A practical approach for predicting fatigue reliability under random cyclic loading

Reliability Engineering and System Safe O, 50 (1995) 7 15 ELSEVIER

0951-8320(95)00072-0

© 1995 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0951-8320/95/$9.50

A practical approach for predicting fatigue reliability under random cyclic loading Jun Tang & Jie Z h a o Center .for Computer-Aided Design, College of Engineering, The University of Iowa, Iowa City, Iowa 52242-1000, USA (Received 2 April 1995: accepted 13 June 1995)

The purpose of this paper is to provide a simple approach for reliability analysis based on fatigue or overstress failure modes of mechanical components, and explain how this integrated method carries out spectral fatigue damage and failure reliability analysis. In exploring the ability to predict spectral fatigue life and assess the reliability under a specified dynamics environment, a methodology for reliability assessment and its corresponding fatigue life prediction of mechanical components using a supply-demand interference approach is developed in this paper. Since the methodology couples dynamics analysis and stochastic analysis for fatigue damage and reliability prediction, the conversion of the duty cycle history for the reliability study of an individual component is also presented. Using the proposed methodology, mechanical component reliability can be predicted according to different mission requirements. For an explanation of this methodology, a probabilistic method of deciding the relationship between the allowable stress or fatigue endurance limit and reliability is also presented.

1 INTRODUCTION A major difficulty in dealing with reliability analysis is that it relies heavily on historical information such as failure modes and mechanisms of component failures. Traditionally, the detection of mechanical component durability failure has relied upon proving durability tests within the industry. When a component or system passes durability tests, however, it is difficult to identify areas of over-design, if there are any. This motivated the development of integrated analysis methodologies which can overcome the abovementioned shortcomings to perform reliability analyses for estimating product life early in the design phase. 1,4 The mechanical component is subjected to cyclic loads in which the component transmits fluctuations over a wide range. It is a difficult problem to decompose a random cyclic loading history into a series of random simple loads. Meanwhile, the allowable stress and fatigue limits are also undeterministic values in the design. In this paper, a probabilistic method of deciding the relationship between the allowable stress or fatigue endurance limit and reliability is presented. An editing algorithm and a cycle counting method of variable amplitude dynamic loading history have also been performed.

Therefore, the reliability of a component that considers the loading, strength, or fatigue limit as random variables can be assessed when these varieties have been specified. The first part of this paper is a discussion of the relationship between the fatigue endurance limit, ultimate stress, and reliability. The reliability is extended to the finite life region of the S-N curves in conjunction with random loading. It gives a function of fatigue limit vs reliability of the component. From this function the component life and its corresponding reliability can be obtained. In the second part of the paper the editing procedure and cycle counting of duty cycle information are discussed. Finally, a numerical example is performed using the developed method which shows reasonable results.

2 FAILURE MODE AND MECHANISM Failure mode of mechanical components Reliability is the probability of survival of a component within a mission period under specified conditions. It is related to failure. However, the failure modes for components are many and varied depending on the operating conditions which include

8

J.

Tang, J.

materials, loads, and environment. Fatigue of engineering materials is commonly regarded as an important deterioration process and a principal mode of failure for various structural and mechanical systems. The operating life of a mechanical component arrangement is not only dependent upon the susceptibility of the materials to corrosion, but also upon the applications which indicate the stress type and level, such as static, cyclic, or dynamic. A component may be in a static, cyclic, or dynamic operating mode. Each load mode will induce a different failure mode. A mechanical component is in a cyclic load mode which can be classified as being unidirectional or reverse and also a constant or variable amplitude. Therefore, only the cyclic loaded components will be considered in this paper. In this case, the following, most common, failure modes for components will be considered. They are: (1) fracture due to fatigue, and (2) broken or wornout materials. Failure mechanisms which cause component failure are overstress, fracture, and fatigue, among others. Since fatigue is the most important parameter to be controlled regarding the life of the component, the relationship between the fatigue life and reliability under specified operating conditions has to be established for component reliability assessment. The following function is then established. 2'a

F(L, R, para.) = 0

(2.1)

where L = fatigue life of component, R = reliability of component during corresponding life period and para. = other parameters, such as environmental conditions and so forth.

For fatigue life analysis, the stress-life approach is the most important method to be used in the real world, especially for high cycle life. Failure is defined as a crack that has been initiated to a certain length. The stress-life curves (S-N curve), were used to characterize the material of component response to cyclic loading. Figure 2.1 shows a typical stress-life curve of metal materials. The ordinate of the S-N diagram just below the stress at failure limit is called the fatigue strength

iiiso=09su 0.41 10E3

Sy. The strength corresponding to the point of S-N curve represented by the horizontal asymptote is called the endurance limit S,,. Suppose N = No at the transition point of the S-N curve. For engineering purposes No is usually considered to be 106, beyond which the curve shows the endurance limit. An approximate power relationship was used to estimate the S-N curve for steel; the expression is as follows: 3

S = lOOnh

(2.2)

where exponents C, b are determined by two designated points (an approximate expression) of the S-N curve. For most cases, C and b are given as: C = log,0 ($100o)~ S,,

(2.3)

Sloo0

b = - ~ loglo S,,

where Smo0=stress at which 1000 cycles' life is achieved, and can be estimated as 0.9 times the ultimate strength, i.e., S~0~) = 0.9 S,. For most kinds of steel, the endurance limit can be estimated using ~0-5 S,,

if

S,, -< 200 kpsi

&=tl00kpsi

if

S,,>200kpsi

(2.4)

The two life related stress parameters are ultimate strength S, and endurance limit S,,. These can be combined in a function of ultimate strength and endurance limit as

Np =f(S, S,,, Se)

(2.5)

where Np is the predicted cycle life of the component, which, considering (2.4), may be rewritten as

Stress-based fatigue life

u=

Zhao

jSe I I I 10EA 10E5 10E6 Life of failure, N (cycles)

Fig. 2.1 Generalized S-N curve.

10E7

Np = f~(S, S,,)

(2.6)

3 RELIABILITY ANALYSIS Reliability and failure probability Reliability analysis depends heavily on the concepts of a failure model. A failure model is an abstract, simplified, mathematical construct related to a failure in a real situation, created for studying mechanisms of failure or making predictions. Two common failure modes exist in mechanical component fracture as mentioned in Section 2. The performance criteria are that the stress must be less than ultimate strength S,, and the fatigue life must be greater than the specified mission time. Therefore, these two criteria are: {S.--o->0},

or{~,>0]~:l=S.-o

"}

{Np > N,,} where Arm is the mission life of the component.

(3.1)

Predicting fatigue reliability S,, Se are obtained from n lot of specimens. These lots should be chosen from a wide selection of conditions, which will be expected to arise in the scattering of the endurance limit and the ultimate strength. From this scattering of failure data, the probability of survival can be calculated for a specified endurance limit and ultimate strength. The endurance limit and ultimate strength are then varied with the reliability and the following functions exist: {Se = h ( R ) S,, = g ( R )

(3.2)

Using eqns (3.2), (2.2) and (2.3), the life of a component with various reliabilities can be predicted. The relationship between the reliability and fatigue life of the component may then be derived. Let reliability function R ( N , o-) be the probability that a component does not fracture until the specified life N under the stress amplitude or. Then the probability, ,0r(N, or), that the component fractures before N, is given by: ~ ~.(N, o-) = 1 - R - (N, o-)

(3.3)

On the other hand, P[Se < o-] is the probability that the endurance limit of the component is lower than o-. Then, if a component with an endurance limit smaller than o- is used under the stress amplitude o-, it tends to fracture before No. Therefore, for the case that N = No, the reliability of the component can be expressed as: R(No, o-) = 1 - P[Se < o-]

(3.4)

For different N, the reliability expression can then be given as: R ( N , o-; N > No) = 1 - P[Se < o-]

(3.5)

From eqns (3.3), (3.4) and (3.5), it is shown that: P[Se < tr] = Py(N, o-; N >- No)

(3.6)

Obviously, a similar relationship can be easily obtained for applied stress and fatigue strength P[Ss > o-] : R ( N , o-) : 1 - Pz(N, o-)

9

where SI is the fatigue strength corresponding to the life N.

Supply-demand interference models The performance conditions (3.1) belong to the interference reliability models. Generally speaking, a failure model called the interference model is one in which the failure occurs when demand or exceeds supply S. The reliability is defined as the probability R = P ( S > o-) = P(S - o- > 0)

Figure 3.1 shows this interference model in both normal distribution functions. It is clear that the unreliability region is the interference part of those two curves. The shaded region in Fig. 3.1 indicates a finite probability of failure whose magnitude is a function of the degree of overlap of the two distributions. As the overlap increases, the shaded area and, consequently, failure probability, increases. By adjusting the failure governing distributions of event o- and event S the desired reliability can be obtained. Thus, the changes in reliability which accompanies the changes in the overlap region will induce the change of the value of strength. Therefore, to evaluate the probability of failure, the performance function, and the nature of the random variability of two factors, nominal demand and nominal supply, that affect the component reliability, have to be known. Figure 3.2 shows the enlarged portion of the interference region. For a given value o-0 of event o-, the probability of o-0 is equal to the area determined by the small interval of width do: 4'7 ( do<< P o-o-~--o--o-o

+do-~ 2 / =f~(o'o) d o - = A l (3.9)

On the other hand, the probability of event S being greater than o-o is P ( S > o-o) = f ~ fs(S) dS = A :

t

Righttail of thestress .distribution

V

I ftt il oftho

• .~ strengthdistribution

(a2_~

region

Fig. 3.1 Failure governing distributions with unreliability.

(3.10)

Since both random variables are statistically independent, the probability that the event o- in the small interval do- and the event S exceeding the event

(3.7)

InterferenceI ~ , S

(3.8)

-

fs( S ) ~

~------~ da ~0

o-, S -...-~

Fig. 3.2 Enlargement of interference region.

J. Tang, J. Zhao

10

o- given by the small interval is simply the product of these two equivalent probabilities A~ and A2. Reliability of the component is the probability of event S being greater than all possible values of applied event ~; thus

`2 = c r - S be the interference random variable. The '2 is also normally distributed under the assumption of normally distributed stress o- and strength S. The mean value and standard deviation of `2 are given by s7 = # - o~

R= f~ f~(~o) x[f[fs.(S)dS]dce

(3.11)

By integrating eqn (3.11) with various distributed events S and s, the reliability can be obtained.

4 RELATIONSHIP BETWEEN RELIABILITY AND ENDURANCE LIMITS The interference approach provides a strategy to evaluate the reliability or probability P[S > or] and P[S,,< or] of a component by adjusting the failure governing strength and stress distributions. Since it is difficult to integrate (3.11) for different distribution functions except for some special cases, the numerical methods play an important role in reliability analysis. In this paper, a reliability index is introduced to simplify the problem of establishing the relationship between reliability and strength or endurance limit. Two kinds of distribution functions, normal and Weibuli, are used as examples to illustrate the methodology.

Normally distributed supply and demand

f(o-) -

1

- - x o-,a,/T~

1

exp[_

f(S)=o_s~xexp

J_,

[ I(S- ~)2] -5,

l

f(`2) - c r ~ x

[ 1(`2-~) 2] ( - z z~)

exp - ~

( - z c , zc)

o-~ / J

f

PIS,, < ' d =

,r

1

cr;X/21r 1 `2__[2

(4.4)

For strength S, using eqn (3.7), it would be o-] = I s L

1

o-S2n

×exp{-~

1 ( ` 2 - ( ) 2 / d ~(r~ / J

(4.4a)

When the value of corresponding probability is given, the endurance limit S,. and strength S are decided from the above equations. The f(`2) is a normal distribution combined by two normally distributed random variables. The corresponding standardized variable z is defined as

(4.1)

o-(

= (~ - s) - (~ - s)

o's / J

(4.5)

x/£, + £,

where ~ = mean value of applied stress, S = mean value of strength, o-,T= standard deviation of applied stress, and O-s = standard deviation of strength. These two populations might be distributed as shown in Fig. 3.1, and these two random variables are independent. Then, the reliability is evaluated by integrating (3.11). For a statistical analysis of the result of combining these two populations, let

(4.3)

From eqn (3.6), the probability that the real endurance limit of a component S,, is smaller than an arbitrarily applied stress cr is the failure probability of these two populations under condition (3.1); it would be

Z--

o-,, J J

(4.2)

Introduce a function f('2) as the joint probability density function. It is defined as the difference density of f(cr) and f(S). Them f(`2) is normally distributed and is given by

P[S >

It is generally accepted that the population of supply and demand, say strength and stress, obey a normal distribution. If the population parameters of normal distribution, i.e., the mean value and the variance are known, the relationship between strength and reliability is readily evaluated. In fact, when the sample mean and sample standard deviation and the confidence intervals are given from n samples, it is proved that applied stress and strength are specified by the following distribution functions:

o~ = X/7-w~ + o'~

The limit state is g" = 0, and the transformation of the random variable z to the standardized normal is (~ z -

-

-

~) (4.6)

Using eqn (4.6), eqns (4.4) and (4.4a) can be rewritten

Predicting fatigue reliability

Based on this index, a reliability factor K r has been defined for both the endurance limit and the ultimate strength.

as

"P[S,. < o'1

dP(Z) =

=

X

(_

X/~r2rT cr~:,/ for S,, Kr =

(\Vo-,~ --2--- ~ I + Ors/

for S

EZ< I] 2 =

0

(4.8)

,49' That is to say, the Cr is defined as an index for identifying the reliability. It shows that the component is in a limit state when G = 1. By solving eqn (4.8), Cr will be given as

1

-

(4.11)

Rk 2~ for S,, 1 - z CTr 1 -

For most types of steel with an ultimate strength below 200kpsi, there is a fatigue ratio, S,,/S,, to identify S, using S,. In this case, the reliability factor can be unified. Also, there is a special case, C,T = Cs, where the reliability factor can also be unified since it is obvious that (Kr), is exactly equal to (Kr),.. Therefore, the ultimate strength S, and the endurance limit S,, can be corrected with reliability using the following two equations: S,* S ,*,

KrS,

(4.12)

K rS e

For the desired reliability, the S-N curve has to be modified using the corresponding ultimate strength S, and the endurance limit S,, from eqn (4.12). The fatigue life at the desired reliability can then be obtained. N o r m a l distributed d e m a n d and W e i b u i l distributed supply

Assume the distribution of the supply random variable is Weibull distributed, whereas the distribution of demand is still normally distributed. The numerical method is used to integrate (3.11). The population parameters of Weibull distribution are the location parameter Y0 (minimum value), scale parameter 0, and the shape parameter b. The Weibull distribution function of strength is shown as

b ( y - y o ~ (b ') f ( S ) - O - yo \ O - yo/ X{exp[-(Y-Y°l~]~

(4.13)

2 2 -

z

C,,

Rk = X/1 - (1 - z2C2)(1

- z2Cs

)2

(4.10)

Obviously, if the strength S is Cr times the stress ~, the reliability corresponding to the standard variable z is assured. Using eqn (4.10) on the S-N curve, the following expressions are given: 1 Cr =

(C~),

2 2 - z C,T forS,, 1 + Rk

\ -0- - ~ o / AJ

1 +Rk Cr

1

(

Define

I

1

~(G),,

where qb(.) is a standard cumulative normal distribution function which can be evaluated by using a table contained in some textbooks (see Ref. 5). It can be seen from eqn (4.7) that each mean value or standard deviation will affect the failure probability. Several possibilities exist as failure probability varies with each mean or standard deviation. For instance, one of these values may be changed with no change in others, or some of them may be held constant and the others varied--even all of them may change. Along with these changes, the probability of failure may vary from a relatively low value to near unity. However, the problem is to evaluate the relationship between the failure probability and random material distribution of a component under a given stress. It can be assumed that the mean stress and two deviations are held constant, and the mean strength can be assumed as the only variable that will vary with reliability. Equation (4.6) can then be solved. Let o-,~= C,~5 and ors = CsS, where C,~ is the standard deviation coefficient of applied stress and Cs is the corresponding coefficient of the material strength. From eqn (4.6), the following equation can be obtained: -

I

(4.7)

P[S > o'] = 1 - dP(Z) = dp ×

11

l+Rk 2 2

for

S,,

- z C,~

1 - Rk 1 2~ for - z C,~

(4.10a)

s,,

The mean Yo of Weibull distribution function is given by

y,, = yo + ( O - yo)F (o1T+ l )

(4.14)

where F(.) is the gamma function. In order to avoid the gamma function, the mode (maximum value) of the Weibull distribution function may sometimes be used for estimating reliability. Assume y, is the mode of Weibull distribution, by differentiating (4.13) with

J. Tang, J. Zhao

12

respect to variable S. The relationship between y,,, yo, 0, b i s

-b(Y"-Y")"+b-l=O \ 0 -Yo/

(4.15)

so the mode of Weibull distribution is expressed as

( y,,=yo+(0-yo)

1~ 1/1' 1-~/

narrow band. For instance, the load amplitude can be dealt with as a random variable and be considered normally distributed, or approximately normally distributed. The probability density function of the toad amplitudes L(T), then is 1

G(L, T)I,,,=,- ~,~(tp,/~

(4.16)

× e x p [ _ ~ ( L ( t ) - L ( t ) ] 2] From the probability density functions of these two populations, the probability that the real endurance limit S,. is smaller than an arbitrarily applied stress o-, is

P(S,, < o-) =

f\ f(o-)

J

W(S) dS do-

\

~L(-t)

/ 3

(5.1)

where

£(t) =-

(4.17)

~:c(t) =

1~ n,=,

ni i

L,(t),

_

(5.2)

] I/2

(L,(t) -- L(t))2]

Since

f~r W(S) dS= I - e x p [ - ( o - - S"l b] ~, L \ 0 - S,/ J

(4.18)

the probability may be expressed as:

aJdo-

(4.19)

Therefore, the reliability can be obtained as:

R=l-f~.f(o-)x{1-exp[ -(o--S'']'-]\OS,] J} do(4.20)

Using the index defined in eqn (4.9), and a numerical method, eqn (4.20) can be evaluated using a given R. Thus, reliability factors K r for different reliability are obtained. A computer code has been developed for evaluating the factors. Table 4.1 shows the results of a Weibull distributed strength, (b = 2, 0 = 60, and Yo = 30), and a normal distributed stress (C,~ = 0"1) problem. For other distribution functions, a similar algorithm may be used to evaluate the reliability factor.

are the mean value and standard deviation of the random amplitude loading of a time step t, and Li(t) are the ith observations of n random variable peak amplitude values at time step t. Therefore, the static loads defined in the reliability analysis of a component are replaced by duty cycle information in a dynamics situation. In particular, for the probabilistic design of a component, the statistics for duty cycle history will be used to replace the statistics only for single load to control the reliability of a component. It is obvious that the definition of a force amplitude in a variable amplitude history is not readily available. In order to perform a reliability assessment on a component subjected to an arbitrary cyclic loading, there must be a technique in which one can break down the variable amplitude history into a series of single loadings statistically, i.e., reducing complex histories yield a number of cycles. Once the cycles are defined from the variable amplitude history, the amplitude, range, and mean of these individual cycles can be used as conventional simple loading to perform reliability. Cycle counting

5 RANDOM

CYCLIC L O A D I N G

From previous discussions, it is clear that the reliability analysis of a component is only concerned with a static load with uncertainty. Mechanical systems, however, are typically exposed to loads of many modes, frequencies, and amplitudes. Amplitudes may be constant or random in which the peak levels may be either constant or random. Frequencies may also vary widely; that is, from broad band to

A simple rainflow counting method is proposed in this paper to define cycles from a variable amplitude history. It is referred to as peak-valley editing and cycle counting procedures. The concept of cycle counting is depicted as a variable amplitude history being equal to a set of constant amplitude cycles. A cycle should contain two reversals of the same size but opposite directions, and it should be closed. This method is based on the identification of local

Table 4.1. Reliability factors R% Kr

60-61 1.000

73-04 0.909

82.57 0.833

89.43 0-769

94-06 0.714

96.97 98.63 0 " 6 6 6 0.625

99.46 0.588

9 9 - 8 2 99.95 0 " 5 5 5 0'526

99.99 0.500

Predicting fatigue reliability

maximum and minimum values. It attempts to define cycles which would correspond to a pair of peaks and valleys by constructing the largest possible cycle using the highest peak and lowest valley, followed by the second largest cycle, etc., until all peak counts are used. The history that is to be cycle counted must be defined in consecutive peak-valley form, i.e., each point defines a change in slope. Peak-valley editing must be applied first. A peak is defined as a local maximum value in the neighbourhood of one position to either side of the current location in the history, whereas a valley is defined as a local minimum. There are no intermediate data points in between consecutive peak-valleys. The conventional triangular waveforms are usually used for representing the real cyclic loading history curve, as shown in Fig. 5.1. Note that the set of cycles are not simply cut out or removed from history. They cannot be pieced back together to form the original history; rather, they are defined based on the particular cycle counting algorithm that is regardless of the time scale; the sequence effects of the original history are lost. Actually, there will not necessarily be a time interval between consecutive data points, since the definition of the cycles has nothing to do with the time scale. A peak-valley variable amplitude history that contains N peak-valley points will produce (N-l)/2 defined cycles. If N is an even number, then N / 2 cycles will be defined. Even after removing unnecessary data between peaks and valleys, the history might contain many cycles. It is obvious that not every peak makes a significant effect on the reliability of a component, and thus some small peaks can be completely neglected. For instance, in Fig. 5.1 (a), there are several small peaks between points E, F, and H, I. Since only the local extremums are important in reliability analysis, these small reversals can be safely removed from the

D (a)

13

history by taking out all the reversals smaller than certain prescribed percentages of the maximum subsequent local extremums range along overall history, i.e., some part of the difference between maximum peak and minimum valley in the history. An editing algorithm that reduces the number of cycles in the duty cycle curve has been developed so that the computational time for component reliability will be reduced substantially. Let AL be the full loading range, as AL = L m a x - L m i n , where L .... and L m i n a r e the maximum and minimum values in the loading history. In reducing the number of cycles, delete those small peak cycles by assigning a range, say k% of the AL, i.e., the cycle will be eliminated when A I < k A L , where AI is the local consecutive loading range. By eliminating the small cycles, the loading history becomes a simple triangular waveform, (see Fig. 5.1 (a) dash line). Figure 5.1 is an example showing the editing algorithm and the cycle-counting procedure. According to the rule, the full range of loading is A L = F o - Fz . Between E and F, H and I, there are some small compression and relaxation. These are negligible because their peak-valley range magnitude is less than k A L , ( k = 0 . 1 5 ) . Therefore, the total number of cycles is four, as shown in Fig. 5.1 (b). The corresponding maximum and minimum loadings in each cycle are: Fo

Fmin = F(

for cycle 1

Fma× = Fn

From= FA

for cycle 2

Fma x --

Fma x =

FF

Fmax = F~

Frnin= F(; for cycle 3 Fmin= FA

for cycle 4

where Fmax and From are used to represent the maximum and minimum loading in each cycle, respectively. The same procedure can be applied to variable amplitude dynamic stress histories. Once a set of constant amplitude cycles have been defined from the variable amplitude history, the reliability for each cycle can be determined from the proposed method. Mean stress effect

c

G

Assume O-max and O'rnin are the maximum and minimum stresses of each cycle, then its amplitude and mean stress are as follows: or,, -

Fig. 5.1 Peak/valley editing and cycle counting.

O'ma x - - O'mi n

2

o'm

O'ma x 4- O"mi n

2

(5.3)

The mean stress, however, may not be zero. Test results indicate that a mean stress has an effect on life-times when added to an alternative stress. The effect of a mean stress in life prediction has then been considered. Typically, the use of the modified Goodman and Gerber relations are recommended

14

J. Tang, J. Z h a o

since actual test data tend to fall between these two curves. They are: ~r,, + ~r,,, = 1 O'.

Su

(5.4)

<,+(o,,,)~= 1 c.r,,

\ S,,

The modified stress 05, can be directly entered on the S - N curve to obtain the corresponding life N. Life prediction

The damage fraction, D, is defined as the fraction of life used up by an event or a series of events. Awareness of these individual cycles' lives, however, still does not immediately indicate the predicted life and reliability for an actual variable amplitude history. To obtain the life and reliability of a whole loading history block, a damage summation model will attempt to combine the individual life found for each defined cycle into the predicted reliability for the whole history. The Palmgren-Miner linear damage rule is the summation model used here. Let N~; = cycles-to-failure at ith cycle. The rule assigns an amount of damage to ith cycle based on the inverse of the life, D = 1/Nt;, and states that the total fatigue damage is obtained by linearly summing the damage for each cycle in the history. The failure will occur when the summation of individual damage values caused by each cycle reaches a value of one, that is, N

D = ~ Di = 1 i

(5.5)

1

where N is the total number of cycles. Then, defining the total accumulated damage for a loading block as Dblock, Palmgren-Miner's rule predicts failure when D b l o c k = 1. After the fatigue damage for a representative segment or block of load history has been determined, the fatigue life for each block is calculated by taking the reciprocal. Using eqn (5.5), Palmgren-Miner's rule can be expressed in a form that directly yields the variable amplitude fatigue life in 'block to failure' Bf as:

3.125-in, 21 active coils, squared and ground ends. The spring is to be assembled with a preload of 10 lbs and will operate to a maximum load of 50 lbs during operation. If the standard deviation of material ultimate stress and endurance limit are 0.08, and the standard deviation of applied load is 0.12, the problems are how large the life and corresponding reliability of the spring will be and what the relationship between the life and reliability is. A loading history is created that contains live load cycles in 0-5 seconds, and every five cycles forms a load block. This loading history feeds the computing model perfectly. The magnitude of load amplitude is assigned to be 10 to 50 lbs, and its peak and valley for each cycle has to be confined within this range. Select the peak and valley values load Fm~,×Fmmof each cycle. The value of these loads are listed in Table 6.1. By using appropriate eqns, 5 the shear stress amplitude v,, and mean stress r,, of each cycle are obtained. Those values are also listed in Table 6.1. To construct the S - N curve family corresponding to different reliabilities, the reliability factor must be calculated. By using eqn (4.11) associated with a standardized variable table of normal distribution functions, the reliability factor Kr can be easily achieved corresponding to a different reliability. The ultimate strength of No. 13 W & M music-wire material is S , , , = A / ( d ) m, where A = 196 and m = 0.146, so S , , = 2 7 8 k p s i . The torsional modulus of rupture, then, is S , , = 186.26kpsi. The shear endurance limit for an unpeened spring is S,. = 45 kpsi. According to eqn (4.12), the corrected ultimate strength and endurance limit for different reliabilities can be achieved. Finally, by using eqns (2.2) and (2.3) with Su~,o= 0"9 S*, . and S , . - S o *, the S - N curve corresponding to this specific spring under certain reliabilities may be constructed. The S - N curve family is depicted in Fig. 6.1. The life under certain reliabilities for each cycle can then be obtained by comparing the shear stress amplitude with the S - N curve. Let N1, N> N~, N4, and Ns be the predicted lives of each cycle, respectively. The life under certain reliability then is 1

1

q

B/ - - -

Dbl"ck

1

(5.6)

~ Di i=1

1

k = -7200 - 1- + - - +1- - + - 1- + - - 1

1

(hours)

N, N2 N3 N4 N, i=1

i

Experience shows that linear damage summation is somewhat of an over-simplification of reality.

6 EXAMPLE

Consider a compression spring s made of No. 13 W & M music-wire material, and a gauge of 0.091-in. The spring has an outside diameter 9/16-in, a free length

Table 6.1. Loading and stress of each cycle in the spring

Fro,,×

F,,m

F.

r,, (kpsi)

50 41 45-5 50 50

10 14-46 14.46 10 10

20 13.27 15.52 20 20

40.85 27.12 31-69 40.85 40.85

Predicting fatigue reliability 150 R = 80%

130 120 ~!10

\~ .~..~

-----

R = 85% R=90%

...............R = 99%

70 50 30

4

6 log life (cycles)

8

Fig. 6.1 The S-N curves associated with reliability.

36

32

15

mission life of components by giving the failure governing stress and strength distributions or their combinations. To evaluate reliability, the variance of fatigue limit strength was extended to finite life as given in the S - N curve. It enables one to obtain the S - N curves associated with the desired reliability. The editing algorithm and cycle counting procedure have been performed to deal with the variable amplitude loading history. The duty cycle history is comprised of the number of cycles. Also, PalmgrenMiner's linear damage rule is applied for accumulating the damage of the whole historical block. The methodology provides an easy and direct way of performing the reliability assessment of mechanical components during the duty time period or the mission life under desired reliability. The relationship between reliability and mission life under other combinations of the failure governing stress and strength distributions are problems awaiting resolution.

~o 28 0a

ACKNOWLEDGEMENT

~ 24

This research was supported by the N S F - A r m y N A S A Industry/University Cooperative Research Center for Simulation and Design Optimization of Mechanical Systems.

20

80

85

90

95

100

Reliability(%) Fig. 6.2 Life vs. reliability curve of the spring

Assume the desired reliability is 95%, and calculate the values of Ni (i = 1. . . . . 5), N1 = 5.87108 x 10 4, N2 = 3"050865 X 105, N3 = 1-629256 x 105, N4 = 5"87108 x 10 4, N 5 = 5-87108 x 104. The total life L is then 22-95 hours. Figure 6.2 shows the life vs reliability curve of this spring. It can be seen that the simulation model of reliability analysis discussed in this paper is feasible, and that it gives reasonable results.

REFERENCES

1. Nelson, J.J., Raze, J.D., et al., Reliability models for mechanical equipment, Proc. Ann. Reliab. & Maint. Symp., 24-26 January 1989, Atlanta, Georgia, pp. 146-153. 2. Tang, J., A general methodology for durability and reliability analysis of mechanical components. Technical Report R - 180, Center for Computer-Aided Design, College of Engineering, The University of Iowa, Iowa City, IA, 1994. 3. Bannantine, J.A., Comer, J.J. & Handroch, J.L., Fundamentals of Metal Fatigue Analysis, Prentice Hall, New Jersey, 1990. 4. Ananda, P., Reliability of mechanical parts. Machine Design, 59 (1987) 152-154. 5. Shigley, J.E. & Mischke, C.R., Mechanical Engineering Design, fifth edition, McGraw-Hill Book Co., New York, 1989.

7 CONCLUSION This paper presents the relationship between the reliability and mission life of a component using the supply-demand interference approach. The equation facilitates the direct calculation of the reliability and

6. Hayashi, K. & Anno, Y., Allowable stress in gear teeth based on the probability of failure. ASME publication 72-PTG-45, 1972. 7. Lewis, E.E., Introduction to Reliability Engineering, John Wiley & Sons Inc., NY, 1981. 8. Yang, L. & Ma, Z.K., A method of reliability analysis and enumeration of significant failure modes for a composite structural system. Computers & Struct., 33 (1989) 337-344.