a statistical investigation of the fatigue lives of q235 steel-welded joints

Fatigue & Fracture of Engineering Materials & Structures 1998; 21: 781–790

A STATISTICAL INVESTIGATION OF THE FATIGUE LIVES OF

Q235 STEEL-WELDED JOINTS

Y.-X. Z, Q. G and X.-F. SInstitute of Applied Mechanics, Southwest Jiaotong University, Chengdu 610031, People’s Republic of China

Received in final form February 1998

Abstract—An investigation into the fitting of six assumed distributions (three-parameter Weibull, two-parameter Weibull, extreme minimum value, extreme maximum value, normal and lognormal distributions)of 23 groups of fatigue life data for Q235 steel-welded joints is performed in terms of linear regressionanalyses. The results reveal that the fatigue life distribution shapes mostly tend to be positively skewed.Therefore, the extreme minimum value and normal distributions are not the most appropriate distributionsto assume for a fatigue life evaluation. The three-parameter Weibull distribution may give misleadingresults in fatigue reliability analyses because the shape parameter is often lesss than 1. This means thatthe hazard rate decreases with fatigue cycling. This is contrary to the general understanding of thebehaviour of welded joints. Reliability analyses may also be affected by slightly non-conservativeevaluations in tail regions of the three-parameter Weibull distribution. The two-parameter Weibulldistribution does not give as good a fit as either the extreme maximum value distribution or thelognormal distribution. On the other hand, the extreme maximum value and lognormal distributions canbe safely assumed in reliability analyses due to the good total fit effects and the conservative evaluationsin tail regions. In addition, the extreme maximum value distribution is in good agreement with thegeneral physical understanding of the structural behaviour of welded joints.

Keywords—Q235 steel; Welded joints; Fatigue lives; Statistical distributions; Linear regression.

NOMENCLATURE

A, B=intercept and slope in linear regression functiona, b, c= location parameter, scale parameter and shape parameter in Weibull distribution function

df1

, df2=fit differences in tail regions

f (N)=probabilistic density functionF(N)=cumulative distribution function

g=coefficient of skewnessk=ordinal of the groups of N dataN=number of cycles to failureN∞=log10 N

N1 , N2=minimum value and sub-minimum value of a group of N dataN9 =mean value of a group of N datan=number of effective specimensns=number of specimensr=linear correlation coefficientS=standard deviation

t(n−2)=t-distribution function on n−2 degrees of freedoma=significance level

e, v=location parameter and scale parameter in extreme value distribution functionw( · )=standard normal distribution function

l(N)=hazard functionm, s=mean value and standard deviation in normal distribution function

m∞, s∞=mean value and standard deviation in lognormal distribution function

INTRODUCTION

It is the requirement of fatigue reliability analyses to accurately determine the statistical distri-bution of a group of fatigue lives. Since the data are generally seldom sufficient to define the

© 1998 Blackwell Science Ltd 781

782 Y.-X. Z et al.

probability distribution, we normally must rely on assumed distributions which arise from con-venience or, preferably, from relevant physical arguments [1]. Two assumed distributions, whichhave been widely used in fatigue studies, are lognormal and Weibull distributions.

The choice of the lognormal distribution has been based primarily on arguments of mathematicalexpendiency [2]. In the central range, the two distributions are often indistinguishable. In contrast,there is a significant difference in the tail regions. What is most important is that the hazardfunction l(N) for the lognormal distribution decreases in the large N range, which violates ourphysical understanding of progressive deterioration resulting from the fatigue process. Relatively,the Weibull distribution is based on more physically convincing arguments [3].

The present work is based on a statistical investigation of 23 groups of fatigue life data on Q235steel-welded joints in terms of linear regression analyses. By comparing the effects of fits to the sixassumed distributions (three-parameter Weibull, two-parameter Weibull, extreme minimum value,extreme maximum value, normal and lognormal distributions), the conclusions listed in this papercan be made.

EXPERIMENTAL PROCEDURES

Material and specimens

The parent metal is Q235 steel, one of the two most widely used engineering structural materialsin China (another is 16 Mn steel ), supplied in the form of rolled plate directly used for production.Its chemical composition (wt%) is 0.17–0.19 C, 0.67–0.68 Mn, 0.18–0.20 Si, 0.017 P, 0.018 S andthe remainder is ferrite. Monotonic mechanical properties are a 0.2% proof strength of 270 MPaand an ultimate tensile strength of 500 MPa.

Seven types of specimens, with dimensions shown in Fig. 1, were, respectively, machined and

(a) (b) (c)

(d) (e) (f)

Fig. 1. Details of loading and weld configurations for six different types of Q235 steel-welded joints;dimensions in mm. Note two alternative designs for type (e).

Fatigue & Fracture of Engineering Materials & Structures, 21, 781–790 © 1998 Blackwell Science Ltd

Fatigue lives of Q235 steel-welded joints 783

prepared using the following three welding procedures: (i) manual electric-arc welding[Fig. 1(a)–(b)]; (ii ) submerged arc auto-welding [Fig. 1(c)–(g), (e∞)]; and (iii) CO2 gas protectivewelding [Fig. 1(c)–(g)].

For the manual electric-arc welding, wires of diameter 4 or 5 mm are used. The weldingprocedures follow Chinese standard-GB. The chemical composition of the weld metal (wt%) is0.14 C, 0.25 Mn, 0.03 Si, 0.018 P, 0.03 S and the remainder is ferrite.

For the submerged arc auto-welding, the ZXG-1000R equipment, H08MnA wires and 431 typeof weld power are used. Welding parameters are a current of 620–740 A, a voltage of 32–35 V anda speed of 42 Cm/min. The chemical composition of the weld metal (wt%) is 0.082 C, 1.21 Mn,0.53 Si, 0.028 P, 0.013 S and the remainder is ferrite.

For the CO2 gas protective welding, the 2-2 pass S-500 equipment and H08MnSiA wires areused. Welding parameters are a current of 130–300 A, a voltage of 23–42 V and a gas input of25 l/min. The chemical composition of the weld metal (wt%) is 0.1 C, 1.28 Mn, 0.47 Si, 0.013 P,0.001 S and the remainder is ferrite.

All specimens were machined in the Taiyuan Huge Machinery Works under normal productionconditions. Local dimensions of the welded joints are in accordance with the Chinese code-GB.

T est conditions

All tests were performed on a 10t high frequency fatigue test machine at the Zhenzhou MechanicalInstitute. The experiments were at room temperature with four-point bending, sine-wave loadingand at a stress ratio of 0.1. Loading frequency was about 80–120 Hz. A group of specimens wastested at a certain loading level. Detailed test situations for the 23 groups of specimens, includingjoint type, welding procedure, number of specimens ns , and number of effective specimens n at acorresponding stress level Ds, are given in Table 1.

Table 1. Details of the testing programme

Ordinal of Type of Welding Loading stress Number of Number of effectivegroup k specimens procedure level Ds (MPa) specimens ns specimens n

1 a i 254.8 11 112 a i 215.6 10 103 a i 176.4 11 104 a i 137.2 13 105 b i 264.8 9 96 b i 215.7 9 97 b i 176.5 8 78 b i 137.3 8 79 c ii 290 8 7

10 c ii 160 8 811 c iii 240 8 712 c iii 140 8 813 d ii 250 7 714 d ii 170 10 1015 d iii 240 7 716 d iii 162 10 1017 e ii 292.5 6 618 e ii 260 6 619 e ii 176.4 11 920 f iii 245 11 1121 f iii 215.6 13 1122 e∞ ii 254.8 6 623 e∞ ii 196 7 5

Fatigue & Fracture of Engineering Materials & Structures, 21, 781–790© 1998 Blackwell Science Ltd

784 Y.-X. Z et al.

Table 2. Cycles to failure for the 23 test conditions

N×105

k 1 2 3 4 5 6 7 8 9 10 11

1 3.990 5.690 3.070 4.500 3.060 1.680 4.670 1.910 2.860 2.930 2.1902 9.030 3.920 3.230 3.850 3.400 2.360 5.200 4.430 4.020 2.7203 16.14 5.450 11.40 7.300 1.039 9.260 9.920 7.770 10.74 7.6004 21.93 22.88 15.30 31.26 27.30 27.50 19.57 10.73 15.27 18.095 1.671 2.010 2.082 2.490 2.582 2.618 3.487 3.524 4.1006 2.540 2.650 2.760 2.860 3.420 3.530 4.430 7.250 7.3907 4.760 6.780 9.540 10.49 11.38 14.13 33.538 12.45 13.75 15.07 16.70 29.39 45.54 46.089 1.788 2.411 2.468 2.534 2.845 3.451 5.789

10 3.640 6.770 8.800 8.970 10.50 10.77 15.08 15.4611 3.837 3.500 5.759 4.247 5.050 4.593 4.00112 11.69 12.24 15.12 16.00 16.67 17.92 23.20 33.3013 3.873 3.905 4.366 4.613 4.967 5.479 5.96214 12.83 15.16 15.66 15.79 16.99 18.67 19.61 24.05 27.63 35.0515 3.470 3.542 3.807 4.430 4.565 5.535 4.25016 51.76 30.72 23.32 21.21 32.46 18.03 15.94 23.41 16.41 18.1817 3.196 5.843 4.230 6.374 7.472 4.03018 2.893 6.234 3.350 2.650 6.140 5.50019 18.50 29.77 12.96 18.96 24.00 12.22 42.48 11.27 15.5620 12.82 20.68 15.61 21.10 11.63 7.330 17.55 5.360 4.120 9.000 5.77021 22.36 25.50 33.48 22.36 20.00 9.740 8.590 14.77 7.720 12.00 16.4122 3.530 5.330 3.320 3.760 7.090 3.86023 12.41 10.79 10.74 12.61 23.42

T est results

The number of cycles to failure, N, is defined as the number of cycles at which the maximumcrack length is one half of the specimen plate thickness. At this instant, the deformation of aspecimen becomes too large to permit continued normal testing, and from safety of structure inpractice, this limitation is also necessary from the point of view of structural safety. Test resultson the 23 groups of specimens are given in Table 2.

DATA FITTING

The investigation into the total fit effect for the six assumed distributions (three-parameterWeibull, two-parameter Weibull, extreme minimum value, extreme maximum value, normal andlognormal distributions) on the 23 groups of N data (see Table 2) is performed in terms of linearregression analyses, and has a unified comparable statistical parameter; i.e. a linear correlationcoefficient for the six distributions.

Linear relationships for the six assumed distributions are given in Table 3. A unified linearregression function can be expressed as

Y=A+BX (1)

where A is the intercept and B the slope of the line of Eq. (1); see Appendix.For a group of N data, a corresponding group of data (X

i, Y

i) (i=1, 2, . . . , n) for a certain as-

sumed distribution can be obtained from Table 3. According to the Pearson statistical parameter [4],



Table 3. Linear regression functions of the six assumed failure distribution curves

Distribution CDF (F(N)) X A BY AF̂(Ni)=

i−0.3

n+0.4BThree-parameter

Weibullln (N−a) −c ln (b) c1−exp C−AN−a

b BcD ln Cln A 1

1−F̂(N)BDTwo-parameter

Weibullln (N) −c ln (b) c1−exp C−ANbBcD ln Cln A 1

1−F̂(N)BDExtreme minimum

valueN −

e

v

1

v1−exp C−exp AN−e

v BD ln Cln A 1

1−F̂(N)BDExtreme maximum

valueN

e

v−

1

vexp G−exp C−AN−e

v BDH ln Cln A 1

F̂(N)BDNormal w−1[F̂(N)] N −

m

s

1

sP N−2

1

E2psexp C− 1

2 AN−m

s B2D dN

Lognormal w−1[F̂(N∞)] N∞= log10 N −m∞s∞

1

s∞P N∞−2

1

E2ps∞exp C− 1

2 AN∞−m∞s∞ B2D dN

the linear correlation coefficient r is given by:

r=

n ∑n

i=1XiYi− ∑

n

i=1Xi∑n

i=1Yi

SCn ∑n

i=1X2i−A ∑

n

i=1XiB2D Cn ∑

n

i=1Y 2i−A ∑

n

i=1YiB2D

(2)

and using the r to t transformation [4], then

t(n−2)=r√n−2

√1−r2(3)

so for a given significance level a, the critical value rc of r can be determined by

rc=ta(n−2)

√(n−2)+t2a(n−2)

(4)

By comparing how close |r | is to 1, we can determine how good the data fits to an assumeddistribution. The r values of the six assumed distributions on the 23 groups of N data are givenin Table 4. The hazard function and probability density function (PDF) curves of the three goodfit assumed distributions for the 5–8 groups of data are given in Fig. 2.

The distribution shape parameters (so-called coefficients of skewness) of the 23 groups of N dataare also given in Table 4. The coefficient g for a group of data is defined as

g=1

n∑n

i=1 ANi−N9S B3 (5)

where N9 and S are, respectively, the mean value and standard deviation of a group of N data.When g is greater than 0, the distribution shape of a group of data tends to be positively skewed.


786 Y.-X. Z et al.

Table 4. The linear correlation coefficients of the six assumed distributions and the coefficients of skewness for the23 groups of N data

Three-parameter Two-parameter Est. min. Est. max. Log-Weibull Weibull value value Normal normal

k g a rc r c r c r r r r

1 0.464714 0.01 0.7348 0.988360 1.45224 0.975393 2.85009 0.936530 −0.986807 0.975726 0.9861682 1.787156 0.01 0.7646 0.977975 1.19338 0.910789 7.73520 0.798375 −0.924635 0.867652 0.9521113 −0.05731 0.01 0.7646 0.903505 1.34896 0.903505 1.34896 0.963238 −0.953382 0.969107 0.8450694 0.064590 0.01 0.7646 0.990896 2.97233 0.990598 3.36955 0.973578 −0.979625 0.990497 0.9819795 0.417285 0.01 0.7977 0.987106 1.56798 0.968035 3.61962 0.937274 −0.979499 0.971233 0.9827366 1.046674 0.01 0.7977 0.991690 0.62245 0.865948 2.30799 0.812776 −0.926321 0.881924 0.9232627 1.616358 0.01 0.8745 0.976221 0.92299 0.934086 1.63959 0.789071 −0.910906 0.854902 0.9630068 0.570792 0.01 0.8745 0.988582 0.60242 0.917076 1.77069 0.874330 −0.939869 0.921460 0.9481149 0.399108 0.01 0.8745 0.956744 1.11958 0.896341 2.61563 0.812837 −0.929248 0.866354 0.938635

10 0.370594 0.01 0.8343 0.977102 2.29655 0.977102 2.29655 0.969547 −0.961931 0.976639 0.94940911 0.588959 0.01 0.8745 0.997476 1.48027 0.962516 6.08591 0.940459 −0.997482 0.978994 0.99090212 1.294577 0.01 0.8343 0.982597 0.87748 0.913791 2.92541 0.844357 −0.957491 0.908879 0.95915913 0.363281 0.01 0.8745 0.972300 1.44808 0.956240 6.29951 0.944277 −0.989677 0.978892 0.98303214 1.114240 0.01 0.7646 0.980496 1.19308 0.918615 3.31417 0.862762 −0.974873 0.929622 0.96674215 0.704056 0.01 0.8745 0.983044 0.86837 0.938531 6.17438 0.916475 −0.982105 0.958992 0.97218516 1.543678 0.01 0.7646 0.997856 0.72289 0.888453 2.68419 0.808223 −0.944925 0.885655 0.94750017 0.173081 0.01 0.9172 0.985730 1.47917 0.976773 3.21324 0.959970 −0.980674 0.980544 0.98211918 0.011919 0.05 0.8114 0.970564 0.64720 0.929828 2.50332 0.917781 −0.916946 0.930079 0.93392619 1.171174 0.01 0.7977 0.996444 0.79435 0.929901 2.32820 0.856568 −0.972239 0.924776 0.97476720 0.261076 0.01 0.7348 0.989914 1.15454 0.979141 1.92902 0.941813 −0.973427 0.975029 0.98002321 0.500849 0.01 0.7348 0.989425 1.09498 0.975885 2.29437 0.934221 −0.989747 0.976476 0.98640022 1.073673 0.01 0.9172 0.988120 0.65680 0.868308 3.13276 0.833642 −0.946064 0.897386 0.92359223 1.400292 0.10 0.8054 0.981174 0.30003 0.789055 2.53494 0.751387 −0.878474 0.819080 0.851989

This implies that the extreme minimum value and normal distributions are not the most appropriatedistributions to assume. In contrast, the extreme maximum value distribution, lognormal distri-bution and Weibull distribution, with a shape parameter greater than 1 may be the more appro-priate. When g is less than 0, the situations are in contrast to those when g is greater than 0.

From Table 4, it can be seen that:

(1) The |r | values of the six assumed distributions are all greater than the critical value rc atthe given significance level. This implies that it is very easy to accept any of the distributions,but it may be misleading to assume that a group of N data follows a distribution withoutcomparing the various degrees of fits of the different distributions.

(2) Except for group 3, the coefficients of skewness for other data are greater than 0. Thisimplies that the distribution shapes tend to be positively skewed, and the extreme minimumvalue and normal distributions are not good distributions to assume.

(3) The order of best fits for the six assumed distributions is the three-parameter Weibull,lognormal or extreme maximum value, two-parameter Weibull, normal and extreme mini-mum value distributions.

(4) There are 10 groups of data in which the shape parameters of the three-parameter Weibulldistribution are less than 1, resulting in the hazard functions decreasing with fatigue cycling[Fig. 2(b)–(d)]. This is contrary to the general behaviour of welded joints. This implies thatin this case the assumed distribution is not good without some improvements being made.

(5) The two-parameter Weibull distribution is not a good distribution to assume, although theshape parameters are all greater than 1, because the fit of this distribution is basically lessthan those of the extreme maximum value and lognormal distributions.



Fig. 2. Curve fitting correlation effects with respect to conditions: (a) K=5; (b) K=6; (c) K=7; (d) K=8;for three assumed distributions: Curve A=three-parameter Weibull distribution; Curve B= lognormal

distribution; Curve C=extreme maximum distribution.


788 Y.-X. Z et al.

(6) The extreme maximum value distribution may be a satisfactory distribution to assume dueto the good fit effects, and from physical arguments, i.e. the good agreement to our generalphysical understanding that fatigue failures possibly result from a ‘weakest link’ condition,e.g. a maximum size crack.

(7) The lognormal distribution may be a good distribution to assume due to the good total fiteffects, although the hazard functions in the large N range decrease with fatigue cycling.

FIT EFFECTS IN THE TAIL REGIONS

Tail regions are the critical areas in fatigue reliability analyses. Therefore, it is important tocheck the fit effects in tail regions before selecting an assumed distribution.

In order to describe the fit effects in the tail regions, the error parameters df1

and df2

betweenthe real value and theoretical value of two life data (so-called fit-differences in the tail regions) are,respectively, defined as:

df1=F̂(N1 )−F(N1 )=

1−0.3

n+0.4−F(N1 ) (6)

df2=F̂(N2 )−F(N2 )=

2−0.3

n+0.4−F(N2 ) (7)

where N1 and N2 are, respectively, the minimum life and sub-minimum life of a group of N data.The smaller the |d

f| value, the better the fit effects in the tail regions. When d

f<0, this implies

a conservative evaluation. In contrast, when df>0, a non-conservative evaluation results. In

addition, by comparing df1

with df2

, the trend of the evaluation can be determined.

Table 5. The fitting parameters df1 and df2

for the 23 groups of N data

Three-parameter Extreme maximumWeibull value Lognormal

k F̂(N1) F̂(N2) d

f1df2

df1

df2

df1

df2

1 0.061404 0.149123 −0.004900 0.018762 −0.018662 0.021692 −0.011689 0.0241512 0.067308 0.163462 0.003906 −0.015629 −0.126450 −0.096579 −0.047912 −0.0297363 0.067308 0.163462 0.025655 −0.164828 0.057371 −0.081452 0.045681 −0.2164604 0.067308 0.163462 0.000962 −0.050946 0.033131 −0.046131 0.023865 −0.0646545 0.074468 0.180851 0.001444 −0.037242 −0.011635 −0.029697 −0.011501 −0.0287976 0.074468 0.180851 0.000979 −0.026649 −0.182017 −0.096289 −0.138233 −0.0576107 0.094595 0.229730 0.003532 −0.038689 −0.153866 −0.092666 −0.030462 −0.0265348 0.094595 0.229730 0.000259 −0.016086 −0.125851 −0.021140 −0.089126 0.0033169 0.094595 0.229730 0.003358 −0.139558 −0.114051 −0.148007 −0.050293 −0.122053

10 0.083333 0.202381 0.014675 −0.053651 0.043819 −0.042146 0.039436 −0.08380011 0.094595 0.229730 0.002403 −0.032025 −0.019204 −0.026330 −0.043848 −0.01976712 0.083333 0.202381 −0.006506 0.039224 −0.121065 −0.029110 −0.077314 0.01167813 0.094595 0.229730 −0.041989 0.077792 −0.047782 −0.075037 −0.059522 0.00490214 0.067308 0.163462 0.003525 −0.089289 −0.083876 −0.107474 −0.055669 −0.08393615 0.094595 0.229730 −0.009621 −0.048361 −0.063706 0.039095 −0.067790 0.03919516 0.067308 0.163462 −0.001100 0.017906 −0.169761 −0.089217 −0.113816 −0.03619717 0.109397 0.265625 −0.001347 −0.031726 −0.005505 −0.014902 −0.007830 −0.02007118 0.109397 0.265625 0.000613 0.006694 −0.067072 0.044016 −0.059709 0.04633819 0.074468 0.180851 −0.000695 −0.004787 −0.126461 −0.053442 −0.077254 0.01223220 0.061404 0.149123 −0.002254 −0.010854 −0.033138 0.001659 −0.010806 0.00080921 0.061404 0.149123 −0.007473 0.022915 −0.039273 0.0184852 −0.024487 0.02686822 0.109397 0.265625 0.001497 −0.043696 −0.151182 −0.046458 −0.124977 −0.02395823 0.129631 0.314850 −0.000119 0.022015 −0.219808 −0.03767 −0.179407 0.002023



The calculated values of df1

and df2

for the 23 groups of data, and the three best fit distributionsare given in Table 5.

From Table 5, it can be seen that:

(1) The order of goodness of fit effects in the tail regions is the three-parameter Weibull,lognormal and extreme maximum value distributions.

(2) For the three-parameter Weibull distribution, more than half of the df1

values are greaterthan 0. This implies that the distribution tends to give slightly non-conservative evaluationsin the tail regions. It is necessary to make some improvements before this distribution isapplied in practice.

(3) Except for three df1

values, all the others are less than zero for the extreme maximum valueand lognormal distributions. This implies that these two distributions tend to give con-servative evaluations. Therefore, these two distributions can be safely used as goodapproximations.

CONCLUSIONS

From this investigation, the following conclusions can be drawn:

(1) The shapes of distributions of fatigue lives of Q235 steel-welded joints tend to be positivelyskewed. The extreme minimum value and normal distributions are not the most appropriatedistributions.

(2) The three-parameter Weibull distribution may give misleading results in fatigue reliabilityanalyses, even though it may give the best total and tail fit. This is because the shapeparameters are often less than 1, resulting in a hazard function which decreases with fatiguecycling. In most cases, this is contrary to real structural behaviour. Also this distributionleads to slightly non-conservative evaluations in the tail regions.

(3) The two-parameter Weibull distribution does not provide a good fit even though the shapeparameters are all greater than 1. This is because the total fit effects are basically less thanthose of the extreme maximum value and lognormal distributions.

(4) The extreme maximum distribution can be used safely as assumed distributions due to thegood fit effects, acceptable physical arguments, and conservative evaluations in the tailregions.

(5) The lognormal distribution can also be used safely as an assumed distribution (althoughone possible drawback is the fact that the hazard functions in the lifelong regime decreasewith fatigue cycling) due to the good total and tail fit effects, the latter giving suitableconservative evaluations.

Acknowledgement—The test material in this study was supplied by the senior engineer Zhang Yuhuai of ZhenzhouMechanical Institute, PRC.

REFERENCES

1. A. M. Freudenthal (1947) Safety of structure. T ransactions of ASCE 112, 125–128.2. The Committee on Fatigue and Fracture Reliability of the Committee on Structural Safety and Reliability

of the Structural Division (1982) Fatigue reliability: introduction. J. Structural Division, ASCE 108 (ST1),3–23.

3. A. M. Freudenthal (1956) Physical and statistical aspects of fatigue. Advances in Applied Mechanics4, 116–159.

4. G. R. Loftus and E. F. Loftus (1988) Essence of Statistics, 2nd edn. Alfred A. Knopf, Inc., New York.


790 Y.-X. Z et al.

APPENDIX

Taking the three-parameter Weibull distribution as an example, the cumulative distribution function F(N) can beexpressed as

F(N)=1−exp C−AN−a

b BcD (A1)

Converting logarithmically the above equation, the following linear equation can be obtained

ln ln1

1−F(N)=−c ln b+c ln (N−a) (A2)

Setting Y= ln ln[1/(1−F/(N))], X= ln (N−a), A=−c ln b, and B=c, respectively, we can obtain the following standardlinear equation

Y=A+BX (A3)

The representative values of X, A and B for the other five distributions in the paper are given in Table 3 using asimilar approach.

For an ordered group of data Ni(from small to large) where i=1, 2, . . ., n, the true failure probability F̂(N

i) of the

ith fatigue life can be approximated by

F̂(Ni)=

i−0.3

n+0.4(A4)

Setting Xi= ln (N

i−a) and Y

i= ln ln[1/(1−F̂(N

i))], a corresponding ordered group of data (X

i/Yi) for the three-parameter

Weibull distribution can be obtained. The corresponding ordered group of data (Xi/Yi) for the five other distributions in

the paper can be obtained by a similar approach.


a statistical investigation of the fatigue lives of q235 steel-welded joints

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