Application of the weakest link analysis to the area of fatigue.pdf

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Application of the weakest link analysis to the area of fatiguedesign of steel welded joints

Ł. Blacha a,⇑, A. Karolczuk a, R. Bański b, P. Stasiuk a

a Department of Mechanics and Machine Design, Opole University of Technology, ul. Mikołajczyka 5, 45-271 Opole, Polandb Department of Materials Science and Chipless Production Engineering, Opole University of Technology, ul. Mikołajczyka 5, 45-271 Opole, Poland

a r t i c l e i n f o

Article history:

Available online 28 June 2013

Keywords:

Welded joints

Fatigue failure

Weakest link concept

Efficient material

a b s t r a c t

A new approach in the area of fatigue life assessment of steel welded joints is being pro-

posed with the following features: (i) methodology of fatigue life calculation is indepen-

dent from geometry of welded element; (ii) fatigue life assessment is based on fatigue

characteristic of introduced efficient material – suitable for different steel welded joints;

(iii) the fatigue life assessment is carried on the desired level of failure probability.

In the proposed method a material volume surrounding the weld is divided into volume

elements and regarded as a serial system having its definition in the reliability theory (the

weakest link concept). Failure probability distribution of the welded structure is character-

ized by the proposed S –N curve for efficient material and the shape parameter introduced to

describe the volume effect.

2013 Elsevier Ltd. All rights reserved.

1. Introduction

Welding is a common method of joining structural elements. However, the use of unmachined welded joints without fur-

ther heat treatment is associated with certain disadvantages: (i) stress concentration resulting from the complex geometry;

(ii) characteristic microstructural heterogeneity of the material in the weld and its vicinity; (iii) presence of residual stresses.

Results from the experimental research [1–3] lead to the following conclusions: (i) a characteristic area with unknown mate-

rial properties arises in a joint as a consequence of the welding process; (ii) welding residual stresses are expected to rise as

the joint dimension increases; (iii) grade of the steel (parent material) is of secondary importance for the fatigue strength.

These features indicate the welded joint as the area where the damage process of the entire structure is initiated.

The approaches for the fatigue assessment of welded joints and components can be divided over the following groups:

(i) Nominal stress approach, preferred for joints classified in a distinctive manner (differing among codes and recommen-

dations, e.g. [4–6]). The idea encapsulated in this approach requires the designer to place the given weld in one of a

number of classes. Fatigue life is estimated based on the nominal stress applied to the joint and separate S –N curves.

The exact fatigue curve to be used is identified by the stress range Dr at 2 106 cycles (FAT class in IIW recommen-

dations [4]).

(ii) Structural hot-spot stress approach, recommended for the cases where the strains can be measured directly at the

weld area or obtained from finite element analysis. Hot-spot stress is determined by the extrapolation from reference

points onto the weld toe. In this way, the structural stress is derived, i.e. including only the geometry as stress raising

effect. Fatigue resistance is evaluated from several S –N curves categorized by the geometry of welded joint [4,7].

1350-6307/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engfailanal.2013.06.012

⇑ Corresponding author. Tel.: +48 77 449 8420; fax: +48 77 449 9934.

E-mail address: [email protected] (Ł. Blacha).

Engineering Failure Analysis 35 (2013) 665–677

Contents lists available at SciVerse ScienceDirect

Engineering Failure Analysis

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c at e / e n g f a i l a n a l

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(iii) Effective notch stress with fictitious notch rounding as an idealization of the real conditions. The corresponding stress

range is determined from finite element analysis, according to the chosen stress criterion. Idea of fictitious notch

rounding derives from the Neuber’s solution for stress averaging based on evaluation of stress in a small material vol-

ume [8,9]. Notch radius (uniform along the weld line) is a function of geometry and loading mode. Fatigue resistance

against effective notch stress is determined from one S –N curve for several geometries, in IIW recommendations a FAT

curve is suggested [4].

(iv) Methods based on linear fracture mechanics, well suited for fatigue life assessment of joints with weld imperfections.

The fatigue parameter is stress intensity factor range tailored to individual notch type by the application of a number

of correction functions. No. of cycles to failure is evaluated from the crack propagation law defined by appropriate

material parameters.

(v) Others, e.g. method that takes into consideration material volume corresponding to the 90% of maximal stresses [10].

The paper describes probabilistic computational model for fatigue life evaluation of steel welded joints. To a certain extent,

the concept behind this model corresponds to the concept as proposed by Sonsino [10]. It is an extension into the wider range

of cycles, as consideration of volume effect and changes in fatigue life scatter are in the assumptions underlying to the model.

From this standpoint, it is suitable for fatigue life evaluation in a wide lifetime range. This is new since the previous research

known to the authors do not take the changing scatter into account [11,12]. From their definition they are applicable only to

one particular stress level – they were developed to predict the fatigue limit (definition according to ASTM [13]).

It can be assumed, that changing scatter phenomena is material dependent. At tensile strength level scatter is non-exis-

tent in terms of fatigue. On the other hand, scatter keeps on growing while stress is decreasing into the endurance limit area

[14], i.e. cumulative failure distribution function is widening.

The introduced model is based on two-dimensional failure probability distribution P f for an element with inhomogeneous

stress fields (i.e. welded), for any given no. of cycles and as a function of stress range and cycles to failure. It is the main con-

tribution of the introduced research, achieved through the implementation of two-parameter Weibull distribution [15]. The

model utilizes the scale parameter (H ) and shape parameter ( p). Scale parameter allows to compare the scatter of fatigue life

on different stress levels. Shape parameter governs the initial shape of failure distribution and covers the volume effect in

welded joints. In this paper they are described in a consistent manner: from their formulation through identification process

and up to their final values.

Weibull introduced his distribution as a result of analysis undertaken on scatter observed during static strength tests. Fa-

tigue life distributions were investigated by a number of researchers, but only in the case of individual stress levels. Schijve

analyzed fitting of log-normal as well as three-parameter Weibull and log-normal distributions [16,17]. Best results were

obtained from the latter two distributions, although the author suggested no physical justification to third parameter –

the location parameter. Normal, log-normal, two- and three-parameter Weibull and extreme value distributions were tested

in [11,12] as the best fit to fatigue life distribution. Again, the three-parameter Weibull distribution was chosen as best fit-

ting. Although, many remarks were raised concerning the values of parameters in the distribution, as they were physically

unjustified.

Several studies were undertaken on two-dimensional P –S –N fatigue failure distribution (probability-stress-cycles to fail-

ure) (e.g. [18,19]). Like in the literature positions, the model presented herein introduced constant shape parameter p. In this

manner it resembles the above mentioned. The difference is elsewhere – here it is tailored to the specific area of steel welded

joints. Shape parameter is determined through comparison of experimental S –N curves for welded elements of different

thicknesses. In this way, the volume effect – a well-known issue of welded joints [2,20,21] – is considered. Scale parameter

is formulated as a material-dependent parameter. It has a form of no. of cycles to failure N f . This value for each loading level

is determined on the basis of introduced S –N curve, corresponding to specific material microstructure in the weld area. The

identification procedure involved numerical optimization with reference to fatigue tests and FAT curves [4].

Calculation algorithm to the model is based on the weakest link concept [22,23]. In this manner calculated fatigue life of a

welded joint is dependent from global failure probability which is a resultant of local failure probabilities derived for each

finite element. The computational method is based on non-local approach to fatigue failure analysis. Effect of geometry is

minimized as throughout the analysis entire stress field in the material volume is taken into account. The weld notch is mod-

eled with a uniform value of root radius. From the standpoint of finite element modeling it is similar to the effective notch

stress approach.

2. Lifetime of welded joints in a weakest link concept formulation

Based on the conclusions drawn from the results of fatigue tests [1,2,20], assumptions to the computational model were

formulated:

– Elements joined by consumable electrode arc welding in shielding gas atmosphere demonstrate characteristic features of

the material surrounding the weld, i.e.: structural inhomogeneity, overheated material sections, welding residual stres-

ses. These properties dominate over the properties of parent material; it leads to the conclusion regarding the versatility

in application of efficient material S –N curve into the area of steel welded joints.

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– It is assumed, that the material extracted from the vicinity of welding point is defined by certain fatigue properties; such a

material is called the efficient material.

– Number of cycles to failure is derived based on predefined cumulative distribution function, according to the weakest link

concept.

– Form of the survival probability distribution P s is influenced by the following factors: inhomogeneous stress field along

the weld, size of the welded element and fatigue properties of efficient material.

– Stress field is obtained through linear elastic finite element analysis.

An analysis carried on the test results reveals the relationship between increasing length of the weld, as well as dimen-

sions of welded element, and decrease in fatigue durability. Such an effect is called the volume effect . The influence of size can

be described by weakest link damage concept [15,22,23]. This kind of approach seems to be especially promising in case of

welded joints. Application of the weakest link concept in the volume of the material [24] defines the cumulative failure dis-

tribution function P f on a certain loading level as a function of number of cycles to failure N :

P f ðN Þ ¼ 1 P sðN Þ ¼ 1 e 1V 0

R V

ðlogN H

Þ pdV

; ð1Þ

where P s(N ) – survival probability as a function of number of cycles to failure N , V 0 – referential volume, H – scale parameter,

p – shape parameter.

Survival probability distribution P s(N ) for an element characterized by referential volume V 0 and homogeneous stress

field is in the following form:

P sðN Þ ¼ eð

logN

H Þ p

:

ð2Þ

The scale parameter H normalizes logN variable for a given loading level. This parameter can be conveniently described in

the form of H = logN f , where N f is the number of cycles to failure of the efficient material, for a certain value of P s:

P sðN Þ ¼ e

logN logN f

p: ð3Þ

An example of P s(N ) distributions for different values of shape parameter is shown in Fig. 1.

The S –N curve for the efficient material takes into account microstructural heterogeneities and the effect of residual stres-

ses. Parameters of such a curve are determined on the basis of experimental data. In order to simplify the notation, it is as-

sumed, that within a certain range of number of cycles to failure this curve can be described by the following relation:

logN f ¼ log C f m f log Dr; ð4Þ

where N f

– number of cycles to failure for efficient material, on the given level of P s, C

f , m

f – fatigue parameters for efficient

material, for the given level of P s, Dr – stress range.

From the comparison between relations (4) and (3), the following formula for P s, Dr–N distribution can be derived:

P sðN ;DrÞ ¼ e

logN log C f m f log Dr

p: ð5Þ

This formula is valid under the assumption that shape parameter p is independent from stress rangeDr. Shape parameter

is considered as a constant value within the certain range of number of cycles to failure. The estimated value of p is decreas-

ing for stress levels below the endurance limit, which comes in connection with widening of the scatter band. The above

relationship becomes clear in view of Fig. 1.

Fig. 1. Simulation of survival probability distributions P s for two values of shape parameter p and N f = 105 cycles.

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The final formula for the proposed failure probability distribution P f as a function of number of cycles to failure N of an

element with inhomogeneous stress field Dr( x, y, z ) can be shown as the following equation:

P f ðN Þ ¼ 1 P sðN Þ ¼ 1 e 1V 0

R V

logN log C f m f log Drð x; y; z Þ

pdV

: ð6Þ

The derived formula is defined through four parameters: V 0, p, C f , m f . The process of identification of these parameters is

presented in the next section.

3. Identification of the parameters proposed in the computational model

3.1. Shape parameter p

The survival probability of an element with ascribed efficient material properties and volume equal to nV (n times larger

than in the Eq. (5) case) can be described as:

P sðn V Þ ¼ en

logN log C f m f log Dr

p: ð7Þ

The simulation of survival probability distributions for different values of n is shown in Fig. 2. For the same value of P s,Dr,

C f , m f the derived fatigue life N differs. This phenomena can be described by the volume effect [25,26]. It should be men-

tioned, that in case of cyclic loading good estimates of this effect can be produced by a continuous, monotonic function [20].

The following notation is proposed:logðN ðV ÞÞ ¼ s logðN ðn V ÞÞ; ð8Þ

where N (V ), N (nV ) – number of cycles to failure for the volume of V and nV , respectively, s – parameter proposed to describe

the volume effect (coefficient of proportionality, in the logarithmic scale).

From the comparison of P s probabilities derived for volume V and nV

e

logN ðV Þ

log C f m f log Dr

p¼ e

n logN ðnV Þ

log C f m f log Dr

p; ð9Þ

and with the consideration of relation (8), the following formula can be derived:

e

slogN ðnV Þ

log C f m f log Dr

p¼ e

n logN ðnV Þ

log C f m f log Dr

p: ð10Þ

After the transformations of

s logN ðV Þ

logC f m f log Dr

p¼ n

logN ðV Þ

logC f m f log Dr

p; ð11Þ

it can be concluded that

s p ¼ n ð12Þ

The resulting formula for the shape parameter can be described by the following equation:

Fig. 2. Simulation of survival probability distributions P s for two values of n with p = 20 and N f = 105 cycles.


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p ¼ log n

log s : ð13Þ

Eq. (13) serves as an input to the identification process. Identification of the shape parameter proceeds through the com-

parison made between fatigue lives obtained for welded joints of different dimensions and described by parameters n and s.

Relation (12) for selected values of shape parameter is shown in Fig. 3. Performed simulation (e.g. for p = 20) pointed to the

conclusion that increase in volume leads to decrease in fatigue life described by the ratio of logarithms of number of cycles to

failure equal to ca. 1.26 (according to Eq. (8)).

The shape parameter p was identified through undertaken experimental research.

3.1.1. Experimental research

The investigated specimens were made of S355J2 steel delivered in the normalized conditions. Chemical composition of

such steel is shown in Table 1. Specimens were manufactured by machining of a large welded plate of a transverse stiffener

geometry. The orientation of the weld direction was perpendicular to the direction of rolling. Plate with the thickness of 5 mm was cut by plasma along the rolling direction to the dimensions of 1300 mm in length and 60 mm in width. The geom-

etry of the specimens is shown in Fig. 4. Final dimensions were obtained by milling.

Plate was welded in one pass using GMAW MAG technique with a mixture of Ar (92%) and CO2 (8%) as shielding gas, by

the same person and in one sequence. Welded joint was fully penetrated. Tests were performed on joints in as-welded con-

dition. Shape and parent material of the specimens were the same as in the tests by Sonsino et al. [1].

Fig. 3. Simulation of the volume effect for two values of shape parameter p.

Table 1

Chemical composition of S355J2 steel (EN 10025-2: 2004).

Chemical element C Si Mn P S Cu

Max amount (%) 0.22 0.55 1.60 0.03 0.03 0.55

Fig. 4. Geometry and dimensions of the specimens investigated in identification process.


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3.1.1.1. Structural tests. For a chosen specimen macroscopic metallographic tests were undertaken, accompanied by hardness

measurements. From the test a macrograph was obtained, together with microhardness distribution across the heat affected

zone (HAZ), joint and parent material (PM).

In the heat affected zone three typical areas were observed: recrystallization zone (500 C – Ac1), normalized zone (Ac1–

Ac3) and the fusion zone (above 1100 C). The boundary between HAZ and material sections that were intact in structure and

properties becomes clear in view of material macrostructure, as shown in Fig. 5. The weld was fully penetrated, without

macro flaws. Microhardness measurements were performed on Leco AHT 2100 hardness tester under the 100 g loading,

according to PN-EN ISO 6507-1. Test results are presented in Fig. 6. The highest increase in the hardness was observed inoverheated area, with value of approx. 200 HV0.1 being considerably higher than 160 HV0.1 in the parent material. Hardness

in the weld area was fluctuating between 181 HV0.1 and 196 HV0.1; this behavior is specific for poured materials without heat

treatment. Recrystallization zone exhibited stable hardness values.

3.1.1.2. Notch radius. Welded joints are regions of complex geometry with variable notch radius q and weld toe angle a,

resulting in stress concentration. A scatter in q radius is considered to be due to the welding technique. In the present paper

q and a (angle between the two lines: (1) tangent to the weld face; (2) tangent to plate) were determined through image

processing. The evaluation of q and a involved:

(1) Making silicone cast for the weld face and its vicinity.

(2) Cutting the cast into sections normal to the weld line (approx. 1 mm cutting scale).

(3) Grayscale scanning of the obtained fragments (0–255 scale) with resolution of 4800 dpi resulting in the precision

range of 0.0053 mm/pixel.

PM HAZ HAZ PM

WeldMaterial: S355J2+N

Thickness: 5 mm

Fig. 5. Material macrostructure in vicinity of the joint, where: PM – parent material, HAZ – heat affected zone.

Fig. 6. Microhardness distribution across the welded joint. Vickers test, indenter load of 100 g.


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(4) Image filtration aimed at differentiation of weld edges (Fig. 7).(5) Identification of lines tangent to plate and weld face – linear regression analysis on 400 extreme image points (Fig. 7).

(6) Identification of circle tangent to obtained lines, through least squares approach and on the basis of E r estimator:

E r ¼Xni¼1

ð yr i yei Þ

2; ð14Þ

where yr i – y coordinate of point i (measured), yei – y coordinate of point i (estimated), n – no. of points.

(7) Computation of radius q and angle a .

(8) Evaluation of the quality estimator P for mean fitting, in accordance with the following relation:

P ¼ ffiffiffiffiffiE r n

r : ð15Þ

Fig. 7. Example of weld contours identified along the weld line: (a) q = 3.26 mm, and (b) q = 4,31 mm.


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(9) Calculation of R, a, P mean values and scatter, defined as being equal to standard deviation.

The derived results are shown in Table 2. An example of digital images together with the determined weld contours are

shown in Fig. 7. Fig. 8 shows the measurement results.

In order to assess the measurement error three cuboidal tiles were cut from the cast. The dimensions of the tiles were

measured by a calliper with an accuracy of 0.02 mm. In the next step similar analysis was undertaken as the one in case

of the cast. Measurement method was evaluated from the comparison between derived results absolute error for image pro-

cessing. It was defined as a square root of mean square value of the deviation and was established at the value of 0.10 mm.

3.1.1.3. Fatigue tests – description. The tests were conducted on specimens of a transverse stiffener geometry (Fig. 4). Each

specimen was subjected to cyclic axial loading and tested under load control (see Table 3). The loading was of a fully reversed

type (R = 1) with stress levels ranging from 80 MPa to 150 MPa of nominal stress amplitude ran.

Table 2

Results from the analysis on weld contours, where: qmin – lowest value of q radius.

q (mm) qmin (mm) a () P (mm) No. of measurements

3.2 ± 0.82 1.6 133.5 ± 2.3 0.026 ± 0.015 24

Fig. 8. Histograms: (a) notch radius q , (b) flank angle a .


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3.1.1.4. Fatigue tests – results. In the tests, fatigue cracks were initiated on the outer edge of the weld, in the heat affected

zone. Cracks propagated along the weld line resulting in fracture surface as shown in Fig. 9. Location of the crack is justified

by the high values of stress concentration factor (e.g. determined by Sonsino et al. [1]). The derived number of cycles to fail-

ure N f for a given value of nominal stress amplituderan are shown in Table 3. The correlation between ran and N f (S –N curve)

is shown in Fig. 10 and described by the following relation:

ran ¼ C N b f ; ð16Þ

where C , b – fatigue parameters of the welded joint.

Fatigue strength computed using Eq. (16) for ran corresponding to 2 106 cycles was determined to be equal to 78 MPa.

The C and b parameters are obtained through least squares regression analysis which yielded the values of C = 7878 MPa and

b = 0.32.

These results were compared to the results available in the literature. Sonsino et al. [1] undertook research on the spec-

imens with the same geometry but different dimensions (Fig. 11) and volume of the joint being eight times larger. Regression

analysis yielded the following values of the fatigue parameters (Eq. (16)): C = 8127 MPa and b = 0.35. The estimate of the

fatigue strength (ran at 2 106 cycles) is equal to 48 MPa (Fig. 10).

In both cases slope of the S –N curve (1/b, Eq. (16)) is considered to be comparable which allows to determine the shape

parameter p. Initial conditions for the identification process are summarized below:

n ¼ 8;

C ðV Þ ¼ 7878;

C ðn V Þ ¼ 8127;

where V – volume of the welded joint.

Difference in fatigue life N (i.e. the volume effect) is described by the coefficient of proportionality s (Eq. (8)). From the

relation s = C (nV ) / C (V ) and Eq. (13) the following values arise:

s ¼ 1:0316; p ¼ 66:82:

3.2. Scale parameter H

As already mentioned, the scale parameter H in distribution (1) is in the form of number of cycles to failure determined

from the efficient materialS –N curve. Such curve is described by two parameters: C f and m f .

The identification process requires the knowledge of the number of cycles to failure for a given geometry and survival

probability P s

. The identification criteria involved in this approach can be ideally met through the application of nominal

stress approach and FAT S –N curves [4]. The FAT number is equal to stress range (Dr = FAT) at 2 106 cycles, derived for

Table 3

Fatigue test results.

Specimen no. ran (MPa) N f , cycles

1 80 2,395,820

2 80 1,557,420

3 100 1,150,270

4 100 644,170

5 150 275,410

6 150 250,450

Fig. 9. Fracture surface.


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95% survival probability P s. The process involves numerical calculation of through thickness stress fields acting along theweld line and corresponding to specified level of nominal loading, applied to joint of a chosen geometry and FAT class.

In the concept, global geometry of the welded joint has no influence on the efficient material S –N curve. Shape of the

welded joint is considered during finite element modeling. In order to avoid stress field singularity, a notch radius q must

be introduced during the modeling. In a welded element, varying q radius together with inhomogeneous residual stress field

affect fatigue life probability distribution. Both features have an impact on the scale of volume effect. In the underlying

assumptions, C f and m f parameters take into account the influence of material inhomogeneity on the fatigue life. It should

be mentioned that in the proposed model notch radius is unrelated to loading type or failure criterion as it is so in the effec-

tive notch stress approach. It is a representative to the real conditions. The value will be different for each S –N curve, but it

should be beared in mind that we are dealing with efficient material, with everything that it implies.

The process of identification of the C f and m f parameters proceeds through Nelder –Mead minimization [27,28] of the

E (C f , m f ) estimator for efficient material parameters:

E ðC f ; m f Þ ¼ 0;95 e log N FAT ð Þ p

R V

1log C f m f log Dr p

dV ; ð17Þ

E ðC f ; m f Þ ! 0;

where N FAT – number of cycles to failure derived from the nominal stress approach.

Referential volume V 0 (Eq. (6)) for efficient material is assumed to be equal to 1 mm3. In the assumptions underlying in

the model, the grade of the steel elements being joined (in fatigue analysis grade of most steels is of secondary importance

[1]), do not affect the efficient material characteristic. Similar situation arises in case of their geometry. Only a slight impact is

forecasted since during the calculations not only the notch but also the surrounding volume has the influence on the distri-

bution (non-local fatigue assessment method).

The identification process should be regarded also as a form of validation. It was carried for two welded elements of a

different geometry: transverse butt weld (type a joint) and a transverse stiffener (type b joint). The geometry of the analyzed

elements is shown in Fig. 12. In the IIW nominal stress approach [4] these elements are classified as structural details of no.

213 and 511, respectively.

Fig. 10. S –N curves derived from the tests by the author and Sonsino et al. [1].

Fig. 11. Geometry and dimensions of specimens investigated by Sonsino et al. [1]




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Finite element analysis was carried out for finite element models of a geometry representative for both welded elementsand with notch radius in ten variants, q = {0.1–1.0} [mm], uniform along the weld line. Global edge length of the elements in

the notch section (section I, Fig 13) was equal to 0.1q, following the IIW recommendations [4]. Plate thickness was equal to

5 mm. The elements were subjected to axial loading.

Fig. 14 briefly describes the process of identification in the form of a serial, iterative algorithm. Detailed description over

the identification process can be found in the literature [29].

The log(C f ), m f parameters extracted for each variation of FE model and joint type are presented in Table 4.

4. Discussion

In the presented model appropriate formulation of the shape and scale parameters is of a major importance. Shape

parameter p is responsible for fitting volume effect into the model. It is an exponent inside the Weibull failure distribution.

As it was pointed out, especially in case of cyclic loadings volume effect can be described by a continuous and monotonic

function. The type of specimens was selected in order to establish the fatigue parameters for volume effect threshold.

Fig. 12. Geometry of the investigated elements: (a) transverse butt weld [4], and (b) transverse stiffener [4].

Fig. 13. Finite element models representative for the transverse butt weld (a) and transverse stiffener (b) welded elements.




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Comparison with the results from another research center demonstrated nearly parallel S –N curves which can imply that the

results are representative.

Scale parameter H describes fatigue properties of a certain material volume – the efficient material volume. To extract the

fatigue parameters of this volume, nominal stress approach was chosen as it can be regarded reliable in case of simple geom-

etries and loading. Results of the identification (Table 4) are to a certain extent affected by mesh refinement. Table 4 should

also be treated as a validation to the underlying assumptions. The number of investigated geometry variations and changes

in significant digits are an estimate of sensitivity to weld geometry. Despite the considerable change of notch radius q, the

obtained values resembled each other.

5. Summary

1. A non-local probabilistic computational model for fatigue life assessment of welded joints has been proposed in this

paper. It allows the evaluation of fatigue life distribution for a variety of steel welded joints. Transferability of the model

is based upon the introduction of proposed form for scale and shape parameters in the Weibull distribution. It is estab-

lished through the application of finite element modeling, introduction of S –N curve for efficient material and simulation

of volume effect. Such an approach governs the issues of geometry and fatigue behavior of welded joints, respectively.

2. A key issue focuses around the identification of the two parameters for efficient materialS –N curve. The process in the pro-

posed form involved finite element analysis undertaken for different finite element models varying in geometry and

notch radius. The obtained results served as an input into Nelder –Mead downhill simplex minimization of the introduced

identification estimator. For the lower radii obtained parameters were identical and for q = 0.1 mm equal to:

logCf = 13.65 and mf = 3.18.

3. The shape parameter provided the necessary insight into the S –N curve behavior. Based on the performed fatigue tests

and further comparison between data series for welded joints of a different volume, the exact value was established

to be equal to p = 66.82.4. Further validation and more refined application procedure are to be reported separately.

Generation of FE model representative for the chosen welded joint classified in

nominal stress approach

Determination of through-thickness stress field along the weld line and

corresponding to the chosen nominal stress ranges

Transformation of stresses according to the chosen multiaxial failure criterion

Generation of the pair of values for C f and m f parameters

Determination of E (C f ,m f ) identification estimator

Comparison of estimator values derived for two levels of nominal loading

Identification of C f and m f parameters according to the results from minimization

of E (C f ,m f ) identification estimator

Fig. 14. Identification algorithm for the parameters of efficient material S –N curve [13].

Table 4

Efficient material log(C f ), m f parameters for different values of notch radius q and joint geometry [13].

q (mm) Type a joint (butt weld, IIW no. 213) Type b joint (transverse stiffener, IIW no. 511)

E , 107 log(C f ) m f E , 107 log(C f ) m f

1,0 6 13.77 2.98 7 13.68 3.11

0,9 6 13.70 3.04 6 13.71 3.11

0,8 7 13.68 3.07 1 13.69 3.12

0,7 2 13.66 3.10 5 13.69 3.13

0,6 5 13.66 3.12 2 13.68 3.13

0,5 6 13.65 3.14 6 13.67 3.14

0,4 7 13.65 3.16 7 13.66 3.14

0,3 4 13.65 3.17 4 13.66 3.15

0,2 9 13.65 3.19 6 13.65 3.15

0,1 3 13.65 3.20 7 13.65 3.16


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13/13

Acknowledgement

This paper was realized as a part of the research Project no. DEC-2011/01/N/ST8/02566 funded by the National Science

Centre in Poland.

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Application of the weakest link analysis to the area of fatigue.pdf