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Some new aspects of stress-strength modelling
Reli~lbilit.v Engineering and System Safety 33 (1991) 131-140
Some New Aspects of Stress-Strength Modelling M. Xie & K. S h e n Division of Quality Technology, Link6ping University,S-58183, Link6ping, Sweden (Received 27 February 1990; accepted 18 April 1990)
A BSTRA C T In industrial development it is essential to improve product reliabilio: In this paper some new methods to model reliability improvement and reliability growth process are suggested. Stress-strength modelling is used in considering some interesting issues of an hnprovement process resulting in reliability growth. The reliability is the probability that the strength of the product is greater than the working as well as environmental stress. An)" improvement of product reliability at the production stage may be considered as the increase of the strength distribution using artificial heredity, as Dr van Otterloo suggested in a recent editorial. The increase in strength may usually be modelled by some changes to the parameters of the strength distribution. Generally the mean strength may be increased to get greater reliability. Also in man)" practical situations an improvement may be achieved by reducing the strength t'ariation; this is in line with Taguchi's ideasfor quality improvement. Some explicit expressions are derived for calculating these effects, assuming normally distributed stress and strength.
1 INTRODUCTION In a recent editorialt Dr van Otterloo introduced AH (artificial heredity) in discussing a connection between stress-strength modelling in reliability and an early view of human ability to survive by Charles R. Darwin3 He pointed out that it is the heredity that results in an increase of component strength relative to the stress. In this paper we note that the artificial heredity is useful in studying the improvement of reliability. We suggest some new methods to model 131 Reliability Engineering and System Safety 0951-8320/91/$03-50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain
132
M. Xie, K. Shen
reliability improvement and reliability growth process. Stress-strength modelling is used in considering some interesting issues of an improvement process resulting in reliability growth. The reliability is the probability that the strength of the product is greater than the environmental stress. An improvement in product reliability during the production as well as the design phase may be considered as the increase of the strength by AH. This may usually be modelled by some changes to the parameters of the strength distribution. Generally we may increase the mean strength to get greater reliability. Also in many practical situations an improvement may be achieved by reducing the strength variation: this is in line with Taguchi's ideas for quality improvement (see, for example, Ref. 3). Such a quality improvement may be achieved by using the robust design methodology by which variation is reduced among important product characteristics (see, for example, Ref. 4). The design and production changes which either increase the mean strength or reduce the strength variation, or both, will have definite effects on the product reliability. Naturally, such effects would depend on what the changes are, how big they are, and on the stress and strength distributions. Using normally distributed stress and strength, some explicit expressions for calculating these effects are now derived.
2 A MODEL OF RELIABILITY GROWTH For a component with stress distribution F(s) and strength distribution G(s), the reliability is given by the probability that the strength is greater than the stress, i.e.
R = p(S > s)
(1)
and it is commonly known that it is expressed by R = f o F(s)dG(s)
(2)
For general discussions and results on stress-strength modelling, we refer to Kapur and Lamberson. 5 Other general references may be found in Dhillon 6 and Loll.v The improvement process is continuous in time, at least it should, as Deming a said, 'improve constantly and forever'. Usually the stress distribution may be considered to be independent of time. For a producer, the stress is assumed to be fixed for his products when they are used by a consumer. An improvement of his products should then be connected to a strengthening of them.
Some new aspects of stress-strength modelling
133
By the above facts, any change in the reliability may be characterized by some changes in the strength distribution, since the stress distribution may usually be treated as unchanged. Any effort should always be devoted to increasing the strength. In line with the comments by Dr van Otterloo in his editorial, t the curve of the strength distribution should be moved away from that of the stress distribution. Improvement during production is a continuous process. The reliability of the components produced is increasing as a function of time, i.e. a component produced later is probably more reliable than a component produced earlier. The production phase may be subjected to direct improvement, for example, by using better materials, other manufacturing methods, other heat treatment procedures, more frequent inspection and maintenance of the production line, etc. The improvement may also be due to the stabilization of the manufacturing process, namely the equipment and the personnel have gradually become used to the production. Human reliability is also strongly connected with any learning process. All the above facts might have greater impact on the strength parameters at earlier stages than at later ones. This may be characterized by the fact that the parameters of the strength distribution are functions of time. We incorporate this by assuming that we have a set of parameters in G(s), 0~, say, and 0 G is a function of time t, denoted by Oa(t). The reliability of a product produced at time t is just R =f(OF, O~(t))
(3)
We now define the failure intensity of products at time t of the improvement process by r(t) = - In R(t)
(4)
A motivation of this is the following relative increase of the reliability (see Fig. 1): R(t + At) - R(t) R(t)
(5)
which is the rate of change of reliability and may be interpreted as the decrease of intensity of failure, Ar(t). Hence r(t) is the solution of the difference equation R(t + At) -- R(t) = -- Ar(t) R(t) or the differential equation dR(t) = -- d(rt)
R(t)
(6)
(7)
134
M, Xie, K. Shen
R
I I t Fig. I.
t+dt
t
The relative increase in the reliability, an illustration of eqns (6) and (8).
If the number of components produced per unit time at t is n and n, is the number of good components among these, the reliability, which may be interpreted as the percentage of good components, is then ng/n. Assuming that the productivity is constant, then only n, varies with t, and the relative increase of the reliability from time t to (t + At) is
R(t
+ At) - R(t) = ng(t + At)/n -- n,(t)/n = ng(t + At) - ng(t)
R(t)
n,(t)/n
(8)
n,(t)
Hence r(t) is then the relative increase of the number of bad components. The solution ofeqn (7), with the boundary condition r(t) = 0 when R(t) = 1, is just
r(t) = - In R(t)
(9)
This may be identified as the intensity function in a non-homogeneous Poisson process that is useful in modelling reliability growth (see, for example, Ref. 9). The next section shows an example in interpreting the intensity function of reliability growth process as the rate of change of product reliability. If R(t) is a constant, then the failure intensity is constant over the whole time interval. In an improvement process, r(t) is usually a decreasing function and its inverse is an increasing function. In our model 1/r(t) may be interpreted as the mean time to failure of an item produced at time t. For a reliability growth process, R(t) is increasing and r(t) is decreasing as a function of t. Some insight into the idea is given in the next section by showing an example using a c o m m o n stress-strength model.
Some new aspects of stress-strength modelling
135
3 AN E X A M P L E - - E X P O N E N T I A L STRESS A N D S T R E N G T H Suppose that both the stress and strength distributions are exponential with parameters 2 r and 2~, respectively. Then the reliability is expressed simply by R =
).f )'f + ,:
(10)
(see, for example, Ref. 5). Suppose that products of a manufacturer are successively improved in such a way that the strength of the products is increased and hence their reliability becomes greater. Assuming that the parameter 2G is a function (hopefully a decreasing one) of time, then by our definition the failure intensity (eqn (4)) is r(t) = -- In ()-f/(';-v + )~6(t)))
(11)
Suppose now that the mean strength is much larger than the mean stress, that is ).c(t) is much less than 2r for all t, and we have approximately r(t) = - l n ( 1 - 2G(t)/(2r + 2~(t))) ~ ).~(t)/(2 r + 2~(t)) ~ ).~(t)/). v
(12)
If 2~(t) is a power function, that is 2G(t) = t', then r(t) = t ~ and this is just the Duane model for reliability growth in our explanation. It can easily be seen that if2G(t) is independent of time, i.e. ~ = 0, then there is no reliability growth and the reliability of the products is independent of the time when they are produced.
4 I N C R E A S I N G THE AVERAGE S T R E N G T H A N D D E C R E A S I N G THE VARIANCE OF THE S T R E N G T H If stress or strength distribution is not exponential, the expression for the reliability in eqn (2) is not ofclosed form (see, for example, Ref. 5). However, there are more possibilities in modelling the change of strength distribution and the reality may be better described. There are many papers dealing with strength-stress modelling problems. In the following some new issues on varying the strength distribution and their effects will be considered. Since for normally distributed stress and strength the reliability expression is quite simple, we use this throughout the rest of the paper. It is well known that for normally distributed stress and strength their difference is also of normal distribution (see, for example, Ref. 5). If the stress
136
M. Aqe, K. Siren
distribution is N(yr, at) and the strength distribution is N(~t6, a~), then the reliability is given by R = 1 - ~(-:)
(13)
= ~(:)
where (14) Since u s u a l l y / ~ > l~r, the larger is the value of : and the larger is the reliability. To increase the value of z, we may increase the mean strength,/~, or decrease the variance of the strength, ~ . In Lewis 1o it is pointed out that design determines the mean strength whereas the degree of quality control in manufacturing primarily influences the strength variation.
4.1 The effect of increasing the mean strength As the first stage in increasing the strength we may try to increase the mean of the strength distribution. It is easily seen that if the mean strength is increased the reliability is also increased. This moves the strength distribution curve directly away from that of the stress distribution, just as Dr van Otterloo I indicated (see Fig. 2). This has been traditionally handled with the help of a so-called safety margin or safety factor (see, for example, Re['. 11). Also in this reference it is pointed out that if it is not economical to improve population reliability by screening out items likely to fail because of widespread stress distribution, the option left may be to increase the mean strength in order to increase the safety margin. The effect of a change in the mean strength may be derived easily. Assume
f(s) I stress /
strength
~ m b .r 0 Fig. 2.
$
Increase in the reliability by increasing the mean strength.
Some new aspects of stress-strength modelling
137
that the improvement is small and suppose that the mean strength is changed from Sto to go + A#. Then the change in the value o f . is
Assuming that the change in the mean strength is small, then it is shown easily by eqn (13) that the change in the reliability is just AR ~ ~
A:
exp (-- ::/2)
(16)
Inserting eqn (15) into eqn (16), we have that
AR
A#
+ od)) exp(--:/2)
(17)
Naturally, we may consider the improvement of the mean strength as being a continuous process. Such a situation may be present, for example, when a systematic error in production decreases in time, resulting in products of higher mean strength. In this case the increase Alt may be treated as a timedependent increasing function, and if the increase is of moderate size the calculations in eqns (15)-(17) are still valid.
4.2 The effect of decreasing the variance of the strength For many practical situations it is too hard or too expensive to increase the average strength. Along with Taguchi's idea, a possible improvement may be reducing the variation of the unknown quantity. Another possibility to increase the reliability is hence to decrease the variance of the strength. Since in practice the strength is always stochastically greater than the stress, it is reasonable to suppose that a decrease in the variance of the strength will result in an increase in the reliability (see Fig. 3). This turns out to be essential, since in fact the unknown strength is also the real problem in practice. Because the strength in unknown, we cannot limit the maximum stress level to ensure that the component will function. The effect of decreasing the variance of the strength under the assumption that the stress distribution is unchanged is hence of great interest. It is easily seen by the expression in eqn (14) that if the variation in the stress is small compared with that of the strength (see, for example, Ref. 7), an increase in reliability can be achieved by reducing the variance of the strength. In this case : ~" (#G - - lav)/a~
(18)
and we see directly that decreasing the variation is a very effective way of increasing the reliability, since ere is a scale parameter.
138
M. Xie, K. Shen
f(s) / stress I
//~
I strength I
0 Fig. 3.
s Increase in the reliability by reducing the variance of the strength.
The change in the value of_- resulting from a small change in the variance of strength is A z ~, (lt F - p~) A a ~ / a 2
(19)
It should be noted that A ~ < 0 if the variation is decreased, and hence Az>0. As before, the change in the reliability due to an infinitesimal change in the value of z is approximately equal to AR ~ ~
Az
exp ( - z2/2)
(20)
and, by inserting eqn (19) into eqn (20), this is equal to AR .,.~-(Pr -- Po)Atro x/(27t)cr2 exp ( - --2/2)
(21)
Similar to the case when Po is increasing with time, we may also consider the case when o'~ decreases with time. This may be achieved by modern quality control so that a~ is not only kept within the tolerance limits but also approaches as close to the nominal value as possible (see, for example, Ref. 3).
5 DISCUSSION It is essential in practice to move the strength curve away from the stress curve in order to increase reliability. In this paper we have shown that it turns out that time dependence may be incorporated. Using stress-strength
Some new aspects of stress-strength modelling
139
modelling and artificial heredity, reliability growth in an improvement process may be modelled. We have also stressed that not only should the average strength be increased but also the variation of the strength must be kept at a minimum in order to have higher reliability. As Professor Pronikov wrote in an early book, 12 the principal methods for increased reliability are the use of wear-resistant materials or increasing the stability of the manufacturing processes in the production of the components. This is also pointed out in Ref. 5, pp. 73-87 (see also Ref. 13). It should be noted that in this paper we have been interested in components that are produced at time t and their failure rates at delivery. A stress-strength model is used here to compare the reliability of two components delivered at different times. Conventionally, stress-strength modelling is a static analysis. For the time-dependent counterpart, i.e. for a fixed component, stress and strength must be modelled by two stochastic processes. The stochastic analysis is not simple (see, for example, Ref. 14, and also an earlier paper by Schatz et al. ~5). However, for a single component, its reliability deterioration as indicated by Loll ~ may also be modelled by noting that the reliability is decreasing in this case. Using stress-strength modelling in considering reliability growth may be advantageous since it is more strongly connected to the failure mechanism than the ordinary lifetime model. The latter is more concerned with the pattern of failures. After a certain failure mechanism is revealed and analyzed, appropriate design or manufacturing actions may be taken to reduce or eliminate the source of unreliability, thus resulting in reliability growth. Normally distributed stress and strength are assumed in Section 4. Of course there are many other possibilities. The effects of varying parameters may be different using other stress-strength models. However, essentially the same procedure may be used in calculating the final effect on the reliability.
ACKNOWLEDGEMENT Part of this research is supported by CENIIT at Link/Sping University under a research project entitled 'system reliability'.
REFERENCES 1. van Otterloo, R. W., Editorial--Artificial heredity. Reliability Engineering and System Safety, 26 (1989) 291-2. 2. Darwin, C. R., On the Origin of Species by Means of Natural Selection or the Preservation of Favoured Races in the Struggle for Life. London, 1859.
140
M. Xie, K. Shen
3. Taguchi, G., Introduction to Quality Engineering. UNIPUB, New York, 1986. 4. Bergman. B., On robust design methodology for reliability improvement. In Reliability Achiecenlent, ed. T. Aven. Elsevier, London, 1989, pp. 72-9. 5. Kapur, K'. C. & Lamberson, L. R., Reliability in Engineering Design. John Wiley & Sons, New York, 1977. 6. Dhillon, B. S., Mechanical reliability: interference theory models. In Proc. Annual Reliability and Maintainability Symp., 1980, pp. 462-7. 7. Loll, V., Load-strength modelling of mechanics and electronics. Quality and Reliability Engineering International, 3 (1987) 149-55. 8. Deming, W. E., Out of the Crisis." Quality, Productit~ity and Competitire Position. Cambridge University Press, Cambridge, 1988. 9. H~irtler, G., The nonhomogeneous Poisson process--a model for the reliability of complex repairable systems. Microelectronics and Reliabilit); 29 (I 989) 381-6. 10. Lewis, E. E., bltro&wtion to Reliability Engineering. John Wiley & Sons, New York, 1987, p. 187. 11. O'Connor, P. D. T., Practical Reliability Engineering. John Wiley & Sons, New York, 1985, pp. 102 and 104. 12. Pronikov, A. S., Dependabilit)' and Durability of Engineering Products. Butterworths, London, 1973, p. 39. 13. Kapur, K. C., Techniques of estimating reliability at design stage. In Handbook of Reliability Engineering and Management, eds W. G. lreson & C. F. Coombs. McGraw-Hill, New York, 1988, Chapter 18. 14. Shen, K., On the relationship between component failure rate and stressstrength distributional characteristics. Microelectronics and Reliability, 28 (1988) 801-12. 15. Schatz, R., Shooman, M. & Shaw, L., Application of time dependent stressstrength models of non-electrical and electrical systems. In Proc. Annual Reliability and Maintainability Syrup., 1974, pp. 540-7.
Some New Aspects of Stress-Strength Modelling M. Xie & K. S h e n Division of Quality Technology, Link6ping University,S-58183, Link6ping, Sweden (Received 27 February 1990; accepted 18 April 1990)
A BSTRA C T In industrial development it is essential to improve product reliabilio: In this paper some new methods to model reliability improvement and reliability growth process are suggested. Stress-strength modelling is used in considering some interesting issues of an hnprovement process resulting in reliability growth. The reliability is the probability that the strength of the product is greater than the working as well as environmental stress. An)" improvement of product reliability at the production stage may be considered as the increase of the strength distribution using artificial heredity, as Dr van Otterloo suggested in a recent editorial. The increase in strength may usually be modelled by some changes to the parameters of the strength distribution. Generally the mean strength may be increased to get greater reliability. Also in man)" practical situations an improvement may be achieved by reducing the strength t'ariation; this is in line with Taguchi's ideasfor quality improvement. Some explicit expressions are derived for calculating these effects, assuming normally distributed stress and strength.
1 INTRODUCTION In a recent editorialt Dr van Otterloo introduced AH (artificial heredity) in discussing a connection between stress-strength modelling in reliability and an early view of human ability to survive by Charles R. Darwin3 He pointed out that it is the heredity that results in an increase of component strength relative to the stress. In this paper we note that the artificial heredity is useful in studying the improvement of reliability. We suggest some new methods to model 131 Reliability Engineering and System Safety 0951-8320/91/$03-50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain
132
M. Xie, K. Shen
reliability improvement and reliability growth process. Stress-strength modelling is used in considering some interesting issues of an improvement process resulting in reliability growth. The reliability is the probability that the strength of the product is greater than the environmental stress. An improvement in product reliability during the production as well as the design phase may be considered as the increase of the strength by AH. This may usually be modelled by some changes to the parameters of the strength distribution. Generally we may increase the mean strength to get greater reliability. Also in many practical situations an improvement may be achieved by reducing the strength variation: this is in line with Taguchi's ideas for quality improvement (see, for example, Ref. 3). Such a quality improvement may be achieved by using the robust design methodology by which variation is reduced among important product characteristics (see, for example, Ref. 4). The design and production changes which either increase the mean strength or reduce the strength variation, or both, will have definite effects on the product reliability. Naturally, such effects would depend on what the changes are, how big they are, and on the stress and strength distributions. Using normally distributed stress and strength, some explicit expressions for calculating these effects are now derived.
2 A MODEL OF RELIABILITY GROWTH For a component with stress distribution F(s) and strength distribution G(s), the reliability is given by the probability that the strength is greater than the stress, i.e.
R = p(S > s)
(1)
and it is commonly known that it is expressed by R = f o F(s)dG(s)
(2)
For general discussions and results on stress-strength modelling, we refer to Kapur and Lamberson. 5 Other general references may be found in Dhillon 6 and Loll.v The improvement process is continuous in time, at least it should, as Deming a said, 'improve constantly and forever'. Usually the stress distribution may be considered to be independent of time. For a producer, the stress is assumed to be fixed for his products when they are used by a consumer. An improvement of his products should then be connected to a strengthening of them.
Some new aspects of stress-strength modelling
133
By the above facts, any change in the reliability may be characterized by some changes in the strength distribution, since the stress distribution may usually be treated as unchanged. Any effort should always be devoted to increasing the strength. In line with the comments by Dr van Otterloo in his editorial, t the curve of the strength distribution should be moved away from that of the stress distribution. Improvement during production is a continuous process. The reliability of the components produced is increasing as a function of time, i.e. a component produced later is probably more reliable than a component produced earlier. The production phase may be subjected to direct improvement, for example, by using better materials, other manufacturing methods, other heat treatment procedures, more frequent inspection and maintenance of the production line, etc. The improvement may also be due to the stabilization of the manufacturing process, namely the equipment and the personnel have gradually become used to the production. Human reliability is also strongly connected with any learning process. All the above facts might have greater impact on the strength parameters at earlier stages than at later ones. This may be characterized by the fact that the parameters of the strength distribution are functions of time. We incorporate this by assuming that we have a set of parameters in G(s), 0~, say, and 0 G is a function of time t, denoted by Oa(t). The reliability of a product produced at time t is just R =f(OF, O~(t))
(3)
We now define the failure intensity of products at time t of the improvement process by r(t) = - In R(t)
(4)
A motivation of this is the following relative increase of the reliability (see Fig. 1): R(t + At) - R(t) R(t)
(5)
which is the rate of change of reliability and may be interpreted as the decrease of intensity of failure, Ar(t). Hence r(t) is the solution of the difference equation R(t + At) -- R(t) = -- Ar(t) R(t) or the differential equation dR(t) = -- d(rt)
R(t)
(6)
(7)
134
M, Xie, K. Shen
R
I I t Fig. I.
t+dt
t
The relative increase in the reliability, an illustration of eqns (6) and (8).
If the number of components produced per unit time at t is n and n, is the number of good components among these, the reliability, which may be interpreted as the percentage of good components, is then ng/n. Assuming that the productivity is constant, then only n, varies with t, and the relative increase of the reliability from time t to (t + At) is
R(t
+ At) - R(t) = ng(t + At)/n -- n,(t)/n = ng(t + At) - ng(t)
R(t)
n,(t)/n
(8)
n,(t)
Hence r(t) is then the relative increase of the number of bad components. The solution ofeqn (7), with the boundary condition r(t) = 0 when R(t) = 1, is just
r(t) = - In R(t)
(9)
This may be identified as the intensity function in a non-homogeneous Poisson process that is useful in modelling reliability growth (see, for example, Ref. 9). The next section shows an example in interpreting the intensity function of reliability growth process as the rate of change of product reliability. If R(t) is a constant, then the failure intensity is constant over the whole time interval. In an improvement process, r(t) is usually a decreasing function and its inverse is an increasing function. In our model 1/r(t) may be interpreted as the mean time to failure of an item produced at time t. For a reliability growth process, R(t) is increasing and r(t) is decreasing as a function of t. Some insight into the idea is given in the next section by showing an example using a c o m m o n stress-strength model.
Some new aspects of stress-strength modelling
135
3 AN E X A M P L E - - E X P O N E N T I A L STRESS A N D S T R E N G T H Suppose that both the stress and strength distributions are exponential with parameters 2 r and 2~, respectively. Then the reliability is expressed simply by R =
).f )'f + ,:
(10)
(see, for example, Ref. 5). Suppose that products of a manufacturer are successively improved in such a way that the strength of the products is increased and hence their reliability becomes greater. Assuming that the parameter 2G is a function (hopefully a decreasing one) of time, then by our definition the failure intensity (eqn (4)) is r(t) = -- In ()-f/(';-v + )~6(t)))
(11)
Suppose now that the mean strength is much larger than the mean stress, that is ).c(t) is much less than 2r for all t, and we have approximately r(t) = - l n ( 1 - 2G(t)/(2r + 2~(t))) ~ ).~(t)/(2 r + 2~(t)) ~ ).~(t)/). v
(12)
If 2~(t) is a power function, that is 2G(t) = t', then r(t) = t ~ and this is just the Duane model for reliability growth in our explanation. It can easily be seen that if2G(t) is independent of time, i.e. ~ = 0, then there is no reliability growth and the reliability of the products is independent of the time when they are produced.
4 I N C R E A S I N G THE AVERAGE S T R E N G T H A N D D E C R E A S I N G THE VARIANCE OF THE S T R E N G T H If stress or strength distribution is not exponential, the expression for the reliability in eqn (2) is not ofclosed form (see, for example, Ref. 5). However, there are more possibilities in modelling the change of strength distribution and the reality may be better described. There are many papers dealing with strength-stress modelling problems. In the following some new issues on varying the strength distribution and their effects will be considered. Since for normally distributed stress and strength the reliability expression is quite simple, we use this throughout the rest of the paper. It is well known that for normally distributed stress and strength their difference is also of normal distribution (see, for example, Ref. 5). If the stress
136
M. Aqe, K. Siren
distribution is N(yr, at) and the strength distribution is N(~t6, a~), then the reliability is given by R = 1 - ~(-:)
(13)
= ~(:)
where (14) Since u s u a l l y / ~ > l~r, the larger is the value of : and the larger is the reliability. To increase the value of z, we may increase the mean strength,/~, or decrease the variance of the strength, ~ . In Lewis 1o it is pointed out that design determines the mean strength whereas the degree of quality control in manufacturing primarily influences the strength variation.
4.1 The effect of increasing the mean strength As the first stage in increasing the strength we may try to increase the mean of the strength distribution. It is easily seen that if the mean strength is increased the reliability is also increased. This moves the strength distribution curve directly away from that of the stress distribution, just as Dr van Otterloo I indicated (see Fig. 2). This has been traditionally handled with the help of a so-called safety margin or safety factor (see, for example, Re['. 11). Also in this reference it is pointed out that if it is not economical to improve population reliability by screening out items likely to fail because of widespread stress distribution, the option left may be to increase the mean strength in order to increase the safety margin. The effect of a change in the mean strength may be derived easily. Assume
f(s) I stress /
strength
~ m b .r 0 Fig. 2.
$
Increase in the reliability by increasing the mean strength.
Some new aspects of stress-strength modelling
137
that the improvement is small and suppose that the mean strength is changed from Sto to go + A#. Then the change in the value o f . is
Assuming that the change in the mean strength is small, then it is shown easily by eqn (13) that the change in the reliability is just AR ~ ~
A:
exp (-- ::/2)
(16)
Inserting eqn (15) into eqn (16), we have that
AR
A#
+ od)) exp(--:/2)
(17)
Naturally, we may consider the improvement of the mean strength as being a continuous process. Such a situation may be present, for example, when a systematic error in production decreases in time, resulting in products of higher mean strength. In this case the increase Alt may be treated as a timedependent increasing function, and if the increase is of moderate size the calculations in eqns (15)-(17) are still valid.
4.2 The effect of decreasing the variance of the strength For many practical situations it is too hard or too expensive to increase the average strength. Along with Taguchi's idea, a possible improvement may be reducing the variation of the unknown quantity. Another possibility to increase the reliability is hence to decrease the variance of the strength. Since in practice the strength is always stochastically greater than the stress, it is reasonable to suppose that a decrease in the variance of the strength will result in an increase in the reliability (see Fig. 3). This turns out to be essential, since in fact the unknown strength is also the real problem in practice. Because the strength in unknown, we cannot limit the maximum stress level to ensure that the component will function. The effect of decreasing the variance of the strength under the assumption that the stress distribution is unchanged is hence of great interest. It is easily seen by the expression in eqn (14) that if the variation in the stress is small compared with that of the strength (see, for example, Ref. 7), an increase in reliability can be achieved by reducing the variance of the strength. In this case : ~" (#G - - lav)/a~
(18)
and we see directly that decreasing the variation is a very effective way of increasing the reliability, since ere is a scale parameter.
138
M. Xie, K. Shen
f(s) / stress I
//~
I strength I
0 Fig. 3.
s Increase in the reliability by reducing the variance of the strength.
The change in the value of_- resulting from a small change in the variance of strength is A z ~, (lt F - p~) A a ~ / a 2
(19)
It should be noted that A ~ < 0 if the variation is decreased, and hence Az>0. As before, the change in the reliability due to an infinitesimal change in the value of z is approximately equal to AR ~ ~
Az
exp ( - z2/2)
(20)
and, by inserting eqn (19) into eqn (20), this is equal to AR .,.~-(Pr -- Po)Atro x/(27t)cr2 exp ( - --2/2)
(21)
Similar to the case when Po is increasing with time, we may also consider the case when o'~ decreases with time. This may be achieved by modern quality control so that a~ is not only kept within the tolerance limits but also approaches as close to the nominal value as possible (see, for example, Ref. 3).
5 DISCUSSION It is essential in practice to move the strength curve away from the stress curve in order to increase reliability. In this paper we have shown that it turns out that time dependence may be incorporated. Using stress-strength
Some new aspects of stress-strength modelling
139
modelling and artificial heredity, reliability growth in an improvement process may be modelled. We have also stressed that not only should the average strength be increased but also the variation of the strength must be kept at a minimum in order to have higher reliability. As Professor Pronikov wrote in an early book, 12 the principal methods for increased reliability are the use of wear-resistant materials or increasing the stability of the manufacturing processes in the production of the components. This is also pointed out in Ref. 5, pp. 73-87 (see also Ref. 13). It should be noted that in this paper we have been interested in components that are produced at time t and their failure rates at delivery. A stress-strength model is used here to compare the reliability of two components delivered at different times. Conventionally, stress-strength modelling is a static analysis. For the time-dependent counterpart, i.e. for a fixed component, stress and strength must be modelled by two stochastic processes. The stochastic analysis is not simple (see, for example, Ref. 14, and also an earlier paper by Schatz et al. ~5). However, for a single component, its reliability deterioration as indicated by Loll ~ may also be modelled by noting that the reliability is decreasing in this case. Using stress-strength modelling in considering reliability growth may be advantageous since it is more strongly connected to the failure mechanism than the ordinary lifetime model. The latter is more concerned with the pattern of failures. After a certain failure mechanism is revealed and analyzed, appropriate design or manufacturing actions may be taken to reduce or eliminate the source of unreliability, thus resulting in reliability growth. Normally distributed stress and strength are assumed in Section 4. Of course there are many other possibilities. The effects of varying parameters may be different using other stress-strength models. However, essentially the same procedure may be used in calculating the final effect on the reliability.
ACKNOWLEDGEMENT Part of this research is supported by CENIIT at Link/Sping University under a research project entitled 'system reliability'.
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