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Geometrical effects on the local joint flexibility of two-planar tubular DK-joints in jacket substructure of offshore wind turbines under OPB loading
Thin-Walled Structures 114 (2017) 122–133
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Full length article
Geometrical effects on the local joint flexibility of two-planar tubular DKjoints in jacket substructure of offshore wind turbines under OPB loading
MARK
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Hamid Ahmadi , Ali Ziaei Nejad Faculty of Civil Engineering, University of Tabriz, Tabriz 5166616471, Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: Local joint flexibility (LJF) Out-of-plane bending (OPB) Two-planar tubular DK-joint Offshore jacket structure Finite element method
The local joint flexibility (LJF) of two-planar tubular DK-joints in offshore jacket structures under the out-ofplane bending (OPB) loads is studied in the present paper. Altogether, 81 finite element (FE) models were generated and analyzed by ANSYS in order to investigate the effects of DK-joint's geometrical parameters on the LJF factor (fLJF) under four types of OPB load cases. Generated FE models were validated against the existing experimental data, FE results, and parametric equations. After the parametric study of geometrical effects, the fLJF in two-planar DK- and uniplanar K-joints were compared. Results showed that the effect of multi-planarity on the fLJF values is considerable. Hence, the application of the equations already available for uniplanar K-joints to determine the fLJF in two-planar DK-joints might result in highly under-/over-predicting outputs. To suggest a solution for this issue, results of 324 analyses carried out on generated FE models were used to develop a new set of parametric formulas for the calculation of the fLJF in OPB-loaded two-planar DK-joints. Applicability of proposed formulas was assessed based on the UK DoE acceptance criteria.
1. Introduction Tubular structures such as the jacket substructure of offshore oil/gas platforms and offshore wind turbines (OWTs) are space frames fabricated by welding the circular hollow section (CHS) members. A tubular joint is the interconnection among the chord member and a number of brace members in a tubular structure (Fig. 1a). Abrupt change in the direction of the load transfer in a tubular joint makes it a fatigue-sensitive structural component subjected to wave induced cyclic stresses. In addition to the fatigue strength, the static capacity and the hysteretic behavior of tubular joints have been the subject of numerous research works. The local joint flexibility (LJF) is one of the factors affecting the behavior of a tubular joint and consequently affects the global static and dynamic responses of a jacket structure. The LJF increases the deflections, redistributes the nominal stresses, reduces the buckling loads and changes the natural frequencies of the structure [1,2]. Hence, the conventional procedures for the analysis and design of tubular structures with the assumption that the tubular joints are rigid might not be rigorous, especially for unstiffened joints. This local deformation lowers the strength requirement for the joint by redistributing the member-end loads and moments compared to a conventional rigid joint that will consequently lower the cost of the tubular structure [3]. Hence, determination of the local joint flexibility of tubular joints with a reliable method is essential.
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The LJF for simple uniplanar tubular joints such as T/Y-, K-, and Xjoints subjected to basic load cases has been extensively studied; based on which a set of parametric formulas have been derived. However, for multi-planar tubular joints which cover the majority of practical applications, the research works in terms of the LJF are very limited and no parametric formulas are currently available to determine the LJF of multi-planar joints. The main reason is perhaps the complexity of geometrical properties and load distribution. Results of previous research works have shown that the stress and strain distributions in multi-planar tubular joints might be quite different from the uniplanar ones [4–8]. Therefore, the effect of multi-planarity on the LJF may also be significant and consequently the use of the equations already available for uniplanar joints to determine the LJF in multi-planar joints may result in highly under-/over-predicting outcomes. Hence, the application of equations developed for uniplanar joints to calculate the LJF in multi-planar connections needs to be validated. In the present paper, the LJF of two-planar tubular DK-joints in offshore jacket structures under the out-of-plane bending (OPB) loads is studied. Altogether, 81 finite element (FE) models were generated and analyzed by ANSYS in order to investigate the effects of DK-joint's geometrical parameters on the LJF factor (fLJF) under four types of OPB loading (Fig. 2). A set of dimensionless geometrical parameters (α, αB, β, γ, τ, and ζ) are used to readily relate the behavior of a tubular joint to its geometrical properties. These parameters are defined in Fig. 1b.
Corresponding author. E-mail address: [email protected] (H. Ahmadi).
http://dx.doi.org/10.1016/j.tws.2017.02.001 Received 31 October 2016; Received in revised form 30 January 2017; Accepted 1 February 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
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Fig. 1. (a) Two-planar tubular DK-joints in an OWT jacket substructure, (b) Geometrical notation for a two-planar DK-joint.
Above discussion indicates that, as mentioned in Section 1, the research works on the LJF of multi-planar tubular joints which cover the majority of practical applications are very limited and no parametric formulas are currently available to calculate the fLJF values for multi-planar joints.
Generated FE models were verified based on the existing experimental data, FE results, and parametric equations. After the parametric study of geometrical effects, the fLJF in two-planar DK- and uniplanar K-joints are compared; and the results of 324 analyses carried out on generated FE models are used to develop a new set of parametric formulas for the calculation of the fLJF in OPB-loaded two-planar DK-joints. Applicability of proposed formulas is verified according to the UK DoE [9] acceptance criteria. Thus, they can be reliably used for the analysis and design of tubular joints commonly found in offshore jacket structures.
3. Definition of the LJF factor (fLJF) for a tubular joint subjected to the OPB loading The LJF of a tubular joint is defined as the displacement attributed to the local chord wall deformation due to a unit applied load. To determine the LJF at the brace/chord intersection under the brace OPB moment loading, the rotation of the joint due to the overall displacement should be omitted from the total measured rotation [23]. In an FE model, the local rotation at the joint can be directly measured without considering the beam bending movement. The LJF of a tubular joint under the OPB loading can be defined as:
2. Literature review DNV [10] and UEG [2] have provided a set of parametric equations to predict the LJF for tubular T/Y-joints. The DNV equations are based on a limited number of FE analyses and the UEG guidelines do not clearly define the validity range. Efthymiou [11] developed a set of equations for T/Y- and K-joints under in-plane bending (IPB) and OPB loads by FE analysis. The database was limited to twelve T-, three Y-, and nine K-joints, five of which were partially overlapped. Fessler et al. [12,13] measured the local deformation of the chord wall under basic loadings within the elastic range based on 27 models and developed a set of parametric formulas for both T/Y- and K-joints. However, their experimental models were made from araldite instead of steel, and they had relatively small scale. Ueda et al. [14] derived a set of formulas to predict the LJF under the axial and IPB loads based on FE analysis of eleven T-joint models. The validity range of these equations was very limited in terms of the β that was restricted to 0.35−0.55. Chen et al. [15] determined the local joint flexibility of T/Y-, and K-joints. By using the semi-analytical approach, Chen et al. [16] and Hoshyari and Kohoutek [17] quantified the LJF for simple gap K- and T/Y-joints, respectively. Buitrago et al. [18] provided the methodology as well as the parametric equations for calculating the LJF in gap and partially overlapped joints based on the FE analysis. Chen and Zhang [19] studied the stress distribution in space frames with the consideration of local flexibility of multiplanar tubular joints. Hu et al. [20] and Golafshani et al. [21] developed equivalent elements representing the local flexibility of tubular joints in structural analysis of offshore platforms. Gao et al. [3,22] and Gao and Hu [23] proposed a set of parametric equations to determine the LJF in completely overlapped tubular joints subjected to axial, IPB, and OPB loading, respectively.
LJF = ϕOPB / MOPB
(1)
where MOPB is the brace OPB moment and ϕOPB is the joint local rotation expressed as:
ϕOPB =
δ − δ3 δ 2 − δ1 sin θ − 4 d−t D−T
(2)
Where D and d are the chord and brace diameters respectively; T and t are the chord and brace wall thicknesses respectively; θ is the brace inclination angle; δ1 and δ2 are the respective deformations at both saddles measured in the direction of brace axis; and δ3 and δ4 are the deformations at the side face of the chord corresponding to δ1 and δ2, respectively (Fig. 3). In order to relate the local joint flexibility to dimensionless geometrical parameters of the joint, a dimensionless coefficient called the local joint flexibility parameter (fLJF) is defined. The fLJF is the LJF multiplied by ED3:
fLJF =
ϕOPB MOPB
ED 3
(3)
where D is the chord diameter and E is the Young's modulus. It should be noted that since a two-planar DK-joint has four brace members, there are four different positions for the application of Eqs. (2) and (3). However, due to the symmetry in the geometry of the joint 123
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Fig. 2. Studied OPB load cases.
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Fig. 3. Positions for measuring the deformation to calculate the LJF under the OPB moment loading.
mesh quantity and quality and avoid badly distorted elements. The mesh generated by this method is shown in Fig. 4. In order to make sure that the results of the FE analysis are not affected by the inadequate quality or the size of the generated mesh, convergence test was conducted and meshes with different densities were used in this test, before generating the 81 models. As an example, results of the convergence test for a joint model (θ=45°, τ=0.7, β=0.5, γ=18) having different mesh densities (Fig. 5) have been presented in Table 1. This table indicates that the difference between the fLJF values in a model with a coarse mesh, i.e. ne=1 (Fig. 5a), and the model with a finer mesh, i.e. ne=3, selected for the present parametric FE study (Fig. 5b) had been nearly 27% meaning that the accuracy of the ne=1 mesh was not adequate. However, the difference between the fLJF values in models with ne=3 and ne=6 was less than 3% implying that such an overly fine mesh (Fig. 5c) was not necessary since it would lead to the increase of the computation time without any considerable improvement of the accuracy. The static analysis of linearly elastic type is suitable for the prediction of LJF in tubular joints [22,23]. The Young's modulus and Poisson's ratio were taken to be 207 GPa and 0.3, respectively.
and either symmetry or antisymmetry in the applied loading, results obtained from all of these four positions would be the same. 4. FE modeling and analysis 4.1. Weld profile Results of the present study's FE verification process (Section 4.4) indicated that the weld sizes must be carefully included in the FE modeling. In this research, the welding size along the brace/chord intersection satisfies the AWS D 1.1 [24] specifications. The dihedral angle (ψ), defined as the angle between the chord and brace surfaces along the intersection curve, is an important parameter in determining the weld thickness. The value of the ψ at three typically important positions along the weld toe equals to: θ (Crown Heel), π – cos–1θ (Saddle), and π – θ (Crown Toe); where θ is the brace inclination angle (Fig. 1b). Details of weld profile modeling according to AWS D 1.1 [24] specifications have been presented by Ahmadi et al. [25]. 4.2. Boundary conditions Fixity condition of the chord ends in tubular joints of offshore structures ranges from almost fixed to almost pinned with generally being closer to almost fixed [26]. In the view of the fact that the effect of the chord end restraints on the stress distribution at the brace/chord intersection is only significant for joints with α < 8 and high β and γ values [27,28], which do not commonly occur in practice, both chord ends were assumed to be fixed, with the corresponding nodes restrained.
4.4. Verification The accuracy of FE results to determine the fLJF in tubular joints should be validated using the experimental test results. As far as the authors are aware, there is no experimental/FE database of fLJF for twoplanar tubular DK-joints currently available in the literature. Considering this issue, in order to verify the FE model, a set of Y-joints were modeled and the fLJF values obtained from these models were compared with the experimental results of Fessler et al. [12], values predicted by Fessler et al. equation [12], and the FE results of Gao et al. [22]. Geometrical properties of the validating Y-joints have been presented in Table 2. The procedure of geometrical modeling (introducing the chord, braces, and weld profiles), the mesh generation method (including the selection of the element type and size), the analysis type, and the method of fLJF calculation are the same for the validating Y-joint models and the DK-joints used in the present research for the parametric study. Hence, the verification of the fLJF values derived from the validating FE models with available corresponding experiment-/FE-/equation-predicted values lends some support to the validity of the fLJF values derived from the DK-joint FE models. Results of the verification process have been presented in Fig. 6 along with Table 2. The effects of the element type and the weld profile were also investigated. A comparison between the results obtained by the solid and shell elements indicated that the solid elements lead to
4.3. Mesh generation and analysis ANSYS Ver. 16 element type SOLID 185 was used to model the chord, braces, and the weld profiles. These elements have compatible displacements and are well-suited to model curved boundaries. The element is defined by eight nodes having three degrees of freedom per node and may have any spatial orientation. Using this type of 3-D brick elements, the weld profile can be modeled as a sharp notch. This method will produce more accurate and detailed stress distribution near the intersection in comparison with a simple shell analysis (See Section 4.4). To guarantee the mesh quality, a sub-zone mesh generation method was used during the FE modeling. In this method, the entire structure is divided into several different zones according to the computational requirements. The mesh of each zone is generated separately and then the mesh of entire structure is produced by merging the meshes of all the sub-zones. This method can easily control the 125
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Fig. 4. The mesh generated by the sub-zone method for (a) a uniplanar tubular K-joint, (b) a two-planar tubular DK-joint.
5. Geometrical effects on the local joint flexibility factor
more accurate fLJF values. Moreover, the comparison of solid models with and without the weld profile showed that the omission of the weld profile results in the increase of the error percentage. As can be seen in Fig. 5, there is a good agreement between the results of previous studies and the predictions of the validating FE model. According to Table 2, the maximum difference between the fLJF of the validating FE model and the experimental results of Fessler et al. [12] is 10.07%; and the maximum difference between the results of the validating FE model and the equation proposed by Fessler et al. [12] is 12%. Hence, generated FE models can be considered to be accurate enough to provide valid results.
5.1. Details of parametric investigation To investigate the effect of geometrical parameters on the fLJF in two-planar DK-joints subjected to four types of OPB loading (Fig. 2), 81 models were generated and analyzed using ANSYS. Different values assigned to the parameters β, γ, τ, and θ are as follows: β=0.4, 0.5, 0.6; γ=12, 18, 24; τ=0.4, 0.7, 1.0; and θ=30°, 45°, 60°. These values cover the practical ranges of dimensionless parameters typically found in tubular joints of offshore jacket structures. Providing that the gap
Fig. 5. Different mesh densities for a joint model: (a) coarse mesh, (b) fine mesh, (c) overly fine mesh.
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Table 1 The effect of the FE mesh size on the fLJF results. Geometrical properties of the joint
θ=45°, τ=0.7, β=0.5, γ=18
fLJF
Difference between fLJF values
ne =3 (Fig. 5b)
Overly fine mesh ne=6 (Fig. 5c)
ne=1 and ne=3
ne=3 and ne=6
2662.52
2590.177
26.68%
2.71%
Coarse mesh ne=1 (Fig. 5a)
Fine mesh
3372.971
ne: The number of elements along the chord thickness
between the braces is not very large, the relative gap (ζ= g / D) has no considerable effect on the stress and strain distribution. The validity range for this statement is 0.2≤ ζ≤0.6 [29]. Hence, a typical value of ζ=0.3 was designated for all joints. Sufficiently long chord greater than six chord diameters (i.e. α≥12) should be used to ensure that the stresses at the brace/chord intersection are not affected by the chord's boundary conditions [26]. Hence, in this study, a realistic value of α=16 was designated for all the models. The brace length has no effect on the stress and strain distribution when the parameter αB is greater than a critical value [30]. According to Chang and Dover [31], this critical value is about 6. In the present study, in order to avoid the effect of short brace length, a realistic value of αB=8 was assigned for all joints.
Fig. 6. Results of the verification process using the validating Y-joint models.
the fLJF under all studied OPB loading conditions. The reason is that the increase of the brace diameter leads to the decrease of the joint local rotation which consequently results in the decrease of the local joint flexibility and then the fLJF. As can be seen in Fig. 7, this conclusion is independent from the values of other geometrical parameters.
5.3. The effect of the brace-to-chord wall thickness ratio on the fLJF Results of investigating the effect of the τ on the fLJF are presented in this section. The interaction of the τ with the other geometrical parameters was also investigated. The parameter τ is the ratio of the brace wall thickness to the chord wall thickness and the γ is the ratio of the section radius to wall thickness of the chord. Hence, provided that the value of the γ remains constant, the increase of the τ leads to the increase of the brace thickness. As an example, nine charts are given in Fig. 8 depicting the change of the fLJF, under the 1st OPB loading condition, due to the change in the value of the τ and the interaction of this parameter with the γ. It can be concluded that the increase of the τ has no considerable effect on the local joint flexibility compared to the other geometrical parameters.
5.2. The effect of the brace-to-chord diameter ratio on the fLJF Results of investigating the effect of the β on the fLJF are presented in this section. The influence of parameters τ, γ, and θ over the effect of the β on the fLJF was also investigated. The parameter β is the ratio of the brace diameter to the chord diameter. Hence, provided that the chord diameter remains constant, the increase of the β results in the increase of the brace diameter. Fig. 7 demonstrates the change of the fLJF, under the 1st OPB loading condition, due to the change in the value of the β and the interaction of this parameter with the γ. Altogether, 81 comparative charts were used to study the effect of the β and only nine of them are presented here for the sake of brevity. Results showed that the increase of the β results in the decrease of Table 2 Details and results of the verification process. Geometrical properties of validating Y-joint models Geometrical parameter Value Chord length L=1888.0 mm Chord diameter D=168.3 mm Chord wall thickness T=7.0 mm Brace length l=623.0 mm Brace diameter d=55.5, 88.2, 127.9 mm Brace wall thickness t=5.5 mm Brace inclination angle θ=60°
Dimensionless parameters α=2 L/D =22.4 γ= D/2 T=12 β= d/D =0.32, 0.53, 0.76 τ= t/T=0.78
The comparison between the results of validating FE model and the experimental data β fLJF Validating FE model Experimental data [12] Shell Solid with weld Solid without weld 0.32 2538.45 1996.40 1525.39 2079.33 0.53 954.39 895.39 1083.61 985.57 0.76 437.98 277.63 360.95 300.48 The comparison between the results of validating FE model and the equation proposed by Fessler et al. [12] β fLJF Validating FE model Fessler et al. Eq. [12] Shell Solid with weld Solid without weld 0.32 2538.45 1996.40 1525.39 2133.41 0.53 954.39 895.39 1083.61 877.40 0.76 437.98 277.63 360.95 312.50
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Error (%) Shell 22.08 3.16 45.75
Solid with weld 4.15 10.07 8.22
Solid without weld 36.31 9.94 20.12
Solid with weld 6.86 2.05 12.00
Solid without weld 39.86 23.50 15.50
Error (%) Shell 18.98 8.77 40.15
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Fig. 7. The effect of the β on the fLJF values in two-planar tubular DK-joints under the 1st OPB load case.
the fLJF, under the 1st OPB loading condition, due to the change in the value of θ and the interaction of this parameter with the γ. Results indicated that the increase of the θ leads to the increase of the fLJF under all studied OPB loading conditions. The reason is that the increase of the brace inclination angle results in the increase of the joint
5.4. The effect of the brace inclination angle on the fLJF Results of studying the effect of the θ on the fLJF and its interaction with the other geometrical parameters are presented in this section. Nine charts are given in Fig. 9, as an example, depicting the change of 128
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Fig. 8. The effect of the τ on the fLJF values in two-planar tubular DK-joints under the 1st OPB load case.
local rotation that then leads to the increase of the local joint flexibility and the fLJF. As can be seen in Fig. 9, this result is independent from the values of other geometrical parameters.
5.5. The effect of the chord slenderness ratio on the fLJF Results of investigating the effect of the γ on the fLJF are presented in this section. The influence of parameters β, τ, and θ over the effect of the γ on the fLJF was also investigated. The parameter γ is the ratio of 129
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Fig. 9. The effect of the θ on the fLJF values in two-planar tubular DK-joints under the 1st OPB load case.
6. The effect of OPB loading mode and the multi-planarity on the fLJF
the section radius to wall thickness of the chord. Hence, provided that the chord diameter remains constant, the increase of the γ means the decrease of the chord thickness. It is evident from Figs. 7–9 that the increase of the γ leads to the increase of the fLJF.. The reason is that the decrease of the chord thickness leads to the increase of the joint local rotation that consequently results in the increase of the local joint flexibility. This conclusion does not depend on the values of other geometrical parameters.
In Fig. 10, the fLJF values under the four considered OPB loading conditions are compared for two sample joints. According to this figure, it can be concluded that the maximum and minimum values of the fLJF occur under the 2nd and 3rd OPB load cases, respectively. It is also evident that the differences among the fLJF values under the four 130
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Fig. 10. Comparing the fLJF values under different OPB loading conditions: (a) θ=60°, τ=0.7, γ=24, β=0.4; (b) θ=30°, τ=0.7, γ=12, β=0.4.
However, under the 2nd loading condition, the uniplanar fLJF is 67% of the corresponding two-planar value (Fig. 11b). These results indicate that the effect of the multi-planarity on the fLJF is quite significant and consequently the use of the formulas already available for uniplanar Kjoints to calculate the fLJF in two-planar DK-joints may lead to highly under-/over-predicting results. To handle this issue, the FE results are used in the next section to develop a set of parametric formulas for the determination of the fLJF values in OPB-loaded two-planar DK-joints.
considered OPB load cases are quite big. This implies that the derivation of an individual parametric formula for each of these load cases is essential; and the use of a single formula to predict the fLJF for all the considered OPB loading conditions may lead to highly under-/ over-predicting results. The fLJF values in uniplanar K- and two-planar DK-joints have been compared in Fig. 11. It can be seen that under the 1st and 3rd OPB load cases, the fLJF values in uniplanar joints are bigger than the corresponding values in two-planar joints. On the contrary, under the 2nd and 4th OPB load cases, the fLJF values in uniplanar joints are smaller. It is also evident that the difference between the fLJF values in uniplanar and two-planar joints can be quite significant. For example, as shown in Fig. 11a, the uniplanar fLJF (for θ=30°, τ=0.4, γ=12, β=0.6) is twice the corresponding two-planar value under the 1st OPB load case.
7. Development of parametric formulas to calculate the fLJF values Results of parametric FE investigation were used to derive four individual parametric formulas for the determination of the fLJF values in two-planar tubular DK-joints subjected to four types of OPB loading
Fig. 11. The comparison of fLJF values in uniplanar K- and two-planar DK-joints (θ=30°, τ=0.4, γ=12).
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The parameter θ in Eqs. (4)–(7) should be inserted in radians. Values obtained for R2 are quite high indicating the accuracy of the fit. The validity ranges of nondimensional geometrical parameters for the proposed formulas are as follows:
Table 3 Assessment of developed formulas based on the UK DoE [9] criteria. Formula
Eq. Eq. Eq. Eq.
(4) (5) (6) (7)
OPB
UK DoE Conditions
Decision
Load Case
%P/R < 0.8
%P/R > 1.5
1st 2nd 3rd 4th
1.23% < 5% OK. 0% < 5% OK. 1.23% < 5% OK. 1.23% < 5% OK.
7.40% < 50% OK. 12.34% < 50% OK. 0% < 50% OK. 1.23% < 50% OK.
0.4 ≤ τ ≤ 1.0 12 ≤ γ ≤ 24 0.4 ≤ β ≤ 0.6 30° ≤ θ ≤ 60°
Accept Accept Accept Accept
The UK Department of Energy (DoE) [9] recommends the following assessment criteria regarding the applicability of the parametric formulas (P/R stands for the ratio of the predicted value from a given formula to the recorded value from the FE analysis):
shown in Fig. 2. Formulas were derived based on multiple nonlinear regression analyses performed by SPSS. Values of dependent variable (i.e. fLJF) and independent variables (i.e. β, γ, τ, and θ) constitute the input data imported in the form of a matrix. Each row of this matrix involves the information about the fLJF value in a two-planar tubular DK-joint having specific geometrical properties. When the dependent and independent variables are defined, a model expression should be built with defined parameters. Parameters of the model expression are unknown coefficients and exponents. The researcher should specify an initial value for each parameter, preferably as close as possible to the expected final solution. Poor initial values may lead to failure in convergence or to convergence to a local (instead of global) solution or even to a physically impossible solution. Various model expressions should be generated to derive a parametric formula with a high coefficient of determination (R2). After performing the nonlinear analyses, following parametric formulas are proposed for the calculation of the fLJF values in twoplanar tubular DK-joints subjected to four types of out-of-plane bending (OPB) moment loading (Fig. 2).
•1
st
•2
nd
•
R 2 = 0.989
(4)
R 2 = 0.991
(5)
LJF values is less than or equal to 25%, i.e. [%P/R < 1.0] ≤ 25%; and the percentage of considerably under-predicting fLJF values is less than or equal to 5%, i.e. [%P/R < 0.8] ≤ 5%, then the equation is accepted. If, in addition, the percentage of considerably overpredicting fLJF values is more than or equal to 50%, i.e. [%P/R > 1.5] ≤50%, then the equation is regarded as generally conservative. If the acceptance criteria is nearly met, i.e. 25% < [%P/R < 1.0] ≤ 30%, and/or 5% < [%P/R < 0.8] ≤ 7.5%, then the equation is regarded as borderline and engineering judgment should be applied to determine the acceptance or rejection. Otherwise the equation is rejected as it is too optimistic.
In the view of the fact that for a mean fit formula, there is always a large percentage of under-prediction, the requirement for joint underprediction, i.e. P/R < 1.0, can be completely removed in the assessment of parametric equations [32]. Assessment results according to the UK DoE [9] criteria are presented in Table 3. As can be seen in this table, all of the proposed formulas satisfy the criteria recommended by the UK DoE. Fig. 12 indicates that, as it is expected from Section 6 and Fig. 10, considerable differences exist among the fLJF values predicted by Eqs. (4)–(7) implying that it was essential to develop an individual fLJF equation for each of the four considered OPB loading conditions. 8. Conclusions
rd
OPB loading condition:
fLJF = 1.189τ −0.108γ 1.931β −2.612θ1.826
•4
•
OPB loading condition:
fLJF = 4.232τ −0.063γ 2.054β −2.011θ 2.141
•3
• For a given dataset, if the percentage of under-predicting f
OPB loading condition:
fLJF = 0.190τ −0.162γ 2.539β −3.545θ1.835
(8)
th
R 2 = 0.995
The LJF of two-planar tubular DK-joints under the OPB loads was studied. Altogether, 81 FE models were generated and analyzed in order to investigate the effects of DK-joint's geometrical parameters on the LJF factor (fLJF) under four types of OPB loading. Results indicated that the change of the τ has no considerable effect on the local joint flexibility compared to the other geometrical parameters; the increase of the θ and/or γ leads to the increase of the
(6)
OPB loading condition:
fLJF = 0.465τ −0.165γ 2.250 β −3.091θ1.585
R 2 = 0.991
(7)
Fig. 12. Comparison of the fLJF values predicted by the proposed equations for 81 tubular DKT-joints (β=0.4, 0.5, 0.6; γ=12, 18, 24; τ=0.4, 0.7, 1.0; and θ=30°, 45°, 60°) under the four considered OPB loading conditions.
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fLJF; and the increase of the β leads to the decrease of the fLJF values under all considered OPB load cases. The local deformation of the joint lowers its strength requirement by redistributing the member-end loads and moments compared to a conventional rigid joint that will consequently lower the cost of the tubular structure. Hence, the enhancement of the joint LJF is recommended during the design stage. Results of the present study showed that, for tubular DK-joints, this design objective can be accomplished through the decrease of the brace-to-chord diameter ratio and the increase of the brace inclination angle and/or chord slenderness ratio. The differences among the fLJF values under the four studied OPB loading conditions are quite big which means that the development of an individual parametric formula for each of these load cases is essential; and the application of a single formula to calculate the fLJF values for all the considered OPB load cases might lead to highly under-/over-predicting results. The difference between the fLJF values in uniplanar and two-planar joints can be quite significant implying that the effect of multi-planarity on the fLJF values is considerable and consequently the application of the formulas already available for uniplanar K-joints to determine the fLJF values in two-planar DK-joints may result in highly under-/overpredicting outcomes. To handle this issue, the FE results were used to develop four individual parametric formulas for the determination of the fLJF values in two-planar DK-joints subjected to four types of OPB loading. Proposed formulas, having high coefficients of determination, were assessed based on the acceptance criteria recommended by the UK DoE and can be reliably used for the analysis and design of tubular joints in offshore jacket structures.
[7] H. Ahmadi, M.A. Lotfollahi-Yaghin, Geometrically parametric study of central brace SCFs in offshore three-planar tubular KT-joints, J. Constr. Steel Res. 71 (2012) 149–161. [8] H. Ahmadi, E. Zavvar, The effect of multi-planarity on the SCFs in offshore tubular KT-joints subjected to in-plane and out-of-plane bending loads, Thin-Walled Struct. 106 (2016) 148–165. [9] UK Department of Energy, Background notes to the fatigue guidance of offshore tubular joints. London, UK, 1983. [10] Det Norske Veritas. Rules for the design, construction and inspection of offshore structures. Norway; 1977. [11] M. Efthymiou, Local rotational stiffness of un-stiffened tubular joints, RKER Rep. (1985) 185–199. [12] H. Fessler, P. Mockford, J. Webster, Parametric equations for the flexibility matrices of single brace tubular joints in offshore structures, in: Proceedings of the Institution of Civil Engineers; vol. 81, 1986, pp. 659−673. [13] H. Fessler, P. Mockford, J. Webster, Parametric equations for the flexibility matrices of multi-brace tubular joints in offshore structures. Proceedings of the Institution of Civil Engineers; vol. 81, 1986, pp. 75−696. [14] Y. Ueda, S.M.H. Rashed, K. Nakacho, An improved joint model and equations for flexibility of tubular joints, Offshore Mech. Arct. Eng. 112 (1990) 157–168. [15] B. Chen, Y. Hu, M. Tan, Local joint flexibility of tubular joints of offshore structures, Mar. Struct. 3 (1990) 177–197. [16] B. Chen, Y. Hu, H. Xu, Theoretical and experimental study on the local flexibility of tubular joints and its effect on the structural analysis of offshore platforms. Proceedings of the 5th International Symposium on Tubular Structures, Nottingham, UK; 1993, p. 543−550. [17] I. Hoshyari, R. Kohoutek, Rotational and axial flexibility of tubular T-joints. Proceedings of the 3rd International Offshore and Polar Engineering Conference (ISOPE), Singapore; 1993, pp. 192−198. [18] J. Buitrago, B. Healy, T. Chang, Local joint flexibility of tubular joints. International Conference on Ocean, Offshore & Arctic Engineering (OMAE), Glasgow, UK; 1993, pp. 405−417. [19] T.Y. Chen, H.Y. Zhang, Stress analysis of spatial frames with consideration of local flexibility of multiplanar tubular joint, Eng. Struct. 18 (6) (1996) 465–471. [20] Y. Hu, B. Chen, J. Ma, An equivalent element representing local flexibility of tubular joints in structural analysis of offshore platforms, Comput. Struct. 47 (6) (1993) 957–969. [21] A.A. Golafshani, M. Kia, P. Alanjari, Local joint flexibility element for offshore platforms structures, Mar. Struct. 33 (2013) 56–70. [22] F. Gao, B. Hu, H.P. Zhu, Local joint flexibility of completely overlapped tubular joints under in-plane bending, J. Constr. Steel Res. 99 (2014) 1–9. [23] F. Gao, B. Hu, Local joint flexibility of completely overlapped tubular joints under out-of-plane bending, J. Constr. Steel Res. 115 (2015) 121–130. [24] American Welding Society (AWS). Structural welding code: AWS D 1.1. Miami (FL): American Welding Society, US; 2002. [25] H. Ahmadi, M.A. Lotfollahi-Yaghin, M.H. Aminfar, The development of fatigue design formulas for the outer brace SCFs in offshore three-planar tubular KT-joints, Thin-Walled Struct. 58 (2012) 67–78. [26] M. Efthymiou, Development of SCF Formulae and Generalized Influence Functions for Use in Fatigue Analysis, 88 Offshore Tubular Joints (OTJ), Surrey, UK, 1988. [27] P. Smedley, P. Fisher, Stress concentration factors for simple tubular joints. Proceedings of the International Offshore and Polar Engineering Conference (ISOPE), Edinburgh, UK, 1991. [28] M.R. Morgan, M.M.K. Lee, Prediction of stress concentrations and degrees of bending in axially loaded tubular K-joints, J. Constr. Steel Res. 45 (1) (1998) 67–97. [29] M.A. Lotfollahi-Yaghin, H. Ahmadi, Effect of geometrical parameters on SCF distribution along the weld toe of tubular KT-joints under balanced axial loads, Int. J. Fatigue 32 (2010) 703–719. [30] E. Chang, W.D. Dover, Parametric equations to predict stress distributions along the intersection of tubular X and DT-joints, Int. J. Fatigue 21 (1999) 619–635. [31] E. Chang, W.D. Dover, Stress concentration factor parametric equations for tubular X and DT joints, Int. J. Fatigue 18 (6) (1996) 363–387. [32] Bomel Consulting Engineers. Assessment of SCF equations using Shell/KSEPL finite element data. C5970R02.01 REV C; 1994.
Acknowledgement The Authors gratefully acknowledge useful comments of anonymous reviewers on draft version of this paper. References [1] J. Bouwkamp, J. Hollings, B. Masion, D. Row, Effect of joint flexibility on the response of offshore structures. Offshore Technology Conference (OTC), Houston, Texas; 1980, p. 455−464. [2] Underwater Engineering Group Design of Tubular Joint For Offshore Structures, UEG/CIRIA London, UK, 1985. [3] F. Gao, B. Hu, H.P. Zhu, Parametric equations to predict LJF of completely overlapped tubular joints under lap brace axial loading, J. Constr. Steel Res. 89 (2013) 284–292. [4] S.A. Karamanos, A. Romeijn, J. Wardenier, Stress concentrations in multi-planar welded CHS XX-connections, J. Constr. Steel Res. 50 (1999) 259–282. [5] H. Ahmadi, M.A. Lotfollahi-Yaghin, M.H. Aminfar, Effect of stress concentration factors on the structural integrity assessment of multi-planar offshore tubular DKTjoints based on the fracture mechanics fatigue reliability approach, Ocean Eng. 38 (2011) 1883–1893. [6] H. Ahmadi, M.A. Lotfollahi-Yaghin, A probability distribution model for stress concentration factors in multi-planar tubular DKT-joints of steel offshore structures, Appl. Ocean Res. 34 (2012) 21–32.
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Full length article
Geometrical effects on the local joint flexibility of two-planar tubular DKjoints in jacket substructure of offshore wind turbines under OPB loading
MARK
⁎
Hamid Ahmadi , Ali Ziaei Nejad Faculty of Civil Engineering, University of Tabriz, Tabriz 5166616471, Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: Local joint flexibility (LJF) Out-of-plane bending (OPB) Two-planar tubular DK-joint Offshore jacket structure Finite element method
The local joint flexibility (LJF) of two-planar tubular DK-joints in offshore jacket structures under the out-ofplane bending (OPB) loads is studied in the present paper. Altogether, 81 finite element (FE) models were generated and analyzed by ANSYS in order to investigate the effects of DK-joint's geometrical parameters on the LJF factor (fLJF) under four types of OPB load cases. Generated FE models were validated against the existing experimental data, FE results, and parametric equations. After the parametric study of geometrical effects, the fLJF in two-planar DK- and uniplanar K-joints were compared. Results showed that the effect of multi-planarity on the fLJF values is considerable. Hence, the application of the equations already available for uniplanar K-joints to determine the fLJF in two-planar DK-joints might result in highly under-/over-predicting outputs. To suggest a solution for this issue, results of 324 analyses carried out on generated FE models were used to develop a new set of parametric formulas for the calculation of the fLJF in OPB-loaded two-planar DK-joints. Applicability of proposed formulas was assessed based on the UK DoE acceptance criteria.
1. Introduction Tubular structures such as the jacket substructure of offshore oil/gas platforms and offshore wind turbines (OWTs) are space frames fabricated by welding the circular hollow section (CHS) members. A tubular joint is the interconnection among the chord member and a number of brace members in a tubular structure (Fig. 1a). Abrupt change in the direction of the load transfer in a tubular joint makes it a fatigue-sensitive structural component subjected to wave induced cyclic stresses. In addition to the fatigue strength, the static capacity and the hysteretic behavior of tubular joints have been the subject of numerous research works. The local joint flexibility (LJF) is one of the factors affecting the behavior of a tubular joint and consequently affects the global static and dynamic responses of a jacket structure. The LJF increases the deflections, redistributes the nominal stresses, reduces the buckling loads and changes the natural frequencies of the structure [1,2]. Hence, the conventional procedures for the analysis and design of tubular structures with the assumption that the tubular joints are rigid might not be rigorous, especially for unstiffened joints. This local deformation lowers the strength requirement for the joint by redistributing the member-end loads and moments compared to a conventional rigid joint that will consequently lower the cost of the tubular structure [3]. Hence, determination of the local joint flexibility of tubular joints with a reliable method is essential.
⁎
The LJF for simple uniplanar tubular joints such as T/Y-, K-, and Xjoints subjected to basic load cases has been extensively studied; based on which a set of parametric formulas have been derived. However, for multi-planar tubular joints which cover the majority of practical applications, the research works in terms of the LJF are very limited and no parametric formulas are currently available to determine the LJF of multi-planar joints. The main reason is perhaps the complexity of geometrical properties and load distribution. Results of previous research works have shown that the stress and strain distributions in multi-planar tubular joints might be quite different from the uniplanar ones [4–8]. Therefore, the effect of multi-planarity on the LJF may also be significant and consequently the use of the equations already available for uniplanar joints to determine the LJF in multi-planar joints may result in highly under-/over-predicting outcomes. Hence, the application of equations developed for uniplanar joints to calculate the LJF in multi-planar connections needs to be validated. In the present paper, the LJF of two-planar tubular DK-joints in offshore jacket structures under the out-of-plane bending (OPB) loads is studied. Altogether, 81 finite element (FE) models were generated and analyzed by ANSYS in order to investigate the effects of DK-joint's geometrical parameters on the LJF factor (fLJF) under four types of OPB loading (Fig. 2). A set of dimensionless geometrical parameters (α, αB, β, γ, τ, and ζ) are used to readily relate the behavior of a tubular joint to its geometrical properties. These parameters are defined in Fig. 1b.
Corresponding author. E-mail address: [email protected] (H. Ahmadi).
http://dx.doi.org/10.1016/j.tws.2017.02.001 Received 31 October 2016; Received in revised form 30 January 2017; Accepted 1 February 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
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Fig. 1. (a) Two-planar tubular DK-joints in an OWT jacket substructure, (b) Geometrical notation for a two-planar DK-joint.
Above discussion indicates that, as mentioned in Section 1, the research works on the LJF of multi-planar tubular joints which cover the majority of practical applications are very limited and no parametric formulas are currently available to calculate the fLJF values for multi-planar joints.
Generated FE models were verified based on the existing experimental data, FE results, and parametric equations. After the parametric study of geometrical effects, the fLJF in two-planar DK- and uniplanar K-joints are compared; and the results of 324 analyses carried out on generated FE models are used to develop a new set of parametric formulas for the calculation of the fLJF in OPB-loaded two-planar DK-joints. Applicability of proposed formulas is verified according to the UK DoE [9] acceptance criteria. Thus, they can be reliably used for the analysis and design of tubular joints commonly found in offshore jacket structures.
3. Definition of the LJF factor (fLJF) for a tubular joint subjected to the OPB loading The LJF of a tubular joint is defined as the displacement attributed to the local chord wall deformation due to a unit applied load. To determine the LJF at the brace/chord intersection under the brace OPB moment loading, the rotation of the joint due to the overall displacement should be omitted from the total measured rotation [23]. In an FE model, the local rotation at the joint can be directly measured without considering the beam bending movement. The LJF of a tubular joint under the OPB loading can be defined as:
2. Literature review DNV [10] and UEG [2] have provided a set of parametric equations to predict the LJF for tubular T/Y-joints. The DNV equations are based on a limited number of FE analyses and the UEG guidelines do not clearly define the validity range. Efthymiou [11] developed a set of equations for T/Y- and K-joints under in-plane bending (IPB) and OPB loads by FE analysis. The database was limited to twelve T-, three Y-, and nine K-joints, five of which were partially overlapped. Fessler et al. [12,13] measured the local deformation of the chord wall under basic loadings within the elastic range based on 27 models and developed a set of parametric formulas for both T/Y- and K-joints. However, their experimental models were made from araldite instead of steel, and they had relatively small scale. Ueda et al. [14] derived a set of formulas to predict the LJF under the axial and IPB loads based on FE analysis of eleven T-joint models. The validity range of these equations was very limited in terms of the β that was restricted to 0.35−0.55. Chen et al. [15] determined the local joint flexibility of T/Y-, and K-joints. By using the semi-analytical approach, Chen et al. [16] and Hoshyari and Kohoutek [17] quantified the LJF for simple gap K- and T/Y-joints, respectively. Buitrago et al. [18] provided the methodology as well as the parametric equations for calculating the LJF in gap and partially overlapped joints based on the FE analysis. Chen and Zhang [19] studied the stress distribution in space frames with the consideration of local flexibility of multiplanar tubular joints. Hu et al. [20] and Golafshani et al. [21] developed equivalent elements representing the local flexibility of tubular joints in structural analysis of offshore platforms. Gao et al. [3,22] and Gao and Hu [23] proposed a set of parametric equations to determine the LJF in completely overlapped tubular joints subjected to axial, IPB, and OPB loading, respectively.
LJF = ϕOPB / MOPB
(1)
where MOPB is the brace OPB moment and ϕOPB is the joint local rotation expressed as:
ϕOPB =
δ − δ3 δ 2 − δ1 sin θ − 4 d−t D−T
(2)
Where D and d are the chord and brace diameters respectively; T and t are the chord and brace wall thicknesses respectively; θ is the brace inclination angle; δ1 and δ2 are the respective deformations at both saddles measured in the direction of brace axis; and δ3 and δ4 are the deformations at the side face of the chord corresponding to δ1 and δ2, respectively (Fig. 3). In order to relate the local joint flexibility to dimensionless geometrical parameters of the joint, a dimensionless coefficient called the local joint flexibility parameter (fLJF) is defined. The fLJF is the LJF multiplied by ED3:
fLJF =
ϕOPB MOPB
ED 3
(3)
where D is the chord diameter and E is the Young's modulus. It should be noted that since a two-planar DK-joint has four brace members, there are four different positions for the application of Eqs. (2) and (3). However, due to the symmetry in the geometry of the joint 123
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Fig. 2. Studied OPB load cases.
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Fig. 3. Positions for measuring the deformation to calculate the LJF under the OPB moment loading.
mesh quantity and quality and avoid badly distorted elements. The mesh generated by this method is shown in Fig. 4. In order to make sure that the results of the FE analysis are not affected by the inadequate quality or the size of the generated mesh, convergence test was conducted and meshes with different densities were used in this test, before generating the 81 models. As an example, results of the convergence test for a joint model (θ=45°, τ=0.7, β=0.5, γ=18) having different mesh densities (Fig. 5) have been presented in Table 1. This table indicates that the difference between the fLJF values in a model with a coarse mesh, i.e. ne=1 (Fig. 5a), and the model with a finer mesh, i.e. ne=3, selected for the present parametric FE study (Fig. 5b) had been nearly 27% meaning that the accuracy of the ne=1 mesh was not adequate. However, the difference between the fLJF values in models with ne=3 and ne=6 was less than 3% implying that such an overly fine mesh (Fig. 5c) was not necessary since it would lead to the increase of the computation time without any considerable improvement of the accuracy. The static analysis of linearly elastic type is suitable for the prediction of LJF in tubular joints [22,23]. The Young's modulus and Poisson's ratio were taken to be 207 GPa and 0.3, respectively.
and either symmetry or antisymmetry in the applied loading, results obtained from all of these four positions would be the same. 4. FE modeling and analysis 4.1. Weld profile Results of the present study's FE verification process (Section 4.4) indicated that the weld sizes must be carefully included in the FE modeling. In this research, the welding size along the brace/chord intersection satisfies the AWS D 1.1 [24] specifications. The dihedral angle (ψ), defined as the angle between the chord and brace surfaces along the intersection curve, is an important parameter in determining the weld thickness. The value of the ψ at three typically important positions along the weld toe equals to: θ (Crown Heel), π – cos–1θ (Saddle), and π – θ (Crown Toe); where θ is the brace inclination angle (Fig. 1b). Details of weld profile modeling according to AWS D 1.1 [24] specifications have been presented by Ahmadi et al. [25]. 4.2. Boundary conditions Fixity condition of the chord ends in tubular joints of offshore structures ranges from almost fixed to almost pinned with generally being closer to almost fixed [26]. In the view of the fact that the effect of the chord end restraints on the stress distribution at the brace/chord intersection is only significant for joints with α < 8 and high β and γ values [27,28], which do not commonly occur in practice, both chord ends were assumed to be fixed, with the corresponding nodes restrained.
4.4. Verification The accuracy of FE results to determine the fLJF in tubular joints should be validated using the experimental test results. As far as the authors are aware, there is no experimental/FE database of fLJF for twoplanar tubular DK-joints currently available in the literature. Considering this issue, in order to verify the FE model, a set of Y-joints were modeled and the fLJF values obtained from these models were compared with the experimental results of Fessler et al. [12], values predicted by Fessler et al. equation [12], and the FE results of Gao et al. [22]. Geometrical properties of the validating Y-joints have been presented in Table 2. The procedure of geometrical modeling (introducing the chord, braces, and weld profiles), the mesh generation method (including the selection of the element type and size), the analysis type, and the method of fLJF calculation are the same for the validating Y-joint models and the DK-joints used in the present research for the parametric study. Hence, the verification of the fLJF values derived from the validating FE models with available corresponding experiment-/FE-/equation-predicted values lends some support to the validity of the fLJF values derived from the DK-joint FE models. Results of the verification process have been presented in Fig. 6 along with Table 2. The effects of the element type and the weld profile were also investigated. A comparison between the results obtained by the solid and shell elements indicated that the solid elements lead to
4.3. Mesh generation and analysis ANSYS Ver. 16 element type SOLID 185 was used to model the chord, braces, and the weld profiles. These elements have compatible displacements and are well-suited to model curved boundaries. The element is defined by eight nodes having three degrees of freedom per node and may have any spatial orientation. Using this type of 3-D brick elements, the weld profile can be modeled as a sharp notch. This method will produce more accurate and detailed stress distribution near the intersection in comparison with a simple shell analysis (See Section 4.4). To guarantee the mesh quality, a sub-zone mesh generation method was used during the FE modeling. In this method, the entire structure is divided into several different zones according to the computational requirements. The mesh of each zone is generated separately and then the mesh of entire structure is produced by merging the meshes of all the sub-zones. This method can easily control the 125
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Fig. 4. The mesh generated by the sub-zone method for (a) a uniplanar tubular K-joint, (b) a two-planar tubular DK-joint.
5. Geometrical effects on the local joint flexibility factor
more accurate fLJF values. Moreover, the comparison of solid models with and without the weld profile showed that the omission of the weld profile results in the increase of the error percentage. As can be seen in Fig. 5, there is a good agreement between the results of previous studies and the predictions of the validating FE model. According to Table 2, the maximum difference between the fLJF of the validating FE model and the experimental results of Fessler et al. [12] is 10.07%; and the maximum difference between the results of the validating FE model and the equation proposed by Fessler et al. [12] is 12%. Hence, generated FE models can be considered to be accurate enough to provide valid results.
5.1. Details of parametric investigation To investigate the effect of geometrical parameters on the fLJF in two-planar DK-joints subjected to four types of OPB loading (Fig. 2), 81 models were generated and analyzed using ANSYS. Different values assigned to the parameters β, γ, τ, and θ are as follows: β=0.4, 0.5, 0.6; γ=12, 18, 24; τ=0.4, 0.7, 1.0; and θ=30°, 45°, 60°. These values cover the practical ranges of dimensionless parameters typically found in tubular joints of offshore jacket structures. Providing that the gap
Fig. 5. Different mesh densities for a joint model: (a) coarse mesh, (b) fine mesh, (c) overly fine mesh.
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Table 1 The effect of the FE mesh size on the fLJF results. Geometrical properties of the joint
θ=45°, τ=0.7, β=0.5, γ=18
fLJF
Difference between fLJF values
ne =3 (Fig. 5b)
Overly fine mesh ne=6 (Fig. 5c)
ne=1 and ne=3
ne=3 and ne=6
2662.52
2590.177
26.68%
2.71%
Coarse mesh ne=1 (Fig. 5a)
Fine mesh
3372.971
ne: The number of elements along the chord thickness
between the braces is not very large, the relative gap (ζ= g / D) has no considerable effect on the stress and strain distribution. The validity range for this statement is 0.2≤ ζ≤0.6 [29]. Hence, a typical value of ζ=0.3 was designated for all joints. Sufficiently long chord greater than six chord diameters (i.e. α≥12) should be used to ensure that the stresses at the brace/chord intersection are not affected by the chord's boundary conditions [26]. Hence, in this study, a realistic value of α=16 was designated for all the models. The brace length has no effect on the stress and strain distribution when the parameter αB is greater than a critical value [30]. According to Chang and Dover [31], this critical value is about 6. In the present study, in order to avoid the effect of short brace length, a realistic value of αB=8 was assigned for all joints.
Fig. 6. Results of the verification process using the validating Y-joint models.
the fLJF under all studied OPB loading conditions. The reason is that the increase of the brace diameter leads to the decrease of the joint local rotation which consequently results in the decrease of the local joint flexibility and then the fLJF. As can be seen in Fig. 7, this conclusion is independent from the values of other geometrical parameters.
5.3. The effect of the brace-to-chord wall thickness ratio on the fLJF Results of investigating the effect of the τ on the fLJF are presented in this section. The interaction of the τ with the other geometrical parameters was also investigated. The parameter τ is the ratio of the brace wall thickness to the chord wall thickness and the γ is the ratio of the section radius to wall thickness of the chord. Hence, provided that the value of the γ remains constant, the increase of the τ leads to the increase of the brace thickness. As an example, nine charts are given in Fig. 8 depicting the change of the fLJF, under the 1st OPB loading condition, due to the change in the value of the τ and the interaction of this parameter with the γ. It can be concluded that the increase of the τ has no considerable effect on the local joint flexibility compared to the other geometrical parameters.
5.2. The effect of the brace-to-chord diameter ratio on the fLJF Results of investigating the effect of the β on the fLJF are presented in this section. The influence of parameters τ, γ, and θ over the effect of the β on the fLJF was also investigated. The parameter β is the ratio of the brace diameter to the chord diameter. Hence, provided that the chord diameter remains constant, the increase of the β results in the increase of the brace diameter. Fig. 7 demonstrates the change of the fLJF, under the 1st OPB loading condition, due to the change in the value of the β and the interaction of this parameter with the γ. Altogether, 81 comparative charts were used to study the effect of the β and only nine of them are presented here for the sake of brevity. Results showed that the increase of the β results in the decrease of Table 2 Details and results of the verification process. Geometrical properties of validating Y-joint models Geometrical parameter Value Chord length L=1888.0 mm Chord diameter D=168.3 mm Chord wall thickness T=7.0 mm Brace length l=623.0 mm Brace diameter d=55.5, 88.2, 127.9 mm Brace wall thickness t=5.5 mm Brace inclination angle θ=60°
Dimensionless parameters α=2 L/D =22.4 γ= D/2 T=12 β= d/D =0.32, 0.53, 0.76 τ= t/T=0.78
The comparison between the results of validating FE model and the experimental data β fLJF Validating FE model Experimental data [12] Shell Solid with weld Solid without weld 0.32 2538.45 1996.40 1525.39 2079.33 0.53 954.39 895.39 1083.61 985.57 0.76 437.98 277.63 360.95 300.48 The comparison between the results of validating FE model and the equation proposed by Fessler et al. [12] β fLJF Validating FE model Fessler et al. Eq. [12] Shell Solid with weld Solid without weld 0.32 2538.45 1996.40 1525.39 2133.41 0.53 954.39 895.39 1083.61 877.40 0.76 437.98 277.63 360.95 312.50
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Error (%) Shell 22.08 3.16 45.75
Solid with weld 4.15 10.07 8.22
Solid without weld 36.31 9.94 20.12
Solid with weld 6.86 2.05 12.00
Solid without weld 39.86 23.50 15.50
Error (%) Shell 18.98 8.77 40.15
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Fig. 7. The effect of the β on the fLJF values in two-planar tubular DK-joints under the 1st OPB load case.
the fLJF, under the 1st OPB loading condition, due to the change in the value of θ and the interaction of this parameter with the γ. Results indicated that the increase of the θ leads to the increase of the fLJF under all studied OPB loading conditions. The reason is that the increase of the brace inclination angle results in the increase of the joint
5.4. The effect of the brace inclination angle on the fLJF Results of studying the effect of the θ on the fLJF and its interaction with the other geometrical parameters are presented in this section. Nine charts are given in Fig. 9, as an example, depicting the change of 128
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Fig. 8. The effect of the τ on the fLJF values in two-planar tubular DK-joints under the 1st OPB load case.
local rotation that then leads to the increase of the local joint flexibility and the fLJF. As can be seen in Fig. 9, this result is independent from the values of other geometrical parameters.
5.5. The effect of the chord slenderness ratio on the fLJF Results of investigating the effect of the γ on the fLJF are presented in this section. The influence of parameters β, τ, and θ over the effect of the γ on the fLJF was also investigated. The parameter γ is the ratio of 129
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Fig. 9. The effect of the θ on the fLJF values in two-planar tubular DK-joints under the 1st OPB load case.
6. The effect of OPB loading mode and the multi-planarity on the fLJF
the section radius to wall thickness of the chord. Hence, provided that the chord diameter remains constant, the increase of the γ means the decrease of the chord thickness. It is evident from Figs. 7–9 that the increase of the γ leads to the increase of the fLJF.. The reason is that the decrease of the chord thickness leads to the increase of the joint local rotation that consequently results in the increase of the local joint flexibility. This conclusion does not depend on the values of other geometrical parameters.
In Fig. 10, the fLJF values under the four considered OPB loading conditions are compared for two sample joints. According to this figure, it can be concluded that the maximum and minimum values of the fLJF occur under the 2nd and 3rd OPB load cases, respectively. It is also evident that the differences among the fLJF values under the four 130
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Fig. 10. Comparing the fLJF values under different OPB loading conditions: (a) θ=60°, τ=0.7, γ=24, β=0.4; (b) θ=30°, τ=0.7, γ=12, β=0.4.
However, under the 2nd loading condition, the uniplanar fLJF is 67% of the corresponding two-planar value (Fig. 11b). These results indicate that the effect of the multi-planarity on the fLJF is quite significant and consequently the use of the formulas already available for uniplanar Kjoints to calculate the fLJF in two-planar DK-joints may lead to highly under-/over-predicting results. To handle this issue, the FE results are used in the next section to develop a set of parametric formulas for the determination of the fLJF values in OPB-loaded two-planar DK-joints.
considered OPB load cases are quite big. This implies that the derivation of an individual parametric formula for each of these load cases is essential; and the use of a single formula to predict the fLJF for all the considered OPB loading conditions may lead to highly under-/ over-predicting results. The fLJF values in uniplanar K- and two-planar DK-joints have been compared in Fig. 11. It can be seen that under the 1st and 3rd OPB load cases, the fLJF values in uniplanar joints are bigger than the corresponding values in two-planar joints. On the contrary, under the 2nd and 4th OPB load cases, the fLJF values in uniplanar joints are smaller. It is also evident that the difference between the fLJF values in uniplanar and two-planar joints can be quite significant. For example, as shown in Fig. 11a, the uniplanar fLJF (for θ=30°, τ=0.4, γ=12, β=0.6) is twice the corresponding two-planar value under the 1st OPB load case.
7. Development of parametric formulas to calculate the fLJF values Results of parametric FE investigation were used to derive four individual parametric formulas for the determination of the fLJF values in two-planar tubular DK-joints subjected to four types of OPB loading
Fig. 11. The comparison of fLJF values in uniplanar K- and two-planar DK-joints (θ=30°, τ=0.4, γ=12).
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The parameter θ in Eqs. (4)–(7) should be inserted in radians. Values obtained for R2 are quite high indicating the accuracy of the fit. The validity ranges of nondimensional geometrical parameters for the proposed formulas are as follows:
Table 3 Assessment of developed formulas based on the UK DoE [9] criteria. Formula
Eq. Eq. Eq. Eq.
(4) (5) (6) (7)
OPB
UK DoE Conditions
Decision
Load Case
%P/R < 0.8
%P/R > 1.5
1st 2nd 3rd 4th
1.23% < 5% OK. 0% < 5% OK. 1.23% < 5% OK. 1.23% < 5% OK.
7.40% < 50% OK. 12.34% < 50% OK. 0% < 50% OK. 1.23% < 50% OK.
0.4 ≤ τ ≤ 1.0 12 ≤ γ ≤ 24 0.4 ≤ β ≤ 0.6 30° ≤ θ ≤ 60°
Accept Accept Accept Accept
The UK Department of Energy (DoE) [9] recommends the following assessment criteria regarding the applicability of the parametric formulas (P/R stands for the ratio of the predicted value from a given formula to the recorded value from the FE analysis):
shown in Fig. 2. Formulas were derived based on multiple nonlinear regression analyses performed by SPSS. Values of dependent variable (i.e. fLJF) and independent variables (i.e. β, γ, τ, and θ) constitute the input data imported in the form of a matrix. Each row of this matrix involves the information about the fLJF value in a two-planar tubular DK-joint having specific geometrical properties. When the dependent and independent variables are defined, a model expression should be built with defined parameters. Parameters of the model expression are unknown coefficients and exponents. The researcher should specify an initial value for each parameter, preferably as close as possible to the expected final solution. Poor initial values may lead to failure in convergence or to convergence to a local (instead of global) solution or even to a physically impossible solution. Various model expressions should be generated to derive a parametric formula with a high coefficient of determination (R2). After performing the nonlinear analyses, following parametric formulas are proposed for the calculation of the fLJF values in twoplanar tubular DK-joints subjected to four types of out-of-plane bending (OPB) moment loading (Fig. 2).
•1
st
•2
nd
•
R 2 = 0.989
(4)
R 2 = 0.991
(5)
LJF values is less than or equal to 25%, i.e. [%P/R < 1.0] ≤ 25%; and the percentage of considerably under-predicting fLJF values is less than or equal to 5%, i.e. [%P/R < 0.8] ≤ 5%, then the equation is accepted. If, in addition, the percentage of considerably overpredicting fLJF values is more than or equal to 50%, i.e. [%P/R > 1.5] ≤50%, then the equation is regarded as generally conservative. If the acceptance criteria is nearly met, i.e. 25% < [%P/R < 1.0] ≤ 30%, and/or 5% < [%P/R < 0.8] ≤ 7.5%, then the equation is regarded as borderline and engineering judgment should be applied to determine the acceptance or rejection. Otherwise the equation is rejected as it is too optimistic.
In the view of the fact that for a mean fit formula, there is always a large percentage of under-prediction, the requirement for joint underprediction, i.e. P/R < 1.0, can be completely removed in the assessment of parametric equations [32]. Assessment results according to the UK DoE [9] criteria are presented in Table 3. As can be seen in this table, all of the proposed formulas satisfy the criteria recommended by the UK DoE. Fig. 12 indicates that, as it is expected from Section 6 and Fig. 10, considerable differences exist among the fLJF values predicted by Eqs. (4)–(7) implying that it was essential to develop an individual fLJF equation for each of the four considered OPB loading conditions. 8. Conclusions
rd
OPB loading condition:
fLJF = 1.189τ −0.108γ 1.931β −2.612θ1.826
•4
•
OPB loading condition:
fLJF = 4.232τ −0.063γ 2.054β −2.011θ 2.141
•3
• For a given dataset, if the percentage of under-predicting f
OPB loading condition:
fLJF = 0.190τ −0.162γ 2.539β −3.545θ1.835
(8)
th
R 2 = 0.995
The LJF of two-planar tubular DK-joints under the OPB loads was studied. Altogether, 81 FE models were generated and analyzed in order to investigate the effects of DK-joint's geometrical parameters on the LJF factor (fLJF) under four types of OPB loading. Results indicated that the change of the τ has no considerable effect on the local joint flexibility compared to the other geometrical parameters; the increase of the θ and/or γ leads to the increase of the
(6)
OPB loading condition:
fLJF = 0.465τ −0.165γ 2.250 β −3.091θ1.585
R 2 = 0.991
(7)
Fig. 12. Comparison of the fLJF values predicted by the proposed equations for 81 tubular DKT-joints (β=0.4, 0.5, 0.6; γ=12, 18, 24; τ=0.4, 0.7, 1.0; and θ=30°, 45°, 60°) under the four considered OPB loading conditions.
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fLJF; and the increase of the β leads to the decrease of the fLJF values under all considered OPB load cases. The local deformation of the joint lowers its strength requirement by redistributing the member-end loads and moments compared to a conventional rigid joint that will consequently lower the cost of the tubular structure. Hence, the enhancement of the joint LJF is recommended during the design stage. Results of the present study showed that, for tubular DK-joints, this design objective can be accomplished through the decrease of the brace-to-chord diameter ratio and the increase of the brace inclination angle and/or chord slenderness ratio. The differences among the fLJF values under the four studied OPB loading conditions are quite big which means that the development of an individual parametric formula for each of these load cases is essential; and the application of a single formula to calculate the fLJF values for all the considered OPB load cases might lead to highly under-/over-predicting results. The difference between the fLJF values in uniplanar and two-planar joints can be quite significant implying that the effect of multi-planarity on the fLJF values is considerable and consequently the application of the formulas already available for uniplanar K-joints to determine the fLJF values in two-planar DK-joints may result in highly under-/overpredicting outcomes. To handle this issue, the FE results were used to develop four individual parametric formulas for the determination of the fLJF values in two-planar DK-joints subjected to four types of OPB loading. Proposed formulas, having high coefficients of determination, were assessed based on the acceptance criteria recommended by the UK DoE and can be reliably used for the analysis and design of tubular joints in offshore jacket structures.
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