An improved method of calculating the peak stress distributionfor a broad-band random process

P E N G M I N L U a n d S H E N G J I E J I AODepartment of Road Construction Machinery, Chang’an University, Xi’an, 710064, China

Received in final form 17 April 2000

A B S T R A C T An improved method of fast fatigue life prediction under broad-band random loadingis proposed, which is based on the power spectral density of stress in the critical pointsof structures and the peak stress distribution of a stationary Gaussian random process.The improved method has higher precision than other existing approximate methodsthat are based on the peak stress distribution.

Keywords broad-band loading, fatigue life, frequency domain analysis.

N O M E N C L A T U R E erf (x)=error functionf=frequency

f0=expected rate of zero crossing with positive slopeG( f )=spectral density functionI1(x)=an approximate series expression of error functionI2(x)=an asymptotic expansion of error function

Ji=ith term of Eq. (14), i=1, 2, 3, 4m=material constant from Paris Law or stress–life curve

Mk=kth moment of spectral density functionn0=expected rate of peaks

p(x)=probability density function of xQ=chi-square functions=stress peak or level

Sh=equivalent stress rangeu(x)=unity function

Z=last term in square brackets in Eq. (5)a=irregularity factorc=incomplete gamma functionC=gamma functione=effective band widths=standard deviation

the S–N curve, da/dN−DK curve, etc.), the fatigue lifeI N T R O D U C T I O N

can be predicted quickly. Consequently, the Sh calcu-lation is very important.Fatigue is one of the most common causes of failure for

components and structures that are subjected to random According to the S–N curve or the Paris formula,along with the Palmgren–Miner linear accumulationloading. Most structures are subjected to broad-band

(BB) random loading. For a fatigue life assessment under damage rule, the equivalent stress range under randomloading can be expressed as:1–3BB loading, the most convenient method is the statistical

approach. By using the power spectral density (PSD) ofthe stress in the critical position of structures, the load

Sh=CP2

0(2s)mp(s) dsD1/m

(1)probability density function (PDF) can be obtained.From PDF, the equivalent stress range Sh under BBloading is calculated. By using experimental data (e.g. Statistical quantities for a fatigue life assessment can

© 2000 Blackwell Science Ltd. Fatigue Fract Engng Mater Struct 23, 581–586 581

582 P. M . L U a n d S . J . J I A O

be obtained through the PSD in the critical zones of and the C function is defined asstructures, as shown in Table 1.2,4

For the stationary BB loading with a Gaussian ampli- C(x)=2 P2

0y2x−1 exp(−y2) dy

tude distribution, the PDF of stress peaks can beexpressed as5,6

for x>0.Because Eq. (6) contains the transcendental function

p(s)=1

(2p)1/2s(1−a2)1/2 exp[−s2(2s2(1−a2))−1] erf (x), it is difficult to obtain an exact solution. Many

researchers have suggested approximate models, includ-ing Chaudhury and Dover,1 Kam and Dover,3 Chow

+s

2s2 a C1+erf A ss(2a−2−2)1/2BD exp A−s2

2s2B and Li,2 and Wirsching.4 The author has proposed aseries solution model7 which has more advantages than(2)any other existing approximate models. In this paper,

where the error function is defined as the series model is further improved and the new modelhas a higher precision.

erf (x)=2√p P x

0exp(−t2) dt (3)

R E V I E W O F T H E S E R I E S M O D E LBy putting The error function erf (x) versus x is shown in Fig. 1,

which can be replaced by a series expression when x isx=

as√2se

in some range. The series expression can be written as

I1(x)=2√p

exp(−x2) ∑k1

k=0

2kx2k+1

(2k+1)!!(7)Eq. (2) can be simplified to

where x<x0.p(x)=e

√2psexp C−Ax

aB2D By investigation in detail, the result shows: whenk1=9, x∏2.0, it is satisfactory that I1(x) is used to

+ex√2s

[1+erf (x)] exp C−Aexa B2D (4) replace erf (x). When x>2.0, erf (x)#1. Therefore, erf (x)

can be expressed by the following model in xµ(0, 2)(Fig. 2)7

Substituting Eq. (4) into Eq. (1) yields

Sh=2√2s Cem+2

2√pC Am+1

2 B+ a

2C Am+2

2 B+aZD1/m

erf (x)=q 2√p

exp(−x2) ∑9

k=0

2kx2k+1

(2k+1)!!x∏2

1 x>2

(8)

(5)

where

Z= P 2

0erf (x) Aex

a Bm+1

exp C−Aexa B2D d Aex

a B (6)

Table 1 Definition and expression of statistical quantities

Definition Expression

kth moment of spectral density function G( f ), Mk ∆2−2f kG( f ) df

Root-mean square stress, s √M0

Expected rate of zero crossing with (+) slope, f0 √M2/M0

Expected rate of peaks, n0 √M4/M2

Irregularity factor, a f0/n x

0 1 3 5

erf

(x)

2 4 6

1.5

1

0.5

0

Fig. 1 The error function erf (x) curve.Spectral width parameter, e √1−a2

© 2000 Blackwell Science Ltd. Fatigue Fract Engng Mater Struct 23, 581–586

A N I M P R O V E D M E T H O D O F C A L C U L AT I N G T H E P E A K S T R E S S D I S T R I B U T I O N F O R A B R OA D - B A N D R A N D O M P R O C E S S 583

new model, two series expansions are used to expresserf (x), i.e.: when xµ[0, 2), I1(x) is used to replace erf (x),and when xµ[2, 2), another asymptotic expansion isused to replace erf (x). This asymptotic expansion can beexpressed as follows

I2(x)=1−exp(−x2)1

√px C1+ ∑k1

k=1

(−1)k(2k−1)!!(2x2)k D

(11)

The curves of I2(x)/erf (x) versus x with different k1

are shown in Fig. 3. It can be seen that, when x�1.5,erf (x) can be expressed by I2(x), and when k1=1, theresult is optimum (Fig. 4).

The expression of erf (x) can be written as

x

0

I 1 (

x,k 1

=9)

,erf

(x)

2 4 6

1

0.6

0.4

0

0.8

0.2

erf (x)

I1 (x,k1=9)

Fig. 2 Curves of the series expansion I1(x, k1=9) and the errorfunction erf (x).

Using Eq. (8), the solution of the equivalent stress rangecan be derived as

Sh=2√2s Gem+2

√pC Am+1

2 B+ a

2C Am+2

2 B×C1+Qx2 A8e2

a2 , m+2BD+

em+2

√p∑9

k=0

2ka2k+2

(2k+1)!!C A2k+m+3

2 B×C1−Qx2 A 8

a2 , m+2k+3BDH1/m

(9) x

0.5

I 2 (

x,k 1

)/er

f (x

)

1 2.5 4

4

0

–2

–5

2

–4

k1=1

2 3.51.5 3

5

1

–1

3

–3

k1=3

k1=2

k1=4

Fig. 3 Curves of the ratio between the asymptotic expansionwhere Qx2(x2a , n) is a chi-square function, in which n is

I2(x, k1) and the error function erf (x).the degree of x2. The term Qx2(x2

a , n) can be calculatedby using the incomplete gamma function. It is definedas

c(v, y2)=2 P2

yx2w−1 exp(−x2) dx

=C(w)Qx2(x2a=2y2, n=2w) (10)

A digital analysis shows that, when eµ(0, 0.9), m=1~9, the maximum relative error between this modeland the numerical solution is less than 0.008%, whichhas a much more suitable range and a higher precisionthan any other existing approximate model.7

E S T A B L I S H M E N T O F T H E I M P R O V E D M O D E L

Although Eq. (9) indicates progress in calculating theequivalent stress range, it still exhibits some error. In x

0

I 2 (

x,k 1

=1)

,erf

(x)

1 2.5 4

1.5

0

0.5

I2(k1)

2 3.51.5 3

1

erf (x)

0.5

fact, only when x�2, erf (x)=1.0, and when x=2, Fig. 4 Curves of the asymptotic expansion I2(x, k1=1) and theerror function erf (x).I2(x)≠1.0. Therefore, it can be improved further. In the

© 2000 Blackwell Science Ltd. Fatigue Fract Engng Mater Struct 23, 581–586

584 P. M . L U a n d S . J . J I A O

erf (x)=G 2√p

exp(−x2) ∑9

k=0

2kx2k+1

(2k+1)!!0∏x<2

1−exp(−x2)1

√px A1−1

2x2B 2∏x<2

(12)

The illustration of this expression is shown in Fig. 5,in which the advantages of I1(x) and I2(x) are used,respectively.

The unity function u(x) is defined as

u(x)=G0 x∏21 x>2 x

0

I 1,I 2

1 4

1.5

0

0.5

I2

2 3

1

I1

Fig. 5 The curve I1(x, k1=9) combining I2(x, k1=1) for the rangeThen erf (x) can be expressed as followsof xµ(0, 2).

erf (x)=2√p

exp(−x2) ∑9

k=0

2kx2k+1

(2k+1)!! because

C1+Ae

aB2D1/2

=1a+u(x) C1−exp(−x2)

1√px A1−

12x2B

and integration of J1 becomes

−2√p

exp(−x2) ∑9

k=0

2kx2k+1

(2k+1)!!D J1=em+2

√p∑9

k=0

2ka2k+1

(2k+1)!!C A2k+m+3

2 B (15)

0∏x<2 (13)

Considering the integrationSubstituting Eq. (13) into Eq. (6) yields

J2= P2

2 Aexa Bm+1

exp C−Aexa B2D d Aex

a BZ= P 2

0 C 2√p

exp(−x2) ∑9

k=0

2kx2k+1

(2k+1)!!D Aexa Bm+1

and using Eq. (10) leads to×exp C−Aex

a B2D d Aexa B

J2= P2

2e/atm+1 exp(−t2) dt

+ P2

2 C1−exp(−x2)1

√px A1−1

2x2B=

12

C Am+22 B Qx2 A8e2

a2 , m+2B (16)

−2√p

exp(−x2) ∑9

k=0

2kx2k+1

(2k+1)!!D Aexa Bm+1

For the same reason

×exp C−Aexa B2D d Aex

a B (14)J3= P2

2

1√px A1−

12x2B exp(−x2) Aex

a Bm+1

With respect to the integration

×exp C−Aexa B2D d Aex

a BJ1= P2

0 C 2√p

exp(−x2) ∑9

k=0

2kx2k+1

(2k+1)!!D Aexa Bm+1

=1√p Cem+2

2aC Am+1

2 B Qx2 A 8a2 , m+1B

×exp C−Aexa B2D d Aex

a B−

em+2

4a3 C Am−12 B Qx2 A 8

a2 , m−1BD (17)

© 2000 Blackwell Science Ltd. Fatigue Fract Engng Mater Struct 23, 581–586

A N I M P R O V E D M E T H O D O F C A L C U L AT I N G T H E P E A K S T R E S S D I S T R I B U T I O N F O R A B R OA D - B A N D R A N D O M P R O C E S S 585

and

J4= P2

2 C 2√p

exp(−x2) ∑9

k=0

2kx2k+1

(2k+1)!!D Aexa Bm+1

×exp C−Aexa B2D d Aex

a B=

em+2

√p∑9

k=0

2ka2k+1

(2k+1)!!C A2k+m+3

2 B×Qx2 A 8

a2 , m+2k+3B (18)

Substituting Eqs (15)–(18) into Eq. (14) yields

x

0

R (

%)

0.2 0.80

0.4 0.6

0.001

0.002

0.003

0.004

0.005

0.006

1

9

7

5m=3

Z=em+2

√p∑9

k=0

2ka2k+1

(2k+1)!!C A2k+m+3

2 B Fig. 6 A comparison of the relative error between the improvedmethod and the numerical solution.

×C1−Qx2 A 8a2 , m+2k+3BD agreement with the NM in the whole calculation range,

and the maximum relative error is less than 0.006%,which makes the calculation precision increase by 12.4%+

12

C Am+22 B Qx2 A8e2

a2 , m+2B in comparison with Eq. (9). The improved method cansatisfy the precision requirement completely with respect

−1√p Cem+2

2aC Am+1

2 B Qx2 A 8a2 , m+1B to engineering calculations.

−em+2

4a3 C Am−12 B Qx2 A 8

a2 , m−1BD (19) C O N C L U S I O N S

An improved method for calculating the equivalent stressTherefore, the solution of the equivalent stress range range under broad-band random loading is proposed in

can be derived as this paper, which is based on the peak stress probabilitydistribution of a stationary Gaussian random process.

Sh=2√2s Gem+2

2√pC Am+1

2 B C1−Qx2 A 8a2 , m+1BD The method is in agreement with the numerical solution

in the whole calculation range, and the maximum relativeerror between it and the numerical solution is less than

+a

2C Am+2

2 B C1+Qx2 A8e2

a2 , m+2BD 0.006%, which makes the precision of the calculatedequivalent stress range increase by 12.4% when usingthe newest series solution method.+

em+2

4a2√pC Am−1

2 B Qx2 A 8a2 , m−1B

R E F E R E N C E S+em+2

√p∑9

k=0

2ka2k+2

(2k+1)!!C A2k+m+3

2 B1 G. K. Chaudhury and W. D. Dover (1985) Fatigue analysis of

offshore platforms of subject to sea wave laoding. Int. J. Fatigue7, 13–19.×C1−Qx2 A 8

a2 , m+2k+3BDH1/m

(20)2 C. L. Chow and D. L. Li (1991) An analytical solution for fast

fatigue assessment under wide-band random loading. Int. J.Fatigue 13, 395–404.

3 J. C. P. Kam and W. D. Dover (1988) Fast fatigue assessmentD I G I T A L A N A L Y S I S procedure for offshore structures under random stress history.

Proc. Inst. Civ. Engng 2(85), 689–700.Verification of the improved method has been carried4 P. H. Wirsching (1980) Fatigue under wide band random stress.

out for a variety of band width and material constants. J. Struct. Div. Am. Soc. Engng Vol.??, 1593–1606.The relative errors between the authors’ method and 5 S. O. Rice (1955) Mathematical analysis of random noise. Selectedthe numerical solution (NM) are shown in Fig. 6 in per Papers on Noise and Stochastic Processes (Edited by N. Wax), Dover,

New York.cent. It can be seen that the improved method is in

© 2000 Blackwell Science Ltd. Fatigue Fract Engng Mater Struct 23, 581–586

586 P. M . L U a n d S . J . J I A O

6 W. B. Huston and T. H. Skopinski (1956) Probability and 7 P. M. Lu, B. H. Zhao and J. M. Yan (1998) Efficient algorithmfor fatigue life calculation under broad band loading based onfrequency characteristics of some flight buffet loads. NACA

TN3733, NACA, Washington, D.C. peak approximation. J. Engng Mech. 124, 233–236.

© 2000 Blackwell Science Ltd. Fatigue Fract Engng Mater Struct 23, 581–586