-_2Q.-_l(Z-ao!
e dl
a
- C2Cl(1- C..j)#I (ao)
-62Q-61(_-a o)
e dZ
C2P2C12_I 2PlaofyaC
(6l + _2 )' e (ao)
-(_l+82)Q-_l _
e d_
where
RESULTS AND DISCUSSION
Figure 4 contains some implications which help formulate a
game plan for the design and use of a structure. Suppose, for
some reason, m is chosen to be 2. All cracks larger than
z = 3.85mm (half-crack length) must be detected Jf the structure
is to survive until the next inspection. If such a crack length
has too low a probability of detection by tNe nondestructive
]0
in._-vl)eci,ion proeedure_ little will be _ined. by ehoozlng a mate-
r i._l w].l:,hl_igher f:,_aeturo _ouf<hness(crack growgh ra,te is reason-
_b!y independ.eni;o.?frt_,eture toughy_es,_);at best, ew_.nif the
mater_al can be madeinfinitely tough any undiscovered crack
larger than a* ;: II.17 mmw_]]. sti.l], cause fail.ere. Instead, the
frequency of [rls])eet]on ,ffn<_;_Id])e ]ner_.:",.z(_d(m chosen ]argot) so
that the curves fo:v z and _:.* ::,)[;n,i.:{tc.(set: Fig. 4).
Figure _ illustrates sometypical relationships amongthe
reliabilities under various conditions. Higher reliabilities are
associated with short inspection periods (large numberof inspec-
tions between t s and tc, Fig. 3). For somevalues of CO,_2,
and m the r_liability after the secondinspection can be lower
than the reliability after the first inspection; this happens
whenthe probability of a crack which will grow to exceed z in
one period is higher than the probability that a crack exists
whoselength is b_ween z and ac at the time of the first
inspection. The analysis is modeling the real ]ife situation
where a er_ck is so likely to propagate in a structure that its
reliability decreases with age.
-990¢i-
I
L
.999 _
1
.vuO , -
0
R
" _ RI2
f
......... I J
10 20
NUMBER OF INSPECTION PERIODS, m
Figure 5. Rcli_oiiitieu bef:;re _nd after the time of the ['_1:'z!,
inspection (R0i and R]2) anu rafter (CO = .2; P[crack betwv(m
a s and a c before firsc inspection] = .2) second inspection
(R23). See test for values of par._meters.
Ii
I
I
H:[gh rcliabilities after inspections are always associated
with structures whose reli_bilities were high before inspection.
Under certain conditions this impl:[es that an optimum reliability
might be obtained by a nonuniform spacing of inspections_ in par-
ticular, three inspections might be better distributed as two
independent inspections at ts and one inspection two periods
later rather than three equally spaced inspections starting at
ts. Additional calculations con Cirmed this.
CONCIUDING REMARKS
An analytic model for reliability was developed which con-
tained the salient features of practical situations where inspec-
tion procedures are less than absolutely perfect, where crack
detectability is a function of crack length, and where undis-
covered cracks grow larger and influence the reliability of
succeeding inspections. The analysis can be used to study the
effects of various schemes for material choice and inspection
intervals.
The relationships between the crack length which grows to
detectable length between inspections, the detectability, and
the frequency of inspections shows that inspection frequencies
may be increased to compensate for imperfect nondestructive
inspection procedures.
Calculations also indicate that the optimum reliability may
be obtained from inspection schedules which are not uniformly
spaced.
REFERENCES
[l]Wolf E!ber and J. R. Davidson, "A Material Selection Method
Based on Material Properties and Operating Parameters," NASA
TN D-7221, April 1973.
[2]R. V. Hogg and A. T. Craig, "Introduction to Mathematical
Statistics,!' 2nd ed. McMillan Co., New York, 1965.
[3]Emanuel Parzen, "Modern ProbabilityTheory and Its Applications,"
John Wiley and Sons, 1962.
12


Reliability and structural integrity

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