Procedia Engineering 101 ( 2015 ) 52 – 60

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Czech Society for Mechanics

doi: 10.1016/j.proeng.2015.02.008

ScienceDirect

Available online at www.sciencedirect.com

3rd International Conference on Material and Component Performance under Variable Amplitude Loading, VAL2015

Strain-based multiaxial fatigue life evaluation using spectral method

Michał Böhma,*, Adam Niesłonya

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

Abstract

To describe the impact of multiaxial service loads a power spectral density matrix of the strain tensor components has been

taken into account. To determine the fatigue life some selected multiaxial fatigue failure criteria have been used. The main focus

of this paper is set on the comparison of the cycle counting method and the spectral method. An algorithm used to calculate the

fatigue life in the time domain as well as in the frequency domain has been introduced. The proposed algorithm has been verified

on the basis of fatigue tests of cruciform specimens made out of the S355J2WP steel.

© 2015 The Authors. Published by Elsevier Ltd.

Peer-review under responsibility of the Czech Society for Mechanics.

Keywords:strain, spectral method, multiaxial fatigue, rain flow;

1. Introduction

The literature presents two methods of determining the fatigue life of operating loads – cycle counting method [1],

where calculations proceed in the time domain, and the spectral method [2-3], where calculations proceed in the

frequency domain. The primary difference in these methods is the way of determining the stress or strain amplitude

distribution. In the cycle counting method the distribution is determined by means of specialized algorithms that while

analyzing the time history of stress or strain courses determinates distribution of their amplitudes. Such is for example

the well know rainflow method. However in the spectral method the amplitude distribution parameters are being

estimated as well as the correction factors that are taking into account the impact of the frequency wide spectrum [4].

The cycle counting method due to the large simplicity and ease of modification is much more frequently used than the

spectral method. In some cases, however, it is desirable to present the structure of the frequency of a load or explicitly

define the load in the frequency domain [5]. Generally for this purpose the Power Spectral Density (PSD) function of

* Corresponding author. Tel.: +48 77 400 84 18; fax: +48 77 449 99 06.

E-mail address:m.bohm@po.opole.pl

© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Czech Society for Mechanics

53 Michał Böhm and Adam Niesłony / Procedia Engineering 101 ( 2015 ) 52 – 60

the stress or strain is being used. This function is used in the spectral methods of determination of fatigue life to

describe the nature of the load. For this purpose the PSD moments are being used and on their basis we determine,

among others, the variance of the random loading history, the expected number of passes through the zero level, the

coefficients of irregularities, etc. In the construction and operations of machines it is rare to find construction elements

that are working in simple loading states (tension-compression, torsion at stress ratio R = −1, etc.). That’s why the

multiaxial fatigue criteria for random loading are being proposed, which allow to transform the spatial condition of

the load to an equivalent simple state [3-11]. Parameter thus obtained is used to determine the fatigue strength, using

methods already verified in the uniaxial states of loading. The aim of this work is the verification of the spectral method

algorithm based on comparison of calculation results with those obtained with the cycle counting method and received

during the experiment. For this, calculations were based on previous works [12, 13]:

� rainflow algorithm (cycle counting method),

� damage accumulation according to Palmgren-Miner rule (cycle counting method),

� Miles formula for predicting the fatigue life (spectral method),

� λ factor taking into account the width of the spectrum (spectral method),

� strain multiaxial fatigue failure criteria (cycle counting and spectral method).

Nomenclature

n factor taking into account the width of the spectrum according to Wirsching and Light, g0(t), g90(t), g45(t) time courses of the strain coming from full strain gage rosette, g(t) strain tensor, gxx(t), gyy(t), gzz(t), gxy(t) components of the strain tensor, gj(t) strain normal to the critical plane, gjs(t) strain tangent component on the critical plane, geq(t) equivalent strain, gai strain amplitude, gae, gap elastic and plastic part of the strain amplitude,

ε’f fatigue ductility coefficient,

E Young’s modulus,

ν Poisson’s ratio,

G(f) power spectral density function,

Gii,ii(f) components of the power spectral density matrix where i = x, y, z,

s vector tangent to the critical plane, j vector normal to the critical plane,

ijs(t) shear stress,

dg, kg, and q constants defining special forms of the criteria,

To observation time,

Trf fatigue life counted by means of the cycle counting method,

TSP fatigue life counted by means of the spectral method,

ni number of cycles with the given amplitude,

Nfi number of cycles calculated numerically from the fatigue characteristic described by the

Manson-Coffin-Basquin formula,

σ’f fatigue strength coefficient,

b fatigue strength exponent,

c fatigue ductility exponent,

M+ expected number of cycles per unit time,

p(ga) amplitude probability density distribution,

Nf(ga) function returning the cycle number from the strain fatigue characteristic,

rX,Y correlation coefficient between X a Y,

54 Michał Böhm and Adam Niesłony / Procedia Engineering 101 ( 2015 ) 52 – 60

oX,Y covariance of variables X and Y, oX , oY variance of variables X and Y,

s , s~ respectively mean value and standard deviation of the scatter parameter scal,exp of the

calculated and experimental results.

2. Experimental research

The tests were conducted using cruciform specimens presented in Fig. 1 made out of the S355J2WP steel. The tests

were conducted by Dr. Będkowski on the MZPK-300L stand in the Department of Mechanics and Machine Design at

the Opole University of Technology and presented in [14-16]. The chemical composition of the material and basic

material constants are given in Table 1 and Table 2.

Table 1. Chemical composition of the S355J2WP steel.

Element C Mn Si P S Cr Cu Ni Fe

% 0.115 0.71 0.41 0.082 0.028 0.81 0.30 0.50 rest

Table 2. The basic material constants (static and fatigue) of the S355J2WP steel.

p E, GPa u’f, MPa b g’

f c

0.29 215 994 -0.095 0.244 -0.464

The cruciform specimens were subjected to a biaxial random Gaussian load. Deformation on the surface in the

middle of the sample was registered with a strain gauge rosette. Due to the long duration of the experiment only

selected parts of the strain histories were registered in intervals of between 130/160 seconds. Number of recorded

strain history blocks was ranged from 2 to 15 depending on the load level and durability of the specimens. Also, the

shapes of the PSD (frequency max value) were adapted to the loading level to avoid thermal effects during the hi-load

tests and ensure the accomplishment of the tests in an acceptable time for low-load tests.

Fig. 1. Shape of the cruciform specimen and its basic dimensions.

3. Computational algorithm

Fig. 2 shows a computation algorithm for predicting the fatigue life using the cycle counting method as well as the

spectral method. The conducted calculations were divided to three extracted steps shown in the algorithm by dotted

lines.

55 Michał Böhm and Adam Niesłony / Procedia Engineering 101 ( 2015 ) 52 – 60

Fig. 2. Computation algorithm of the cycle counting and spectral method [16].

3.1. Values characterizing the load

It was assumed that in the calculation, the values that describe the load will be non-zero components of the strain

tensor. A formula has been defined:

)](),(),(),([)( ttttt xyzzyyxx gggg?ε . (1)

The components are based on deformation g0(t), g90(t) (along the arms of the specimen) and g45(t) (deformation

under the angle of 45o to g0(t), g90(t) obtained directly from the strain gauge rosette). Following equations were used

for calculating the particular strain tensor components

] _] _)()(5.0)()(

)()(1

)(

)()(

)()(

90045

900

90

0

tttt

ttt

tt

tt

xy

zz

yy

xx

ggggggp

pggggg

-/?-/

/???

. (2)

Fragment of the strain time course is shown in Fig.3.

Fig. 3. Fragment of the strain history.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3

t, s

g(t)

gxx

(t)

gyy

(t)

gzz

(t)

gxy

(t)

56 Michał Böhm and Adam Niesłony / Procedia Engineering 101 ( 2015 ) 52 – 60

The spectral method of fatigue life determination uses to describe the load by the spectral power density matrix of

the deformation

ÙÙÙÙÙ

Ú

×

ÈÈÈÈÈ

É

Ç?

)()()()(

)()()()(

)()()()(

)()()()(

)(

,,,,

,,,,

,,,,

,,,,

fGfGfGfG

fGfGfGfG

fGfGfGfG

fGfGfGfG

f

xyxyzzxyyyxyxxxy

xyzzzzzzyyzzxxzz

xyyyzzyyyyyyxxyy

xyxxzzxxyyxxxxxx

G . (3)

The components of the matrix were determined by the Welch method [17] with additional segmental averaging

estimators that give smoother PSDs.

3.2. Cycle counting method

For the purpose of calculations the known multiaxial fatigue criteria based on the parameter of deformation

proposed by Macha [18] and further developed inter alia by Ogonowski and Łagoda [19, 20] has been used. These

criteria can be derived from the following assumptions: ‚ fatigue life is dependent on the linear combination of strain normal to the critical plane gj(t) and strain tangent

component on the critical plane gjs(t) in the s direction, on the critical plane of normal j ,

‚ direction s on the critical plane coincides with the average direction of the maximum shear )}({max tst

ji ,

‚ In the boundary state the material is defined by the maximum linear value of the combination of the gj(t) and

gjs(t) strains, which in the multiaxial random loading conditions satisfies the equation:

} ’ qtktb st

?- )()(max jgjg gg (4)

where: bg, kg, and q are constants defining the particular form of the criteria. For the calculation purposes three forms

of the criteria which were used to determine the equivalent strain have been chosen:

‚ Criterion of maximum normal strain [19] (K1),

)()( tteq jgg ? . (5)

‚ Criterion of maximum normal and shear strain parameter in the plane of maximum normal strain [19] (K2),

)()(2

2)( ttt seq jj ggg -? . (6)

‚ Criterion of maximum normal and shear strain parameter in the plane of maximum shear strain [19] (K3),

)(1

)1(2)()( t

btbt seq jgjg gp

pgg /-/-? , (7)

where for the S355J2WP steel the bg = 1.68 and results from the ratio of fatigue limits for tension-compression and

torsion [17]. The position of the critical plane has been determined by the variance method [21]. It is based on finding

such a position of the critical plane (position of the vectors j and s ), for which the strain variance geq(t) reaches the

maximum value. The history of the equivalent deformation geq(t) is subjected to cycle counting using the rain flow

algorithm. Fatigue life was determined using the hypothesis of damage accumulation according to Palmgren-Miner

rule.

57 Michał Böhm and Adam Niesłony / Procedia Engineering 101 ( 2015 ) 52 – 60

Â?i i

if

orfn

NTT , (8)

where To – is the observation time, ni – number of cycles with the amplitude εai and Nfi is the number of cycles

calculated numerically from the fatigue characteristic described by the Manson-Coffin-Basquin formula

* + * +cffb

f

f

apaea NNE

22 guggg |-|?-? . (9)

3.3. Spectral method

In this method the switch from the multiaxial state to an equivalent simple uniaxial state is being done in the

frequency domain by means of criteria (K1)-(K3). For the fixed position of critical plane, the criteria are linear, which

allows to determine the PSD of the equivalent deformation directly from the PSD matrix (3). This operation has been

shown in many earlier papers [11, 13, 22]. Examples of power spectral density functions of the equivalent deformation

process are shown in Fig. 4. It is easy to notice that we have obtained deformation with a broad frequency spectrum.

To determine the fatigue life in terms of the spectral method a formula has been used [2, 5]:

Т-?0

)(

)(

1

aaf

a

SP

dN

pM

T

gggn

, (10)

where the λ coefficient includes the impact of broad frequency spectrum on the fatigue life [5, 14], M+ is the expected

number of cycles in unit time, p(ga) is the amplitude probability density distribution, and Nf(ga) is a function giving

back the cycle number of the fatigue characteristic (9).

Fig. 4. Examples of power spectral density functions of the equivalent deformation process obtained by using the (K1) criteria

for 6 chosen cruciform specimens.

4. Results and conclusions

The calculated fatigue life by means of the cycle counting and spectral method with three chosen criteria has been

compared with the experimental results in Fig. 5. It is easy to notice that the best results were obtained for the criteria

based on the normal strain to the critical plane (K1) and (K2) (Fig. 5a and 5b). Because of the small result differences

between the durability Trf and Tsp it has been proposed, beside the graphic presentation of results, a comparison of the

correlation and scatter coefficients of the results in Table 3.

0 10 20 30 40 50 60 70 800

0.5

1

1.5

2

2.5

3

3.5x 10

-8

f, Hz

G(f

), 1

/Hz

specimen 2

specimen 3

specimen 5

specimen 8

specimen 15

58 Michał Böhm and Adam Niesłony / Procedia Engineering 101 ( 2015 ) 52 – 60

Fig. 5. Comparison of the calculated durability according to the three criteria (a) (K1), (b) (K2), (c) (K3) with the experimental results.

The correlation coefficients between the calculated X = log(Tcal) and experimental Y = log(Texp) durability have

been calculated from the formula:

YX

YXYXr oo

o ,, ? , (11)

where oX,Y is the covariance of variables X and Y, where oX and oY are the appropriate variances. The analysis of the

scatter of data was conducted with a mean value s and a standard deviation s~ of the s parameter defined as:

ÕÕÖÔ

ÄÄÅÃ?

exp

calcal

T

Ts logexp, . (12)

It’s easy to notice that for a perfect match of the calculated and experimental durability scal,exp = 0. The study

indicates the following conclusions: Table 3. Values of parameters for assessing the accuracy of methods of determining the fatigue life

rX,Y

X = log(Trf)

Y = log(Texp)

rX,Y

X = log(Tsp)

Y = log(Texp)

exp,~

rfs exp,~

sps exp,ˆrfs exp,

ˆsps

K1 0.9193 0.8414 0.1711 0.2299 0.2994 0.1612

a) b)

c)

59 Michał Böhm and Adam Niesłony / Procedia Engineering 101 ( 2015 ) 52 – 60

1. The spectral method of fatigue life assessment gives comparable results with the cycle counting method, but it

should be noticed that a slightly bigger scatter appears for results, it’s probably caused by the different frequency

characteristics of the courses in relation to the fatigue tests.

2. The comparison of the calculated results with the obtained experimentally has shown, that for the S355J2WP steel

we receive the best results when using the criterion of maximum normal and shear strain parameter in the plane of

maximum normal strain (K2).

Acknowledgements

The Project was financed from a Grant by National Science Centre (Decision No. 2013/09/N/ST8/04332).

References

[1] T. Łagoda, E. Macha, A. Niesłony,: Comparison of the rain flow algorithm and the spectral method for fatigue life determination under

uniaxial and multiaxial random loading, Journal of ASTM International, 1 (2004) 544-556

[2] A. Niesłony, E. Macha: Spectral Method in Multiaxial Random Fatigue, Springer-Verlag, Berlin Heidelberg 2007

[3] C. Braccesi, F. Cianetti, L. Tomassini, Random fatigue. A new frequency domain criterion for the damageevaluation of mechanical

components, International Journal of Fatigue 70 (2014) 417-427

[4] P.H. Wirsching, M.C. Light, Fatigue under wide band random stresses using rainflow method, Journal of Structural Division, 106 (1980)

1593–1607

[5] A. Niesłony: Comparison of some selected multiaxial fatigue failure criteria dedicated for spectral method, Journal of Theoretical and Applied

Mechanics 48 (2010) 233-254

[6] A. Carpinteri, R. Brighenti, E. Macha, A. Spagnoli: Expected principal stress directions under multiaxial random loading. Part II: numerical

simulation and experimental assessment through the weight function method, International Journal of Fatigue, 21 (1999) 89-96

[7] A. Carpinteri, E. Macha, R. Brighenti, A. Spagnoli: Expected principal stress directions under multiaxial random loading. Part I: theoretical

aspects of the weight function method, International Journal of Fatigue, 21 (1999) 83-88

[8] D. Benasciutti, A. Cristofori, R. Tovo: A stress invariant based spectral method to estimate fatigue life under multiaxial random loading,

International Journal of Fatigue 33 (2011) 887–899

[9] D. Benasciutti, A. Cristofori, R. Tovo: Analogies between spectral methods and multiaxial criteria in fatigue damage evaluation, Probabilistic

Engineering Mechanics, 31 (2013) 39–45

[10] A. Carpinteri, A. Spagnoli, S. Vantadori: Reformulation in the frequency domain of a critical plane-based multiaxial fatigue criterion, Int. J.

of Fatigue, 67 (2014) 55–61

[11] A. Niesłony, M. Ruzicka, J. Papuga, A. Hodr, M. Balda, J. Svoboda: Fatigue life prediction for broad-band multiaxial loading with various

PSD curve shapes, International Journal of Fatigue 44 (2012) 74–88

[12] T. Łagoda, E. Macha, A. Niesłony,: Comparison of the rain flow algorithm and the spectral method for fatigue life determination under

uniaxial and multiaxial random loading, Journal of ASTM International, 1 (2004) 544-556

[13] A.Niesłony, M. Böhm: Application of spectral method in fatigue life assessment – determination of crack initiation, Journal of Theoretical

and Applied Mechanics, 50 (2012) 819-829

[14] W. Będkowski, Determination of critical plane and effort criterion in fatigue life evaluation for materials under multiaxial random loading.

Experimental verification based on fatigue tests of cruciform specimens, Fourth International Conference on Biaxial/Multiaxial Fatigue,

Paris, (1994) 435-447

[15] Kenmeugne, B., Weber, B., Carmet, A., & Robert, J.L. A stress-based approach for fatigue assessment under multiaxial variable amplitude

loading, in proceedings of ICBMFF5, Cracow, Poland, (1997) 557-573.

[16] W. Będkowski, E. Macha, A. Niesłony, Verification of the spectral method of fatigue life calculation on the basis of 10HNAP steel, Przegląd

Mechaniczny 12 (2004) 13–16 (in Polish).

[17] Signal Processing Toolbox, User's guide, Copyright 1988-2002 by The MathWorks, Inc., Version 6

[18] E. Macha: The spectral method of fatigue life calculation under random multiaxial loading, Physicochemical Mechanics of Materials, 32

(1996) 86-96

K2 0.9090 0.8455 0.1885 0.2262 0.2639 0.1224

K3 0.8830 0.8233 0.1953 0.2365 -0.4994 -0.4587

60 Michał Böhm and Adam Niesłony / Procedia Engineering 101 ( 2015 ) 52 – 60

[19] T. Łagoda, P. Ogonowski: Criteria of multiaxial random fatigue based on stress, strain and energy parameters of damage in the critical plane,

Mat-wiss. u. Werkstofftechnik, 36 (2005) 429-437

[20] T. Łagoda, P. Ogonowski: Fatigue life estimation of notched specimens under bending and torsion with strain energy density parameter,

Journal of Theoretical and Applied Mechanics, 45 (2007) 349-361

[21] W. Będkowski, B. Weber, E. Macha, J.-L. Robert, Comparison of Variance Method and Damage Indicator Methods for Prediction of the

Fracture Plane Orientation in Multiaxial Fatigue, in Multiaxial Fatigue and Fracture, Ed. E. Macha, W. Będkowski, T. Łagoda, ESIS

Publication 25, Elsevier – Netherlands (1999) 147-165

[22] A. Bannvillet, T. Łagoda, E. Macha, A. Niesłony, T. Palin-Luc, J.-F. Vittori, Fatigue life under non Gaussian random loading from various

models, International Journal of Fatigue, 26 (2004) 349-363