The Finite Element Analysis on the Submarine Pipeline under the Seismic Loading

Yifei Yan1,a, Lufeng Cheng2,b

1College of Electromechanical Engineering, China University of Petroleum , Qingdao, Shandong, 266555, China

2China petroleum Engineering and Construction Corp, Drum Tower Street 28 in Dongcheng District, Beijing,100120,China

ayanyf163@163.com , boilcae@126.com

KEYWORDS :Submarine Pipeline; Seismic Loading; Finite Element Method; Numerical Simulation

ABSTRACT. Seismic loading is one of the most important factors of submarine pipeline damage,

so the research on submarine pipeline failure mechanism is still lifeline engineering frontier topics.

According to Biot consolidation theory, considering the interaction of submarine pipelines with the

soil medium under earthquake action, the model of the seabed-pipeline interaction is established.

The influences of wall thickness, radius and cover layer thickness on submarine pipeline strain

response are studied under El Centro seismic wave based on this model. The calculating results

show that effective stress and axial strain of the submarine pipeline increases with wall thickness,

radius and cover layer thickness increasing.

INTRODUTION

In recent years, with the development of offshore oil and gas, submarine pipeline has been widely

used all over the world, which becomes an important part petroleum development engineering of

oil-gas field engineering. However, the safety problem of submarine pipeline is aroused extensive

attention at home and abroad for its prohibitive costs, complex maintenance and environmental

pollution. Locating in the joint of two seismic belts, China is one of the countries in the world that

earthquake occurs frequently. The Bohai Bay region abound in natural gas resources, but this place

is the earthquake easy-happening area. So we must consider the impact of the earthquake in the

process of laying submarine pipelines. The seismic stress calculation is an important content of

submarine pipelines design. Currently, because of the experimental equipment restriction and the

practical problem complexity[1][2][3], the research of dynamic response analysis of submarine

pipelines under seismic load is very limited[4][5][6]. According to the Biot consolidation theory, the

model of the seabed-pipeline interaction is established. The influence and variation of submarine

pipeline strain response are studied with influence factors such as wall thickness, radius and cover

layer thickness under El Centro seismic wave based on this model. The examples show that

effective stress and axial strain of the submarine pipeline increase with wall thickness and cover

layer thickness increasing.

FINITE ELEMENT DYNAMIC EQUATION AND SOLUTION

Finite element dynamic equation. According to the Biot consolidation theory[7][8], when

considering the compressibility of fluid, mass conservation equation of saturated pore fluid is as

follows:

( ) 01

=∇−∇+∂

∂+

∂∂

pKTtt

p

K

n ii

ωω γε

(1)

Advanced Materials Research Vols. 490-495 (2012) pp 2977-2981Online available since 2012/Mar/15 at www.scientific.net© (2012) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.490-495.2977

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Where ∇ is the Laplacian; K is the matrix of permeability of soils; iiε is the volume strain of soil

skeleton; n is the porosity factor; ωK is the bulk modulus of fluid; p is the pore pressure; ωγ

is the weight per unit.

When the relative acceleration effect between the pore fluid and the soil skeleton is ignored, and

excluding the compressibility of soil particles, the dynamic equations of the saturated sea-bed is as

follows:

)3,2,1,('

, ==++ jiiibp iiijjjij ρρδσ (2)

Where '

, jijσ is the effective stress; ijδ is Kronecker symbol; ρ is the density of soil; ib is the

acceleration of volume force; iii is the acceleration of soil skeleton.

According to the elastic dynamics theory[9], the governing equation of the submarine pipeline is:

)3,2,1,(, ==+ jiiib pippipjpij ρρσ (3)

Where jpij ,σ is the submarine pipeline internal stress; pρ is the mass density of submarine

pipeline; pib is the volume force acceleration of submarine pipeline; iii is the acceleration of

pipeline.

Finite element equation of saturated seabed. According to the Galerkin method[10], after

applying finite element discrete to the equation (1), the coupled finite element equation can be

obtained:

=

+

+

+

+

+

+

+

++

+

+

++

+

+

++

pf

∆tt

u

∆tt

f

∆tt

∆tt

)(

pfpf

∆tt

upf

∆tt

uu

∆tt

f

∆tt

∆tt

)(

pfpf

∆ttT

upf

∆tt

∆tt

f

∆tt

∆tt∆tt

R

R

P

U

K -

K K

P

U

K K

C

P

U

M

2

1

0

0

00

0

�

�

��

��

(4)

Where:

∑∫ ∆+

∆+∆+∆+∆+∆+ =m

mtt

mttm

u

ttmttm

u

tt

uu

tt

v

T

vdBDBK)(

)()()()(

∑∫ ∆+

∆+∆+∆+∆+ =m

mtt

mttm

pf

ttmm

u

tt

upf

tt

v

T

vdHIBK)(

)()()()(

∑∫ ∆+

∆+∆+∆+∆+∆+ =m

mtt

mttm

pf

ttm

pf

ttmtt

f

pfpf

tt

v

T

vHHnK

K)(

)()()()()1( 1

∑∫ ∆+

∆+∆+∆+∆+∆+ =m

mtt

mttm

pf

ttmttm

pf

tt

f

pfpf

tt

v

T

vdBBBK)(

)()()()()2( 1

γ

( )∑∫ ∆+

∆+∆+∆+∆+∆+ =m

tSt

m

q

ttmttT

mtst

pf

tt

pf

tt

mq

mq SdqHR

)(

)()()()(

( )∑∫∑∫ ∆+

∆+∆+∆+∆+

∆+

∆+∆+∆+∆+ +=m

tSt

m

q

ttmttT

mtst

u

tt

mmtt

mttmttm

u

tt

u

tt

mq

mq

v

T

SdfHvdfHR)(

)()()()(

)(

)()()(

Where U is the nodal displacement vector of soil; fP is the pore pressure vector; M is the mass

matrix of soil; C is the damping matrix of soil; D is the elastic coefficient matrix of soil; f

and q are load vectors; uB and pfB are the geometry gradient matrix of soil nodal displacement

and hole hydraulic pressure respectively; uH is the nodal displacement of soil; pfH is the

interpolation function matrix of pore pressure; I is the identity matrix.

2978 Mechatronics and Intelligent Materials II

Numerical solution. After using the equation (4) , the numerical solution can be obtained by using

step-by-step integral scheme of the Newmark-β:

++−=

⋅

++

+++

+

+

+

+

++++

f

t)(

pfpf

∆tttT

upf

∆tt

pf

∆tt

d

u

∆tt

f

∆tt

∆tt

)(

pfpf

∆ttT

upf

upf

∆tt∆tt∆tt

uu

∆tt

PKUKR∆t

R

P

U

K K

KC aMaK

1

1

10

(5)

( )p

tt

p

tt

uup

tt

p

ttd

up

tt CaMaKMR ∆+∆+∆+∆+∆+ ++= 10 (6)

Where:

0 2

1a

tα=

∆; 1a

t

βα

=∆

; 2

1a

tα=

∆; 3

11

2a

α= − ; 4 1a

βα

= − ; 5 12

a tβα

= ∆ −

.

ANALYSES OF CALCULATION RESULTS

In order to get the influence of wall thickness, pipe radius and cover layer thickness on the strain

response of submarine pipelines under seismic load, the proposed model in this paper is used to

gain the variation of submarine pipelines strain by taking the different values of these parameters.

The model parameter values are as follows: submarine pipeline diameter is 529 mm; wall thickness

is 7mm; pipe material is 16Mn; material yield limit sσ is 343MPa; ultimate strength bσ is

510MPa;pipeline pressure is 5MPa; sea-bed thickness is 40m;seismic intensity is 9 degrees; φK =0.4,

site soil type is III. Taking ak as 12 N/mm2; the vibration period of the soil T is 0.8s.and the other

parameter values are respectively pC =1.5×106mm/ s; or =1.4 g / cm

3; n= 60%. The calculated

results are shown in Fig.1 to 4.

The influence of radius-thickness ratio on submarine pipelines strain response. The

relationship between the radius-thickness ratio and effective stress and the one between the

radius-thickness ratio and axial strain are shown in Fig.1 and Fig.2 respectively.

35 40 45 50 55 60 65 70 75 8090

95

100

105

110

115

120

125

130

135

sM

ises

(MP

a)

radius-thickness ratio

t=7mm

Figure 1. Relationship curve between radius-thickness ratio and effective stress

Advanced Materials Research Vols. 490-495 2979

35 40 45 50 55 60 65 70 75 80

-0.210

-0.205

-0.200

-0.195

-0.190

-0.185

radius-thickness ratio

ax

ial

stra

in(%)

t=7mm

Figure 2. Relationship curve between radius-thickness ratio and axial strain

The radius-thickness ratio may affect the interaction between submarine pipeline and soil, control

the integral deformation and critical load of the pipeline to some extent. With the increase of

radius-thickness ratio (decrease of thickness), the value of Misesσ also increases when other

conditions remained constant.

The influence of cover layer thickness on submarine pipelines strain response. The relationship

between cover layer thickness and effective stress and the one between the cover layer thickness

and axial strain are shown in Fig.3 and Fig.4 respectively.

0.5 1.0 1.5 2.0 2.5 3.080

100

120

140

160

180

200

220

240

sM

ises

(MP

a)

cover layer thickness(m)

t=7mm

Figure 3. Relationship curve between cover layer thickness and effective stress

0.5 1.0 1.5 2.0 2.5 3.0-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

axia

l st

rain(%

)

cover layer thickness(m)

t=7mm

Figure 4. Relationship curve between cover layer thickness and axial strain

2980 Mechatronics and Intelligent Materials II

As illustrated in Fig.3 and Fig.4, the influence of cover layer thickness on strain response of

submarine pipelines can not be ignored. With the increase of cover layer thickness, σMises stress and

axial strain of submarine pipelines become larger obviously. The calculating results also show that

shallow-buried submarine pipeline can improve the ability of resisting fault, which mainly because

the resisting fault ability of submarine pipeline increases with the decrease of buried depth.

Shallow- buried submarine pipeline can decrease the pressure of soil and vertical friction, which

can lighten the damage of fault movement.

CONCLUSION

According to the model of the seabed-pipeline interaction, the influence and variation of submarine

pipeline strain response are studied with influence factors such as wall thickness, radius and cover

layer thickness under El Centro seismic wave based on this model. The examples show that

effective stress and axial strain of the submarine pipeline increases with wall thickness and cover

layer thickness increasing.

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[3] K.Matsubara & M. Hoshiya. “Soil spring constants of buried pipelines for seismic design.”

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[5] G.W.Housner. “Bending vibration of a pipeline containing fluid.” J. Journal of Applide

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Advanced Materials Research Vols. 490-495 2981

Mechatronics and Intelligent Materials II 10.4028/www.scientific.net/AMR.490-495 The Finite Element Analysis on the Submarine Pipeline under the Seismic Loading 10.4028/www.scientific.net/AMR.490-495.2977