An analytical solution to lateral buckling control of subsea pipelines by distributed buoyancy sections

Thin-Walled Structures 107 (2016) 221–230

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An analytical solution to lateral buckling control of subsea pipelines by distributed buoyancy sections Gang Li n, Lichao Zhan, Hao Li State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116023, China

art ic l e i nf o

a b s t r a c t

Article history: Received 21 January 2016 Received in revised form 3 June 2016 Accepted 8 June 2016

Lateral buckling is the primary buckling form of the subsea pipelines. A promising recent practice is to the distributed buoyancy sections to control the lateral buckling of subsea pipeline. In this study, an analytical solution is deduced for the lateral buckling of the pipeline with buoyancy sections in the entire design region. However, numerical difficulty was encountered during the solution of the design equations. To overcome this difficulty, we adopt a response surface method (RSM) for the solution of nonlinear equation system. A strategy of multi-response-surface is proposed by dividing the entire design region into several partial domains and establishing different kinds of response surfaces for each region. A framework for lateral buckling control of subsea pipelines using distributed buoyancy sections is presented. Several illustrative examples demonstrate that the proposed method has a high efficiency and accuracy in solving the problem of the lateral buckling control. & 2016 Elsevier Ltd. All rights reserved.

1. Introduction Energy resources exploitation and transportation in deep water require long pipelines operated at high temperature and high pressure (HT/HP). The axial expansion caused by the increase in temperature and pressure may lead to lateral or upheaval buckling along the pipeline. Such uncontrolled global buckling would cause serious damage to the safety of the pipelines. Consequently, the industry preferred to restrain the global buckling problem by trenching and burying of pipeline. Trenching for the subsea pipelines is uncommon due to high cost in deep water [1,2]. Since the vertical resistance is larger than the lateral soil resistance against the pipeline, the lateral buckling becomes the primary buckling form and must be considered in the design [3]. On the positive side, lateral buckling can release the axial expansion effectively. Thus the solution to control the lateral buckles is considered to be more elegant than preventing them. Moreover, when the temperature and pressure increase further, the controlled lateral buckling is deemed to be the only economic solution [4,5]. Based on the concept of controlling the lateral buckling, the new design method is to install the buckle triggers along the pipeline route to initiate planned buckle [5]. The key challenge is to control the buckle behavior when exciting the lateral buckling. As a typical technique to initiate the lateral buckling under control, the use of n

Corresponding author.

http://dx.doi.org/10.1016/j.tws.2016.06.003 0263-8231/& 2016 Elsevier Ltd. All rights reserved.

distributed buoyancy sections have attracted more and more attention in the design of subsea pipelines [6]. This technique reduced the submerged weight of the pipeline by installing distributed buoyancy at the planned sites. It means that the buckle initiation force is reduced caused by the reduction in the lateral soil restraint. Finally, the likelihood of the buckle as planned is increased. Many investigators have studied the pipeline buckling problem by experimental, numerical and analytical method in past decades [7,8]. Kerr [9,10] derived the analytical solution for the buckling problem of the thermal track modeled as an Euler beam. Based on Kerr's study, Hobbs [11] investigated the buckling problem of the ideal straight heated pipeline on the rigid seabed and obtained the concise analytical solution of the buckling behavior. The studies of Kerr and Hobbs laid the foundation of theoretical analysis of the global buckling of pipelines. When using the buoyancy sections, the lateral resistance along the pipeline is no longer uniform, and Hobbs's solution is no longer applicable. The buoyancy sections can be regarded as a means of initial imperfections. Antunes [12] presented the governing equations and the boundary conditions for the lateral buckling of subsea pipelines with distributed buoyancy sections. In his study, the length of the buoyancy section and the submerged weight ratio between the buoyancy section region and other region were chosen as the design parameters. However, the solution was not given in his paper. A number of investigators analyzed the global buckling of a pipeline on a rigid seabed with the initial imperfections on basis of the studies of Kerr

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and Hobbs. The influence of the initial imperfections on the global buckling behavior is investigated in their works [13–16]. Ralf Peek [17,18] investigated the effect of the flotation on the pipeline buckling form, and found that the flotation will cause the pipe to buckle laterally not upwards on the flat seabed. A scaling technique was proposed by the author to calculate lateral buckling behavior of the inelastic pipeline quickly. The analysis of pipeline buckling by finite element method (FEM) and numerical match method has been paid more attentions [3,19]. Zeng [20] studied the upheaval buckling behavior of a buried pipe by dimensional analysis and FE analysis. The approximation formula of critical axial force for imperfections was further developed through numeric fitting in his paper. Furthermore, the influence of the soil on the pipeline buckling behavior has also attracted the researchers' attention in recent years [21,22]. However, there is less theoretical analysis for the pipeline lateral buckling with distributed buoyancy sections at present, which hinders the cognition on the physical essence. This paper deduces an analytical solution for the lateral buckling of the subsea pipelines with distributed buoyancy sections. In the analytical solution, the length ratio β between the buoyancy section region and the buckling region and the submerged weight ratio γ between the buoyancy section region and other region are chosen as the design parameters. Benefiting from the dimensionless design parameters with the values from 0 to 1, a closed design region is obtained. This analytical solution, covering the entire design region, is no longer limited to the upper bound of the distributed buoyancy sections length. In order to overcome the numerical instability of the solution of nonlinear equation system and to improve the calculation efficiency, the relationship between the buckling behavior and the design parameters of distributed buoyancy sections is modeled by using RSM based on the analytical solution. RSM consists of a set of mathematical and statistical techniques introduced by Box and Wilson in 1951 [23]. This method can be used to predict an approximate response for the known variables by establish a relationship between them. The examples demonstrate the efficiency and accuracy of the proposed RSM based solution.

2. Analytical model for lateral buckling of a pipeline with buoyancy sections Kerr [10] and Hobbs [11] deduced the analytical solutions for lateral buckling of track-beam and pipelines, respectively. In Hobbs's analytical model, the pipeline was assumed to be placed on rigid seabed. The axial compression force P0 in the pipeline before buckling, the axial compression force P in the buckling region, the axial coefficient of friction μA and the lateral coefficient of friction μL are assumed to be constants in their solution. The lateral buckling configuration and the force distribution are shown in Fig. 1, in which the pipeline is divided into lateral buckling region and axial slip region. The axial and lateral displacements are 0 at the point x=L 0 , defined as the virtual anchor point. In this paper, we adopt the same assumptions with the referred authors to derive the analytical solution of the symmetrical lateral buckling modes for the pipeline with buoyancy sections. In this study only the reducing in the submerged weight is considered and the contribution to the stiffness of the buoyancy sections is neglected. According to the symmetry of the lateral buckling configuration, half of the model is to be analyzed in the following sections. Because of the design of buoyancy sections, the submerged weight per unit of length along the pipeline is nonuniform, with Wsb in the region of the buoyancy section and Ws in other regions. The submerged weight ratio γ between Wsb and Ws, the length ratio β between the buoyancy section region and the

Fig. 1. Classical lateral buckling configuration and the axial force distribution.

buckling region are chosen as the design parameters in this paper. And the length of distributed buoyancy sections are assumed to be smaller than the buckle lobe, so a closed design region 0 ≤ γ ≤ 1 and 0 ≤ β ≤ 1 is obtained. Previous researches indicated the symmetrical modes (mode 1 and mode 3) are more dangerous than the antisymmetrical modes (mode 2 and mode 4) [10,11]. 2.1. Analysis for lateral buckling mode 1 of problem The general lateral buckling configuration of mode 1 is shown in Fig. 2. The distributed buoyancy sections region is from 0 to LB in which the axial displacement u, lateral displacements v and Lagrangian strain ε are identified by the subscripts B. The buckle lobe without distributed buoyancy sections is from LB to L1 in which the axial displacement u, lateral displacement v and Lagrangian strain ε are identified by the subscripts 1. The axial sliding region is from L1 to L 0 in which the axial displacement u and Lagrangian strain ε are identified by the subscripts 2. The point at L0 is considered as the virtual anchor point. According to the equilibrium of forces and moments, the following equations are obtained:

EIvB″″ + PvB″ = − γμL Ws

0 ≤ x ≤ LB

(1)

EIv1″″ + Pv1″ = − μL Ws

LB < x ≤ L1

(2)

P0 − EAεB = P

0 ≤ x ≤ LB

Fig. 2. Lateral buckling configuration-mode1.

(3)

G. Li et al. / Thin-Walled Structures 107 (2016) 221–230

P0 − EAε1 = P

LB < x ≤ L1

EAu2′ ′ = − μA Ws

(4)

L1 < x ≤ L 0

(5)

where E is the Young Modulus, I is the pipe second moment of area, A is the cross-sectional area and εi can be expressed by the following expression:

εi = ui′ +

1 2 vi′ 2

i = B, 1, 2

(6)

The general solutions of Eqs. (1) and (2) are

vB(x) = A1 cos(λx) + A2 sin(λx) + A3x + A 4 −

v1(x) = A5 cos(λx) + A6 sin(λx) + A7 x + A8 −

γω 2 x 2λ 2

ω 2 x 2λ 2

(7)

(8)

solved as:

P λ = ; ω= EI 2

(P0 − P ) 1 x− EA 2

u1(x) =

(P0 − P ) 1 (x − LB ) − EA 2

(10)

(P0 − P ) 1 − v1′ 2(x) EA 2

(11)

The general solution of Eq. (5) is

μA Ws x2 EA 2

vB′ 2(x)dx

∫L

x

v1′ 2(x)dx + uB (LB )

B

(23)

(24)

(25)

Finally, by utilizing the boundary conditions (21), the following equation is obtained:

⎡ P0 = P + μA WsL1⎢ −1 + ⎢⎣

1+

⎤ EA ⎥ S μA WsL12 ⎥⎦

(26)

where

∫0

LB

vB′ 2(x)dx +

∫L

L1

v1′ 2(x)dx

B

(27)

(9)

EI

(P − P ) 1 uB′(x) = 0 − vB′ 2(x) EA 2

u2(x) = −

∫0

⎡ μ Ws P − P0 ⎤ u2(x) = ⎢ A (x + L 0 − 2L1) − ⎥(L 0 − x) EA ⎦ ⎣ 2EA

S=

Eqs. (3) and (4) are rewritten by utilizing the Eq. (6) as follows

u1′(x) =

x

uB (x) =

where

μL Ws

223

+ B1x + B2

(12)

The boundary and matching conditions in lateral direction are presented as follow:

vB′(0) = 0, vB″″(0) = 0

(13)

vB(LB ) = v1(LB ), vB′(LB ) = v1′(LB )

(14)

vB′′(LB ) = v1′′(LB ), vB′′′(LB ) = v1′′′(LB )

(15)

v1(L1) = 0, v1′′(L1) = 0

(16)

v1′(L1) = 0

(17)

The boundary and matching conditions in axial direction are presented as follow

uB (LB ) = u1(LB ), uB (LB ) = u1(LB )

(18)

u1′(L1) = u2′(L1), u2(L 0) = 0

(19)

uB′(LB ) = u1′(LB )

(20)

u1(L1) = u2(L1), u2′(L 0) = 0

(21)

2.2. Analysis for lateral buckling mode 3 of problem Different from the mode 1, the buckling region in general lateral buckling form of mode 3 is divided into the primary lobe 2L1 and secondary lobe 2L2 as shown in Fig. 3. Similar to mode 1, the distributed buoyancy sections region is from 0 to LB in which the axial displacement u, lateral displacement v and Lagrangian strain ε are identified by the subscripts B. The primary lobe without distributed buoyancy sections is from LB to L1 in which the axial displacement u, lateral displacement v and Lagrangian strain ε are identified by the subscripts 1. The secondary lobe is from L1 to L2 in which the axial displacement u, lateral displacement v and Lagrangian strain ε are identified by the subscripts 2. The axial sliding region is from L2 to L 0 in which the axial displacement u and Lagrangian strain ε are identified by the subscripts 3. The point at L0 is considered as the virtual anchor point. Similar to the analysis above, the following equations are obtained:

EIvB′′′′ + PvB′′ = − γμL Ws

0 ≤ x ≤ LB

(28)

EIv1′′′′ + Pv1′′ = − μL Ws

LB < x ≤ L1

(29)

EIv2′′′′ + Pv2′′ = μL Ws

L1 < x ≤ L 2

P0 − EAεB = P

0 ≤ x ≤ LB

(31)

P0 − EAε1 = P

LB < x ≤ L1

(32)

P0 − EAε2 = P

L1 < x ≤ L 2

(33)

The constants A1 to A8 are obtained (see Appendix) by using the boundary conditions (13)–(16). By utilizing the boundary condition (17), λL1 can be solved through the equation below:

( β − 1)⎡⎣ γ ( λL1)cos( λL1) −

sin(γλL1)⎤⎦ + (λL1)cos(λL1) − sin(λL1) = 0

(22)

At this point, the displacements vi(x ) are a series of known functions in different regions of the pipeline. Then using the boundary conditions (18) and (19), (Eqs. (10), (11) and 12) are

(30)

Fig. 3. Lateral buckling configuration-mode3.

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G. Li et al. / Thin-Walled Structures 107 (2016) 221–230

EAu3′ ′ = − μA Ws εi = ui′ +

1 2 vi′ 2

L2 < x ≤ L 0

(34)

i = B, 1, 2

(35)

The general solutions of (Eqs. (28), (29), 30) are

vB(x) = A1 cos(λx) + A2 sin(λx) + A3x + A 4 −

γω 2 x 2λ 2

By utilizing the boundary conditions (50), we can solve λL1, λL2 through the equations below:

2⎡⎣ (γ − 1)cos(λL1 − λL 2) − γ + 1⎤⎦sin(βλL1) + 4 cos(λL1)sin(λL1 − λL 2) + 4⎡⎣ sin(λL 2) − sin(λL1)⎤⎦ + 2 ⎡⎣ ( γ − 1)β + 2⎤⎦⎡⎣ cos(λL 2) − cos(λL1)⎤⎦ λL1

{

(36)

}

− 2⎡⎣ cos(λL 2) − cos(λL1)⎤⎦λL 2 − ( 2γβ − 2β + 3)(λL1)2 sin(λL 2) v1(x) = A5 cos(λx) + A6 sin(λx) + A7 x + A8 −

ω 2 x 2λ 2

+ 2(γβ − β + 2)λL1λL 2 sin(λL 2) − (λL 2)2 sin(λL 2) = 0

(55)

(37) (γ − 1)sin(βλL1) − sin(λL 2) + 2 sin(λL1)

v2(x) = A 9 cos(λx) + A10 sin(λx) + A11x + A12 +

ω 2 x 2λ 2

− ⎡⎣ (γβ − β + 2)λL1 − λL 2⎤⎦cos(λL 2) = 0

(38)

where

μ Ws P ; ω= L EI EI

Following the procedure in the analysis of mode 1, we obtain the expressions of the axial displacement as below: x

(39)

uB (x) =

(P0 − P ) 1 x− EA 2

(Eqs. (31), (32) and 33) are rewritten by utilizing Eq. (35) as follows:

u1(x) =

(P0 − P ) 1 (x − LB ) − EA 2

∫L

(P0 − P ) 1 (x − L1) − EA 2

∫L

λ2 =

uB′(x) =

(P0 − P ) 1 − vB′ 2(x) EA 2

(40) u2(x) =

(P − P ) 1 u1′(x) = 0 − v1′ 2(x) EA 2

(41)

(P − P ) 1 u2′(x) = 0 − v2′ 2(x) EA 2

(42)

The general solution of Eq. (34) is

u3(x) = −

μA Ws x2 EA 2

+ B1x + B2

(43)

The boundary and matching conditions in lateral direction are presented as follows:

vB′(0) = 0, vB′′′(0) = 0

(44)

vB(LB ) = v1(LB ), vB′(LB ) = v1′(LB )

(45)

vB′′(LB ) = v1′′(LB ), vB′(LB ) = v1′(LB )

(46)

v2(L1) = 0, v1′(L1) = v2′(L1)

(47)

v1′′(L1) = v2′′(L1), v1′′′(L1) = v1′′′(L1)

(48)

v2′(L 2) = 0, v2′′(L 2) = 0

(49)

v1(L1) = 0, v2(L 2) = 0

(50)

The boundary and matching conditions in axial direction are presented as follows:

uB (0) = 0, uB (LB ) = u1(LB )

(51)

uB′(LB ) = u1′(LB ), u1(L1) = u2(L1)

(52)

u1′(L1) = u2′(L1), u2(L 2) = u3(L 2)

(53)

u2′(L 2) = u3′(L 2), u3(L 0) = 0, u3′(L 0) = 0

(54)

The constants A1 to A12 are obtained (see Appendix) by using the boundary conditions (44)–(49).

(56)

∫0

vB′ 2(x)dx x

(57)

v1′ 2(x)dx + uB (LB )

B

x

v2′ 2(x)dx + u1(L1)

1

⎡ μ Ws P − P0 ⎤ u3(x) = ⎢ A (x + L 0 − 2L1) − ⎥(L 0 − x) EA ⎦ ⎣ 2EA

(58)

(59)

(60)

Finally, by utilizing the last two boundary conditions (54), we obtain

⎡ P0 = P + μA WsL 2⎢ −1 + ⎢⎣

1+

⎤ EA S⎥ 2 ⎥ μA WsL 2 ⎦

(61)

where

S=

∫0

LB

vB′ 2(x)dx +

∫L

L1

v1′ 2(x)dx +

B

∫L

L2

1

v2′ 2(x)dx

(62)

At this moment of the analysis, substitute the λL1 and λL2 ( λL1 for mode 1) solved from (Eqs. (55), 56) (Eq. (22) for mode 1) into integration S, we can solve the half buckling length L1 from the Eq. (61) (Eq. (26) for mode 1). Then, the maximum amplitude vmax and the maximum bending moment Mmax can be obtained from the following expressions:

vmax = vB(0)

(63)

Mmax = − EIvB″(0)

(64)

Similar to Hobbs's research, we obtain the spacing between the anchor points (VAS) as the below expression:

⎡ ( P − P) ⎤ 0 VAS = 2⎢ + L1⎥ ⎢⎣ μA Ws ⎥⎦

(65)

For the case of γ = 1 or β = 0, the effect of distributed buoyancy sections disappears and the solutions deduced in this paper is the same to those proposed for a long straight pipeline by Kerr [10]. Noting that the length of the distributed buoyancy sections is less than the buckle length in practice, this analytical solution covers the entire variation range of the distributed buoyancy sections length, with the length ratio β between the buoyancy section region and the buckling region as the design parameter.

G. Li et al. / Thin-Walled Structures 107 (2016) 221–230

3. The lateral buckling control

k3 =

3.1. RSM of the analytical solution In general, the stability of solution for the nonlinear equation system is associated with the boundary conditions and external disturbance. It is found that the numerical instability exists when solving the nonlinear equations of (Eqs. (55) and 56) by numerical methods. Some unsuitable initial values may lead to physically non-plausible solutions that will be discussed in detail in the following numerical example. In order to correct the analytical solution and improve the computational efficiency, the response surface method (RSM) is performed in this section. RSM is a collection of mathematical and statistical techniques to be used in testing, modeling, data analysis and optimization [24]. A main application of RSM is modeling the relationship between the responses (output) and the parameters (input) to be a polynomial equation by using regression analysis. The residual error between the estimated and the actual responses is minimized by the least square technique. And then the corresponding coefficients of the polynomial equation can be obtained. For mode 3, when λL1 and λL2 are obtained from (Eqs. (55) and 56) with given the design parameters γ and β , the integral term S, vB(0), vB′′(0) can be written as

⎛ μ Ws ⎞2 S = C1(γ , β )L 27⎜ L ⎟ ⎝ EI ⎠

vB(0) = C2(γ , β )L 24

vB″(0) = C3(γ , β )L 22

(66)

μL Ws (67)

EI

μL Ws (68)

EI

where C1(γ , β ),C2(γ , β ),C3(γ , β ) are the constants only associated with the design parameters γ and β . Noting the difference of the analytical solutions between mode 1 and mode 3, the entire buckle length L is selected to construct the equation. Then the unified equations of mode-1 and mode 3 are obtained as follows:

⎡ P0 = P + k3μA WsL⎢ −1 + ⎢ ⎣

vmax = k 4L4

(75)

⎛ L ⎞4 k 4 = C2(γ , β )⎜ 2 ⎟ ⎝ 2L 1 ⎠

(76)

⎛ L ⎞2 k5 = C3(γ , β )⎜ 2 ⎟ ⎝ 2L 1 ⎠

(77)

L = 2L 1

(78)

It should be noted that Eq.(55) and Eq.(56) are dimensionless equations, and the coefficients k i(i = 1, 2, 3, 4, 5) are dependent only upon the design parameters γ and β . Then the design region of ( 0 ≤ γ ≤ 1, 0 ≤ β ≤ 1) is discretized into 2500 design points uniformly, with the increment of γ and β by 0.02 between the adjacent design points. The coefficients k i(i = 1, 2, 3, 4, 5) are calculated on these 2500 design points. Using these data, 5th degree polynomial RSM are fitted for the coefficients k i(i = 1, 2, 3, 4, 5). Due to the high nonlinear degree of the equation, it is difficult to establish a unified response surface with the required accuracy in the design region, such as the region of γ ≤ 0.3 and β ≥ 0.7. In this paper, the strategy of multi-response-surface is proposed, in which the entire design region is divided into several partial domains and different kinds of response surfaces can be formulated in these domains. The response surfaces of the adjacent domains share the boundary to ensure the continuity of the multi-response surfaces. Then based on the modified results, the expressions of k i(i = 1, 2, 3, 4, 5) are obtained in the different design domains by using the multi-response-surface (see Appendix). For the case of γ = 1, the RSM is equivalent to which proposed by Hobbs [11]. The RSM no longer needs to calculate the integral term S or solve the nonlinear equation system which composed of Eq.(55) and Eq.(56) (Eq.(22) for mode 1). So it improves calculation efficiency greatly, meanwhile, filters the numerical instability solutions caused by unsuitable initial values. 3.2. Framework of control for lateral buckling of subsea pipelines A framework for the lateral buckling control of subsea pipelines using distributed buoyancy sections is proposed, as illustrated in

(69)

μL Ws EI

Mmax = − k5L2μL Ws

P = k1

⎤ EAL5(μL Ws )2 ⎥ 1 + k2 μA Ws(EI )2 ⎥⎦

L2 2L 1

225

EI L2

(70)

(71)

(72)

where

k1 = (2λL1)2

(73)

⎛ L ⎞5 k2 = C1(γ , β )⎜ 2 ⎟ ⎝ 2L 1 ⎠

(74)

Fig. 4. The framework for controlling lateral buckling of the pipeline by using distributed buoyancy sections.

226

G. Li et al. / Thin-Walled Structures 107 (2016) 221–230

Fig. 4. Firstly, RSM is used to calculate the coefficients ki in the entire design region. And the axial load P0 is obtained according to the load and structure parameters. Then the length of the buckle L is solved from the lateral buckling Eq. (69) and the lateral buckling behavior is analyzed by (Eqs. (70)–72). Finally, the design parameters γ and β can be determined after the parameter’s study.

4. Numerical example In this section, an engineering example is employed to demonstrate the effectiveness of the proposed method for the control of lateral buckling of subsea pipeline with distributed buoyancy sections. The comparison analysis among the results from the analytical solution, RSM and finite element analysis (FEA) is performed. The parameters are illustrated in Table 1.

Fig. 5. The maximum errors of buckle lobe length between analytical solution corrected and RSM.

4.1. Comparison analysis between the analytical solution and RSM The camparison analysis will be performed on the 2500 design points described above. Noting (Eqs. (70)–72), the maximum bending moment Mmax, the maximum lateral placement vmax and the axial load in the buckle region P0 are all calculated by the length of buckle region L, thus the solutions of L are chosen to be compared for the sake of simplicity. The results show that the majority of analytical solutions are on a smooth curved surface, while some solutions are outside the surface even though they also satisfy the nonlinear (Eqs. (55) and 56). However, it is inconsistent with the physical essence obviously that the solution changes abruptly with the slight increments of the design parameters of 0.02. The instability in the solution of the nonlinear equations may lead to failure in design. RSM can filter the numerical instability successfully. As described in the previous section, the analytical solution is corrected by exhaustive method. The entire design region is divided into several partial domains according to the parameter γ . Then the maximum errors of different domains between the analytical solution corrected and RSM is shown in Fig. 5. It can be seen that the errors are far less than 0.05 in the region γ > 0.05, and γ > 0.1 is usually adopted on practical projects [12]. That means the accuracy of RSM is acceptable in engineering application.

Fig. 6. The design points selected for comparing analysis.

4.2. Comparison analysis between FEA and RSM A comparison analysis between FEA and RSM is performed in this section. The values of the parameters in Table1 are adopted in the FEA. 9 groups of design parameters are chosen in the closed design region, as shown in Fig. 6. Considering the symmetry of the lateral buckling configuration, half of the model is employed in FEA. The displacement of

Fig. 7. The FE model.

Table 1 Physical and mechanical parameters of the numerical example. Parameter

Value

Unit

Outside diameter Wall thickness Young Modulus Poisson's ratio Pipeline submerged weight Design temperature Thermal coefficient Axial friction factor Lateral friction factor

609.6 20.6 207 0.3 3303.52 50 0.0000117 0.4 0.5

mm mm GPa – KN/m °C 1/°C – –

Fig. 8. The lateral displacement along the pipeline of mode 1 of the design point 3.

G. Li et al. / Thin-Walled Structures 107 (2016) 221–230

Fig. 9. The lateral displacement along the pipeline of mode 3 of the design point 3.

the virtual anchor point is 0, which is subjected to axial load and the static friction force. However, the static friction force cannot be simulated accurately by FEM. In this paper, an infinite length pipeline is analyzed by RSM firstly. Then The FE model is built in the commercial software ABAQUS as shown in Fig. 7, in which the length of pipeline is the spacing between the anchor points (VAS) calculated by RSM above. Noting the FE model is half and symmetric, the end of the pipeline is fixed and the axial displacement of the middle point is constrained. Finally other parameters such as the length of the buckle lobe, the maximum bending moment at the buckle crown, etc. are compared. The pipeline is meshed by pipe31 elements, and the seabed is modeled by discrete rigid elements. The FEA is performed by the Riks method in ABAQUS/Standard. Figs. 8 and 9 illustrate the lateral displacements along the pipeline of design point 3 with γ ¼0.25 and β ¼0.75. Fig. 10 shows the errors of the buckle length L, the maximum lateral displacement vmax, the axial load in the buckle region P and the maximum bending moment at the buckle crown Mmax evaluated by RSM, compared with the results by FEA. It can be seen that the errors of L and Mmax of mode 1 are about 12%, and the other are less than 10%. The errors of mode1 are higher than mode 3 on the whole. Noting that the axial load in the buckle region P is assumed to be uniform in RSM, which is different form the result of FEA. Moreover, FEA can not excite the buckle of mode 1, and the buckling shape is mode 3 although the secondary lobe is small, as shown in Fig. 8. When the effects of the buoyancy section is significant, such as design point 3 ( γ = 0.25 and β = 0.75), the buckling behavior of mode 3 is similar to that of mode 1, shown in Fig. 9. In fact, the primary buckle length reaches 80% of the overall buckle lobe in the region of γ ≤ 0.3and 0.8 ≤ β . This explains why the errors of design point 3 are different from other design points, which is consistent to the result by Autunes [12] and Solano [25]. 4.3. Lateral buckling control analysis using RSM Distributed buoyancy technique decreases the critical buckling force by reducing submerged weight to achieve the aim of initiating lateral buckling at the planed site of the pipeline. It changes not only the critical buckling force but also the buckle shape, the mechanical properties, such as the buckle length, the maximum displacement at the buckle crown, etc. In order to initiate lateral buckling effectively, the critical buckling force must be decreased remarkably in the lateral buckling control design. At the same time, the buckling behavior should be fully considered. This example illustrates the framework for the lateral buckling control of subsea pipeline by using distributed buoyancy sections,

227

meanwhile, the influence of the design parameters on the behavior is discussed. The study case adopts the data presented in Table1. Noting that when γ = 1 or β = 0, the pipeline is without buoyancy sections, and the solution using RSM is equivalent to which proposed by Hobbs [11]. In this example, the critical buckling force, the length of the buckle lobe and the maximum bending moment are considered to determine the design parameters. The critical buckling force is a key factor of initiating lateral buckling as planed. It can be seen as the minimum value of P0 which makes the Eq. (69) has solution. The results of the critical buckling force show that the presence of distributed buoyancy reduces the critical buckling force compared to the solution proposed by Hobbs. With the decreasing of γ form 1-0 and the increasing of β form 0 to 1, the critical buckling force decreases increasingly. In the region where γ ≤ 0.4 and β ≥ 0.6, compare to the pipeline without buoyancy sections, the critical buckling force reduces to below 50% and the influence of buoyancy section is significant. The length of buckle lobe is very important for lateral buckling control, which changes obviously in the region of γ ≤ 0.4 and β ≥ 0.6. The buoyancy section increases the length of buckle, for example, more than 300 m in the region of γ ≤ 0.2 and β ≥ 0.8. And the influence of design parameters increases remarkably with the decreasing of γ and the increasing of β . At buckle lobes, the lateral resistance leads to the high bending moments with the highest value at buckle crown. The decrease of maximum bending moment is beneficial to increasing the tolerable VAS and buckle formation reliability [12]. The results of the example show that the small length of the buoyancy sections ( β ≤ 0.3 for mode 3 and β ≤ 0.2 for mode 1) lead to a negative effect on the maximum bending moment at the buckle crown. The buoyancy concentring at a short region leads to a bigger curvature and higher bending moment at the buckle crown than Hobbs solution. This behavior coincides with that described by Autunes [12]. In the region where 0.1 ≤ γ ≤ 0.4 and 0.5 ≤ β ≤ 1 the maximum bending moment is reduced significantly by the buoyancy sections. Based on the above results, the reasonable design region is 0.2 ≤ γ ≤ 0.4 and 0.6 ≤ β ≤ 0.8, in which the critical buckling force and the maximum bending moment decrease remarkably and the length of lateral buckling is acceptable, compared to those of the pipeline without buoyancy sections.

5. Conclusions An analytical solution for the lateral buckling of the subsea pipeline with the buoyancy sections is presented in this paper. In order to overcome the numerical instability solutions of the nonlinear equations, a response surface model (RSM) is adopted in the entire closed design region, based on which, a framework for the lateral buckling control using distributed buoyancy sections is performed. And a numerical example is employed in this study. The comparison analysis between analytical solution and RSM demonstrates that the numerical instability can be overcome effectively by RSM, and the error between them is very small. The comparison analysis between RSM and FEA indicates that it is difficult to simulate the mode 1 lateral buckling of subsea pipeline by FEA accurately. Indeed, the buckle shape in FEA is mode 3. And the referred analysis shows that when the influence of buoyancy is remarkable, the mode 3 lateral buckling is similar to mode 1. The lateral buckling control analysis verifies the efficiency of the framework proposed in this paper. The analysis demonstrates that the design parameters of the buoyancy sections can be

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G. Li et al. / Thin-Walled Structures 107 (2016) 221–230

Fig. 10. Errors between the RSM and the FEA. (a) mode 1. (b) mode 3.

determined efficiently in the preliminary design phase by using the RSM and the framework proposed in this paper. The seabed is assumed to be rigid in this study. The influence of interaction between pipeline and soil and the plasticity model are still under development.

⎧ ⎪ ω{ [ 2 cos(λL1) + (γ − 1)cos(βλL 2)]sin(λL 2)} ⎨ A1 = − ⎪ λ4 sin(λL 2) ⎩ +

ω ⎡⎣ 2 sin(λL1) + (γ − 1)sin(βλL1)⎤⎦cos(λL 2) − [ (γ − 1)β + 2]λL1 + λL 2

{

}

λ4 sin(λL 2)

A2 = 0 A3 = 0

{ [ (γ − 1)β(β − 2) − 1](λL )

Acknowledgments

ω A4 = −

The authors thank the support provided by the National Basic Research Program of China (2014CB046803). Thank Dr. Liu Xiang for fruitful discussions. Thank Professor Bo Ping Wang of University of Texas at Arlington for English revision.

+ +

A5 =

}

A6 =

A4 =

A5 = − A6 =

A7 = −

A8 =

λ4 ωβL1(γ − 1)

2λ4 sin(λL 2)

}

ω{ +2 cos(λL1)sin(λL 2) + (γ − 1)βλL1 + 2λL1 − λL 2} λ4 sin(λL 2)

{ [ −2(γ − 1)β − 1]( λL ) sin(λL ) + 2(γ − 1)sin(βλL )cos(λL − λL )} 2

A9 =

A12 =

2

1

1

2

ω{ 4 sin(λL1 − λL 2)cos(λL 2) − 2[ (γ − 1)βλL1 + 2λL1 − λL 2]cos(λL1)} 2λ4 sin(λL 2)

ω ⎣⎡ 2 sin(λL1) + (γ − 1)sin(βλL1)⎤⎦cos(λL 2) − [ (γ − 1)β + 2]λL1 + λL 2

{

}

λ4 sin(λL 2) ω⎡⎣ (γ − 1)sin(βλL1) + 2 sin(λL1)⎦⎤ λ4 ω(γβ − β + 2)L1

A11 = −

4

1

2λ4 sin(λL 2) −

A10 =

2λ

ω{ 2(2λL1 − λL 2)cos(λL1) + 2(γ − 1)sin(λL 2)}

λ2

λ2

2 2 ⎡ ⎤ ω⎣⎢ 2β ( λL1) ( γ − 1) + ( λL1) + 2⎥⎦

2λ4 sin(λL 2)

ω

4

λ cos(λL1)

ω{ 2(γ − 1)cos(λL1)cos(λL 2)sin(βλL1)2(γ − 1)βλL1 cos(λL1)}

λ4 ωβL1(γ − 1)

A8 = −

2λ4 ω⎡⎣ (γ − 1)sin( βλL1)sin(λL1) + 1⎤⎦

ω(γ − 1)sin( βλL1)

}

2λ4 sin(λL 2)

ω(γ − 1)sin( βλL1)

A3 = 0 2 ⎡ ⎤ ω⎣⎢ ( λL1) ( − γβ 2 + β 2 + 2γβ − 2β + 1) + 2γ ⎥⎦

{

λ4 sin(λL 2)

A7 = −

A2 = 0

ω 4⎡⎣ sin(λL 2 − λL1)⎤⎦cos(λL1) − 2(γ − 1)sin(βλL1)sin(λL1)sin(λL 2)

{

−

{

}

sin(λL 2)

ω ⎡⎣ 2 sin(λL1) + (γ − 1)sin(βλL1)⎤⎦cos(λL 2)

A-1. Functions A1~A8 for mode 1 (used in (Eqs. (7) and 8)) ⎧ ⎪ ω (γ − 1)⎡⎣ (cos(βλL1)cos(λL1) + sin(βλL1)sin(λL1)⎤⎦ + 1 ⎨ A1 = − ⎪ λ4 cos(λL1) ⎩

2

2λ sin(λL 2)

+

Appendix

1

4

λ2

{

}

ω 2[ (γ − 1)β + 2]λL1λL 2 sin(λL 2) − (λL 2)2 sin(λL 2) 2λ4 sin(λL 2)

A-2. Functions A1~A12 for mode 3 (used in (Eqs. (36) to 38)) +

ω 2⎡⎣ ( γβ − β + 2)λL1 − λL 2⎤⎦cos(λL 2) − 4 sin(λL1) − 2(γ − 1)sin(βλL1)

{

}

2λ4 sin(λL 2)

G. Li et al. / Thin-Walled Structures 107 (2016) 221–230

A-3. Functions k i(i = 1, 2, 3, 4, 5) for mode 1.

⎧ 1 1 1 2 ⎪ 8.087 × 10 − 8.21γ − 4.533 × 10 β + 6.868 × 10 γ ⎪ 2 1 2 3 2 ⎪ +1.888 × 10 γβ − 5.597 × 10 β − 1.220 × 10 γ β ⎪ ⎪ −4.858 × 10 2γβ 2 + 2.126 × 10 2β 3 + 5.053 × 10 3γ 2β 2 ⎪ ⎪ +4.762 × 10 2γβ 3 − 3.182 × 10 2β 4 − 6.123 × 10 3γ 2β 3 ⎪ ⎪ +4.648 × 10 2γβ 4 + 1.822 × 10 2β 5 0< γ ≤ 0.1, 0 < β ≤ 0.8; ⎪ ⎪ 6.5 × 101 + 4.6γ + 9.59 × 101β + 2.817 × 10 3γ 2 ⎪ ⎪ +1.788 × 10 2γβ − 3.928 × 10 2β 2 + 1.549 × 104β 3 ⎪ ⎪ 3 2 2 2 2 3 0< γ ≤ 0.1, 0.8 < β ≤ 0.9; ⎪ −8.608 × 10 γ β + 5.205 × 10 γβ + 2.98 × 10 β ⎪ 2 5 2 4 ⎪ 7.0 × 10 + 3γ + 6β + 2.57 × 10 γ − 9.815 × 10 γβ ⎪ 3 2 3 2 5 2 ⎪ +8.037 × 10 β − 7.212 × 10 γ β + 3.067 × 10 γβ ⎪ 4 3 5 2 2 5 3 ⎪ −2.685 × 10 β + 6.356 × 10 γ β − 3.133 × 10 γβ ⎪ ⎪ +2.942 × 104β 4 − 1.713 × 105γ 2β 3 + 1.047 × 105γβ 4 ⎪ ⎪ −1.06 × 104β 5 0< γ ≤ 0.1, 0.9 < β ≤ 1; ⎪ ⎪ 8.022 × 101 + 7.692γ − 4.156 × 101β − 2.532 × 101γ 2 ⎪ ⎪ ⎪ −4.277 × 101nβ − 9.131β 2 + 2.346 × 101γ 3 + 3.091 × 10 2γ 2β k1 = ⎨ ⎪ +2.078 × 10 2γβ 2 − 7.37 × 101β 3 − 2.747 × 10 2γ 3β ⎪ ⎪ −7.747 × 10 2γ 2β 2 + 3.027 × 10 2γβ 3 + 1.488 × 10 2β 4 ⎪ ⎪ 2 3 2 1 2 3 2 4 ⎪ +6.009 × 10 γ β + 2.114 × 10 γ β − 3.179 × 10 γβ ⎪ 1 5 ⎪ −2.938 × 10 β 0.1 < γ ≤ 0.5, 0 < β ≤ 0.8; ⎪ 1 −1 −1 2 ⎪ 8.0 × 10 + 9.8 × 10 γ + 2.2β − 3 × 10 γ − 1γβ ⎪ ⎪ −4.671 × 10 2β 2 + 6.339 × 10 3γ 3 − 9.357 × 10 3γ 2β ⎪ ⎪ +5.295 × 10 3γβ 2 + 2.254 × 10 2β 3 − 3.477 × 10 3γ 4 ⎪ ⎪ −7.423 × 10 3γ 3β + 1.58e × 104γ 2β 2 − 9.909 × 10 3γβ 3 ⎪ ⎪ +8.935 × 10 2β 4 + 3.515 × 10 3γ 4β + 1.028 × 10 3γ 3β 2 ⎪ ⎪ −6.413 × 10 3γ 2β 3 + 4.607 × 10 3γβ 4 − 6.526 × 10 2β 50.1 < γ ≤ 0.5, 0.8 < β ≤ 1; ⎪ ⎪ 1 1 −1 ⎪ 8.076 × 10 − 4.966 × 10 γ − 3.774 × 10 β ⎪ 1 2 2 −1 2 ⎪ +6.102 × 10 γ + 3.763 × 10 γβ − 1.095 × 10 β ⎪ 2β + 1.905 × 10 2γβ 2 + 2.913 × 10 2β 3 1.733 γ − ⎪ ⎪ ⎪ −7.194 × 101γ 2β 2 − 3.863 × 10 2γβ 3 − 1.378 × 10 2β 4 ⎪ 1 2 3 2 4 5 ⎪ ⎩ +7.729 × 10 γ β + 1.521 × 10 γβ − 3.721β 0.5 < γ ≤ 1, 0 < β ≤ 1

k3 = 0.5

229

0 < γ ≤ 1, 0 < β ≤ 1

k 4 = 2.365 × 10−3 + 5.818 × 10−5γ − 1.452 × 10−3β + 1.211 × 10−3γβ − 3.43β 2 + 4.454γβ 2 + 1.668β 3 − 3.298 × 10−3γβ 3 + 8.461 × 10−4 β 4

0<γ

≤ 1, 0 < β ≤ 1

⎧ 6.937 × 10−2 − 7.283 × 10−4 γ − 2.89 × 10−2β ⎪ ⎪ + 1.733 × 10−3γ 2 ⎪ ⎪ +3.945 × 10−2γβ − 1.652 × 10−1β 2 − 2.805 × 10−2γ 2β ⎪ ⎪ +1.56 × 10−1γβ 2 + 8.184 × 10−2β 3 + 9.021 × 10−2γ 2β 2 ⎪ −1 3 −1 4 −2 2 3 ⎪ −1.72 × 10 γβ + 1.008 × 10 β − 6.359 × 10 γ β ⎪ +4.703 × 10−2γβ 4 − 5.794 × 10−2β 5 0 ⎪ ⎪ k5 = ⎨ < γ ≤ 0.4; 0 < β ≤ 1; ⎪ ⎪ 6.886 × 10−2 + 7.42 × 10−4 γ − 1.934 × 10−2β ⎪ −2 ⎪ + 1.456 × 10 γβ ⎪ −1 2 −1 2 −1 3 ⎪ −2.059 × 10 β + 2.377 × 10 γβ + 1.452 × 10 β ⎪ −1 3 −2 4 −2 4 ⎪ −2.309 × 10 γβ + 5.157 × 10 β + 4.664 × 10 γβ ⎪ −3.986 × 10−2β 5 0.4 ⎪ ⎪ ⎩ < γ ≤ 1; 0 < β ≤ 1

A-4. Functions k i(i = 1, 2, 3, 4, 5) for mode 3.

k2 ⎧ 6.385 × 10−5 + 1.095 × 10−7γ − 1.131 × 10−4 β ⎪ ⎪ + 6.286 × 10−7γ 2 ⎪ ⎪ +1.069 × 10−4 γβ + 3.401 × 10−5β 2 − 1.051 × 10−5γ 2β ⎪ ⎪ +6.845 × 10−6γβ 2 − 1.936 × 10−4 β 3 + 8.114 × 10−5γ 2β 2 ⎪ −4 3 −4 4 −5 2 3 ⎪ −2.207 × 10 γβ + 4.44 × 10 β + 4.222 × 10 γ β ⎪ −5 4 −4 5 0< γ ⎪ +4.791 × 10 γβ − 2.308 × 10 β ⎪ ≤ 0.4, 0 < β ≤ 0.6; ⎪ ⎪ 5.893 × 10−5 − 1.468 × 10−4 γ − 4.04 × 10−6β ⎪ ⎪ + 1.806 × 10−5γ 2 ⎪ 4 − = ⎨ +9.351 × 10 γβ − 5.205 × 10−4 β 2 − 1.762 × 10−4 γ 2β ⎪ ⎪ −1.638 × 10−3γβ 2 + 9.86 × 10−4 β 3 + 4.978 × 10−4 γ 2β 2 ⎪ −3 3 −4 4 −4 2 3 ⎪ +1.052 × 10 γβ − 6.882 × 10 β − 2.76 × 10 γ β ⎪ −4 4 −4 5 0< γ ⎪ −2.026 × 10 γβ + 1.678 × 10 β ⎪ ≤ < ≤ β 0.4, 0.6 1; ⎪ ⎪ 6.55 × 10−5 − 6.537 × 10−6γ − 1.309 × 10−4 β + 5.87 × 10−6γ 2 ⎪ ⎪ +1.966 × 10−4 γβ + 5.062 × 10−5β 2 − 8.254 × 10−5γ 2β ⎪ ⎪ −2.83 × 10−4 γβ 2 − 4.726 × 10−5β 3 + 3.19 × 10−4 γ 2β 2 ⎪ −5 3 −4 4 −4 2 3 ⎪ +3.772 × 10 γβ + 1.251 × 10 β − 1.764 × 10 γ β ⎪ −5 4 −5 5 ⎩ +5.176 × 10 γβ − 6.175 × 10 β 0.4 < γ ≤ 1, 0 < β ≤ 1

⎧ 3.401 × 101 + 2.19γ + 3.207 × 101β − 1.135 × 101γ 2 ⎪ ⎪ −5.676 × 101γβ − 3.145 × 101β 2 + 1.386 × 102γ 2β ⎪ ⎪ +1.204 × 102nβ 2 − 2.927 × 101β 3 − 3.719 × 102γ 2β 2 ⎪ 0< γ ⎪ +6.892 × 101γβ 3 + 4.487 × 101β 4 ⎪ ≤ 0.2; 0 < β ≤ 0.7; ⎪ ⎪ 2 3.256 10 1.489 − × + × 103γ + 1.763 × 103β ⎪ ⎪ +1.149 × 104 γ 2 − 1.003 × 104 γβ − 2.729 × 103β 2 ⎪ ⎪ −1.526 × 102γ 3 − 2.818 × 104 γ 2β + 1.813 × 104 γβ 2 ⎪ k1 = ⎨ +1.372 × 103β 3 − 7.923 × 103γ 4 + 6.182 × 103γ 3β ⎪ ⎪ +1.521 × 104 γ 2β 2 − 9.531 × 103γβ 3 0< γ ⎪ ≤ 0.2; 0.7 < β ≤ 1; ⎪ ⎪ 1 3.425 10 3.647 × − γ + 3.999 × 101β + 1.072 × 101γ 2 ⎪ ⎪ 1 1 2 3 ⎪ −4.268 × 10 γβ − 8.767 × 10 β − 7.545γ ⎪ −5.136 × 101γ 2β + 2.836 × 102γβ 2 + 3.227 × 101β 3 ⎪ ⎪ +5.594 × 101γ 3β − 1.432 × 102γ 2β 2 − 2.642 × 102γβ 3 ⎪ ⎪ +8.829 × 101β 4 − 4.57 × 101γ 3β 2 + 1.873 × 102γ 2β 3 ⎪ 1 4 1 5 ⎩ −2.351 × 10 γβ − 2.88 × 10 β 0.2 < γ ≤ 1; 0 < β ≤ 1

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⎧ 1.658 × 10−4 + 6.041 × 10−6γ − 5.577 × 10−4 β ⎪ ⎪ − 3.73 × 10−6γ 2 ⎪ ⎪ +4.04 × 10−4 γβ + 9.194 × 10−4 β 2 + 1.345 × 10−4 γ 2β ⎪ ⎪ −8.885 × 10−4 γβ 2 − 9.845 × 10−4 β 3 − 1.325 × 10−5γ 2β 2 ⎪ −4 3 −4 4 0< γ ⎪ +5.485 × 10 γβ + 4.853 × 10 β ⎪ ≤ < ≤ 0.4; 0 β 0.6; ⎪ ⎪ 1.38 × 10−4 − 3.361 × 10−5γ − 2.907 × 10−4 β ⎪ ⎪ − 1.406 × 10−4 γ 2 ⎪ ⎪ +5.785 × 10−4 γβ + 2.707 × 10−5β 2 − 1.141 × 10−5γ 3 k2 = ⎨ ⎪ +4.993 × 10−4 γ 2β − 1.014 × 10−3γβ 2 + 2.638 × 10−4 β 3 ⎪ ⎪ +7.183 × 10−5γ 3β − 3.004 × 10−4 γ 2β 2 + 4.682 × 10−4 γβ 3 ⎪ −4 4 0< γ ≤ 0.4; 0.6 < β ≤ 1; ⎪ −1.381 × 10 β ⎪ −4 −4 −4 ⎪ 1.839 × 10 − 1.04 × 10 γ − 5.639 × 10 β −4 2 ⎪ + 1.823 × 10 γ ⎪ ⎪ +8.587 × 10−4 γβ + 4.484 × 10−4 β 2 − 9.613 × 10−5γ 3 ⎪ ⎪ −8.226 × 10−4 γ 2β − 5.653 × 10−4 γβ 2 − 1.587 × 10−4 β 3 ⎪ −4 3 −4 2 2 −5 3 ⎪ +5.28 × 10 γ β + 1.518 × 10 γ β + 7.61 × 10 γβ ⎪ −5 4 ⎩ +4.779 × 10 β 0.4 < γ ≤ 1; 0 < β ≤ 1

k3 = 1.291 − 9.634 × 10−3γ − 1.11β + 7.375 × 10−2γ 2 + 0.997γβ + 0.5287β 2 − 9.34 × 10−2γ 3 − 4.851 × 10−2γ 2β − 0.5716γβ 2 − 0.2089β 3 + 2.928 × 10−2γ 4 + 0.1915γ 3β − 1.958 × 10−2γ 2β 2 + 0.2439γβ 3

0<γ

≤ 1; 0 < β ≤ 1

k 4 = 1.024 × 10−2 + 4.254 × 10−4 γ − 2.247 × 10−2β − 1.353 × 10−3γ 2 + 1.872 × 10−2γβ + 2.092 × 10−2β 2 + 184 × 10−3γ 3 + 8.675 × 10−3γ 2β − 3.01 × 10−2γβ 2 − 1.689 × 10−2β 3 − 8.821 × 10−4 γ 4 − 8.46 × 10−3γ 3β + 2.169 × 10−2γ 2β 2 + 2.026 × 10−2γβ 3 + 9.496 × 10−3β 4 + 3.59 × 10−3γ 4β + 7.415 × 10−3γ 3β 2 − 5.299 × 10−3γ 2β 3 − 6.726 × 10−3γβ 4 − 1.312 × 10−3β 5

0<γ

≤ 1; 0 < β ≤ 1 k5 = − 0.1432 + 5.202 × 10−3γ + 0.1476β − 1.52 × 10−2γ 2 − 0.1724γβ + 0.1438β 2 + 1.423 × 10−2γ 3 + 4.85 × 10−2γ 2β − 5.326 × 10−2γβ 2 − 0.1483β 3 − 4.405 × 10−3γ 4 − 2.279 × 10−2γ 3β − 9.552 × 10−2γ 2β 2 + 0.153γβ 3 0 < γ ≤ 1; 0 < β ≤ 1

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