- Email: [email protected]
Numerical study on lateral buckling of pipelines with imperfection and sleeper
Applied Ocean Research 68 (2017) 103–113
Contents lists available at ScienceDirect
Applied Ocean Research journal homepage: www.elsevier.com/locate/apor
Full length article
Numerical study on lateral buckling of pipelines with imperfection and sleeper Zhe Wang a,b , Zhihua Chen a,b , Hongbo Liu a,b,∗ , Zechao Zhang b a b
State Key Laboratory of Hydraulic Engineering Simulation and safety, Tianjin University, Tianjin 300072, China Department of Civil Engineering, Tianjin University, Tianjin 300072, China
a r t i c l e
i n f o
Article history: Received 11 December 2015 Received in revised form 30 July 2017 Accepted 14 August 2017 Keywords: Subsea pipelines Lateral buckling Sleeper Finite element Genetic programming
a b s t r a c t Lateral buckling is an important issue in unburied high-temperature and high-pressure (HT/HP) subsea pipelines systems. The imperfection–sleeper method is one of the most well-known methods used to control lateral buckling of HT/HP pipelines. Pipelines–sleeper–seabed numerical models are established and verified to analyze the buckling behavior of pipelines using the imperfection–sleeper method. The influence of six main factors on lateral buckling behavior is investigated in details based on the numerical results. Equations of buckling displacement (buckling displacement is defined by the final displacement of the middle point of the pipelines), critical buckling force, and buckling stress (Mises stress) are proposed using the gene expression programming technique. These equations show good accuracy and can be used to assist in the design of sleepers and assess the compressive and stress levels of pipelines. © 2017 Published by Elsevier Ltd.
1. Introduction Subsea pipelines operated under high temperature and high inner pressure will develop compressive force because of seabed soil restraining its axial expansion. If the compressive force is sufficiently high, then global buckling will occur, which is similar to the buckling of a steel bar. For unburied subsea pipelines, which are common in deep-sea conditions, global buckling usually occurs in the lateral direction [1,2]. Lateral buckling may lead to fracture, collapse, or buckling propagation [3]. With the increase in operation temperature and pressure, controlling lateral buckling becomes an essential problem in the design of subsea HT/HP pipelines. A number of methods are used, such as snaked lay (imperfection) method [4], sleeper method [5], and distributed buoyancy method [6], to control lateral buckling responses. All of these methods are aimed at reducing the compressive force levels and buckling responses. This study focuses on the method that combines imperfections and sleepers. For pipelines with imperfections, a number of studies on lateral buckling of imperfect pipelines have been conducted in recent years. Miles and Calladine [7] provided a design formula to calculate the maximum buckling strain on the basis of the experimental and
∗ Corresponding author at: State Key Laboratory of Hydraulic Engineering Simulation and safety, Tianjin University, Tianjin 300072, China. E-mail address: [email protected] (H. Liu). http://dx.doi.org/10.1016/j.apor.2017.08.010 0141-1187/© 2017 Published by Elsevier Ltd.
numerical results. Karampour et al. [3] derived an analytical solution for pipelines with half-wavelength sinusoidal imperfection. Hong et al. [8] used the energy method to calculate the analytical solution for pipelines lateral buckling with a single-arch initial imperfection and indicated that a small single-arch initial imperfection is associated with snap buckling. Most of the lateral buckling studies have a reasonable two-dimensional assumption. However, the buckling problem becomes a three-dimensional (3-D) problem when a sleeper is considered. For pipelines with sleepers, Sinclair et al. [5] indicated that buckling mode 2 may occur in pipelines with sleepers, which should be ignored. One effective way to avoid mode 2 is to combine the sleeper method with the imperfection method. The initial imperfection not only reduces the critical buckling force but also guarantees that only mode 1 occurs, which will be discussed in Section 3. The proposed critical buckling force and maximum strain formulas cannot be applied directly to the pipelines with imperfections and sleepers. Therefore, formulas of critical buckling force, buckling displacement, and buckling stress are necessary to improve the design of pipelines with imperfections and sleepers. The lateral buckling behavior of subsea pipelines with imperfection and sleeper has been investigated numerically. First, a 3-D finite element model of a 2000 m-long pipelines–sleeper–seabed system is developed in Abaqus. Then, six factors which influence the buckling response are investigated to analyze the lateral buckling behavior. Finally, on the basis of the numerical results, three formulas (critical buckling force, buckling displacement, and buckling
104
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Fig. 1. The finite element model.
stress) related to pipelines with imperfections and sleepers are proposed by the gene expression programming (GEP) method. These equations can be used to design pipelines with both imperfection and sleeper.
Table 1 Pipelines property for verification.
2. Numerical model 2.1. Finite element model The finite element model provides a convenient method to calculate the lateral buckling of pipelines. The finite element model of the pipelines with imperfection and sleeper is described on the basis of the following four aspects: First, the element types are selected. The beam element is suitable for simulating pipelines structures because the subsea pipelines is an ultra-slender structure, which indicates that the length of one direction (pipelines axial direction) is more than the length of two other directions (pipelines cross-section). As a result, PIPE31 elements in Abaqus, i.e., two-node linear beam elements, are selected to simulate the pipelines [9]. It is noteworthy that thinwalled PIPE31 element is selected here; therefore the hoop stress is constant across the cross-section of the pipe. This assumption needs to be taken care when the section of pipelines is extremely thick, in which case the hoop stress varies across the section. The seabed is assumed to be a rigid surface, which is a common assumption in the traditional analytical method. Thus, rigid surface elements (R3D4 element in Abaqus) are selected to simulate the seabed [10]. The sleeper is also assumed to be a rigid surface with a semi-circle section, as shown in Fig. 1. Second, the constraint and load of the finite element model are determined. Most parts of the pipelines lie on the seabed and constrained by the seabed through contact elements. The middle point of the pipelines is held by a sleeper, which results in free segments on both sides of the middle point (Fig. 1). The ends of the pipelines are both far from the middle point (1000 m). Therefore, the pin constraint at the ends of the pipelines has only a slight influence on the buckling area. The seabed and sleeper are both fixed. The contact pressure–overclosure relationship of the interaction between pipelines and seabed (sleeper) is presented as a hard contact model. The Coulomb friction model is used to represent friction. 1 and 2 represent the friction coefficients of seabed and sleeper, respectively. A uniform temperature field, constant inner pressure, and distributed load in the z-direction are applied to the pipelines to simulate working conditions. Third, the imperfection is selected. A sinusoidal imperfection is used to achieve lower buckling force, as shown in Fig. 2. The imperfect segment of the pipelines can be expressed in Eq. (1), as follows:
h = h0 sin
(l + l0 /2) l0
Fig. 2. The sinusoidal imperfection.
(1)
Diameter (d) (mm)
15
Wall thickness (t) (mm) Young’s modulus (E) (MPa) Poisson’s ratio () Thermal coefficient of expansion (␣) (/◦ C) Submerged weight (q) (N/m) Sleeper height (h2 ) (mm) Seabed friction coefficient (1 ) Sleeper friction coefficient (2 ) Temperature (◦ C) Pressure (MPa)
0.9 191,000 0.3 1.73 × 10−5 3.94, 0.0 30 0.7 0.1, 0.2 35 12
where h0 and l0 are the amplitude and length of the imperfection, respectively. Finally, the quasi-static dynamic method is selected as the calculation method to simulate the lateral buckling of the pipelines, which means dynamic. Geometric nonlinearity switch in Abaqus is turned on at the beginning of the simulation, which means geometric stiffness is considered. The stiffness matrix of the model is updated during the simulation process.
2.2. Verification of the finite element model The same pipelines model used by De Oliveira et al. [13] in their experimental study is simulated in this section to test the accuracy of the proposed finite element model. Three of the experiments models are selected here to verify the proposed numerical model. The pipelines properties are shown in Table 1. Fig. 3 shows the axial compressive force and displacement results of the buckling apex obtained by the experiments and simulation. Only the representative data point is selected because many fluctuations exist in the original axial force experimental data. In Fig. 3, the temperature increase is shown along the x-axis, whereas the compression force of the pipelines is shown along the y-axis. All the three experiments show a similar pattern. The compression force keeps increasing with the increase in temperature until the temperature reaches approximately 5 ◦ C (critical buckling temperature). Then, the axial force decreases rapidly, which indicates that lateral buckling occurs. The apex of the compression force is referred to as the critical buckling force of the pipelines. Simulated critical buckling forces are 530 N, 462 N and 551 N with relative error −8.0%, −7.5% and − 9.2%. It is assumed that the differences between the simulated and experimental results are mainly due to exclusion of the pipelines coating.
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
105
Fig. 3. Axial force at the buckle apex.
3. Lateral buckling behavior Factors that influence the buckling behavior of the pipelines are studied in order to derive the formulas that control the buckling behavior of pipelines with imperfections and sleepers. The influences of different parameters on buckling responses are investigated in detail. Six parameters are selected, namely, bending stiffness (EI), imperfections (h0 /l0 ), submerged weight (q), friction coefficient of the seabed (1 ), friction coefficient of the sleeper (2 ), and sleeper height (h2 ). The values of the different parameters are shown in Table 2. First, the buckling process of the pipelines with imperfection and sleeper is analyzed from the point of lateral displacement along the length of the pipelines. Fig. 4a shows that buckling mode 1 occurs in the pipelines. The free span segment of the pipelines moves slightly toward the positive direction of y at the initial stage (temperature of 14 ◦ C). This phenomenon is different from the lateral buckling process of the pipelines without sleeper [12]. Afterward, the center lobe grows rapidly; at the same time, the lobes (side lobes) beside the center lobe grow toward the opposite direction (temperature of 39 ◦ C). Then, with the increase in temperature, secondary side lobes occur beside the side lobes. The final buckling region is as long as 200 m, which is symmetrical with the
Table 2 Pipelines properties discussed in Section 3. Diameter (d) (mm)
254, 381, 323.9, 445 [14]
Wall thickness (t) (mm) Young’s modulus (E) (MPa) Poisson’s ratio () Thermal coefficient of expansion (␣) (/◦ C) Imperfection (h0 /l0 ) Submerged weight (q) (N/m) Seabed friction coefficient (1 ) Sleeper friction coefficient (2 ) Sleeper height (h2 ) (m)
12.7 206,000 0.3 1.01 × 10−5 0.004–0.020 1000–3000 0.3–0.9 0.1–0.6 0.3–0.9
central point. Fig. 4b shows von Mises stress along the length of the pipelines, each point represents the average von Mises of the four integration points of that section. As shown in this figure, the maximum stress occurs near the midpoint area and all the stress peaks appear at the lobes. Fig. 5 shows the Mises stress distribution of the pipelines at 100 ◦ C. Here the Mises stress has three components: axial stress, hoop stress, and radial stress [11]. All the buckling processes, which will be discussed in the subsequent sections, follow the buckling pattern discussed in the previous section. As a result, only the compression
106
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Fig. 4. Buckling process of the pipelines with imperfection and sleeper (only the buckling region is shown in the figure).a) Lateral buckling shape along the length of the pipelines b) von Mises stress along the length of the pipelines.
Fig. 5. Stress distribution of the pipelines at 100 ◦ C.
force–displacement relationship, which is one of the most important properties that deal with buckling problems, is discussed in this section. All following calculations terminate at temperature 100 ◦ C. The base case of the parametric analysis is d = 254 mm, h0 /l0 = 0.012, q = 1500 N/m, 1 = 0.5, 2 = 0.3, and h2 = 0.5 m. Fig. 6. The influence of bending stiffness (EI).
3.1. Bending stiffness Bending stiffness of the pipelines is represented by the outer diameter. Four diameters are selected, i.e., D = 254, 381, 323.9, and 445 mm. Fig. 6 shows that the critical buckling force and post-buckling force clearly increase with the increase in bending stiffness. As shown in Fig. 4, the maximum displacement always shows in the middle of the pipelines. The final displacement also increases with the increase in bending stiffness because a large section area results in a large section force under the same temperature. 3.2. Imperfection (out-of-straightness) Fig. 7 shows that imperfection is a significant element in the analysis of buckling responses. The critical buckling force increases with the decrease in imperfection. Imperfections hardly influence the post-buckling force and final buckling displacement of the pipelines. Notably, the axial compressive force decreases rapidly after the critical buckling point if the imperfection is small. In such cases, the buckling process is always intense and dynamic, which we believe should be avoided in lateral buckling design. If the imperfection is large, then the axial compressive force decreases
Fig. 7. The influence of imperfection h0 /l0 .
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Fig. 8. The influence of submerged weight (q).
Fig. 10. The influence of seabed friction (2 ).
Fig. 9. The influence of seabed friction (1 ).
Fig. 11. The influence of sleeper height (h2 ).
107
gradually after the critical buckling point and the critical buckling force is similar to the post-buckling force.
3.5. Friction between the pipelines and sleeper
3.3. Submerged weight
Fig. 10 shows that the critical buckling force increases with the increase in sleeper friction, whereas the buckling displacement decreases with the increase in sleeper friction.
Submerged weight is also an important factor influencing the buckling process. Fig. 8 shows the influence of submerged weight on the buckling response of pipelines. The figure shows that a large submerged weight implies a high critical buckling force and a high post-buckling force. However, the final buckling displacement decreases with the increase in submerged weight. Notably, the axial compressive force with large submerged weight decreases faster than those with less weight. 3.4. Friction between the pipelines and seabed Fig. 9 shows that seabed friction only has a slight effect on critical buckling force and final buckling displacement compared with other factors because the critical buckling force and maximum buckling displacement are calculated at the area on the sleeper. Normally, the friction of the sleeper is lower than seabed friction. As a result, the buckling force and displacement mainly depend on the sleeper friction, as will be discussed in Section 3.5.
3.6. Sleeper height The sleeper height has only a slight influence on the critical buckling force, as shown in Fig. 11. We believe that the imperfection reduces the effect of sleeper height. The final buckling displacement increases with the increase in sleeper height. Notably, the free spanning segment induced by the sleeper is harmful to the pipelines through the phenomenon of vortex-induced vibration [15]. As a result, the imperfection can reduce the sleeper height in the lateral buckling design. In summary, for critical buckling force, bending stiffness, imperfections, submerged weight, and sleeper friction are the most important factors. For buckling displacement, bending stiffness, submerged weight, seabed friction, and sleeper friction combined with temperature and pressure (obviously) are the most significant factors. The results presented in this section serve as an important guidance in the subsequent section, which intends to derive
108
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Table 3 Statistical parameters of the input data.
EI (MNm2 ) d (m) t (m) h0 /l0 q (kN/m) 1 2 h2 T (◦ C)
Table 4 GEP parameters.
Range
Parameter
14–39 0.25–0.35 0.0127 0.009–0.015 1–3 0.3–0.7 0.1–0.6 0.1–1 80–100
Uniform Uniform Fixed Normal = 0.012, = 0.05 Uniform Normal, = 0.5, = 0.03 Normal, = 0.3, = 0.03 Normal, = 0.5, = 0.2 Uniform
Model 1 Model 2 Model 3
Item
Head size
Link function
Mutation
Number of variables
disp Pcr max
8 8 10
+ * *
0.009 0.009 0.005
7 5 7
Chromosomes = 30, number of genes = 5, range of constants = −10 to 10, constants per gene = 2, inversion = 0.05, transposition = 0.1.
The coefficient of determination (R2 ) is applied as another indicator to derive a high-performance model, which is obtained by using the following equation:
⎛
the formulas of buckling displacement, critical buckling force, and maximum buckling von Mises stress.
⎜ ⎜ R =1−⎜ ⎜ ⎝ 2
n
⎞
(ti − oi )
2
i=1 n
o2 i
⎟ ⎟ ⎟ ⎟ ⎠
. (3)
i=1
4. Genetic programming analysis In the design of lateral buckling of pipelines with imperfections and sleepers, buckling displacement, critical buckling force, and buckling stress are the key parameters. Buckling displacement indicates the length of the sleeper and critical buckling force indicates the axial force level along the pipelines, whereas buckling stress is used to assess the integrity of the pipelines. As a result, in this section, we generate three GEP [16] models to derive the formulas of the aforementioned key parameters. GEP is an extension of genetic programming (GP) [17], which is an evolutionary algorithm that automatically creates computer programs. GEP and GP have been successfully applied to many structural problems including pipelines problems (i.e., the shear strength of reinforced concrete beams is predicted using GP [18], upheaval buckling displacement of offshore pipelines buried in clay soil is predicted using GEP [19], and other applications [20,21]). Details of GEP can be found in [17] and [18].
4.1. Gene expression programming modeling Based on the results presented in Section 3, seven parameter categories, six parameters presented in Section 3, and temperature are selected in this section to comprise the formulas. A set of 400 randomly simulated results were selected to develop the prediction model using GEP. Table 3 shows the statistical properties of the parameters. Fig. 12 shows the frequencies of different parameters. Bending stiffness (EI), diameter (d), weight (q) and temperature (T) are sampled from a uniform distribution, the rest of the parameters are generated from a normal distribution. The datasets are divided randomly to 350 and 50 and are used to train and validate the model. The GEP modeling process is developed on the basis of the Java GEP library gep4 j [22]. Root mean square error (RMSE) is used as the fitness function in GEP according to the following equation:
n 1 RMSE = (ti − oi )2 n
The other GEP parameters are shown in Table 4. As shown in the table, head size means the length of gene’s head [17]. Link function is the function links different segments of the chromosome, we found that plus function works better for displacement model and manipulate function performs better for critical buckling force model and maximum stress model. Mutation is the probability of mutation of each model. The number of variables is the quantity of dependent variables in each model. The rest of the parameters are all the same among the models. Every generation has 30 chromosomes (individuals), each chromosome contains 5 genes. The range of constants in the model is between −10 to 10. The probability of inversion and transposition are 0.05 and 0.1 respectively. Models 1–3 represent the models of buckling displacement, critical buckling force, and buckling stress, respectively. Notably, temperature and sleeper height are not used as input parameters in Model 2 because these two factors have only a slight influence on the critical buckling force on the basis of the results presented in Section 3. Several trials were conducted to derive a high-performance model with the highest coefficient of determination (R2 ) and the highest fitness in all training and validating phases. Fig. 13 shows the distribution of the simulated results. 4.2. Equations for buckling force, buckling displacement, and maximum von mises stress The equations of all three proposed GEP models (simplified) are summarized in Table 5. Fig. 14 shows the correlation between simulated and predicted results in training and validation. All of the three models show good agreement with the simulated response. In Table 6, Model 1 shows the highest performance with the maximum training error of 3.5% and validation error of 3.0%. Although model 2 shows the lowest performance, only two data points have an error of more than 10.0%, as shown in Fig. 15e. Most of the errors of model 2 are less than 10.0%. The maximum error of model 3 is 6.0%. All of the three models have a high R2 value (larger than 95% in this study), which is considered high accuracy prediction [19]. 4.3. Parameter analysis
(2)
i
Where, n is the number of data points in the training set. ti is the result of the model and oi is the real (training) value.
Parametric analysis is conducted based on the proposed equations. Fig. 15 shows the results of the parametric analysis. The trends of the calculated curves are similar to simulated results presented in Section 3. Fig. 15a and b only show the equations of displacement and stress because temperature and sleeper height is not included in the equation (model 2). Notably, buckling stress
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Fig. 12. Distribution of the input parameters.
109
110
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Fig. 13. Distribution of the output results.
Table 5 Equations of the GEP models. Equation
Displac-ement
d=
abs
q+
tan q1/3
−
T − EI −5.57
1/3 +
abs
1 + EI −
h0 − 2 − 5.8593 0.3093l0
1/3
+ h2
− q − 2
+ arctan (h2 ) + T 1/3 −(tan (2 ))2+ 11.14392 EI h0 ∗ 2 ∗ tan −7.22454 + 79.68093 tanh(sin(2 )) Pcr = arctan arctan 1 − tan − 26.13666 l0
Critical Axial Force ∗ sin (tanh (q + 1 ) ∗ tan (1 − 1.22835)) ∗
max =
Maxim-nm Stress
T − 6.9519 (2 − h2 ) − tan
tanh
earctan(T )
arctan
q2 +
h0 l0
sin
q−q3
e
1 2
h0 l0
1/3
1/3 ∗ tanh
−
h0 + q − 10.4446 l0
1/3
2 − q − 2.06349q arctan (h2 − 8.37604)
h0 e l0 +
∗ tanh
q+
EIq (h2 − 0.30453) (q + 5.50048)
3.02842 − T + 58.16547 arctan (T )2
1/3
∗
∗
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Fig. 14. Error of the output results.
111
112
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Fig. 15. Parametric analysis of the effect of the input parameters based on the proposed equations.
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
113
Table 6 Performances of the GEP models.
Model 1 (Displacement) Model 2 (Critical Axial Force) Model 3 (Maximum Stress)
Maximum training error
Maximum validation error
R2
3.40% 13.00% 5.90%
3.00% 9.80% 6.00%
0.991 0.977 0.955
of the pipelines mainly depends on temperature (pressure), submerged weight, and sleeper height. Fig. 15e shows that seabed friction has only a slight influence on all three dependent variables, as expected. In other words, buckling response of the pipelines with sleeper and imperfection is relatively independent of seabed friction, which improves the probability of triggering lateral buckling on the complex marine soil. 5. Conclusion A numerical model of pipelines with imperfections and sleepers is generated to analyze the lateral buckling behavior of the pipelines. The numerical model shows good consistency with the existing experiment. Based on the simulated results, the GEP technique is used to establish three equations to calculate buckling displacement, critical buckling force, and buckling stress. The following main conclusions are drawn to summarize this study: Bending stiffness, submerged weight, seabed friction, and sleeper friction are the most significant factors influencing buckling displacement. Bending stiffness, imperfections, submerged weight, and sleeper friction are the most important factors influencing critical buckling force. Submerged weight and sleeper height are the most important factors influencing buckling stress. The sleeper height has only a slight influence on critical buckling force when an imperfection is considered. If the height of sleeper in a range between 0.3 m and 0.9m, the critical buckling force mainly depends on initial bending stiffness, imperfection, weight and friction. While the height of sleeper influences the buckling displacement and maximum stress dramatically. As a result, a small height leads to the decrease in free span length. Consequently, the response of vortex-induced vibration, buckling displacement and stress are reduced. Once the buckling force level determined, we can select a proper friction of sleeper using proposed equations. The lateral initial imperfection guarantees that only mode 1 occurs in pipelines with sleepers. The proposed buckling displacement equation exhibits the highest accuracy and can help design the length of the sleeper. The accuracy of buckling force and stress equations are acceptable for engineering use and can be applied to assess the compressive force and stress level of the pipelines. Acknowledgment The authors are grateful for the support provided by the National Basic Research Program of China (NO. 2014CB046801).
References [1] R.E. Hobbs, In-service buckling of heated pipelines, J. Transport Eng. 110 (2) (1984) 175–189. [2] Wang Zhe, et al., Overview of global buckling of deep-sea pipelines, J. Tianjin Univ. Sci. Technol. 47 (2014) 17–23. [3] H. Karampour, et al., On lateral and upheaval buckling of subsea pipelines, Eng. Struct. 52 (2013) 317–330. [4] Y.X. Liu, et al., Postbuckling analysis of snaked-Lay pipelines based on a new deformation shape, J. Offshore Mech. Arct. 135 (2013) 1–9 (031704). [5] F. Sinclair, et al., Design challenges and experience with controlled lateral buckle initiation methods, Proceedings of 28th International Conference on Ocean, Offshore and Arctic Engineering. Hawaii : American Society of Mechanical Engineers (2009) 319–330. [6] R. Shi, L.Z. Wang, Single buoyancy load to trigger lateral buckles in pipelines on a soft seabed, J. Eng. Mech. 141 (5) (2014) 0401415. [7] D.J. Miles, C.R. Calladine, Lateral thermal buckling of pipelines on the seabed, J. Appl. Mech. 66 (4) (1999) 891–897. [8] Z. Hong, et al., Study on lateral buckling characteristics of a submarine pipelines with a single arch symmetric initial imperfection, Ocean Eng. 108 (2015) 21–32. [9] Z. Wang, et al., Numerical study on upheaval buckling of pipe-in-pipe systems with full contact imperfections, Eng. Struct. 99 (2015) 264–271. [10] Z. Wang, et al., Static and dynamic analysis on upheaval buckling of unburied subsea pipelines, Ocean Eng. 104 (2015) 249–256. [11] ABAQUS, Analysis User Guide Version 6 14., Hibbit, Karlsson and Sorensen, Inc., RI, USA, 2014. [12] Z. Wang, et al., On Lateral Buckling of Subsea Pipe-in-pipe Systems International Journal of Steel Structures, 2017, 2017.09.(Accepted). [13] C. De Oliveira, et al., Performed of triggers to control thermal buckling of subsea pipelines using reduced scale model, The Twenty-fifth International Offshore and Polar Engineering Conference (2015) 281–290 (International Society of Offshore and Polar Engineers). [14] American Petroleum Institute (API), Recommended Practice for Planning Designing and Constructing Fixed Offshore Platforms—Working Stress Design. API RP 2A, 21st edition, American Petroleum Institute, Washington D.C, 2000. [15] H.A. Sollund, et al., Modal response of short pipelines spans on partial elastic foundations, Ocean Eng. 105 (2015) 217–230. [16] C. Ferreira, Gene expression programming in problem solving, in: Soft Computing and Industry, Springer, London, 2002, pp. 635–653. [17] J.R. Koza, Genetic Programming: on the Programming of Computers by Means of Natural Selection, MIT Press, Cambridge, MA, USA, 1992. [18] J.L. Pérez, et al., Optimization of existing equations using a new Genetic Programming algorithm: application to the shear strength of reinforced concrete beams, Adv. Eng. Softw. 50 (2012) 82–96. [19] A. Nazari, et al., Modelling of upheaval buckling of offshore pipelines buried in clay soil using genetic programming, Eng. Struct. 101 (2015) 306–317. [20] H.M. Azamathulla, et al., Genetic programming to predict bridge pier scour, J. Hydraul. Eng. 136 (3) (2010) 165–169. [21] H.M. Azamathulla, et al., Linear genetic programming to scour below submerged pipelines, Ocean Eng. 38 (8) (2011) 995–1000. [22] gep4j. https://code.google.com/archive/p/gep4j/.
Contents lists available at ScienceDirect
Applied Ocean Research journal homepage: www.elsevier.com/locate/apor
Full length article
Numerical study on lateral buckling of pipelines with imperfection and sleeper Zhe Wang a,b , Zhihua Chen a,b , Hongbo Liu a,b,∗ , Zechao Zhang b a b
State Key Laboratory of Hydraulic Engineering Simulation and safety, Tianjin University, Tianjin 300072, China Department of Civil Engineering, Tianjin University, Tianjin 300072, China
a r t i c l e
i n f o
Article history: Received 11 December 2015 Received in revised form 30 July 2017 Accepted 14 August 2017 Keywords: Subsea pipelines Lateral buckling Sleeper Finite element Genetic programming
a b s t r a c t Lateral buckling is an important issue in unburied high-temperature and high-pressure (HT/HP) subsea pipelines systems. The imperfection–sleeper method is one of the most well-known methods used to control lateral buckling of HT/HP pipelines. Pipelines–sleeper–seabed numerical models are established and verified to analyze the buckling behavior of pipelines using the imperfection–sleeper method. The influence of six main factors on lateral buckling behavior is investigated in details based on the numerical results. Equations of buckling displacement (buckling displacement is defined by the final displacement of the middle point of the pipelines), critical buckling force, and buckling stress (Mises stress) are proposed using the gene expression programming technique. These equations show good accuracy and can be used to assist in the design of sleepers and assess the compressive and stress levels of pipelines. © 2017 Published by Elsevier Ltd.
1. Introduction Subsea pipelines operated under high temperature and high inner pressure will develop compressive force because of seabed soil restraining its axial expansion. If the compressive force is sufficiently high, then global buckling will occur, which is similar to the buckling of a steel bar. For unburied subsea pipelines, which are common in deep-sea conditions, global buckling usually occurs in the lateral direction [1,2]. Lateral buckling may lead to fracture, collapse, or buckling propagation [3]. With the increase in operation temperature and pressure, controlling lateral buckling becomes an essential problem in the design of subsea HT/HP pipelines. A number of methods are used, such as snaked lay (imperfection) method [4], sleeper method [5], and distributed buoyancy method [6], to control lateral buckling responses. All of these methods are aimed at reducing the compressive force levels and buckling responses. This study focuses on the method that combines imperfections and sleepers. For pipelines with imperfections, a number of studies on lateral buckling of imperfect pipelines have been conducted in recent years. Miles and Calladine [7] provided a design formula to calculate the maximum buckling strain on the basis of the experimental and
∗ Corresponding author at: State Key Laboratory of Hydraulic Engineering Simulation and safety, Tianjin University, Tianjin 300072, China. E-mail address: [email protected] (H. Liu). http://dx.doi.org/10.1016/j.apor.2017.08.010 0141-1187/© 2017 Published by Elsevier Ltd.
numerical results. Karampour et al. [3] derived an analytical solution for pipelines with half-wavelength sinusoidal imperfection. Hong et al. [8] used the energy method to calculate the analytical solution for pipelines lateral buckling with a single-arch initial imperfection and indicated that a small single-arch initial imperfection is associated with snap buckling. Most of the lateral buckling studies have a reasonable two-dimensional assumption. However, the buckling problem becomes a three-dimensional (3-D) problem when a sleeper is considered. For pipelines with sleepers, Sinclair et al. [5] indicated that buckling mode 2 may occur in pipelines with sleepers, which should be ignored. One effective way to avoid mode 2 is to combine the sleeper method with the imperfection method. The initial imperfection not only reduces the critical buckling force but also guarantees that only mode 1 occurs, which will be discussed in Section 3. The proposed critical buckling force and maximum strain formulas cannot be applied directly to the pipelines with imperfections and sleepers. Therefore, formulas of critical buckling force, buckling displacement, and buckling stress are necessary to improve the design of pipelines with imperfections and sleepers. The lateral buckling behavior of subsea pipelines with imperfection and sleeper has been investigated numerically. First, a 3-D finite element model of a 2000 m-long pipelines–sleeper–seabed system is developed in Abaqus. Then, six factors which influence the buckling response are investigated to analyze the lateral buckling behavior. Finally, on the basis of the numerical results, three formulas (critical buckling force, buckling displacement, and buckling
104
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Fig. 1. The finite element model.
stress) related to pipelines with imperfections and sleepers are proposed by the gene expression programming (GEP) method. These equations can be used to design pipelines with both imperfection and sleeper.
Table 1 Pipelines property for verification.
2. Numerical model 2.1. Finite element model The finite element model provides a convenient method to calculate the lateral buckling of pipelines. The finite element model of the pipelines with imperfection and sleeper is described on the basis of the following four aspects: First, the element types are selected. The beam element is suitable for simulating pipelines structures because the subsea pipelines is an ultra-slender structure, which indicates that the length of one direction (pipelines axial direction) is more than the length of two other directions (pipelines cross-section). As a result, PIPE31 elements in Abaqus, i.e., two-node linear beam elements, are selected to simulate the pipelines [9]. It is noteworthy that thinwalled PIPE31 element is selected here; therefore the hoop stress is constant across the cross-section of the pipe. This assumption needs to be taken care when the section of pipelines is extremely thick, in which case the hoop stress varies across the section. The seabed is assumed to be a rigid surface, which is a common assumption in the traditional analytical method. Thus, rigid surface elements (R3D4 element in Abaqus) are selected to simulate the seabed [10]. The sleeper is also assumed to be a rigid surface with a semi-circle section, as shown in Fig. 1. Second, the constraint and load of the finite element model are determined. Most parts of the pipelines lie on the seabed and constrained by the seabed through contact elements. The middle point of the pipelines is held by a sleeper, which results in free segments on both sides of the middle point (Fig. 1). The ends of the pipelines are both far from the middle point (1000 m). Therefore, the pin constraint at the ends of the pipelines has only a slight influence on the buckling area. The seabed and sleeper are both fixed. The contact pressure–overclosure relationship of the interaction between pipelines and seabed (sleeper) is presented as a hard contact model. The Coulomb friction model is used to represent friction. 1 and 2 represent the friction coefficients of seabed and sleeper, respectively. A uniform temperature field, constant inner pressure, and distributed load in the z-direction are applied to the pipelines to simulate working conditions. Third, the imperfection is selected. A sinusoidal imperfection is used to achieve lower buckling force, as shown in Fig. 2. The imperfect segment of the pipelines can be expressed in Eq. (1), as follows:
h = h0 sin
(l + l0 /2) l0
Fig. 2. The sinusoidal imperfection.
(1)
Diameter (d) (mm)
15
Wall thickness (t) (mm) Young’s modulus (E) (MPa) Poisson’s ratio () Thermal coefficient of expansion (␣) (/◦ C) Submerged weight (q) (N/m) Sleeper height (h2 ) (mm) Seabed friction coefficient (1 ) Sleeper friction coefficient (2 ) Temperature (◦ C) Pressure (MPa)
0.9 191,000 0.3 1.73 × 10−5 3.94, 0.0 30 0.7 0.1, 0.2 35 12
where h0 and l0 are the amplitude and length of the imperfection, respectively. Finally, the quasi-static dynamic method is selected as the calculation method to simulate the lateral buckling of the pipelines, which means dynamic. Geometric nonlinearity switch in Abaqus is turned on at the beginning of the simulation, which means geometric stiffness is considered. The stiffness matrix of the model is updated during the simulation process.
2.2. Verification of the finite element model The same pipelines model used by De Oliveira et al. [13] in their experimental study is simulated in this section to test the accuracy of the proposed finite element model. Three of the experiments models are selected here to verify the proposed numerical model. The pipelines properties are shown in Table 1. Fig. 3 shows the axial compressive force and displacement results of the buckling apex obtained by the experiments and simulation. Only the representative data point is selected because many fluctuations exist in the original axial force experimental data. In Fig. 3, the temperature increase is shown along the x-axis, whereas the compression force of the pipelines is shown along the y-axis. All the three experiments show a similar pattern. The compression force keeps increasing with the increase in temperature until the temperature reaches approximately 5 ◦ C (critical buckling temperature). Then, the axial force decreases rapidly, which indicates that lateral buckling occurs. The apex of the compression force is referred to as the critical buckling force of the pipelines. Simulated critical buckling forces are 530 N, 462 N and 551 N with relative error −8.0%, −7.5% and − 9.2%. It is assumed that the differences between the simulated and experimental results are mainly due to exclusion of the pipelines coating.
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
105
Fig. 3. Axial force at the buckle apex.
3. Lateral buckling behavior Factors that influence the buckling behavior of the pipelines are studied in order to derive the formulas that control the buckling behavior of pipelines with imperfections and sleepers. The influences of different parameters on buckling responses are investigated in detail. Six parameters are selected, namely, bending stiffness (EI), imperfections (h0 /l0 ), submerged weight (q), friction coefficient of the seabed (1 ), friction coefficient of the sleeper (2 ), and sleeper height (h2 ). The values of the different parameters are shown in Table 2. First, the buckling process of the pipelines with imperfection and sleeper is analyzed from the point of lateral displacement along the length of the pipelines. Fig. 4a shows that buckling mode 1 occurs in the pipelines. The free span segment of the pipelines moves slightly toward the positive direction of y at the initial stage (temperature of 14 ◦ C). This phenomenon is different from the lateral buckling process of the pipelines without sleeper [12]. Afterward, the center lobe grows rapidly; at the same time, the lobes (side lobes) beside the center lobe grow toward the opposite direction (temperature of 39 ◦ C). Then, with the increase in temperature, secondary side lobes occur beside the side lobes. The final buckling region is as long as 200 m, which is symmetrical with the
Table 2 Pipelines properties discussed in Section 3. Diameter (d) (mm)
254, 381, 323.9, 445 [14]
Wall thickness (t) (mm) Young’s modulus (E) (MPa) Poisson’s ratio () Thermal coefficient of expansion (␣) (/◦ C) Imperfection (h0 /l0 ) Submerged weight (q) (N/m) Seabed friction coefficient (1 ) Sleeper friction coefficient (2 ) Sleeper height (h2 ) (m)
12.7 206,000 0.3 1.01 × 10−5 0.004–0.020 1000–3000 0.3–0.9 0.1–0.6 0.3–0.9
central point. Fig. 4b shows von Mises stress along the length of the pipelines, each point represents the average von Mises of the four integration points of that section. As shown in this figure, the maximum stress occurs near the midpoint area and all the stress peaks appear at the lobes. Fig. 5 shows the Mises stress distribution of the pipelines at 100 ◦ C. Here the Mises stress has three components: axial stress, hoop stress, and radial stress [11]. All the buckling processes, which will be discussed in the subsequent sections, follow the buckling pattern discussed in the previous section. As a result, only the compression
106
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Fig. 4. Buckling process of the pipelines with imperfection and sleeper (only the buckling region is shown in the figure).a) Lateral buckling shape along the length of the pipelines b) von Mises stress along the length of the pipelines.
Fig. 5. Stress distribution of the pipelines at 100 ◦ C.
force–displacement relationship, which is one of the most important properties that deal with buckling problems, is discussed in this section. All following calculations terminate at temperature 100 ◦ C. The base case of the parametric analysis is d = 254 mm, h0 /l0 = 0.012, q = 1500 N/m, 1 = 0.5, 2 = 0.3, and h2 = 0.5 m. Fig. 6. The influence of bending stiffness (EI).
3.1. Bending stiffness Bending stiffness of the pipelines is represented by the outer diameter. Four diameters are selected, i.e., D = 254, 381, 323.9, and 445 mm. Fig. 6 shows that the critical buckling force and post-buckling force clearly increase with the increase in bending stiffness. As shown in Fig. 4, the maximum displacement always shows in the middle of the pipelines. The final displacement also increases with the increase in bending stiffness because a large section area results in a large section force under the same temperature. 3.2. Imperfection (out-of-straightness) Fig. 7 shows that imperfection is a significant element in the analysis of buckling responses. The critical buckling force increases with the decrease in imperfection. Imperfections hardly influence the post-buckling force and final buckling displacement of the pipelines. Notably, the axial compressive force decreases rapidly after the critical buckling point if the imperfection is small. In such cases, the buckling process is always intense and dynamic, which we believe should be avoided in lateral buckling design. If the imperfection is large, then the axial compressive force decreases
Fig. 7. The influence of imperfection h0 /l0 .
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Fig. 8. The influence of submerged weight (q).
Fig. 10. The influence of seabed friction (2 ).
Fig. 9. The influence of seabed friction (1 ).
Fig. 11. The influence of sleeper height (h2 ).
107
gradually after the critical buckling point and the critical buckling force is similar to the post-buckling force.
3.5. Friction between the pipelines and sleeper
3.3. Submerged weight
Fig. 10 shows that the critical buckling force increases with the increase in sleeper friction, whereas the buckling displacement decreases with the increase in sleeper friction.
Submerged weight is also an important factor influencing the buckling process. Fig. 8 shows the influence of submerged weight on the buckling response of pipelines. The figure shows that a large submerged weight implies a high critical buckling force and a high post-buckling force. However, the final buckling displacement decreases with the increase in submerged weight. Notably, the axial compressive force with large submerged weight decreases faster than those with less weight. 3.4. Friction between the pipelines and seabed Fig. 9 shows that seabed friction only has a slight effect on critical buckling force and final buckling displacement compared with other factors because the critical buckling force and maximum buckling displacement are calculated at the area on the sleeper. Normally, the friction of the sleeper is lower than seabed friction. As a result, the buckling force and displacement mainly depend on the sleeper friction, as will be discussed in Section 3.5.
3.6. Sleeper height The sleeper height has only a slight influence on the critical buckling force, as shown in Fig. 11. We believe that the imperfection reduces the effect of sleeper height. The final buckling displacement increases with the increase in sleeper height. Notably, the free spanning segment induced by the sleeper is harmful to the pipelines through the phenomenon of vortex-induced vibration [15]. As a result, the imperfection can reduce the sleeper height in the lateral buckling design. In summary, for critical buckling force, bending stiffness, imperfections, submerged weight, and sleeper friction are the most important factors. For buckling displacement, bending stiffness, submerged weight, seabed friction, and sleeper friction combined with temperature and pressure (obviously) are the most significant factors. The results presented in this section serve as an important guidance in the subsequent section, which intends to derive
108
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Table 3 Statistical parameters of the input data.
EI (MNm2 ) d (m) t (m) h0 /l0 q (kN/m) 1 2 h2 T (◦ C)
Table 4 GEP parameters.
Range
Parameter
14–39 0.25–0.35 0.0127 0.009–0.015 1–3 0.3–0.7 0.1–0.6 0.1–1 80–100
Uniform Uniform Fixed Normal = 0.012, = 0.05 Uniform Normal, = 0.5, = 0.03 Normal, = 0.3, = 0.03 Normal, = 0.5, = 0.2 Uniform
Model 1 Model 2 Model 3
Item
Head size
Link function
Mutation
Number of variables
disp Pcr max
8 8 10
+ * *
0.009 0.009 0.005
7 5 7
Chromosomes = 30, number of genes = 5, range of constants = −10 to 10, constants per gene = 2, inversion = 0.05, transposition = 0.1.
The coefficient of determination (R2 ) is applied as another indicator to derive a high-performance model, which is obtained by using the following equation:
⎛
the formulas of buckling displacement, critical buckling force, and maximum buckling von Mises stress.
⎜ ⎜ R =1−⎜ ⎜ ⎝ 2
n
⎞
(ti − oi )
2
i=1 n
o2 i
⎟ ⎟ ⎟ ⎟ ⎠
. (3)
i=1
4. Genetic programming analysis In the design of lateral buckling of pipelines with imperfections and sleepers, buckling displacement, critical buckling force, and buckling stress are the key parameters. Buckling displacement indicates the length of the sleeper and critical buckling force indicates the axial force level along the pipelines, whereas buckling stress is used to assess the integrity of the pipelines. As a result, in this section, we generate three GEP [16] models to derive the formulas of the aforementioned key parameters. GEP is an extension of genetic programming (GP) [17], which is an evolutionary algorithm that automatically creates computer programs. GEP and GP have been successfully applied to many structural problems including pipelines problems (i.e., the shear strength of reinforced concrete beams is predicted using GP [18], upheaval buckling displacement of offshore pipelines buried in clay soil is predicted using GEP [19], and other applications [20,21]). Details of GEP can be found in [17] and [18].
4.1. Gene expression programming modeling Based on the results presented in Section 3, seven parameter categories, six parameters presented in Section 3, and temperature are selected in this section to comprise the formulas. A set of 400 randomly simulated results were selected to develop the prediction model using GEP. Table 3 shows the statistical properties of the parameters. Fig. 12 shows the frequencies of different parameters. Bending stiffness (EI), diameter (d), weight (q) and temperature (T) are sampled from a uniform distribution, the rest of the parameters are generated from a normal distribution. The datasets are divided randomly to 350 and 50 and are used to train and validate the model. The GEP modeling process is developed on the basis of the Java GEP library gep4 j [22]. Root mean square error (RMSE) is used as the fitness function in GEP according to the following equation:
n 1 RMSE = (ti − oi )2 n
The other GEP parameters are shown in Table 4. As shown in the table, head size means the length of gene’s head [17]. Link function is the function links different segments of the chromosome, we found that plus function works better for displacement model and manipulate function performs better for critical buckling force model and maximum stress model. Mutation is the probability of mutation of each model. The number of variables is the quantity of dependent variables in each model. The rest of the parameters are all the same among the models. Every generation has 30 chromosomes (individuals), each chromosome contains 5 genes. The range of constants in the model is between −10 to 10. The probability of inversion and transposition are 0.05 and 0.1 respectively. Models 1–3 represent the models of buckling displacement, critical buckling force, and buckling stress, respectively. Notably, temperature and sleeper height are not used as input parameters in Model 2 because these two factors have only a slight influence on the critical buckling force on the basis of the results presented in Section 3. Several trials were conducted to derive a high-performance model with the highest coefficient of determination (R2 ) and the highest fitness in all training and validating phases. Fig. 13 shows the distribution of the simulated results. 4.2. Equations for buckling force, buckling displacement, and maximum von mises stress The equations of all three proposed GEP models (simplified) are summarized in Table 5. Fig. 14 shows the correlation between simulated and predicted results in training and validation. All of the three models show good agreement with the simulated response. In Table 6, Model 1 shows the highest performance with the maximum training error of 3.5% and validation error of 3.0%. Although model 2 shows the lowest performance, only two data points have an error of more than 10.0%, as shown in Fig. 15e. Most of the errors of model 2 are less than 10.0%. The maximum error of model 3 is 6.0%. All of the three models have a high R2 value (larger than 95% in this study), which is considered high accuracy prediction [19]. 4.3. Parameter analysis
(2)
i
Where, n is the number of data points in the training set. ti is the result of the model and oi is the real (training) value.
Parametric analysis is conducted based on the proposed equations. Fig. 15 shows the results of the parametric analysis. The trends of the calculated curves are similar to simulated results presented in Section 3. Fig. 15a and b only show the equations of displacement and stress because temperature and sleeper height is not included in the equation (model 2). Notably, buckling stress
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Fig. 12. Distribution of the input parameters.
109
110
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Fig. 13. Distribution of the output results.
Table 5 Equations of the GEP models. Equation
Displac-ement
d=
abs
q+
tan q1/3
−
T − EI −5.57
1/3 +
abs
1 + EI −
h0 − 2 − 5.8593 0.3093l0
1/3
+ h2
− q − 2
+ arctan (h2 ) + T 1/3 −(tan (2 ))2+ 11.14392 EI h0 ∗ 2 ∗ tan −7.22454 + 79.68093 tanh(sin(2 )) Pcr = arctan arctan 1 − tan − 26.13666 l0
Critical Axial Force ∗ sin (tanh (q + 1 ) ∗ tan (1 − 1.22835)) ∗
max =
Maxim-nm Stress
T − 6.9519 (2 − h2 ) − tan
tanh
earctan(T )
arctan
q2 +
h0 l0
sin
q−q3
e
1 2
h0 l0
1/3
1/3 ∗ tanh
−
h0 + q − 10.4446 l0
1/3
2 − q − 2.06349q arctan (h2 − 8.37604)
h0 e l0 +
∗ tanh
q+
EIq (h2 − 0.30453) (q + 5.50048)
3.02842 − T + 58.16547 arctan (T )2
1/3
∗
∗
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Fig. 14. Error of the output results.
111
112
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
Fig. 15. Parametric analysis of the effect of the input parameters based on the proposed equations.
Z. Wang et al. / Applied Ocean Research 68 (2017) 103–113
113
Table 6 Performances of the GEP models.
Model 1 (Displacement) Model 2 (Critical Axial Force) Model 3 (Maximum Stress)
Maximum training error
Maximum validation error
R2
3.40% 13.00% 5.90%
3.00% 9.80% 6.00%
0.991 0.977 0.955
of the pipelines mainly depends on temperature (pressure), submerged weight, and sleeper height. Fig. 15e shows that seabed friction has only a slight influence on all three dependent variables, as expected. In other words, buckling response of the pipelines with sleeper and imperfection is relatively independent of seabed friction, which improves the probability of triggering lateral buckling on the complex marine soil. 5. Conclusion A numerical model of pipelines with imperfections and sleepers is generated to analyze the lateral buckling behavior of the pipelines. The numerical model shows good consistency with the existing experiment. Based on the simulated results, the GEP technique is used to establish three equations to calculate buckling displacement, critical buckling force, and buckling stress. The following main conclusions are drawn to summarize this study: Bending stiffness, submerged weight, seabed friction, and sleeper friction are the most significant factors influencing buckling displacement. Bending stiffness, imperfections, submerged weight, and sleeper friction are the most important factors influencing critical buckling force. Submerged weight and sleeper height are the most important factors influencing buckling stress. The sleeper height has only a slight influence on critical buckling force when an imperfection is considered. If the height of sleeper in a range between 0.3 m and 0.9m, the critical buckling force mainly depends on initial bending stiffness, imperfection, weight and friction. While the height of sleeper influences the buckling displacement and maximum stress dramatically. As a result, a small height leads to the decrease in free span length. Consequently, the response of vortex-induced vibration, buckling displacement and stress are reduced. Once the buckling force level determined, we can select a proper friction of sleeper using proposed equations. The lateral initial imperfection guarantees that only mode 1 occurs in pipelines with sleepers. The proposed buckling displacement equation exhibits the highest accuracy and can help design the length of the sleeper. The accuracy of buckling force and stress equations are acceptable for engineering use and can be applied to assess the compressive force and stress level of the pipelines. Acknowledgment The authors are grateful for the support provided by the National Basic Research Program of China (NO. 2014CB046801).
References [1] R.E. Hobbs, In-service buckling of heated pipelines, J. Transport Eng. 110 (2) (1984) 175–189. [2] Wang Zhe, et al., Overview of global buckling of deep-sea pipelines, J. Tianjin Univ. Sci. Technol. 47 (2014) 17–23. [3] H. Karampour, et al., On lateral and upheaval buckling of subsea pipelines, Eng. Struct. 52 (2013) 317–330. [4] Y.X. Liu, et al., Postbuckling analysis of snaked-Lay pipelines based on a new deformation shape, J. Offshore Mech. Arct. 135 (2013) 1–9 (031704). [5] F. Sinclair, et al., Design challenges and experience with controlled lateral buckle initiation methods, Proceedings of 28th International Conference on Ocean, Offshore and Arctic Engineering. Hawaii : American Society of Mechanical Engineers (2009) 319–330. [6] R. Shi, L.Z. Wang, Single buoyancy load to trigger lateral buckles in pipelines on a soft seabed, J. Eng. Mech. 141 (5) (2014) 0401415. [7] D.J. Miles, C.R. Calladine, Lateral thermal buckling of pipelines on the seabed, J. Appl. Mech. 66 (4) (1999) 891–897. [8] Z. Hong, et al., Study on lateral buckling characteristics of a submarine pipelines with a single arch symmetric initial imperfection, Ocean Eng. 108 (2015) 21–32. [9] Z. Wang, et al., Numerical study on upheaval buckling of pipe-in-pipe systems with full contact imperfections, Eng. Struct. 99 (2015) 264–271. [10] Z. Wang, et al., Static and dynamic analysis on upheaval buckling of unburied subsea pipelines, Ocean Eng. 104 (2015) 249–256. [11] ABAQUS, Analysis User Guide Version 6 14., Hibbit, Karlsson and Sorensen, Inc., RI, USA, 2014. [12] Z. Wang, et al., On Lateral Buckling of Subsea Pipe-in-pipe Systems International Journal of Steel Structures, 2017, 2017.09.(Accepted). [13] C. De Oliveira, et al., Performed of triggers to control thermal buckling of subsea pipelines using reduced scale model, The Twenty-fifth International Offshore and Polar Engineering Conference (2015) 281–290 (International Society of Offshore and Polar Engineers). [14] American Petroleum Institute (API), Recommended Practice for Planning Designing and Constructing Fixed Offshore Platforms—Working Stress Design. API RP 2A, 21st edition, American Petroleum Institute, Washington D.C, 2000. [15] H.A. Sollund, et al., Modal response of short pipelines spans on partial elastic foundations, Ocean Eng. 105 (2015) 217–230. [16] C. Ferreira, Gene expression programming in problem solving, in: Soft Computing and Industry, Springer, London, 2002, pp. 635–653. [17] J.R. Koza, Genetic Programming: on the Programming of Computers by Means of Natural Selection, MIT Press, Cambridge, MA, USA, 1992. [18] J.L. Pérez, et al., Optimization of existing equations using a new Genetic Programming algorithm: application to the shear strength of reinforced concrete beams, Adv. Eng. Softw. 50 (2012) 82–96. [19] A. Nazari, et al., Modelling of upheaval buckling of offshore pipelines buried in clay soil using genetic programming, Eng. Struct. 101 (2015) 306–317. [20] H.M. Azamathulla, et al., Genetic programming to predict bridge pier scour, J. Hydraul. Eng. 136 (3) (2010) 165–169. [21] H.M. Azamathulla, et al., Linear genetic programming to scour below submerged pipelines, Ocean Eng. 38 (8) (2011) 995–1000. [22] gep4j. https://code.google.com/archive/p/gep4j/.