The weibull stress-life, log-log stress-life, and the overload-stress reliability models in accelerated life testing

Rehabdzty Engmeermg 11 (1985) 109-120

The Weibull Stress-Life, Log-Log Stress-Life, and the Overload-Stress Reliability Models in Accelerated Life Testing Dimitri Kececioglu Professor of Aerospace and Mechamcal Engineering, The Umversity of Arizona, Tucson, Arizona 85721, USA

and

Julie A. Jacks Naval Ship Weapons Systems, Engineering Station,

Systems Assurance Department, Port Hueneme, Cahforma 93043, USA (Received 8 June 1984)

ABSTRACT Thzs zs the second in a series of papers on Accelerated Lzfe Testing by the authors. 1 In thzs paper, the followmg three accelerated testmg models are covered and thezr apphcatwn zs zllustrated by examples 1. 2 3.

Wezbull stress-hfe; log-log stress-hfe, and overload-stress rehabihty

These models may be apphed to any type of components and equzpment

1

INTRODUCTION

Accelerated hfe testing is achieved by testing units at stress levels higher than those experienced under normal use conditions. Such stresses may be temperature, voltage, pressure, vibration level, cycling rate, physical loads, and combinations thereof. 109 Reliabdzty Engineering 0143-8174/85/$03 30 © Elsevier Apphed Soence Publishers Ltd, England, 1985 Printed m Great Britain

110

D Ke~e~mglu, J ,4 Jacks

When the time-to-failure distribution is Welbullian, and the Welbull shape parameter value remains constant when the stress level is increased, the Wetbull Stress-Life accelerated life test model is an excellent one to use to determine the life and reliability of components and equipment. The m a j o m y of components and equipment have Weibull life distributions, for this reason this model has wide applicability When components and equipment are subjected to fluctuating loads and the resulting stresses, the L o g - L o g Stress-Life accelerated hfe test model is an appropriate one to use to determine the life and the rehability of these components and equipment When the reliability of components and equipment at different stress levels needs to be determined, the Overload-Stress Reliabdity accelerated life test model can be used to determine the reliability at a desired life of these components and equipment. It requires knowledge of the slope of the log-log stress life hne and the slopes of the Welbull times-to-failure distributions at different stress levels, assuming the slopes remain essentially constant. These three accelerated life testing models are discussed in detail in this paper and are illustrated by examples.

2

WEIBULL STRESS-LIFE MODEL

When the time-to-failure distribution is Weibullian, the reliability and hfe can be determined for several stress levels as follows. 2 For each of at least five stress levels, test at least five units to failure Plot the failure data on Welbull paper using standard probability plotting techniques. The data plots are shown in Fig I. From this figure, life can be found as a function o f stress level. Also, from Fig. 1, If any two of three quantities (i.e unreliability, stress level, and life) are given, the third can be determined. Figure 2 presents a plot of the hfe versus stress level, at various constant unrehablhties, on log-log scale. The analysis assumes that the shape parameter, fl, is the same at all stress levels and that the location parameter is zero. In Fig. 2, if the lives are extended to stress levels below the accelerated stress levels, or to the low-use stress level, then the unreliabllitles and the associated lives corresponding to the use stress level can be determined. Thus, the use stress level line can be plotted in Fig. l and the life for a desired unreliability can be found, as illustrated by the next example.

Models m accelerated h.]e testing

111

.or) 99 9 99

90 80 70 60 SO 40 / ' / 30

t-

E t-

a_

.g e.

~ ' ~ ",.~

WelbuLL

95

ii #¢11¢" ¢ r

Stress

/

/

20 10

I--

o ,'1

50 40 30

/

t-

z

/

/

////'y

ZO

0

1-0 c-

-SO 40 30 20

10

05 04 03

I

i0 z Fig. 1.

I

103 104 Lde,t~me-to- fadure

105

Tlme-to-fadure data for various stress levels on Weibull paper

Example 1 The times-to-breakdown of an insulating fluid at various voltage levels are presented in Table 1. Do the following. I. 2.

Plot these data on Weibull probability plotting paper. Determine the 0.01 ~ up to the 99 ~o unreliability lives for the insulating fluid.

85 29 29 07 60

10 0 30.0 50,0 70,0 90 0

68 108 110 426 I 067

5 79 1 579 52 2 323 70

16 3 50 0 83.3

Obser- Plottmg v a t : o n pOSlHOtl

28ki/"

Obser- Plotting l¥1Hol'l poslllOtt

26kV

774 17 05 2046 21 02 22 66 43 40 47'30 139 07 144 12 175 88 194'90

Observatton 45 136 227 31 8 40 9 50 0 59 1 68 2 77 3 86 4 95 5

Plotting positron

30kV

027 040 069 0 79 2 75 3 91 9'88 13 95 15 93 27 80 53 24 82 85 89 29 100'58 215 10

33 100 167 23 3 30 0 36 7 43 3 50 0 56 7 63 3 70 0 76 7 83 3 90'0 96 7

Obser- Plotting ~'atlon positron

32kV

019 078 096 1 31 2 78 3 16 4 15 4 67 4 85 6 50 7 35 8 01 8 27 12'06 31 75 32 52 33 91 36 71 72 89

Obserration 26 79 13'2 18 4 23 7 28 9 34.2 39 5 44 7 50 0 55 3 60 5 65 8 71 1 76 3 81 6 86 8 92 1 97 4

Plotting pos~t~on

34kV

035 059 096 0 99 1 69 1 97 2'07 2 58 2 71 2 90 3 67 3 99 5 35 13.77 25 50

33 100 167 23 3 30 0 36 7 43 3 50 0 56 7 63 3 70 0 76-7 83 3 90 0 96 7

Obser- Plottmg ration positron

36kV

009 039 047 0 73 0 74 1 13 I 40 2 38

62 18 7 31 2 43 7 56 2 68 7 81 2 93 7

Obset- Plotting L'allOtt po~tllon

38kV

TABLE 1 D a t a and Plotting P o s m o n s for the Times to B r e a k d o w n (m man) at Various C o n s t a n t Elevated Voltage Stresses 2 for Example 1

Models m accelerated hfe testmg

113

105

10 4

103

102

,-J

10

10

01

001 Stress |ever

Fig. 2.

3. 4. 5.

Rellabll~ty as a funcuon of stress level and hfe, log-log scale

Find the 20kV use stress level line and plot it in Figs 3 and 4. Find the breakdown hfe of the insulating fluid for a reliabdlty of 95 % and 99 %. F m d the reliability of the insulating fired for a life of 1000 mln and 500 mm, both at the use stress level.

S o l u t i o n to E x a m p l e I

1. A failure plotting pos)tion Q,, in %, is calculated for each failure of rank t and the total number, n s, of observations at that stress level from Q, = [1000 - 0-5)]/nj

D Ke~e~loglu, J ,4 Ja~k~

114

O0

Small beta esbn'~tor }

~(t=melstr~38kV

36kV

34kV 30kV

28kV

05

10

,oo 15 5-~~-~

YW

20 10

•

,

i

01

~

10

10

0

A

0

x

0

38kV36kV 34kV 32kV30kV 28kVZ6,kV 100

1000

10000

Minutes to breakdown,T

We=bull plot of the d a t a o f E x a m p l e 1

Fig. 3.

The data for the seven stress levels m Table 1 are plotted this way in Fig. 3 The same data are presented m Fig. 4 using a c o m p r o m i s e c o m m o n slope. 2 2 The unrehability, Q = 1 - R, is read o f f F l g . 4 to obtain the life and stress level for Q --- 0.01 °/o up to 99 ~o- These data are then plotted as stress oofS---Srna[[ beta estimator Q

05

-o

O

//,///,//.

/

40< 10

20kV

~

.//

"

15

1

L

01 Fig. 4.

1

10

100 1000 Minutes to breakdown

10000

T i m e s - t o - b r e a k d o w n data at various stress levels for Example I

100003

Models m accelerated hfe testing

115

107

106

105

104 c E t-"

,o ~ 0

102 .0

q

0-1

001 10

2O

30

40

50

60 7 0 8 0 9 O 1 0 0

Test voltQge ,n KV

Fig.

5.

Unrehablhty as a function of time-to-breakdown and voltage level for the insulating fired of Example 1

versus hfe for a constant unreliabihty, as shown in Fig. 5, on log-log scale, where it plots as a straight line. 3. Extend the lines in Fig. 5, obtained in the previous Case 2, towards the upper left, past the 20 kV use voltage level. Read off the lives, T,, for the different Q, values at the 20 kV use voltage level and plot these Q, values above the corresponding T, values in Figs 3 and 4. Draw the best straight line through these points at the compromise slope in Fig. 3 and parallel to the other lines in Fig. 4, thus obtaining the Q versus T hne for the use stress level using accelerated test results. 4. From Fig. 4, the breakdown life of the insulating fluid for a reliability of 95 9/0 is 1600 min, and for a reliability of 99 9/0 it is 200 min at the use stress level of 20 kV.

116

D Kece~loglu, J A Jacks

5. For a life of 1000rain at the 2 0 k V use stress level the reliablhty is 96,5 °'o, and for a life of 500 min at the 20 kV use stress level the reliability 1S 98 ° o. 3

LOG-LOG STRESS-LIFE MODEL

Traditional mechamcal engineering design employs the strength-cyclesto-failure ( S - N ) diagram when designing components to withstand stresses that fluctuate over time. The S - N diagram can be improved u p o n by making it distributional, as in Fig. 6, 3 and determining the cycles-tofailure distribution at each fixed stress level. Figure 6 is constructed as follows For each of at least five stress levels, several Identical components (group) are tested to failure. The times-to-failure of this group are recorded. There should preferably be at least 35 components in each group. The mean of the distribution is found for each stress level, and is plotted on log-log scales The straight line passing through these mean points at each stress level is 4 l o g S = m I log T + b 1

=

mlT' + b I

(1)

where S is the maximum value of the fluctuating stress level, m I is the slope of the mean S - N l i n e on a log-log plot, log T = T' = mean of the life of the c o m p o n e n t at that stress level calculated by averaging the logarithms of the cycles-to-failure data at that stress level, and b 1 is the stress required to cause failure when the stress does not fluctuate. Least-squares analysis can be used to find the best-fit straight line to the mean hves at various stress levels. Similarly the _ 3a T, hmlt hne equations on the life are found from logS

=

m 2 ( T ' + 3at, ) + b 2

(2)

m 3 ( T ' - 3at,) + b 3

(3)

and logS

=

where m, and b, (i = 2, 3) are the empirically determined parameters of the T' + 3o-r, limit hnes, and tr r, is the standard deviation of the logarithms of the cycles-to-failure data at the respective stress levels. F r o m least-squares analysis, b

( ~ log S) [ ~ (log T) 21 - ( ~ l o g T ) [ ~ ( l o g S . log T)] = n ~ ( l o g T) 2 ( ~ l o g T) 2 - -

(4)

Models m accelerated hfe testing

117

~ ~~:.-~-, I 16,400

psi

too F

|

80

so

u~

~ "'"

-

~

-

--------..

_

.

40

-

30

20

i

tt 4

I

i

I

5

6

In C y c l e s

I

))l

i

I

I

I

i

I I i J 10 s

I

I

I

I

I

L I I I 106

10 °

kklO.

6.

Statistical S - N surface for SAE 4340 steel wire---cold drawn and annealed, at fixed stress levels 3

Cycles 1o failure (in) Fig.

and m=

n ~ (log T. log S) - ( ~ log T ) ( ~ log S) n ~ ( l o g T) 2 - ( ~ l o g T) 2

(5)

where n is the number of stress levels, and T becomes T' + 3tr r, for the limits. Example

2

Using Fig. 6, find the 95 ~ rehable hfe, T O95, for SAE 4340 steel wire subjected to a 80 ksi fluctuating stress. F r o m Fig. 6, at this fluctuating stress level, T' = Iog(8'5 × 104cycles) Since it is assumed that the logarithms of the cycles-to-failure data are normally distributed, the standard deviation is given by eqn (6), where the

118

D Kecec.~g/u, J A J a t k

T' +_3a r, are read off the corresponding experimental hmlts at 80 ksl stress, or (T' + 3at, ) - (T' - 3at, ) O'T' =

6

1og2-5 x 105 - 1 o g 2 - 2 x 104

(6)

6 = 0 1759 The reliability is given by the probability, P, that the cycle to failure is equal to or greater than the T O 95 life, or

R(To 95) = P(T>_ T O~5) = 0 - 9 5 =

J(Ts=aok~,)dT

(7)

095

or, utihzlng the fact that the T's are normally distributed, o-95 =

J(Ts=8ok~,)dT'

(8)

og To 95

and finally, utilizing the standardized normal distribution, 0 95 =

q~(z) dz

(9)

ITo 45)

where.l(Ts=80ks,) is the cycles-to-failure distribuUon at the fluctuating stress level of 80 000 p s i , f ( T s = 80 k~,) is the cycles-to-failure distribution of the logarithms of the cycles to failure at the fluctuating stress level of 80 000 psi, with parameters T' and at,, and

z(To 95) -

To95

__T

r

(10)

GT

F r o m the standardized normal distribution area tables for R = 95 ~ , the corresponding value of z(T o 95) is 1.645 The 95 ~ reliable life IS then found from eqn (10), or -

1-645 = TO 95 - 1 ° g ( 8 5 0 1759

x 104)

or

To 95 = 4"64 and To 95 = 4-37 × 104 cycles

Models in accelerated liJe testmg

4

119

OVERLOAD-STRESS RELIABILITY MODEL

A binomial test is applicable when there are only two possible test results, often taken to be success and failure. For this type of test, a unique equation 5 relating the reliability at accelerated stress level to the reliabdity at use stress level is RNw = ( Rw) ~w?/"

(11)

where R u w and R w are the reliabilitles with no overload and with overload, respectively, W is the overload factor = $ 2 / S 1 > 1 (S, being the specific stress level; t = 1,2), m Is the slope o f the stress-life equation, such as that of eqn (1), and fl is the Weibull slope of the times-to-failure distribution at a specific stress level, such as that obtained from Fig. 1. The average reliability at accelerated stress is calculated from

_Us+l NT+ 2

(12)

where N s is the number of successful trials and N r is the total number of trials undertaken.

Example 3 Umts of a particular design are tested at 15 ~ overload with the following results' N s = 14 successes

N r = 20 total units tested Then, from eqn (12), / ~ = 1 4 + 1 =0.682 20 + 2 It IS known that and

m = - 1/6

from prior testing. Since the umts are tested at 1 5 ~ overload the overload factor is W=l.15

120

D Kece~rog/u, d A Jail,

T h e n the r e h a b l h t y at the use. or n o - o v e r l o a d (NW). stress level f r o m eqn (11) is RN,. = (0.682)11

js~i-'0,

,6,j

or

RN~ = 0 9309

REFERENCES 1 Kececloglu, D and Jacks, J_ A The Arrhemus, Eyrlng, reverse power law, and combmauon models m accelerated life testing, Rellab Engng, 8(1) (1984), pp 1-9_ Nelson, W B Statistical methods for accelerated life test d a t a - - t h e reverse power law, General Electrtc Research and Development Techmcal lnJormatwn Series Report 71-C-011, December, 1970 (109pp), pp. 5-5 and 5-6 3. Kececloglu, D and Haugen, E. Interactions among the various phenomena involved in the design of dynamic and rotary machinery and their effects on rehablhty, 2nd Techmcal Report to the ONR, Washington, DC, July, 1969, 242 pp 4 Kececloglu, D , Chester, L. B and Dodge, T M. Combined bending torsion fatigue rehabdlty of AISI 4340 steel shafting with K, - 2 34, Trans ASME, J Engng Indy (May 1975), pp 748 61 5 Kececloglu, D Rehablhty Testmg, January, 1983, 854pp., p_ 507 [Available from the author at 7340N La Oesta Avenue, Tucson, AZ 85704, USA ]