Structural reliability of rocket motor hardware—A probabilistic approach

Reliability Engineering 12 (1985) 193-203

Structural Reliability of Rocket M o t o r H a r d w a r e - - A Probabilistic Approach

S. Rajagopalant and M. Sundaresan~. t Quality Assurance Division, Launch Vehicle Systems Division, Vikram Sarabhai Space Centre, Trivandrum-695 022, India

(Received: 5 December 1984)

ABSTRACT The classical design oJ mechanical hardware is based on the saJetyjactor (sometimes reJerred to as the "index oJ ignorance'); this does not consider the statistical nature oJ the design variables and leads to overdesign, which is usually reflected in excess weight. The problem is tackled by using the Stress Strength lnterJerence theory, wherein we take into account the statistical distributions oj the environmental stress acting on the component and the component's ability to withstand that stress. In a rocket motor, the structural reliability is given by R = P[(S

- s) > 0]

where P, S and s stand jor the probability, strength and stress, respectively. Equations were developed for expressing S and s in terms c~[ basic' design parameters such as shell thickness, weld eJficiency, uniaxial ultimate strength, biaxial gain, radius oJ motor ca'~e, weld mismatch, etc., and the reliability at a spec!fied confidence level is obtained using a Monte Carlo simulation technique.

1

BG

NOMENCLATURE

Biaxial gain.

S e m i - c r a c k length. D i s t r i b u t i o n o f s t r e n g t h S. 193 Reliability Engineering 0143-8174/85/$03.30 ( Elsevier Applied Science Publishers Ltd. England, 1985. Printed in Great Britain

C f(S)

194

F

g(s) k K1 K=

K,(, m S m s

MEOP n Pull r

R s

S SSI t

V 8 V Gf O'uu O'u~ (7S

0

S. Rq]agopalan. .14, Sundaresat~

Safety factor. Distribution of stress s. Confidence coefficient. Plastic correction factor for secondary bending stresses. Correction factor for through cracks. Plane strain fracture toughness. Mean of the distribution of strength S. Mean of the distribution of stress s. Maximum Expected Operating Pressure. Test sample size. Ultimate design pressure. Radius of motor case. Reliability. Stress. Strength. Stress Strength Interference. Shell thickness. Coefficient of variation of both strength S and stress ~s. Longitudinal weld mismatch. Weld efficiency. Poisson's ratio. Allowable fracture stress for through crack. Uniaxial ultimate strength. Uniaxial yield strength. Standard deviation of distribution of stress s. Standard deviation of distribution of strength S. Standard normal probability density function. Standard normal distribution function.

2

INTRODUCTION

Reliability analysis is an absolute necessity in modern day science, where complex devices are utilised for military and scientific purposes. Unreliability has consequences in cost, time wasted, psychological effect of inconvenience and, in certain instances, personal and national security. While reliability analysis procedures are more or less well organised and aptly documented in the form of military handbooks, etc., for electronic systems, the work done with regard to mechanical systems does not seem

Structural reliability ~/ rocket motor hardware

195

to be appreciable. This is mainly due to the fact that mechanical components do not conform to conventional mathematical tractability, i.e. a constant failure rate, so that failure rates are not well documented. However, with the advent of the Stress Strength Interference (SSI) theory, the evaluation of the reliability of mechanical systems, such as rocket motor hardware, is now possible. The classical approach to design does not take into account the statistical nature of the design variables and leads to overdesign, which is usually reflected in excess weight. The object of the present paper is to make a brief overview of the probabilistic approach to the design of mechanical systems based on the Stress Strength Interference (SSI) theory I - 13 and to develop a generalised mathematical model to arrive at the reliability of a mechanical system, say, rocket motor hardware, taking into account its fracture criteria.

3

MATHEMATICAL MODEL

The Stress Strength Interference (SSI) theory is concerned with the problem of determining the probability that a system of strength S does not fail when subjected to a stress s. The reliability of the system is given by the probability that S is greater than s, where S and s follow certain statistical distributions, whose parameters are computed from actual test data. i.e. R=P(S>s)

(l)

R = P [ ( S - s) > 01

(2)

If the strength and stress distributions are given b y j ( S ) and g(s), we have R =

; IfJ, 1

S) d S g(s) ds

(3)

where a and b are respectively the minimum and maximum values the stress can assume in its probability density function (pdf) and c is the maximum value the strength can assume in its pdf. In particular, for the lognormal, Weibull and gamma pdf's, a is the location parameter, b = + oc and c = + vc,. Table l presents the various models available in the literature. In the rocket motor casing, the strength is measured by the burst pressure and the stress is measured by the Maximum Expected Operating

196

S. Rajagopalan, M. Sundaresan

Pressure (MEOP). In the case where the burst pressure values are available from the burst pressure test results, the values can be fitted to a certain standard statistical distribution; its 'goodness' of fit can be tested by statistical tests and SSI theory can be employed to obtain the reliability of the m o t o r case. In the case where burst pressure values are not available, its distribution can be obtained as described below: The strength of the m o t o r case, as measured by burst pressure, is given by t~/*a*BG*

,4,

For fracture-based design (i.e. o-1 < auy),

cos

5

/(taut) TrCK 2

a* = af = Kic

j

(5)

q*= 1

t6)

BG* = 1

(7)

K~'= 1 with

K2, the

correction factor for t h r o u g h cracks, given by (5rc22)(4 - r,"~

K~ = ! + \ 5 ~ - ) \ - 5 - - ) 22

. . . C2 .

x/[12(l

(9)

- v2)]

(1o)

C = 0.5 (through crack)

(12)

rl

where v is Poisson's ratio, and ~'=3 -4v is the plane strain correction factor, and

For non-fracture-based design (i.e. af > auy), (7'i' __ O'uu

t/* = t / = weld efficiency BG* = 1.1

(13) (14)

(15)

197

Structural reliability oj rocket motor hardware

depending upon the stress-strain curve of the material and the factor for secondary plastic bending stresses: K* --2

(16)

From the distributions of the basic parameters (namely, shell thickness, weld efficiency, uniaxial ultimate strength, radius of motor case, longitudinal weld mismatch, ultimate design pressure, etc.), the TABLE 1

Various Models of SS| Theory Serial number

Category

1 2 3

I. CIosed/orm solution available

4 5 6 7

8 9 10 11 12 13 14

II. Values tabulated

Ill. Numerical integration needed

Strength distribution J(S)

Stress distribution g(s)

Exponential Normal Gamma Weibull Power series Normal Lognormal Gamma Weibull Weibull

Exponential Exponential Exponential Power series Power series Normal Lognormal Gamma Weibull Type-I largest extreme value Weibull

Type-i smallest extreme value Type-I smallest extreme value Type I smallest extreme value Type-III smallest extreme value

Type-I largest extreme value Type-II largest extreme value Type-III largest extreme value

distribution of the burst pressure can be obtained by using Monte Carlo simulation with the various parameters as described below. A set of 'n' random numbers following normal distribution with the respective parameters is generated for each of the design variables, and thus from eqn (4), 'n' values of the strength are obtained. Normal distribution is fitted to this set of values and statistically tested for 'goodness' of fit. The mean and standard deviations of the resulting normal distribution are taken as those for the distribution of strength of

198

S. Rajagopalan, M. Sundaresan

the motor case. The number of Monte Carlo replications, viz. 'n', is fixed as follows: Start with some value, say 100 for 'n'. Then, by the procedure described below, obtain the reliability of the motor case. Next, increment 'n' by a number (say, 50) and again arrive at the reliability value. Now, these two values of the reliability will be different, since the number of replications is different in each case. Now, again increment the number of replications and continue the process until the difference in values of the reliability between the previous and present trials is less than a preassigned small value. If the distributions of the strength S and stress s, given byj(S) and g(s) respectively, are normal distributions, then, they can be expressed as

1

{ (S-ms)2~ 2~ j

1

{

j(S) - asx/(2rt) exp and

g(s)-a~x/(27t ) exp

5~s2

j

17)

18)

where s=individual stresses; m s = m e a n of the individual stresses. Reliability can be defined as the probability that the strength S exceeds the stress s, or, that R = P [ ( S - s ) > 0] (19) If we designate z= S- s

(20)

then the reliability is the probability that z > 0. Since j(S) and g(s) are normally distributed, h(z), the distribution of z is also normal, and is given by

1 J h(z) = o-:x/(2rt) exp [

2a 2

(21)

where o: = x / ( a s2 + o~2 )

(22)

m: = m s - m,

(23)

and The reliability would then be given by all the probabilities of z being a positive value.

Structural reliabiliO, oJ rocket motor hardware

199

Therefore R = P ( z > O) =

R - °':x/ (2~)

h(z) dz

f[

exp [

~2

j

(24)

By transforming the integration variables, we obtain (25)

R = O(u)

where m s - rn~

u - / ( ~ + ~)

(26)

where • is the standard normal distribution function. To find the lower confidence limit on R, we first compute the variance of R as follows: (ms--m~) 2 ] 02 I =W.r R , 2 {1 + ( ~ } 3 ] (27) \a~/ where 0 ~ 0

m s _ ms / "2 --~2

J

(28)

0 = standard normal probability density function and n = test sample size. Then, the lower confidence limit (RL), with a confidence coefficient k on R, is given by R E = R - k V12

(29)

Essentially, the equation for R Eindicates the minimum reliability that can be assumed, associated with the stipulated confidence. The various steps in the process to be followed in calculating the minimum reliability (lower confidence limit) are (i) Determine the reliability R using eqn (25). (ii) Determine the variance Vat R using eqn (27). (iii) Calculate the minimum reliability RE, associated with the stipulated confidence using eqn (29).

200

S. R a j a g o p a l a n ,

M. Sundaresan

So far only normal distribution has been considered. In fact, the whole analysis is not restricted to normal distribution; any other distribution (such as those listed in Table 1) can be treated in exactly the same fashion. Of course, eqn (26) has to be replaced by an appropriate expression. In some cases, such as when both the strength as well as stress variables follow Weibull distributions, a closed form solution is not available; here, one has to resort to some numerical techniques.

4

RELIABILITY VERSUS SAFETY F A C T O R

The structural reliability of the rocket motor, as given by eqn (25), is

R--,

( ms - m~ ~

(30)

Now, the safety factor Fis the ratio of the burst pressure to the maximum operating pressure. Therefore, F = m~-s rn~

(3i)

Using eqn (31), eqn (30) can be written as

+{ Fms-m~+ ) \4( s It is seen practically that the percentage variations in operating pressure and burst pressure are equal, i.e. their coefficient of variation values are equal. Thus GS

-

~7s

ms m~

-

V

(33)

where Vis the coefficient of variation of strength as well as stress (assumed equal), i.e. ms C7S = ¢7s - ms

= a~F using eqn (31).

(34)

Structuralreliability~ff"rocketmotorhardware TABLE

201

2

Reliability vs Safety Factor for Various Values of the Coefficient of Variation of Strength and Stress

Reliability R ~C

raO¢lficiento/ riation V SaJety ~ 1.00 1.02 1-04 1-06 1.08 1.10 1-12 1.14 1.16 1-18 1.20 1.25 1.30 t.40 1.50 2-00

7. ¢ o

0.50 0.75804 0-91621 0.98024 0.99671 0.99962 0.999967 0.999998 0.999998 0.999998 0.999998 0.999998 0.999998 0.999998 0.999 998 0.999 998

0.50 0.712 26 0-86520 0.95021 0.985 15 0-99643 0.99931 0.99989 0-999989 0.99999 0.99999 0.99999 0.99999 0.99999 0.999 99 0.999 99

3 ""

4 '!i,

5 "i,

10",,

0.50 0.67964 0.82121 0.91505 0.96499 0.98753 0.99614 0-99868 0.99975 0.999948 0.999990 0.999990 0.999990 0.999990 0.999 990 0.999 990

0.50 0.63683 0.75490 0.84835 0.91291 0.95369 0.97715 0.98950 0.99550 0.998 19 0.999 32 0-999953 0-999998 0.999998 0.999 998 0.999 998

0-50 0.61026 0.70953 0.79489 0-86150 0.910 74 0.94504 0.96758 0.98166 0-99003 0.99478 0.99911 0.99987 0-99987 0-99987 0.999 87

0.50 0.55567 0.60873 0-65970 0.70664 0.74943 0.78357 0.82205 0.85192 0.87775 0-89980 0.94084 0.966 31 0-98996 0.997 23 0.999 96

Using eqn (34), eqn (32) b e c o m e s R= R =

@J. m~(F-

1) } [ ~ 7 , ~ / ( r 2 + 1)

@F V(F-1) 2 +

1

(35)

(36)

T h u s , the structural reliability o f a r o c k e t m o t o r can be estimated for various values o f safety f a c t o r ( F ) a n d coefficient o f variation (V) o f the strength a n d stress, as given in T a b l e 2 (taken f r o m ref. 14).

5

CONCLUSIONS

A generalised m e t h o d for stress S t r e n g t h I n t e r f e r e n c e (SSI) t h e o r y has been described in this p a p e r in o r d e r to estimate the structural reliability

202

S. Rajagopalan, M. Sundaresan

of a rocket motor chamber from known distributions of burst pressure and M E O P of the chamber using the Monte Carlo simulation approach. In the case where the actual distribution of burst pressure is not known, this can be found from the specification values of the basic parameters. The approach can be adopted for both non-fracture-based as well as fracture-based designs.

ACKNOWLEDGEMENT The authors are grateful to Shri K. Chandragupta, Head, Quality Assurance Division, Vikram Sarabhai Space Centre, Trivandrum for his kind encouragement in the preparation of this paper.

REFERENCES I. Kapur, K. C. and Lamberson, L. R. Reliability in Engineering Design, John Wiley & Sons, New York, 1977. 2. Smith, C. O. Introduction to Reliability in Design, McGraw-Hill, New York, 1976. 3. Kececioglu, D. Reliability analysis of mechanical components and systems, Nuclear Engineering and Design, 19 (1972), pp. 259-90. 4. Haugen, E. B. Probabilistic Approaches to Design, John Wiley & Sons, New York, 1968. 5. Disney, R. and Lipson, C. The determination of the probability of failures by stress strength interference theory, Proc. Ann. Syrup. Reliab., Boston, Mass., 16-t8 January 1968, pp. 417 22. 6. Shigley, J. E. Mechanical Engineering Design, McGraw-Hill, New York, 1963. 7. Juvinall, R. C. Engineering Considerations o[ Stress, Strain and Strength, McGraw-Hill, New York, 1967. 8. Lipson, C. and Juvinall, R. C. Handbook oJStress and Strength, Macmillan. New York, 1963. 9. Cable, C. W. and Virene, E. P. Structural reliability with normally distributed static and dynamic loads and strength, Proc. Ann. Symp. Reliab., Washington, DC, 1967. 10. Lemon, C. H. and Manning, S. D. Literature survey on structural reliability, IEEE Trans. Reliab., 23 (October 1974). 11. Dhillon, B. S. Mechanical reliability:interference theory models, Proc. Ann. Reliab. Maintainab. Symp., 1980, pp. 462 -7. 12. Dhillon, B. S. Stress strength reliability models, Microelectron. Reliab., 20 (1980), pp. 153-60.

Structural reliability q[ rocket motor hardware

203

13. Johnson, W. and Maxwell, R. E. Reliability analysis of structure a new approach, Proc. Ann. Reliab. Maintainab. Symp., Las Vegas, Nevada, 20 22 January 1976, pp. 213-17. 14. Debnath, N. K. Structural reliability of a rocket chamber, The QR Journal. (January 1978), pp. 17-20. 15. Lloyd, D. K. and Lipow, M. Reliability Management, Methods and Mathematics, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1962, pp. 237 8.