Buckle propagation of offshore pipelines under external pressure

Marine Structures 29 (2012) 115–130

Contents lists available at SciVerse ScienceDirect

Marine Structures journal homepage: www.elsevier.com/locate/ marstruc

Buckle propagation of offshore pipelines under external pressure Shunfeng Gong, Bin Sun, Sheng Bao*, Yong Bai Institute of Structural Engineering, Zhejiang University, Hangzhou 310058, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 13 April 2012 Received in revised form 27 August 2012 Accepted 12 October 2012

Accidental damage of offshore pipelines in the form of local buckles induced by excessive bending deformation during deepwater installation may severely lead to local collapse of the tube and consequent buckle propagation along the pipeline. The present paper describes experimental and numerical research conducted to predict the magnitude of buckle propagation pressure of offshore pipelines under external pressure. The experiments of buckle propagation for pipe specimens with different initial geometric imperfections using 316 grade stainless steel tubes are carried out under quasi-static steady-state conditions in a sealed hyperbaric chamber. The stress–strain characteristics in the axial tensile test are measured for the tube material, and then used to numerically calculate the buckle propagation pressure of the pipe. The comparisons between experimental and numerical results are conducted to establish the precise numerical simulation technique. Based upon experimental and extensive numerical results, a more reasonable empirical formula for buckle propagation pressure of offshore pipeline with various values of diameterto-thickness ratio as well as different strain hardening modulus and yield stress is proposed.
2012 Elsevier Ltd. All rights reserved.

Keywords: Offshore pipeline Buckle propagation Collapse External pressure

1. Introduction In recent years, the increasing demand for oil and natural gas resources has prompted the industry to extend exploration and production to deepwater or ultra-deepwater regions. Pipelines play a very

* Corresponding author. Tel.: þ86 571 88208728. E-mail address: [email protected] (S. Bao). 0951-8339/$ – see front matter
2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.marstruc.2012.10.006

116

S. Gong et al. / Marine Structures 29 (2012) 115–130

Nomenclature D E E0 J2 n Pco Pp t D/t V0 dV w0 x, y, z

D0 ε

q n s s0 sy a

tube outer diameter Young’s modulus strain hardening modulus second invariant of deviatoric stress tensor Ramberg–Osgood hardening parameter collapse pressure buckle propagation pressure tube wall thickness diameter-to-thickness ratio of the tube initial internal volume of the tube volume change of the tube initial imperfection coordinates imperfection amplitude, D0 ¼ (Dmax  Dmin)/(Dmax þ Dmin) strain angular coordinate Poisson’s ratio stress API yield stress (0.5% strain offset) Ramberg–Osgood yield parameter strain hardening parameter, a ¼ E/E0

important role in these challenging activities, and are the most efficient and economical means of gathering and transporting oil and natural gas from subsea wells to offshore or coastal storage facilities. The current maximum laying water depth has exceeded 3000 m [1], and with ever increasing water depths, the pipes have to be designed to withstand the high loads dictated by the extreme water depth during installation as the pipeline is empty and not subject to any internal pressure. Accordingly, the development of higher strength and toughness steel pipes with thicker wall is required to resist ambient hydrostatic pressure. In general, the diameter-to-thickness ratios (D/t) of offshore pipeline range from 10 to 60. However, for deepwater scenarios, the pipe with a value of D/t ranging from 10 to 35 is more suitable. In addition, the yield stress of steel for typical offshore pipelines is commonly between 276 and 448 MPa. Local buckle is prone to occur in offshore pipelines due to excessive bending deformation nearby the touch-down point in the process of deepwater installation. If the external hydrostatic pressure is high enough, the buckle could catastrophically propagate along the pipelines, or even threaten their structural integrity [2,3]. The lowest pressure required to sustain the buckle propagation, is referred to as the propagation pressure Pp, which is a characteristic pressure of the pipe and only 15% to 30% of collapse pressure Pco of the intact pipe. In general, collapse pressure replaced by propagation pressure to guide the design of offshore pipeline is deemed to be very conservative and uneconomical. Therefore, a better choice is to install buckle arrestors at regular intervals along the pipeline, with the purpose of confining the extent of pipeline damage to a limited length between two adjacent arrestors. Buckle arrestors are the most commonly used devices that locally increase the wall bending rigidity of the pipe in the circumferential direction, and thus provide an obstacle in the path of the propagating buckle. There are many different types of arrestors presented, respectively, by Johns et al. [4], and Kyriakides and Babcock [5]. For deepwater applications, the integral buckle arrestor with superior efficiency is preferred to weld on to the pipe. It consists of a ring with certain length that has the same internal diameter but is thicker than the pipe. The modified empirical design formula for arrestor efficiency is proposed by Lee et al. [6] based on extensive experimental and numerical results. Using

S. Gong et al. / Marine Structures 29 (2012) 115–130

117

this formula to design buckle arrestors should predetermine buckle propagation pressure of the pipe. In the past few decades, many researchers have addressed in such predictions and put forth some empirical formula determined from theoretical and experimental results. Palmer and Martin [3] firstly put forth an equation for propagation pressure as a function of both diameter-to-thickness ratio D/t and material yield stress s0 based on the assumptions of the deformation energy of the collapsed pipe cross-section equating to the work done by external pressure, and rigid-perfectly plastic material behavior of the pipe,

PP

s0

¼ p


2 t : D

(1)

The results underestimated the observed values of tests in that the mode of deformation considered is purely inextensional so as to all deformation is assumed to be concentrated on four plastic hinges equally spaced around the perimeter, and the surface stretching and strain hardening of the pipe are neglected, thus it is only suitable for thin-walled pipes with larger values of D/t. As to deepwater thickwalled pipes, a propagating buckle will inevitably result in plastic and extensional circumferential deformations as well as surface stretching. Johns et al. [4] firstly conducted the experimental studies on the arresting capability of different types of buckle arrestors conceived to stop a propagating buckle along the pipeline. Based upon the experimental results of the pipes with the values of D/t ranging from 21 to 71, an empirical formula was proposed to evaluate the propagation pressure as follows:

PP

s0

¼ 6


2:5 2t : D

(2)

The dynamic behavior of a propagating buckle for the purpose of designing effective arresting devices was initially investigated through experiments by Kyriakides and Babcock [7], who subsequently implemented theoretical study in association with experimental results for aluminum tubes and steel alloy tubes to develop an empirical formula for buckle propagation pressure of offshore pipeline, taking into account the effects of post-yield characteristics of the pipe material behavior represented by the strain hardening modulus E0 and geometric parameter represented by the diameter-to-thickness ratio D/t of the pipe [8, 9],


0 
2:25 E t ¼ 10:7 þ 0:54 : s0 s0 D PP

(3)

The theoretical study was conducted to predict the propagation pressure of offshore pipeline based on internal energy dissipation of a quadrant of ring model including the influence of the material strain hardening behavior by Steel and Spence [10], which developed the following expression to evaluate the propagation pressure through some simplified assumptions,

PP

s0

¼


0 0:12
0:35 
2 E 2t 2t 1:0 þ 2:07 : s0 D D 4

p

(4)

DNV [11] in the Offshore Standard DNV-OS-F101 defined the calculating expression of the propagation pressure for the pipe with the values of D/t less than 45 as follows:

PP

s0

¼ 35


2:5 t : D

(5)

Similarly, API [12] in the Offshore Pipeline Criterion RP 1111 recommended the following formula for the propagation pressure,

PP

s0

¼ 24


2:4 t : D

(6)

118

S. Gong et al. / Marine Structures 29 (2012) 115–130

Lately, Albernami et al. [13] proposed a modification to the lower bound solution of Eq. (1) by accounting for the circumferential membrane and flexural effects in the pipe wall, and obtained the following expression:


2  t ¼ 1:193 p : s0 D PP

(7)

Through finite element analysis they further discovered that a faceted pipe has a substantial increase in initiation and propagation buckling capacities in contrast with the cylindrical pipe with the same value of D/t. The indeed great differences exist in predicting the buckle propagation pressures of offshore pipeline from the foregoing formula. The theoretical study [3,8–10] based upon the ring model generally neglects the effects of stretching and bending along the length of the profiles of propagation, and entirely concentrates on the ring-bending. In addition, the results predicted from Eqs. (2), (5) and (6) are almost very close for the values of D/t larger than 35. However, the drawbacks are very obvious as these equations only take into account the diameter-to-thickness ratio and overlook the post-yield characteristics of the pipe material behavior. Especially for the pipes with the values of D/t less than 35, the plastic buckle will be predominant. Therefore, a more reasonable empirical formula for predicting the buckle propagation pressure of offshore pipelines with various values of D/t should be developed further. The numerical simulation is no doubt a very effective means to predict the buckle propagation pressure for offshore pipelines. Jensen [14] firstly carried out theoretical and numerical study for a long circular cylindrical shell to obtain the propagation pressure based on thin shell theory and small strain approximation. The comparison of the calculated results for different elastic–plastic material descriptions shows that the J2 flow theory [15] predictions of propagation pressure are in better agreement with the experimental results. Dyau and Kyriakides [16] systematically reviewed the experimental aspects of propagation pressure of the tubes, then established a numerical analysis model based upon the Sander’s nonlinear shell kinematics with small strains and large displacements, and the elastic–plastic material behavior was modeled through the J2 flow theory [15] with isotropic strain hardening. The results as compared with experimental values from small-scale aluminum tubes and steel tubes with the values of D/t between 18.2 and 37.2 show that the proposed numerical analysis method may provide good estimates of the propagation pressure. Pasqualino and Estefen [17] developed a three-dimensional model based on the thin shell theory proposed by Sanders [18] with finite strains and finite displacements to predict the propagation pressure of deepwater pipelines, and the equilibrium equations were solved numerically using the finite difference method combined with the explicit dynamic relaxation technique. The correlation between numerical and experimental results for six small-scale steel pipe models with the values of D/t, respectively, equal to 16, 21 and 24 presented relatively good agreements. Xue and Hoo Fatt [19], respectively, conducted rigid-plastic theoretical analysis and finite element analysis using the software ABAQUS for the steady-state buckle propagation scenario in a corroded pipeline subjected to external hydrostatic pressure, and revealed the buckle propagation pressure and the buckling mode depending on the depth and angular extent of the corrosion. Furthermore, the buckle propagation phenomenon in pipe-in-pipe systems under external pressure was studied through combined experimental and analytical efforts by Kyriakides [20], and Kyriakides and Vogler [21], which put forth an empirical formula for propagation pressure of the system and design recommendations. Lately, in combination with experiments and theoretical analyses, Zara [22] investigated buckling propagation and failure causes of the pressurized double-piping system under the thermoconditioning installation in the sewage treatment plant. Showkati and Shahandeh [23] carried out the experimental studies on the buckling behavior of ring-stiffened thin-walled pipelines under external hydrostatic pressure, and appraised the influence of ring stiffeners on buckling strength, buckling propagation, development of yield line and final collapse of the pipeline. In association with practical pipe-laying circumstances, the buckling responses of thick-walled tubes under complicated load combinations were systematically investigated by Gong et al. [24,25]. In the present study, the buckle propagation experiments of steel pipes with the value of D/t equal to 15 are conducted in a sealed hyperbaric chamber to identify the mechanism of a propagating buckle

S. Gong et al. / Marine Structures 29 (2012) 115–130

119

in a long thick-walled pipe, which is suited to deepwater pipelines. In combination with experimental observations and results, a three-dimensional finite element model for the experimental steel pipe within the frame of ABAQUS is developed to model the buckle propagation phenomenon. The finite element model is validated by comparing numerical results with experimental observations. Subsequently, the parametric analysis for the propagation pressure is performed. Based on the extensive numerical simulations and experimental results, a more reasonable empirical formula for predicting the buckle propagation pressure of offshore pipeline is proposed taking into account geometric parameter with the values of D/t between 10 and 60, and material properties such as yield stress s0 and strain hardening modulus E0 . 2. Experiments 2.1. Experimental set-up and procedure The purpose of this experiment is to initiate a propagating buckle in a long thick-walled pipe under quasi-static steady-state conditions to measure the propagation pressure and observe the postbuckling configuration of the pipe. For a propagating buckle to occur in a long pipe, two important conditions are necessary. One is that the exterior surface of the pipe has to be subjected to high enough external pressure, and the other is that a large initial geometric imperfection at one end of the pipe should be imposed to initiate the propagating buckle. The experiments are conducted in a specially designed and fabricated pressure cylinder shown in Fig. 1(a). The cylinder has an inner diameter of 0.2 m and a length of 1.75 m, and its work pressure may attain 50 MPa. The external pressure applied on the pipe specimen is achieved by using a single-piston manual pump shown in Fig. 1(b) to pressurize water into the cylinder, in which the pressure is real-time monitored by an electrical transducer shown in Fig. 1(c) as well as the digital pressure gage shown in Fig. 1(b) with a precision of 0.01 MPa.

Fig. 1. Experimental facility.

120

S. Gong et al. / Marine Structures 29 (2012) 115–130

A section of pipe, welded two solid end-plugs on each end, usually more than 20 diameters long, is completely sealed at one end and placed into a stiff pressure cylinder, and the other end with a hole in the end-plug is connected with a container placed on the electronic scale through a rigid conduit, as shown in Fig. 1(d) and (e). Therefore, the internal pressure of the pipe specimen is all along maintained atmospheric pressure. The pressure cylinder, container, pipe specimen, and rigid conduit are completely filled with water. It is ensured by vertically placing the pressure cylinder in a specially built steel box fixed at the underground base with 2.0 m depth and venting water at the upper hole in the lid. The gap between pressure cylinder and steel box is filled up with fine sand so as to guarantee the stability of experimental set-up. The watertight test of the system is conducted by pressurizing water into the cylinder, and keeping a relatively long time and then observing the pressure change inside the cylinder through the digital pressure gage. To initiate a propagating buckle under external pressure, the pipe specimen over a short length about 2–3 diameters away from the end-plug is intentionally inflicted to form a dent. In the process of experiments, the pumping rate is always maintained at a relatively low, constant velocity about 1.0 MPa per minute, therefore any material rate effects may be negligible. Along with gradual increase of the pressure in the cylinder, local collapse is sure to firstly occur at the imperfection, which will lead to a sudden drop of the pressure in the cylinder to the level below the buckle propagation pressure of the pipe. Then, the buckle will gradually spread along the length while the pressure in the cylinder remains at Pp. From a hole in the upper end-plug of the pipe specimen the displaced water is directed to the container, the weight variation in the electronic scale is proportional to the displaced water and therefore to the variation of the specimen inner volume. A schematic diagram of the experimental apparatus assembly is shown in Fig. 2.

2.2. Experimental results The experiments are carried out using seamless stainless steel tubes, and the steel grade is SS316, which has a very good elongation. In such case, a propagating buckle will not result in fracturing of the wall or flooding of the tube, and the buckle can be propagated in tubes with much lower values of D/t. Therefore, the steel pipes with lower diameter-to-thickness ratio (D/t) and larger initial geometric imperfection can be adopted in experiments. The geometric characteristics of the pipes tested as well

Container

Rigid conduit

Electronic scale Water pump

Transducer

Pressure gage

Lid End-plug Initial imperfection

Pipe specimen

Pressure cylinder

Fig. 2. Schematic diagram of experimental apparatus.

S. Gong et al. / Marine Structures 29 (2012) 115–130

121

Table 1 Geometric parameters and experimental results No.

L (mm)

D (mm)

t (mm)

D/t

D0

Pco (MPa)

Pp (MPa)

s0 (MPa)

Pp/s0  103

1 2

1200 1200

60.0 60.0

4.0 4.0

15 15

2.0% 4.7%

27.6 21.4

12.3 11.6

319.2 319.2

38.5 36.3

as their propagation pressures measured are listed in Table 1. The local collapsed and final deformed configurations of the steel pipe specimen No. 2 are shown in Fig. 3, and the pipe after buckle propagation is completely flattened. In addition, the axial test coupon in Fig. 4 cut from the pipe is used to measure the engineering stress–strain response of the material. The measured stress–strain response is further transformed into true stress–logarithmic strain curve, as shown in Fig. 5. The elastic modulus E of the material is 188.9 GPa. The API [26] definition of yield stress s0, i.e., the stress at a strain of 0.005, is used. To conveniently conduct numerical simulation for different strain hardening parameter in predicting the propagation pressure, it is required a good representation of the material stress–strain behavior for the complete strain range of interest. Apparently, a bilinear approximation of the stress– strain behavior is too rough. A more accurate, three-parameter fit of the stress–strain behavior is provided by the well-known Ramberg–Osgood model [15] given by

ε ¼

    3 s n1 1 þ   : E 7 sy

s

Fig. 3. Deformed configurations of steel pipe specimen No. 2.

(8)

122

S. Gong et al. / Marine Structures 29 (2012) 115–130

Fig. 4. Material property test.

where sy is the effective yield stress, and n is the material hardening parameter. This material model in nature cannot fit the typical experimental results for strain levels larger than 1–2%. Fig. 5 similarly shows the fitted stress–strain curve through Eq. (8). As can be seen, a good fit is achieved for strain levels less than 0.015. However, for higher strains, it will cause a larger deviation. Due to this inadequacy of Eq. (8), the following modification is adopted in subsequent parametric analyses. The Ramberg–Osgood model is still adopted for strain levels less than 0.015. Whereas for strain levels larger than 0.015, the stress–strain curve is approximated with a straight line having a slope given by

700 600

σ / MPa

500 400 300

Modified R-O Fit R-O Fit Experimental

200 100 0 0.00

0.03

0.06

ε

0.09

0.12

0.15

Fig. 5. Measured and fitted uniaxial stress–strain curves for SS316.

S. Gong et al. / Marine Structures 29 (2012) 115–130

E0 ¼

 ds ¼ dε ε¼0:015

E :   3  s n1 1 þ n  7 s

123

(9)

y

where E0 , in fact, is the strain hardening modulus. The modified R–O fit is also drawn in Fig. 5, and is seen to be very close to the experimental curve. The fit parameters for the material SS316 are given as sy ¼ 291 MPa, n ¼ 10.1, E0 ¼ 2680 MPa. 3. Comparison of experimental and numerical results 3.1. Finite element model A finite element model is particularly developed in parallel with the experimental effort within the framework of ABAQUS to simulate numerically the quasi-static steady-state buckle propagation scenarios in the steel pipe specimen No. 2 presented in Table 1 under external pressure. The actual geometric characteristics and material behavior of the pipe are used to establish the numerical model. A three-dimensional, eight-node brick element with incompatible mode, C3D8I, is chosen to model the pipe. Since this type of element is enhanced by incompatible modes to bending behavior, it is well suited for the present large deformation problems [27,28]. The general geometric characteristics of the model are shown in Fig. 6. Guided by the experimental observations, the deformation of the pipe cross-section is assumed to have two planes of symmetry, namely, planes x-y and x-z. A local imperfection is added at the location of x ¼ 2D–3D. This is the only way of initiating local collapse without affecting subsequent buckle propagation. The tubes are assumed to be circular and the wall thickness to be uniform along the length. The local imperfection in the form of ovality is described in the following form:

Fig. 6. Mesh and imperfection of finite element model.

124

S. Gong et al. / Marine Structures 29 (2012) 115–130

8

   > D 2:25D  x 2 > > D b  cos2q; x˛ð2:0D; 2:25DÞ exp  > 0 > D > >
2 < D cos 2q; x˛ð2:25D; 2:75DÞ : w0 ðqÞ ¼ D0 > 2 > >

 2  > > D x  2:75D > > : D0 cos 2q; x˛ð2:75D; 3:0DÞ exp  b 2 D

(10)

where w0 is the radial displacement, q is the polar angular coordinate measured from y axis, and x is the axial coordinate. The imperfection parameters used are the initial ovality D0, and the constant b ¼ 0.5D, which characterizes the extent of the imperfection. The pipe model is discretized into 4 parts through the thickness, 20 parts around the quarter circumference, and 240 parts along the length, which is found to be adequate through trial analysis. The symmetrical boundary conditions are applied at planes x-y and x-z, i.e., in plane x-y, z-direction displacement of the nodes is only constrained, and in plane x-z, y-direction displacement of the nodes is only constrained. However, at both ends of the tube, all displacements of the nodes are fixed. Since only one quarter of the cross-section is analyzed, an imaginary rigid surface is set along plane x-z using rigid elements (R3D4) to simulate the contact of inner surfaces of the tube wall. Contact between the walls of the collapsing tube is modeled through the surface-based contact pair. In this scheme, the rigid surface is defined as the specified master surface, and the inner surface of the tube wall is defined as the slave surface. The contact direction is always normal to the master surface, and

Fig. 7. A sequence of deformed configurations for steel pipe specimen No. 2.

S. Gong et al. / Marine Structures 29 (2012) 115–130

125

the slave nodes are constrained not to penetrate into the master surface. In addition, the collapsing walls of the tube do not allow separation after contact, and the hard contact and finite sliding options are used. The J2 flow theory [15] of plasticity with isotropic strain hardening and finite deformation is adopted to describe the elastoplastic behavior of the pipe material. The multi-linear approximations of true stress–logarithmic strain curve and the modified Ramberg–Osgood model are used to characterize the material properties, respectively. As similar calculations have been conducted by Kyriakides and Vogler [21], a “volume controlled” loading procedure is adopted using the hydrostatic fluid elements of ABAQUS (a combination of F3D3 and F3D4). This type of elements can indicate the change in volume inside a control region defined around the structure. Thus, the pressure becomes an additional unknown while the volume change is enforced as a constraint via the Lagrange multiplier methods. 3.2. Simulation of a propagating buckle The pipe model is loaded by external pressure. Buckling and collapse are initiated by a small, local, initial geometric imperfection in the neighborhood of x ¼ 0. The response of such systems often exhibits limit load and turning point instabilities, thus Riks’ path-following method (arc length method) is used to follow the loading history. Nlgeom option is also selected for this problem characterized by several nonlinearities which severely varies during the loading history. An incremental solution scheme with variable loading increments depending on the stage of the loading history is developed empirically. Fig. 7 shows that a sequence of deformed configurations of the steel pipe specimen No. 2 is calculated by numerically simulating a propagating buckle. The main characteristics of the calculated configurations are similar to those seen in the experiments shown in Fig. 3. The initial configuration of the structure is identified by the numbered I. The configuration II represents a pipe of local collapse at the region of imperfections. When the opposite walls of the pipe come into contact in configuration III, the collapse is arrested locally and the buckle starts to propagate along the downstream pipe. The configuration IV illustrates the profile of buckle propagation of such a pipe, and the area of the pipe wall in contact is seen to have increased. Eventually, as the buckle propagation travels to both ends of the pipe and has to be terminated, the pipe is completely flattened in configuration V. Fig. 8 shows experimental and calculated pressure–change in volume responses for steel pipe specimen No. 2 (V0 is the initial internal volume of the tube, and dV is the absolute value of the change of volume evaluated for each deformed configuration). As can be seen in the figure, local collapse of the 30

Experiments FEM(Multi-linear Model) FEM(Modified R-O Model)

25

P/MPa

20 15 10 5 0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

δ V/V0 Fig. 8. Pressure–change in volume responses for steel pipe specimen No. 2.

126

S. Gong et al. / Marine Structures 29 (2012) 115–130

pipe results in a precipitous decrease in pressure, and the subsequent pressure plateau represents steady-state propagation of the buckle. Finally, two ends of the pipe are engaged by the buckle, the pressure in the closed vessel once again starts rising. It is noted that these responses have a very similar varying trend. The predicted and experimental results for collapse pressure and buckle propagation pressure are listed in Table 2. The errors are larger for the collapse pressure Pco than ones for the propagation pressure Pp, because the collapse pressure is more sensitive to initial imperfections and material properties of the tubes. In fact, it is unimportant in the present study and becomes negligible for the buckle propagation. As to propagation pressure, the errors are minor and the maximum difference is less than 8% for the specimen No. 2. However, for the specimen No. 1 listed in Table 1, the maximum difference reduces to less than 3%. The sources of errors are mainly attributed to variations of the tube thickness along the length, differences in material properties in the circumferential and longitudinal directions, differences in material properties in tension and compression, etc. In addition, the response calculated from the multi-linear approximation material model is compared to the one obtained from the modified R–O material model in Fig. 8. The predicted responses for two material models are almost always identical, and the observed differences for collapse pressure and propagation pressure listed in Table 2 are both less than 2%. Therefore, the modified R–O model is suitably used to conduct parametric study on buckle propagation pressure of the tube. In conclusion, it has been demonstrated that the aforementioned numerical model may offer a very satisfactory prediction on the buckle propagation of a long tube with various geometric characteristics and material properties. 4. Results and discussion The validity of numerical model proposed in the present study has been demonstrated in the foregoing section. In what follows, a parametric study of the propagation pressure is carried out numerically in order to illustrate the influence of different material parameters on the results and establish a more reasonable empirical formula. Previous research findings have revealed that the propagation pressure is primarily related to yield stress and post-yield characteristics of the pipe material behavior as well as diameter-to-thickness ratio. The analyses are, respectively, performed for the pipe models based the parameters of L ¼ 6096 mm, D ¼ 304.8 mm (12 inch), E ¼ 206 GPa, n ¼ 0.3, D/t ¼ 10, 12, 15, 18, 20, 22, 25, 30, 35, 40, 50 and 60, s0 ¼ 272, 350, 400 and 500 MPa, and the strain hardening parameter a ¼ E/E0 ¼ 25, 50, 102, 103 and 104. a ¼ 25 represents a very strong strain hardening material behavior, and a ¼ 104 represents the elastic perfectly plastic case. If the nominal yield stress s0 and the strain hardening parameter a are given, using Eqs. (7) and (8) the Ramberg–Osgood model parameters sy and n may be solved, thus the stress–strain curve for the modified Ramberg–Osgood model is completely determined, and then used to model the pipe material behavior. Fig. 9 shows the predicted propagation pressures including a total of 240 data points with varying diameter-to-thickness ratio and material property through extensive numerical simulations based on the software ABAQUS. It is clearly observed that for a given geometry, the tubes with smaller strain hardening parameter a (stronger post-yield behavior) always possess larger propagation pressure, and the effect of having a harder material on propagation pressure is essentially a shift of the straight line upwards. Similarly, it is seen that the tubes with larger material yield stress s0 characterize larger propagation pressure. In addition, it is worth noting that for the tubes with smaller values of D/t, the effect of yield stress is more pronounced compared with the case of larger values of D/t. The values of D/ t between 10 and 60 are carefully examined, because this is range of interest for pipeline application. The predicted results for various values of D/t indicate that the propagation pressure is a power law

Table 2 Comparison between experimental and numerical results for specimen No. 2. Parameter

Experimental value

FEM multi-linear model

Difference

FEM modified R–O model

Difference

Pco (MPa) Pp (MPa)

21.4 11.6

25.8 12.4

17.05% 6.45%

26.1 12.6

18.01% 7.94%

S. Gong et al. / Marine Structures 29 (2012) 115–130

127

dependence on this parameter. Summarizing the above-mentioned results, the propagation pressure is heavily affected by the strain hardening modulus E0 and the yield stress s0. Therefore, it can be concluded that the propagation pressure PP has a parametric behavior as follows:


0 
b 
g E t t ¼ aþb : s0 s0 D D PP

(11)

where the coefficients a and b, and the exponents b and g are to be fitted. It is more useful for design purposes to have a simple, single equation for the propagation pressure. Based upon the numerical results, the unknown parameters in Eq. (11) are determined by nonlinear least-squares estimation procedures that used the Levenberg–Marquardt algorithm. Accordingly, the expression of propagation pressure can be written as:


0 

2:4 E t t ¼ 23 þ 13 s0 s0 D D PP

(12)

All curves shown in Fig. 9 are calculated by Eq. (12), it is seen that these data points are very close to the corresponding curve, the difference of results between numerical simulation and empirical formula is within 10% other than a few data points, but all differences are all less than 12%.

a 100

α

1.0 ×10

b

1.0 ×10 Pp/MPa

3

1.0 ×10 50 25

10

3

2

1

10

20

30 D/t

40

50

1.0 ×10 1.0 ×10 50 25

30 D/t

40

50

60

σ 0 = 350MPa 100

α

4

4

1.0 ×10

3

1.0 ×10

1.0 ×10 Pp/MPa

20

d α

10

10

2

1.0 ×10 50 25

10 Pp/MPa

100

0

60

σ 0 = 272MPa

c

3

2

1

1

0

4

1.0 ×10

2

1

0

α

1.0 ×10

Pp/MPa

1.0 ×10 50 25

10

100

4

10

20

30 D/t

40

σ 0 = 400MPa

50

60

0

10

20

30 D/t

40

50

60

σ 0 = 500MPa

Fig. 9. Predicted propagation pressures as a function of (D/t) for various values of strain hardening factor and yield stress.

128

S. Gong et al. / Marine Structures 29 (2012) 115–130

20

PP(Experimental Results)

15

10 Kyriakides Gong Estefen

5

0

0

5

10

15

20

2.4 ⎡ ⎛ E ' ⎞ ⎛ t ⎞⎤ ⎛ t ⎞ Pp = σ 0 ⎢23 + 13 × ⎜⎜ ⎟⎟ ⎜ ⎟⎥ ⎜ ⎟ ⎝ σ 0 ⎠ ⎝ D ⎠⎦ ⎝ D ⎠ ⎣

Fig. 10. Comparison between experimental results and predicted values from empirical formula.

Fig. 10 shows the comparison between the experimental results collected from previous limited information and the predicted values via Eq. (12), the abscissa of data points is the calculated value of Eq. (12), and the ordinate represents the corresponding experimental result. Most of the data points locate at the vicinity of the diagonal line, and their differences are very small and within 10% other than several exceptional data points. The good correlation between predicted and experimental results validates the empirical expression and the proposed numerical simulation technique. Illustrated in Fig. 11 is the propagation pressure of the tubes with varying values of D/t predicted from different empirical formula. It is clearly shown that the predicted results through Eq. (4) proposed by Steel and Spence [10] generate a relative larger deviation. For larger values of D/t, this empirical formula overestimates the propagation pressure, whereas for lower D/t values, it contrarily underestimates the propagation pressure. For the tubes with the values of D/t larger than 35, the predicted results from Eq. (5) [11], Eq. (6) [12] and Eq. (12) are comparatively consistent. However, the empirical

100 Gong Kyriakides Steel API DNV

P p/MPa

10

1

0

10

20

30

40

50

60

D/t Fig. 11. Comparison of the results predicted from different empirical formula. (s0 ¼ 310 MPa, E0 ¼ 2.06 GPa).

S. Gong et al. / Marine Structures 29 (2012) 115–130

129

formula, respectively, proposed by Kyriakides and Babcock [8], and Steel and Spence [10] apparently overestimate the propagation pressure for this range of D/t values. As to the values of D/t between 15 and 35, the empirical formula, respectively, proposed by Kyriakides and Babcock [8], and DNV [11], as well as Eq. (12) provide very close predictions of propagation pressure. But the results, respectively, obtained from Eqs. (4) and (6) actually underestimate the propagation pressure. In addition, when the values of D/t are between 10 and 15, the predicted results for propagation pressure by Eq. (12) are larger than ones of other empirical formula. In conclusion, through aforementioned comparative analysis of predicted results for propagation pressure from different empirical formula, Eq. (12) proposed in the present study is able to provide very credible results. 5. Conclusions The experiments have been successfully conducted to identify the mechanism of a propagating buckle in a long thick-walled pipe under quasi-static steady-state conditions in a sealed hyperbaric chamber. In combination with experimental observations and results, a three-dimensional finite element model for the experimental steel pipe within the frame of ABAQUS is developed to model the buckle propagation phenomenon. The good correlation between numerical and experimental results demonstrates that the numerical model can accurately predict the total buckle propagation process of the tube under external pressure. To study the parametric dependence of the propagation pressure, the modified Ramberg–Osgood model is used to characterize the stress–strain behavior of the pipe material, and then a series of parametric study on the propagation pressure of offshore pipelines with the values of D/t ranging from 10 to 60 is conducted adopting the numerical technique developed. The following conclusions can be drawn. (1) If the material and geometric parameters of the tube are carefully established, the presented numerical technique can offer an accurate prediction of the buckle propagation pressure. (2) The material property has an important effect on the buckle propagation pressure. The paramount influencing factors are the strain hardening modulus E0 and the yield stress s0. The tubes with larger strain hardening modulus and yield stress always possess larger propagation pressure. For lower D/t values, the effects of strain hardening modulus and yield stress are more pronounced compared with the case of larger values of D/t. Therefore, higher strength and toughness steel should be preferred to enhance the post-buckling behaviors in practical engineering, especially for deepwater pipes. (3) Based on experimental and extensive numerical results, an empirical formula for buckle propagation pressure Pp of offshore pipeline with various values of D/t as well as different strain hardening modulus and yield stress is proposed in the present study. The close agreement between predicted and experimental results shows that the empirical formula can provide very reasonable estimates of the propagation pressure.

Acknowledgments The authors express their gratitude to the National Natural Science Foundation of China (Grant No. 51009122), the Fundamental Research Funds for the Central Universities (Grant No. 2010QNA4030), and the National Science and Technology Major Project of China (Grant No. 2011ZX05027-002-005011) for the financial support to this study. References [1] Hillenbrand HG, Graef MK Grob-Weege J, Knauf G, Marewski U. Development of linepipe for deepwater applications. In: Proceedings of the 12th international offshore and polar engineering conference, Kitakyushu, Japan; 2002, pp. 287–294. [2] Mesloh RE, Sorenson JE, Atterbury TJ. Buckling and offshore pipelines. Gas Mag 1973;7:40–3. [3] Palmer AC, Martin JH. Buckle propagation in submarine pipelines. Nature 1975;254:46–8. [4] Johns TG, Mesloh RE, Sorenson JE. Propagating buckle arrestors for offshore pipelines. In: Offshore technology conference, Houston, Texas, USA, OTC2680; 1976, pp. 721–730.

130

S. Gong et al. / Marine Structures 29 (2012) 115–130

[5] Kyriakides S, Babcock CD. On the slip-on buckle arrestor for offshore pipelines. ASME J Pressure Vessel Technol 1980; 102(2):188–93. [6] Lee LH, Kyriakides S, Netto TA. Integral buckle arrestors for offshore pipelines: enhanced design criteria. Int J Mech Sci 2008;50:1058–64. [7] Kyriakides S, Babcock CD. On the dynamics and the arrest of the propagating buckle in offshore pipelines. In” Offshore technology conference, Houston, Texas, USA, OTC3479; 1979, pp. 1035–1045. [8] Kyriakides S, Babcock CD. Experimental determination of the propagation pressure of circular pipes. ASME J Pressure Vessel Technol 1981;103:328–36. [9] Kyriakides S, Yeh MK, Roach D. On the determination of the propagation pressure of long circular tubes. ASME J Pressure Vessel Technol 1984;106:150–9. [10] Steel WJM, Spence J. On the propagating buckles and their arrest in sub-sea pipelines. Proc Inst Mech Eng 1983;197A:139–47. [11] DNV (Det Norske Veritas). Offshore standard. DNV-OS-F101: Submarine Pipeline Systems. Høvik: DNV; 2007. [12] API (American Petroleum Institute). API recommended practice 1111: design, construction, operation, and maintenance of offshore hydrocarbon pipelines (limit state design). 4th ed. Washington: API Publishing Services; 2009. [13] Albermani F, Khalilpasha H, Karampour H. Propagation buckling in deep sub-sea pipelines. Eng Struct 2011;33:2547–53. [14] Jensen HM. Collapse of hydrostatically loaded cylindrical shells. Int J Solids Struct 1988;24(1):51–64. [15] Chakrabarty J. Theory of plasticity. Oxford, UK: Elsevier Butterworth-Heinemann; 2006. [16] Dyau JY, Kyriakides S. On the propagation pressure of long cylindrical shells under external pressure. Int J Mech Sci 1993; 35(8):675–713. [17] Pasqualino IP, Estefen SF. A nonlinear analysis of the buckle propagation problem in deepwater pipelines. Int J Solids Struct 2001;38:8481–502. [18] Sanders JL. Nonlinear theories of thin shells. Q Appl Math 1963;21(1):21–36. [19] Xue JH, Hoo Fatt MS. Symmetric and anti-symmetric buckle propagation modes in subsea corroded pipelines. Mar Struct 2005;18:43–61. [20] Kyriakides S. Buckle propagation in pipe-in-pipe systems. Part I: experiments. Int J Solids Struct 2002;39:351–66. [21] Kyriakides S, Vogler TJ. Buckle propagation in pipe-in-pipe system. Part II: analysis. Int J Solids Struct 2002;39:367–92. [22] Zaras J. Analysis of an industrial piping installation under buckling propagation. Thin Walled Struct 2008;46:855–9. [23] Showkati H, Shahandeh R. Experiments on the buckling behavior of ring-stiffened pipeline under hydrostatic pressure. ASCE J Eng Mech 2010;136(4):464–71. [24] Gong SF, Yuan L, Jin WL. Buckling response of offshore pipelines under combined tension, bending, and external pressure. J Zhejiang Univ Sci A (Appl Phys Eng) 2011;12(8):627–36. [25] Gong SF, Ni XY, Yuan L, Jin WL. Buckling response of offshore pipelines under combined tension and bending. Struct Eng Mech 2012;41(6):805–22. [26] API (American Petroleum Institute). API specifications 5L: specifications for line pipe. 43rd ed. Washington: API Publishing Services; 2004. [27] Hibbitt HD, Karlsson BI, Sorensen P. ABAQUS theory manual, version 6.3. Pawtucket, Rhode Island, USA; 2006. [28] Simo JC, Armero F. Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int J Numer Methods Eng 1992;33(7):1413–49.