E~/.quthtg Fractur~ Mec/um~s Vol. 11, pp. $39-499 oo13-7944r79/0601..e839/$ozoo/o Persam~ Press Ltd., 1979 Printed m Great Britain

FATIGUE RELIABILITY UNDER MULTIPLE- AMPLITUDE LOADS

RAMESH TALREIM

Department of Solid Mechanics, The Technical University of Denmark, Lyngby, Denmark

Almtract--A method to determine the fatigue reliability of structures subjected to multiple-amplitude loads is presented. Unlike the more common cumulative damage methods, which are usually based on fatigue life data, the proposed method is based on tensile strength data. Assuming the Weibull distribution for the initial tensile strength and the fatigue life, the probability distributions for the residual tensile strength in both the crack initiation and the crack propagation stages of fatigue are determined. The method is illustrated for two-amplitude loads by means of experimental results obtained by testing specimens of a structural steel and is shown to be more accurate than the Palmgren-Miner cumulative damage method.

1. INTRODUCTION PROBLEMS OF probability-based structural design have recently attracted considerable interest. For structures subjected to fatigue loads, the probabilistic considerations are essential----even under deterministic loads--due to the statistical nature of fatigue.

The assessment of fatigue reliability of a structure depends on its design requirements. If the structure is required to perform safely for a specified life, without inspection, then it must maintain an acceptably low probability of failure throughout its life. Most civil engineering structures fall into this category. However, requirements of weight and economy necessitate in some instances, e.g. in aircraft and spacecraft structures, to allow some accumulation of fatigue damage in the components before they can be inspected and repaired or replaced. In this case, the probability of the strength of structure remaining above a specified limit is required to be acceptably high until the time of inspection.

In the reliability analysis dealing with the first case, only the terminal event in the fatigue process, i.e. fatigue life, is considered [1-4]. The probability of failure Pr at a given number of cycles N is given by

PdN) = P(Nr <- N) (1)

where N~ is the random number of cycles to failure. In the second case, the strength of the structure is taken as a function of the number of

cycles and failure is assumed when the strength equals the maximum stress applied. Thus, if R is the resisting (residual) strength after N cycles, then the probability of failure is given by

PI(N) = P [ R ( N ) <- S] (2)

where S is the applied maximum stress. The second approach is more general in the sense that it considers not just the terminal

event, failure, but also the development of fatigue through the deterioration in strength. In most previous work using the second approach[5-8], it is assumed that the strength

remains unchanged until a fatigue macro-crack initiates. In a recent work [9], however, it has been demonstrated by proper statistical analysis of test data that the strength also decreases in the crack initiation stage. The reasons for deterioration of strength in the crack initiation stage are not clear, but it is thought that this may be caused by microcracks generated at strain incompatibility centers (voids, grain boundaries, etc.) in the bulk of material. Negligence of this strength degradation could, however, lead to erroneous reliability calculations if the crack initiation period occupied a significant portion of the fatigue life. In some structural components, where sharp notches or severe surface defects exist, fatigue cracks might be

tPresent address: Metallurgy Department, Research Establishment Rise, Roskilde, Denmark.

839

840 RAMESH TALREJA

expected to initiate early in the fatigue life. However, in most structural components, fatigue cracks are not found to appear until late in the fatigue life. It was therefore thought appropriate to formulate a reliability procedure where the change in strength in both the crack initiation and the crack propagation stages of fatigue are considered. This was done in a previous paper[9]. In the present work, the procedure formulated therein will be exploited further and extended to the case of multiple-amplitude loads. As an illustration, the procedure will be applied to test results obtained under two-amplitude loading on specimens of a high-strength steel.

2. FATIGUE RELIABILITY To evaluate the probability of fatigue failure given by eqn (2), the probability distribution of

the residual strength will be considered in both the crack initiation and the crack propagation stages of fatigue.

2.1 Crack initiation stage Extensive microscopic studies of fatigue have revealed several aspects of this very complex

phenomenon. Although a good deal of qualitative understanding of the crack initiation mechanism has been gained, a quantitative description of the fundamental mechanism is still under development. At present, therefore, the deterioration of strength in the crack initiation stage cannot be expressed in terms of the physical material parameters. Consequently, we resort to a fatigue "damage" parameter D, which is defined such that its value is zero initially and increases monotonically to unity when a macro-crack initiates.

Equating the normalized strength difference to the damage parameter, we have

R o - R - D (3) Ro-R~

where Ro is the initial tensile strength and R~ is the residual strength when a macro-crack has just appeared.

Re, thus, marks the end of the crack initiation period, and is related to the macro-crack length Co by the Griffith-Irwin relation

R¢ = KcotCo -112 (4)

where K~ is the material parameter known as fracture toughness and a is a parameter dependent on geometry. Co is thus the minimum crack length beyond which the decrease in strength is primarily due to crack propagation.

It is further assumed that the rate of fatigue damage is given by

d---N = n ~ (5)

where n and m are material parameters. Integrating eqn (5) and using eqn (3), the following relation between the initial strength and

the residual strength is obtained

R = R~ + (Ro- Rc)(1 - N[N~) m' (6)

where

1 1 m'=l+---m and N c = n ( l + m ) S m"

From eqn (6) it is seen that R will be a random variable as a consequence of randomness in Ro, Ro Nc and m'. Thus, the distribution of R can be evaluated from the knowledge of the distributions of these parameters. It is assumed that the 3-parameter Weibull distribution can be taken for the initial strength Ro and the number of cycles to crack initiation No. For a given type of specimen (representing a fatigue-sensitive site in a structural component) the strength

Fatigue reliability under multiple-amplitude loads 841

parameter Rc can be expected to have little scatter. Test results, to be described later, support this supposition. The scatter in the exponent m (eqn 5), and hence in m', will depend on the material. There is, at present, no physical basis for a priori specification of the distribution of m.

If, however,_only the strength parameters R and R0 are assumed to be random, then it can be seen from eqn (6) that, given a Weibull distribution for Ro, the distribution for R will also be Weibull. Thus, if

P(Ro <- ro) = 1 - e-[(r°-a)/b]c (7)

then

where

and

P ( R <- r) = 1 - e -l¢'-°~/~c (8)

a' = Rc + (a - g~)(l - NINe) m' (9)

b'= b(I - NINe)". (10)

2.2 Crack propagation stage Once a macro-crack has initiated, further fatigue loading will make this crack grow to

failure. At any given time during the crack propagation stage, the residual strength will be related to the instantaneous crack length C by the Griflith-Irwin relation:

R = KcaC -u2, C>-Co. (11)

For most of the crack propagation period, the rate of crack growth is given by the Paris-Erdogan relation:

dC d---N = A(AK)n (12)

where AK is the range of stress intensity factor and A and n are constants. By using eqn (11), eqn (12) can be rewritten as

dC = BC,a2 dN (13)

where/~ is a function of stress. Differentiating eqn (11) w.r.t. N and using eqn (13), we have

where

dR / 1 ,~n-3 dN = - ~ , ~ ) (14)

= ~(Kca) n-2. (15)

The crack growth rate exponent n is found to be equal to 2 for many steels and aluminum alloys, except at low fracture toughness, where higher values are obtained [10]. In the following, the residual strength distribution is considered when n -- 2 and n > 2.

n = 2 . Substituting n = 2 in eqn (14), it is seen that the rate of residual strength decrease becomes

proportional to the instantaneous strength. Thus,

d R = - ~ n = - ~ R . 06) dN

842 RAMESH TALREJA

Integrating exln (16), a relation between the residual strength at the start of crack pro- pagation and its instantaneous value is obtained:

R = R c e -(a/2XN-Nc). (17)

and

where

The insensitivity of the constant/3 to material parameters has recently been demonstrated for steels [l l]. Therefore, assuming no scatter in/3, the probability distribution of R can be obtained if either that of N~ [eqn (17)] or that of Nr [eqn (18)] is specified. Experimentally, it is easier to determine N i than it is to determine N~. Therefore, assuming the Weibull distribution for N t, given by

and rewriting eqn (18) as

P ( N r <- N) = 1 - e-[(N-a")lb"F

N t= ~ I n R - ~ I n S + N

the following distribution of R is obtained

P ( I n R < - I n r ) = P [ N , <2_-~ In r - ~ ln S+ N ]

= 1 -- e -[(r*-a*)lb*lcn

r*= ln r (22)

a * = l n S - ~ ( N - a . ) (23)

b*=~b . . (24)

This is the so-called log-Weibull distribution. Taking R = Rc and N = Nc in eqn (20), the distribution of N~ can be similarly obtained.

Thus,

where

P(Nc <- N) = 1 - e -ffu-a;~/b.lc,

2 , Rc a ~ = a. - ~ ln -~ (26)

(25)

n > 2 . Integrating eqn (14) between N and N I the following relation is obtained

R "-2 = S "-2 + ~ ( n - 2 ) ( N I - N ) . (27)

Assuming once again the Weibuil distribution for N s, we get

P(R n-2 <- r n-2) = 1 - e-ttr'-aTO/bT'lc" (28)

(19)

(20)

(21)

Similarly, R and S are related by

R = Se(~2xNt -m. (18)

Fatigue reliability under multiple.amplitude loads 843

where

r " = r "-2 (29)

a~ = S "-2 + ?(n - 2)(a, - N) (30)

and

bE: 3,(n - 2)b,. (31)

Thus, the parameter R "-2 is Weibull-distributed with its location and scale parameters given by eqns (30) and (31) and its shape parameter equal to that of the fatigue life distribution.

For the special case of n = 3, the rate of residual strength decrease increases linearly with the number of cycles. In this case the residual strength is Weibull-distributed with its parameters given by

1 at : S + ~ai~Kc(a, - N ) (32)

and

1 b: = ~ ~/3K~bn (33)

c~ = cn. (34)

Figure 1 shows the change in residual strength with the number of cycles in the crack propagation stage for various values of n.

R

Rc

S o i t I

i i

N c Nf 'N

Fig. !. Residual strength vs number of cycles in the crack propagation stage for different crack growth rate exponents.

3. MULTl l~AMPLIT UDE LOADS

For given mean value and amplitude of an applied stress, the probability distribution of the residual strength in both the crack initiation and the crack propagation stages can be determined from the distributions of the initial strength and the fatigue life. For a given probability, then, a plot of the residual strength against the number of cycles can be drawn. Such a plot will be called a P - R - N diagram here. Figure 2 shows P - R - N diagrams for two maximum stresses S~ and $2. This figure also illustrates the reliability procedure for two stress amplitudes. Thus, the strength decreases from its initial value Ro to r on application of either Nt cycles at stress St or N2 cycles at stress $2. Now, if we assume no stress interaction effects, then the number of cycles to failure at S2, following Nt cycles at St, will be Nt2-N2. Similarly, the number of cycles to failure at St, following N2 cycles at $2, will be Nr, - Ni.

844 RAMESH TALREJA

R c . . . . . . . . .

f

s 2

Ncl N I Nf I Nc 2 N 2 N;2

Fig. 2 P-R-N diagrams at two stress amphtudes illustrating the reliabihty procedure

For a load spectrum containing multiple amplitudes, the reliability procedure consists of plotting the P - R - N diagrams at a selected probability for all the amplitudes involved. The number of cycles to failure at the selected probability is then found by following the residual strength in these diagrams until the maximum stress being applied is reached.

The stress interaction effects, such as the delay in the decrease of residual strength caused by high stress amplitudes, can in principle be incorporated in the reliability procedure. However, this requires quantification of the delay effects, which is at present at the develop- mental stage. A P - R - N diagram brings out clearly the two-stage characteristics of fatigue. In contrast, the conventional P - S - N diagram (plotting stress vs number of cycles to failure at selected probabilities) reflects only the final event--failure. Therefore, a reliability procedure based on a P - S - N diagram and using a cumulative damage method, such as the Palmgren-Miner method would be expected to be inaccurate unless the P - R - N diagram showed one of the two fatigue stages to be insignificant and a fairly linear decrease in the residual strength.

In the following, a test program to generate the strength and the fatigue life data required to plot P - R - N and P - S - N diagrams at two stress amplitudes is described. The predictions of the reliability procedure based on the P - R - N diagrams and of that based on the P - S - N diagram and using the Palmgren-Miner failure criterion are then compared.

4. TEST PROGRAM

4.1 Material and specimen The specimens were prepared from 22 mm dia. bars of a Cr-Mo--V steel by machining

longitudinally along four surfaces to reduce the cross section smoothly to 10 x 15 mm dimen- sions. These were then heat-treated at 1050~ to an average Rockwell hardness value of 51. The surfaces of the heat-treated specimens were ground to give an average peak-to-peak roughness of 3 ttm. Flaws 0.05 mm deep and 0.15 mm on the surface were introduced in one 15 mm face of the specimens by electro-discharge machining. The processes of roaching the specimens and introducing flaws were carried out in random order.

The chemical composition of the steel alloy used is shown in Table 1.

Table I. Chemical composition

Element C Sl Mn P S Cr Mo V Content in% 0.35 102 0.47 0.02 0.009 5.2 1.42 096

Fatigue reliabihty under multiple-amplitude loads 845

4.2 Test series The test program consisted of 10 test series. In each series a preselected number of

specimens were drawn by use of random numbers from the lot containing all the specimens. The specimens in Series Nos. 1 and 2 were tested to failure under cyclic stress S~ = (amplitude 666.7 MPa, mean 200 MPa) and $2 = (amplitude 533.3 MPa, mean 133.3 MPa), respectively. The specimens in Series No. 3 were pulled in tension to determine their initial strength. The specimens in Series Nos. 4 and 5 were subjected to prespecified numbers of cycles at stress Sin, after which the unfailed specimens were pulled in tension to determine their residual strength. The same was done with specimens in Series Nos. 6--8 at stress $2. In Series No. 9 the specimens were first subjected to 29.04 kilocycles at stress Sin, whereupon the unfailed specimens were fatigued to failure at stress 82. In Series No. 10 the stress sequence was reversed, applying first 59.13 kilocycles at stress $2 and then subjecting the unfailed specimens to stress S~ until failure.

The fatigue testing was carried out on a servo-hydraulic (20 tons MTS) testing machine. The tensile tests were carried out on a 40 ton universal testing machine.

4.3 Test results Fatigue life data of specimens subjected to stress S~ (Series No. 1) and stress $2 (Series No.

2) are listed in Table 2. Table 3 shows the initial tensile strength of specimens in Series No. 3 and the residual strength of the unfailed specimens after 29.04 kilocycles (Series No. 4) and after 32.15 kilocycles (Series No. 5) at stress Sin. The fatigue lives of the failed specimens are also shown in this table. Similarly, Table 4 shows the residual strength and the fatigue lives corresponding to stress $2 at different numbers of cycles.

The fatigue life data for two-amplitude loading in Series Nos. 9 and 10 are shown in Table 5. As will be noticed in Tables 2-5, all the test data are listed in increasing order of magnitude.

Table 3. Initial tensile strength Ro, and residual strength R, at the end of different number of cycles at stress S~. Strength units in MPa. N~ =

number of cycles to failure

Series No. 3 4 5 Table 2 Number of cycles to failure N~ Stress level -- St $1

at stresses $1 and $2 No. of cycles - - 29,040 32,150 i Ro, R, N~ R,

Series No. 1 2 I 1781.5 -- 27,060 --

Stress level S] $2 2 1784.8 -- 27,200 -- 3 1817.5 -- 27,780 --

i N~ N~ 4 1833.8 - - 27,940 -- 5 1850.2 -- 28,930 --

1 29,040 59,130 6 1852.1 925.1 2 29,460 70,230 7 1863.3 934.9 -- 3 31,710 78,330 8 1869.8 944.7 -- 4 31,740 80,430 9 1876.3 1059.1 -- 5 32,150 89,950 10 1902.5 1340.2 -- 6 35,620 104,940 1 ! 1905.8 1484.1 889.1 7 36,350 132,760 12 1915.6 1634.4 1013.4 8 37,750 134,070 13 1935.2 1696.6 1016.6 9 46,270 140300 14 1938.4 1752.1 1046.0

10 66,870 141,690 15 1948.3 1794.6 1049.3 16 1958.1 1850.2 1235.6 17 1961.3 1876.3 1268.3 18 1967.9 1892.7 1333.7 19 1974.4 1938.4 1627.9 20 1980.9 1987.5 1837.1

25,160 27,610 28,650 29,020 29,190 30,720 31,140 31,160 31,180 31,460

5. RELIABILITY ANALYSIS

5.1 Distribution parameters The strength and fatigue life data listed above are analysed to estimate the probability

distribution parameters. For a Weibull distribution a more efficient estimation method than the usual method of plotting on a Weibull paper has been described earlier [9]. This method plots

846 RAMESH TALREJA

Table 4. Residual strength R, in MPa at the end of different number of cycles at stress $2 N~ -- number of cycles to failure

Series No. 6 7 8 Stress ~vel $2 $2 $2

No of cycles 59,130 80,000 89,950 i R, N~ R, N~ R,

I -- 48,200 -- 45,700 -- 54,330

2 -- 53,000 -- 54,970 -- 57,610

3 993.7 -- 57,160 -- 68,000

4 1033.0 -- 60,000 -- 68,130

5 1157 2 -- 65,130 -- 71,II0

6 1274.9 -- 67,580 -- 71,360

7 1542.9 - - 67,980 - - 73,000 8 1784 8 - - 69,950 - - 74,300 9 1827.3 - - 72,440 - - 76,430

10 1827 3 - - 75,040 - - 76,650 I I 1833.8 - - 76,280 - - 76,980 12 1843 7 693 0 - - 77,150 13 1850.2 706.1 - - 79,100 14 1863.3 823.8 - - 80,000 15 1866.5 836.8 -- 82,880

16 1902.5 1346.8 -- 85,720

17 1909.0 1601.8 885.9

18 1909.0 1784.8 1811.0

19 1954.8 1850.2

20 2000.6 1902.5

Table 5 Fatigue hfe data for two-amplitude loads

t Nt2, N2t,

1 ? t 2 1" 10 3 t 10 4 t 1670 5 3940 1680 6 5770 3320 7 7380 3860 8 8200 6010 9 8310 7030

10 12,440 7250 I 1 14,040 7570 12 14,260 7650 13 15,890 8070 14 16,450 8450 15 16,660 9730 16 19 ,710 10,240 17 19 ,850 10,580 18 27,980 11,350 19 38,100 11,830 20 62,430 18,600

of Failed at the first stress level

Nm = number of cycles to failure at stress $2 follow- ing 29.04 kilocy- cles at stress S~.

N2t, = number of cycles to failure at stress St follow- ,ng 59 13 kilocy- cles at stress $2

the order statistics x, of a Weibull-distributed variable X vs the expected values Ez, of the standardized variable Z given by

X - a z=--- K - (35)

for different values of the shape parameter c. The value of c that minimizes the sum of the squared deviations of the data points from the fitted line is taken as the estimate of c. The x-intercept and the slope of the best-fit line then give the estimates of a and b, respectively.

The results of the statistical analysis are shown in Tables 6-8. Table 6 shows the distribution parameters for fatigue lives at stresses S~ and $2. At stress $2

the fatigue life distribution has two components. Table 7 shows the distribution parameters for the initial tensile strength and the crack

initiation components of the residual strength after 29.04 kilocycles at stress S~ and after 59.13 kilocycles at stress $2. As seen in this table, the initial strength distribution has two

Table 6. Estimated Weibull parameters for fatigue life distribu-

tions at two stress levels

S a, b, cn SI 23.240 10,087 3 5

33.404 51.537 3.0 P --- 0.3 $2 47.255 32 106 3 0 P ->0.3

S~: amplitude 666.7MPa, mean value 200 MPa. $2: amplitude 533.3MPa, mean value 133 3 MPa.

Table 7. Estimated Weibuil parameters for initial strength and the crack initiation components of the

residual strength. Strength units in MPa

N S a b c r

1719.8 209.6 2.5 ~ 1958.5 0

1904.4 47.5 2.5 --- 1958.5 29.04 St 1501.8 299.3 2.5 >- 1850.2

59.13 $2 1647 8 210.1 2.5 -> 1811.5

Fatigue reliability under multiple-amplitude loads

Table 8. Estimated Iog-Weibull parameters for the crack propagation components of the residual strength assuming c. given by the estimated

values in Table I

N S a* b*

29.04 Si 5.947 i.421 32.15 $1 5,501 1.421 59.13 $2 5.625 2.311 80.00 $2 4.218 2.271

847

components. An inspection of the failed specimens revealed no apparent cause of this, e.g. two failure modes. Some scatter in the hardness of the specimen material was found, but no direct correlation between this and the two-component initial strength distribution could be established. It must, however, be pointed out that the two components in the initial strength distribution are not very different, as can be seen from the distribution parameters, and it appears likely that a larger sample might make the two components indistinguishable.

The residual strength distributions for each of the two cases listed in Table 7 show two very distinct components. An inspection of the failure surfaces showed that the specimens belonging to the higher component either had no fatigue cracks or had fatigue cracks less than 0.5 mm in length. All specimens in the lower component had fatigue cracks greater than 0.5 mm in length. Thus, the fall in strength in the lower components appears to be due to crack propagation, while that in the upper component may be due to initiation of microcracks. The strength at which the separation in the two components occurs appears to be fairly constant.

In estimating the distribution parameters for the crack initiation components of the residual strength, the shape parameter is assumed, in accordance with eqn (8), to be given by that of the initial strength distribution.

Table 8 shows the distribution parameters for the crack propagation components of the residual strength. In cases of the residual strength after 32.15 kilocycles at stress St and after 80 kilocycles at stress $2, an inspection of the specimens showed that all failures had been from fatigue cracks greater than 0.5 mm in length. Thus, in these two cases, all specimens had been fatigued into the crack propagation stage. The distribution parameters have been estimated assuming a Iog-Weibuli distribution in accordance with eqn (21). This implies assuming that the crack growth rate exponent is equal to 2. For steel specimens with spark-machined surface flaws, it has been confirmed that n = 2 holds for most of the fatigue life[12].

As seen in Table 8, only two distribution parameters have been estimated assuming, in accordance with eqn (21), that the shape parameters are given by those of the corresponding fatigue life distributions. The two distribution parameters thus estimated confirm the validity of their relations to the corresponding fatigue life distribution parameters through eqns (23) and (24), as these equations are very nearly satisfied by the estimated parameters when the unknown constant/3/2 is eliminated.

5.2 Fatigue life prediction From the distribution parameters listed above, the P - R - N diagrams were constructed for

the two stresses. Figure 3 shows the P - R - N diagram for stress $2 at three probabilities and Fig. 4 shows the P - R - N diagrams at P = 0.5 for both the stresses.

The number of cycles to failure at stress $2 following Nm cycles at stress St, denoted Nt2, is determined as illustrated by Fig. 4 for P = 0.5. Thus, N,, the number of cycles at stress $2 equivalent to Nt cycles at stress S~, is determined at different probabilities. Nt2 at a given probability is then given by Nl2 = N/2- N2, as seen in Fig. 4.

The experimental values of Nt,, shown in Table 5 for Nt = 29.04 kilocycles, were statistic- ally analysed assuming Weibull distributions, and their percentiles at different values of P are shown in Table 9. The corresponding values of Nt, predicted by the method described above are also listed in this table as N1'2. The percentiles of Ns2 predicted by the Palmgren-Miner method using the percentiles of Nlz and Nrz are listed as Mr2 in this table. The equation used to

BPM Vd. II, No 4--0

848 RAMESH TALREJA

R

2000- M~ 1500-

1000-

=

I S2[

5001 I, I

I I

,, O/ ' o 2; 6'o 8'0 lO'O.1 ;

Fig. 3. P-R-N diagrams at stress $2 for three different probabilities

R j

2000-

M PO

1500-

S 2 500 i

O/ 0 I

I

-~ 0.5

N10'2 0'4 1"6N12_ ~.8 N2

1'o .los "N

Fig. 4. P - R - N diagrams at stresses S], and $2 for P = 0.5.

Table 9. Comparison of the experm~ntal fatigue lives with those predmted by the proposed method and the Palmgren~

Miner method

P Nt2 N*2 M12 N*2 MI.....22 Nt2 NI2

01 0856 0 0 0 0 0.2 3.825 5.192 1.672 1.357 0.437 0.3 6.442 7.052 3.900 1.095 0.605 04 8.977 8.655 5.835 0.964 0.650 0.5 I1581 10.151 7.688 0.877 0664 06 14.394 11.639 9.575 0.809 0.665 0.7 17.617 13 .214 11.622 0750 0.660 0.8 21.641 15 .432 14.041 0.713 0.649 0.9 27 620 18.273 17.415 0.662 0631

P = probability of fatigue failure. N12 = number of kilocycles to failure at $2 following 29.04

kilocycles at $1 N*2 = N~2 as predicted by the proposed method MI~ = Nt2 as predicted by the Palmgren~Miner method.

determine M~, is as follows:

Nl q. MI2 Nfl Ni2 1 (36)

where Nrl and Nrz are the percentiles of the fatigue lives at stresses $1 and $2, respectively, for a given value of P.

The last two columns in Table 9 show the ratios of the predicted to the experimental values of N~2 for the two methods. As seen here, the proposed method is more accurate than the Palmgren-Miner method.

The corresponding data for N2~, the number of cycles to failure at stress $1 following N2 = 59.13 kilocycles at stress $2, are shown in Table 10. Once again, the predictions baled on the P-R-N diagrams prove to be better than those of the Palmgren-Miner method.

It should be pointed out here that the proposed method does not account for the stress interaction effects. The specimen material for the present experimental investigation was a high strength steel with low ductility. Hence, the plastic zone at the fatigue crack tip would be small and would therefore cause small delay effects in the crack propagation stage. The error in the

Fatigue reliability under multiple-amplitude loads

Table 10. Comparison of the experimental fatigue lives with those predicted by the proposed method and Miner's method

P N21 N~I M2, N ~.__~1 M2......!l N21 N21

0.1 0 0 0 - - - - 0.2 1.683 !.863 2.551 1.107 1.516 0.3 3.843 3.546 4.784 0.923 1 245 0.4 5.742 5 026 5.969 0.875 1.040 0.5 7.365 6.135 7.065 0.833 0.959 0.6 8 227 7 920 8.143 0.963 0.990 0.7 9.152 9.123 9.271 0.997 I 013 0.8 10 232 9.919 10.556 0.969 1.032 0.9 11.718 12 016 12.270 i.025 1.047

P = probability of fatigue failure. N2, = number of kilocycles to failure at St following 59.13

kilocycles at S2. N~, = Nzt as predicted by the proposed method. M2t = N2, as predicted by the Palmgren-Miner method.

849

proposed method due to neglect ing the s t ress interact ion effects may therefore be small . The method could, however , be ex tended to incorpora te the s t ress interact ion effects when these are

quantified.

6. CONCLUSION

The es t imat ion of the fat igue life under mul t ip le-ampl i tude loads at a given fai lure p rob- abi l i ty can be based on the plots of the residual s t rength vs the number of cyc les appl ied. This method has the advan tage of account ing for both the c rack init iat ion and the c rack propagat ion stages of fat igue, as the probabi l i ty d is t r ibut ion of the residual s trength reflects the two fat igue stages as two dis t inct components .

Fo r two-ampl i tude loads the p roposed method gives be t te r es t imates of the percent i les of the fat igue life than the P a l m g r e n - M i n e r cumula t ive damage method.

Acknowledgements--The author wishes to express his gratitude to Prof. Walnddi Weibull for providing guidance in carrying out the statistical analysis for this work. The financial support by the Danish Council for Scientific and Industrial Research is also gratefully acknowledged.

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99(EM6), 1165--I 179 (1973). [4] P. H. Wirsching, B. Stahl and K. G. Nolte, Probabilistic fatigue design for ocean structures. J. Structural Division,

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(Received 6 June 1978; received for publication 21 September 1978)