parametric study of scf on tripod joint

Parametric Study of SCF onTripod Joint

Yanxin Qin

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PARAMETRIC STUDY OF SCF ON TRIPODJOINT

by

Yanxin Qin

in partial fulfillment of the requirements for the degree of

Master of Sciencein Offshore and Dredging Engineering

at the Delft University of Technology,to be defended publicly on Thursday September 28, 2016 at 10:00 AM.

Supervisor: A. Prof. dr. ir. A. RomeijnThesis committee: A. Prof. dr. ir. Sape A. Miedema, TU Delft

Dr. ir. X. Jiang, TU DelftDr. ir. Henk den Besten, TU Delft

An electronic version of this thesis is available at http://repository.tudelft.nl/.

http://repository.tudelft.nl/

ABSTRACT

Circular Hollow Section (CHS) are widely utilized as a structure element due to its excellent property of com-pression, tension, bending and torsion resistance as well as the less consumption of materials. However,the connection joint of the CHS is the place where we should pay attention because of the stress concen-tration effect, which may lead to the fatigue damage and even structure fail. For offshore tubular structuresin the North Sea, the fatigue failure mode must be taken into consideration because of the arduous oceanconditions. The work present in this master thesis focus on the SCF on tripod joint due to both referenceand carry-over loading in order to make a precise determination of the geometry stress on this multi-planartubular joint.

In the first part of the master thesis, the 20 node solid element modelled weld profile according to AWSspecification is introduced because previous studies have shown that the SCF is sensitive to the weld shape.Apart from this, mesh generation method which include a fine mesh around the stress concentration areaand a coarse one in the far field area is introduced to trade-off between computation efficiency and resultreliability.

Then SCF on the reference brace due to axial loading, in-plane bending and out-of-plane bending is inves-tigated through ANSYS based on the surface linear extrapolation method. Extensively parametric FE analysisis carried out and database is established to compare the correlation of SCF between the FE analysis resultsand existing T/Y joint equation calculated values.

In the final part of the thesis, carry-over effect induced SCF’s, related to the three different types of refer-ence loadings are achieved through FE analysis in ANSYS. Besides, the corresponding equations to calculatethe carry-over induced SCF value are proposed.

iii

ACKNOWLEDGEMENTS

I owe the successful completion of this thesis to several people involved it. I would like to extend my sincerethanks to all of them.

Firstly, I would like to express my sincere gratefulness to Arie Romeijn, my supervisor at TU Delft, forgiving me the precious opportunity to work on this challenging topic, guiding me with his vast knowledgeand providing me with encouragement and inspiration.

I also want to thank all my committee members. Especially, I wanna give my thanks to Henk den Bestenfor helping me with the academic suggestion about my thesis and giving me useful comments on the report.And, the sincere gratefulness goes to Xiaoli Jiang for her valuable advice, guidance and assessment.

Finally, I would like to thank my family for being always supportive.

Yanxin QinDelft, 17th September 2016

v

CONTENTS

Abstract iii

Acknowledgements v

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Fatigue damage assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 Intact geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Crack damaged geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Literature Review 52.1 Uni-planar simple joint Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Uni-planar simple joint Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Loading Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 Literature review about Uni-planar simple joint SCF. . . . . . . . . . . . . . . . . . . . 72.1.4 DNV C203 RP for T/Y joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.5 Refined Lloyd’s Register Equations for T/Y joints . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Multi-planar joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Carry-over effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Hot Spot Stress Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.1 Surface Extrapolation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.2 Determination of the Geometry Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Geometrical Modelling of Weld 153.1 Double Mapping Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Dihedral Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Weld Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.1 Projection Angle β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.2 The original contact thickness T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.3 The outer thickness T2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.4 The inner thickness T3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.5 Chord Side curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.6 Brace Side curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Verification of the Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4.1 Requirement of AWS Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Finite Element Analysis 314.1 Parameter Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Element Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.1 Shell Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2.2 Solid Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3 Weld Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 Mesh Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.5 Convergence Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.6 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.7 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

vii

viii CONTENTS

5 Comparison with T/Y joint Parametric Equations 375.1 Stress Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2 Modelling of Tubular Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2.1 Comparison of Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2.2 Stress Distribution in Solid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2.3 Surface Extrapolation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2.4 Through Thickness Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.3 Loading Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.3.1 Axial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.3.2 Out of Plane Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.3.3 In Plane Bending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.4 Carry over effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4.1 Axial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4.2 Out-of-Plane Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.4.3 In-Plane Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6 Parametric Study 496.1 Axial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2 In-plane Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.3 Out-of-plane Bending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.4 conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7 Carry Over Effect & Formula 557.1 Out of plane bending moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.2 Axial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

8 Conclusion 61

9 Appendix 639.1 DNV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639.2 Refined LR equations for T/Y joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659.3 Weld Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Bibliography 67

1INTRODUCTION

1.1. BACKGROUNDLast decades have already witnessed the rapid growth of the demand on traditional energy as well as the greenhouse gas emission, which leads the whole society moves from the traditional energy to the renewable one.

In October 2014, the European Council set a target for the EU to meet at least 27% of its energy needs fromrenewable energy by 2030[1]. Considering the fact that capital cost of offshore wind is much more economicthan nuclear power (2550 and 3600 $/Kw[2] respectively) and the catastrophic results caused the Chernobyland Fukushima nuclear disaster, the clean and safe wind energy has a very promising future with the dramaticassumption of the resources in the modern society and the great demand for the clean and economic energy.

Tripod structure is used widely as the offshore wind turbine supporting structures and will be more andmore constructed on the sea with the huge demand of the economic and environmentally friendly windenergy. And fatigue is an essential and inevitable problem that should be treated for the tripod structure inthe design stage.

Fatigue is the progressive and permanent structural damage and can never be avoided because the dam-age happens when the structure is subjected to fluctuated stress which is much less than the material tensilestrength. And it has caused the catastrophic 123 casualty in the Alexander L. Kielland platform accident[3]because of the fatigue crack of the one brace on the platform.

Even though fatigue is inevitable, it could still be predicted due to the fact that the fatigue time can becalculated based on the SN curve which follows from the SN curve C = Nσm[4] proposed by August Wöhler.What is more, this means that the determination of the stress range is very essential to the reliability of theestimated fatigue lifetime. For example, if the stress range is 10% larger than the real state, this will result in a25% shorter fatigue lifetime for the power m equals 3.

The tripod joint of the supporting structure is the essential place where the stress concentration occursand this means that the stress is much larger than the nominal stress because of the structural discontinuitybetween the chord and the brace. This will also lead to the high probability of failing due to fatigue at the tri-pod joint,which means that having a accurate determination of the stress at this place is of great importance.

Usually the hot spot stress approach is utilized to have the determination of the geometry stress and theprediction of structure lifetime. And the hot spot stress is calculated based on the SCF(Stress ConcentrationFactor) as Eq. 1.1.

σh = SC F ·σn (1.1)

Thus the task of this graduation thesis is to have a determination of the parametric tripod SCF (stress con-centration factor) based on the justify model in ANSYS and have a comparison with the existing parametricformulas of T joints. Thus leading to a more reliable and accurate prediction of the SCF on the tripod joint.

1.2. FATIGUE DAMAGE ASSESSMENTThree methods: nominal stress concept, structural hot-spot stress concept and effective notch stress conceptare widely utilized when making the determination of the intact geometry far field stress [5] [6]. If there existcracks in the geometry, then fracture mechanics approach is considered.

1

2 1. INTRODUCTION

1.2.1. INTACT GEOMETRYSummation of cumulative damage approach is utilized when determining the fatigue damage for a intactgeometry. However, when considering the fatigue resistance in the S-N curves, three approaches are providedby the IIW.

• Nominal Stress Approach

This approach only considers the effect of macro-geometrical effects due to the design of the compo-nent, but excludes stress concentrations due to the welded joint itself.

• Hot Spot Approach

This approach has taken not only the factors considered in nominal stress approach, but also the struc-tural discontinuities due to the structural detail of the welded joint. However, the notch effect of theweld toe transition are excluded.

• Notch Stress Approach

Due to the consideration of the notch stress concentration because of the weld bead notches, effectivenotch stress approach is the most accurate method in these three methods. In the meanwhile, it has ahigh requirement for the FEM modelling details.

Figure 1.1: Stress distribution at the hot spot

1.2.2. CRACK DAMAGED GEOMETRYApart from the intact geometry situations considering above, fatigue may also be present due to the stressintensity at crack tip. In this situation, the fatigue resistance is determined by the crack propagation lawinstead of the S-N curves, and the fracture mechanics is utilized.

• Crack propagation Approach

based on linear elastic fracture mechanics(LEFM) theory, assumes a crack is already available and usesspecial parameters such as J-integral or the range of the stress intensity (∆k) to determine the increasein the crack length per cycle (or: crack propagation rate da/dN).

Generally speaking, the comparison of the accuracy and efforts made during the prediction of the struc-ture lifetime is illustrated in Fg.1.2. In the end, a trade off between the efficiency and the reliability of the tasklead to the adoption of the hot spot stress approach in this master thesis.

1.3. STRUCTURE OF THE THESIS 3

Figure 1.2: Qualitative comparison of accuracy and effort for common fatigue assessment methods for welded structures,G. Marquis, J.Samuelsson(2005)

1.3. STRUCTURE OF THE THESISIn this master thesis, the stress concentration factor(SCF) on tripod tubular joint connection is investigatedin FEM software and then compared with the existing T/Y joint formulas results. By comparing the SCF fromthese two groups, the verification of the design formulas are proved and then can be utilized in the design ofthe tripod joint structures.

In the first part of the master thesis, the 20 node solid element modelled weld profile according to AWSspecification is introduced because previous studies have shown that the SCF is sensitive to the weld shape.Apart from this, mesh generation method which include a fine mesh around the stress concentration areaand a coarse one in the far field area is introduced to trade-off between computation efficiency and resultreliability.

Then SCF on the reference brace due to axial loading, in-plane bending and out-of-plane bending is inves-tigated through ANSYS based on the surface linear extrapolation method. Extensively parametric FE analysisis carried out and database is established to compare the correlation of SCF between the FE analysis resultsand existing T/Y joint equation calculated values.

In the final part of the thesis, carry-over effect induced SCF’s, related to the three different types of refer-ence loadings are achieved through FE analysis in ANSYS. Besides, the corresponding equations to calculatethe carry-over induced SCF value are proposed.

2LITERATURE REVIEW

Circular Hollow Section(CHS) are widely utilized as a structure element due to its excellent property of com-pression, tension, bending and torsion resistance as well as the less consumption of materials. However, theconnection joint of the CHS is the place where we should pay attention because of the stress concentrationeffect, which may lead to the fatigue damage and even structure fail.

In this chapter, different categories of CHS as well as the SCF equations are introduced. Uni-planar simplejoint type is firstly described as it has been most widely applied in the structures and has been investigatedsince 1980s, which leads to a reliable and accurate SCF calculation equations. Then the multi-planar jointsare described. Finally, the hot spot stress approach which is applied in this master thesis to make the deter-mination of the SCF of the tripod joint is fully introduced.

2.1. UNI-PLANAR SIMPLE JOINT TYPE

Fg. 2.1 illustrates the basic types of the uni-planar joints configurations, i.e. T, X, K, N joint, which means thatthe axes of all members lay in the same plane.

Figure 2.1: Uni-planar Tubular Joint Types (DESIGN GUIDE1 for CHS,2009)

5

6 2. LITERATURE REVIEW

2.1.1. UNI-PLANAR SIMPLE JOINT PARAMETER

To investigated the SCF at the uni-planar tubular joint in simple T & Y joints,the following four parameterswere proposed by J. P. Kuang and A. B. Potvin[7] in 1975.

• The chord length to chord diameter ratio α=2L/D.

• The chord diameter to chord wall thickness ratio γ=D/2T, which has a influence on the radial flexibilityof the chord.

• The brace diameter to chord diameter ratioβ=d/D, which ia a governing factor in the stress distribution.

• The brace wall thickness to chord wall thickness ratio τ=t/T, which is an indication of the relative bend-ing stiffness.

• The angle of inclination of the brace to chord θ.

The geometry parameters are illustrated in the Fg.2.2, in which, D is the chord diameter, T is the chordthickness, L is the chord length, d is the brace diameter, t is the brace thickness and θ is the brace to chordinclination, g is the gap between brace weld toe separation for the K and KT joints.

Figure 2.2: T Joint

2.1.2. LOADING TYPE

The tubular joint is subjected to three types of loading modes, i.e. axial load, in-plane bending moment andout-of-plane bending moment as illustrated in the Fig.2.3.

Figure 2.3: Loading Mode

2.1. UNI-PLANAR SIMPLE JOINT TYPE 7

2.1.3. LITERATURE REVIEW ABOUT UNI-PLANAR SIMPLE JOINT SCFIn 1967 Toprac and Beale[8] conducted a series of experiments on ten different steel tubular joints in orderto determine the stress magnitude distribution at the joints subjected to brace axial load. And they firstlyproposed the parametric SCF equations for simple tubular joints.

However, the various types of tubular joints in practical and high cost of the scaled steel model experi-ments makes people have to find an alternative which is economic as well as reliable.

W. Visser [9] in 1975 made the determination of the stress in the tubular joint using the finite elementcomputer program called SATE by employing the thin shell element. A good agreement between experimen-tal data and the FEM program calculated results has been observed in several joint geometries.

The first parametric SCF equations covering simple tubular joints to be comprehensively used in the fa-tigue life estimation were those derived by J. P. Kuang[7][10] based on modified thin shell finite elements andby Wordsworth and Smedley[11][12]using acrylic model specimens.

J. P. Kuang and A. B. Potvin[7] proposed the formulas for the estimation of the SCF in simple non-reinforcedT/Y, K and KT joint configurations by utilising a modified thin-shell FE program. However, the tubular con-nections were modelled without a weld fillet, and stresses were measured at the mid-section of the memberwall. So the accuracy is somewhat affected.

Wordsworth and Smedley[11][12] utilized the acrylic model test results following the DEn recommenda-tions for the derivation of the hot-spot stress. Because of the inclusion of both cut-back (Fig.2.4) used in β=1and carry-over effect of multi-axial joint, the equation of Wordsworth is considered to gave a good predictionof the SCF[13].

Figure 2.4: Weld cut-back at the saddle on β= 1 joints

J. Buitrago[14] in 1984 found a way to calculate the combined hot-spot stress for tubular K and Y jointsunder the combined brace loading through the development of a influence function.

M. R. Morgan [15] investigated the stress concentration of the K joints due to the axial stress loading, in-plane bending and out-of-plane bending respectively in 1997 through finite element parametric study. Basedon the data acquired in the FE simulation, a series of parametric equations of the SCF on the interactions ofthe joints. Validation of these equations are also proved with comparison of existing text results.

Efthymiou and Durkin[16][17] has made great contribution to the accuracy of the parametric equationsby employing 3-D shell FE analyses and analysed over 150 joints by the FE program. Not only the weld filletprofile was included in the FE model, the carry-over effect of the K, KT and multi-axial joints was also consid-ered with the influence function. The accuracy of the Efthymiou formulas is proved to be the best one thenby Smedley[13].

UEG[18] equations put forward the equations based on the Wordsworth equations[11][12] with a modifi-

cation factor applied to joint configurations with high β (β > 0.6)or high γ (γ > 20) values. The factor√

Q ′β

and√

Q ′γ are both applied under axial load or OPB(out-of-plane bending), and only

√Q ′

γ ia applied under

IPB(in-plane bending) as the Eq.2.1.


Q ′β = 1.0, f orβ≤ 0.6

Q ′β = 0.3/

[β

(1−0.833β

)], f orβ≥ 0.6

Q ′γ = 1.0, f orγ≤ 20

Q ′γ = 480/

[γ

(1−0.833γ

)], f orγ≥ 20

(2.1)

P. Smedley and P. Fisher[13] in 1991 made the comparison of the widely used SCF parametric formulassuch Kuang Equation[7], Wordsworth/Smedley equations[19][12], UEG equations[18] and Efthymiou/Durkinequations[16][17]. By comparing the anomalies between these equations and the established database andanalysing the reason of these, the new Lloyd’s Register parametric SCF estimation formulas[13] were put for-ward finally.

In 1997 Lloyd’s Register of Shipping modified the original database[13] and refined formulas of SCF atsimple tubular joints was presented[20].

For the simple T joint which is similar to the tripod tubular joint, the SCF formulas are listed below andthe comparison of SCF between the simple T joint and multi-planar tripod joint will be carried out in thisthesis by the author.

2.1.4. DNV C203 RP FOR T/Y JOINTS

The stresses are calculated at the crown and the saddle points, see Figure 2.5. Then the hot spot stress at thesepoints is derived by summation of the single stress components from axial, in-plane and out of plane action.The hot spot stress may be higher for the intermediate points between the saddle and the crown. The hotspot stress at these points is derived by a linear interpolation of the stress due to the axial action at the crownand saddle and a sinusoidal variation of the bending stress resulting from in-plane and out of plane bending.Thus the hot spot stress should be evaluated at 8 spots (Fig. 2.6) around the circumference of the intersection.

Figure 2.5: Geometrical definitions for tubular joints

Figure 2.6: Superposition of stresses

2.2. MULTI-PLANAR JOINT 9

σ1 = SC FAC ·σx +SC FM I P ·σmy

σ2 = 1

2(SC FAC +SC FAS ) ·σx + 1

2

p2 ·SC FM I P ·σmy − 1

2

p2 ·SC FMOP ·σmz

σ3 = SC FAS ·σx −SC FMOP ·σmz

σ4 = 1

2(SC FAC +SC FAS ) ·σx − 1

2

p2 ·SC FM I P ·σmy − 1

2

p2 ·SC FMOP ·σmz

σ5 = SC FAC ·σx −SC FM I P ·σmy

σ6 = 1

2(SC FAC +SC FAS ) ·σx − 1

2

p2 ·SC FM I P ·σmy + 1

2

p2 ·SC FMOP ·σmz

σ7 = SC FAS ·σx +SC FMOP ·σmz

σ8 = 1

2(SC FAC +SC FAS ) ·σx + 1

2

p2 ·SC FM I P ·σmy + 1

2

p2 ·SC FMOP ·σmz

(2.2)

Here σx ,σmy and σmz are the maximum nominal stresses due to axial load and bending in-plane andout-of-plane respectively. SC FAS is the stress concentration factor at the saddle for axial load and the SC FAC

is the stress concentration factor at the crown. SC FM I P is the stress concentration factor for in plane momentand SC FMOP is the stress concentration factor for out of plane moment.

The parametric formulas due to each single loading mode for the simple T and X joint utilized by the DNVis attached in the appendix. And they are the same as the formulas proposed by Efthymiou and Durkin [16][17]

Based on experience it is not likely that fatigue cracking from the inside will occur earlier than from theoutside for simple T and Y joints and K type tubular joints.

2.1.5. REFINED LLOYD’S REGISTER EQUATIONS FOR T/Y JOINTS

Apart from the DNV equations mentioned above, the LR Equations[20] for SCF of T/Y joints under axial load,in-plane bending and out-of-plane bending are listed in the Appendix and are used to make the make thecomparison with the FEM results.

2.2. MULTI-PLANAR JOINTOver the past half century, significant effort has been devoted to the study of SCFs in various uni-planartubular joints such as T/Y, K, X joint. Through the experimental steel or acrylic model and the FE model,a large database of the simple uni-planar joint has already been established and refined. Based on such areliable database, parametric SCF equations are also proposed by engineers and these are also proved tohave a good fit with the service value.

Muti-planar tubular joints, which means the axes of brace members lay in the different plane with thechord member one. What is more, multi-planar joints (Fig.2.7) such as KK, XX and TT joints instead of thesimple uni-planar ones cover the majority of practical applications in offshore engineering. So the thesecomplex joints are what we should pay attention and effort during the design stage. What is more, the carry-over effect induced stress concentration should be considered to calculate the geometry stress at the hot spotinstead of using the existing T/Y joint SCF equations directly.

Figure 2.7: Multi-planar KK, XX, TT Tubular Joint


2.2.1. CARRY-OVER EFFECTCarry-over effect is defined as the stress concentration at a certain location at the weld toe, due to a load (axialor bending) on another brace. Referring to the joint of Fig.2.8, the local stress at a weld location of brace (a)due to a load on brace (b) is a carry-over effect. In such a case, braces (a) and (b) are called the reference andthe carry-over brace respectively[21].

Figure 2.8: Carry-over Effect

2.2.2. LITERATURE REVIEWWordsworth and Smedley[11][12] proposed the equations for K and KT joints generally utilise carry-over func-tions applied to the T joint expressions. Therefore the influence of adding further braces to a simple T jointcan clearly be determined.

Efthymiou[17] extend the equations for the estimation of SCF in uni-planar T, Y, and K joints to the multi-planar X and KT joints through the proposition of generalized influence functions on the basis of previousdata.

Lloyd’s Register[22] also employed 12 acrylic joint configurations to investigate the carry-over effect foraxial loading and out-of-plane bending.

A. Romijn[23] considered the both reference effect and carry-over effect when investigating the multi-planar tubular joints. The reference effect is caused by the loads on the reference brace due to the axialstress, in-plane bending and out-of-plane bending loaded on the brace itself. And the carry-over effect is theinfluence of the SCF due to the loads on the other brace members in this multi-planar joint.

In order to incorporate the bending moment, the parameterα=2L/D is included in the previous paramet-ric SCF equations. For the simple uni-planar joint, P. Smedley, P. Fisher[13] also proposed the chord in-planebending term (BO) from the brace axial force applied to a simply supported centrally located brace.

B0 = Cτ(β−τ/

(2γ

))(α/2−β/si nθ

)si nθ(

1−3/(2γ

)) ≈Cτβ (α/2) si nθ (2.3)

While BO gives a good estimate of the bending stresses in a test specimen, the stresses in the chord wallin an offshore structure are not simply related to the brace axial force.

A. Romeijn[23] put forward the concept of compensating moment, which means it is possible to deriveSCFs independently of chord bending stresses and, thus the parameter α should not be taken into consider-ation during the parametric SCF determination.

2.3. HOT SPOT STRESS APPROACHConsidering the balance between the computation efficiency and the result reliability, the hot spot stressapproach is adopted in this master thesis.

The structural hot spot stress approach is a relatively new approach for fatigue assessment of welds. Themethod is advantageous compared to the traditional nominal stress method mainly because of its ability toassess more types and variations of the structural details.

• Hot spot

A point in structure where a fatigue crack may develop due to the combined effect of structural stressfluctuation and the weld geometry or a similar notch.

2.3. HOT SPOT STRESS APPROACH 11

• Hot spot stress

The value of structural stress on the surface at the hot spot (also known as geometric stress or structuralstress).

The stress distribution through the plate thickness in vicinity of the weld toe is disturbed by notch effectof the weld. This is shown in Fig.2.9. The notch stresses are much higher than nominal stresses in the weldtoe. As a result, they control the fatigue cracking in the plate. Considering the non-linear stress distributionshown in Fig.2.9, three stress components can be distinguished:

• membrane stress σmem

• shell bending stress σben

• non linear stress peak σnl p

Figure 2.9: Notch Stress Distribution

2.3.1. SURFACE EXTRAPOLATION METHOD

Extrapolation region is the region chosen to calculate the hot spot stress through surface extrapolation method.This region should be a distance from the weld toe to eliminate the effect of the weld notch. Apart from that,it must be also smaller than a distance from the weld toe in order to take the effect of the stiffness due to weldinto consideration.

According to the recommend practice[6][20], the hot spot stress in tubular joints can be calculated by thelinear extrapolation of the stresses obtained from analysis at positions at distances a and b from the weldtoe as indicated in Fig.2.10. As to the extrapolation region, it is chosen as Eq.2.4, Eq.2.5 and Eq.2.6. Forextrapolation of stress along the brace surface normal to the weld toe.

a = 0.2p

r t

b = 0.65p

r t(2.4)

For extrapolation of stress along the chord surface normal to the weld toe at the crown position. Forextrapolation of stress along the brace surface normal to the weld toe.

a = 0.2p

RT

b = 0.44p

r tRT(2.5)

For extrapolation of stress along the chord surface normal to the weld toe at the saddle position.

a = 0.2p

RT

b = 2πR5

360= πR

36

(2.6)


Figure 2.10: Derivation of hot spot stress in tubular joints

However, a more simplified and less sensitive extrapolation region can be used for both chord and bracemember locations according to Romeijn[23].

• Chord member: a = 0.4 ·T,b = 1.4 ·T

• Brace member: a = 0.4 · t ,b = 1.4 · t

In this master thesis, the latter extrapolation region is selected in this master thesis to determine of thehot spot stress. The extrapolation region Fig.2.11 extends from 0.4 times the member thickness to 1.4 timesthe member thickness[24]. Another important issue concerns the type of stress to be used. Up to now, SCFcalculations were based mostly on the maximum principal stress[20]. Alternatively, the primary stress asdefined below is used as the extrapolation stress.

• The maximum stress normal to the weld for the chord side,

• The maximum stress in the direction of the brace axis for the brace side.

Figure 2.11: Surface Extrapolation Method

Compared with the maximum principal stress, primary stress is chosen mainly because of the followingreasons :

2.3. HOT SPOT STRESS APPROACH 13

• The direction of the principal stress changes inside the extrapolation region, and the experimental ob-servation that the developed cracks are normal to the primary stresses[21].

• The use of principal stress and its extrapolation will result in a lower value of SCF than the primarystress[25].

2.3.2. DETERMINATION OF THE GEOMETRY STRESSHaving obtained the hot spot stressσh through surface extrapolation method, SCF due to each single loadingis then calculated according to Eq.1.1. However, when the members of a joint are subjected to a combinationof axial and bending loads on all members, the geometric stress at a specific location around the weld iscalculated by superimposing the contributions of the nominal stresses from each loading type (k) consideringthe corresponding SCF values:

S′ =∑k

(SC F )k∆σknom (2.7)

In which, S′ is the geometric stresses and k means loading type axial stress, in-plane bending and out-of-plane bending.

3GEOMETRICAL MODELLING OF WELD

In the graduation thesis, solid element is utilized to simulate the weld profile of the tripod joint according tothe AWS specification(Table3.2). However, if the weld thickness exactly equals to the AWS specification. theweld path will not be smooth. Thus the following modelling procedure is adopted to satisfy the smoothnessof the weld path.

3.1. DOUBLE MAPPING PROCEDUREThe double mapping method is firstly proposed by Cao [26]. The mesh for a tubular structure is generatedin a plane first and then be mapped onto the tubular according to the formulae transferring the space loca-tion between a plane and corresponding tubular surface. However, the double mapping procedure is mainlyutilized to calculate the location of the points on intersection curve.

Figure 3.1: Coordinate Sytem

The first step of mapping is to map the point in the local coordinate to the plane coordinate.

X 2 +Y 2 = R2

x2 +x2 = r 2(3.1)

15

16 3. GEOMETRICAL MODELLING OF WELD

In which, (x,y,z)are the coordinate in the brace local coordinate system, whereas (X,Y,Z) is that of thechord. θ is the inclination between the these two tubulars. R and r are the radius of the chord and bracerespectively.

The chord can be also expressed in the parametric form.

u2 + v2 = r 2

u = r · sinα

v = r ·cosα

(3.2)

For a plane coordinate system {X ′,Y ′, Z ′} which is tangential to the chord outer surface, the relationshipbetween {X ′,Y ′, Z ′} and [{X ,Y , Z } is given:

X = R ·cos

(Y ′

R

)Y = R · sin

(Y ′

R

)Z = Z ′

(3.3)

The relationship between these local coordinate system based on the chord{X,Y,Z} and brace{x,y,z} is givenby:

X = x ·cosθ+ z · si nθ+R

Y = y

Z = z ·cosθ−x · si nθ

(3.4)

It can be also expressed in another way:

x = (X −R) ·cosθ−Z · sinθ

y = Y

z = (x −R) · sinθ+Z ·cosθ

(3.5)

Thus the intersection curve can be written as

x2 + y2 = r 2

[(X −R) ·cosθ−Z · sinθ]2 +Y 2 = r 2(3.6)

Substitute the Eq. 3.3 into Eq.3.6, the intersection curve yields:

[R · (1−cos

Y ′

R) ·cosθ+Z ′ · sinθ

]2

+R2 · sin2 Y ′

R= r 2 (3.7)

The first step of the double mapping procedure is to transfer from the parametric space (u,v) to the planecoordinate system (X’,Y’,Z’). This relationship can be derived by the combination of Eq. 3.2and Eq.3.6.

u = R · sinY ′

R

v = R · (1−cosY ′

R) ·cosθ+Z ′ · sinθ

(3.8)

It can be also expressed by:

Y ′ = R ·ar si n( u

R

)Z ′ =

[v −R · (1−cos

Y ′

R) ·cosθ

]1

sinθ

(3.9)

3.2. DIHEDRAL ANGLE 17

The second step is to transfer from the plane coordinate system (X’,Y’,Z’) to that on the tubular (X,Y,Z), therelationship can be obtained through considering the Eq.3.4 and Eq.3.9,

X = R ·cos(ar si n

( u

R

))Y = R · sin

(ar si n

( u

R

))= u

Z =[

v −R · (1−cosY ′

R) ·cosθ

]1

sinθ

(3.10)

Figure 3.2: Coordinate Sytem

Figure 3.3: fig:5.1.3

The angle ϕ corresponding to the driving angle α in the parametric coordinate system.

ϕ= ar tan

(Y

Z

)(3.11)

3.2. DIHEDRAL ANGLE

According to the American Welding Society(AWS) Code[27], the method of welding and thickness of the weldare determined by the dihedral angle along the joint perimeter. The definition of the dihedral angle is illus-trated in Fig.3.4. And the following procedures are proposed by Nguyen[28].


Figure 3.4: Coordinate System between Brace And Chord

n1 and n2 are the two normal vectors of the tangential planes corresponding to the chord and brace,respectively. The dihedral angle can be calculated as:

γ=π−ar cos

[n1n2

‖n1‖‖n2‖]

(3.12)

The tangential plane to the outer surface can be expressed in the XY plane in the form of:

X = mt Y +k (3.13)

In which mt is the surface gradient of the tangential plane, it can be derived through the outer surfacefunctions of the chord in XYZ coordinate, considering the point A(X A ,YA , ZA) in the chord outer surface:

X 2 +Y 2 = R2

mt = d X

dY=− YA

X A

(3.14)

Substitute Eq.3.14 into Eq. 3.13, yields,

X =− YA

X A·Y +k (3.15)

In which

k = X A + Y 2A

X A= R2

X A(3.16)

Thus the tangential plane at the point A in the (X,Y,Z)coordinate can be expressed.(X A

R2

)X +

(YA

R2

)Y = 1 (3.17)

The tangential plane can be written in the general form:

A1 ·X +B1 ·Y +C1 ·Z = D1 (3.18)

In which,

A1 = X A

R2 ;B1 = YA

R2 ;C1 = 0;D1 = 1 (3.19)

3.3. WELD JOINT 19

Similarly, the tangential plane at the point A in the (x,y,z) coordinate based on the brace can be expressed:( xA

r 2

)x +

( y A

r 2

)y = 1 (3.20)

Substitute Eq.3.5 into the Eq. 3.20, both Eq. 3.20 and Eq. 3.18 can be expressed in the (X,Y,Z) coordinate.( xA

r 2

)cosθ ·X +

( y A

r 2

)Y −

( xA

r 2

)sinθ ·Z =

( xA

r 2

)R ·cosθ+1 (3.21)

It can be also written in the general form as Eq3.18.

A2 ·X +B2 ·Y +C2 ·Z = D2 (3.22)

In which,

A2 =( xA

r 2

)cosθ;B2 = y A

r 2 ;C2 =−( xA

r 2

)sinθ;D2 =

( xA

r 2

)R ·cosθ+1 (3.23)

The normal vector n1 and n2 of these two tangential plane can be derived.

n1 = A1i +B1 j +C1k

n2 = A2i +B2 j +C2k(3.24)

Thus the dihedral angle which is calculated based on Eq.3.12 can be expressed according to the value ofvector multiplication and the vector length of n1 and n2.

γ=π−ar cos

A1 · A2 +B1 ·B2 +C1 ·C2√A2

1 +B 21 +C 2

1

√A2

2 +B 22 +C 2

2

(3.25)

3.3. WELD JOINTThe weld has a great influence on the stress concentration, thus it should be modelled as close possible asthe real weld profile. According to the AWS specification[27], the minimum weld thickness TAW S is calculatedthrough Eq.3.26

TAW S = kAW S × tb (3.26)

The value of kAW S is listed in the Table 3.1.

Dihedral Angle, γ Minimum K API Minimum K AW S

50◦−135◦ 1.25 1sinγ

35◦−50◦ 1.5 1sinγ

Below 35◦ 1.75 2.0 (for γ≤ 30◦)

Over 135◦ Build out to full thickness Build out to full thicknessbut need to not exceed 1.75 but need to not exceed 1.75

Table 3.1: Specification of API and AWS

NTU model is firstly proposed by Woo[29] on the basis of the AWS specification[27]. It also follows thefollowing assumptions[28].

• The thickness of brace member tb is small compared to the outer radius of the chord R.

• The intersecting angle between the brace and chord member θ is at least greater than 30◦.

• The weld would be extended with extra thickness at the joint when γ is greater than 135◦.

• The material property of the weld is the same as the tubular.


The welded model is obtained by modifying the original inner and outer intersecting curves. The originalcontact thickness T1 is defined as the thickness at a particular section normal to the intersection at the joint.To model the weld toe W0, a shift of a distance T2 from A0 is made (Fig.3.5, Fig.3.6, Fig.3.8, Fig.3.9).

Figure 3.5: Intersection Curve

3.3.1. PROJECTION ANGLE β

As to the projection angle β on the (Y’,Z’) plane, the normal line can be obtained by partial differentiatingEq.3.22.

y A

r 2 − ∂Z

∂Y

( xA

r 2

)sinθ = 0

∂Z

∂Y=

(y A

xA · sinθ

) (3.27)

The gradient of the normal to the intersecting curve at the point A, mnor mal , in the (Y’,Z’) plane can becalculated as:

mnor mal =− 1

∂Z /∂Y=−

(xA · sinθ

y A

)(3.28)

As a result, β can be given by:

β= π

2−ar tan

(−xA · sinθ

y A

)(3.29)

3.3.2. THE ORIGINAL CONTACT THICKNESS T1

As illustrated in Fg. 3.6, the inner and outer of the dihedral angle are different. However, based on the as-sumption, The thickness of brace member tb is small compared to the outer radius of the chord R. Thusthese two angles are thought to be the same, i.e., γi = γo = γ (Fg.3.7). The original contact thickness follows.

3.3. WELD JOINT 21

Figure 3.6: Inner and outer dihedral angle

Figure 3.7: Comparison of Inner and Outer Dihedral Angle

T1 = k1 · tb (3.30)

In which

k1 = 1

sinγ(3.31)

3.3.3. THE OUTER THICKNESS T2T2 is the modified outer thickness as illustrated in Fg.2.8 and Fg.2.9, it can be calculated based on the outerdihedral angle.

T2 = k2 · tb (3.32)

In which

k2 = Fosouter

[1−

(γo −θs

π−θs

)n2]

(3.33)

3.3.4. THE INNER THICKNESS T3T3 is the shift distance used to determined the inner curve of the weld, and it is expressed as.

T3 = k3 · tb (3.34)


In which

k3 = Fosi nner

[1−

(γi −θs

π/2−θs

)n3]

(3.35)

The factor Fosouter ,Fosi nner ,n2 and n3 in Eq.3.30 and Eq.3.32 are constant, and θs in Eq.3.30 and Eq.3.32is the smallest intersection angle 30◦ as described in the assumption.

Figure 3.8: Intersection curves of a brace on the chord and the weld paths

Figure 3.9: Cross-section of the weld model at Section 1–1 in Fg.3.8 (30◦ ≤ γ≤ 90◦)

3.3. WELD JOINT 23

Figure 3.10: Cross-section of the weld model at Section 2–2 in Fg.3.8 (90◦ ≤ γ≤ 180◦)

In general,when a brace with thickness tb intersects a chord, the final contact thickness of the weld ob-tained from the model can be expressed as.

Tw = T1 +T2 −T3 (3.36)

In which, Tw is the thickness of the weld, T1 is original contact thickness due to pure geometrical inter-section, T2 and T3 are the fill-out and the cut-in of the weld,respectively.

3.3.5. CHORD SIDE CURVEAs illustrated in Fg.3.9, the location of point Wi should be first calculated by the β on the (X’,Y’,Z’) plane, andthe be mapped onto the (X,Y,Z) coordinate system. Thus the parametric equation for the outer weld on thechord side is.

Z ′W o = Z ′

Ao +T2 ·cosβo

Y ′W o = Y ′

Ao +T2 · sinβo

XW o = R cosY ′

W o

R

YW o = R sinY ′

W o

RZW o = Z ′

W o

(3.37)

ZW o = ZAo +T3 ·cosβo

YW o = YAo +T3 · sinβo

XW o =√

R2 −Y 2W o

(3.38)

Similarly, the parametric equation for the inner weld on the chord side can be derived.

Z ′W i = Z ′

Ai +T3 ·cosβi

Y ′W i = Y ′

Ai +T3 · sinβi

XW i = R cosY ′

W i

R

YW i = R sinY ′

W i

RZW i = Z ′

W i

(3.39)


ZW i = ZAi +T3 ·cosβi

YW i = YAi +T3 · sinβi

XW i =√

R2 −Y 2W i

(3.40)

The weld toe and weld root profile is illustrated in Fg. 3.11 and Fg.3.12.

Figure 3.11: Weld Intersection Line

Figure 3.12: Weld Intersection Line

3.3.6. BRACE SIDE CURVE

As to the intersection curve on the brace side, as is illustrated in Fg. 3.8, Fg. 3.9,Fg. 3.10. The points Bo and Bi

can be determined by firstly transferring the points Ao and Ai from the (X,Y,Z)coordinate to (x,y,z)coordinateon the brace, and then shifting these by a distance of T2 and T4 respectively along the brace axis.

3.4. VERIFICATION OF THE MODELLING 25

X Ao = xAo ·cosθ+ zAO · si nθ+R

YAo = y Ao

ZAo = zAo ·cosθ−xAo · si nθ

(3.41)

The points on the brace side inner curve can be expressed by.

X Ai = xAi ·cosθ+ zAi · si nθ+R

YAi = y Ai

ZAi = zAi ·cosθ−xAi · si nθ

(3.42)

For both Ao and Ai , a shift of a distance T2 and T4 means the (x,y,z+T2) and (x,y,z+T4) respectively. ThusBo and Bi can be expressed by.

XBo = xAo ·cosθ+ (zAO +T2) · si nθ+R = X Ao +T2 · si nθ

YBo = y Ao = YAo

ZBo = (zAO +T2) ·cosθ−xAo · si nθ = ZAo +T2 ·cosθ

(3.43)

The points on the brace side inner curve can be expressed by.

XBi = xAi ·cosθ+ (zAi +T4) · si nθ+R = X Ai +T4 · si nθ

YBi = y Ai = YAi

ZBi = (zAi +T4) ·cosθ−xAi · si nθ = ZAi +T4 ·cosθ

(3.44)

3.4. VERIFICATION OF THE MODELLING

3.4.1. REQUIREMENT OF AWS SPECIFICATION

The definition and detailed selections for the prequalified CJP (complete joint penetration) T,Y and K tubularjoints is illustrated in the Fg.3.13.

Figure 3.13: Range of details

The weld profile according to AWS is shown in Fg.3.14. The detailed regulations of the configurations isshown in the Table 3.2


Figure 3.14: Weld Profile

ParameterDetail A Detail B Detail C Detail D

γ= 180◦−135◦ γ= 150◦−50◦ γ= 75◦−30◦ γ= 40◦−15◦

Root Opening or Fit-up 2-5mm 2-6mm 3-13mmJoint induced angle φ 45◦−90◦ 37.5◦−60◦ 0.5γ−40◦

L≥ tp /sinγ ≥ tp for γ≥ 90◦ ≥ tp /sinγ ≥ 2tp

not exceed 1.75tp ≥ tp /sinγ for γ≤ 90◦ not exceed 1.75tp

Table 3.2: example of table

The weld model proposed by Wong[29] and Nguyen[28] shared the same equation like Eq.3.30, Eq.3.31,Eq.3.32, Eq.3.33, Eq.3.34, Eq.3.35.However, the factor such as Fosouter , Fosi nner , n2 and n3 have the differentvalues. The values are listed in the Table 3.3.

Parameter Wong’s Equation Nguyen’s EquationFosouter 0.3 0.55Fosi nner 0.25 0.6

n2 2.0 2.0n3 0.4 0.4

Table 3.3: example of table


The values which will be calculated in the master thesis are listed in the Table 3.4 with the diameter of thechord d0=2000mm.

Rangeγ 10 ∼ 50β 0.2 ∼ 0.9τ 0.2 ∼ 1.0θ 35◦ ∼ 60◦

Table 3.4: Parameter Range

The calculated value of Kt w according to Nguyen[28] and Wong[29] are illustrated in the Fg 3.15 and 3.16respectively.

Figure 3.15: K According to Nguyen


Figure 3.16: K According to Wong

In order to validate which factor is better, a comparison is made on the assumption that D=2000mm ,d=1000mm, T=100mm and t=50mm(γ = 10,β = 0.5,τ = 0.5). Based on the Wong’s Equation (Fosi nner =0.25,Fosouter =0.3), the results is illustrated in the Fg.3.17. It is required that the KT W should be larger than 2 whenthe dihedral angle is smaller than 40◦ and no more than 1.75 when the dihedral angle is larger than 135◦,which is only considered in the case with the inclination θ = 35◦. However, with a comparison of the twoequation in Fg.3.17 and Fg. 3.18, it can be found that Wong’s equation is better than Nguyen’s with just a littleover-provision in large dihedral angle range.

Figure 3.17: K According to Wong


Figure 3.18: K According to Nguyen

When it comes to the root opening or fit-up requirements of AWS, both two equations have their limita-tion. As illustrated in Fg.3.19, Fg.3.20, Nguyen’s equation overestimates the root opening or fit-up comparedwith Wong’s Equation. So Wong’s Equation is much better and it satisfies the requirement by AWS.

Figure 3.19: Wong’s Equation with t=50mm


Figure 3.20: Nguyen’s Equation with t=50mm

As to the joint induced angle φ, a shift distance of T1 +T4 from point A0 to B0 according to Wong meansthe too large angle, thus the distance is chosen to be T2 as Nguyen’s model. The detailed weld modelling isillustrated in Fg.3.21. The factor are listed in the Table 3.5. The application of these factors are also validatedby S.T. Lie[30].

Parameter Wong’s EquationFosouter 0.3Fosi nner 0.25

n2 2.0n3 0.4

Table 3.5: Wong’s Parameter

Figure 3.21: Illustration of the weld

4FINITE ELEMENT ANALYSIS

4.1. PARAMETER SCALEIn order to describe the geometry of a tripod joint parametrically, the following dimensionless joint parame-ters are defined:

• The thickness-to-diameter ratio γ=D/2T, which has a influence on the radial flexibility of the chord.

• The brace to chord diameter ratio β=d/D, which ia a governing factor in the stress distribution.

• The brace to chord thickness ratio τ=t/T, which is an indication of the relative bending stiffness.


Due to the inclination of the compensating moment, the parameter chord length-to-diameter ratioα=2L/Dis not considered in this thesis. And the range of these parameters is illustrated in the Table4.1 with the chorddiameter D=2000 mm.

Table 4.1: Parameters

Parameter βInclination Angle θ

35◦ 40◦ 45◦ 50◦

0.2 * * * *0.3 * * * *0.4 * * * *0.5 * * * *0.6 * * * *γ 10; 15; 20τ 0.5; 0.75; 1.0

In the master thesis, there are 5 values of β(0.2, 0.3, 0.4, 0.5, 0.6), 4 values of θ(35◦, 40◦, 45◦, 50◦), 3 valuesof τ( 0.5, 0.75, 1.0) and 3 values of γ(10, 15, 20). So a total of 180 tripod joint cases will be analysed, whichcover a wide range of practical applications.

4.2. ELEMENT TYPEUsually the 2D shell element and the solid element are selected as the FEM element. Both of them can give areliable result, it is just a trade-off between the accuracy of joint description and the computation time. Andthe comparison of these two type of element is made by E. Chang[31].

31

32 4. FINITE ELEMENT ANALYSIS

4.2.1. SHELL ELEMENT

Shell element is widely utilized as the element to model the structure especially in early days mainly becauseit relative accurate results and the poor performance of the early computers. Using these shell elements,tubular joints are modelled as intersecting cylindrical tubes at the mid-surfaces of the walls (Fig. 4.1). Themid-plane stress is equal to the membrane stress, and the top and bottom surface stresses are superimposedmembrane and shell bending stresses. The hot spot stresses at the weld toe are estimated from the valueobtained directly at the brace/chord intersection. Combining relatively high accuracy with low consumptionof time, thin-shell elements are generally recognised as a good means for calculating the SCFs in the tubularjoints.

Figure 4.1: 2D Shell Element

4.2.2. SOLID ELEMENT

The 3D solid element modelling is proposed mainly due to the fact that the lack of weld detail modelling inthe thin shell model and some details of the 3D stresses in the joint are lost because of this, which leads tohot spot stress locations are different to steel models, especially for the brace(Fig. 4.1). This is master thesis,solid element will be chosen as the FEM model element.

4.3. WELD PROFILEEmploying 90 acrylic model Xjoints, Smedley [19] produced a weld fillet correction factor based on the chord-side weld fillet leg length.

SC FW el dToe = SC FNo W eld / 3

√1+ x

T(4.1)

In which x is the weld fillet leg length on the chordside.

However, this formula only suits for the determination of the chordside stress at 90° Xjoints. As to theinclined brace joint and the stress of the braceside, it is not suitable.

Alternatively, Marshall[32] proposed an equation, based on FE analysis of Kjoints.

SC FW el dToe = 1+ [SC FMi d−sec t i on −1] ·exp

(−0.5T + tp

r t

)(4.2)

This equation, applied to the braceside mid-section stress, uses an exponential decay function to approx-imate the SCF at the braceside weld toe, and is generally used in conjunction with the Kuang[10] parametricequations.

An investigation into the effect upon the SCF of a weld fillet was undertaken by Efthymiou[16], using a3-D FE analysis on one T joint configuration with and without a weld fillet. The cinclusion can be also drownthat the inclusion of the weld detail will have an influence on the SCF. It can be seen in Table 4.2, that for thejoint geometry modelled by Efthymiou, the Wordsworth weld reduction factor gives the closest agreement tothe measured weld fillet reduction factor.

4.4. MESH SIZE 33

Table 4.2: Reduction in SCF due to the inclusion of weld fillet

Measured Factor Smedley Factor Marshall Factor Wordsworth FactorChordside 0.95 0.87 1.00 0.95Braceside 0.86 0.87 0.69 0.88

In this master thesis, the weld profile is model using the solid model in order to increase the accuracyof the SCF parametric equation instead of using the reduction factors directly. The weld profile is modelledbased on the dihedral angle (ψ in Fg.4.2) along the joint perimeter according the AWS regulation[27], and theweld shape details is illustrated in the Appendix Fig.9.3.

Figure 4.2: Mesh Refinement at Tubular Joint

50 key points are used to describe each intersection line, and the key point coordinate is calculatedthrough MATLAB based on the equations in Chapter 3. Segmented spline through these key points is gener-ated and the weld profile is modelled. The weld profile modelled according to AWS specification is illustratedin the Fig. 3.9 and Fig.3.10. The whole geometrical model of the tripod joint in the is illustrated in Fig.4.3

Figure 4.3: Geometrical Model

4.4. MESH SIZEInitial numerical results have shown that the computed value of geometric stress is sensitive to the weldprofile, as well as to the mesh near the weld (Fig.4.4). Considering both the computation efficiency and theresults accuracy, a dense mesh is used near the weld where stress concentrations are to be computed, whereas


the coarse mesh is generated in the far field area. What is more, the size of an element at the vicinity of theweld is selected based on the convergence study.

Figure 4.4: Division of Tubular Joint

4.5. CONVERGENCE STUDY

Convergence study is conducted to choose the suitable mesh size at the weld joints where stress concen-tration occurs. As illustrated in Fig.4.5, the minimum stress in the x direction σx is investigated against thenumber of elements along the weld seam. Finer meshes lead to a more accurate result. However, it alsocomes with more computation time and higher requirement of computer memory. In this case, 100 elementsfor the weld seam is enough to get a accurate result.

Element Numbers20 40 60 80 100 120 140

Min

imum

σ

x

-3.7

-3.6

-3.5

-3.4

-3.3

-3.2

-3.1

-3

-2.9

-2.8

Convergence

Figure 4.5: Convergence Study

4.6. MESH GENERATION

Mesh generation is critical in the FEM, usually a fine mesh is generated around the stress concentration zonewhereas a coarser mesh is chosen far from the intersection zones as shown in Fig.4.6, Fig.4.7. In this way,both a relative accurate result and a short computation consumption time will be balanced and guaranteed.As to which element size should be chosen, a convergence study is conducted.

4.7. BOUNDARY CONDITIONS 35

Figure 4.6: Mesh

Figure 4.7: Mesh

4.7. BOUNDARY CONDITIONSAforementioned concept of compensating moment proposed by A. Romeijn[23] is used in this master thesisin order to make sure that the SCF is independently of chord bending stresses and eliminate the effects ofboundary conditions of the chord.

An example (Fg.4.8) of the compensating moment for the axially loaded brace of a T joint is given for thelocations of interest(crown, saddle and place in between) based on the following equations.

• Location 1 and 5(crown):

Mcompensati ng = 0.5 ·Faxi al ×x1

• Location 2,6 and 4,8(place in between):

Mcompensati ng = 0.5 ·Faxi al ×x2


• Location 3 and 7(saddle):

Mcompensati ng = 0.25 ·Faxi al ×L

Figure 4.8: An example of compensating moment

5COMPARISON WITH T/Y JOINT PARAMETRIC

EQUATIONS

The parameters of the sample geometry is listed in Table 5.1, and these parameters are also valid for theEfthymiou/Durkin, DNV T/Y joint parametric equations.

Parameter Value[mm] Parameter ValueD 2000.0 α 8.0d 1000.0 γ 10.0T 100.0 β 0.5t 50.0 τ 0.5L 8000 θ 45◦

θ 45◦

Table 5.1: Parameter

5.1. STRESS DETERMINATIONSurface extrapolation method is utilized to make the determination of the hot spot stress. However, as tomake the choice between primary stress which is perpendicular to the weld toe or the maximum principalstress, different specification has different views about this. In this thesis, the former stress is utilized mainlybecause the experiment results shows the developed crack are normal to the primary stress, whereas thedirection of the maximum principal stress changes along the extrapolation path.

However, in order to extrapolate the hot spot stress at the saddle, crown and in-between points, the pri-mary stress which is perpendicular to the weld toe as well as along the extrapolation path (Fig.5.1) is utilizedto make the determination of the primary stress.

Figure 5.1: Local Coordinate System

The calculation of the primary stresses of the nodes in the extrapolation region can be done by using dotproduct between stress tensor and normal vector, as shown in the formula below.

37

38 5. COMPARISON WITH T/Y JOINT PARAMETRIC EQUATIONS

σp =−→n T ·σ ·−→n (5.1)

The unit vector −→n is the vector normal to the weld. And the stress tensor which is written in the form:

σ= σxx σx y σxz

σy x σy y σy z

σzx σz y σzz

(5.2)

5.2. MODELLING OF TUBULAR JOINT

The configurations of the tripod joint are listed in Table5.1. Firstly, the element type is investigated in orderto have a choice of the suitable element. The SCF around the weld on both chord and brace will be calculatedwith the modelling of three types of element:

• Shell element.

• Solid element without weld.

• Solid element with weld.

5.2.1. COMPARISON OF RESULT

From the comparison of clamped shell element model and clamped solid element model illustrated in Fg.5.2,Fg.5.3 and Fg.5.4.

Figure 5.2: Clamped Shell Element

5.2. MODELLING OF TUBULAR JOINT 39

Figure 5.3: Clamped Solid Element without Weld

Figure 5.4: Clamped Solid Element with Weld

The result of these three models are listed in Table5.2

Model Maximum Displacement[mm]Shell Element 0.052

Solid Element Without Weld 0.052Solid Element With Weld 0.052

Table 5.2: Comparison of Result

From the results listed on the Table5.2, the maximum displacement is are all the same for all models. Asto the maximum principal stress, the solid element without weld will lead to a larger stress than that withweld, which means the inclusion of weld have a positive effect on the stress reduction. What is more, thesolid element will give a more accurate prediction on the stress concentration area. Thus the solid element is


selected.

5.2.2. STRESS DISTRIBUTION IN SOLID MODEL

The maximum principal stress and primary stress in clamped solid element are illustrated in Fg.5.5. Theprimary stress and maximum principal stress are plotted against the distance from the weld toe.

Distance, [mm]0 10 20 30 40 50 60 70 80 90 100

Stre

ss, [

MP

a]

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Maximum Principal Stress VS Primary Stress on Brace In-Between2

Maximum Principal StressPrimary StressExtrapolation Region

Figure 5.5: Clamped Shell Element

It can be seen that notch stress within the extrapolation region ( distance < 0.4∗ thi ckness) is effectedby the geometry of the weld and the biggest difference occurs on the weld toe. However, maximum principalstress and primary stress are almost the same within the extrapolation region where 0.4 and 1.4 times of thethickness from the weld toe.

5.2.3. SURFACE EXTRAPOLATION METHOD

A curve is utilized to fit the extrapolation points within the extrapolation region(0.4 ∗ t to 1.4 ∗ t ), then alinear extrapolation method to used to calculate the hot spot stress through the boundary of the extrapolationregion. The method is illustrated in Fg.5.6.

5.2. MODELLING OF TUBULAR JOINT 41

Distance, [mm]0 10 20 30 40 50 60 70 80 90 100

Stre

ss, [

MP

a]

0

1

2

3

4

5

6Maximum Principal Stress VS Primary Stress on Brace In-Between1

Primary StressExtrapolation Region

Figure 5.6: HSS Extrapolation through Primary Stress

The extrapolated hot spot stress based on primary stress and maximum principal stress in the conditionof solid element are listed in Table5.3. The extrapolated hot spot stress based on primary stress is within thestress range calculated by two specifications. Apart from this, primary stress is selected because maximumprincipal stress may change the direction along the extrapolation path.

Position MPS PS DNV LRCrown 2.0670 2.586 2.366 2.519

In-between 2.4848 2.794 - -Saddle 2.5193 2.491 3.007 2.219

In-between 2.0062 1.466 - -Crown 1.1525 0.901 - -

Table 5.3: Comparison of Hot Spot Stress on Chord

5.2.4. THROUGH THICKNESS METHOD

Through thickness method is also utilized to determine the hot spot stress at the weld toe. It can be assumedthat for a given local through thickness stress distribution as show in the figure, there exist a correspondingsimple structural stress distribution stress as show the figure. With this simple linear stress curve obtain, SCFis nothing but hot spot stress divided by nominal stress. For single side joint, the stress at the bottom of thethickness is assumed undisturbed from the hot spot, for the double side joint, the stress in the middle of thethickness is assumed undisturbed, this stress for the joint can be treat as nominal stress.

• The equivalence of external force σm

• The equivalence of external moment σb

σm = 1

t

∫ y=t

y=0σ(y) ·d y

σb = 6

t 2

(∫ y=t

y=0σ(y) ·

(t

2− y

)·d y −σm · t 2

2

) (5.3)


Figure 5.7: Multi-planar KK, XX, TT Tubular Joint

5.3. LOADING TYPE

5.3.1. AXIAL STRESS

The SCF distribution on both brace and chord under axial stress is illustrated as Fg.5.8. It can be seen that themaximum difference of SCF between brace and chord occurs in the saddle position. However, the SCF aremore or less the same in other positions.

Angle, [°]0 45 90 135 180 225 270 315 360

SC

F, [-

]

-2

-1

0

1

2

3

4

5

6

SCF Distribution around Brace & Chord, β=0.6,γ=20,τ=0.5,θ=45

Brace SCFChord SCF

Figure 5.8: Stress Distribution along the weld

5.3.2. OUT OF PLANE BENDING

Out of plane bending moment is also considered in this master thesis. Th nominal stress of OPB-loaded bracewhich is calculated as follows:

σn = 32 ·D ·M

π · (D4 −d 4) (5.4)

In which: σn is the nominal stress; D is the outer diameter of the brace; d is the inner diameter of thebrace; M is the bending moment at the end of the brace;

The stress distribution distribution on the brace and chord are illustrated in Fg.5.9. The maximum differ-ence of SCF between brace and chord occurs in the saddle position. And the SCF on chord has a larger valuethan that on the brace. What is more, a comparison is also conducted between the FEM results and existingparametric equations in Table5.4 .

5.3. LOADING TYPE 43

Angle, [°]0 45 90 135 180 225 270 315 360

SC

F, [-

]

-4

-3

-2

-1

0

1

2

3

4


Brace SCFChord SCF

Figure 5.9: SCF distribution due to reference Out-of-Plane Bending

Position SCF DNV LRBrace Saddle 2.630 3.095 2.571Chord Saddle 3.771 4.232 3.699

Table 5.4: Comparison of Hot Spot Stress

5.3.3. IN PLANE BENDING

As to the in plane bending moment on the brace, the nominal stress is calculated as Eq.5.4. The SCF distribu-tions in Fg.5.10 shows that the stress concentration occurs in the brace crown position. Apart from this, theSCF at brace and chord crown are listed in the Table5.5 with the comparison with DNV and LR specification.

Angle, [°]0 45 90 135 180 225 270 315 360

SC

F, [-

]

-1.5

-1

-0.5

0

0.5

1

1.5

2SCF Distribution around Brace

Brace SCFChord SCF

Figure 5.10: Distribution of SCF on Brace and Chord


Position SCF DNV LRChord Crown 1.853 2.516 1.401Brace Crown 1.025 1.442 1.183

Table 5.5: Comparison of Hot Spot Stress on Brace

5.4. CARRY OVER EFFECT

The stress concentration effect due to the reference loading are discussed ahead. However, the carry-over ef-fect of the multi-planar tripod joint and the corresponding SCF formulas are to be investigated in this masterthesis under the these three loading mode. As Karamanos [21] has stated during the investigation of DT joint,the out of plane bending moment will cause stress concentration on the other brace apart from the referenceone.

5.4.1. AXIAL STRESS

As shown in Fg.5.11 and Fg.5.12, the SCF distribution for one brace and three braces tripod joint configura-tions have very small margin on both brace and chord.

Angle, [°]0 45 90 135 180 225 270 315 360

SC

F, [

-]

0

1

2

3

4

5

6

SCF Distribution around Brace, β=0.6,γ=20,τ=0.75,θ=50

Tripod Brace SCFOne Brace SCF

Figure 5.11: Comparison of SCF on Brace of Tripod and Single one Brace Joint

5.4. CARRY OVER EFFECT 45

Angle, [°]0 45 90 135 180 225 270 315 360

SC

F, [

-]

0

1

2

3

4

5

6

7

8

9

10

SCF Distribution around Chord, β=0.6,γ=20,τ=0.75,θ=50

Tripod Chord SCFOne Chord SCF

Figure 5.12: Comparison of SCF on Chord of Tripod and Single one Brace Joint

As to the SCF distribution due to reference axial stress and carry-over effect are shown in Fg.5.13 andFg.5.14. It can be seen that:

• The maximum SCF difference occurs in the saddle position on the reference brace.

• The maximum SCF due to carry-over effect on the other brace occurs in the saddle position close to thereference brace, which is called the near brace saddle, this is also the place where the maximum SCFdifference between brace and chord exists.

This means that the carry over effect under axial stress is so significant that this phenomenon must betaken into consideration in order to get an reliable geometry stress under a combination loading.

Angle, [°]0 45 90 135 180 225 270 315 360

SC

F, [-

]

0

1

2

3

4

5

6

7

8

9

10


Brace SCFChord SCF

Figure 5.13: SCF Distribution on Reference Brace and Chord under Axial Stress


Angle, [°]0 45 90 135 180 225 270 315 360

SC

F, [-

]

-3

-2

-1

0

1

2

3

Carry-over SCF Distribution around Brace & Chord, β=0.6,γ=20,τ=0.75,θ=50

Brace SCFChord SCF

Figure 5.14: SCF Distribution on Carry-over Brace and Chord under Axial Stress

5.4.2. OUT-OF-PLANE BENDING

The SCF distribution due to reference out-of-plane bending and carry-over effect are shown in Fg.5.15 andFg.5.16. It shares the same characteristics as that due to axial stress.

Angle, [°]0 45 90 135 180 225 270 315 360

SC

F, [-

]

-4

-3

-2

-1

0

1

2

3

4


Brace SCFChord SCF

Figure 5.15: SCF due to reference Out-of-Plane Bending

5.4. CARRY OVER EFFECT 47

Angle, [°]0 45 90 135 180 225 270 315 360

SC

F, [-

]

-1.5

-1

-0.5

0

0.5

1

1.5


Brace SCFChord SCF

Figure 5.16: SCF due to carry-over Out-of-Plane Bending

5.4.3. IN-PLANE BENDING

The SCF distribution due to reference in-plane bending and carry-over effect are shown in Fg.5.17 and Fg.5.18.It illustrates that:

• The maximum SCF due to carry over effect also occurs in the near saddle position.

• The SCF due to carry-over effect is so small that it is negligible in this master thesis.

Angle, [°]0 45 90 135 180 225 270 315 360

SC

F, [

-]

-5

-4

-3

-2

-1

0

1

2

3

4

5


Brace SCFChord SCF

Figure 5.17: SCF due to reference Out-of-Plane Bending


Angle, [°]0 45 90 135 180 225 270 315 360

SC

F, [

-]

-3

-2

-1

0

1

2

3


Brace SCFChord SCF

Figure 5.18: SCF due to carry-over Out-of-Plane Bending

6PARAMETRIC STUDY

The investigated parameters are listed in Table6.1, the involved parameters are listed as the following:

• The thickness-to-diameter ratio γ=D/2T, which has a influence on the radial flexibility of the chord.

• The brace to chord diameter ratio β=d/D, which ia a governing factor in the stress distribution.

• The brace to chord thickness ratio τ=t/T, which is an indication of the relative bending stiffness.


Due to the inclination of the compensating moment, the parameter chord length-to-diameter ratioα=2L/Dis not considered in this thesis. And the range of these parameters is illustrated in the Table6.1 with the chorddiameter D=2000 mm. Thus 180 (4*5*3*3) cases are investigated in this master thesis.

Parameter βInclination Angle θ

35◦ 40◦ 45◦ 50◦

0.2 * * * *0.3 * * * *0.4 * * * *0.5 * * * *0.6 * * * *γ 10; 15; 20τ 0.5; 0.75; 1.0

Table 6.1: Parameters

6.1. AXIAL STRESS

Finally the results of 180 cases are compared with the existing DNV T/Y joint equation calculated values,the results are illustrated in Fg.6.1, Fg.6.2, Fg.6.2, Fg.6.3. It can be seen that the largest difference exists inthe brace crown. However, the SCF in other positions keep in accordance with the DNV T/Y joint equation.Considering the fact that the SCF on brace crown is much smaller compared with other 3 positions, thisdifference on brace crown is negligible.

49

50 6. PARAMETRIC STUDY

SCF DNV0 0.5 1 1.5 2 2.5 3 3.5 4

SC

F FE

M

0

0.5

1

1.5

2

2.5

3

3.5

4SCF of Brace Crown

Figure 6.1: SCF Comparison On Brace Crown

SCF DNV0 1 2 3 4 5 6 7

SC

F FE

M

0

1

2

3

4

5

6

7SCF of Brace Saddle

Figure 6.2: SCF Comparison On Brace Saddle

The value on the brace saddle, chord saddle and chord crown from the FEM result and DNV equationslay on the line of y=x , which means that there exists a good correlation between these two groups. And theexisting DNV T/Y joint equations can be utilized directly for the calculation of SCF on the reference brace.As to the scatter points for the SCF on the brace crown shows the mismatch with the DNV equation. Thisis mainly due to the inclination of the compensating moment on the chord end and the exclusion of theparameter α during the parametric study, which is considered originally in these equations. However, dueto the fact that the maximum SCF occurs on the saddle position. This equation is considered reliable for theSCF prediction under axil stress.

6.2. IN-PLANE BENDING 51

SCF DNV0 1 2 3 4 5 6 7

SC

F F

EM

0

1

2

3

4

5

6

7SCF of Chord Crown

Figure 6.3: SCF Comparison On Chord Crown

SCF DNV0 2 4 6 8 10 12 14

SC

F F

EM

0

2

4

6

8

10

12

14SCF of Chord Saddle

Figure 6.4: SCF Comparison On Chord Saddle

6.2. IN-PLANE BENDING

Under in-plane bending loading mode, the crown is the position where maximum SCF occurs and it is alsothe position we focus on. The final results of these 180 cases are compared with LR T/Y joint parametricequation results and they are illustrated in Fg.6.5 and Fg.6.6.


SCF LR0 1 2 3 4 5 6 7

SC

F F

EM

0

1

2

3

4

5

6

7SCF of Chord Crown due to IPB

Figure 6.5: SCF Comparison On Chord Crown

For the SCF on the chord crown is overestimated by the LR equation. Whereas that on the brace crown isstill within the range of the LR equation value.

SCF LR0 0.5 1 1.5 2 2.5 3 3.5 4

SC

F F

EM

0

0.5

1

1.5

2

2.5

3

3.5

4SCF of Brace Crown due to IPB

Figure 6.6: SCF Comparison On Brace Crown

6.3. OUT-OF-PLANE BENDINGUnder out-of-plane bending loading mode, the brace is the position where maximum SCF occurs and it isalso the position we focus on. The final results of these 180 cases are compared with LR T/Y joint parametricequation results and they are illustrated in Fg.6.7 and Fg.6.8.

6.4. CONCLUSION 53

SCF LR0 1 2 3 4 5 6 7

SC

F F

EM

0

1

2

3

4

5

6

7SCF of Brace Saddle

Figure 6.7: SCF Comparison On Brace Saddle

For these two focused positions where stress concentration occurs, the value from FEM is collinear withthe LR equation results.

SCF LR0 2 4 6 8 10 12 14

SC

F F

EM

0

2

4

6

8

10

12

14SCF of Chord Saddle

Figure 6.8: SCF Comparison On Chord Saddle

6.4. CONCLUSIONThe above figures indicate that the SCF value in the chord is larger than those in the brace under all the threeloading modes(axial stress, in-plane bending and out-of-plane bending). For the axial stress loading mode,there is a acceptable correlation for the saddle position in both brace and chord as illustrated in Fg.6.2 andFg.6.4. As to the SCF value on crown positions, the correlation is acceptable because the value is still among


the range and this location is not the maximum SCF occurs, which means that the fatigue crack have lessprobability to initiate in crown position.

When it comes to the SCF value due to IPB, the equation of LR overestimates the value of SCF whencompared with the FEM calculated ones in the chord crown. However, it is acceptable in the preliminaryfatigue design, for the reason that these simplified equations gives an overestimated SCF for a safer design.The values in the brace is also acceptable.

As to the SCF value comparison due to OPB, the correlation on the chord is much more satisfied than thebrace one, this is also the location where maximum SCF occurs. And the points all locate around the line ofy=x. In any case, the overall correlation is considered to be satisfactory for the purposes of the present study,considering the simplicity of the proposed equations.

7CARRY OVER EFFECT & FORMULA

7.1. OUT OF PLANE BENDING MOMENTFor the out-of-plane bending moment induced carry-over effect illustrated in Fg.5.16,the stress concentrationeffect occurs in the near saddle and far saddle position as illustrated in Fg. 7.1. However, the SCF in far saddleposition is so small that it is negligible.

carry-over brace

reference brace

near saddle

far saddle

carry over

Figure 7.1: Location of Stress Concentration Points

Karamanos [21] has investigated the carry-over effect in DT joint and proposed the corresponding for-mulas for this joint. In similar, the formulas for the SCF at the near saddle position is proposed to keep inconsistence with the DT equation .

SC F chor dcov =

(β

0.5

)A1

·( γ

12

)A2·( τ

0.5

)A3· (sinθ)A4 ·ω(

ϕ)

(7.1)

log(SC F chor dcov_i ) = A1 · log

(βi

0.5

)+ A2 · log

(γi

12

)+ A3 · log

( τi

0.5

)+ A4 · log(sinθi )+ log

(ω

(ϕ

))(7.2)

Using standard regression to find the parameters A1, A2, A3, A4. And the SCF on the near chord saddledue to carry-over OPB effect is proposed as following:

SC F chor dcov = (−0.7) ·

(β

0.5

)3.3( γ12

)0.9( τ

0.5

)1.2(sinθ)1.7 (7.3)

55

56 7. CARRY OVER EFFECT & FORMULA

In which, w(ϕ

)is the function which shows the variation of the carry-over effect induced SCF in terms of

the out-of-plane angle ϕ. For the tripod joint with an angle of 120◦, this function equals to -0.7.In similar, And the SCF on the far chord saddle due to carry-over OPB effect is proposed as following:

SC F chor dcov = (−0.4) ·

(β

0.5

)2.3( γ12

)−0.8( τ

0.5

)1.3(sinθ)2.0 (7.4)

SCF Formula-5 -4 -3 -2 -1 0 1

SC

F F

EM

-5

-4

-3

-2

-1

0

1SCF of Chord Near Saddle

Figure 7.2: Carry Over Effect Chord Near Saddle SCF induced by OPB

The SCF at brace saddle induced by out-of-plane bending moment is calculated bt the correspondingchord saddle formula. The comparison of the calculated SCF according to Eq.7.3, Eq.7.5 and the FEM resultsare illustrated in Fg.7.2 and Fg.7.3.

SC F br acecov =

(3

1+4τ

)·SC F chor d

cov (7.5)

SCF Formula-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

SC

F F

EM

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1SCF of Brace Near Saddle

Figure 7.3: Carry Over Effect Brace Near Saddle SCF induced by OPB

7.2. AXIAL STRESS 57

SCF Formula-2 -1.5 -1 -0.5 0 0.5 1

SC

F F

EM

-2

-1.5

-1

-0.5

0

0.5

1SCF of Chord Far Saddle

Figure 7.4: Carry Over Effect Chord Far Saddle SCF induced by OPB

SCF Formula-2 -1.5 -1 -0.5 0 0.5 1

SC

F F

EM

-2

-1.5

-1

-0.5

0

0.5

1SCF of Brace Far Saddle

Figure 7.5: Carry Over Effect Brace Far Saddle SCF induced by OPB

Through Fg.7.2, Fg.7.3, Fg.7.4, Fg.7.5, the validation of the proposed formulas is proven.

7.2. AXIAL STRESSFor the axial stress induced carry-over effect which is illustrated in Fg.??, similar formulas are also establishedin consistence with the existing DT equation to estimate the carry-over effect induced SCF on near chordsaddle and far chord saddle.

SC F chor dcov = (−8) · (β2.8 −0.88 ·β6) ·(2.4(γ−7)

12

)0.1( τ

0.5

)1.1(sinθ)0.6 (7.6)

SC F chor dcov = 1.8 · (β1.3 −0.88 ·β6) ·(2.4(γ−7)

12

)0.5( τ

0.5

)1.0(sinθ)1.4 (7.7)

As to the SCF on the brace saddle, the same simple SCF equation is proposed as the OPB carry-over situ-ation.

58 7. CARRY OVER EFFECT & FORMULA

SC F br acecov =

(3

1+4τ

)·SC F chor d

cov (7.8)

As illustrated in Fg.7.6, Fg.7.7, Fg.7.8, Fg.7.9, the correlation between the estimated SCF according to pro-posed SCF equations and FEM results are satisfactory.

SCF Formula-5 -4 -3 -2 -1 0 1

SC

F F

EM

-5

-4

-3

-2

-1

0

1SCF of Chord Near Saddle

Figure 7.6: Carry Over Effect Chord Near Saddle SCF induced by Axial Stress

SCF Formula-1 -0.5 0 0.5 1 1.5 2

SC

F F

EM

-1

-0.5

0

0.5

1

1.5

2SCF of Chord Far Saddle

Figure 7.7: Carry Over Effect Chord Far Saddle SCF induced by Axial Stress

7.2. AXIAL STRESS 59

SCF Formula-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

SC

F F

EM

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1SCF of Brace Near Saddle

Figure 7.8: Carry Over Effect Brace Near Saddle SCF induced by Axial Stress

SCF Formula-1 -0.5 0 0.5 1 1.5 2

SC

F F

EM

-1

-0.5

0

0.5

1

1.5

2SCF of Brace Far Saddle

Figure 7.9: Carry Over Effect Brace Far Saddle SCF induced by Axial Stress

8CONCLUSION

Based on the large database acquired through FEM analysis, the various parameter affected SCF values onall essential positions due to reference loading are compared with the existing DNV, LR equation ones. Thevalidation of the existing T/Y joint equation are proven on the reference brace.

As to the SCF values on the carry-over brace due to axial stress and OPB, the corresponding equations areproposed and the validation of these are also proven, which extends the design equations to the multi-planartripod joint.

61

9APPENDIX

9.1. DNVIn 1985, Efthymiou and Durkin[16] published a series of parametric equations covering T/Y and gap/overlapK joints. Over 150 configurations were analysed via the PMBSHELL finite element program using 3-dimensionalshell elements, and the results were checked against the SATE finite element program for one T joint and 2 Kjoint configurations. The hot-spot SCFs were based on maximum principal stresses linearly extrapolated tothe modelled weld toe, in accordance with the HSE recommendations, with some consideration being givento boundary conditions (ie short chords and chord end fixity).

In 1988, Efthymiou[17] published a comprehensive set of simple joint parametric equations covering T/Y,X, K and KT simple joint configurations. These equations were designed using influence functions to describeK, KT and multi-planar joints in terms of simple T braces with carry-over effects from the additional loadedbraces.

With respect to the Efthymiou/Durkin equations, the following points may be noted:(i) It has been shown by Efthymiou that the saddle SCF is reduced in joints with short chord lengths, due to

the restriction in chord ovalisation caused by either the presence of chord end diaphragms or by the rigidityof the chord end fixing onto the test rig. Therefore, the measured saddle SCFs on joints with short chordsmay be less than for the equivalent joint with a more realistic chord length. Factors have been included in theEfthymiou parametric equations to cover short chords.

(ii) The T/Y joint equation for the saddle under axial load includes a short chord correction factor for eitherfixed or pinned ends. The short chord effect at the saddle is due to the presence of chord end diaphragms,therefore, it is unclear why the chord end fixity should be a factor. The equation for X joints at the bracecrown under axial load does not equal the corresponding T/Y joint equation excluding chord bending terms,as would be expected.

(iii) The Efthymiou equations give a comprehensive coverage of all the parametric variations and are de-signed to be mean fit equations. Due to the greater correlation with steel models by the Efthymiou FE models,and the fewer conservative assumptions made, these equations tend to be nearest to a mean fit and conse-quently more underpredictions are frequently observed.

(iv) Under unbalanced OPB, the Efthymiou equations give a good fit to symmetric K joints or the outerbraces in KT joints, but consistently appear to underestimate the SCF in the branch with θ in non-symmetricK joints.

The equations for T/Y joints are generally valid for joint parameters within the following limits:

• 0.2 ≤β≤ 0.8

• 8 ≤ γ≤ 32

• 0.2 ≤ τ≤ 1.0

• 20◦ ≤ θ ≤ 90◦

• 4 ≤α≤ 20

63

64 9. APPENDIX

Figure 9.1: Efthymiou/Durkin equations for T/Y joint

9.2. REFINED LR EQUATIONS FOR T/Y JOINT 65

9.2. REFINED LR EQUATIONS FOR T/Y JOINT

Figure 9.2: Refined LR equations for T/Y joint

66 9. APPENDIX

9.3. WELD SHAPE

Figure 9.3: Weld Shape(AWS)

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[17] M. Efthymiou, Development of scf formulae and generalised influence functions for use in fatigue analysis,in Recent Developments in Tubular Joints Technology (Surrey, 1988).

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parametric study of scf on tripod joint

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