Prediction of the upheaval buckling critical force for imperfect submarine pipelines

Ocean Engineering 109 (2015) 330–343

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Prediction of the upheaval buckling critical force for imperfect submarine pipelines Xinhu Zhang a,b,n, Menglan Duan a,b a b

College of Mechanical and Transportation Engineering, China University of Petroleum-Beijing, Beijing 102249, China Offshore Oil and Gas Research Center, China University of Petroleum-Beijing, Beijing 102249, China

art ic l e i nf o

a b s t r a c t

Article history: Received 16 June 2015 Accepted 14 September 2015

Upheaval buckling behavior of submarine pipelines under high temperature and high pressure conditions is a primary concern for structural integrity. The critical axial force is a key factor governing the buckling behavior. There have been already some formulas to calculate critical axial force for some particular initial imperfection shapes. However, there is no universal formula to express the effects of initial imperfection shape and Out-of-Straight (OOS) on the critical axial force. In this paper, the upheaval buckling behaviors of eight groups of pipeline segments with different imperfection shapes and different OOS have been studied using the finite element method. A new parameter is defined to express the differences of imperfection shapes. An approximation and universal formula is proposed to calculate the critical axial force which covers the new parameter and the OOS of pipeline. A case study is presented which illustrates the application of the formula. Finally, comparison between this study and previous research results is conducted, and it manifests that this formula has a greater precision. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Upheaval buckling Critical axial force Imperfection Curvature radius

1. Introduction The safe running of submarine pipelines has been paid more attention especially with the development of ocean petroleum industry in recent years. While for submarine pipelines, one of the key issues in engineering design is the potential for global instability-buckling. When a submarine pipeline is operated at a high temperature and high pressure conditions, it will try to expand. In general, the pipeline is not free to expand because of being axially restrained, for instance by the friction of the surrounding soil. So an axial compressive force is produced in pipeline. If the force exerted by pipeline on the soil exceeds the vertical restraint against uplift movement created by the pipeline's submerged weight, its bending stiffness and the resistance of the soil cover, the pipeline will tend to move upward and considerable vertical displacements may occur (Palmer et al., 1990). This phenomenon is called upheaval buckling. This buckling mode may lead to the final failure such as fatigue or fracture (Det Norske Veritas, 2007). Then, it will bring disastrous results to marine creature, marine environment and human. Many researchers have investigated the upheaval buckling problem. Allan (1968) conducted analytical and experimental n Corresponding author at: Offshore Oil and Gas Research Center, China University of Petroleum-Beijing, Beijing, China. Tel.: þ 86 13811754215. E-mail address: [email protected] (X. Zhang).

http://dx.doi.org/10.1016/j.oceaneng.2015.09.022 0029-8018/& 2015 Elsevier Ltd. All rights reserved.

studies on upheaval buckling of an axially compressed frictionless strip. He found the sensitivity of the buckling problem to initial imperfections. Hobbs (1984) analyzed the upheaval and lateral buckling of submarine pipeline on the basis of related work on railroad track. He assumed the pipeline is an ideal straight and perfect elasticity pipe with a small slope when it reached critical buckling condition. So the governing equation is a second-order differential equation. By solving the governing equation he obtained a theoretical solution for the critical force. He proposed that for normal coefficients of friction, the lateral buckling occurs at a lower axial load than upheaval buckling and is dominant in pipeline unless the pipeline is trenched or buried. At last he pointed out that the initial out-of-straightness (OOS) of pipeline should be taken into account in analysis of upheaval buckling of submarine pipeline. A series of studies including some theoretical analysis and experiments on upheaval and lateral buckling problem of submarine pipelines were conducted by Taylor and Gan (1986, 1987), Taylor and Tran (1993, 1996). They focused on the effects of structural imperfections and deformation-dependent axial friction resistance on submarine pipelines buckling. For initial imperfection, they proposed three theoretical imperfection models, analyzed their characteristics of buckling and predicted the critical axial forces. They pointed out that the amplitude and wave length are very important parameters for buckling of submarine pipeline and the two parameters are covered in their equations. Ju and Kyriakides (1988) studied the effect of localized and small initial geometric imperfection on the response and

X. Zhang, M. Duan / Ocean Engineering 109 (2015) 330–343

stability of pipeline. They pointed out that the critical force is sensitive to form and magnitude of the imperfections as well as the pipe material inelastic characteristics. As for critical temperature, they believed that it is dependent on the curvature of the imperfection at the horizontal position at which it occurs. Subsequently, Richards (1990) studied the effect of imperfection shape on upheaval buckling behavior. He obtained the conclusion again that the critical force is sensitive to imperfection shape of pipeline. Palmer et al. (1990) studied upheaval buckling problem of pipeline. The initial imperfection of pipeline is expressed by two parameters height and wave length. They believed the pipeline follows from elementary beam-column theory and the initial specific shape only affects the coefficients and not the general form of the equation. By defining two dimensionless quantities they obtained a semi-empirical design method. Maltby and Calladine (1995) studied the upheaval buckling of buried pipeline by experiments and theoretical analysis. They presented a formula for predicting the critical force of buried pipeline which involves three parameters-the flexural stiffness of the pipe, the initial imperfection amplitude of the pipe and the “plateau” value of the soil resistance curve. Also, they pointed out initial uplift temperature is inversely proportional to the curvature of initial imperfection at x ¼0. Croll (1997) provided a simple means for predicting the nonlinear response displayed by geometrically imperfect pipelines. Also, he presented two equations for predicting the critical axial force aimed at two initial imperfection models that Taylor presented. The two equations are similar to those presented by Taylor. After that, Wang et al. (2011a, 2011b), Shi et al. (2013), Liu et al. (2013, 2014), Zhao and Feng (2015) also studied this problem. But they all did not provide an explicit and universal formula to express the effects of initial imperfection shape, wave length, and amplitude on the critical axial force of pipeline. Taking into account the imperfection out-of-straightness as a whole, Zeng et al. (2014) presented a new formula of critical force based on dimensional analysis. The formula is P L ¼ gðw0 =L0 Þðq2 EIÞ1=3

ð1Þ

He defined a coefficient function gðw0 =L0 Þ to express the effect of initial imperfection on the critical force. Here w0 =L0 represents the OOS of pipeline. He pointed out that different initial imperfection shapes correspond to different coefficient values. Specific to three different initial imperfections, he obtained three formulas by finitely element analysis. Karampour et al. (2013) provided some analytical solutions based on a long heavy elastic beam resting on a rigid frictional foundation. The shape influence on the critical force was verified again in his work. Their works made the effect of initial imperfection on critical force more explicit. But they did not explain why different initial imperfection shapes correspond to different critical forces. And they did not give a united formula to express the effect of the initial perfection shape of pipeline on the critical axial force. Many researchers found that the pipe/soil interaction characteristics also affect the upheaval buckling behavior of submarine pipeline (Ellinas et al., 1990; Schaminee and Zorn, 1990; Palmer, 2003; Newson and Deljoui, 2006; Cheuk et al., 2007; Merified et al., 2008). The uplift resistance and the mobilization distance are two important parameters which affect the critical axial. Nonetheless, the current study mainly discusses the effect of initial imperfection on critical force. Previous researches have shown that both initial imperfection and pipe/soil interaction characteristics affect the upheaval buckling behavior. For initial imperfection, the critical axial force of upheaval buckling is sensitive to the imperfection shape and OOS of pipeline. Although many researchers have presented some formulas to calculate critical axial force for some particular initial imperfect shapes, there is no united formula to express the effects

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of the initial perfection shape and OOS of pipelines on the upheaval buckling critical axial force. This paper focuses on the effect of initial imperfection on the critical axial force. In this paper, eight different initial imperfections are analyzed using the finite element software ABAQUS. The reason why different initial imperfection shapes have different critical values is presented. A new parameter is defined to express the differences of initial imperfection shapes. Finally, based on the works of previous researchers especially Zeng et al. (2014) and Karampour et al. (2013), a simple and united formula to calculate critical axial force is presented which covers both the effects of initial imperfection shape and the OOS. It can be used to predict the critical axial force for any known initial imperfection.

2. Analytical model According to the research of Taylor and Tran (1993), an initial configuration of a submarine pipeline can be illustrated in Fig. 1. The pipeline is laid on an uneven sea bed. The soil of the seabed is very hard, so the sea bed can be treated as a rigid foundation. It is assumed that the system is symmetric on the w axis, as shown in Fig. 1. A pipeline segment with a length of L is selected for research. The pipeline segment has an imperfection due to the uneven sea bed. L0 denotes the wave length of imperfection. And w0 denotes the maximum height of imperfection. Because of downward loads, for instances of the pipeline submerged selfweight and covering soil, the pipeline is closely in contact with the rigid foundation. The axial force P caused by temperature is (Hobbs, 1984) P ¼ EAαT

ð2Þ

The axial force P caused by internal pressure is (Hobbs, 1984) P¼

ApD ð0:5  υÞ 2t

ð3Þ

where parameters A, D and t denote pipeline cross-section area, external diameter and thickness, respectively. E and υ denote Young's Modulus and Poisson ratio of pipeline material, respectively. T, α and p denote temperature change, thermal expansion coefficient of pipeline and internal pressure, respectively. It is assumed that the initial imperfections have some forms as follows. 2.1. Initial imperfection forms Based on elementary beam-column theory of idealized pipeline (Palmer et al., 1990), the equilibrium equation is 4

EI

2

d w d w þP 2 þ q ¼ 0 dx4 dx

ð4Þ

Tore Soreide et al. (2005) pointed out that the equation has a general solution wðxÞ ¼ A0 þ A1 cos ðkxÞ 

q Ux2 2P

Fig. 1. Analytical model.

ð5Þ

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so it is rational that the functions of the initial imperfections are composed of polynomial functions or sinusoidal functions or both. Actually, those functions of initial imperfections adopted by Taylor and Tran (1993), Karampour et al. (2013), and Zeng et al. (2014) are all in this list. In this paper the following eight initial imperfections are assumed in the upheaval buckling analysis. Among them three are adopted by Karampour et al. (2013) and Zeng et al. (2014). They are 8 8 2x 2 2x 2x 3 >  L20 r x o 0 < w0 ð3ð L0 Þ  3 L0 þ 1Þð1 þ L0 Þ f 1 ðxÞ ¼ ð10 Þ 8 2x 2 2x 2x 3 > 0 r x r L20 : w0 ð3ð L0 Þ þ3 L0 þ1Þð1  L0 Þ f 2 ðxÞ ¼ w20 ð1 þ cos ð2Lπ0xÞ

f 3 ðxÞ ¼

 L20 r x r

L0 2

8 4 2x 2x > <  w0 ð4 L0 1Þð L0 þ 1Þ

 L20 rx o0

> :

0 r x o L20

4 2x w0 ð42x L0 þ1Þð L0  1Þ

ð20 Þ Fig. 2. Profiles of eight different initial imperfections.

ð30 Þ

Two initial imperfections are assumed and meet the following conditions: 8 2 df d f < x ¼ 0; f ðxÞ ¼ w0 ; dx ¼ 0; dx 2 ¼ 0 : x ¼ 7 L0 ; 2

f ðxÞ ¼ 0;

df ¼ dx

0

The two functions of initial imperfections could be 8 3 3 2x 5 < w0 ð1 þ 52ð2x  L20 r x o 0 L0 Þ  2ð L0 Þ Þ f 4 ðxÞ ¼ 3 3 2x 5 : w0 ð1  52ð2x 0 r x o L20 L0 Þ þ 2ð L0 Þ Þ

f 5 ðxÞ ¼

8 2x 3 2x 4 > < w0 ð1 þ 4ð L0 Þ þ 3ð L0 Þ Þ 2x 3 2x 4 > : w0 ð1  4ð L0 Þ þ 3ð L0 Þ Þ

ð40 Þ

f ðxÞ ¼ 0;

df ¼ dx

 L20 x r L20

0

 L20 x r L20

ð8Þ 02

1 ð1 þf Þ3=2 ¼ KðxÞ f″

ð9Þ

ð60 Þ

ð70 Þ

In the following, the Finite Element Method is been used to study the relationship between different upheaval buckling behaviors and different initial imperfections.

ð80 Þ

The profiles of these eight different initial imperfections are as shown in Fig. 2. In Fig. 2, these eight different initial imperfections from No. 1 to No. 8 have same maximum height w0 and wave length L0. Set δ denotes OOS. It is (Zeng et al., 2014)

δ ¼ w0 =L0

f″ ð1 þ f 02 Þ3=2

If ρ(x) have a negative value it denotes that the curvature center of that position is under the profile of imperfection. In this way, we have two parameters to express the initial imperfection. So the coefficient g in Eq. (1) (Zeng et al., 2014) is a function which contains two independent parameters. It can be written as   w0 ρðxÞ g ¼ gðδ; ϕÞ ¼ g ; ð10Þ L0 L0

The eighth function of initial imperfection is also from the research of Karampour et al. (2013). It is 2 2

ð7Þ

where K(x) and ρ(x) denote curvature and curvature radius at the position x on a specific initial imperfection, respectively. But here, the curvature and curvature radius have a little difference from their general definition. They are allowed to be negative values. They are

ρðxÞ ¼

The two functions of initial imperfections could be 8 L0 2x 2 2x 3 > < w0 ð1  3ð L0 Þ  2ð L0 Þ Þ  2 r x o 0 f 6 ðxÞ ¼ L 2x 2 2x 3 > : w0 ð1  3ð L0 Þ þ 2ð L0 Þ Þ 0 r x r 20

πx f 8 ðxÞ ¼ w0 ð0:707  0:2617πL2x þ 0:293 cos ð2:86 L0 ÞÞ

1 ρðxÞ ¼ KðxÞL0 L0

ð50 Þ

0 r x r L20

0

2 2x 4 f 7 ðxÞ ¼ w0 ð1 2ð2x L0 Þ þ ð L0 Þ Þ

ϕ¼

KðxÞ ¼

 L20 r x o 0

Two initial imperfections are assumed and meet the following conditions: 8 df < x ¼ 0; f ðxÞ ¼ w0 ; dx ¼0 : x ¼ 7 L20 ;

researches (Ju and Kyriakides, 1988; Maltby and Calladine, 1995; Karampour et al., 2013) manifested that the temperature rise required for initial uplift or critical force is related to the curvature of the imperfection at x¼ 0. Here we define a dimensionless parameter to depict their differences. It is

ð6Þ

so these different initial imperfections of pipeline segment have same OOS. According to Zeng et al. (2014), those three imperfections (from No. 1 to No. 3) have different critical axial forces even when they have same OOS. From Fig. 2, we can see these eight initial imperfections all have different heights except points A–C. Besides, they have different curvature at x. Some

2.2. Finite element modeling According to the research of Palmer et al. (1990), the finite element analysis is a convenient and effective tool for simulation of upheaval buckling and the results can be used to obtain the approximation function. In this paper, we used the uplifting of a heavy beam on rigid foundation (Terndrup Pedersen and Juncher Jensen, 1988) to model the upheaval buckling. 2.2.1. Pipeline We select a pipeline segment with its length 200 m, external diameter 0.457 m, and thickness 0.0143 m. The wave lengths of those eight different initial imperfections above have the same value 100 m. The maximum heights of those eight different initial imperfections above are all same and vary from 0.1 m to 1.0 m with increment of 0.1 m. So the OOS values also vary from 1/1000 to 1/100 with the maximum heights. API X-65 steel is used for the

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pipeline with yield stress 448 Mpa, Young's modulus 207 Gpa and Poisson’s ratio 0.3. It is modeled by Hooke's law of elasticity theory and J2 flow theory of plasticity associated with Mises yielding criteria and isotropic hardening law (An et al., 2012). The thermal expansion coefficient of steel is selected as 1.17E  5/°C. Linear Timoshenko beam element B21 (ABAQUS, 2014) is used to model the pipeline segment. The pipeline segment is divided into 200 elements. The ends are fixed in both the X and Y directions. The downward force per unit applied on pipeline segment is modeled by identical and evenly distributed linear loads with its value 1500 N/m (Zeng et al., 2014). The change of temperature is 200 °C which is used to model the axial loads caused by both change of temperature and internal pressure . 2.2.2. Foundation As mentioned before, the sea bed is treated as a rigid foundation. It is assumed that the foundation and pipeline segment have same profiles of initial imperfections and they are completely in contact with each other. Discrete rigid element R2D2 (ABAQUS, 2014) is adopted to model the foundation. The whole foundation is divided into 200 elements. The interaction between pipeline and ambient soil is treated as surface to surface contact with Coulomb friction acting tangentially and hard contact acting normally. The research of Karampour et al. (2013) manifested that the effect of friction coefficient on upheaval buckling is only evident in the post-buckling response. So the friction coefficient does not affect much the upheaval prebuckling behavior. In this research, the friction coefficient is set to a constant of 0.3. 2.2.3. Solution method The upheaval buckling of submarine pipelines is an unstable nonlinear and localized problem. The results are often not convergent by pure Newton–Raphson method. This kind of problem has to be solved by dynamic damping method (Liu et al., 2014) or by static damping method (Zeng et al., 2014) or by the Modified Risk method. In this study, an automatic stabilization method specifying dissipated energy fraction (ABAQUS, 2014) has been used and Newton–Raphson method is the numerical technique for solving the nonlinear equilibrium equations. Two general static steps have been set in the problem accounting for geometric nonlinearity. In the first step the distributing load applies on the pipeline segment and in the second step the temperature increases by modifying the predefined field. 2.2.4. Verification of solution method The same pipeline model used by Karampour et al. (2013) in their analytical studies is simulated in this section to test the accuracy of the proposed finite element model. The pipeline properties are shown in Table 1. The imperfection No. 1 is adopted as the initial imperfection of pipeline. The maximum height w0 and wave length of imperfection is 0.1 m and 100 m, respectively. The response of pipelines simulated by both static and dynamic damping methods is shown in Fig. 3. The horizontal axis is the vertical displacement of the middle point of the pipeline, while, the vertical axis is the temperature. As shown in Fig. 3, the critical buckling temperature of this method is 147.2 °C which is very close Table 1 Pipeline properties for verification. Pipe mean diameter Pipe wall thickness Young's modulus Poisson's ratio Thermal expansion coefficient

D t E ν α

0.214 m 0.0143 m 207 GPa 0.3 1.1E  5/°C

Fig. 3. The comparison of static and dynamic damping method.

to the result Karampour et al. (2013) gave, i.e., 147.8 °C. This manifested that this model and method are very effective to solve the critical value of pipelines buckling. The reason why the simulation curve shows a different post-buckling path is that an automatic stabilization method is applied in this study. The finite element model and automatic stabilization method are used in this study to obtain the critical force.

3. Numerical results and discussion The upheaval buckling behavior of each pipeline segment has been simulated successfully. In total, 80 sets of results are obtained. All upheaval buckling behaviors happen within the elastic limit range. 3.1. The position of upheaval buckling A set of typical stress and deformation results of the pipeline segment is plotted in Fig. 4. The stress and deformation of the pipeline segment just before and after upheaval buckling are shown in Fig. 4 (a) and (b), respectively. Fig. 4(c) shows the stress and deformation of the pipeline segment in final state when the temperature reaches to 200 °C. From them we can see clearly that when temperature exceeds some critical value, the upheaval buckling of pipeline segment happens. Different positions of the pipeline segment have different stress values. Before buckling, the maximum stress value happens at the midpoint of the pipeline segment. After buckling, the maximum displacement also happened at the midpoint of the pipeline segment. The groups of Nos. 1–3 and Nos. 6–8 with different OOS from 1/1000 to 1/100 all have such results. These are consistent with the results from the previous investigation by Zeng et al. (2014). But there are two groups Nos. 4 and 5 which have different results from other six groups. While, the pipeline segments with different OOS values in sets Nos. 4 and 5 show different upheaval buckling behaviors. Totally, they can be divided into three types. When the OOS of the pipeline segment is 1/100, the upheaval buckling behavior of pipeline segment is shown in Fig. 5. The stress and deformation contour plot before upheaval buckling is shown in Fig. 5(a). In Fig. 5(b), upheaval buckling firstly happens on the positions (we call them buckling points) at the left and right side with a certain distance to the midpoint of the pipeline segment. With the increasing of temperature or axial force, vertical displacement of buckling points increases and the midpoint begins to move upward, at the same time the maximum stress begins to decline. Until the maximum stress attaining a minimum value, the heights of buckling points and the midpoint are almost

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Fig. 4. The typical Mises stress and deformation contour plot.

equal. These are shown in Fig. 5(c) and (d). Then the upheaval buckling suddenly happens again at the midpoint of pipeline segment which is shown in Fig. 5(e). The stress and deformation contour plot of pipeline segment in final state when the temperature reaches to 200 °C is shown in Fig. 5(f). From them, we can clearly see that the pipeline segment happened upheaval buckling with coupled modes firstly. Then with the increasing of temperature or axial force, the mode of upheaval buckling jumped from coupled modes to single mode. In this evolutionary process, a local transfer of strain energy occurred from the buckling points to the midpoint of pipeline segment. This belongs to post-buckling behavior. Fig. 6 shows the vertical displacements versus temperature at the pipeline midpoint and buckling points which displays the evolution process of upheaval buckling. But, in this paper we focus on the positions and axial force where the upheaval buckling firstly happens, as shown by point A in Fig. 6. When the OOS of the pipeline segment is from 1/200 to 1/900, the upheaval buckling behavior of pipeline segment is shown in Fig. 7. The pipeline segments just before and after upheaval buckling are shown in Fig. 7(a) and (b), respectively. Fig. 7 (c) displays the stress and deformation contour plot of pipeline segment in final state when the temperature reaches to 200 °C. From them, we can clearly see that the pipeline segment happened upheaval buckling with coupled modes. Then with the increasing of temperature, the upheaval buckling moved towards the left and right side rather than middle. The mode jumping did not happen. In this process, a local transfer of strain energy

occurred from the buckling points to two sides of pipeline segment. It also belongs to post-buckling behavior. In the whole process the buckling keeps coupled modes. Take the pipeline segment with OOS 1/200 for example, Fig. 8 shows the vertical displacements versus temperature at the pipeline midpoint and buckling points. Other pipeline segments with OOS from 1/300 to 1/900 are similar to this. Likewise, we focus on the position and axial force where the upheaval buckling firstly happened, as shown by point B in Fig. 8. When the OOS of the pipeline segment is 1/1000, the upheaval buckling behavior of pipeline segment is shown in Fig. 9. When temperature reaches to 134.0 °C, 188.5 °C and 200 °C, the Mises stress and deformation contour plot are shown in Fig. 9(a)–(c), respectively. From them, we can clearly see that the upheaval buckling actually does not occur. Maybe in this case, the critical force of pipeline segment is extremely large. The axial force produced by temperature exerted on the pipeline segment does not reach the critical value of the upheaval buckling. So the upheaval buckling does not occur in this situation. In general, these results are very exceptional. They are very different from previous research results. Previous research results manifest that the upheaval buckling happens at the midpoint or the maximum height of initial imperfection on the pipeline. While in this case, initial upheaval buckling does not happen at the midpoint of pipeline segment but at left and right side with a certain distance of the midpoint. More specifically, the buckling positions of the set No. 4 are at node 80 and node 122 (the midpoint of pipeline segment is

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Fig. 5. Mises stress and deformation contour plot of pipeline segment with the OOS 1/100.

node 101). The buckling positions of the set No. 5 are at node 83 and node 119. Comparing to the typical results, the maximum stresses and displacements before buckling and after buckling all happens at those

nodes which are node 80 and node 122 for the set of No. 4 and node 83 and 119 for the set of No. 5, but at the midpoint of pipeline segment. So the maximum stress and the position of upheaval buckling

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X. Zhang, M. Duan / Ocean Engineering 109 (2015) 330–343

are not always happening at the midpoint or the maximum height of initial imperfection. In order to find out why the special behaviors could be produced, we plotted the curvatures graphs of different imperfections on different horizontal positions of pipeline segment in the range of initial imperfections, as shown in Fig. 10. In order to clearly show their features, they are divided into two groups. Although they are plotted when the imperfection maximum height w0 is 0.1 m and the imperfection wavelength L0 is 100 m, their shapes

and features will not change when the maximum imperfection height varies from 0.01 to 0.09. As mentioned before, here we allowed these curvatures and radii of curvatures having negative values. It denotes that the curvature center of that position is under the profile of imperfection or the profile near that position is convex. From Fig. 10(a) we can see that the minimum curvature values of Nos.1–3 are all at the position x ¼0 which denotes the midpoint

Fig. 6. Vertical displacements of midpoint and buckling points with the OOS 1/100.

Fig. 8. Vertical displacements of midpoint and buckling point with the OOS 1/200.

Fig. 7. Mises stress and deformation contour plot of pipeline segment with the OOS from 1/200 to 1/900.

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Fig. 9. Mises stress and deformation contour plot of pipeline segment with the OOS 1/1000.

of pipeline segment, as shown by points A1–A3 respectively. However the minimum curvature value of No. 4 is approximately at the position x ¼ 21 and x ¼21, as shown by points A4. And yet, the two positions are consistent with those of upheaval buckling points (nodes 80 and 122) on the pipeline segment. Similarly, in Fig. 10(b) the minimum curvature values of Nos. 6–8 are all at the position of x¼ 0, as shown by points A6–A8. The minimum curvature value of No. 5 is at the approximate positions x¼  18 and x ¼18 which are consistent with the positions of upheaval buckling points (nodes 83 and 119), as shown by points A5. From the above discussions we can obtain a conclusion that upheaval buckling of a pipeline segment with an initial perfection will probably happen firstly at the position where the curvature has a minimum value in the range of negative curvatures values. That is to say, upheaval buckling of a pipeline segment with initial perfection will probably happen firstly at the position where the radius of curvature has a minimum absolute value |ρ|min in the convex region of initial imperfection. Here we make a brief explanation and it will be verified later. Take the imperfection No. 1 for example, others are similar with it. As shown in Fig. 11, the blue curve denotes the profile of imperfection of No. 1 and the left coordinate axis denotes the imperfection heights. The green curve denotes the curvature corresponding to the nodes on the profile of imperfection and the right coordinate axis denotes curvatures values. The red dashed line is a cutting line which denotes that curvature is zero at that place. These curvatures above the cutting

line are all positive, as shown by arc segments of B1B4 and B2B5 respectively, which denote the centers of curvature of these pipeline segments (shown by arc C1C4 and C2C5) are all above the profile of imperfection, or it denotes these pipeline segments are concave. When the axial force is applied to this area, these pipeline segments have a tendency of downward movement but because of the rigid foundation, this movement will be restrained. So upheaval buckling will not happen at this area. Excessive axial force will be transmitted to other position. Instead, those curvatures below the cutting line are all negative values, as shown by arc segment of B1B3B2, which denotes that the centers of curvature of this pipeline segment (shown by arc C1C3C2) are all below the profile of imperfection, or it denotes that these pipeline segments are convex. When the axial force is applied to this area, this pipeline segment will have a tendency of upward movement and because the restraint for upward is smaller than the restraint for downward and with the increase of axial force being transmitted from concave area, this movement will happen more easily. What is more, the smaller the absolute value of curvature radius is, the greater the upward movement tendency will be. Due to restraint of ambient parts on pipeline, the lager axial force will be produced in this place, and the upheaval buckling will happen more suddenly. So upheaval buckling of pipeline segment with imperfections of Nos. 4 and 5 will happen at those positions. This can qualitatively explain the above conclusion. And this conclusion will be verified later.

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X. Zhang, M. Duan / Ocean Engineering 109 (2015) 330–343

the OOS increases, the snap occurs at lower temperatures and lessens dramatically; (3) after upheaval buckling, the maximum vertical displacement of pipeline segments with imperfection Nos. 4 and 5 is obviously much smaller than others; and (4) for these imperfections with the same OOS, the critical temperature is always No. 3 oNo. 8 oNo. 1 oNo. 6o No. 2 oNo. 7 oNo. 4o No. 5. Table 2 lists the dimensionless quantity ρmax =L0 with different imperfections as we defined before. If we change them into their absolute values, these values have following relationship: No. 3o No. 8 oNo. 1 oNo. 6 oNo. 2 oNo. 4 oNo. 7 oNo. 5. We can see that this order except No. 4 is consistent with the order of critical temperature. The values of Nos. 4, 5 and 7 are very close. Their critical temperatures corresponding to them are also very close, especially the smaller OOS, as shown in Fig. 12. This manifests they have some common relationships. Their relationships will be discussed in Section 4. These critical temperatures are listed in Table 3. Strictly speaking, for the imperfections with larger OOS, there is no critical temperature, instead to the temperature of pipeline first-lift-off (Zeng et al., 2014). However, it doesn’t matter that these values become a part of data. When the OOS of pipeline segment is 1/ 1000, the upheaval buckling with the imperfection Nos. 4, 5 and 7 actually does not occur, as shown in Fig. 12(j). However, using linear interpolation algorithm we obtained these three critical temperatures. They cannot change the whole rules. So it does not matter that these three values become a part of data. According to Eq. (2), a series of critical axial forces are obtained, as shown in Table 4. According to Eq. (1), a series of dimensionless quantity P L =ðq2 EIÞ1=3 are obtained, as shown in Table 5.

4. Approximation formula of critical axial force Fig. 10. Curvature graphs of different imperfections. (a) Curvature graphs from No. 1 to No. 4. and (b) curvature graphs from No. 5 to No. 8.

Fig. 11. Imperfection profile and its curvatures of No. 1.

3.2. Critical temperature and critical axial forces The temperature versus vertical displacement curves of buckling points on each pipeline segment with different OOS are all shown in Fig. 12. As mentioned before, we focus on the prebuckling behavior. The buckling points of initial imperfections Nos.1–3, 6–8 are all at the midpoints of pipeline segment. For initial imperfection No. 4 the buckling point is the node 80 or 122. For initial imperfection No.5 the buckling point is the node 83 or 119. From these figures we know that: (1) the smaller the OOS of a pipeline segment is, the more obvious a large snap occurs; (2) as

Now, we have obtained eight sets of critical axial force values which correspond to different OOS from 1/1000 to 1/100 with different initial imperfection shapes from No. 1 to No. 8. As mentioned before, the differences among these eight initial imperfection shapes are that they have different negative minimum curvature radius values. And these values affect their upheaval buckling behaviors. So the coefficient function gðw0 =L0 ; ρðxÞ=L0 Þ can be written as gðw0 =L0 ; ρmax =L0 Þ. To determine the coefficient function, the relationship between the dimensionless quantity P L =ðq2 EIÞ1=3 and the OOS for same ρmax =L0 needs to be determined firstly. The dimensionless quantity P L =ðq2 EIÞ1=3 versus OOS of pipeline segments with same initial imperfection shapes are shown in Fig. 13, which is very similar with the research of Zeng et al. (2014). But if we plot the dimensionless quantity versus the inverse of OOS of pipeline segments with same initial imperfection shapes, we can see clearly that they have very good linear relationship, as shown in Fig. 14. By linear correlation analysis, the results show that the fitting curves of the linear correlation coefficient are all greater than 99.72%. So it is reasonable to use linear relationship to express the relationship between the dimensionless quantity P L =ðq2 EIÞ1=3 and the inverse of OOS of pipeline segments. Then, the relationship between the dimensionless quantity P L =ðq2 EIÞ1=3 and the initial imperfection shapes parameter ρmax =L0 with same OOS values of pipeline segments needs to be determined. Table 6 lists the initial imperfection shapes parameter ρmax =L0 of initial imperfections on pipeline segments at buckling points. The dimensionless quantity P L =ðq2 EIÞ1=3 versus the initial imperfection shapes parameter ρmax =L0 with same OOS of pipeline segment is shown in Fig. 15. By linear correlation analysis, the results show that the fitting curves of the linear correlation coefficient are all greater than 95.70% except one set 94.52%. The fitting

X. Zhang, M. Duan / Ocean Engineering 109 (2015) 330–343

339

Fig. 12. Temperature versus vertical displacement curve of buckling point of each pipeline segment with different OOS.

curves are shown in Fig. 16. So it is feasible using linear relationship to express the relationship between the dimensionless quantity P L =ðq2 EIÞ1=3 and the initial imperfection shapes parameter

ρmax =L0 of initial imperfections of pipeline segments. By analysis, one form of the formula is obtained to express the relationship between the coefficient function gðw0 =L0 ; ρmax =L0 Þ and the two

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parameters w0 =L0 and ρmax =L0 . It can be written as  g

w0 ρmax ; L0 L0



L0 ρ ¼ a þ b max þc w0 L0

ð11Þ

Now, multiply linear regression method is adopted to determine the above coefficients a–c. By multiply linear regression analysis, the results show that the fitting curve surface of the linear correlation coefficient is 99.39%. The fitting plot, residuals plot and contour plot are shown in Fig. 17. It manifests that the fitting results are reliable. The values of coefficients are obtained: a ¼0.0320, b¼  2.003, c¼ 1.404. An approximation universal formula of upheaval buckling critical axial force is obtained which contains both the parameter of Table 2 Dimensionless quantity ρmax =L0 of different imperfections. Imperfections

ρmax =L0

Imperfections

ρmax =L0

No. 1 No. 2 No. 3 No. 4

 0:0375L0 =w0  0:0507L0 =w0  0:0125L0 =w0   0:0613L0 =w0

No. No. No. No.

  0:0629L0 =w0  0:0417L0 =w0  0:0625L0 =w0  0:0347L0 =w0

5 6 7 8

OOS and the parameter of initial imperfection shape. It is written as   L0 ρ P L ¼ 0:032  2:003 max þ1:404 ðq2 EIÞ1=3 ð12Þ w0 L0 Here, it must be noticed that ρmax is in the range of negative values, as mentioned before; it means that |ρ| has a minimum value |ρ|min in convex area of initial imperfection. This formula manifests that if a pipeline segment has an initial imperfection, the effect of initial imperfection on the upheaval critical axial force can be expressed by three parameters wave length L0, maximum height w0 and maximum value of negative radius of curvature ρmax. The upheaval buckling critical axial force is inversely proportional to the OOS if ρmax/L0 is kept constant. And the critical axial force is directly proportional to ρmax/L0 if we keep OOS constant. Some researchers such as Ju and Kyriakides (1988), Maltby and Calladine (1995), and Karampour et al. (2013) pointed out that the critical force is inversely proportional to the curvature of initial imperfection at x ¼0. Because in their research, the maximum absolute value of curvatures in convex area is obtained at x ¼0. In this situation, our conclusion is consistent with theirs. While, from this study, we know that not all of initial imperfections are obtained maximum absolute value of curvatures in

Table 3 Critical temperature of pipeline segment. OOS

Imp No. 1

Imp No. 2

Imp No. 3

Imp No. 4

Imp No. 5

Imp No. 6

Imp No. 7

Imp No. 8

1/1000 1/900 1/800 1/700 1/600 1/500 1/400 1/300 1/200 1/100

127 116 103 91.5 80 67.5 55 41 28 14

164.5 148.5 132.5 116.5 101 84 68.5 50 34.5 17.5

74 68 61 54.5 48 41 33 25 17 9

202.5a 182.5 163.5 144 125 105 85 65 44 22

204a 185 166.5 147 127 107 88 66 45 23

152 138 124 109 95 80 65.5 49 33 17

200a 180.5 160 140.5 121 101 81 62 41.5 21

120 108 96 84 73 61 49 37 25 12.5

Table 4 Critical axial force of pipeline segment. OOS

Imp No. 1

Imp No. 2

Imp No. 3

Imp No. 4

Imp No. 5

Imp No. 6

Imp No. 7

Imp No. 8

1/1000 1/900 1/800 1/700 1/600 1/500 1/400 1/300 1/200 1/100

6.12E þ06 5.59E þ06 4.96E þ06 4.41E þ06 3.85E þ06 3.25E þ06 2.65E þ06 1.97Eþ06 1.35Eþ06 6.74E þ05

7.92E þ06 7.15E þ06 6.38E þ06 5.61E þ06 4.86E þ06 4.05E þ06 3.30E þ06 2.41E þ06 1.66E þ06 8.43E þ05

3.56E þ 06 3.28E þ 06 2.94E þ 06 2.63E þ 06 2.31E þ 06 1.97E þ 06 1.59E þ 06 1.20E þ 06 8.19E þ 05 4.34E þ 05

9.75E þ06 8.79E þ06 7.88E þ06 6.94E þ06 6.02E þ06 5.06E þ06 4.09E þ06 3.13E þ06 2.12E þ06 1.06E þ06

9.83E þ 06 8.91E þ 06 8.02E þ 06 7.08E þ 06 6.12E þ 06 5.15E þ 06 4.24Eþ 06 3.18E þ 06 2.17E þ 06 1.11E þ 06

7.32E þ 06 6.65E þ 06 5.97E þ 06 5.25E þ 06 4.58E þ 06 3.85E þ 06 3.15E þ 06 2.36E þ 06 1.59E þ 06 8.19E þ 05

9.63Eþ 06 8.69Eþ 06 7.71E þ 06 6.77Eþ 06 5.83Eþ 06 4.86Eþ 06 3.90Eþ 06 2.99Eþ 06 2.00E þ 06 1.01E þ 06

5.78E þ 06 5.20E þ 06 4.62E þ 06 4.05E þ 06 3.52E þ 06 2.94E þ 06 2.36E þ 06 1.78E þ 06 1.20E þ 06 6.02E þ 05

Table 5 The dimensionless quantity P L =ðq2 EIÞ1=3 of upheaval buckling points of pipeline. OOS

Imp No. 1

Imp No. 2

Imp No. 3

Imp No. 4

Imp No. 5

Imp No. 6

Imp No. 7

Imp No. 8

1/1000 1/900 1/800 1/700 1/600 1/500 1/400 1/300 1/200 1/100

100.26 91.57 81.31 72.23 63.15 53.29 43.42 32.37 22.10 11.05

129.86 117.23 104.60 91.97 79.73 66.31 54.07 39.47 27.23 13.81

58.42 53.68 48.15 43.02 37.89 32.37 26.05 19.74 13.42 7.10

159.86 144.07 129.07 113.68 98.68 82.89 67.10 51.31 34.73 17.37

161.04 146.04 131.44 116.04 100.26 84.47 69.47 52.10 35.52 18.16

119.99 108.94 97.89 86.05 74.99 63.15 51.71 38.68 26.05 13.42

157.88 142.49 126.31 110.91 95.52 79.73 63.94 48.94 32.76 16.58

94.73 85.26 75.78 66.31 57.63 48.15 38.68 29.21 19.74 9.87

a

These results are obtained from linear interpolation algorithm.

X. Zhang, M. Duan / Ocean Engineering 109 (2015) 330–343

convex area at x¼ 0. In this case, the upheaval buckling happens at the position with maximum absolute value of curvatures in convex area, and the critical force is related to the maximum curvature value at that position. That is to say that their conclusion is a special example. Our conclusion is more extensive and universal. If we extend the formula (12) to the range of initial imperfection, that is P Lx ¼ ð0:032

  L0 ρðxÞ L0 L0 rxr  2:003 þ1:404Þðq2 EIÞ1=3 L0 w0 2 2

ð13Þ

341

force applied on the structure exceeds the maximum capacity of axial force at here, a large downward displacement will happen at here, but due to the restraint of foundation, actually the buckling will not happen. So upheaval buckling will probably happen when ρðxÞ is in the range of negative values. And in this range, when ρðxÞ has a maximum value ρmax, by Eq. (13), PLx has a minimum value which is just PL, as shown Eq. (12). That is to say, PL is the minimum value among the maximum capacity of axial force on the pipeline. So it verifies the conclusion presented in Section 3.1 about the position of upheaval buckling.

Suppose PLx denotes the maximum capacity of axial force at the position of x on a structure where upheaval buckling cannot occur. As mentioned before, if ρðxÞ has a positive value, when an axial

Fig. 13. Dimensionless quantity P L =ðq2 EIÞ1=3 versus OOS.

Fig. 15. Dimensionless quantity P L =ðq2 EIÞ1=3 versus the dimensionless quantity ρmax =L0 .

Fig. 14. Dimensionless quantity P L =ðq2 EIÞ1=3 versus the inverse of OOS.

Fig. 16. Results of linear regression.

Table 6 The initial imperfection shapes parameter ρmax =L0 of initial imperfections of pipeline segments at buckling points. OOS

Imp No. 1

Imp No. 2

Imp No. 3

Imp No. 4

Imp No. 5

Imp No. 6

Imp No. 7

Imp No. 8

1/1000 1/900 1/800 1/700 1/600 1/500 1/400 1/300 1/200 1/100

 37.5000  33.7500  30.0000  26.2500  22.5000  18.7500  15.0000  11.2500  7.5000  3.7500

 50.6606  45.5946  40.5285  35.4624  30.3964  25.3303  20.2642  15.1982  10.1321  5.0661

 12.5000  11.2500  10.0000  8.7500  7.5000  6.2500  5.0000  3.7500  2.5000  1.2500

 61.3143  55.1829  49.0515  42.9200  36.7886  30.6572  24.5257  18.3943  12.2629  6.13143

 62.9026  56.6123  50.3221  44.0318  37.7415  31.4513  25.161  18.8708  12.5805  6.29026

 41.6667  37.5000  33.3333  29.1667  25.0000  20.8333  16.6667  12.5000  8.3333  4.1667

 62.5000  56.2500  50.0000  43.7500  37.5000  31.2500  25.0000  18.7500  12.5000  6.2500

 34.6973  31.2276  27.7578  24.2881  20.8184  17.3486  13.8789  10.4092  6.9395  3.4697

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Fig. 17. Results of multiply linear regression.

(4) The maximum radius of curvature is obtained as

5. Application In the following paragraphs, the formula (12) is used to determine the upheaval buckling critical axial force of pipeline segment with an initial imperfection. We use the case Karampour et al. (2013) presented to calculate the critical axial force. According to Karampour et al. (2013) the suspended pipeline segment resting on a rock and infilling with soft soils will have the following equilibrium profile:

f 9 ðxÞ ¼

8 qL20 x2 qx4 qL0 x3 >  48EI < w0  24EI  18EI 3 qL20 x2 qx4 > 0x : w0  24EI þ qL 18EI  48EI

where L0 ¼ 2

0 r x r L20

72EIw0 q

Likewise, we presume that a pipeline has an imperfection like the above profile and its height w0 ¼0.3 m and the other parameter values are the same as that used in above FE simulating. 0 1=4 (1) L0 can be determined by L0  ¼ 2ð72EIw ¼ 69:4477 m q Þ L0 =w0 can be determined L0 w ¼ 231:4924. 0 (2) Take a half of the imperfection to analyze due to its symmetry, for example, we take the left half of the imperfection. The curvature can be determined as

K¼

ðqL20 =24EIÞ þ ðqL0 x=3EIÞ þ ðqx2 =2EIÞ ð1 þ ððqL20 x=24EIÞ þ ðqL0 x2 =6EIÞ þ ðqx3 =6EIÞÞ2 Þ3=2

(3) When x ¼0, K has a minimum value: K min ¼  q L20

1 24EI ¼ 2: K min qL0

(5) The dimensionless quantity ρmax =L0 can be determined as

ρmax L0

¼

24EI qL30

¼  4:8228:

(6) The critical axial force PL can be determined as   L0 ρ P L ¼ 0:032  2:003 max þ1:404 ðq2 EIÞ1=3 ¼ 1; 127; 000ðNÞ: w0 L0

 L20 r x o 0

1=4



ρmax ¼

The result is smaller than the results Zeng et al. (2014) estimated: PL ¼160,000 N. In the same way, we determined the critical axial force of pipeline with initial imperfection shape Nos. 1–3 with wave length L0 ¼69.4449 m and height w0 ¼0.3 m: PL1 ¼ 1,598,600 N, PL2 ¼1,972,000 N and PL3 ¼ 891,300 N. The three results are also a little different from the results Zeng et al. (2014) obtained: PL1 ¼1,713,400 N, PL2 ¼1,964,000 N and PL3 ¼1,234,600 N. While, according to the results of Karampour et al. (2013) for imperfection No. 2, the real upheaval critical force is close to 2,000,000 N. Although exact critical force of imperfections No. 9 is unknown, it is apparent our results are more accurate than that Zeng et al. (2013) gave.

:

. 24

6. Conclusions

EI. So we

can determine that if the upheaval buckling will happen, it must happen at the midpoint of the pipeline segment firstly.

In this paper, the finite element method is employed to study the upheaval buckling behaviors of pipelines with initial imperfections. Eight groups of pipeline segments with different imperfection shapes and different OOS values have been simulated by

X. Zhang, M. Duan / Ocean Engineering 109 (2015) 330–343

ABAQUS. Two groups of special upheaval buckling behaviors are obtained. A new parameter is defined to express the differences of different imperfection shapes. An approximation and universal formula which covers the new parameter and the parameter of OOS is proposed to calculate the critical axial force. This formula can be used to explain those two groups of special upheaval buckling behaviors. Some conclusions can be drawn as follows:

 The upheaval buckling does not always happen at the maximum

 





amplitude of an initial imperfection. Sometimes it will occur at two sides of the maximum amplitude. Then, the upheaval buckling may probably move to two sides or middle. The position of upheaval buckling which firstly happens on a pipeline has a relationship with the minimum values of negative curvature or maximum values of negative curvature radius. The parameter of OOS w0/L0 and the new parameter ρmax/L0 are extremely important parameters for upheaval buckling. For same imperfection shape or same ρmax/L0, the larger the OOS is, the smaller the critical axial force will be. For same OOS, the larger the new parameter is, the smaller the critical axial force will be. It is the minimum absolute value of curvature radius in convex area of initial imperfection that lead to different imperfection shape has different critical force even though they have same initial wave length L0 and maximum height w0. Increasing the radius of curvature in the convex area of the initial imperfection can dramatically increase the critical axial force. This can improve the buckling capacity of pipeline. It will have an important meaning for practical engineering.

Acknowledgments This research is supported by the National Basic Research Program of China (No. 2011CB013702) and the National Natural Science Foundation of China (No. 51379214).

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