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APPLICATION OF A 2-DIMENSIONAL RAINFLOW METHOD


Fatigue Fract. Engng Mater. Struct. Vol. 17, No. 12, pp. 1405-1416, 1994 Printed in Great Britain. All rights reserved

8756-758U94 %6.00+0.00 Copyright 0 1994 Fatigue & Fracture of Engineering Materials & Structures Ltd

FATIGUE LIFE ESTIMATION OF WELDED JOINTS OF AN ALUMINIUM ALLOY UNDER SUPERIMPOSED RANDOM LOAD WAVES: APPLICATION OF A 2-DIMENSIONAL RAINFLOW METHOD
I. TAKAHASHI, MAENAKA T. MIYAMOTO H. and
Ship Research Institute, 6-38-1 Shinkawa, Mitaka, Tokyo 181, Japan

Received infrnal form 31 July 1994
Abstract-Fatigue tests were carried out on welded T-joints of JIS A5083P-0 Al-Mg alloy under constant amplitude and superimposed random load waves. Each random wave was generated by superimposing a secondary zero-mean random process on a constant amplitude pulsating trapezoidal wave, which simply simulated the GAG (ground-air-ground) stress cycles, or on a constant d.c. (direct current) component. It is proved that the 1-dimensional rainflow method is insufficient as the mean stress effects are not considered. From a procedure using a 2-dimensional rainflow method and the original and a modified Goodman diagram, it was concluded that the mean stress effects on secondary waves are larger in the case of GAG than d.c. Considering this, a life estimation procedure is proposed that gives consistent and acceptable estimation results.

NOMENCLATURE
A, B D
fpear

= constants in power spectral density S(w) = cumulative fatigue damage factor ( = Z n / N )

= peak frequency of secondary random wave K , = linear stress concentration factor at weld toe N , = number of cycles of primary wave N , = failure life R = load ratio or stress ratio p = toe radius S(o) power spectral density of broad banded secondary random wave = u* = fatigue strength (stress amplitude) uatR= = fatigue strength (stress amplitude) for R = - 1 urn= mean stress uT = true fracture stress uu = ultimate tensile strength uo,2 0.2% offset proof stress = Au = stress range Aceq = equivalent stress range Aun = nominal stress range ij = reduction of area (YO) o = circular frequency of secondary random wave oo= peak circular frequency of secondary random wave

INTRODUCTION

In some structures and machines, e.g., railway bridges, aircraft, high speed ships, and gas turbines, engineering components experience characteristic stress cycles in which secondary stress waves with relatively small amplitudes and short periods are superimposed on primary stress waves with large amplitudes and long periods [1,2]. As those stress waves generally have broad banded spectra and large irregularity factors, ordinary fatigue life estimation procedures for narrow banded
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I. TAKAHASHI et al.

waves, such as simple Rayleigh distribution approximation, are not really applicable. In such cases, it is necessary to analyze the amplitude distribution characteristic of each stress wave. Furthermore, since the primary stress waves can act on the secondary stress waves as mean stresses, then mean stress effects should be considered in a fatigue life estimation. In Japan, two types of high speed cargo ships are now under development based on new concepts (loads of 1000 ton and 50 knots in speed), and the construction of the prototype models for experiments in the sea has been already completed. When they cruise, their hulls are lifted above the water surface using an air cushion or a hydrofoil, and the GAG (ground-air-ground) loads act on the structural members during every cruise. In order to reduce the weight of the ships, JIS A5083 Al-Mg alloy is mainly used as the structural material and the structural members are welded by a TIG or MIG welding. A standard for the construction of such new types of high speed craft has not yet been established, and many discussions are now held both inside and outside of Japan, for example at the IMO (International Maritime Organization). In this study, fatigue tests were carried out on welded T-joints of JIS A5083P-0 A1-Mg alloy under constant amplitude and superimposed random load waves. A 10mm thick rib plate was attached to a 10 mm thick main plate by a MIG welding. After welding, the weld toes were ground with a pencil grinder. For fatigue life estimation, three types of wave count methods were comparatively examined, namely, the range count method, the 1-dimensional rainflow method (only stress ranges are counted) and the 2-dimensional rainflow method (both stress ranges and mean stresses are simultaneously counted), in combination with a modified Miner rule, in which the fatigue limit for Miner sum calculation [3,4] is set to zero and the S-N diagram is extrapolated below the original fatigue limit. Each stress range obtained by the 2-dimensional rainflow method was adjusted using the original and a modified Goodman diagrams and the Gerber curve according to the mean stress value prior to the application of the modified Miner rule.
MATERIALS AND SPECIMENS

The chemical composition (wt.%) of the JIS A5083P-0 A1-Mg alloy was as follows: Si 0.09, Fe 0.24, Cu 0.02, Mn 0.70, Mg 4.70, Cr 0.06, Zn 0.01, Ti 0.01, remainder aluminium. Similarly, the mechanical properties were: tensile strength 325 MPa, 0.2% offset proof stress 160 MPa, and percentage elongation 27.7. Specimens were non-load-carrying fillet welded T-joints, consisting of 10 mm thick main and rib plates. The specimen design is given in Fig. 1. The welding was performed by an automatic

in the fatigue tests

100
c

-? F10 _ I _ 190 300

?

: strain gauge locatbn

Fatigue life estimation of welded joints of an aluminium alloy under superimposed random load waves

1407

MIG welding, and the welding conditions were as follows: number of passes = 3 passes for each side; filler metal = JIS A5183WY (4 = 1.2 mm); welding current, d.c./reversed polarity; interpass temperature less than 60°C; heat input, 4.5-5.1 kJ/cm; and target leg length, 12 mm. The welding was done 3 times and the specimens were grouped due to the welding order, as series A, B, and C . The weld toes on the main plate were ground using a pencil-grinder with a target toe radius of p = 9 mm. The residual stresses perpendicular to the weld line, measured by strain gauges with stress relief, were of small values scattering from - 9 to 31 MPa. The reason for these small values was considered to be due to the relatively small restraint of the joint and the residual stresses could have been relaxed by the toe-grinding.
FATIGUE TESTS

A servo-hydraulic type testing machine was used for the fatigue tests. Fatigue tests were made under 3-point bending with a span of 200mm (see Fig. l), and the loads were controlled by a personal computer. Fluctuations of concentrated strains at weld toes measured by strain gauges were recorded at appropriate intervals by the personal computer through an A/D converter.

Superimposed random load waves Each random load wave in the fatigue test was generated by superimposing a secondary zeromean random process having a specified power spectral density on a primary wave or a constant direct current (d.c.) component. Both broad banded and narrow banded spectra were used for the secondary random waves, while constant amplitude pulsating trapezoidal waves (which simply simulated the ground-air-ground (GAG) loads) were used as the primary wave. If the value of the random load wave was going to be negative, the negative part was clipped to zero, because a negative load could not be applied by the test set up. The following equation was used for the broad banded spectrum:
S(w) =

AB B2 + (O - COO)’

The peak circular frequency is wo, and the spectrum shape is symmetric about w = wo (see Fig. 2). In the fatigue tests, the amplitugof the secondary wave was controlled by the value of A under the condition of the peak circular frequency wo= 15.2 rad (the peak frequency fpeak = 2.42 Hz) and B=5.0. The irregularity factor of the secondary wave expressed by Eq. (1) was about 0.82 [ I S ] .

Circular Frequency, w [rad]

Fig. 2. Power spectral density for broad banded random load waves.

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I. TAKAHASHIal. et

300

k

2
z
??
C .-

300
200

2
L

200
100

W

E 2 .g 2

E a
10

100

0

0

20

2

g

o

0

10

20

Time [sec]
(c) Trapezoidal wave shape

Time [sec]

C

Time
Fig. 3. Examples of superimposed random stress waves, and the trapezoidal wave shape (a) GAG + random (broad banded), Acrl =159.0 MPa, uzR,=30.3 MPa; (b) GAG + random (narrow banded), Aul = 108.5 MPa, uZRMS26.8 MPa; and (c)the trapezoidal wave shape. =

On the one hand, the narrow banded load wave was generated by superimposing three wave components with the same amplitude but different frequencies of 3.83,3.98, and 4.14 Hz respectively. Examples of superimposed random stress waves are shown in Figs. 3(a) and (b), for both broad banded and narrow banded spectra. Figure 3(c) also shows the shape of the primary trapezoidal wave: period, 25 s (frequency of 0.04 Hz); time at maximum stress, 22.5 s; and time at minimum stress, 0 s. In both cases of broad banded (Fig. 3(a)) and narrow banded (Fig. 3(b)) spectra, approximately 100 cycles of secondary random waves are superimposed on 1 cycle of the primary trapezoidal wave. In order to acquire the basic data for life estimation procedure constant amplitude fatigue tests were conducted using pulsating triangular and sinusoidal load waves (load ratio R s 0). In the constant fatigue test results, any difference due to the wave type (triangular or sinusoidal) was not observed.
RESULTS OF FATIGUE TESTS

Stress-strain responses at weld toes Prior to the fatigue tests, a static loading test was carried out to decide the location of maximum stress concentration using strain gauges spaced equally at 1 mm intervals (see Fig. 1). As the maximum stress concentration occurred on the weld metal, approximately 2 mm distant from the weld toe, strain gauges for the fatigue tests were attached at that location. Figure 4 shows examples of stress-strain responses at weld toes during the superimposed random fatigue tests with the number of cycles of the primary wave being N , = 1 in Fig. 4(a), and N1= 84 in Fig. 4( b) respectively. A non-linear response with plastic strain can be observed immediately (Fig. 4(a)) after loading, whereas strain hardening of the material occurred with the stress cycles finally producing a linear response; see Fig. 4 b). Similar phenomena of strain hardening were ( commonly observed in all the fatigue tests.

Fatigue life estimation of welded joints of an aluminium alloy under superimposed random load waves
300
3OOL, I
. ,

;.
z 0

0)

.I,Bl
100

1409

,,

~

, .

, . , . . , . . . , , . , , ,,

0 0.0

0
0.1

0.2

0.3

0.4

0.5

0.0

L

I , , . . ,

....
0.4

~

0.5

0.1

0.2

0.3

Strain at Weld Toe [%]

Strain at Weld Toe

[%I

Fig. 4. An example of stress-strain response during a fatigue test; broad banded, A~J,= 159.0 MPa, gZRMS = 30.3 MPa. (a) N, = 1, (b) N , = 84.

Results of the constant amplitude fatigue tests Figure 5(a) shows the results of the constant amplitude fatigue tests arranged by the nominal stress range, Ao,, and the failure life. Some differences in fatigue strength are observed between the series of specimens. It can be considered that the value of the stress concentration factor, K,, slightly varies among the specimen series, therefore the mean value of K , was calculated for each series from the measured values of concentrated strain at the weld toes and the values of nominal stress, and the following results were obtained.

Series A: K,= 1.12 Series B: K , = 1.18 Series C : K , = 1.08 Then, the failure lives were rearranged using the estimated values of the local stress ranges at the weld toe, K , x Ac,,, and these results are shown in Fig. 5(b). It can be seen that the differences among the specimen series decrease in comparison to Fig. 5(a), proving that the three series can be handled together using K , x do,. The line in the figure was fitted to the data, except the run out values, by the least squares method, and is expressed by the following equation.
K , x Acn= 1359 x N;0.'521

(2)

z
a

a

; a

d

z-

Failure Life [cycles]

Failure Life [cycles]

Fig. 5. Results of constant amplitude fatigue tests. (a) nominal stress range, Agn, vs. failure life, (b) local stress range, K,Aa,, vs. failure life.

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I. TAKAHASHI et al.

The run out data were also excluded in the application of the modified Miner rule, in which the fatigue limit is set to zero and the S-N diagram is extrapolated below the original fatigue limit. Results of the superimposed random fatigue tests The test conditions and the results of the superimposed random fatigue tests are shown in Table 1. As mentioned above, both broad banded and narrow banded waves were used as the secondary random waves. The stress waves can be categorized according to the types of the primary waves as follows.
0

0

0

0

The primary wave does not fluctuate and kept a constant value in tension (A-8, C-3). Zeromean secondary random waves were superimposed on constant tensile mean stresses for this type. This type is abbreviated to “d.c.” in this paper. In Table 1, the failure life is expressed by the corresponding number of GAG cycles, which is obtained by dividing the total time to failure by the GAG period (25 s except C-2), for the sake of comparison to the case of the GAG type. Constant amplitude trapezoidal waves with a period of 25 s were used for the GAG stress cycles (A-6, A-9, B-5, B-8). A constant amplitude trapezoidal wave with a period of 12.5 s was used for the GAG stress cycles (C-2). A trapezoidal wave with a period of 25 s was used for the GAG stress cycles, and the amplitude was alternately varied approximately 20% every GAG cycle (B-7).

When random loading fatigue tests are carried out on welded joints, the quantity of the specimens and tests tends to be restricted because of the problem of time and cost, as compared to aircraft studies where simple small sized plate specimens are often used. Therefore it should be noted that the number of data has not yet been ample in this study to obtain some general and decisive conclusions. Although the difference in Kt value should be considered in the interpretation of Table 1 data, the following observations can be immediately extracted from the table. The failure lives decrease in the case of GAG as compared with d.c., and the life reduction ratio is from 0.2 to 0.3 (A-8 with A-9, C-3 with A-6). When the GAG period is 12.5 s (C-2), the failure life expressed by the GAG cycles increases as compared with the case of a period of 25 s (A-9). The reason for this is considered to be due to the fact that the fatigue damage caused by the secondary random waves per GAG cycle is reduced to one-half. However, the time to failure of C-2 is approximately 1.3 times that of A-9, showing a relatively small difference. Therefore it may be concluded that the GAG stress cycles contribute to the total fatigue damage not only as independent cyclic stresses but also as mean stresses on the secondary random stress waves. For comparative purposes, a simple equivalent stress range at the weld toe, doeq, calculated is by the following equation

(3) where Aol is the stress range of the primary wave (mean stress in the case of d.c.), and ACT~RMS is the RMS value of the secondary random wave. In the equation, it is considered that the amplitude times as large as the RMS value. In the case of the constant of a sinusoidal wave is equal to amplitude test, Aoeq agrees with the value of K,Ao,. The failure life as a function of Aoeq is shown in Fig. 6. In Fig. 6, the random fatigue lives are
A‘eq

= Kt(Aol

+ .\/?I‘2RMS)

.\/?I

Table 1. Testing conditions and results of random fatigue tests and life estimation Estimated Nf based on the modified Miner rule and values of D ( = Z n / N )
Aai ( MPa) 158.5

Actual Nf (expressed by aZms numbexof (MPa) GAG cycles) 29.1 18,587*

+

+
137.7 26.3 30.3 37.5 27.3 26.8 30.3 30.4 6624 14,355** 33,179 9335 2915 5603 46,589* 159.0 160.0 138.8 108.5 159.0 158.7 127.2

Specimen No. Type of stress wave A-8 d.c. +random (broad banded) c -3 d.c. random (narrow banded) A-9 GAG +random (broad banded) B-8 GAG +random (broad banded) A-6 GAG +random (narrow banded) GAG + random B-5 (narrow banded) c -2 GAG (half period) +random (broad banded) GAG (alternate amplitude) B-7 +random (broad banded) Range count 165.850 [D=0.1 121 402,482 C0.1161 155,924 C0.0361 27,439 [0.106] 240,758 C0.0391 192,842 C0.1721 393,793 C0.0371 167,247 [0.040] 1-D rainflow 107.524 [0.173] 380,595 [O. 1221 29,853 C0.1881 9485 C0.3071 83,242 c0.1121 111,817 C0.2971 45,002 C0.3191 30,600 C0.2171

2-D rainflow Goodman (using au) 5748 [3.2341 26,034 C1.7901 5648 C0.9921 1488 [ 1.959) 14,740 C0.6331 25,208 C1.3161 15,213 [0.944] 4995 C1.3261

2-D rainflow +modified Goodman (using UT) 20.021 c0.9281 78,351 C0.5951 13,655 [0.410] 3752 C0.7771 34,931 C0.2671 48,241 C0.6881 27,728 [OSlS] 12,648 [0.524]

2-D rainflow + Gerber (using a) , 13.259 r1.4021 76,203 [0.611] 10,659 C0.5261 2498 C1.1671 32,689 C0.2861 50,610 [0.656) 24,680 [0.582] 9393 C0.7051

d.c. (direct current) =constant mean stress. GAG (ground-air-ground) =trapezoidal stress wave. * Corresponding Nf was calculated as: N,= [Time to failure]/[GAG period (25 s)] . ** GAG period was 12.5 s in this case.

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h

300

1

+

Failure Life [cycles]
Fig. 6 . Relation between equivalent stress range and failure life.

all expressed by the corresponding number of GAG cycles, which is obtained by dividing the total time to failure by the GAG period (25 s). In Fig. 6, the results of the superimposed random fatigue tests with the GAG cycles are located on a line with approximately the same slope of the constant amplitude tests (denoted by the dotted line), showing no significant difference due to the band width of the secondary random waves, nor due to the GAG period. As the constant amplitude data can be regarded as the results of the GAG stress cycles only, it can be concluded that the fatigue life is reduced to approximately one twentieth by superimposition of the secondary random waves. Additionally the results for d.c. (solid marks) show a little longer fatigue lives than for GAG, suggesting that the GAG stress cycles are more harmful than the d.c. stress component, when combined with the same secondary random waves.

The methods of fatigue life estimations In this study, three types of wave count methods were used, namely, the range count method, the 1-dimensional ( 1-D) rainflow method, and the 2-dimensional (2-D) rainflow method. Although the rainflow method originally gave 2-dimensional information consisting of the stress range and the mean stress (An, om), here only the information on stress range is utilized and is named “1-dimensional” for convenience in this paper. Such a case is generally used in fatigue life estimation procedures. On the one hand, each stress range obtained by the 2-D rainflow method was modified according to the mean stress value of the original Goodman diagram, a modified Goodman diagram, and the Gerber diagram (see Fig. 7). When constant amplitude stress cycles are applied under a mean stress, the original Goodman diagram (using the ultimate tensile strength 6,) often gives excessively conservative results, and there has been a case in which the true fracture strength oT gave more reasonable results [71. Therefore a modified Goodman diagram (using the true fracture strength cT) was also applied, and the results were comparatively examined. The value of oTwas obtained in this study by the following equation:
0~=100/(100-tj)0, ~ 1 . 5 6 0 , (4) where tj denotes the percentage reduction of area, and has a value of 36%. The three kinds of diagrams are shown in Fig. 7 with some actual data, and they are expressed by the following equations:

Fatigue life estimation of welded joints of an aluminium alloy under superimposed random load waves

1413

Original Modified Goodman line
U r n
00.2

0“

6 ,

Fig. 7. Original Goodman line, modified Goodman line, and Gerber curve. Solid marks denote some is actual data ( N = 2 x lo6)for A5083-0 plate specimens [ 6 ] , and o , , ~ the 0.2% offset proof stress.
0

Original Goodman line using cru

0

Modified Goodman line using cT

Gerber curve

where cra denotes the fatigue strength (stress amplitude) under the mean stress om, and (z,(~= - 1 ) denotes the fatigue strength (stress amplitude) for the stress ratio R = - 1. Although these equations are simply mathematical ones and do not have any physical formulation, they are used in this paper as temporary expedients from a practical standpoint of design. In order to eliminate the effects of the local weld toe shapes, the fatigue life estimation in this paper was done using the estimated values of the local stress range at the weld toe, &do,, and Eq. (2) was used for the constant amplitude fatigue test data. As mentioned above, the fatigue limit for Miner sum calculation is set to zero and the S-N diagram is extrapolated below the original fatigue limit (Eq. (2) is applied for all stress levels), and this is named “modified Miner rule” in this paper. Results of fatigue life estimations for each method are as follows. Results of life estimation using the range count method and the modiJied Miner rule The results are shown in Table 1 and Fig. 8(a). The value of Miner’s cumulative fatigue damage factor [ ] D = b / N , at failure ranges from 0.036 to 0.172, showing that this method is unsuitable 4, for the present superimposed random stress waves and gives very unconservative estimations. The reason for this is considered to be due to the fact that the range count method cannot count the maximum stress range during a GAG stress cycle and that the mean stress effects by the GAG primary waves or the d.c. component are not taken into consideration by this method. Results of life estimation using the 1-D rainjow method and the modiJied Miner rule The results are shown in Table 1 and Fig. 8(b). The accuracy is fairly improved when compared with the range count method especially in the case of the GAG primary waves. However, the

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I. TAKAHASHIal. et

Actual Failure Life

Actual Fallure Life

Fig. 8. Results of life estimation by (a) the range count method and the modified Miner rule, and (b) the 1-D rainflow method and the modified Miner rule.

estimation results are all unconservative and the value of D at failure ranges from 0.112 to 0.319 and is still too small. When this method is applied to the superimposed random stress waves in a high mean stress region (as in this study), a safety factor of approximately one order in life ratio is considered necessary. Results of l$e estimation using the 2-0 rainjlow method, Goodman diagrams and the modified Miner rule
, ( a ) Using the original Goodman diagram (using a) The results are shown in Table 1 and Fig. 9(a). The value of D for the GAG primary waves ranges from 0.633 to 1.959, resulting in very much improved estimations when compared with the 1-D rainflow method. Here a safety factor of 2 suffices in this case. On the other hand, the values of D for the d.c. are 1.790 and 3.234, resulting in slightly over-conservative estimations.

( b ) Using the modified Goodman diagram (using o T ) The results are shown in Table 1 and Fig. 9(b). The value of D for the GAG primary waves ranges from 0.267 to 0.777 resulting in unconservative estimations whereas those for the d.c. ranges from 0.595 to 0.928 resulting in fairly good estimations, suggesting that oT is sufficient for the d.c. From the results of (a) and (b), it can be derived that the mean stress effects on the secondary random waves are more significant in the case of the GAG primary waves than the d.c. component.

Results o l$e estimation using the 2-0 rainjlow method, the Gerber diagram and the modified Miner f rule The results are shown in Table 1 and Fig. 9(c). The results are almost equivalent to the case of the modified Goodman diagram using oT (as expected from Fig. 7), being good for the d.c. and unconservative for the GAG cycles. The life estimation results by each method have now been presented, and as a conclusion the following procedure is recommended for a more accurate life estimation. (1) Using the 2-D rainflow method, store the value of the mean stress for each stress range. (2) Modify each stress range according to the mean stress value, using the original Goodman diagram (using au)for the GAG cycles and the modified Goodman diagram (using aT)for the d.c. component respectively. (3) Apply the modified Miner rule to the obtained histogram of stress ranges.

Fatigue life estimation of welded joints of an aluminium alloy under superimposed random load waves
10'

1415

10 '

10 '

10'

10.

1' 0

10 '

10'

1om

10 '

Actual Failure Life
10-

Aclual Failure Life

f

d
10'

1 o1

I' 0 Actual Failure Life

Actual Failure Life

Fig. 9. Results of life estimation by the 2-D rainflow method and involving (a) the original Goodman diagram (using o"), and the modified Miner rule, (b) the modified Goodman diagram (using uT), and the modified Miner rule, (c) the Gerber diagram (using uu), and the modified Miner rule, (d) the original ,) Goodman diagram (using o , for GAG, and the modified Goodman diagram (using +) for d.c., and the modified Miner rule.

The results of a life estimation by the above procedure are shown in Fig. 9(d). The values of D for all the cases are within the range from 0.5 to 2.0, showing that the applicability of the rainflow method and the modified Miner rule is acceptable when using the above procedure.
CONCLUSIONS

The major conclusions of this paper are as follows. 1. The ground-air-ground (GAG) stress cycles are more harmful than the direct current (d.c.) stress component when combined with the same secondary random waves. The reason for this is considered to be that the GAG stress cycles act, not only as independent stress cycles, but also as a mean stress of the secondary random waves and that the mean stress effects on the secondary random waves are more significant in the case of the GAG cycles than the d.c. component. 2. The estimated results of lifetime were all unconservative in the case of the 1-D rainflow method and a modified Miner rule. When this method is applied to the superimposed random stress waves (in such a high mean stress region as in this study), a safety factor of approximately one order in life ratio is considered necessary. 3. The most accurate and consistent life estimation results were obtained by (a) using the 2-D rainflow method, (b) applying the original Goodman diagram (using a,) for the GAG cycles and the modified Goodman diagram (using gT)for the d.c. component respectively, and (c) applying a modified Miner rule to the modified stress ranges.

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I. TAKAHASHI al. et

Acknowledgements-The authors are much obliged to the Hitachi Zosen Corporation for their kindness in providing the materials and the specimens.

REFERENCES
1. 0. Part1 and J. Schijve (1990) Reconstitution of crack growth from fractographic observations after flight simulation loading. Int. J. Fatigue 12(3), 175-183. 2. D. Aliaga, A. Davy and H. Shaff (1987) A simple crack closure model for predicting fatigue crack growth f under flight simulation loading. In: Mechanics o Fatigue Crack Closure, ASTM STP 982, pp. 491-504. 3. A. Palmgren (1924) Die Lebensdauer von Kugallagern. VDI-Zeitschrft 68(14), 339-341 (in German). 4. M. A. Miner (1945) Cumulative damage in fatigue. J . Appl. Mech. 12(3 ) , A159-164. 5. K. Iida and I. Takahashi (1988) Power spectrum density effects on strain cycling fatigue life. Naval Arch. Ocean Engng 26, 183-193. 6. Y. Takesima and M. Ito (1978) Technical Report of Sumitomo Light Metal Ind., Ltd. No. 19, p. 74. 7. JIW Commission XI11 (1992) Position o JIW Commission XI11 on IlWInternational Joint Research. f


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methodofmakingalgorithmforcalculatingthisequivalentwithrainflowmethod.Theequivalent...第2章结构疲劳理论 第2章结构疲劳理论本文研究的是车辆耐久性试验载荷谱,并建立...

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The water falls as rain. It…from clouds. The clouds wool and they drop water as rain. Flow ...

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2007 nCode Deterministic and Stochastic 30 0.4 20 0.2 Acceleration (g) ...2007 nCode block loading What is the method based on? ? Rainflow cycle ...

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MethodRainflow Counting 及 Level Crossing Counting,本文中使用 Rainflow ...图 2 MotionView 计算 RLDA 节点力与实际量测节点力之比对:弹簧行程与避振器...

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? Apply a method to one interval of a single ... including the one dimensional histogram display ...Mark the two Rainflow matrix files and all of ...

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The name rainflow results from a comparison of this method to the flow ...1 1 2 3 4 5 6 7 8 9 from from 2 3 4 5 6 7 8 to 9 10 11...

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to using an irregular wave approach employing the rainflow counting method. ...Pipe bending moments about two principle bending axes of pipe cross-section ...

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In this work methods are developed for computation...The RainFlow Cycle (RFC) was invented 7 8 ...a 2-dimensional histogram of the cycles and ...

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rainflow domain method and discuss some differences between the two methods....Hence, in a fatigue test for the car application, instead of repeating ...

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