Frequency-domain fatigue life estimation with mean stress correction

Accepted Manuscript Frequency-domain fatigue life estimation with mean stress correction Adam Niesłony, Michał Böhm PII: DOI: Reference:

S0142-1123(16)00080-3 http://dx.doi.org/10.1016/j.ijfatigue.2016.02.031 JIJF 3868

To appear in:

International Journal of Fatigue

Received Date: Revised Date: Accepted Date:

31 July 2015 15 February 2016 19 February 2016

Please cite this article as: Niesłony, A., Böhm, M., Frequency-domain fatigue life estimation with mean stress correction, International Journal of Fatigue (2016), doi: http://dx.doi.org/10.1016/j.ijfatigue.2016.02.031

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Frequency-domain fatigue life estimation with mean stress correction Adam Niesłonya*, Michał Böhma a

Opole University of Technology, Faculty of Mechanical Engeeniering, Department of Mechanics and Machine Design, Mikołajczyka 5, Opole 45-271, Poland

Abstract Two frequency-domain fatigue life calculation methods are presented which take into account the impact of the mean stress effect. The emphasis is set on the algorithm for fatigue life assessment of the method proposed by the authors. It is supplemented with a mean stress effect correction. Correction method is based on the direct transformation of the zero mean stress power spectral density (PSD) due to mean stress. The method is verified on the basis of own results for the S355JR steel. The authors analyze five models for the designation of the probability density function used in the calculation process. The results are presented in the form of probability distributions after PSD transformation and the calculated fatigue life is being compared with the experimental life in fatigue comparison graphs. An analysis of the choice of a mean stress correction model is also widely discussed and a fatigue life estimation is also performed. The method proposed by the authors is being compared with the Kihl Sarkani method for mean stress correction in frequency domain. Keywords:mean stress, spectral method, random loading, PSD, fatigue;

1. Introduction The phenomenon of material fatigue, which occurs due to the impact of time varying forces is one of the main reasons for material failure [1,2]. The variable forces which are the main reason for this effect can be divided into two groups with constant amplitude loading and random loading. Those loads can be described with the use of deterministic formulas or with the use of stochastic theory [3–5]. The mean stress effect in fatigue life assessment is a well-known issue discussed widely in the literature [6–10]. The mean stress is an extra static load in the form of an additional time independent load applied to the construction e.g. as a results of self-weight, prestressing of springs or compressive stress in a screw connection [11,12]. Engineers have to take into account those kind of extra loads and prevent early fatigue failure or other construction defects. Although the literature presents solutions for the correction of mean stress in the time domain (cycle counting methods) it is rare to find a solution in the frequency domain (spectral method). The authors have presented theoretical backgrounds of mean stress correction method that can be used in the frequency domain [13]. For this purpose a Power Spectral Density (PSD) transformation is used. The transformation process is realized with the use of well-known mean stress compensation models. The fatigue life is being calculated with the use of Probability Density Functions (PDF) of stress amplitudes as well as Palmgren-Miner damage accumulation hypothesis. The presented correction method is verified with own test results obtained for the S355JR steel under narrowband and broadband loading. The method is being compared with the method proposed by Kihl and Sarkani [14]. The verification is being done also with the use of experimental results * Corresponding author. Tel.: +48 77 400 83 99; fax: +48 77 449 99 06. E-mail address:[email protected]

Adam Niesłony, Michał Böhm

2

and calculations performed by Kihl and Sarkani for cruciform shaped welded specimens. The proposed calculation procedure can be used for narrowband as well as for broadband loading characteristics, independently from the spectral method for determination of the PDF of amplitudes. Nomenclature A and φ K(σm, P) G( f ) H(f) λ σ(t) σa σm σmax σmin σ’f Δσ p(σa) R Re Rm αk t Tcal Texp

i

scale functions, mean stress compensation coefficient, power spectral density of a centered stress course, frequency response function, damping factor, stress history, stress amplitude, mean stress, maximum stress, minimum stress, fatigue strength coefficient, stress range, stress amplitude probability density function, stress ratio, yield strength, tensile strength, material-dependent parameter for the Kwofie method, time, calculated fatigue life, experimental fatigue life, moments obtained from power spectral density for i=0,..,4 ,

 

variance, coefficient calculated with the use of spectral moments, w, α, β factors calculated for the Zhao-Baker probability density function, C, m constants from the Basquin curve, b weight function dependent from the PSD, νp factor calculated for the Benasciutti-Tovo probability density function. The mean stress value used in the process of fatigue life assessment is presented as the static component of the stress history according to the formula: T

1  (t )dt . T  T

 m  lim



(1)

0

For the constant amplitude loading the mean stress value is usually defined as the algebraic mean of the maximum and minimum stress value in one loading cycle. Following that, some basic formulas can be presented:  Stress range

Adam Niesłony, Michał Böhm

   max   min ,

3

(2)

where σmax and σmin are respectively maximum and minimum stress.  Stress amplitude a 

 max   min 2

.

 Mean stress    min .  m  max 2

(3)

(4)

 Stress ratio R

 min .  max

(5)

All these parameters describe the stress course as shown in Fig.1a. Some possible values of the stress ratio R are presented in Fig.1b.

1.1. Mean stress correction methodology for time-domain The mean stress effect is still a problem in regards to fatigue life assessment in the time domain for variable loading. Nevertheless there are existing solutions to this problem which refer to the transformation of stress amplitudes with the use of linear and nonlinear mean stress compensation models. As presented by Łagoda et al [15] three paths can be taken while calculating the fatigue life:  without compensating for the mean stress, not preferred in most practical cases, usually gives an overestimation of the lifetime,  transformation of stress amplitude in regards to each stress cycle and its mean stress value after cycle counting,  transformation of stress history with global mean stress value before cycle counting. Following the third path we can transform the stress history with the appropriate formula:

 T (t )   (t )   m  K ( m , P) ,

(6)

where: [(t) − m] is the fluctuating part of the loading history; K(σm, P) is the transformation coefficient that can be calculated out of majority compensation models for mean stress m and material parameter P, for example: For Goodman’s model [16] KGo 

1 1

m Rm

,

(7)

Adam Niesłony, Michał Böhm

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For Morrow’s model [17] KM 

1

 1 m 'f

,

(8)

For Gerber’s model [18] KGe 

1   1   m   Rm 

2

,

(9)

For Kwofie’s model [19] KK 

1 ,  m   exp    K  Rm  

(10)

For Niesłony-Böhm model [12]  1  R   aN , R  1   aN , R , K NB  1   m    aN,R 2  R 1

(11)

where: KGo, KM, KGe, KK, KNB – coefficients determined on the basis of appropriate models, Rm – tensile strength, σ’f – fatigue strength coefficient, α – material-dependent parameter for the model proposed by Kwofie, aN, R=1 and aN, R – stress amplitudes read off from S-N curves for alternating and for a chosen stress ratio. Łagoda et al in the paper [15] claim that the last two paths are equivalent in terms of fatigue calculations. Each path requires the damage accumulation procedure with chosen hypothesis e.g. Palmgren-Miners or Haibachs [20]. Then the final stage of the calculation algorithm will be the fatigue life estimation. In the case of uniaxial fatigue these procedures will be good enough. For the case of multiaxial fatigue we have to add an extra step which will be the selection of an multiaxial fatigue failure criterion with mean stress compensation inside [21–24]. The chosen criterion will allow us to obtain an equivalent uniaxial stress history out of a group of stress histories which define the multiaxial stress state.

1.2. Mean stress correction methodology for frequency-domain Going through the literature there are very few solutions regarding the problem of mean stress compensation in the frequency domain. The first is proposed by Kihl and Sarkani [14]. They were analysing the effect of mean stress on the fatigue life estimation of welded cruciform shaped steel joints. They have tested their specimens in cyclic and random loading conditions with zero and non-zero mean stress values. The authors have derived a formula for fatigue life calculation. It calculates the expected number of cycles to initiation of fatigue cracks in the case of narrow-band Gaussian random load:

Adam Niesłony, Michał Böhm

 m N cal  1   Rm

   

B

2 

B 2

A ,  B  1    2

5

(12)

where: Ncal – number of cycles to fatigue crack initiation, A and B – life axis and slope of the constant amplitude SN curve,  – is the variance of the stress history, Γ() – is the gamma function, σm – global mean stress value of the random stress history, Rm – tensile strength. It is easy to notice that in the formula (12) the part being responsible for taking into account the mean value is (1 – m/Rm)–B, which modifies the cycle number till the initiation of the fatigue crack, is determined by the well-known narrow-band formula [25]. The second approach that we can find in the literature is presented by Niesłony and Böhm [13]. They have proposed a simple yet reliable calculation procedure for the frequency domain that operates directly on the power spectral density function. The approach as presented in their paper proposes the transformation of a zero mean stress PSD with the use of linear and nonlinear compensation models K(σm, P). The assumptions for this method have their underlying in the stochastic theory, presented for example by Bendat and Piersol for PSD transformation using linear objects and the use of the frequency response function H(f) [26]: 2

G y ( f )  H ( f ) Gx ( f ),

(13)

where: Gx(f) and Gy(f) - respectively PSD input and output. The formula for the transformed PSD presented by Niesłony and Böhm takes the form of:

GT ( f )  K ( m , P)2 G ( f ),

(14)

where: G( f ) is the power spectral density of a centered stress course [σ(t) − σm]. It is important to mention that some mean stress compensation models are working in such a form:

 T (t )   a  K ( a , m , P) ,

(15)

Due to the fact that σa appears in some models inside the coefficient K e.g. Smith-Watson-Topper, it is not possible to perform a linear transformation for the power spectral density based on this models. The main advantage in regards to Kihl and Sarkani method is the direct transformation of the PSD, which allows us to choose any fatigue calculation formula in the later stages of fatigue life assessment. After the transformation it is needed to calculate the probability density of amplitudes. 2. Experimental research An experimental investigation of fatigue life has been made for the S355JR steel. The tests have been performed on the test stand SHM250 for uniaxial fatigue tests for the case of tension-compression. The tests have been performed for 12 smooth cylindrically shaped test samples shown in Fig. 2. The basic strength properties of the tested steel have been presented in Table 1. The aim of the tests was the obtaining of results for narrowband and broadband signals to prove, that the method proposed by the authors can be used for both cases. The authors have

Adam Niesłony, Michał Böhm

6

generated two signals that have been uploaded to the test stand control system. Both signals had a normal distribution. The first signal has a narrowband signal characteristic with a dominating frequency of 13Hz as shown in Fig. 3a. The second signal has a broadband signal characteristic with the frequency band from 5 to 25Hz as shown in Fig. 3b. The obtained experimental results for narrowband and broadband signals are presented in Fig.4 in the form of S-N curves. The exact values used for the plots are presented in Table 2. The authors have also analyzed experimental results provided by Kihl and Sarkani [14]. They have tested the HSLA-80 cruciform welded test specimens under constant and random loading conditions. The shape and dimensions of the specimens are presented in Fig.5. The experimental fatigue results for constant amplitude tests for two stress ratios are presented in Fig.6. Data used for the plots are presented in Table 3. The random tests have been performed using peaks and troughs distributed according to a Rayleigh probability distribution what is an equivalent to loading scenario with a stationary narrowband Gaussian signal.

3. Fatigue life assessment The proposed method has not been applied and verified yet on the basis of experimental results. Therefore the authors have calculated the fatigue life of the S355JR steel using their algorithm for spectral method as presented in Fig.8. For the PSD transformation the Morrow correction coefficient KM has been chosen. Due to the fact that the correction is performed with the use of fatigue strength coefficient, the Morrow model is not limited to a certain group of materials as it is in the case of many mean stress correction models. The calculations required the selection of a model for estimation of probability density function of amplitudes. The authors have tested four models for estimation of PDF of rainflow amplitudes that are being frequently used in spectral method: Rayleigh, Dirlik, ZhaoBaker, Benasciutti-Tovo, and Lalanne [27–32] Rayleigh p( a ) 

  a2  a , exp  2 0  0  

(16)

where  0 - zero order PSD moment equal to the variance  obtained from PSD of transformed stresses according to the following equations 



 k  GT ( f ) f k df

for k = 0, 1, 2, 3

(17)

0

Dirlik Z Z 2 Z 2   K1 K 4 K 2  Z 2R 2  p( a )   e  e  K 3  Ze 2 ,  R2 2 0  K 4  

1

where K1, K2, K3, K4, and Z are model coefficient described in details for example in [28,33]

(18)

Adam Niesłony, Michał Böhm

7

Zhao-Baker





  2 p( a )  w     1 exp   b  1  w exp  2 

 ,  

(19)

where , w, α, β are factors described as: 2  a ,

(20)



1 

w

1

,

(21)

2  1  1 1         8  7 ,

(22)

1.1;   0.9    1 . 1  9   0 . 9 ;   0.9 

 

 

(23)

2 ,  0 4

(24)

Benasciutti-Tovo p( a )  b

   a2  a exp  2 0 

2 2      1  b   a exp   a  0  , 2    2  0   

(25)

where b - weight function dependent from the PSD. Lalanne

p( a ) 

erf (u ) 

  a2 1  2 exp  2 1   2 20  0



2

u

e  

t 2

dt .



   a    a 2   a   1  erf  exp   2 0  2 0  2     2 0 1  





    ,  

(26)

(27)

0

These four PDF of amplitudes have been shown against the rainflow amplitude distribution of the signal. The obtained PDF have been calculated for the case with transformation of the PSD and presented in Fig. 9 for the

Adam Niesłony, Michał Böhm

8

narrowband signal and in Fig.10 for broadband signal. The obtained PDF values have been used to calculate the fatigue life with the use of the Palmgren-Miner hypothesis with the formula [25]: 1

Tcal 



M

 0

p( a ) d a N ( a )

,

(28)

where: p(σa) – probability density function, N(σa) – number of cycles calculated from the S-N curve for a given stress amplitude σa, M+ the expected number of peaks within a unit of time

M 

4 . 2

(29)

As it can be noticed in the Figs. 9 and 10, most of the formulas for PDF calculation are describing the rainflow amplitude distribution in a reliable way. The transformation of the PSD due to mean stress has changed the distribution of the stress amplitudes. We can notice that the Rayleigh function is overestimating for the broadband characteristic. The best description is obtained for both cases with the Dirlik and Zhao-Baker PDF. Because of that only the most known Dirlik model for probability density function of amplitudes has been used in the calculation process. The obtained fatigue life results for the used algorithm have been compared with the experimental results and have been presented in Fig. 11. As it can be noticed all results are within the acceptable scater bandwith of 3. It seems that the role of the proper selection of a mean stress compensation model has to be analyzed. An analysis of use of some of the mean stress compensation models has been performed for both the narrowband and broadband cases. For this purpose the authors have also used their mean stress compensation model presented in paper [12]. The fatigue results have been presented in Fig.12. 4. Comparison with experimental results The method presented in present paper has been compared with the experimental and fatigue calculations that have been presented in the paper by Kihl and Sarkani [14]. The calculations have been performed for all stress ratio cases. The experimental results together with calculations have been presented in Table 4. The calculation results have been compared with experimental and presented in Fig.13. The compensation of mean stress has been performed with the same compensation models as shown in the earlier section. To fully compare the methods a comparison of the values of the mean square scatter error has been performed. The root mean square scatter error has been calculated with the use of the formula proposed by Walat and Łagoda [34]: n

ERMS 

 log

2

i 1

n

N expi N cali

,

(29)

where Nexp and Ncal are the experimental and calculated number of cycles. Finally we obtain the mean deviation from the expected number of cycle to failure value with the use of the formula:

Adam Niesłony, Michał Böhm

N RMS  10ERMS .

9

(30)

The obtained results have been presented in Table 5. The lower the value, the better the calculation result against the experimental values. We can notice that the results with the use of the proposed method give very close RMS error results in comparison to the original Kihl-Sarkani calculation results.

5. Observations and conclusions

It is undoubtedly that the proper mean stress compensation in the process of assessment of fatigue life for variable amplitude loading using the spectral method is crucial. It can be noticed that most of the standard spectrums have a notable nonzero mean stress value. The algorithm can be smoothly implemented into existing spectral methods for fatigue life prediction. It operates directly on the power spectral density of a signal, therefore can be used independent from model for PDF estimation. The theoretical assumed universal application has been confirmed for both narrowband and broadband signal stress characteristics. The method has been tested on experimental results performed by Kihl and Sarkani for cruciform shaped welded test samples. It can be summarize that:  the proposed method for correction of mean stress in the frequency domain has been successful verified,  it can be used for both narrow and broadband signals,  proposed method operate directly on the power spectral density of the signal – calculation are performed in frequency domain,  not every probability density function describes the amplitude distribution in a correct way in comparison with the rainflow amplitude distribution,  nevertheless every probability density function that has been used for calculations gave satisfying fatigue calculation results,  the mean stress compensation model has a large influence on the fatigue life results,  a comparison of the Kihl-Sarkani methods and of the authors gave satisfying results,  the mean deviation from the expected number of cycle to failure for both methods gave similar results. Acknowledgements The Project was financed from a Grant by National Science Centre (Decision No. DEC-2012/05/B/ST8/02520). References [1] [2] [3] [4] [5] [6] [7]

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[34] Walat K, Łagoda T. Lifetime of semi-ductile materials through the critical plane approach. International Journal of Fatigue 2014;67:73–7. doi:10.1016/j.ijfatigue.2013.11.019. [35] Wahl NK. Spectrum Fatigue Lifetime and Residual Strength for Fiberglass Laminates. PhD Thesis. Montana State University, 2001.

Figure captions: Fig. 1. (a) basic parameters describing the constant amplitude stress course, (b) some possible values of the stress ratio R [35]. Fig. 2. Shape and dimensions of the specimens used in the experiments for the S355JR steel. Fig. 3. (a) PSD of the narrowband stress course; (b) PSD of the broadband stress course Fig. 4. Experimental fatigue results for (a) narrowband load; (b) broadband load. Fig.5. Shape and dimensions of the specimens used in the experiments by Kihl and Sarkani [14]. Fig.6. Experimental fatigue tests for constant amplitude tests of HSLA-80 steel (a) R=-1 (b) R=0 by Kihl and Sarkani [14]. Fig. 7. Fatigue life assessment algorithm for the mean stress compensation in the frequency domain, calculations for the uniaxial case. Fig. 8. PDF of amplitudes of the narrowband stress course with PSD transformation. Fig. 9. PDF of amplitudes of the broadband stress course with PSD transformation. Fig. 10. Comparison of experimental results with calculation results (a) narrowband; (b) broadband. Fig. 11. Comparison of experimental results with calculation results (a) narrowband; (b) broadband with the use of different mean stress compensation models Fig. 12. Comparison of experimental results with calculation results for the fatigue tests performed by Kihl and Sarkani (a) R= (b) R=5 (c) R=0.33 (d) R=0.66

Adam Niesłony, Michał Böhm

a

b

Fig. 1. (a) basic parameters describing the constant amplitude stress course, (b) some possible values of the stress ratio [35].

Fig. 2. Shape and dimensions of the specimens used in the experiments for the S355JR steel.

12

Adam Niesłony, Michał Böhm

a

b

Fig. 3. (a) PSD of the narrowband stress course; (b) PSD of the broadband stress course.

b





a

Fig. 4. Experimental fatigue results for (a) broadband load; (b) narrowband load.

13

Adam Niesłony, Michał Böhm

Fig.5. Shape and dimensions of the specimens used in the experiments by Kihl and Sarkani [14].

b





a

Fig.6. Experimental fatigue tests results for constant amplitude tests of HSLA-80 steel (a) R=-1 (b) R=0 by Kihl and Sarkani [14].

14

Adam Niesłony, Michał Böhm

Fig.7. Fatigue life assessment algorithm for the mean stress compensation in the frequency domain, caluclations for the uniaxial case.

15





Adam Niesłony, Michał Böhm







Fig. 8. PDF of amplitudes of the narrowband stress course with PSD transformation.

 Fig. 9. PDF of amplitudes of the broadband stress course with PSD transformation.

16

Adam Niesłony, Michał Böhm

a

b

Fig. 10. Comparison of experimental results with calculation results (a) narrowband; (b) broadband.

a

b

Fig. 11. Comparison of experimental results with calculation results (a) narrowband; (b) broadband with the use of different mean stress compensation models

17

Adam Niesłony, Michał Böhm

18

b

a      

c

d

Fig. 12. Comparison of experimental results with calculation results for the fatigue tests performed by Kihl and Sarkani (a) R=

(b) R=5 (c) R=0.33 (d) R=0.66

Table captions: Table 1. Mechanical properties of the S355JR steel Table 2. Data used in the stress fatigue diagrams for the S355JR steel Table 3. Data used in the stress fatigue diagrams for the HSLA-80 steel for constant amplitude tests Table 4. Experimental and calculation results for Kihl and Sarkanis fatigue tests. Table 5. Root mean square scatter error for the experimental results.

Adam Niesłony, Michał Böhm

21

Table 1. Mechanical properties of the S355JR steel Re , MPa

Rm , MPa

Elongation , %

Necking , %

E , GPa

ν

394

611

20

51

213

0,31

Table 2. Data used in the stress fatigue diagrams for the S355JR steel i

Steel S355JR R = 0 broadband

1 2 3 4 5 6

298 298 298 278 278 278

amaxi

Ni 131820 159120 59280 351780 400920 486720

298 298 298 278 278 278

R = 0 narrowband Ni 124800 78000 152100 390000 540540 324792

amaxi

Adam Niesłony, Michał Böhm

Table 3. Data used in stress fatigue diagrams for the HSLA-80 steel for constant amplitude tests i

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Steel HSLA-80, Kihl and Sarkani [14] R=0 R = 1 Ni Ni ai ai 69 12634300 69 958300 69 2903700 69 1323200 69 26195300 69 11182800 69 16805600 69 2451800 69 25000000 103 281500 83 775600 103 255200 83 3732900 103 236600 83 1118600 103 318100 83 810800 207 25500 83 1392000 207 21300 103 572000 207 60800 103 779500 207 22800 103 515000 103 229200 103 1071600 207 66500 207 61900 207 70600 207 82800 207 79100 310 14500 310 14800 310 16300 310 23200 310 18000

22

Adam Niesłony, Michał Böhm

23

Table 4. Experimental and calculation results for Kihl and Sarkanis fatigue tests. σamax MPa

σm MPa

Nexp, cycles

NMorrow, cycles

NGerber, cycles

NGoodman, cycles

NKwofie, cycles

NNiesłony-Böhm, cycles

34.4

-34.4

3131400

5559159

4706977

5567915

5151129

7403787

NKihlSarkani, cycles 7245200

34.4

-34.4

35742100

5559159

4706977

5567915

5151129

7403787

7245200

34.4

-34.4

16763400

5559159

4706977

5567915

5151129

7403787

7245200

34.4

-34.4

50016900

5559159

4706977

5567915

5151129

7403787

7245200

68.9

-103.4

6712500

806619

475406

810101

655682

2482309

3238300

68.9

-103.4

4191400

806619

475406

810101

655682

2482309

3238300

68.9

-103.4

2326800

806619

475406

810101

655682

2482309

3238300

68.9

-103.4

1679800

806619

475406

810101

655682

2482309

3238300

68.9

137.8

263100

250605

447574

248604

370072

259552

225100

68.9

137.8

183100

250605

447574

248604

370072

259552

225100

68.9

137.8

245200

250605

447574

248604

370072

259552

225100

68.9

137.8

217500

250605

447574

248604

370072

259552

225100

68.9

275.7

72500

111487

287020

109735

268027

153824

144500

68.9

275.7

161700

111487

287020

109735

268027

153824

144 500

68.9

275.7

123100

111487

287020

109735

268027

153824

144 500

68.9

275.7

301700

111487

287020

109735

268027

153824

144 500

Table 5. Root mean square scatter error for the experimental results. R

NRMSMorrow, cycle

NRMSGerber, cycle

NRMSGoodman, cycle

NRMSKwofie, cycle

NRMSNiesłony-Böhm, cycle

NRMSKihl, cycle

4.8

5.44

4.79

5.08

3.95

4.00

5

4.43

7.33

4.41

5.39

1.8

1.7

0.33

1.19

2.02

1.18

1.67

1.22

1.15

0.66

1.78

2.36

1.79

2.23

1.68

1.67