The application of a statistical fatigue life prediction method to agricultural equipment

IntJ Fatigue9 No 2 (1987) pp 115-118

The application of a statistical fatigue life prediction method to agricultural equipment B. B. Harral

The derivation of the statistical expression for the fatigue life at a welded connection is outlined which requires only constant amplitude stress/life data and the standard deviation and ruling frequency of the stress history. The expression has been used to predict fatigue lives from wide-band stress histories from an agricultural soil cultivator. Standard deviations varied from 5 to 26 MPa and ruling frequencies from 10.7 to 20.1 Hz. A comparison with the rainflow cycle-by-cycle prediction method shows good agreement. Key words: fatigue; life prediction; statistical method; cycle-by-cycle method; welded structures; agricultural cultivator

The frames of m o d e m agricultural machines, like many modern engineering structures, are welded fabrications. Although the machinery is often complicated in terms of the number of parts bolted and welded together, it is usually unsophisticated in terms of design and material usage. Generous margins on material and welds were traditionally included in an attempt to avoid failures. Despite these allowances structural failures still occurred, with the majority being at welded details. 1 Commercial pressures are now forcing manufacturers to reduce the margins on materials and welds, and unless stress levels are taken into account the inevitable consequence will be more failures. The common method of fatigue life prediction on welded structures is to use rainflow analysis to extract cycles from a stress or strain history, and then calculate the damage caused by each cycle from data given in BS 5400. 2 Miner's rule is then used to sum the damage cycle by cycle, and failure is assumed when the damage equals unity. The problem for the designer is that the cycle-by-cycle approach requires a time history to operate on, and this is not known in enough detail at the design stage. However, the statistical properties of the history can be estimated at the design stage since they depend on quantities such as frequency content of inputs and structural response. This paper describes a statistical method of fatigue life prediction and its application to welded joints on agricultural machinery. Many studies 3'4 have investigated the predictive accuracy of cycle-by-cycle methods and more work is in progress, particularly at long lives) The work described here has concentrated on a comparison between the statistical method and the cycle-by-cycle method when applied to agricultural stress histories.

Statistical method Statistical methods have been proposed by several authors 6,7 for stationary random signals with a Gaussian amplitude

distribution. Both narrow-band and wide-band signals were considered and expresssions for damage were derived involving the standard deviation and the irregularity factor of the history. The advent of rainflow analysis, s which takes account of the material's cyclic stress/strain behaviour, has provided a basis for further work on the statistical technique to include the new cycle counting method. In a comprehensive study, Tunna 9 has derived damage expressions for both narrowand wide-band histories, applied to welded and non-welded details, based on range-mean and rainflow methods of cycle counting. O f particular relevance in the present work is the expression for damage per cycle at a welded detail using rainflow cycle counting..The derivation was based on the statistics of a narrow-band history, but the expression was then found to hold for wide-band signals providing the irregularity factor, ~ was greater than 0.2. The expressions for damage per cycle and fatigue life are given in the Appendix, with an outline of their derivation. The expressions show that only two statistical properties are required for fatigue life prediction, namely standard deviation and ruling frequency. Both parameters can be conveniently found from the spectral density function of the signal. The irregularity factor, which should be greater than 0.2 for the statistical method to be valid, can also be found from the spectral density function (see Appendix). Thus, if the spectral density function is known or can be estimated, it is then a simple matter to calculate fatigue life.

A g r i c u l t u r a l stress histories To test the validity of the statistical approach, stress histories were taken from a tractor-powered soil cultivator and analysed using both the rainflow cycle-by-cycle method and Equations (5-10). The histories were obtained from strain gauges attached to the main structural member of the cultivator. Although the gauges were fixed near load-carrying fillet welds, fatigue failures were not expected. The gauges were intended as indicators of the relative severity of different

0142-1123/87/02115-04 $3.00 © 1987 Butterworth & Co (Publishers) Ltd Int J Fatigue April 1987

115

field tests rather than a means of predicting when failure would occur. Most importantly, the gauges enabled typical strain histories from this type of agricultural machine to be recorded. The signals from four positions were recorded in a series of 14 field tests covering a range of machine variables. Results

and discussion

Basic signal statistics were extracted from the stress histories, either directly or via spectral density functions, to check standard deviation, normality, irregularity factor, etc. The statistics from a typical test are shown in Table 1. The skewness and kurtosis figures show that the amplitude distributions are nearly Gaussian, hence one of the requirements for the statistical life prediction method is satisfied. Spectra of the histories analysed to give the results in Table 1 are shown in Fig. 1. These are typical of the spectra obtained in the tests and show peaks at harmonics of either the input shaft speed or the speed of the multi-bladed rotor that cultivates the soil. Values of irregularity factor a in Table 1 are above or very close to 0.2, the lower limit of applicability suggested by Tunna for the statistical fatigue life expression. Over all tests, a varied from 0.19 to 0.41. The standard deviations of the stress histories varied from

0.75

T a b l e 1. S t r e s s s i g n a l p r o p e r t i e s f r o m a t y p i c a l test on a rotary cultivator

Strain gauge position Property

1

Mean (MPa) Standard deviation, (MPa) Maximum (MPa) Minimum (MPa) Skewness Kurtosis Ruling frequency, N, (Hz) Irregularity factor, a

47

2

3

-22

-31

6.2 7.7 10.1 39 15 26 -17 -63 -93 0.056 - 0.101 - 0.207 3.001 3.176 3.190 14.5 0.37

4

16.7 0.37

47 23.2 177 -62 0.199 2.981

10.7 0.19

11.0 0.23

5 to 26 MPa. A variety of ruling frequencies (Nr) was also obtained but at any one gauged position the variation was relatively small. Table 1 shows typical values of N, at the four gauged positions. These values generally varied by less than 20% despite an 80% change in machine operating speed. This implies that the structural responses at the gauged positions on the cultivator are not simply linear static but have 0,75

m

0.60

0.60

m

D

0.45

g 0.45

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0.30

0.30

0.15

0.15

0.00

5

0

10

15

20

25

30

m

0.00

0

a

b

0.75

0.75

~

5

10

15

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20

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I

25

30

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0,60

0.60

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~0.45-

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5

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15 20 Frequency (Hz)

25

30

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d

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10

16 20 Frequency (Hz)

I

25

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30

Fig. 1 Typical power spectra from the four strain gauge positions on the cultivator frame: (a) gauge 1, (b) gauge 2, (c) gauge 3 and (d) gauge 4

116

Int J Fatigue April 1987

a significant dynamic component which is narrow band and consequently filters the input frequency spectrum. Over all histories, the ruling frequency varied from 10.7 to 20.1 Hz. A fatigue life was calculated for each gauged position in each test using the statistical expression and the standard deviation and ruling frequency found for each stress history. Mean life, constant amplitude data for a class F2 weld from BS 5400 were used, with a change of slope from - 3 to - 5 at 107 cycles. A value of 50 MPa was taken for S 0. Since failure data did not exist for the machine, fatigue lives were calculated using the cycle-by-cycle method to compare with the lives calculated by the statistical method. The range of cycle-by-cycle fatigue lives is shown in Fig. 2. To show the comparison between the fatigue lives calculated by the two methods, the ratio of the statistical life to the cycle-bycycle life has been plotted in Fig. 3; clearly, the two methods are equivalent if the ratio is unity. Although Fig. 3 shows some scatter, the agreement is good, considering the accuracy generally expected from fatigue life prediction methods. More importantly, the estimate by the statistical method is consistent and conservative at standard deviations greater than 10 MPa. For agricultural purposes agreement above a standard deviation of about 10 MPa is desirable because, io 2

M Q.

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101 4.+

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as Fig. 2 shows, this represents fatigue lives of less than 1000 hours. A mean life of 1000 hours implies a 50% chance of failure in a usage of less than 10 hours/day on 20 days/year for five years. The close correlation between standard deviation of stress and cycle-by-cycle fatigue life, shown in Fig. 2, suggests a point of practical significance to agricultural field testing. It is often difficult to judge the relative severity of different operating conditions in terms of fatigue damage without a full calculation of life. When time histories are available Fig. 2 shows that for this type of cultivator a judgement can be based on the standard deviation alone. Since standard deviation can be calculated using very inexpensive equipment, this is a cheaper and quicker alternative to the full cycle-by-cycle analysis. In all the foregoing analyses a change of slope has been assumed in the constant amplitude stress/life data at 10v cycles, rather than a cut-off. At high standard deviations ( > 20 MPa) there is little difference between the lives predicted under either assumption, while at low values the difference is very large. Tunna 5 and Pook 1° have suggested that the statistical approach predicts actual life more accurately with a cut-off than a slope change, particularly when the life is long. Since failure data do not exist for this machine the point cannot be investigated. The results from this kind of agricultural machine are very encouraging. Now, histories from other machines need to be analysed to establish the generality of the statistical approach. The normality of the amplitude probability density functions needs to be checked. This study has shown that some non-normality can be tolerated. Some machines, particularly passive soil cultivators, may give stress histories with irregularity factors of less than 0.2. In this situation a correction factor based on a may be necessary, as proposed by Wirsching and Light 11 for non-welded details. Conclusions

too 101

I

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I

1111111

I

I IIIIIll

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103 104 Cycle-by-cycle fatigue life (h)

102

105

Fig. 2 Relationship between standard deviation of stress and cycleby-cycle fatigue life 2.0 1.8 o=

B

1.6

A statistical expression for fatigue life has been tested on broad-band stress histories from an agricultural cultivator. The histories were almost Gaussian and had irregularity factors of at least 0.19. Ruling frequencies depended on position and varied between 10.7 and 20.1 Hz. Standard deviations varied from 5 to 26 MPa. In the absence of failure data for this machine, the fatigue life calculated from the statistical expression was compared with the life calculated by the rainflow cycle-by-cycle method. Agreement was good, particularly in the important standard deviation range above 10 MPa.

u I

1.4

B

4.

References

1.2-

1.

Richardson, R. C. D. et al 'A pilot survey of the durability of farm machinery" Proc Symp Inst Agric Engrs, 1967 paper no 3/F/26

2.

'Steel, concrete and composite bridges. Part 10. Code of practice for fatigue" British Standard BS 5400 (British Standards Institution, London, UK, 1980)

3.

Tilly, G. P. and Nunn, D. E. 'Variable amplitude fatigue in relation to highway bridges" Proc IME194 No 27 (1980) pp 259-267

4.

Gurney, T. R. Fatigue of Welded Structures (Cambridge University Press, UK, 1979)

5.

Tunna, J. M. "Random load fatigue: theory and experiment' Proc I M E I ~ No C3 (1985)

6.

Bendat, J. S. 'Probability functions for random responses:

+ ÷ ++ #~q4l.~++ 0.8

+

.~ 0.6 0.4 0.2

0.0

+

f

O

I

I 5

I

"

I I I I I 10 15 20 Standard deviation (MPa)

I

I 25

I 30

Fig. 3 Comparison between the statistical and cycle-by-cycle prediction methods

Int J Fatigue April 1 9 8 7

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prediction of peaks, fatigue damage, and catastrophic failure' NASA paper no CR-33 (April 1964) 7.

Kowalewski, J. "On the relation between component life under irregularly fluctuating and ordered load sequences, Part I1' DVL Reoort No 249 (1963) MIRA Translation No 60/66

8.

Matsuishi, i . and Endo, T. 'Fatigue of metals subjected to varying stress' presented at Japan Society of Mech Engrs, Fukuoka, Japan, 1968

9.

Tunna, J. U. 'Theoretical and experimental aspects of two cycle counting methods for fatigue life prediction" Fatigue of Engng Mater and Structures (to be published)

10.

11.

Peek, L. P. 'The effect of mean-stress on fatigue crack growth in cruciform welded joints under non-stationary narrow band random loading' NEL Report No 690 (National Engineering Laboratory, UK, December 1983) Wirsching, P. H. and Light, i . C. 'Fatigue under wide band random stresses' J Struct Div, ASCEI06 No 8T7 (July 1980)

where or is the standard deviation of the narrow-band Gaussian random process. When the expressions for ]ife, Equations (1-3), and cycle probability, Equation (4), are combined and integrated over all possible stress ranges (0o > S > So), the damage per cycle, D, is given by: D = 1 [ F ( I - m/2) Q(Z212 _ m) a ] o'm2 3m/2 L-

0..23./2

Dr Harral is with the Machine Dynamics and Reliability Group, AFRC Institute of Engineering Research, Wrest Park, Silsoe, Bedford MK45 4HS, UK.

F(n) -- g a m m a function Q(tA[v) = chi-squared function and Xc - 2 a

An outline derivation of the damage and life formulae is given below. The complete derivation of the damage equation is given by Tunna. 9 A constant amplitude stress range/life relationship is assumed of the form given in BS 5400 for welded details, namely

N(S) = aS m

(1)

where N(S) is the allowable number of constant amplitude cycles with stress range S, and a and m are constants depending on the weld classification. For random loading of welded structures the design code calls for a change in the slope of the curve from m to n at a stress range So. In some cases a designer might want to include an endurance limit at a cut-off stress, S¢, so:

N(S) = aS0m-ns n

for So /> S >t S~

(2)

N(S) =00

for Sc >/ S

(3)

and

The normalized probability, p(S), of a stress cycle of range S is given by:

118

The life in cycles, L o is the reciprocal of the damage per cycle: 1

L¢ - D

(6)

If the frequency of the narrow-band process is N o the life in seconds, Ls, is:

Appendix

p(S) = ~ - ~ exp

(5)

where

Z0 =

Author

(Z~212 - n ) - Q ( Z ~ I 2 - n

(J)

1

Ls =N¢D

When the damage expression is used for broad-band processes with irregularity factors greater than 0.2, N¢ is replaced by the ruling frequency Nr. The statistical properties required for a fatigue life prediction, namely a, o" and Nr, are most conveniently found from the spectral density function S(f) of the signal. These are given by:

6to-=

R,,2 R R ....

(8)

~

(9)

Nr = !2 z V~ R

(lO)

where

R = R"

S(f)df = (2rc)2f : f : s(f)df

R .... = (2z)4 (4)

(7)

f?

4

0 a n d f i s the frequency.

Int J Fatigue April 1987