THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING CUMULATIVE FATIGUE DAMAGE DUE TO SPECTRAL LOADING John P. Harris Charles Lipson June, 1964 IP-675

TABLE OF CONTENTS Page LIST OF TABLES o. o o o o o o 0.4 o 0. o o o ~ o o 0 o o a o.. ^o a a iii LIST OF FIGURESo., o o.... o, o o o o..O O o o o o iV CUMULATIVE FATIGUE DAMAGE DUE TO SPECTRAL LOADING...o......... 1 REFERENCES....... o. o o....... a. a a o o o 0........ 0....0 14 ii

LIST OF TABLES Table Page 1 Applied Load Distribution..........o.....o.. 0.. 9 2 Calculation of ci(Si/Sld.o...................... 11 iii

LIST OF FIGURES Figure Page 1 S - N Curve and Miners Line....oo... o. oo.... o o. 2 2 S - N Curve Compared to Corten and Dolan's Line,..... 2 3 Effect of Incipient Fatigue Failure on a Specimen with and without and Initial Stress Concentration... 6 4 S - N Curve Comparing MinersT, Freudenthals' and Corten and Dolan's Line....,,,,,..................... 8 5 Ni Locus Lines for Proposed Relation for Various Kf.o 8 6 S - N Curve and Ni Locus Line for Sample Problem,... 12 iv

CUMULATIVE FATIGUE DAMAGE DUE TO SPECTRAL LOADING The fatigue life of a metal part subjected to a spectrum of load intensities cannot be adequately evaluated from the results of a constant stress vs. cycles (S - N curve) experiment alone. Laboratory tests have been conducted by several investigators to determine an appropriate means of calculating the number of stress cycles to failure with a known applied stress spectrum. There have been many papers written which relate to this basic problem and some of them give a new insight to previous data,(6 - 11) In 1945 M. Ao Miner(l) proposed the relation: M ni Z = A (1) i=l -i where: M = the number of different stress levels ni = the actual number of stress cycles applied at stress'i' Ni = the limiting life at the stress level'i' Figure 1 illustrates the meaning of these symbols on S - N coordinates. The S - N curve can be constructed empirically as a straight line on a log-log cycle from a point at.9 Su and 103 cycles to a point at a stress equal to the endurance strength at 106 cycles. This relationship was orginally evaluated by Miner using Alclad 24 S - T aluminum sheet specimens with up to 4 applied stress levels, ranging between.32 and.72 Su (ultimate strength) for which -1

-2I,_ ______ ^^s^ 100 n N, 60 40 n n2 V)N (J) 30In3 Ns 20 10 10 10 106 LIFE - CYCLES Figure 1. S - N Curve and Miners Line. 100 80 60 - I S - N Curve -- S0 t__________ e Corten & Dolan's Line 40- n N I 30 20 n3 10 10 10 10 LIFE - CYCLES Figure 2. S - N Curve Compared to Corten and Dolan's Line.

-3he obtained values of A between.161 and 1.49, with an average value based on 22 items of 1.02 and a standard deviation of.24. He thus concludes that A equals 1.0 will yield a good estimate of fatigue life. Several points should be noted with regard to these tests. They were made with plates containing stress concentrations with a net effect of Kf (fatigue strength reduction factor due to geometrical notch in specimen) equal to 2.5 combined with mode of loading effects which reduce the endurance limit to.11 Su.(5) Miner's experiments used reversed axial loading combined with static axial loading. The applied loads were never less than the experimentally found endurance limit, and only aluminum alloys were tested. In 1956 A. M. Freudenthal(2) tested 3/16 inch diameter aluminum specimens of 2024 and 7075 aluminum alloys in bending. He obtained values of A in Equation (1) between.13 and.99. He concluded that the test results did not support the cumulative damage based on linearity of accumulation theory (Miner's rule with A = 1), but that his data would approximately fit a relation similar to Miner's, except with a value of A equal to.58o It will be noted that these tests were also based on the performance of aluminum alloys. Freudenthal tested at 6 stress levels applied for a known percentage of the time for each specimen. The stress levels ranged from 033 to o91 of the ultimate strength. He does not explain that S - N curve was used to obtain his values of Ni, nor does he give any listing of his results, but

-4it is clear that the specimens had no initial stress concentration, and that they were loaded in a combination of shear and bending. As soon as the specimen is cracked, a stress concentration factor will appear and reduce the magnitudes of the locus of Ni appreciably. It is this effect which is important in explaining the value of A =.58 in Freudenthal's experiments. In 1956 Corten and Dolan(3) developed a new theory as the result of tests that they conducted using hardened steel wire of.050 inch diameter on a total of 497 specimens. Each specimen was tested at two stress levels with a known percentage of the cycles at each level. The stresses ranged from.32 to.67 Su. Rather than evaluating their data by Miner's linear cumulative damage rule, Corten and Dolan related the total number of stress cycles to the portion of total cycles at each stress level as follows (see Figure 2): M NG = N1/ Z (Si/S) (2) i =1 where: NG = the total number of stress cycles to failure for spectrum loading N1 = the number of cycles to failure at S1 M = the number of different stress levels S. = the stress at level'it S1 = the maximum stress applied al = the fraction of the total cycles at stress Si, i = C1, = i ~~~~~~~~~ T~G

-5d =.87 m (d = 6.57 for steel wire) m = the inverse slope of the S - N curve ni = actual stress cycles at Si This relation is in many ways similar to Miner's rule except that it does not use the S - N curve as a reference for the locus of Ni, but rather a line of slope d which is.87 of the inverse slope of the S - N curve, and extends as a straight line to the lowest stress levelo Corten and Dolan's specimens fit this curve, shown in Figure 2, very wello Next we should examine the influence of using a specimen with a initial stress concentration such as a notch or corner, A stress concentration of this type will generally be the source of fatigue failure. Shown in Figure 3a is the member with a stress concentration of 2.5. When this member is damaged by fatigue the physical structure at the notch is rearranged, producing a microscopic crack, and the stress concentration increases toward Kf = 350, a relatively small change from 2~5~ A member which has no notch or corner to act as an initial stress concentration will be designed on the basis of Kf = 10, but when damage begins the fatigue strength reduction factor Kf increases to Kf = 350, and the part will then approach failure at a greatly increased rate (12) Considering the differences between the test conditions and material properties used in the experiments of Miner, Freudenthal, and Corten and Dolan, it is possible to postulate a logical homogeneous

-6(a) (b) (c) Bar with Notch, Bar with Notch and Bar with Crack Kf = 2.5 Crack, Kf = 3.0 Kf = 3.0 Figure 3. Effect of Incipient Fatigue Failure on a Specimen with and without and Initial Stress Concentration.

-7relationship which is consistent with the results obtained by each. The three theories each take the summation of i; the difference Ni between them is in the line they choose to represent the locus of values of Ni (see Figure 4). That is: lo Miner chose the S - N curve using a part with an initial stress concentration of 2~5 for the locus of his values of Ni o 2. Freudenthal chose a line of ~58 of the S - N curve using a specimen with no initial stress concentration. 35 Corten and Dolan chose a line of slope d on S - N co-ordinates which is equivalent to a line of 87% of the inverse slope of their S - N curve using a specimen with, no stress concentration. In Figure 4 the line is drawn for an arbitrary value of SI = o8 Su As can be seen from Figure 4, these lines are not as deviant as the theories would seem. In order to unify the above three relations the following equation is suggested: M Nx a N/z d/s' (5) d c-'/E i (Si/sl. ) i=1 where the symbols have the same meaning as in the Corten and Dolan Equation (2) except that~ dr = d(o79 + o08 Kf) (4)

-8100 80- r S - N curve and Miners line / for Kf = 1 Freudenthals line for Kf = 1 0 60 Corten & Dolan/s ~~~~I <^^^ line for Kf = 1 50 -.8 Su C) 40 - 30 10 3 104 10 106 LIFE -CYCLES Figure 4. S - N Curve Comparing Miners', Freudenthals' and Corten and Dolan's Line. 100 / S - N Curve for Kf = 1 Same as Corten & Dolan's 60 800 r ^^^ ^.. / line for Kf = 1 S1 =.9 Su C) 40Line for Kf = 2.5 \ss Kf = 2 C) u) w 30 Kf = 1.5 20 10,,,104 I0e I0e LIFE - CYCLES Figure 5. Ni Locus Lines for Proposed Relation for Various Kf.

-9where Kf = the fatigue strength reduction factor. (5) This equation results in Ni locus lines as shown in Figure 5. As an example of the above approach, consider the following problem. Select a steel with a tensile strength adequate to withstand the following spectrum of loading for 105 cycles with a reliability of 99.9o, The applied load is symmetrical bending with a random loading sequence as shown in Table 1 and an estimated fatigue strength reduction factor, Kf = 2.0, TABLE 1 APPLIED LOAD DISTRIBUTION nt Si Spectrum Test Load-KSI Cycles 70 300 6o 4oo 40 1000 20 1000 10 2000 4700 = Nt = total number of cycles in spectrum test

-10Solution: 1. The maximum value of Si = S1 = 70 KSI 2. The value of d in Equation (4) can be calculated by either of two methods. If the S - N curve has been found by previous testing d is equal to.87 of the inverse slope of this curve0 If the S - N curve is not known the values at the end points of the straight line are sufficient to determine the line and the value of d. The left end point of the S - N curve at 103 cycles is equal to o9 Su o The right end point at 106 cycles is calculated with the following empirical correlations for this example.* S106 = S x K1 x K2 x /Kf K1 =.5, endurance limit correction for bending K2 = o8, reliability correction for 99.9% reliability from the S - N curve value of 50% Kf = 2.0, fatigue strength reduction factor due to stress concentration s@106 = ~5 x.8 x 1/2.0 x Su =.2 Su Taking the ratios 106/103 = (.9 Su/.2 Su)m = 4.5m; solving yields m = 4.6, and since d = o87m, d = 4.0. 3. Using Equation (4); d' = 4.0 (,79 +,08 x 200) = 3.8 4. To find Z ai(Si/Sl)d' (refer to Table 1) recalling that Nt = 4700, S1 = 70 KSI, NG = 105 and d' = 3e8 *For further information on obtaining the S - N curve empirically see Reference 5, Chapter 11o

-11TABLE 2 CALCUIATION OF Z ci(Si/Sl)d' Si i nt, i(si/ldd Load-KSI N = (/Sl) 70 ~ o64.064 60.085.047 40.212.027 20.212.002 10.414.000 6o Referring to Figure 6, drawing a line of slope d' from point N1S1 to the right gives the locus of values of Ni o 7o Drawing a line of slope m = 4 6 from N1S1 to the left yields the value of o9 Su at N = 103 cycles which yields S@103 =.9 Su = 124 KSI. 8. Thus the required ultimate strength, Su = 124/79 = 138 KSI to meet the given requirements.

-12200 0.9 Su124 KSI ~S - N Curve for Bending with Kf = 2 and 99.9% 100 Reliability (Drawn through N1,S1 with Inverse Slope 80 - (N1,S1) m = 4.6) 60 40 - 30 Aln 0 | Locus of Ni Cn i W (Drawn through r_- N, S, with Inverse C) 20 Slope of d' R 3.68) I I i 100 103 104 10 106 LIFE- CYCLES Figure 6. S - N Curve and Ni Locus Line for Sample Problem.

-13This paper has presented a method of modifying Corten and Dolan's original equation with the additional information of the fatigue strength reduction factor so that the resulting equation is capable of predicting the results obtained by Miner, Freudenthal, and Corten and Dolan,

REFERENCES 1. Mo A, Miner "Cumulative Damage in Fativgue, Jo Apple Mechanics, vol. 12, September, 1945. 2. A, M. Freudenthal, "Cumulative Damage under Random Loading," The International Conference in Fatigue of Metals, IME and ASME, 1956. 53 Ho T, Corten and To Jo Dolan "Cumulative Fatigue Damage," The International Conference on Fatigue of Metals, IME and ASME, 1956. 4. C. Lipson, Jo Kerawalla, LO Mitchel, A. Krafve, Engineering for Reliability, The University of Michigan, College of Engineering Summer Conferences, 1962. 5. C. Lipson and Ro Juvinall, Handbook of Stress and Strength, The Macmillan Co., 1963. 6. J. Waisman and Go Sines. Metal Fatigue, McGraw-Hill, 1959. 7 R R, Ro Gatts, "Application of a Cumulative Damage Concept to Fatigue " ASME paper no, 60-WA144, 1960. 8* A. K. Head, "The Propagation of Fatigue Cracks," J. Appl. Mechanics, Vol. 23, ASME trans V78, 1956. 9~ J, B Kommers, "The Effect of Overstressing the Understressing in Fatigue," Proco ASTM Volo 43, 1943. 10o Ho Jo Grover, So Mo Bishop and Lo Ro Jackson,"Fatigue Strength of Aircraft Materials," NACA TN 2324 1951, 11a Jo Ao Graham, "Use of Cumulative Damage in Designing to Resist Fatigue," SAE paper 572F, September, 1962. 12. Do McLean, Mechanical Properties of Metals, Wiley, N.Yo -14