A new method for calculating hot-spot stress in gap DKK CHS joints under arbitrary loading

Engineering Structures 210 (2020) 110366

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A new method for calculating hot-spot stress in gap DKK CHS joints under arbitrary loading

T

Ernest O. Oshogbunu, Yong C. Wang , Tim Stallard ⁎

Department of Mechanical, Aerospace and Civil Engineering, Faculty of Science and Engineering, University of Manchester, Manchester, UK

ABSTRACT

This paper presents a new method to calculate stress concentration factors (SCF) and hot-spot stresses in gap double K (DKK) circular hollow section (CHS) joints used in construction of offshore wind turbine jacket support structures. The focus of this paper is to overcome important limitations of existing analytical methods due to their unsuitability for application under arbitrary loading and their inability to account for carryover stresses. The proposed new analytical method consists of two steps: in the first step, an arbitrary brace loading condition in any DKK CHS joint is decomposed into symmetric and anti-symmetric axial, in-plane bending and outof-plane bending unit load cases. In the second step, 11 key SCFs are identified and used to calculate SCF distributions around the four weld perimeters of the bracechord connections under the aforementioned unit load cases, including carryover SCFs. These SCF distributions can be combined to calculate stress distributions and hot-spot stresses under any arbitrary brace loading condition in the DKK joint. Based on the results of an extensive numerical parametric study, regression equations have been found for calculating the 11 key SCFs. The proposed method has been demonstrated to calculate hot-spot stresses much more accurately than the most widely current used method.

1. Introduction The offshore wind sector has been undergoing rapid development in recent years. To 2017, around 20 GW of offshore wind turbine capacity had been installed globally with major deployments concentrated in European waters [1,2]. The rate of installation is forecast to increase with deployment of larger turbines and exploitation of deeper water sites. For depths greater than around 30 m, the traditional monopile support structure is impractical and either jacket structure or floating system is required. Support structure represents a significant part of the total capital cost of each offshore wind turbine and so it is important to reduce its weight and cost by optimizing its structural design, in particular fatigue design. The hot-spot stress S-N curve method is the widely accepted simplified method for assessing fatigue damage of welded tubular joints [3–7]. This method requires prediction of the maximum stress (hot-spot stress) at the weld perimeter of the joint and this stress is then used to quantify the fatigue life of the joint by comparison against the S-N curve. For a given combination of brace and chord loads, the hot-spot stress at each weld is determined by multiplication of the nominal stress by the respective stress concentration factor (SCF). The nominal stress can be easily calculated according to the member force obtained from using any validated simple frame structural analysis method. However, determining the SCFs under different loading conditions is a major

challenge yet to be satisfactorily resolved for multiplanar CHS joints due to complexity of carryover stresses [8–12] and variable SCF distributions [13–17] under different loading conditions in different structural configurations. The SCF of a joint is dependent on both the intrinsic geometric features of the joint and the loading condition (e.g. axial load, in-plane bending (IPB), out-of-plane bending (OPB) and their combinations). Although SCFs can be accurately predicted with 3D Finite Element Analysis (FEA) numerical models, the use of 3D FEA models requires extensive time and high computational cost. This is often not feasible during the initial design stage where key design decisions are made. Therefore, analytical equations to calculate SCFs are needed. A widely used joint type in jacket foundations for offshore wind turbines is gap double K (DKK) joint. A gap DKK joint is configured with a chord and four braces inclined at angles in the range 30–60°along the longitudinal axis of the chord, as shown in Fig. 1. For DKK joints, the existing analytical equations, including those of Karamanos et al [8] and van Wingerde et al [9] as compiled in the CIDECT fatigue design guide [5], and those of Efthymiou and Durkin [11] and Efthymiou [12] which are recognized in DNV and other standards [7,18,19], can only be used to calculate hot-spot stresses under limited individual loading conditions (Axial, IPB, and OPB) that are not suitable for arbitrary combination of load conditions experienced by the joint. Also, these equations do not calculate stresses at the weld perimeters of the

Corresponding author. E-mail addresses: [email protected] (E.O. Oshogbunu), [email protected] (Y.C. Wang), [email protected] (T. Stallard). ⁎

https://doi.org/10.1016/j.engstruct.2020.110366 Received 8 September 2019; Received in revised form 4 February 2020; Accepted 10 February 2020 0141-0296/ © 2020 Elsevier Ltd. All rights reserved.

Engineering Structures 210 (2020) 110366

E.O. Oshogbunu, et al.

can be a substantial fraction of the stresses in the loaded braces (e.g. see Fig. 1). Therefore, a new analytical method should be developed for DKK joints under arbitrary loading combinations in the brace members. Recognising the nature of arbitrary loading, the new analytical method consists of two steps. In the first step, the arbitrary loading condition is resolved into six basic loading cases (symmetrical and anti-symmetrical brace axial load, in-plane bending and out-of-plane bending). In the second step, stress concentration factor (SCF) distributions under all the basic loading cases are calculated. In order to minimise the number of analytical equations, the number of SCF points (key SCFs) necessary for quantifying the SCF distributions is minimised based on an assessment of SCF distributions under different individual basic loading cases. Multiplication of SCF distributions under each loading condition with the nominal stress of the loading condition gives stress distributions in the joint. Summation of the stress distributions under the different individual loading cases then yields the hot-spot stress for the joint under an arbitrary load combination. For this purpose, a large amount of SCF distributions, under different loading conditions and with different DKK joint geometries are necessary, and this is best carried out by numerical simulation. This paper presents the results of this detailed study where the numerical simulations were carried out using the general finite element software ABAQUS [20].

(a) Stress in Joint Brace 1 (B1) Brace 3 (B3)

25

Brace 2 (B2) Brace 4 (B4)

20

SCF

15 10 5 0 -5 -10 -15

0

0.25

0.5

0.75

1

2. Numerical modelling and validation

Normalised weld toe perimeter

(b) SCFs distribution around all four weld perimeters

For validation of the authors’ ABAQUS model, the relevant CIDECT test [21] on welded DKK gap joints in a girder was simulated and the test and simulation results were compared for two DKK gap joints, each selected from two different girders.

Fig. 1. Example of a DKK CHS joint under single brace axial load.

unloaded members (carryover stresses). These limitations can be significant, because (1) the location of the maximum stress under different loading conditions do not coincide, hence it is not appropriate to add up hot-spot stresses under individual loading conditions to give the total hot-spot stress under combined loads, and (2) the carry-over stresses

2.1. Experiments The experiments were conducted on four triangular lattice steel girders (refereed to girders 5, 6, 7, 8) each with four bays of DKK joints

Fig. 2. Global structural dimensions of girders 6 and 7 triangular lattice girders tested in CIDECT [21].

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Fig. 3. Test set up for girder 7: (a) overall view of test girder, (b) & (c): gap and overlap joints (GR07GP01) [21]. Table 1 Structural sizes of brace and chord members [21].

Table 2 Girders 6 and 7 steel mechanical properties [21].

Girder

Chord (mm)

Brace (mm)

Gap

Overlap

Girders

Member

6 7

193.7 × 16 193.7 × 8.0

76.1 × 4 114.3 × 8

0.44D 0.17D

100% 50%

Yield strength (MPa)

Ultimate Strength (MPa)

Ultimate strain %

6

Top Chord Bottom Chord Braces Top Chord Bottom Chord Braces

260.5 257

405.5 393

34 32

283 270 262

413 400 392

35.7 33 36

388.5

564.75

30

[21]. Girders 6 and 7 (see Fig. 2) have been selected for the numerical validation study, where a gap joint in each girder tagged GR06GP01 and GR07GP01 was considered as the referenced joint. Fig. 3 shows the set-up for girder 7 which is typical of the four tested girders. Table 1 lists the structural sizes of each joint in girders 6 and 7. A typical tested girder consisted of two gap and two overlap DKK joints as illustrated in Fig. 4. The tests investigated 50% and 100% overlapped joints. The gap joints in girder 6 had no vertical eccentricity while those in girder 7 had a vertical eccentricity of 48 mm from the chord’s longitudinal axis as shown on the right side in Fig. 2. The mechanical properties of steel of the different members of girders 6 and 7 are different and are given in Table 2. Girders 6 and 7 were supported at both ends of the two top chord members by hinges. Strain gauges were placed at the weld toe sides and regions further away from the weld toes of the selected members. Fig. 5 sketches the strain gauge arrangement for the failed brace (tagged as brace 2 in Fig. 3) in joint GR07GP01. A total of eight strain gauges were positioned at this brace chord intersection at designated distances from the weld toes for joint GR07GP01. All strain gauge arrangements at the weld toes in joints GR06GP01 and GR07GP01 were as recommended by the ECSC Technical working party on tubular joints [22], which is illustrated in Fig. 6 with details in Table 3. The rutile electrode used in the welding process for the selected

7

girders was in accordance with AWS specification A5.01 [6]. Table 4 gives the leg lengths of the as-built welds of joint GR06GR01 and joint GR07GP01 illustrated in Fig. 7. As shown in Fig. 2, two point loads were applied at the mid span of the girder at the top chord members. Cyclic loads, with a sinusoidal constant amplitude load ratio of R = Fmin/Fmax of 0.1 at 1Hertz were applied to the lattice steel girder until through thickness crack was observed on the joints’ walls. However, the SNCF before initial crack propagation were examined under a static load of F = 100kN, which is used in this validation study. 2.2. FEA model of DKK joint GR06GP01 and joint GR07GP01 2.2.1. Assemblage of numerical DKK joint models in ABAQUS Because the internal member forces of the joint of interest (GR06GP01 and GR07GP01) in girders 6 and 7 were not recorded, the whole test girder was modelled. To minimise computational effort, the

Fig. 4. Typical DKK joints in the lattice girders investigated by CIDECT [21].

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Fig. 5. Strain gauge arrangement for brace 2 of joint GR07GP01.

Fig. 7. Locations of weld profile measurements for GR06GP01 and GR07GP01 joints.

in Fig. 8. The same approach was used for girder 6.

Fig. 6. Strain measuring positions for calculating hot-spot strain using linear extrapolation, recommended by ECSC WG III [22] and CIDECT guide 8 [5].

2.2.2. Boundary conditions Fig. 8 shows the boundary conditions and interfaces between solid and beam elements. At the physical supports of the test girder, displacements of the simulation model nodes were prevented. At the intersections of the beam/3D solid elements, the nodes of 3D solid elements were restrained by coupling, as shown in the insert of Fig. 8.

Table 3 Extrapolation locations at brace and chord sides (R = chord radius, r = brace radius, T = chord thickness, t = brace thickness) [5]. Distance from Weld Toe

Chord

Brace

Saddle A

Crown

Saddle

0.2 RT

B

Crown

2.2.3. Meshing sensitivity analysis 3D brick element types C3D8R (eight node with one integration point) and C3D20R (twenty node with eight integration point) were used for the members of the joint of interest away from the brace-chord intersection region and for the brace-chord intersection region respectively [23]. Mesh sensitivity studies were performed in three different directions in two steps. In the first step, mesh sizes in the extrapolation zone around the intersection of chord-brace joint and in the extrapolation direction were varied. Once convergence was achieved, a second mesh analysis through the tube thickness was then carried out. Fig. 9 shows different mesh sizes in different zones. Based on the results in Table 5, the number of mesh element around the brace-chord intersection can be taken as 20 and 40 for joint GR06GP01 and joint GR07GP01 respectively. In the thickness direction, it is adequate to use two and three elements through the brace and chord thickness respectively.

0.2 RT 0.65 rt

0.4 4 rtRT

Table 4 Measured weld leg lengths for joints GR06GP01 and GR07GP01 [21]. Weld leg length (mm) Girder

Brace

Near Saddle

Far Saddle

Crown Toe

Crown Heel

H

V

H

V

H

V

H

V

Girder 6

1 2 3 4

8.0 6.9 6.0 5.1

11.5 11.2 9.5 10.0

5.0 5.8 6.0 5.2

8.6 8.5 8.4 9.2

4.8 5.5 6.6 6.0

11 10.8 10.5 10.5

10.1 11.3 9.2 12.7

11.8 11.0 11.0 14.2

Girder 7

1 2 3 4

9.5 10.0 9.5 10.8

6.9 7.7 11.8 10.3

3.6 5.3 5.4 5.4

8.7 10.3 7.6 11.0

11.5 7.1 9.2 8.2

8.4 9.9 9.7 10.4

16.6 15.8 17 18.4

15.3 16.8 18.1 17.1

2.2.4. Comparison between simulation and test results for hot-spot strain In the test report of CIDECT [21] nominal strains, were determined at the hot-spot zones by linear extrapolation using distances three times greater than the brace diameter. In order to make comparison for the same strain quantity, the ABAQUS model strain magnitudes at the same positions were obtained, and they were then used to obtain the SNCFs perpendicular to the weld toes around the weld perimeter of the joints.

test girders were modelled using a hybrid model in ABAQUS with a mixture of solid elements around the joint of interest to obtain detailed strain concentration factors (SNCFs) and beam elements for other members. A schematic of the hybrid model setup for girder 7 is shown

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Fig. 8. Hybrid model; beam elements for members and 3D brick elements for the joint of interest.

Fig. 9. Parts of a meshed typical gap DKK CHS joint (solid element type). Table 5 Results of mesh sensitivity study for joints GR06GP01 and GR07GP01. Load

Total load on girder F = 100 kN

Joint

GR06GP01

GR07GP01

Member (No of Through Thickness Elements)

Brace(1) Chord (1) Brace(1) Chord (1) Brace(1) Chord (1) Brace (1) Chord(2) Brace (2) Chord(3) Brace(1) Chord (1) Brace(1) Chord (1) Brace(1) Chord (1) Brace (1) Chord(2) Brace (2) Chord(3)

Number of Elements around brace-chord intersection

40 26 20 20 20 78 52 40 40 40

5

Maximum SNCFs Numerical

Test

1.61 1.58 1.57 1.57 1.55 2.43 2.39 2.35 2.34 2.32

1.53

2.26

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numerical simulation model. 3. Development of a new SCF calculation method 3.1. Definition of parametric study variables To develop a new analytical method to calculate SCF distributions in gap DKK joints, an extensive numerical parametric study was conducted, covering different gap DKK joint geometries and different loading conditions for offshore wind turbine support structures. Table 6 summarises the parametric study ranges for the geometric variables and explains their rationales for selection. The external loading conditions are: – Sea wind speeds; 12 m/s and 25 m/s – Wave: Fully developed by corresponding wind speeds according to Stoke 5th order wave theory – Wind turbine Specification: 5 MW (12 m/s peak rate speed).

Fig. 10. SNCF distribution for failed connection Brace 2 chord side. Test

1.6

FE (Perpendicular SNCF)

The simulation weld profile was according to AWS specification [6]. The minimum weld thickness in AWS specification was used. Changing the weld thickness has little effect on SCFs and using the minimum AWS weld thickness ensures safe results are generated. Fig. 12 shows an example weld profile according to AWS specification and its implementation in ABAQUS simulation model for a joint with 60° inclined braces.

1.4

SNCF

1.2 1 0.8

3.2. Outline of the new method

0.6

A two-step strategy is developed. In the first stage, the arbitrary combined brace loading of a joint in a jacket structure is decomposed into six basic brace loading cases. In the second step, a limited number of key SCFs are calculated for the six basic loading cases and the resulting stress distributions are combined to give the hot-spot stress of the joint.

0.4 0

0.2

0.4

0.6

0.8

1

Crown toe to crown toe location of joint intesrection (Chord side) Fig. 11. SNCF distribution for failed connection Brace 2 brace side.

3.2.1. Decomposition of arbitrary internal load conditions Fig. 13 shows an arbitrary combination of a DKK joint, with axial (Ax) load, in-plane bending (IPB) and out-of-plane bending (OPB) in all brace members. Instead of obtaining hot-spot stress distributions under each load in each brace member, it is proposed to deal with pairs of brace members and make use of symmetry and anti-symmetry principles. The pair of braces can either be those around the chord (e.g. brace pairs 1&3 and 2&4 in Fig. 14) or along the chord (e.g. brace pairs 1&2 and 3&4 in Fig. 14). It is better to use the latter pairing because the

The extrapolated perpendicular SNCFs were determined using the extrapolation method described in Fig. 6 and Table 3 as in the test. Figs. 10 and 11 compare the ABAQUS simulation and test results for SNCF distributions perpendicular to the weld toes of the failed brace (Brace 2) in joint GR07GP01. Furthermore, Table 5 compares numerical simulation and test results of the maximum SNCFs for both tests. These comparisons indicate very good agreement; in particular, the locations and values of the maximum SNCFs were well predicted using the

Table 6 Summary and commentary on the variables selected for numerical parametric study of DKK CHS joints in offshore jacket foundations. Geometric Parameters

Selected values

Commentary

Brace out of plane angle, φ Brace in- plane angle, θ Brace-chord thickness ratio, τ = t/T Chord slenderness ratio, γ = D/2T Brace-chord diameter, β = d/D

90° 30°, 45°, 60° 0.25, 0.5, 1.0 12, 18, 24 0.4 and 0.6

Typical out-of-plane angle of a four legged jacket structure Identified to have significant influences on SCFs by existing FLS design guides [5,7]. Therefore three values were selected to cover practical ranges in jacket construction.

Joint gap parameter (in-plane brace gap), ξ = g/D Chord length-Diameter ratio αc = 2L/D Weld Specification

0.4

Although the range of β in CIDECT guide 8 [5] is from 0.3 to 0.5, the practical range of β for gap DKK joints in offshore jackets is from 0.4 to 0.6 [24]. The effect of this parameter on SCF distributions is small according to Efthymiou [11] for uniplanar K joints.

6D

Any longer length will have minimal effect [25].

External Load Conditions

It is necessary to include weld profile of the joint in modelling SCFs; however, the weld thickness has modest effects on SCFs as long as the minimum weld thickness specified by the AWS [6] specifications (shown in Fig. 12) is used. Selected to generate different combinations of brace axial load, in-plane bending and out-of-plane bending.

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Fig. 12. Exemplar weld profile, for a joint with 60° inclined brace, in accordance with American Welding Society [6] prescription and its implementation in ABAQUS simulation.

Fig. 13. An arbitrary combination of internal loads of a DKK CHS joint.

angle between braces along the chord is arbitrary whilst that around the chord is fixed at 90°. Considering one pair of brace members, e.g. 1 & 2, the six basic loading cases are:

the six basic loading cases. Each basic loading case can be obtained by multiplying the unit load case by a load coefficient (μ). The coefficient for symmetrical loading is the average value of the loads in the two braces, and the coefficient for anti-symmetrical loading is half of the difference in loads in the two braces, as expressed in Eq. (1), where P1 and P2 are internal loads in braces 1 and 2 respectively.

• Symmetrical and anti-symmetrical axial loads; • Symmetrical and anti-symmetrical IPB; • Symmetrical and anti-symmetrical OPB.

I ¼12, sym =

Furthermore, using the latter pairing would enable using a minimal number of key SCFs to obtain complete SCF distributions. This way, there are six basic brace loading cases, as shown in Fig. 15, and any arbitrary loading condition in Fig. 13 can be constructed by combining

P1 + P2 P2 - P1 , I ¼12, asym = 2 2

(1)

Under the loading cases in Fig. 14, the primary SCFs in braces 1 and 2 and the carryover SCFs in braces 3 and 4 can be calculated. The same methodology can be applied to braces 3 and 4 to calculate the primary SCFs in braces 3 and 4 and the carryover SCFs in braces 1 and 2 under

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Fig. 14. Six basic unit load cases and their load multipliers for the joint in Fig. 13.

Fig. 15. Basic unit brace load cases.

Fig. 16. Possible hot-spot stress locations under all basic loading cases (LC).

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Fig. 17. Normalised SCF distributions for primary SCF distributions for two reference joints under unit axial symmetric basic load (the SCFs selected to determine the key SCF values of the 30° joint are marked with circular pointers).

Fig. 18. Comparison of carry over proposed SCF distribution with simulation results of all joints.

loads in braces 3 and 4. Owing to symmetry, the primary and carryover SCFs under unit loads in braces 1 and 2 (Fig. 15) are the same as those under unit loads in braces 3 and 4, but with corresponding changes in brace references. The total stresses in brace pairings 1&2 and 3&4 are as follows: hs 12

= µ12, sym .

µ34, sym . hs 34

34, sym .SCFc . ov sym

= µ34, sym.

µ12, sym .

12, sym .SCFsym

+ µ12, asym .

+ µ34, asym .

34, sym .SCFsym

34, asym.

+ µ34, asym .

12, sym .SCFc . ovsym + µ12, asym .

12, asym.

SCF asym+

SCFc . ov asym

34, asym.

weld perimeters of the loaded braces under symmetrical and antisymmetrical unit brace load cases respectively. SCFc.ov sym and SCFc.ov asym are the carryover SCFs generated under unit symmetrical load in the adjacent braces and carryover SCFs under unit anti-symmetrical loads in the adjacent braces respectively (e.g. SCFs at the weld perimeters of unloaded braces 3 and 4 due to loading of braces 1 and 2). 12, sym and 34, sym are nominal stresses under symmetric paired unit forces in braces 1 & 2 and braces 3 & 4 respectively, and 12, asym and 34, asym are nominal stresses under antisymmetric paired unit forces in braces 1 & 2 and braces 3 & 4 respectively. The nominal stresses under unit values of axial load, IPB and OPB in a brace member are calculated as follows:

(2)

SCF asym+

12, asym. SCFc . ov asym

(3)

In whichSCFsym and SCFasym are the primary SCFs generated at the

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Fig. 19. Comparison of proposed SCF distribution and normalised primary SCF distributions of ABAQUS simulations for two reference joints under unit axial antisymmetric basic load.

Fig. 20. Comparison of carry over proposed SCF distribution with simulation results of all joints. bending

axial

= 1kNm/braceelasticmodulus

= 1kN/bracecrosssectionalarea

on hot-spot stress in weld on the chord side because this was found to be higher than that on the brace side. Under chord load, stresses in the chord side welds are similar to nominal stresses, and stresses in the brace side welds are negligible, as found by [26] for uniplanar K joints.

(4) (5)

For each loading condition, the SCFs at the weld toes under each of the six basic load cases where calculated as follows:

principalstressatweldtoeofloadedbrace bracenominalstress

(6)

principalstressatweldtoeofunloadedbrace bracenominalstress

(7)

SCF , primary = SCF , c . ov =

3.2.2. Identification of key SCFs Fig. 16 shows 16 possible locations of the maximum hot-spot stress and carryover stress, consisting of 4 locations (crown toe, crown heel, far saddle and near saddle) around 4 weld perimeters of the joint. Combining this with 6 basic load cases, a total of 96 SCFs would be required. This number of SCF equations is not manageable and this section will explain how this number can be massively reduced based

Principal stress, instead of stress perpendicular to weld profile, was used because principal stress is the highest stress. This research focused

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Fig. 21. Comparison of proposed and normalised primary SCF distributions of ABAQUS simulations for 30° referenced joint geometry under unit IPB symmetric and anti-symmetric basic load. (SCFs selected to determine keys SCF values are marked with circular pointers).

Fig. 22. Comparison of proposed and normalised primary SCF distributions of ABAQUS simulations for 60° referenced joint geometry under unit IPB symmetric and anti-symmetric basic load. (SCFs selected to determine keys SCF values are marked with circular pointers).

on symmetry/anti-symmetry considerations and an examination of simulation results. This was done by identifying 1 or 2 key SCFs for each basic brace load case. For clarification of presentation of results, SCF distributions under a load case are normalized by dividing SCFs by the maximum SCF of the load case. If one key SCF is sufficient for a particular load case, the normalised SCF distributions for different joint geometries should be banded together. If two key SCFs are necessary to draw the approximate SCF distribution for a load case, the approximate normalised SCF distribution will be different for different joints. The results of two joint geometries (θ = 30°, β = 0.6, τ = 0.25, γ = 12 and θ = 60°, β = 0.4, τ = 0.25, γ = 24) will be used to demonstrate how to obtain the two

key SCFs. It should be pointed out that the purpose of this exercise is to obtain the hot-spot stress of the joint. Therefore, as long as SCFs near the location of the maximum SCF are closely calculated, no refinement was attempted to accurately calculate SCFs in regions with low values. In Figs. 17–28, each bold number corresponds to the relevant key SCF marker in Fig. 29. 3.2.2.1. Axial symmetric load. For this loading case, three key SCFs are needed; two to draw SCF distributions (primary SCFs) at the weld toes of the loaded braces and one to draw carryover SCF distributions of the unloaded braces.

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Fig. 23. Carryover stress distributions normalised to the hot-spot stress for each joint.

Fig. 24. Comparison of proposed SCF distribution and normalised primary SCF distributions of ABAQUS simulations for several joint geometries under OPB symmetric basic load.

giving a value for key point 2 of 0.52. For the 60° reference joint, these two values are 1.0 and 1.0 respectively, giving a value for key point 2 of 1.0. The SCF distributions for the two reference cases are completely different (higher SCFs around the crowns, lower SCFs around saddles for the 30° reference joint, and opposite trend for the 60° referenced joint). Other joint geometries display SCF distributions between such two opposing trends. However, because two key SCFs (marker 1 and marker 2) are used, the trends for all joints can be captured by the proposed approximate SCF distributions. For carryover SCFs, the saddle regions tend to have higher SCFs than the crown regions. For simplification, it is assumed that the SCF at the crown regions is zero. Fig. 18 shows that all simulation results are within a band around the proposed SCF distribution. The SCF value for

Fig. 17 compares the normalised ABAQUS SCF distributions using two key SCF values for the primary SCF distributions of the two reference joints. Key value 1 is the average of the crown toe and crown heel maximum SCFs if these SCFs are higher than those in the saddle regions (dashed lines in Figs. 17, 30° reference joint), or the minimum SCF at the crown toe if the SCFs in the saddle regions are higher than those in the crown toe (solid lines in Fig. 17, 60° reference joint). For example, for the former (30° reference joint), the value (0.744) for key point 1 is the average of 1 for the crown toe and 0.487 for the crown heel respectively. For the latter (60° reference joint), the value (0.35) for key point 1 is the minimum SCF at the crown toe only. The value for key point 2 is the average of the near and far saddle maximum SCFs for both joints. For the 30° reference joint, these two values are 0.5 and 0.54 respectively,

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Fig. 25. Comparison of proposed SCF distribution and normalised carryover SCF distributions of ABAQUS simulations for several joint geometries under OPB symmetric basic load.

Fig. 26. Trends of OPB symmetric carryover SCF distributions as β changes for 0.4, 0.5 and 0.6 with their respective proposed SCF distribution.

marker 3 is the average of the maximum value at the far saddle and the near saddle regions. The proposed approximate SCF distributions are symmetric about both the crown-crown and saddle-saddle axes, whilst the ABAQUS simulation results are slightly different, especially for the cases of

β = 0.6. The slight non-symmetrical SCF distributions from ABAQUS simulation are mainly caused by the presence of brace members on only one side of the loaded braces. As will be presented later when comparing combined SCF distributions, this slight non-symmetry of SCF distribution has little effect on the maximum hot-spot stress under

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Fig. 27. Comparison of proposed SCF distribution and normalised primary SCF distributions of ABAQUS simulations for several joint geometries under OPB antisymmetric basic load.

Fig. 28. Comparison of proposed SCF distribution and normalised carryover SCF distributions of ABAQUS simulations for several joint geometries under OPB antisymmetric basic load.

combined loads.

ABAQUS simulation results. The ABAQUS simulation results are banded around the proposed SCF distribution.

3.2.2.2. Axial anti-symmetric load. Fig. 19 compares the proposed and ABAQUS simulation results for primary SCF distributions. Under axial anti-symmetrical load, SCFs around the crown and saddle regions are different, so two key SCFs (markers 4 and 5) are needed as under axial symmetric load. The values for key SCF markers 4 and 5 are determined in the same way as described for markers 1 and 2 under axial symmetric basic load case. Fig. 19 shows that these two key SCFs are able to represent the contrasting trends of ABAQUS simulation results for the two reference joints. For carryover SCFs, the values are almost zero near the far saddle and almost constant in other regions. Therefore, only one key SCF (marker 6) can be used. The value of marker 6 is the average of the two peak SCFs within the crown toe to crown heel regions. These two peaks SCFs were selected from either the crown toe, near saddle or crown heel regions depending on the ABAQUS modelling results. Fig. 20 compares the proposed carryover SCF distribution with all

3.2.2.3. Under IPB load. Figs. 21 and 22 compares the proposed approximate SCF distribution with ABAQUS results for primary stress for the two reference joints under both symmetrical and antisymmetrical in-plane bending (IPB). Two key SCFs (markers 7 and 8) are necessary to represent SCFs at the crown heel and crown toe locations and the SCF distribution is almost linear between these two locations. The values for marker 7 are the same under symmetrical and antisymmetrical IPB but change directions according to the change in directions of IPB. The values of marker 8 follow the same trend. The values of markers 7 and 8 are the averages of the ABAQUS SCF values in their respective quadrants. For example, for the 30° reference joint in Fig. 21, the value of 0.874 for marker 7 was obtained as the average of 1.0 and 0.748, and the value of 0.95 for marker 8 was obtained as the average of 1.0 and 0.898. For the 60° reference joint in Fig. 22, the value of 0.917 for marker 7

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Fig. 29. Summary of approximate SCF distributions constructed using 11 key SCFs.

was obtained as average of 0.931 and 0.907, and the value of 0.917 for marker 8 was obtained as average of 0.931 and 0.907. The carryover stresses are negligible in all cases, as shown in Fig. 23.

far saddle regions have similar values but opposing directions. Refer to Fig. 26(a), the explanation is as follows. Under symmetric OPB, the inward displacement of the loaded braces makes the chord member of the DKK joint to change from a circular to an oval shape (A), thus causing the far saddle regions of the loaded and unloaded braces to experience tensile stresses (B). This tensile stresses generated could reach the near saddle of the unloaded brace, depending on how far the caved in point (C) is from the unloaded braces. When the brace diameter is small, the near saddle SCFs follows those at the far saddle, which are tensile. As the brace diameter increases, the near saddle SCFs are primarily controlled by the nearby primary SCFs are compressive just as shown in Fig. 24. In this research, it is assumed that the OPB symmetric load case carryover SCF distribution pattern of similar SCFs in both the near and far saddle regions applies to β < 0.5, whilst opposing trends in these two regions are applicable to β ≥ 0.5. The absolute value for Marker 10 is determined as the average of the maximum carryover SCF in the saddle regions under OPB symmetric load case and maximum carryover SCF at the near saddle region under anti-symmetric load case. This was so

3.2.2.4. Under OPB load. Under symmetrical out-of-plane bending (OPB), the saddle regions experience high stresses whilst the crown regions have very little stress, for both the primary stress and carryover stress. Therefore, one key SCF each is needed for the primary stress and the carryover stress respectively, as shown in Figs. 24 and 25. Because there is only one proposed SCF distribution curve for primary stress the same proposed SCF distribution can be compared to all ABAQUS simulations. For primary SCFs, the value of marker 9 is the average of the maximum SCF at the far and the near saddle regions. However, in Fig. 25, the carryover SCF distribution in the near saddle region may be opposite in directions depending on the brace to chord diameter ratio, β. As shown in Fig. 26, when β is 0.4, the carryover SCF distributions in the near and far saddle regions follow the same trend. When β is 0.5 or 0.6, the carryover SCF distributions in the near and

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because the calculated maximum carryover SCF for OPB symmetric load case and OPB anti-symmetric load case were very close. The comparisons of OPB anti-symmetric load case in Figs. 27 and 28 confirm that the proposed SCF distributions for both primary SCF (using key value 11) and carryover SCF (using value 10) are adequate to represent ABAQUS simulation results because the simulation results are banded around the respective proposed distribution. In summary, the 11 key SCFs and the assumed SCF distributions using the 11 key SCFs are able to generate SCF distribution profiles close to ABAQUS simulation results in terms of the maximum values as

well as approximate locations of the maximum values under different unit load cases. The joint geometries in Table 6 have a total of 54 combinations. Table 7 lists the values of the 11 key SCFs for all 54 parametric study cases. These values will be used in regression analyses to derive analytical equations. These approximate profiles and the key SCFs required to define each are summarised in Fig. 29. The thick linear lines in the figure are the approximate SCF distributions for primary SCF distributions at the weld toes of the loaded braces and the broken lines are for carryover SCF

B1 Axial Symmetric LC B3 Axial Symmetric LC ABAQUS Modelling B3

60

B2 Axial Symmetric LC B4 Axial Symmetric LC ABAQUS Modelling B1

40

Stress MPa

20 0 -20 -40 -60 -80 -100

0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

Crown toe to crown toe normalised weld toe perimeter

(a) Axial symmetric load case for all four brace-chord weld perimeters of exemplar joint in Figure 13. B1 Axial Anti-symmetric LC B3 Axial Anti-symmetric LC ABAQUS Modelling B3

150

B2 Axial Anti-symmetric LC B4 Axial Anti-symmetric LC ABAQUS Modelling B1

100

Stress M Pa

50 0 -50 -100 -150

0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

Crown toe to crown toe normalised weld toe perimeter

(b) Axial anti-symmetric load case for all four brace-chord weld perimeters of exemplar joint in Figure 13.

Fig. 30. (a)–(g): Results of individual and combined stress distributions in all four braces of exemplar joint in Fig. 13 under combined load cases.

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B1 IPB Symmetric LC B3 IPB Symmetric LC ABAQUS Modelling B3

50

B2 IPB Symmetric LC B4 IPB Symmetric LC ABAQUS Modelling B1

40 30

Stress MPa

20 10 0 -10 -20 -30 -40 -50

0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

Crown toe to crown toe normalised weld toe perimeter

(c) IPB symmetric load case for all four brace-chord weld perimeters of exemplar joint in Figure 13.

B1 IPB Anti-symmetric LC B3 IPB Anti-symmetric LC ABAQUS Modelling B1

15

B2 IPB Anti-symmetric LC B4 IPB Anti-symmetric LC ABAQUS Modelling B3

10

Stress MPa

5 0 -5 -10 -15

0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

Crown toe to crown toe normalised weld toe perimeter

(d) IPB anti-symmetry load case for all four bracechord weld perimeters of exemplar joint D. Fig. 30. (continued)

distributions at the weld toes of the unloaded brace members.

The exemplar joint in Fig. 13 is first presented to show the hot-spot stress calculation procedure using the 11 key SCFs. Table 8 lists the values of force multiplication factors (load coefficient, μ) for the 6 basic loading cases and the 11 key SCFs extracted directly from ABAQUS simulation results under unit loads for the six basic brace loading cases. Using the values in Table 8 and the proposed approximated SCF distributions in Fig. 29, stress distributions around the weld perimeters of the four braces under all 6 loading cases can be generated. These are

3.2.3. An example of comparison between FE and simplified method of constructing SCF distributions To validate the proposed approach of superposing a set of simplified SCF distributions that are defined using only 11 key SCF values, this section presents an example of comparisons of stress distributions between ABAQUS simulation results and the proposed approximations.

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B1 OPB Symmetric LC B3 OPB Symmetric LC ABAQUS Modelling B1

15

B2 OPB Symmetric LC B4 OPB Symmetric LC ABAQUS Modelling B3

10

Stress MPa

5 0 -5 -10 -15

0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

Crown toe to crown toe normalised weld toe perimeter

(e) OPB symmetry load case for all four brace-chord weld perimeters of exemplar joint in Figure 13. B1 OPB Anti-symmetric LC B3 OPB Anti-symmetric LC ABAQUS Modelling B1

200

B2 OPB Anti-symmetric LC B4 OPB Anti-symmetric LC ABAQUS Modelling B3

150

Stress MPa

100 50 0 -50 -100 -150 -200

0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

Crown toe to crown toe normalised weld toe perimeter

(f) OPB anti-symmetry load case for all four brace-chord weld perimeters of exemplar joint in Figure 13. Fig. 30. (continued)

shown in Fig. 30(a) to (f) and are compared against ABAQUS results. For clarity, only ABAQUS results for braces (B1) and (B3) are shown since results are the same for braces (B2) and (B4) respectively for symmetric load cases and opposite in sign convention for braces (B2) and (B4) respectively for anti-symmetric load cases.

Summation of the stress distributions under the 6 basic loading cases of each brace-chord weld perimeter respectively gives the final stress distributions under the combined loads as shown in Fig. 30(g). In Fig. 30(g), the ABAQUS numerical simulation results are also plotted. Comparison of the two sets of results indicates that the magnitude and

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(g) Comparison of approximate combined stress distributions against ABAQUS Modelling results for braces 1-4 of the exemplar joint in Figure 13 (Maximum stresses marked with circles) Fig. 30. (continued)

location of the maximum stresses in all brace-chord intersection weld perimeters are close. The same procedure was repeated for all the 54 parametric study cases under the two different external loading conditions, giving a total of 432 hot-spot stresses (54 × 2 × 4 brace members). Fig. 31 compares the ABAQUS hot-spot stresses with the proposed method of using 11 key SCFs to plot stress distributions of a joint. For this comparison, the key SCFs were directly extracted from ABAQUS simulation results. These results are also compared to the hot-spot results using Efthymiou’s equations [11,12]. Although the Efthymiou equations were not derived for DKK joints under combined loading, due to a lack of available method, they have been used for such cases [7,18,19]. For the Efthymiou results, the hot-spot stresses under different individual loading cases were added together respectively at weld toe locations were stresses can be calculated using the Efthymiou’s equations [11,12]. This comparison confirms that using the proposed 11 SCFs calculates hot-spot stresses very close to the ABAQUS simulation results, with a mean value of P/R (calculation/ABAQUS value) of 0.99 and standard deviation of 0.0711, and with the P/R within a range of 0.87 to 1.13. In contrast, the Efthymiou equation calculation results are widely different in most cases, with the P/R value ranging from 0.50 to 2.24. In summary, it is acceptable to use the 11 key SCFs and the proposed SCF distributions to find hot-spot stresses for DKK CHS joints in offshore wind turbine lattice structures. The next section will perform regression analyses to directly calculate these 11 key SCFs.

4. Regression analyses Based on research studies by others [5,11,12,27,28], a power equation form can be used to relate the SCF to the geometry parameters of joints. The power equation form is adopted in this study and the power equation is expressed in Eq. (8) where x1, x2, x3 and x4 are to be determined by regression analysis.

SCFpn = a0

x 1 x2 x3 x 4

(8)

The regression analysis was carried out using an optimizer tool, based on the generalized reduced gradient nonlinear solver [29]. The optimizing process is outlined below:

• Initial values (preferably smallest values of respective geometry properties) were input as the tentative exponents. • The tentative non-revised SCFs are predicted. • The percentage differences between ABAQUS modelling results and • •

the above are squared for each of the 54 DKK joints and then summed up to get a total percentage difference. The summed percentage difference is minimized. Iterating if necessary until the results converge.

The optimization process yielded the power values in Table 9 for Eq. (8). Using Eq. (8) with the values in Table 9, the 432 hot-spot stresses were recalculated using the 11 key SCFs for the 54 joint geometries. Fig. 32 compares the calculation results of combined hot-spot

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Table 7 Key SCFs extracted from ABAQUS Models for all 54 DKK CHS joints under unit basic loads.

Note: The shaded rows are for the two reference joints.

stresses using the calculated key SCFs (SCFpn) and calculation results using Efthymiou’s equation [11,12] against the ABAQUS results. The new calculation results using the proposed SCF equations give P/R values ranging from 0.76 to 1.20, with a mean P/R value of 1.0 and standard deviation of 0.093. These calculation results are slightly less accurate than from directly obtaining the 11 SCFs from ABAQUS

simulations (Fig. 32), but given complexity of the problem under consideration, and much superior performance compared to the Efthymiou’s equations [11,12], the level of accuracy of the author’s method is acceptable.

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Table 8 Multipliers, braces nominal stresses under unit forces and key SCF values for the exemplar joint in Fig. 13. Locations wrt Basic load Cases

Key SCFMarkers Multipliers (load coefficients, μ, defined in Equation (1)) Brace 1

Brace 2

Brace 3

Brace 4

Nominal stress under Joint D unit kN or kNm SCFs brace load (MPa)

1 2 3 4 5 6

−89.66

−89.66

−599.59

−599.59

599.59 200.15

599.59 −200.15

89.66 1763.64

89.66 −1763.64

−1763.6

1763.64

−200.15

200.15

IPB

Crown Toe (symmetric/anti-symmetric) Crown Heel (symmetric/anti-symmetric)

7 8

8.8/−44.76 −8.8/44.76

8.8/44.76 −8.8/−44.76

−145.1/6.01 145.1/−6.01

−145.1/−6.01 145.1/6.01

OPB

Symmetric Carryover (symmetric/anti-symmetric) Anti-symmetric

9 10 11

−12.73 −12.73 −13.22 −13.22 0.10641 −13.22/109.76 −13.22/−109.76 −12.73/330.15 −12.73/−330.15 −330.15 330.15 −109.76 109.76

Methods Stresses/Abaqus Results for Braces 1-4

Axial Symmetric Toe Symmetric Saddle Symmetric Carryover Anti-symmetric Toe Anti-symmetric Saddle Anti-symmetric Carryover

Using 11 Key SCFs from Abaqus Model P/R=0.87

2.25

0.02214

3.73 5.95 3.45 2.442 2.66 1.15

0.10641

1.67 1.605 4.29 0.74 2.68

Efthymiou (1985;1988) P/R=1.13

2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25

0

50

100

150

200

250

300

350

Abaqus Model Stresses for Braces 1-4 (MPa) Fig. 31. Comparison of hot-spot stresses (maximum stress) between the author’s proposal of using 11 key SCFs (obtained from ABAQUS results) and Efthymiou equations.

4.1. Conclusions

Table 9 Eq. (8) constants and geometry variables exponent. Basic Load Cases wrt Weld Locations (see Fig. 29)

Markers

This paper has provided the results of a numerical study to develop a simple method to obtain the maximum stress (hot-spot stress) for gap DKK CHS joints in offshore wind turbine lattice structures under arbitrary loading conditions. The following conclusions can be drawn:

Constants and Exponents a₀

x1

x2

x3

x4

Axial Symmetric Toe Saddle Carryover (C.Ov)

1 2 3

4.07 0.49 0.67

0.12 1.08 0.91

0.97 1.09 1.15

−0.32 −0.59 0.03

0.23 1.43 1.28

• Any arbitrary brace loading case can be decomposed into six basic

Axial Anti-symmetric Toe Saddle Carryover (C.Ov)

4 5 6

1.17 0.65 0.77

0.49 0.79 0.62

0.90 1.08 1.07

−0.16 −0.21 0.55

0.76 1.30 0.87

IPB Toe IPB Heel

7 8

0.63 0.96

0.60 0.47

0.69 0.90

0.03 −0.05

0.94 0.23

• •

OPB Symmetric Saddle OPB Carryover (C.Ov) OPB Anti-symmetric Saddle

9 10 11

1.03 1.01 0.64

0.97 0.78 0.84

0.96 1.10 0.89

0.57 1.79 0.02

1.21 1.21 1.19

21

brace load cases (symmetric and anti-symmetric loading cases for brace axial, IPB and OPB forces). Complete stress distributions around all four weld perimeters of the gap DKK CHS joint can be constructed by using only 11 key SCFs. Using the 11 key SCFs extracted from ABAQUS simulation results to construct SCF distributions and combining them to calculate hotspot stresses for the 432 cases of the numerical parametric study (54 joint geometries, 2 external loading cases, 4 brace members), the calculated hot-spot stress (P) to the ABAQUS simulation hot-spot stress (R) ratio has a mean value of 0.99 and standard deviation of 0.0711. In contrast, the respective statistical data using the

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E.O. Oshogbunu, et al.

Methods Stresses/Abaqus Results for Braces 1-4

Using SCFpn P/Rmin=0.76

Efthymiou (1985;1988) P/Rmax=1.20

2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25

0

50

100

150

200

250

300

350

Abaqus Model Stresses for Braces 1-4 (MPa) Fig. 32. Comparison of hot-spot stresses (maximum stresses) between the authors’ proposal of using 11 key SCFs (obtained from Eq. (8)) and Efthymiou equations.

•

Efthymiou equations [11,12] are 1.12 and 0.335 respectively. Power forms, relating the 11 key SCFs to relevant joint geometries, were adopted to develop equations to predict the 11 key SCFs and non-linear regression analyses were carried out to obtain the coefficients of the power forms using the generalized reduced gradient nonlinear solver [29]. Using the regression equations to calculate the 11 key SCFs to construct SCF and stress distributions and combining them to calculate hot-spot stresses for the 432 cases, the P/R ratios have a mean value of 1.00 and a standard deviation of 0.093. Although this is slightly worse than the above, the method is now entirely predictive.

[4] IIW-XV-E. Recommended Fatigue Design Procedure for Welded Hollow Section Joints, IIW Docs, XV-1035–99/XIII-1804-99. International Institute of Welding, France; 1999. [5] Zhao, X.-L., et al. Design guide for circular and rectangular hollow section welded joints under fatigue loading, Comité International pour le Développment et l'Etude dela Construction Tubulaire (CIDECT), editor; 2001. [6] American Welding Society (AWS). Structural welding code: AWS D 1.1. Miami (FL), US, 2002. [7] Det Norske Veritas Recommended Practice DNV-RP-C203. Fatigue design of offshore steel structures. Høvik, Norway: DNV; 2010. [8] Karamanos SA, Romeijn A, Wardenier J. Stress concentrations and joint flexibility effects in multi-planar welded tubular connections for fatigue design, Stevin Report 6-98-05, CIDECT Report 7R-17/98, Delft University of Technology, Delft, The Netherlands; 1997. [9] van Wingerde AM, Wardenier J, Packer JA. Simplified design graphs for the fatigue design of multiplanar K-joints with gap, Final report, Stevin report 6-98-34, Delft, The Netherlands; 1998b. [10] Wingerde AM, Packer JA, Wardenier J. Simplified SCF formulae and graphs for CHS and RHS K- and KK-connections. J Constr Steel Res 2001;57:221–52. [11] Efthymiou M. Development of SCF formulae and generalized influence functions for use in fatigue analysis. In: Recent developments in tubular joint technology: OTJ’88. 1988: London, UK; 1988. [12] Efthymiou M, Durkin S. Stress concentrations in T/Y and gap/overlap K joints. Delft, The Netherlands: Elsevier; 1985. [13] Karamanos SA, Romeijn A, Wardenier J. Stress concentrations in multi-planar welded CHS XX-connections. J Constr Steel Res 1999;50:259–82. [14] Woghiren CO, Brennan FP. Weld toe stress concentrations in multi planar stiffened tubular KK Joints. Int J Fatigue 2009;31:164–72. [15] Lotfollahi-Yaghin MA, Ahmadi H. Effect of geometrical parameters on SCF distribution along the weld toe of tubular KT-joints under balanced axial loads. Int J Fatigue 2010;32:703–19. [16] Ahmadi H, Lotfollahi-Yaghin MA, Aminfar MH. Geometrical effect on SCF distribution in uni-planar tubular DKT-joints under axial loads. J Constr Steel Res 2011;67:1282–91. [17] Ahmadi H, Lotfollahi-Yaghin MA, Aminfar MH. Distribution of weld toe stress concentration factors on the central brace in two-planar CHS DKT-connections of steel offshore structures. Thin-Walled Struct 2011;49:1225–36. [18] America Bureau Shipping (ABS). Guide for fatigue assessment of offshore structures. ABS Plaza, 16855 Northchase Drive, Houston, Texas 77060, USA; 2018. [19] OTH 91 353. Stress concentration factors for tubular complex joints. U. Lloyd’s Register of Shipping, UK Health and Safety Executive; 1992. [20] ABAQUS. ABAQUS documentation. Providence, RI, USA: Dassault Systèmes; 2016. [21] CIDECT report 7J-92. Fatigue strength of multi-planar welded, unstiffened hollow section joints and reinforcement measures for in-plane and multi planar joints in repair. Stevin report No S.6.92.03/AI./12.06, Delft University of Technology. The Netherlands; 1992. [22] Back J de, Vaessen GHG. Fatigue behaviour and corrosion fatigue behaviour of offshore structures. Final report ECSC Convention 7210-KB/6/602 (1.7. I f/76). Foundation for materials research in the sea. Delft Apeldoom, April 1981. The Netherlands; 1981. [23] Gho WM, Gao F. Parametric equations for stress concentration factors in completely overlapped tubular K(N)-joints. J Constr Steel Res 2004;60:1761–82. [24] Muglitz J, Weise S, Hermann J, Muckenheim U, Buscher KA. Tubular based support structures for offshore wind turbines. In: Proceedings of the 15th international symposium on tubular structures, Rio de Janeiro, Brazil; 2015.

CRediT authorship contribution statement Ernest O. Oshogbunu: Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Validation, Visualization, Writing - original draft, Writing - review & editing. Yong C. Wang: Conceptualization, Formal analysis, Methodology, Project administration, Supervision, Visualization, Writing - review & editing. Tim Stallard: Supervision, Writing - review & editing. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.engstruct.2020.110366. References [1] Wind in power 2017 – Annual combined onshore and offshore wind energy statistics. [Online] 2018 [cited 2018 29.05.2018]; Available from: https://windeurope. org/wp-content/uploads/files/about-wind/statistics/WindEurope-AnnualStatistics-2017.pdf. [2] Global Wind Energy Council- Global Wind Statistics; 2017 [Online] 2018 [cited 2018 29.05.2018]; Available from: http://gwec.net/wp-content/uploads/vip/ GWEC_PRstats2017_EN-003_FINAL.pdf. [3] United Kingdom Department of Energy (UK DEn). Background to new fatigue design guidance for steel joints in offshore structures. London: Department of Energy; 1993.

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E.O. Oshogbunu, et al. [25] Yong-Bo S, Zhi-Fu D, Seng-Tjhen L. Prediction of hot-spot stress distribution for tubular K-joints under basic loading. Singapore J Constr Steel Res 2009;65:2011e26. [26] Karamanos SA, Romeijn A, Wardenier J. Stress concentrations in tubular gap Kjoints: mechanics and fatigue design. Eng Struct 2000;22:4–14. [27] Beale LA, Toprac AA. Analysis of in-plane T, Y and K welded tubular connections.

New York: Welding Research Council; 1967. [28] Kuang JG, Potvin AB, Leick RD. Stress concentration in tubular joints. In: Seventh annual offshore technology conference. Houston, Texas; 1975. p. 593–612. [29] Lasdon LS, Fox RL, Ratner MW. Nonlinear optimization using the generalized reduced gradient method. Revue française d’automatique, informatique, recherché opérationnelle. Recherché opérationnelle, tome 1974;8(V3):73–103.

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