Propagation Buckling of Subsea Pipelines and Pipe-in-Pipe Systems

Hassan Karampour and Mahmoud Alrsai

## Abstract

This chapter investigates buckle propagation of subsea single-walled pipeline and pipe-in-pipe (PIP) systems under hydrostatic pressure, using 2D analytical solutions, hyperbaric chamber tests and 3D FE analyses. Experimental results are presented using hyperbaric chamber tests, and are compared with a modified analytical solution and with numerical results using finite element analysis for single-walled pipelines and PIPs. The experimental investigation is conducted using commercial aluminum tubes with diameter-to-thickness (D/t) ratio in the range 20–48. The comparison indicates that the modified analytical expression presented in this work provides a more accurate lower bound estimate of the propagation buckling pressure of PIPs compared to the existing equations, especially for higher Do/to ratios. A 3D FE model is developed and is validated against the experimental results of the propagation bucking. A parametric FE study is carried out and empirical expressions are provided for buckle propagation pressures of PIPs with (Do/to) ratio in the range 15–25. Moreover, empirical expressions are proposed for the collapse pressure of the inner pipe (Pci), the proposed empirical equation for Pci, is shown to agree well with the experimental results of the tested PIPs.

Keywords: collapse pressure, external pressure, offshore pipelines, pipe-in-pipe, propagation buckling

### 1. Propagation buckling of single pipe

#### 1.1 Introduction

Deep and ultra-deep water pipelines are vulnerable to propagation buckling due to the high external pressures. The pipeline may collapse due to the local dents, imperfections and ovalizations in the pipe-wall. This collapse will change the cross-section of the pipeline from a circular shape into a dog-bone or even flat shape. The buckle may then propagate along the pipeline and cause the pipeline to be shut down. A typical propagation buckle scenario is shown in Figure 1, which is triggered by impact on the pipeline from an anchor dropped from a passing vessel.

Different stages of the buckle are shown in Figure 1 in terms of the external pressure versus change in volume of the pipe. The dent caused by the impact can initiate the buckle due to high external pressure. The elastic buckling is followed by a plastic collapse and change in the cross-section of the tube from circular to oval

length, forcing the flow line to be shut. The lowest pressure that maintains propagation is known as the propagation pressure, and is much smaller than the collapse pressure. To account for the difference between the collapse and the propagation

As shown in Figure 1, the propagation pressure is much less than initiation pressure (peak pressure in Figure 1). The initiation pressure is significantly affected by the size of the local dent. Local dents may also occur during the installation period. The most common types of offshore pipeline installation are S-lay method, J-lay method, Reel-lay method and Towing method. A combination of bending and external pressure happens in the sag bend length of the pipe. Normally high tension is applied to the pipe to maintain its stiffness during installation. If for any reason this tension is released, high bending in the sag bend region may cause local buckling which may be followed by propagation buckling. Apart from the foretold

loading sources, manufacturing imperfections in pipe such as non-uniform

thickness, varying elastic modulus, local ovalization, and also erosion and corrosion

As stated before, a local dent or ovalization in the pipe wall can cause a local collapse as in the pipe-wall. It is well-known that the collapse pressure of a 2D

4 1 � ν<sup>2</sup> ð Þ

where E is the modulus of elasticity, ν is the Poisson's ratio, t is the pipe wall thickness and r is the mean radius of pipe. As shown in Figure 1 prior to the collapse pressure no significant change in cross section of pipe is observed. Note that for

exaggerated. During the propagation buckling the pipe endures substantial change

A typical buckle propagation response is characterized by the pressure at which the snap-through takes place (the initiation pressure PI) and the pressure that maintains propagation (the propagation pressure Pp) which is a small

Palmer and Martin [4] suggested a 2D approximation for propagation buckling of subsea pipelines Eq. (2). Their solution is based on a 2D ring collapse (plane strain) mechanism, and accounts for the circumferential bending effect of the pipe wall (see Figure 2). The Palmer and Martin (PM) solution underestimates the propagation pressure when compared to experimental results. This difference increases as D/t decreases. The propagation pressure from the PM solution, PPM,

t r <sup>3</sup>

(1)

arch (similar to a single pipeline (Pcr)), under lateral pressure can be

Pcr <sup>¼</sup> <sup>E</sup>

sake of clarity the slope of line ending to collapse pressure in Figure 1 is

1.2 Analytical solution of propagation pressure of single pipe

Many researchers have investigated various aspects of this problem since it was first presented by Mesloh et al. [3] and Palmer and Martin [4]. Most notably is the extensive work of Kyriakides [5, 6], Kamalarasa [7] and Albermani et al. [2]. Recent books by Kyriakides [1] and Palmer and King [8] provide comprehensive review of this problem and the associated literature. The work done by Xue et al. [9] investigates the effect of corrosion in the propagation buckling of subsea pipelines. Buckle arrestors [1, 18], pipe-in-pipe system [10–14], sandwich pipe system [15] and ring-stiffened pipelines [16], are used to confine the propagation buckling in

pressures in design, a thick-walled pipeline is required [1, 2].

Propagation Buckling of Subsea Pipelines and Pipe-in-Pipe Systems

DOI: http://dx.doi.org/10.5772/intechopen.85786

may cause local buckling in pipelines.

subsea pipelines.

in its shape.

fraction of PI.

is given by:

117

approximated by [17]:

and finally a dog-bone shape. If the pressure is maintained, the buckle will propagate quickly along the length of the pipe. Offshore pipelines normally experience high service external pressure; therefore the buckle will propagate through the

#### Propagation Buckling of Subsea Pipelines and Pipe-in-Pipe Systems DOI: http://dx.doi.org/10.5772/intechopen.85786

length, forcing the flow line to be shut. The lowest pressure that maintains propagation is known as the propagation pressure, and is much smaller than the collapse pressure. To account for the difference between the collapse and the propagation pressures in design, a thick-walled pipeline is required [1, 2].

As shown in Figure 1, the propagation pressure is much less than initiation pressure (peak pressure in Figure 1). The initiation pressure is significantly affected by the size of the local dent. Local dents may also occur during the installation period. The most common types of offshore pipeline installation are S-lay method, J-lay method, Reel-lay method and Towing method. A combination of bending and external pressure happens in the sag bend length of the pipe. Normally high tension is applied to the pipe to maintain its stiffness during installation. If for any reason this tension is released, high bending in the sag bend region may cause local buckling which may be followed by propagation buckling. Apart from the foretold loading sources, manufacturing imperfections in pipe such as non-uniform thickness, varying elastic modulus, local ovalization, and also erosion and corrosion may cause local buckling in pipelines.

Many researchers have investigated various aspects of this problem since it was first presented by Mesloh et al. [3] and Palmer and Martin [4]. Most notably is the extensive work of Kyriakides [5, 6], Kamalarasa [7] and Albermani et al. [2]. Recent books by Kyriakides [1] and Palmer and King [8] provide comprehensive review of this problem and the associated literature. The work done by Xue et al. [9] investigates the effect of corrosion in the propagation buckling of subsea pipelines. Buckle arrestors [1, 18], pipe-in-pipe system [10–14], sandwich pipe system [15] and ring-stiffened pipelines [16], are used to confine the propagation buckling in subsea pipelines.

As stated before, a local dent or ovalization in the pipe wall can cause a local collapse as in the pipe-wall. It is well-known that the collapse pressure of a 2D arch (similar to a single pipeline (Pcr)), under lateral pressure can be approximated by [17]:

$$P\_{cr} = \frac{E}{4(1 - \nu^2)} \left(\frac{t}{r}\right)^3 \tag{1}$$

where E is the modulus of elasticity, ν is the Poisson's ratio, t is the pipe wall thickness and r is the mean radius of pipe. As shown in Figure 1 prior to the collapse pressure no significant change in cross section of pipe is observed. Note that for sake of clarity the slope of line ending to collapse pressure in Figure 1 is exaggerated. During the propagation buckling the pipe endures substantial change in its shape.

#### 1.2 Analytical solution of propagation pressure of single pipe

A typical buckle propagation response is characterized by the pressure at which the snap-through takes place (the initiation pressure PI) and the pressure that maintains propagation (the propagation pressure Pp) which is a small fraction of PI.

Palmer and Martin [4] suggested a 2D approximation for propagation buckling of subsea pipelines Eq. (2). Their solution is based on a 2D ring collapse (plane strain) mechanism, and accounts for the circumferential bending effect of the pipe wall (see Figure 2). The Palmer and Martin (PM) solution underestimates the propagation pressure when compared to experimental results. This difference increases as D/t decreases. The propagation pressure from the PM solution, PPM, is given by:

and finally a dog-bone shape. If the pressure is maintained, the buckle will propagate quickly along the length of the pipe. Offshore pipelines normally experience high service external pressure; therefore the buckle will propagate through the

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Figure 1.

116

Buckle propagation scenario [1].

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$$P\_{\rm PM} = \frac{\pi}{4} \sigma\_{\rm \y} \left(\frac{t}{r}\right)^2 \tag{2}$$

Δl ¼ 0:626r (6)

� �<sup>2</sup> � � <sup>¼</sup> <sup>1</sup>:193pPM (8)

<sup>4</sup> (7)

mp ¼ σ<sup>y</sup>

π 4 σy t r

(Eq. (8)) is 19% higher than the PM prediction Eq. (2), regardless of t/r ratio. However, it should be noted, that by adopting plane strain conditions, the tensile coupon yield stress can be augmented by a factor of (2/√3) in (Eq. (8)) which

1.3 Experiments on propagation buckling of single-walled pipelines

<sup>p</sup><sup>~</sup> <sup>¼</sup> <sup>3</sup> 2:515

Propagation Buckling of Subsea Pipelines and Pipe-in-Pipe Systems

DOI: http://dx.doi.org/10.5772/intechopen.85786

results in an additional 15% increase in <sup>e</sup>p.

ovalization ratio Ω (Eq. (9)) around 0.46–0.67%

Figure 3.

119

Substituting Eqs. (5)–(7) into (4), the propagation pressure, <sup>e</sup>p, is obtained as:

Experimental observations confirm that the propagation pressure predicted by

A stiff 4 m long hyperbaric chamber rated for 20 MPa (2000 m water depth) internal pressure was used for testing (Figure 3a). Three meter long aluminum pipes were used in the hyperbaric chamber tests [2]. Ovalization measurements along the pipe samples before testing were carried out that gave an average

The experimental set-up: (a) the hyperbaric chamber, high-pressure pump, scales, pressure gauge and vents,

(b) pipes and fittings, (c) failed pipes tested in the hyperbaric chamber.

t 2

for a pipe with radius, r, wall thickness, t, and material yield stress, σy. Based on experimental observations from hyperbaric chamber tests, the top and bottom hinges in Figure 2a move towards each other while the left and right hinges move laterally away from each other. This deformation continues until touchdown (Figure 2b), the lateral movement seizes and flattening of the resulting four arch segments commence (Figure 2c).

Accordingly, a modification to the lower bound PM solution is proposed [2], by accounting for the circumferential membrane as well as flexural effects in the pipe wall

$$\mathcal{W}\_{\rm ex} = (\mathcal{W}\_{\rm in})\_{\rm f} + (\mathcal{W}\_{\rm in})\_{\rm m} \tag{3}$$

where Wex is the external work done by the net hydrostatic pressure and Win is the internal work due to circumferential flexure, f, and membrane, m, effects. The initially circular cross section of the pipe (Figure 2a) will deform into a dog-bone (Figure 2b) and eventually into a nearly flat segment. Accordingly, (Eq. (3)) can be written as:

$$p(\Delta A) = \mathfrak{Z}m\_p + (pr)(\Delta l) \tag{4}$$

where ΔA is the change in the cross section area, Δl is the change in the circumferential length and mpis the plastic moment, these are given by:

$$
\Delta A = \pi r^2 \tag{5}
$$

#### Figure 2.

A schematic of 2D deformation stages in propagation buckling of single pipe; (a) the initial circular cross section of the single pipe; (b) dog-bone deformed shape; (c) flat segment of the pipe.

Propagation Buckling of Subsea Pipelines and Pipe-in-Pipe Systems DOI: http://dx.doi.org/10.5772/intechopen.85786

PPM <sup>¼</sup> <sup>π</sup>

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experimental observations from hyperbaric chamber tests, the top and bottom hinges in Figure 2a move towards each other while the left and right hinges move laterally away from each other. This deformation continues until touchdown (Figure 2b), the lateral movement seizes and flattening of the resulting four arch

segments commence (Figure 2c).

wall

written as:

Figure 2.

118

4 σy t r <sup>2</sup>

for a pipe with radius, r, wall thickness, t, and material yield stress, σy. Based on

Accordingly, a modification to the lower bound PM solution is proposed [2], by accounting for the circumferential membrane as well as flexural effects in the pipe

where Wex is the external work done by the net hydrostatic pressure and Win is the internal work due to circumferential flexure, f, and membrane, m, effects. The initially circular cross section of the pipe (Figure 2a) will deform into a dog-bone (Figure 2b) and eventually into a nearly flat segment. Accordingly, (Eq. (3)) can be

where ΔA is the change in the cross section area, Δl is the change in the circum-

ΔA ¼ πr

A schematic of 2D deformation stages in propagation buckling of single pipe; (a) the initial circular cross section

of the single pipe; (b) dog-bone deformed shape; (c) flat segment of the pipe.

ferential length and mpis the plastic moment, these are given by:

Wex ¼ ð Þ Win <sup>f</sup> þ ð Þ Win <sup>m</sup> (3)

pð Þ¼ ΔA 3πmp þ ð Þ pr ð Þ Δl (4)

<sup>2</sup> (5)

(2)

$$
\Delta l = 0.626r \tag{6}
$$

$$m\_p = \sigma\_\gamma \frac{t^2}{4} \tag{7}$$

Substituting Eqs. (5)–(7) into (4), the propagation pressure, <sup>e</sup>p, is obtained as:

$$\bar{p} = \frac{3}{2.515} \left[ \frac{\pi}{4} \sigma\_\circ \left( \frac{t}{v} \right)^2 \right] = 1.193 p\_{\text{PM}} \tag{8}$$

Experimental observations confirm that the propagation pressure predicted by (Eq. (8)) is 19% higher than the PM prediction Eq. (2), regardless of t/r ratio. However, it should be noted, that by adopting plane strain conditions, the tensile coupon yield stress can be augmented by a factor of (2/√3) in (Eq. (8)) which results in an additional 15% increase in <sup>e</sup>p.

#### 1.3 Experiments on propagation buckling of single-walled pipelines

A stiff 4 m long hyperbaric chamber rated for 20 MPa (2000 m water depth) internal pressure was used for testing (Figure 3a). Three meter long aluminum pipes were used in the hyperbaric chamber tests [2]. Ovalization measurements along the pipe samples before testing were carried out that gave an average ovalization ratio Ω (Eq. (9)) around 0.46–0.67%

#### Figure 3.

The experimental set-up: (a) the hyperbaric chamber, high-pressure pump, scales, pressure gauge and vents, (b) pipes and fittings, (c) failed pipes tested in the hyperbaric chamber.

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$$\Omega\_0 = \frac{D\_{\text{max}} - D\_{\text{min}}}{D\_{\text{max}} + D\_{\text{min}}} \tag{9}$$

modified analytical solution p~ (Eq. (8)) vary from 1.428 to 1.9 depending on D/t ratio. Mesloh et al. [19] suggested similar relations. The ratio of PI/PP from the hyperbaric chamber tests is also shown in Table 2 and varies from 3.5 to 4.0. The results represented in Table 2 highlight the susceptibility of deep and ultra-deep subsea pipelines to propagation buckling. To confine the buckle propagation, external ring stiffeners are exploited intermittently on the pipeline. These buckle

Sample D/t Hyperbaric chamber Finite element

D50 25 1.720 4.01 1.253 1.453 D60 20 1.900 3.58 1.011 1.437 D76 47.5 1.428 3.77 1.234 1.167

PP=P<sup>e</sup> PI=PP PI=PIFE PP=PPFE

1.4 Finite element study on propagation buckling of single-walled pipelines

FE models were created in ANSYS [20] to investigate the response of the pipe to propagation buckling. Thin 4-noded shell elements (181) were used to model the pipe. SHELL181 is suitable for analyzing thin to moderately-thick shell structures. It is a four-noded element with six degrees of freedom at each node: translations in the x, y, and z directions, and rotations about the x, y, and z axes. Hydrostatic pressure can be applied as surface loads on corresponding surface. Pipe wall thickness is defined using section data command. A convergence study was performed and five integration points was found to be adequate for propagation buckling of cylindrical pipes. Frictionless contact and target elements (ANSYS elements 174 and 170) are used to define the contact between the inner surfaces of the pipe wall. These elements are created on the surface of the existing shell elements using ESURF command. The 3D contact surface elements CONTA174 are associated with the 3D target segment elements TARGE170 via a shared real constant set. Contact stiffness can be controlled by normal penalty stiffness factors and tangent penalty stiffness factor. Normal penalty stiffness factor of 0.1 was selected based on a convergence study performed that ensures both real contact behavior and reasonable computational time. Tangent stiffness factor appeared not to affect the results significantly. A von-Mises elastoplastic material definition with isotropic hardening was adopted based on material properties shown in Table 1. Total of 40 shell-181 elements in circumference were utilized for modeling the pipe. Local ovalizations were introduced to FE model by applying external pressures symmetrically on 8 elements on top of the pipe along a length equal to diameter of the pipe. Geometry is then updated using UPGEOM command and nonlinear geometric and material analysis is carried out. The FE model is 3 m long and is restrained against translation

The initiation and propagation pressures obtained from FE analysis (PIFE and PPFE respectively) are summarized in Table 1 and are in reasonable agreement with the experimental results from the hyperbaric chamber. Unlike buckle initiation pressure (PI), buckle propagation pressure (PP) is independent of curvature or ovalization of pipe. Palmer and Martin prediction PPM estimates a lower bound for propagation pressure. The FE predictions of initiation and propagation pressures on

arrestors can only confine the pressure between two stiffeners.

Comparison of experimental, analytical and numerical results.

Propagation Buckling of Subsea Pipelines and Pipe-in-Pipe Systems

DOI: http://dx.doi.org/10.5772/intechopen.85786

at all nodes at both ends.

121

Table 2.

where Dmax and Dmin are the maximum and minimum measured outer diameters along the pipe length.

The hyperbaric chamber test procedure is as follows. Thick discs are welded at both ends of 3 m pipeline. The pipeline is then filled with water and inserted inside the chamber (Figure 3b). The bolts at the chamber lid are tightened using a pneumatic torque wrench and the chamber is sealed. Using a control-volume analogy, the water inside the chamber is pressurized at a slow rate, using a hand pump. When the pressure reaches the initiation pressure PI of the pipeline, a section along the pipe sample collapses. This leads to a substantial drop in chamber pressure and is followed by water flowing from within the pipe sample through vent. Then, the chamber pressure is stabilized at the propagation pressure, Pp, with the buckle longitudinally propagating along the pipe sample accompanied by uniform water flow from the vent. The failed samples are sown in Figure 3c.

The average pressures of the 19 pipes tested in the hyperbaric chamber are represented in Table 1. A typical pressure-volume change response obtained from the hyperbaric chamber tests is shown in Figure 4. In Figure 4, the pressure inside the chamber is normalized by the propagation pressure, PPM, and the change in the pipe volume ΔV is normalized by the initial volume of the pipe, V. As stated before, the buckle initiation pressure, PI, is sensitive to imperfections (such as a dent in the pipe wall). However, the buckle propagation pressure, Pp, is not affected by the imperfection.


The analytical, experimental and numerical pressures are compared in Table 2. The ratio of propagation pressure from the hyperbaric chamber tests PP to the

#### Table 1.

Summary of experimental, analytical and numerical results.

Normalized pressure-volume response (experimental and numerical results) for D50.


Propagation Buckling of Subsea Pipelines and Pipe-in-Pipe Systems DOI: http://dx.doi.org/10.5772/intechopen.85786

Table 2.

<sup>Ω</sup><sup>0</sup> <sup>¼</sup> <sup>D</sup>max � <sup>D</sup>min Dmax þ Dmin

where Dmax and Dmin are the maximum and minimum measured outer diame-

The hyperbaric chamber test procedure is as follows. Thick discs are welded at both ends of 3 m pipeline. The pipeline is then filled with water and inserted inside the chamber (Figure 3b). The bolts at the chamber lid are tightened using a pneumatic torque wrench and the chamber is sealed. Using a control-volume analogy, the water inside the chamber is pressurized at a slow rate, using a hand pump. When the pressure reaches the initiation pressure PI of the pipeline, a section along the pipe sample collapses. This leads to a substantial drop in chamber pressure and is followed by water flowing from within the pipe sample through vent. Then, the chamber pressure is stabilized at the propagation pressure, Pp, with the buckle longitudinally propagating along the pipe sample accompanied by uniform water

The average pressures of the 19 pipes tested in the hyperbaric chamber are represented in Table 1. A typical pressure-volume change response obtained from the hyperbaric chamber tests is shown in Figure 4. In Figure 4, the pressure inside the chamber is normalized by the propagation pressure, PPM, and the change in the pipe volume ΔV is normalized by the initial volume of the pipe, V. As stated before, the buckle initiation pressure, PI, is sensitive to imperfections (such as a dent in the pipe wall). However, the buckle propagation pressure, Pp, is not affected by the

The analytical, experimental and numerical pressures are compared in Table 2.

Sample/material D/t Coupon tests Analytical (MPa) Hyperbaric chamber (MPa)

=E(%) PPM

D50 T591 25 122 440 1.5 0.778 0.93 6.42 1.6 5.12 1.1 D60 T4 20 81 716 1.9 1.011 1.21 8.24 2.3 8.15 1.6 D76 T5 47.5 156 367 0.4 0.205 0.245 1.32 0.35 1.07 0.3

Eq. (2)

Pe Eq. (8) Experiment Finite element PI PP PIFE PPFE

The ratio of propagation pressure from the hyperbaric chamber tests PP to the

flow from the vent. The failed samples are sown in Figure 3c.

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ters along the pipe length.

imperfection.

Table 1.

Figure 4.

120

ID Al-6060 σ<sup>Y</sup> (MPa) E=σ<sup>Y</sup> E<sup>0</sup>

Summary of experimental, analytical and numerical results.

Normalized pressure-volume response (experimental and numerical results) for D50.

(9)

Comparison of experimental, analytical and numerical results.

modified analytical solution p~ (Eq. (8)) vary from 1.428 to 1.9 depending on D/t ratio. Mesloh et al. [19] suggested similar relations. The ratio of PI/PP from the hyperbaric chamber tests is also shown in Table 2 and varies from 3.5 to 4.0. The results represented in Table 2 highlight the susceptibility of deep and ultra-deep subsea pipelines to propagation buckling. To confine the buckle propagation, external ring stiffeners are exploited intermittently on the pipeline. These buckle arrestors can only confine the pressure between two stiffeners.

#### 1.4 Finite element study on propagation buckling of single-walled pipelines

FE models were created in ANSYS [20] to investigate the response of the pipe to propagation buckling. Thin 4-noded shell elements (181) were used to model the pipe. SHELL181 is suitable for analyzing thin to moderately-thick shell structures. It is a four-noded element with six degrees of freedom at each node: translations in the x, y, and z directions, and rotations about the x, y, and z axes. Hydrostatic pressure can be applied as surface loads on corresponding surface. Pipe wall thickness is defined using section data command. A convergence study was performed and five integration points was found to be adequate for propagation buckling of cylindrical pipes. Frictionless contact and target elements (ANSYS elements 174 and 170) are used to define the contact between the inner surfaces of the pipe wall. These elements are created on the surface of the existing shell elements using ESURF command. The 3D contact surface elements CONTA174 are associated with the 3D target segment elements TARGE170 via a shared real constant set. Contact stiffness can be controlled by normal penalty stiffness factors and tangent penalty stiffness factor. Normal penalty stiffness factor of 0.1 was selected based on a convergence study performed that ensures both real contact behavior and reasonable computational time. Tangent stiffness factor appeared not to affect the results significantly.

A von-Mises elastoplastic material definition with isotropic hardening was adopted based on material properties shown in Table 1. Total of 40 shell-181 elements in circumference were utilized for modeling the pipe. Local ovalizations were introduced to FE model by applying external pressures symmetrically on 8 elements on top of the pipe along a length equal to diameter of the pipe. Geometry is then updated using UPGEOM command and nonlinear geometric and material analysis is carried out. The FE model is 3 m long and is restrained against translation at all nodes at both ends.

The initiation and propagation pressures obtained from FE analysis (PIFE and PPFE respectively) are summarized in Table 1 and are in reasonable agreement with the experimental results from the hyperbaric chamber. Unlike buckle initiation pressure (PI), buckle propagation pressure (PP) is independent of curvature or ovalization of pipe. Palmer and Martin prediction PPM estimates a lower bound for propagation pressure. The FE predictions of initiation and propagation pressures on

and Li [12] carried out a finite element study of propagation buckling of PIPs with carrier pipes having Do/to values of 25, 20 and 15 and inner tubes having Di/ti of 15 and 20. Although both studies [11, 12] covered similar Do/to range of the carrier

Numerous analytical solutions have been suggested to estimate the propagation pressure of a single pipe. Unlike propagation pressure, the initiation pressure is very sensitive to initial imperfection such as local dents or ovalizations. The propagation pressure is related to plastic properties of the pipe and is only a fraction of the buckle initiation pressure. Both buckle initiation pressure and buckle propagation pressure are related to the diameter to wall-thickness ratio of the pipe, however previous studies suggest that there is no evident relationship between the two [2, 3]. The simplest propagation pressure model was established by Palmer and Martin [4], which only considered the initial and final configurations of the cross-section of the pipe. Figure 6 shows the four plastic hinges developed in the pipe at different stages of propagation buckling on subsea pipelines and pipe-in-pipe systems. By adopting plane strain analogy, Kyriakides and Vogler [11] proposed the following expression for the propagation pressure of the PIP system. Their formulation accounts for development of four plastic hinges in each of the carrier and the

2.2 Analytical solution of propagation pressure of pipe-in-pipe systems

pipe, two different empirical expressions were suggested.

Propagation Buckling of Subsea Pipelines and Pipe-in-Pipe Systems

DOI: http://dx.doi.org/10.5772/intechopen.85786

<sup>P</sup>^<sup>p</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> ffiffiffi <sup>3</sup> <sup>p</sup> <sup>σ</sup>Yo

the membrane and flexural effects of the outer and the inner pipes:

to Do � �<sup>2</sup>

where subscripts o and i denote the outer pipe and inner pipe, respectively. The analytical lower bound solution to propagation buckling of a single pipe given by (Eq. (8)), can be extended to the pipe-in-pipe systems by accounting for

A schematic of deformation stages in propagation buckling of a single pipe (stages a–c) and a pipe-in-pipe

<sup>1</sup> <sup>þ</sup> <sup>σ</sup>Yi σYo

ti to

Wex ¼ Winð Þ<sup>f</sup> þ Winð Þ <sup>m</sup> (11)

� �<sup>2</sup> " # (10)

inner pipes (Figure 6d-f).

Figure 6.

123

system (stages d–f).

Figure 5. FE model of 3 m long D50 showing the onset of propagation buckling.

average represent 87 and 74%, respectively, of the experimental results. A typical FE result for D50 pipe is shown in Figures 4 and 5.
