a traction force approach for fatigue assessment of

This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.

Powered by TCPDF (www.tcpdf.org)

This material is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user.

Rautiainen, Matti; Remes, Heikki; Niemela, AriA traction force approach for fatigue assessment of complex welded structures

Published in:Fatigue and Fracture of Engineering Materials and Structures

DOI:10.1111/ffe.13548

Published: 01/11/2021

Document VersionPublisher's PDF, also known as Version of record

Published under the following license:CC BY

Please cite the original version:Rautiainen, M., Remes, H., & Niemela, A. (2021). A traction force approach for fatigue assessment of complexwelded structures. Fatigue and Fracture of Engineering Materials and Structures, 44(11), 3056-3076.https://doi.org/10.1111/ffe.13548

https://doi.org/10.1111/ffe.13548


OR I G I N A L AR T I C L E

A traction force approach for fatigue assessment of complexwelded structures

Matti Rautiainen1,2 | Heikki Remes1 | Ari Niemelä2

1Department of Mechanical Engineering,Aalto University School of Engineering,Espoo, Finland2Structural Design, Meyer Turku Oy,Turku, Finland

CorrespondenceMatti Rautiainen, Department ofMechanical Engineering, Aalto UniversitySchool of Engineering, P.O. Box 15300,Espoo 02150, Finland.Email: [email protected]

Funding informationMeyer Turku Oy

Abstract

A new traction force approach (TFA) for weld root fatigue analysis of complex

cruciform connections is introduced. The approach defines the local nominal

weld stress by using a solid element model and connecting the welded parts

only from the toe and root lines. This enables the peak weld forces to be dis-

tinctly defined. The TFA approach was validated with a fully connected solid

element model. As a comparison, the same analyses were performed for a shell

element model. The study shows that the approach that is introduced here

gives a good estimate for the local weld force with a maximum error of 10%

while the commonly used shell element model greatly overestimates the peak

weld force, particularly for complex connections. The analysis of fatigue-tested

connections showed that by using TFA, instead of an approach based on over-

all nominal weld stress, the scatter range index is reduced from 3.71 to 1.98.

The TFA approach is much more efficient compared to conventional

approaches because of the low pre- and post-processing effort.

KEYWORD S

fatigue life prediction, fatigue of welds, finite element analysis, welded structures

1 | INTRODUCTION

In welded structures, where scantlings are defined on thebasis of stability or stiffness criteria, e.g., buckling,welded connections do not necessarily require matchingstrength with the connected member. If a high numberof such joints exists, considerable production savings canbe achieved by reducing the number of welded stiffenersand by using single bead fillet welds instead of, e.g., fullpenetration welds. This often means that the jointbecomes unsymmetrical and the fatigue strength of theconnection is seriously affected as the fatigue initiationsite changes from the weld toe to the weld root. As a con-sequence, fatigue analysis becomes more challenging,

especially in large structures such as a ship hull or a steelbridge, where the connection between the structuralmembers is complex. Complex connections have a lack ofsymmetry planes, eccentricity of welds, thin intermediateplates, and stress concentrations as a result of theintersecting welds. One example of such a connectioncan be found in cruise ships, where a rectangular pillar iswelded to the deck and sides of the RHS profile arealigned with an underlying web, girder, and flat bar; seeFigure 1. With such a light structural arrangement, suffi-cient strength can be achieved since the pillar loading isnot significant. However, at present, the time-efficientfatigue design methods needed for the structural optimi-zation and for the approval process with classification

Received: 17 June 2021 Accepted: 10 July 2021

DOI: 10.1111/ffe.13548

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided

the original work is properly cited.

© 2021 The Authors. Fatigue & Fracture of Engineering Materials & Structures published by John Wiley & Sons Ltd.

Fatigue Fract Eng Mater Struct. 2021;1–21. wileyonlinelibrary.com/journal/ffe 1

https://orcid.org/0000-0001-6504-9188

mailto:[email protected]


http://creativecommons.org/licenses/by/4.0/

http://wileyonlinelibrary.com/journal/ffe

http://crossmark.crossref.org/dialog/?doi=10.1111%2Fffe.13548&domain=pdf&date_stamp=2021-08-03

societies do not exist for such complex connections.Considering that structural analysis is carried out usingthe finite element (FE) method, an efficient designmethod should be insensitive to element type, size, andshape. Additionally, the post-processing of the resultsshould be simple and straightforward in order to makethe method suitable for structural optimization.

According to present fatigue design standards andrecommendations, the nominal weld stress approach isthe most commonly adapted approach for the fatigueanalysis of a weld root.1–4 The nominal weld stress isdefined as the averaged stress at the weld

σw ¼ FΣ awlð Þ , ð1Þ

where F is the total transmitted force, aw the throat thick-ness, and l the total weld length l. In fillet welds aroundloaded attachment ends, the effective area and forceshould be used.5 For more complex filled welded joints,the definition of the nominal weld stress is a challenge,and thus, the use of local approaches such as effectivenotch stress (ENS) or structural weld stress isrecommended.3,4

The most common local approach is ENS. In ENS, awelded joint is modeled with solid or 2D elements,

including the weld geometry with a 1-mm radius at theweld toe and root. Since a very small element size, i.e., aquarter of the radius for higher order elements, isrequired, this approach is very time-consuming for com-plex structures. Furthermore, it is observed that ENS issensitive to the element mesh and the applied sub-modeling technique in the case of 3D structures.6,7 Toavoid the modeling of the weld radius, the peak stressmethod8 can be used. It uses solid FE models to calculatethe notch stress intensity factor at the weld root or toe,and thus, the weld root can be modeled with a zeroradius, making the geometry modeling straightforward.In the peak stress method, the element size can vary from0.4 to 2 mm. The disadvantage is that the method is sen-sitive to the selected element type and FE software.Fatigue assessment of a weld root can also be performedusing the strain energy density (SED) approach.9,10 Itderives from the theory that the high cycle fatigue ofsteels is driven by the critical value of the averaged strainenergy density at the control volume around the notchtip. When applying SED to the FE method, the controlvolume, with a radius of, e.g., 0.28 mm, is meshed with afine or coarse mesh. As in the peak stress approach, SEDuses a zero radius to model the notch tip, making thegeometry modeling straightforward. However, in largeand complex structures, challenge is that the required

FIGURE 1 An example of complex fillet

welded joints, where weld root fatigue is the

dominant failure mode [Colour figure can be

viewed at wileyonlinelibrary.com]

2 RAUTIAINEN ET AL.

http://wileyonlinelibrary.com

element size can be very small compared to overalldimensions of the structure. As a large structure such asshown in Figure 1 cannot be modeled with so smallelement size, it is challenging to control the element sizeto be changed from micrometer scale to millimeter orcentimeter scale. To achieve a smooth transition betweenthese element sizes, the number of elements can increasebeyond what is reasonable to be analyzed. This issueconcerns also all crack initiation and propagationapproaches.11,12

Using a structural weld stress approach, the elementsize requirement can be relaxed. In a structural weldstress approach, the stress is calculated at the weld rootby considering both the weld membrane stress and thelinearly distributed bending stress. For defining the struc-tural weld stress using FE modeling, different approacheshave been presented. Sorensen et al.13 used the linearextrapolation of the principal stress at the fillet weldthroat section to capture the bending effect, whileFricke14 suggests that the weld leg section perpendicularto the load should be used to calculate the weld rootstructural stress as the fatigue crack path is closer to that.To avoid a stress singularity problem at the sharp weldnotch, the traction structural stress approach, based onnodal forces, is also introduced.15 Although all thesestructural weld stress approaches are highly appreciated,they are still challenging in terms of the post-processingof the nodal forces to define the stress at the weld leg sec-tion. In complex geometries using automatic FE meshing,nodes are often positioned inconsistently along the weldsurface, making automated post-processing difficult. Fur-thermore, the critical position, i.e., the place where thereis the maximum stress, is impossible to predict before FEanalysis. Therefore, there is a need for local nominal orstructural stress approaches that can efficiently analyzeall the fillet welded joints in complex structures.

In this paper, a new approach is introduced for calcu-lating the local nominal weld stress of complex load-carrying cruciform connections. The modeling techniqueuses nodal traction forces. Differently from the existingmethods, fillet welds are connected only from their rootand toe borderlines. As a result, the weld forces are easilycaptured, and the local nominal stress at the welded jointis defined with minor post-processing work. In this work,the traction force approach (TFA) is applied for variousfilled weld configurations. The approach is shown to beaccurate and independent of element size, making it suit-able for automated FE modeling and optimizing struc-tural arrangement in welded connections.

This paper has the following structure and content.First, the method which TFA is based on is introduced inSection 2. Section 3 explains the complex connectionsselected for validation. The TFA approach is validated

with a fully connected solid element model and com-pared to the shell modeling approach. The results,e.g., the accuracy of the TFA, is shown in Section 4. Thissection also includes a comparison in terms of fatigue testdata. Sections 5 and 6 include the discussion andconclusions.

2 | TRACTION FORCE APPROACH

2.1 | Modeling principle

The purpose of the TFA approach is to solve the forcecomponents (i.e., normal and shear forces) acting locallyon a weld in a way that is easy to apply in FE modeling.These force components are used for calculating thenominal weld stress for the local nominal weld stressfatigue approach.

In a general case, a fillet welded connection is loadedby external forces and bending moments as shown inFigure 2A. Considering the weld leg section, the externalloading is transmitted from one structural member toanother by the normal force Nz, longitudinal shear forceQx, transverse shear force Qy, and bending moment Mx.The normal force Nz and bending moment Mx inducenonlinear normal stress distribution in the weld leg sec-tion. The shear stress τxy is induced by the longitudinalshear force Qx and the shear stress τyz by the transverseshear force Qy. The stress state is shown schematically inFigure 2B. In general, a welded connection can also beloaded by the longitudinal normal force. However, sincethis loading component is not transmitted from onestructural member to another by a weld, it is notconsidered here.

In order to get distinct weld force readings, apointwise (or linewise in 3D) connection is neededbetween the weld and the connected plate. Also, to solvethe correct transmission of the force from the welds onone side of an intermediate plate to another, the bendingstiffness of the welds needs to be included in the connec-tion. This is also required to solve the correct force distri-bution along the weld length in complex welded joints.

The principle in TFA is to model the contact of theweld leg section with two pivot points, i.e., the root andtoe point. As a result, the pivot points will have threeforce components acting on them, i.e., the normal forcesNz,t and Nz,r, the shear forces Qy,t and Qy,r, and the shearforces Qx,t and Qx,r (see Figure 2C). TFA is used for find-ing the normal force Nz, the shear forces Qx and Qy, andthe bending moment Mx.

Nz ¼Nz,tþNz,r , ð2Þ

RAUTIAINEN ET AL. 3

Qx ¼Qx,tþQx,r , ð3Þ

Qy ¼Qy,tþQy,r , ð4Þ

Mx ¼ 0:5Lw Nz,t�Nz,rð Þ: ð5Þ

2.2 | FE implementation

Applying TFA to FE modeling is explained here. In astructural design process, a 3D geometry model of thestructure is typically created, making the implementationof TFA particularly simple. The geometry model shouldinclude the weld geometry, and it should be modeledwith solid elements in order to model correctly the forcetransmission in complex welded connections. Therefore,3D geometry is not modified into a mid-surface model asin shell modeling. Since hexahedron meshing is challeng-ing for complex welded connections, in TFA, theconnection is modeled with tetrahedron elements. Beforeautomated mesh generation is used, the node specifica-tion (number or spacing) is only added to the borderlinesof the weld toe and root. In the model, the abutting parts

will have a continuous mesh with the fillet welds, ashighlighted in gray in Figure 3A. The abutting structureis then connected to the intermediate plate by mergingthe nodes at the toe and root borderlines. To prepare thegeometry model for meshing, the surfaces of the interme-diate plate and weld leg section need to have imprintedlines at the weld toe and root. This can be done byutilizing the geometry of the abutting structure. In orderto correctly sum up the nodal forces at the root and toelines, as per Equations 2–4, an equal number of nodesneeds to be set on these lines. The element size is selectedto be such as to maintain good element quality inautomated mesh generation. The following bullet pointsgive guidelines for FE modeling with the traction forceapproach.

• Obtain a 3D model of the structure, including weldgeometries.

• Set a continuous mesh for the abutting structuralmember and the weld.

• Utilizing the geometry model of the abutting structure,imprint weld root and toe lines on the intermediateplate and weld leg surface.

• Set a constant element size over the connectionregion.

FIGURE 2 General loading of welded

connection (A) and stress state at weld leg

section (B). Force components for TFA (c)

[Colour figure can be viewed at

wileyonlinelibrary.com]

FIGURE 3 Connected

nodes in TFA (A) and

corresponding nodal forces with

linear elements (B) and

parabolic elements (C) [Colour

figure can be viewed at


4 RAUTIAINEN ET AL.



• If needed, adjust the mesh so that an equal number ofelements is created along the root and toe line.

• After meshing, utilizing the imprinted line geometries,merge the nodes along the weld root and toe lines.

In the post-processing phase, the forces and momentas per Equations 2–5 are calculated element-wise alongthe weld length, using nodal forces Fi (Figure 3B,C), thedistances Di between nodes and the weld leg length Lw.Considering here only the normal force (Equation 2) andbending moment (Equation 5), the element-wise solutionfor 3D-structures is the line traction force fl and linebending moment ml shown in Table 1. The line tractionforce is calculated by dividing the combined elementalforce at weld toe and root line by the average elementlength at weld toe and root line. For linear elements, theelemental force is the average of element corner nodeforces (e.g., Fr2 and Fr3 for weld root line). With parabolicelements, the mid-node force is added. From the linetraction force, elemental nominal weld stress σw is calcu-lated according to the definition of IIW.4 For the linebending moment ml, bending moment calculated fromelemental weld toe and root forces is divided by the aver-age element length at weld toe and root line. All equa-tions, fl, ml, and σw, are presented for both linear andparabolic elements in Table 1.

3 | VALIDATION OF THETRACTION FORCE APPROACH

3.1 | Selected welded connections

The proposed TFA approach is validated with the ninecomplex load-carrying welded connections reported inTable 2. The selected connections are arranged in orderof increasing complexity. The first three cases are sym-metrical connections, for which fatigue tests are availablefrom the previous investigation.14,16 Cases 4 to 9 are morecomplex unsymmetrical connections with a more pro-nounced 3D effect. As shown in Figure 1, such a connec-tion can be found on cruise ships, where a rectangularpillar is welded to the deck, aligning the sides of the RHSprofile with an underlying web, girder, and flat bar. It is

possible to put such a complex structure with a thinintermediate plate into practice when the pillar load isnot significant. Moreover, cases 7 and 9 represent situa-tions where the pillar is positioned eccentrically becauseof manufacturing tolerances. A graphic representation ofthese connections can be seen in Figure 4, while detailedtest specimen dimensions for cases 1–3 can be foundfrom the original papers. The material of all the case con-nections is steel with a Young's modulus E = 210 GPaand Poisson's ratio ν = 0.3. A more thorough descriptionof the connections is given below.

In case 1, RHS-120x80x6 profiles are welded onto a15-mm intermediate plate using fillet welds with a meanthroat thickness of 4.7 mm, as shown in Figure 4A.14 Theconnection is loaded with an X-axis three-point bendingload. In such a connection, only a slight weld force con-centration is induced to the corners of the RHS profile,making the structural analysis with beam theory possiblewith reasonable accuracy. In cases 2 and 3, the level ofcomplexity is increased as they have welds intersecting ata 90� angle, creating a severe weld force concentrationat the intersection point. In case 2, the axial load isapplied on 6-mm-thick plates welded onto a 6-mm-thickintermediate plate and a 10-mm-thick cover plate(Figure 4B).14 The loaded plates are positioned at a 90�

angle. The failed fillet welds between the cover plate andintermediate plate have a mean throat thickness of4.3 mm. In case 3, a Ø76.1 � 2.9 tube is welded betweenHEA200 profiles, which are cut in half (Figure 4C).16 Anaxial load is applied to the side surfaces of the HEA webplate as the force is transmitted to the HEA profilethrough the friction force. A weld stress concentration iscreated at the point where the tube weld intersects withthe web of the HEA profile. The weld profile is a partialpenetration weld reinforced with a fillet weld with a totalmean throat thickness of 6.0 mm and a weld leg length of8.2 mm. Case 3 has a more severe stress concentrationthan case 2 because of the longer weld length.

In addition to symmetrical fatigue-tested connections(cases 1 to 3), the proposed TFA is validated with sixunsymmetrical and more complex connections. Cases 4 to9 have an RHS-150 � 100 � 12.5 profile welded to an inter-mediate plate supported by an unsymmetrical 7-mm-thickplate structure. The edges of the intermediate plate and the

TABLE 1 Equations for line traction force fl, line bending moment ml and nominal weld stress σw

Linear elements Parabolic elements

f li,iþ1 ¼ FriþFriþ1þFtiþFtiþ1Dri,iþ1þDti,iþ1

f li,iþ2 ¼ Friþ2Friþ1þFriþ2þFtiþ2Ftiþ1þFtiþ2Dri,iþ2þDti,iþ2

mli,iþ1 ¼ Lw FriþFriþ1�Fti�Ftiþ1ð ÞDri,iþ1þDti,iþ1

mli,iþ2 ¼ Lw Friþ2Friþ1þFriþ2�Fti�2Ftiþ1�Ftiþ2ð ÞDri,iþ2þDti,iþ2

σwi,iþ1 ¼ f i,iþ1�1mmaw

σwi,iþ2 ¼ f i,iþ2�1mmaw

RAUTIAINEN ET AL. 5

TABLE

2Caseconnection

sforvalid

ation

Case

Profile

Supporting

stru

cture

IPth

ickness

[mm]

Profile

weld

Supporting

stru

cture

weld

Numbe

rof

symmetry

planes

inco

nnection

Loa

dingtype

Profile

positioned

symmetrica

llywith

supportingstru

cture

1RHS-120x80x6

RHS-120x80x6

15VariedSSFW

VariedSSFW

24k

Nload

,3-point

bending

YES

2Plate-80x80x6

Plate-80x80x6

6an

d10

VariedDFW

VariedDFW

21k

Naxialload

YES

3CHS-76.1x2.6

HEA200web

plate,

t=6.5mm

10VariedSSPP

+FW

-1

2kN

axialload

NO,1

.95mm

offset

4RHS-150x100x12.5

Plate

intersection

,t=

7mm

10a5

SSFW

a4DFW

01k

Nm

bending

mom

ent

YES

5RHS-150x100x12.5

Plateintersection

,t=

7mm

5a5

SSFW

a4DFW

01k

Nm

bending

mom

ent

YES

6RHS-150x100x12.5

Plateintersection

,t=

7mm

10a5

SSFW

a4DFW

01k

Naxialload

YES

7RHS-150x100x12.5

Plateintersection

,t=

7mm

10a5

SSFW

a4DFW

01k

Naxialload

NO,7

mm

offset

8RHS-150x100x12.5

Plateintersection

,t=

7mm

5a5

SSFW

a4DFW

01k

Naxialload

YES

9RHS-150x100x12.5

Plateintersection

,t=

7mm

5a5

SSFW

a4DFW

01k

Naxialload

NO,7

mm

offset

6 RAUTIAINEN ET AL.

FIGURE 4 Geometry of case connections 1 to 9 [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 5 TFA models with connection lines for cases 1–9 (A–D) and principle of shell element modeling (E) [Colour figure can be

viewed at wileyonlinelibrary.com]

RAUTIAINEN ET AL. 7



supporting structure have symmetrical surface constraints.A load is applied to the end of the RHS profile. The RHSprofile and the supporting structure are connected to theintermediate plate with fillet welds. The connection conceptis the same for all of cases 4–9. The varied parameters arethe thickness of the intermediate plate, the alignment ofthe welds, i.e., the eccentricity and loading type as shownin Table 2. The connection geometry and alignment of thewelds are visualized in Figure 4D.

In cases 4 and 5, the RHS profile is loaded with anX-axis bending moment, inducing peak weld forces onthose corners of the RHS profile where the adjacentshorter edge is supported by a girder. Since the local loadis divided over the plate intersection area, there is no sig-nificant stress concentration at the weld. The intermedi-ate plate thicknesses in cases 4 and 5 are 10 and 5 mm,respectively. In cases 6 to 9, the RHS profile is loadedwith a tensile axial force. Because of the unsymmetricalsupporting structure, a secondary bending moment is cre-ated on the RHS profile. Therefore, high local weld forcesare induced on the corners of the RHS profile adjacent tothe un-supported shorter edge. In cases 6 and 7, thethickness of the intermediate plate is 10 mm, thus some-what enabling the local forces to be distributed. Com-pared to case 6, in case 7, the RHS profile is moved 7 mmin the XY-plane in order to create eccentricity in the con-nection. In cases 8 and 9, the complexity is increasedeven further, with a 5-mm-thick intermediate platewhich does not have the capacity to distribute the localweld forces, thereby causing severe stress concentrationson the weld. The level of complexity is highest in case9, where a thin intermediate plate is combined with anunsymmetrically positioned RHS profile according to thesupporting structure. Such an arrangement causesthe forces to distribute highly unsymmetrically to doublefillet welds of the supporting structure.

3.2 | FE modeling

FE models are created for all nine cases with three differ-ent modeling approaches. First, TFA models are created

in which the root and toe lines are connected(Figure 3A). Then, shell element models, which are con-ventionally used for such structures, are created for com-parison (Figure 5E). Lastly, for validation, fullyconnected FE models that have a continuous mesh overthe weld region are created (Figure 6A).

In Figure 5, the arrows show the loading directionand the surfaces highlighted in red show the symmetryboundary conditions that were applied (s.b.c). In all theFE models, Femap 11.4.217 was used for the pre- andpost-processing and NX-Nastran 11.0.218 as a solver.

For cases 1 to 3, structural symmetry is utilized bymodeling only one quarter of the structure. The boundaryconditions and loading are applied according to the testdescriptions.14,16 In case 1, a rigid link is created at the endof the RHS profile for suppressing Y-direction translation.A Y-direction load is applied to the top face of the interme-diate plate. Symmetry boundary conditions are set in theYZ and XY planes as shown in Figure 5A. In case 2, anaxial load is applied to the end faces of the 6-mm plates,which are free to deform. The symmetry boundary condi-tions are set in the XZ and YZ planes as shown inFigure 5B. In case 3, a Z-direction load is applied to theclamping surface around the hole in the HEA200 profileweb. The symmetry boundary conditions are set in the XYand XZ planes as shown in Figure 5C. In cases 4 to 9, arigid link is modeled to the free end of the RHS profile forapplying bending loads for cases 4 and 5 and an axial ten-sile load for cases 6 to 9. As shown in Figure 5D, symmetryboundary conditions are set in five planes of the supportingstructure to enable the structure to deform as the structurewould be continuous in the x, y, and z directions. The num-ber of elements and nodes and the element type from theNX Nastran element library18 used for each model aregiven in Table 3. In the remaining part of this section, theelement mesh is described for each case.

3.2.1 | Models for traction force approach

In order to study the effect of element size, two mesheswith 2.5- and 1.25-mm elements are created with linear

FIGURE 6 Connected nodes in full

connection model and corresponding

nodal force sum from sliced surface



8 RAUTIAINEN ET AL.


TABLE

3Num

berof

elem

ents(nod

es)in

FEmod

elswithaveraged

values

ofcases4-9markedwith*

Case1

Case2

Case3

Cases

4–9*

Elemen

tsize

andtype

TFA

Full

Shell

TFA

Full

Shell

TFA

Full

Shell

TFA

Full

Shell

2.5-mm

linear

elem

ents

98,456

(22,927)

-6,214

(6,339)

20,042

(5,132)

-1,584

(1,700)

79,051

(18,007)

-4,173

(4,324)

502,659

(113,846)

-27,734

(27,361)

TFA:4

-nod

eCTETRA

Shell:4-nod

eCQUAD4

2.5-mm

parabo

licelem

ents

98,456

(157,029)

99,193

(157,565)

-20,042

(33,765)

20,100

(33,548)

-79,051

(124,604)

77,018

(120,892)

-502,659

(793,315)

533,401

(834,916)

-

TFA:1

0-nod

eCTETRA

Full:10-nod

eCTETRA

1.25-m

mlin

ear

elem

ents

640,806

(129,908)

--

144,887

(31,393)

--

505,330

(102,952)

--

2,674,120

(539,166)

--

TFA:4

-nod

eCTETRA

1.25-m

mpa

rabo

licelem

ents

640,806

(946,841)

--

144,887

(221,916)

--

505,330

(748,949)

--

2,674,120

(3,386,432)

--

TFA:1

0-nod

eCTETRA

RAUTIAINEN ET AL. 9

and parabolic element types. Cases 2 and 3 are modeledwith a single element size, whereas in the other cases,the areas outside the connection region are modeled witha coarser mesh using a 10-mm element size. Because ofthe complexity of the connections, tetrahedron elementsare used to maintain good element quality. As describedin Section 2, a mesh is created so that the nodes are coin-cident at the root and toe lines and an equal number ofnodes is created on both lines. The FE mesh used for theanalysis and the connected node lines are highlighted inFigure 5.

3.2.2 | Shell element model

In the shell model, elements are created in the mid-planeof the wall thickness using the principle shown inFigure 5E. For weld elements, the thickness is set to beequal to the actual weld throat thickness, and elementsare modeled at half the distance of the throat thickness,as recommended by Radaj et al.19 Linear shell elementswith an element size of 2.5 mm are applied. Similarly tothe TFA models, cases 2 and 3 are modeled with a singleelement size, whereas in other cases, areas outside theconnection region are modeled with a coarser mesh usinga 10-mm element size. The averaged line traction force fland bending moment ml are calculated as

f li,iþ1 ¼0:5 FiþFiþ1ð Þ

Li,iþ1, ð6Þ

mli,iþ1 ¼ 0:5 MiþMiþ1ð ÞLi,iþ1

: ð7Þ

3.2.3 | Full connection model

The full connection model is considered as an accuratemodel used for validation. The difference from TFA isthat all the nodes along the weld leg are connected to theintermediate plate, as shown in Figure 6A. Otherwise,the meshes for cases 1–9 are as shown in Figure 5. Inorder to get a distinct value for the weld force along theweld line, the connecting surfaces are sliced into 5-mm-long sections before meshing (Figure 6B). An elementsize of 2.5 mm with parabolic tetrahedra is used. Again,cases 2 and 3 are modeled with a single element size,whereas in the other cases, areas outside the connectionregion are modeled with a coarser mesh using a 10-mmelement size. In the full connection model, the nodalsum force within the sliced sections at the connection

surface is divided by the average length of thesection (Figure 6B). A section-wise graph is created afterall the sum forces are read. For each section, the otheredge nodes are excluded from the nodal sum, thus creat-ing the correct sum force for the given weld length. Thebending moment for the section is calculated fromthe node forces to half the width of the weld throatsection.

4 | RESULTS

4.1 | Validation of traction forceapproach for complex connections

The traction force approach (TFA) is validated by com-paring the weld force distribution resulting from TFA tothat from the full connection model. In addition to theweld force distribution plots, numerical values forthe peak weld force and error percentage are shown. Theanalyses of the results for TFA include the linear and par-abolic element models with element sizes of 1.25 and2.5 mm. The comparisons are shown for the most rele-vant, i.e., heavily loaded welds, divided into single-sidedand double fillet welds. All the weld force distributionplots are presented in Appendix A (Figures A1–A3),including also the degree of bending δb calculated as

δb ¼ σwbj jσwbj jþ σwmb c¼

6 mlj jLw2

1Lw

f lj jþ 6 mlj jLw

� �¼6 mlj jLw

f lj jþ 6 mlj jLw

: ð8Þ

4.1.1 | Single-sided fillet welds

The single-sided welds that were analyzed, together withthe plotting paths, are shown in Figure 7. Case 1 is thecorner of the hollow section that failed in the fatiguetests. For case 2, the fatigue-critical location is the weldbetween the cover plate and intermediate plate. For case3, the failed weld in the fatigue tests was the Ø76.1 � 2.9tube weld. Cases 4–9 have single-sided fillet welds onRHS for which the weld forces are plotted for two weldsegments separately, as shown in Figure 7.

Figure 8 shows an example of a weld force distribu-tion comparison for the Case 8 RHS weld. It can be seenthat TFA can capture the force distribution and peakforce value, whereas the shell element model overesti-mates the peak weld force value, predicting force concen-trations also in unrealistic locations. The reason for thisis the force concentration that occurs at each locationwhere the shell elements intersect. Such areas are found

10 RAUTIAINEN ET AL.

on those corners of the RHS profile where the adjacentshorter edge is supported by a girder (at the 30-mm loca-tion in Figure 8) and on those corners of the RHS profileadjacent to the unsupported shorter edge (at the 170-mmlocation in Figure 8). There is only a minor differencebetween using 2.5- or 1.25-mm elements, showing thatthe TFA is mesh-insensitive. Similar findings areobserved for all the cases that are studied.

A summary of all the results for single-sided filletwelds is given in Table 4. Zero error is considered if thepeak force along a weld line averaged for a 5-mm lengthis the same with the full connection model. In cases 4–9,the error is calculated for the �Y-side segment that has ahigher peak force. For TFA, the maximum error is 10%and the average error 4%. For the shell element model,the maximum error is 34% and the average error 13%,with only an element size of 2.5 mm being considered. Inaddition to proper estimation of the weld force, the trac-tion force can be used to indicate the degree of bendingδb. The example of the δb values at the peak weld forcelocation are shown for a fully connected model resultand TFA with a 2.5-mm linear element model in bracketsin Figure 8. All the results are provided in Table 4, wherethe error for δb is given in percentage points. Theaccuracy of the bending estimation is very good in thesymmetrical cases 1 to 3, with an average error of 2 pp,and reasonably good in cases 4 and 5, with an averageerror of 5 pp. As the complexity increases in cases 6 to

9, so does the error in the accuracy of the degree of bend-ing, being 14 pp on average. In general, it is observed thatwhen the actual value of degree of bending is below 0.55,the error increases to above 10 pp.

4.1.2 | Double fillet welds

Double fillet welds concern cases 4–9, namely, web, flatbar, and girder welds to an intermediate plate. The weldforces are plotted individually for the inner and outerwelds of the web and flat bar, as shown in Figure 9. Sincethe forces at the girder welds are low, they are not shown.In the web-to-girder and flat bar-to-girder inter-section area, the total force at the corner segment cannotbe distinctly defined because of the intersecting welds. Toovercome this issue, it is considered here that it is appro-priate to define the local fatigue failure in the inter-section area as a failure of the corner segment and itsfirst adjacent segments together, as shown in Figure 9.The selected area is comparable to the one used in thecase of a fillet weld around an attachment end in earlierstudies.20 The starting point for the weld distributionplots is set after the intersection area. In all cases, thepeak forces were outside the intersection area.

Figure 10 shows an example of a comparison of theweld force distribution of an inner fillet weld of a weband flat bar in case 8; all the results are given in

FIGURE 7 Direction with start and stop

point for plotting results for single-sided

welds [Colour figure can be viewed at


FIGURE 8 Example of weld force

and degree of bending of single-sided

fillet welds [Colour figure can be viewed

at wileyonlinelibrary.com]

RAUTIAINEN ET AL. 11



Appendix A. It is observed that TFA can capture the forcedistribution and peak weld force in all fatigue-criticalwelds. For the shell element model, the peak weld forceis highly overestimated for inner welds in cases 6–9,while the accuracy of the outer welds is good in all cases.The location of the peak weld force is not predictedcorrectly in cases 4–5 with the shell element model.

Table 5 shows the error in the peak weld force for allthe double fillet welds. When the error is being defined,both welds in a double fillet weld are considered as onewelded connection. Therefore, the peak force foundeither from the inner or outer weld defines the error ofthe double fillet weld. Zero error is considered if the peakforce of the web and flat bar double fillet welds averagedto a 5-mm distance is the same as in the full connectionmodel. Considering the results of a 2.5-mm linear ele-ment mesh, the maximum error for TFA is 8% and themean error 3%. When all four different meshes for TFAare considered, the maximum error is 10% and the aver-age error 4%. For the shell element model, the maximumerror is 74% and the average error 29%. For double filletwelds, the accuracy of the degree of bending from TFA isgood in all cases, with the error being on average 3 pp.All the results for δb are given in Table 5, where the errorfor δb is given in percentage points.

4.2 | Influence of structural complexity

As the structural complexity increases, the accuracy ofthe shell element model significantly decreases, while theaccuracy of TFA remains same. This effect is shown inFigure 11 for single (A) and double fillet welds(B) separately. As can be seen in Figure 11A, case 1 hasgood accuracy in both models since it does not containsevere weld force concentrations. Case 3 already has con-siderable structural discontinuity since the weld of theØ76.1 � 2.9 tube intersects with the web plate of theHEA200 profile. In case 5, in addition to intersectingwelds, the connection is unsymmetrical, thus concentrat-ing forces on those corners of the RHS profile where theadjacent shorter edge is supported by the girder. Suchcomplexity creates a highly overestimated force concen-tration at the RHS corner using the shell element model.

As shown in Figure 11B for double fillet welds, theaccuracy is good for both models for the outer weldssince they do not contain intersecting welds that wouldcause severe force concentrations (see also Figure 4D).The inner welds, however, intersect with the RHS filletweld. Also, as cases 4–9 are highly unsymmetrical, weldforces are unsymmetrically distributed along theweld length. Here, the influence on the shell elementmodel is demonstrated for the inner web welds in casesT

ABLE

4Error

(e)(%

)of

fatigu

ecritical

weldforce(N

/mm)an

derror(pp)

ofdegree

ofbendingwithvaried

elem

entsize

andtype

insingle-sidedfille

twelds

Case

12

34

56

78

9

Value

eValue

eValue

eValue

eValue

eValue

eValue

eValue

eValue

e

Fullcon

nection

mod

elForce

60.9

013.9

020.9

088.0

090.8

04.6

04.7

05.4

06.4

0

δ b0.70

00.70

00.73

00.58

00.52

00.47

00.53

00.30

00.55

0

Shellm

odel,2

.5-m

mlin

ear

Force

57.2

616.0

1525.5

2285.2

3121.8

344.1

105.0

55.9

97.1

10

TFA,2.5-m

mlin

ear

Force

60.3

114.2

219.7

687.9

088.8

24.3

54.5

55.2

45.8

9

δ b0.73

30.73

30.74

10.64

60.61

90.62

150.58

50.54

240.58

3

TFA,2.5-m

mpa

rabo

licForce

61.9

214.3

319.2

885.4

385.5

64.4

44.4

75.2

36.1

5

TFA,1.25-mm

linear

Force

61.0

014.4

319.3

887.7

087.8

34.4

44.5

45.2

36.2

4

TFA,1.25-mm

parabo

lic

Force

60.4

114.3

218.8

1085.6

383.7

84.2

74.3

85.1

56.1

6


5 and 8. The shell element model highly overestimatesthe peak force and predicts a wrong position for it in case5, where intersecting welds at the +X edge (see Figure 9)are estimated to have a higher local load than the web-girder intersection at the �X edge. On one hand, this isbecause the complex load distribution on each weldis not correctly captured in the web-girder inter-section area. On the other hand, the intersecting welds atthe +X edge create an overestimated local weld force,which is a typical effect in shell element models.

4.3 | Fatigue strength of complex load-carrying cruciform connections

Fatigue test results were reviewed for cases 1–314,16 bycalculating the S-N curves, standard deviation, and scat-ter index on the basis of the nominal weld stress andlocal nominal weld stress. The load ratio in the tests wasR = 0.5 for case 1 and R = 0 for cases 2–3. The R = 0 testresults are corrected to R = 0.5 by multiplying the stressvalues by a factor of 1/1.1, according to the mean

stress correction.21 In the nominal weld stress approach,the weld stress is calculated with beam theory, consider-ing the actual weld throat thickness for each specimen.In the local nominal weld stress approach, the averagedweld force for a 5-mm length is calculated for each spec-imen using the TFA approach. In each case, 1–3, thesame FE model is used for all the specimens, using theaverage weld throat thickness of all the specimens. Thelocal nominal weld stress is then calculated by dividingthe local weld force by the actual weld throat thicknessof the corresponding specimen. To compare the effect ofthe complexity of the connections to the resulting S-Ncurves, the force concentration factor (FCF) was calcu-lated by dividing the peak weld force by the nominalweld force.

From the results (Figure 12), it can be seen that theFAT36 design curve given in IIW, DNV-GL, and EC3 isun-conservative for the nominal weld stress approach butwell suitable for the local nominal weld stress approach.In cases 1–3, the weld dimensions are as defined for theFAT36 curve in IIW, i.e., ratio of fillet weld throatthickness aw to the thickness of a loaded plate t larger

FIGURE 9 Direction with

start and stop points for plotting

results for double fillet welds



FIGURE 10 Example of weld force

and degree of bending of double fillet

welds [Colour figure can be viewed at





TABLE

5Error

(e)(%

)of

fatigu

ecritical

weldforce(N

/mm)an

derror(pp)

ofdegree

ofbendingwithvaried

elem

entsize

andtype

indo

ublefille

twelds

Case

4web

4FB

5web

5FB

6web

6FB

Value

eValue

eValue

eValue

ee

Value

e

Fullcon

nection

mod

elForce

33.7

035.2

053.9

053.6

00

2.0

0

δ b0.68

00.70

00.68

00.68

00

0.65

0

Shellm

odel,2

.5-m

mlinear

Force

31.7

633.0

650.8

646.4

1320

2.2

11

TFA,2.5-m

mlin

ear

Force

33.7

034.9

155.1

251.9

32

1.9

4

δ b0.74

60.73

30.73

50.73

53

0.70

5

TFA,2.5-m

mpa

rabo

lic

Force

33.9

035.4

153.5

152.2

37

1.9

3

TFA,1.25-mm

linear

Force

33.5

135.2

054.2

052.4

25

1.9

3

TFA,1.25-mm

parabo

licForce

33.3

134.9

152.0

451.1

57

1.9

4

TABLE

5(Con

tinued)

Case

7web

7FB

8web

8FB

9web

9FB

Value

eValue

eValue

eValue

eValue

eValue

e

Fullcon

nection

mod

el2.9

01.9

03.6

02.5

04.8

02.9

0

0.58

00.63

00.71

00.70

00.62

00.68

0

Shellm

odel,2

.5-m

mlin

ear

4.9

693.3

745.2

443.6

436.5

343.5

22

TFA,2.5-m

mlin

ear

2.8

51.9

13.4

52.6

24.5

82.7

4

0.65

70.67

40.70

10.72

20.63

10.67

1

TFA,2.5-m

mpa

rabo

lic2.7

81.9

13.4

62.6

34.5

72.8

2

TFA,1.25-mm

linear

2.7

71.9

03.4

52.6

34.5

72.8

3

TFA,1.25-mm

parabo

lic2.6

101.8

23.4

52.7

64.4

92.8

2


than 1/3. In case 3, in addition to the fillet weld, weldpenetration was also present.

The scatter range index Tσ was calculated accordingto Radaj et al.19 using a 10% and 90% probability of sur-vival with an S-N curve slope m = 3. The resulting scatterrange index is Tσ = 3.71 for the nominal weld stress andTσ = 1.98 for the local nominal weld stress. It is worthnoting that these results for the local nominal weld stressmatch well with the results calculated with full connec-tion models, which result in a scatter range indexTσ = 1.89. From the nominal weld stress results, it can beseen that there is a high scatter between each case with adifferent level of FCF, whereas in the case of the localnominal weld stress, this scatter is clearly smaller.

5 | DISCUSSION

The cases to validate the traction force approach (TFA)were selected to be such that the complexity of the caseconnection gradually increased. The simplest connection,i.e., case 1, did not contain any structural discontinuitiesthat would influence the nominal weld stress. At the nextlevel of complexity, i.e., cases 2–3, symmetrical connec-tions included a sharp stress concentration resulting fromintersecting welds over the intermediate plate. Thehighest level of complexity, i.e., cases 4–9, contained anunsymmetrical arrangement of scantlings, eccentricity,and intersecting welds over the intermediate plate. Cases4–9 have great significance since research is typicallyfocused on simplified and symmetrical connections, andyet cases 4–9 represent a realistic structural design in a

steel structure industry such as shipbuilding. In actualsteel structures, there is always also a certain amount ofeccentricity between connected plates. In previousresearch, this behavior has been studied with 2D casesfor which analytical formulae can also be found forfatigue analysis.22 In the present paper, a rather severe7-mm eccentricity was included in cases 7 and 9.

When the effect of the thickness of the intermediateplate is studied, it can be seen that the peak weld force offatigue-critical welds increases up to 37% in single-sidedfillet welds and up to 67% in double fillet welds whenthickness is decreased from 10 to 5 mm. On the otherhand, when the effect of eccentricity is studied, it can beseen that the peak weld force of fatigue-critical weldsincreases by up to 35% when the eccentricity is increasedfrom 0 to 7 mm.

The traction force approach introduced in this paperrequires small pre-processing effort in FE analysis as,typically, 3D geometry with fillet welds is available forFE modeling. The required modeling routine is first tomerge the weld geometry with the abutting structuralmember and to imprint the weld toe and root lines tothe surfaces of the weld leg and intermediate plate. Afterthis, an equal number of nodes is set to the toe and rootlines and coincident nodes merged on these lines. As allthe nodes to be merged are associated with the root andtoe line geometries, the merging of the correct nodes byusing the line geometries is easy. In the post-processingphase, the location of the peak weld force is easy todetect from the nodal forces of the two lines and thepeak force value is easy to define from the node dis-tances (see the equations in Table 1). If the nodal forces

FIGURE 11 Shell element

model and TFA error for

increasing complexity of

connection, single-sided fillet

welds (A) and double fillet

welds (B) [Colour figure can be

viewed at wileyonlinelibrary.

com]




and node coordinates along the weld length are providedin tabular format, it is rather easy to create a weld forcedistribution plot using these equations. A weld force dis-tribution plot reveals the weld force gradients, which arean important measure for the designer in optimizing theconnection.

Fully connected modeling with 3D solid elements ora 2D element is a well-known approach, as used for weldroot analysis, e.g., by Fricke.14 For weld toe structuralstress analysis, it has been widely used, e.g., by Dong.23

Another approach utilizing nodal forces and fully con-nected modeling is the traction structural stressapproach.15 With that approach, meshing needs to becontrolled by mapped meshing to create element linesacross the weld throat thickness and across the platethickness at the weld toe. The mesh also needs to be con-trolled in the longitudinal direction of the weld to haveplanar element lines in each cross-section along the weld.Therefore, the disadvantage is that in a complex 3D con-nection such as that presented in Figure 4D such meshmapping is extremely time-consuming. In the proposedTFA approach, the mesh controlling is not so strict as theonly requirement is to have an equal number of nodes onthe weld toe and root lines.

In complex 3D connections, including many weldspre- and post-processing becomes challenging for allapproaches that require fully connected modeling tech-nique. In the pre-processing phase, special attention isrequired in merging the nodes because of the unfusedroot surfaces. The other challenge is that for detectingthe peak load location and defining a distinct force/mmvalue the weld leg surface needs to be sliced into sectionswith a specific length, e.g., 5 mm, as was done in thispaper (Figure 6B). Creating weld force plots also requiresa more complex code than for TFA as in addition to theweld line axis, the axis in the thickness direction alsoneeds to be included. Moreover, compared to straightwelds, the pre- and post-processing work becomes evenmore challenging when the weld line is curved. Since inthe connection optimization process multiple conceptsare typically tested, significant savings in the total

analysis time can be achieved by using the traction forceapproach.

Similarly to other traction force-basedapproaches,15,23 the TFA presented in this paper wasfound to be mesh-insensitive since with the four differentmeshes; i.e., varying element size and types, on average,the maximum difference between the peak weld force ofthe fatigue-critical welds was 3%. Considering thefatigue-critical welds in cases 1–9, the maximum error inthe TFA approach was 10%. For single-sided fillet welds,the highest error was found in cases 3 and 9. In thesecases, there was also a severe force concentration.Looking at cases 1–3, the highest force concentrationfactor (FCF) was in case 3, with an FCF of 2.5. For cases4–9, the highest FCF of 2.8 is found for case 9. In thedouble fillet welds, the highest error was at the web-to-IPweld in cases 7 and 9, with eccentricity of 7 mm. Thehighest peak force was found for case 9, and yet the con-nection between the magnitude of error and the magni-tude of the peak force is not as clear as with single-sidedfillet welds. In addition to the correct peak force of thefatigue-critical weld, i.e., a weld with a higher loading,TFA is able to capture the load ratio of the inner andouter welds.

A comparison between the nominal and local nomi-nal weld stress for the fatigue test results in cases 1–3showed that the local nominal weld stress approachresults in more reliable fatigue life estimations with asmaller scatter range index (Tσ = 1.98). This is a well-expected result since fatigue failures originate from thestress concentrations where the stress is highest. Stressconcentrations are not included in the nominal weldstress approach, which is reflected as a higher scatter infatigue test results. These test points are also logically dis-tributed over the S-N plot since the points in case 3 withthe highest FCF, i.e., 2.5, have the lowest fatigue strengthwhen analyzed with the nominal weld stress approach.Case 2, with FCF = 2.2, has higher fatigue strength, andcase 3, with FCF = 1.02, has clearly the highest fatiguestrength. Despite the clear improvement in scatter fromthe nominal to the local nominal weld stress approach,

FIGURE 12 Literature fatigue test

data, nominal weld stress (A) and local

nominal weld stress (B) [Colour figure

can be viewed at wileyonlinelibrary.

com]




the scatter is still somewhat higher than for simplewelded connections where Tσ = 1.5 has been observedfor weld toe failures by, e.g., Meneghetti.24 This is asimilar observation to that of Meneghetti et al.8 using apeak stress approach. Higher scatter is observed forthree-dimensional welded connections than for simplesmall-scale specimens of welded joints. The results alsoshow that FAT36 found from standards and recommen-dations for analysis based on the nominal weld stressmatch the local nominal weld stress results well. On theother hand, over half of the test data falls below FAT36with the nominal weld stress approach. It is worth notingthat all the test series had almost equal degrees of bend-ing, varying between 0.70 and 0.73.

The accuracy of the shell element models was verypoor, as error up to 74% was detected. Only the outerdouble fillet welds that do not intersect with RHS weldswere found with good accuracy. One example of theinstability of shell element models is the RHS-to-IP weldsin cases 4 and 5. Case 4 has good peak weld force accu-racy even though the force distribution is not correct.However, just as a result of changing the thickness of theintermediate plate from 10 to 5 mm in case 5, the peakweld force error increases drastically. This is becausemodeling with mid-plate elements creates the weld linesat a different position in the two cases. Thus, in case5, the RHS welds more clearly intersect with a doublefillet weld, thus creating an unrealistically high force con-centration. Generally, in welds that intersected with RHSwelds, the error was high. In such cases, there were alsoweld force peaks in unrealistic locations. The validity ofthe shell element's plane stress assumption was verifiedin the cases that were studied since the stress in thethickness direction is negligible compared to the stressesin the loading direction. The aforementioned challengesin the stress calculations make the utilization of the shellmodel impossible in the automated analysis routinesrequired for structural optimization. Using TFA, thesechallenges are removed, and there is still analyticalefficiency and the possibility of automated meshing andanalysis. It is also worthy of note that TFA can predictthe degree of bending in the weld leg section. Thisinformation can be used as an indicator in optimizing theconnection since a lower degree of bending results inhigher fatigue strength through structural stress analysisif the local weld force is kept the same.

In the simplest case that was studied, case 1, the con-nection type was a single-sided load carrying a cruciformconnection with only a very light weld force concentra-tion along the weld. This suggests that the approach isalso suitable for analyzing 2D cross-sections of simpleload-carrying cruciform connections which also do notconsider any force concentrations along the weld.

However, compared to a full connection model, the timesaving with TFA in such connections is not as significantas with complex 3D connections. In such connections, inaddition to weld root fatigue, another possible failuremode is weld toe fatigue, which can be analyzed withthe well-accepted structural hot spot approach.1,3 Alter-natively, the TFA introduced here can also be validatedfor the analysis of weld toe failure. This is beyond thescope of current study, and thus, this is left for futurework.

The connections studied in this paper were axiallyloaded, and thus, the effect of longitudinal shear stressesis small compared to stress components in the loadingdirection. However, the traction force approach can alsobe used to calculate the longitudinal shear stress for caseswhere shearing effects need to be considered. This topicis left for future work.

6 | CONCLUSION

This paper introduced a new traction force approach(TFA) for efficient fatigue analysis of load-carrying cruci-form connections in large complex structures. The valida-tion of TFA was performed by comparing the results witha full connection solid element model that had a realisticconnection over the welds. Furthermore, the fatigue scat-ter was compared using a nominal weld stress approachand a local nominal weld stress approach obtained byTFA. Based on the present study, the following conclu-sion can be drawn:

• The TFA, connecting the welded parts only from thetoe and root lines, can correctly and efficiently definelocal peak weld forces for the complex welded connec-tions. The maximum peak force error of fatigue-criticalwelds was 10%, with the average error being 4%. Fur-thermore, the approach was found to be independentof element size and formulation.

• For complex welded connections, traditionally usedshell element models show severe challenges in cap-turing the correct weld forces, as the maximum peakforce error of fatigue-critical welds was 74%, with theaverage error being 22%. This was due to shell ele-ment's improper modeling of local stiffness in the areaof intersecting welds.

• The fatigue analysis based on local weld forces usingTFA reduced the scatter range index from 3.71 to 1.98when compared to a nominal weld stress approach,showing that TFA can result in reliable fatiguestrength predictions.

• When TFA is compared to the alternative full connec-tion model, significant analysis time savings can be


achieved in the pre- and post-processing phases forcomplex connections.

In this paper, the TFA was validated for complex filletwelded joints under axial loading. In order to validate theapproach for other joint types and weld profiles,e.g., fillet welds with significant weld penetration, furtherstudy is needed. Furthermore, the shear-loaded connec-tions need to be considered as future work.

ACKNOWLEDGMENTSThis research was funded by Meyer Turku Oy. The financialsupport is gratefully acknowledged. Sincere thanks to Prof.Jani Romanoff for his valuable comments on the paper.

DATA AVAILABILITY STATEMENTThe authors elect to not share data.

ORCIDMatti Rautiainen https://orcid.org/0000-0001-6504-9188

REFERENCES1. Eurocode 3, Design of steel structures, Part 1-9, Fatigue, EN

1993-1-9, Brussels, CEN, 2006.2. DNV-GL. DNVGL-RP-C203. Fatigue design of offshore steel

structures. 2014 06/:201.3. DNV-GL. DNVGL-CG-0129. Fatigue assessment of ship struc-

tures. 2018 01/:236.4. Hobbacher AF. Recommendations for Fatigue Design of Welded

Joints and Components. Cham: Springer; 2016.5. Fricke W. IIW guideline for the assessment of weld root

fatigue. Weld World. 2013;57(6):753-791.6. Fricke W, Bollero A, Chirica I, et al. Round robin study on

structural hot-spot and effective notch stress analysis. ShipsOffshore Struct. 2008;3(4):335-345.

7. Fricke W, Codda M, Feltz O, et al. Round robin study on localstress and fatigue assessment of lap joints and doubler plates.Ships Offshore Struct. 2013;8(6):621-627.

8. Meneghetti G, Guzzella C, Atzori B. The peak stress method com-bined with 3D finite element models for fatigue assessment of toeand root cracking in steel welded joints subjected to axial or bend-ing loading. Fatigue Fract Eng Mater Struct. 2014;37(7):722-739.

9. Foti P, Berto F. Fatigue assessment of high strength weldedjoints through the strain energy density method. Fatigue FractEng Mater Struct. 2020;43(11):2694-2702.

10. Foti P, Santonocito D, Risitano G, Berto F. Fatigue assessmentof cruciform joints: Comparison between strain energy density

predictions and current standards and recommendations. EngStruct. 2021;230:111708.

11. Carpinteri A, Ronchei C, Scorza D, Vantadori S. Fracturemechanics based approach to fatigue analysis of welded joints.Eng Fail Anal. 2015;49:67-78.

12. Remes H, Gallo P, Jelovica J, Romanoff J, Lehto P. Fatiguestrength modelling of high-performing welded joints. Int JFatigue. 2020;135:105555.

13. Sorensen J, Tychsen J, Andersen J, Brandstrup R. Fatigueanalysis of load-carrying fillet welds. J Offshore Mech Arct Eng.2006;128(1):65-74.

14. Fricke W. Fatigue assessment of root cracking of fillet weldssubject to throat bending using the structural stress approach.Weld World. 2006;50(7):64-74.

15. Wang P, Pei X, Dong P, Song S. Traction structural stress analy-sis of fatigue behaviors of rib-to-deck joints in orthotropicbridge deck. Int J Fatigue. 2019;17:11-22.

16. Krenzel M. Fatigue strength of a welded tube-to-plate joint atan offshore structure [dissertation]. University of SouthernDenmark; 2019.

17. Femap. User Guide Version 11.4.218. NX Nastran. 11.0.2 Release Guide19. Radaj D, Sonsino C, Fricke W. Fatigue Assessment of Welded

Joints by Local Approaches. 2nd ed. Cambridge: WoodheadPublishing; 2006.

20. Fricke W, Doerk O. Simplified approach to fatigue strengthassessment of fillet-welded attachment ends. Int J Fatigue.2006;28(2):141-150.

21. Sonsino CM. A consideration of allowable equivalent stressesfor fatigue design of welded joints according to the notch stressconcept with the reference radii rref = 1.00 and 0.05 mm. WeldWorld. 2009;53(3–4):64-75.

22. Xing S, Dong P. An analytical SCF solution method for jointmisalignments and application in fatigue test data interpreta-tion. Mar Struct. 2016;50:143-161.

23. Dong P. A structural stress definition and numerical imple-mentation for fatigue analysis of welded joints. Int J Fatigue.2001;23(10):865-876.

24. Meneghetti G. The peak stress method applied to fatiguestrength assessments of load carrying transverse fillet weldswith toe or root failures. Struct Durability & Health Monit.2012;8(2):111-130.

How to cite this article: Rautiainen M, Remes H,Niemelä A. A traction force approach for fatigueassessment of complex welded structures. FatigueFract Eng Mater Struct. 2021;1-21. https://doi.org/10.1111/ffe.13548


https://orcid.org/0000-0001-6504-9188

https://orcid.org/0000-0001-6504-9188

https://orcid.org/0000-0001-6504-9188



APPENDIX A: WELD FORCE PLOTS

FIGURE A1 Weld force

distribution plots for cases 1–5[Colour figure can be viewed at




a traction force approach for fatigue assessment of

Documents