effects of elastic joints on 3-d nonlinear responses of a deep-ocean pipe_modeling and boundary...

8/11/2019 Effects of Elastic Joints on 3-D Nonlinear Responses of a Deep-Ocean Pipe_Modeling and Boundary Conditions, Jin

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International Journal of Offshore and Polar Engineering

Vol. 6, No.3, September 1996 ISSN 1053-5381

Copyright by The International Society of Offshore and Polar Engineers

Effects of Elastic Joints on D Nonlinear Responses of a Deep Ocean Pipe:

Modeling and Boundary Conditions

Jin S. Chung*

Department of Engineering, Colorado School of Mines, Golden, Colorado, USA

B.-R. Chengt

Tsinghua University, Beijing, China

ABSTRACT

Pipe vibration is often excited in the deep ocean by ship motions, wave forces and vortex shedding. Elastic joints along the

pipe are modeled in an attempt to move the resonance frequencies away from the pipe system. The numerical examples

focus on the investigation of single and multiple elastic joints along a long pipe and their effect on three-dimensional 3-D

nonlinear coupled pipe responses, including torsional coupling. The multi-substructure technique is introduced in order to

get the governing equation of the entire pipe system. The pipe is subjected to a vertically varying current flow in establishing

the static equilibrium configuration. Dynamic responses are excited by large-amplitude horizontal as well as vertical ship or

pipe-top motion. Ocean-mining pipes 4,000 ft and 18,000 ft in length are used to investigate the effects of the joint stiffness

and position on the pipe responses. The bending stiffness can affect the bending moments along the pipe and the associated

maximum values, but has little influence on the bending deflection. However, the axial stiffness of the joint can greatly

change the axial fundamental frequency, as well as static axial displacement, while it has little effect on the static internal

axial force. The appropriate position ofjoints can have a greater influence on the static responses. The dynamic responses to

the external excitation of a pipe with multiple elastic joints can be greatly reduced. The results are presented for both free

and pinned bottom-end conditions of the pipe.

INTRODUCTION

The importance of the axial stress of a long pipe for design was

first pointed out by Chung and Whitney 1981,1983 with uncou

pled oscillatory axial motions of an 18,000-ft vertical pipe and

later with 3-D coupled responses of 4,000-ft and 18,000-ft pipes

Chung, Cheng and Huttelmaier, 1994; Cheng, Chung and

Huttelmaier, 1994 . Among many possible problems, the oscillat

ing axial stresses have been found to be a critical design parame

ter for such a deep-ocean pipe Chung and Whitney, 1981 .

Changes in or control of the axial or bending resonance frequen

cies are often desired in design and ocean operations. One of the

methods applied here is to change the fundamental axial frequen

cy and the static equilibrium state of a pipe. A concept of elastic

joints on a marine riser Caldwell et aI., 1976, and Ortloff et aI.,

1976 was previously tested for the purpose of reducing the bend

ing stress. It was applied in actual design. However, the paper

does not present substantiating technical data, and it can only be

used as qualitative information.

In the previous paper Chung and Cheng, 1995 , the pipe eigen

frequencies are calculated with different arrangements of elastic

joints along a vertical pipe in the ocean. This was investigated as

a means to control or change the resonance frequencies of the

axial and bending vibrations. Extending this work, actual respons

es of the pipe with joints to the hydrodynamic forces are present-

*TSOPE Member.

A visiting research scholar at Colorado School of Mines. Golden,

Colorado, USA.

Unit conversion: 1m

3.281 ft, I ftls

0.305 m/s.

Received September ]5, 1995: revised manuscript received by the editors

December 18, 1995. The manuscript was submitted directly to the

Journal.

KEY WORDS: Nonlinear finite element, elastic joints, modeling, static

and dynamic responses, vertical pipe, coupled axial, bending and tor

sional responses, pipe boundary conditions, deep-ocean mining.

ed in this paper, and favorable nonlinear 3-D coupled responses of

a deep-ocean pipe can be obtained, using elastic joints on a long

vertical pipe.

The two-substructure technique is successfully adopted to treat

an eigenvalue problem of a pipe with examples of multiple elastic

joints Chung and Cheng, 1995 . This technique is extended in

this paper to multi-substructures and is used for solving static and

dynamic nonlinear responses of a pipe with multiple joints. The

vertically varying, unidirectional steady current flow influences

the static equilibrium configuration of the pipe and its static stress

state. In practice, the pipe vibration is often excited by horizontal

as well as vertical ship or pipe-top motion and the vortex shed

ding. This is the first modeling and technical analysis about

effects of elastic joints along a long pipe on 3-D nonlinear cou

pled static and dynamic responses. A previous paper Chung,

Cheng and Zheng, 1995 is updated with new examples of multi

ple joints and a case of pinned bottom-end condition of the pipe

such as for a deep-ocean marine riser.

MODELING OF VERTICAL PIPE WITH MULTIPLE

JOINTS

Let a pipe system be divided into N-segments with N-l elastic

joints. Every segment is considered as a substructure. Every elas

tic joint is modeled by a special pin with springs in the direction

of bending-rotation and axial displacements, as shown in Fig. I.

The stiffness of these springs may be determined by material

properties of the joints. For the modeling, the assumptions are

made as follows:

Length of the joint is very short, as compared to the pipe ele

ment length, and can be neglected.

Two adjacent pipe segments, connected by an elastic joint,

have equal transverse displacements, as well as an equal twist


2/9

204 Effects of Elastic Joints on 3-D Nonlinear Responses of a Deep-Ocean Pipe: Modeling and Boundary Conditions

ELASTIC JOINTS

(5)

6

Transformation of Eqs. 2-4 with Eq. 5 leads to:

or in an expanded form as:

Similarly to the entire pipe system, the generalized coordinate

vector,

x,

of the substructure is divided into 3 subvectors,

Xi, xb

and xr Introducing an index matrix,

LV ,

a relationship between

xv

and X is established as:

SEAFLOOR

'l~yA~

c Elastic Joint Model

8

9

7

(13)

10

11

12

m J)

=

L J)Tm J) L J)

k~J) = L J)Tk J) 1j1)

J J) = rJJ)T j J)

T J) = iT m J)i

, 2

U J) = XTk J) X

, 2 '

W J) = X T J J)

where kr is the submatrix, which includes the axial and bending

stiffness of the elastic joint.

Transformation ofEq. 13 with Eq. 6 leads to:

where:

Let the p-th and q-th substructure be connected by the e-th elas

tic joint, where all joints are made of elastic material, and the

elastic potential energy of the joint can be expressed by:

(I)

bl Pinned Bottom End

al Free Bottom End

I

C5f) ELASTIC JOINT 1

I

c{) ELASTIC JOINT 2

I

Gf) ELASTIC JOINT 3

I

Let X denote an independent vector in the generalized coordi

nate of the entire pipe system. It is divided into 3 sub vectors as:

angle at a joint. But the bending rotational angles and axial dis

placements can differ.

Ability of the joint to support the internal axial force and the

bending moment depends on the stiffness of the joint material.

Fig. I (a) free bottom end, (b) pinned bottom, and (c) elastic joint

model

where

Xi =

the inner coordinate vector, and

Xb

and

Xr =

the inter

face coordinate vectors without and with relative motion, respec

tively.

The kinetic energy, T}j), potential energy,

uy),

and work done

by the nonconservative force,

Wi),

of the j-th substructure may be

expressed as:

(IS)

14

o

o

Summation of all substructures and elastic joints gives the total

kinetic energy, potential energy and work by the applied noncon

servative force of the entire pipe system as follows:

where

2

(3)

4

T J)

I .

J)T J). J)

-x m x

, 2

U J) = x J) T k J ) x J)

.< 2 '

W J) = X J) T j J)

where the superscript j denotes the j-th substructure. m, ks x

and j are the mass matrix, stiffness matrix, generalized coordinate

vector, and external load vector, respectively.

N N

N-I

N

T= T J) U= U J) +~ ute) and W= W J) Js ~s ir

j=1

j=l e= =

16


3/9


Based on the principle of virtual work, the equation of motion

of the entire pipe system may be expressed as:

205

where

d

= the outer diameter of pipe,

Pw

= mass density of water,

CM, CD and Cf= hydrodynamic inertia, drag and friction coeffi

cients, respectively. The relative velocity, VR, is defined by:

MX+KX=F

(17)

(22)

where

N

M =

I,

LW

m J

IY

j=l

N N-[

K= I,I5j)Tk~j)L(j) + I,k;e)

j=1 e=

N

F= I,I5j)T f(J)

=1

(18)

where

vp

= pipe velocity, Vc = steady current velocity,

vw=

water

particle velocity due to wave, vRN

=

the normal component of vR

vRT = the tangential (along pipe axis) component of vR, v RN = the

normal component of the relative acceleration, v WN, and = nor

mal component of the water particle acceleration.

The excitation motion of the pipe top is caused by the ship

motion, which, in turn, is induced by waves. For numerical exam

ples, it is represented for the horizontal (x-) and axial or vertical

(z-) motions of the pipe top, respectively, as follows:

If the damping is accounted for, the equation of motion can be

written as:

x =

Xo

sin

0Jt

and z =

Zo

sin

0Jt

Initial Condition and Static Equilibrium State

23

SOLUTION OF DYNAMIC RESPONSE INVOLVING

GEOMETRIC NONLINEARITY

Governing Equation of Motion

Incremental updated Lagrangian formulation, implicit time

integration and Newton-Raphson iteration technique are adopted.

Details are provided in Chung, Cheng and Huttelmaier (1994).

The governing equation of motion at time step, H.1t, is written

as:

MX+CX+KX=F

where C = the damping matrix.

19

For the linear system, the zero position of the generalized coor

dinate is usually placed at the static equilibrium position, as the

static stress has no effect on amplitude of dynamic response.

However, when the dynamic response is calculated at the first

time step for the deep-ocean pipe system involving geometrical

nonlinearity, the effect of the static axial stress must be consid

ered. This is because it produces nonlinear stiffness matrix and

equivalent force vector.

The static load includes self-weight and buoyancy of the pipe,

external normal drag due to steady current, and steady f1ow

induced torsional moment caused by asymmetric arrangement of

cable and buffer to the flow. The static equilibrium configuration

of the pipe and the element static stress are obtained by solving

the static equation, Eq. 20, without the mass and damping terms.

In order to link the start of the dynamic response with the static

equilibrium, the undeflected or vertical pipe configuration is taken

as zero position of the generalized coordinates.

where ( +

d iiU

and ( + d ti ( = acceleration and velocity vectors at

H.1t, .1Ji)

=

the incremental displacement vector at iteration i ,

H.1tF = the external force vector at H.1t, M and C = time-inde

pendent mass and damping matrices, respectively, :KL and :KNL

=

linear- and nonlinear-strain incremental stiffness matrices, and

~:~R HI= nodal force vector equivalent to element stresses at

H.1t at iteration (i-I). For the beam element, formulations of :KL,

:KNL and

~:~~RU-IJ

re given by Bathe (1982).

Among several kinds of integration schemes that may be used

for solving Eq. 20, the Newmark scheme is effective and used, as

it was for the previous papers (Chung, Cheng and Huttelmaier,

1994).

External Forces and Excitation

Dynamic excitation considered in the paper includes the exter

nal hydrodynamic forces that consist of the wave forces on the

pipe and the forces induced by the pipe-top motion, which is

caused by wave-induced ship motion. The hydrodynamic force

consists of the normal component,

F

N and the tangential compo

nent,

FT,

as follows (Chung, Whitney and Loden, 1981):

NUMERICAL EXAMPLES

Pipe Models

In order to study the effects of elastic joints on the nonlinear

static and dynamic responses, 4 pipe models are chosen (Chung,

1994; Cheng, Chung and Zheng, 1995). Their properties and prin

cipal dimensions are presented in Table I.

The properties and effect of the elastic joints are embodied in

the solution for the static equilibrium and in the internal force and

moments of the joints along the pipe. The axial and bending stiff

ness of the joint have a different contribution.

Pipe A

ipeBipeCipeD

Length (ft)

4,000,0008,0008,000

Outer Diameter (ft)

1.7397

.7397

.6667.6667

Inner Diameter (ft)

1.6667

.6667.5608

.5608

Young s Modules

4.32x109

4.32x109.32x109.32x109

Pipe Weight in Air (lb/ft)

95.0

5.026.4

26.4

Pipe Weight in Water (lb/ft)

83.0

3.014.314.3

Bottom Buffer Weight (L.T.)

20000

Number of Elements

16688

Bottom Boundary Conditions

freereeree

pinned

FN

=-

Jr /4)Pwd2CMV

RN

+

Pwd2VWN

(Pw

/2)dCDlv RNlv RN

FT

=-

(JrPw /2)dCflv

Rlv

RT

21

Table I Principal dimensions and properties of 4 pipe models.

Top end is pinned (no rotation about pipe axis allowed).


4/9

206 Effects of Elastic Joints on 3-D Nonlinear Responses of a Deep-Ocean Pipe: Modeling and Boundary Conditions

Fig. 2 Effect of bending-rotational stiffness of a joint on maxi

mum static bending moment of pipe A free bottom end and

4,000 ft): a buffer; I joint at z] = -1,750 ft;

ka

= 00

512J12J12J

1 JOINT

Z

=

-1,750 FT

512J12J12J

BENDING MOMENT, M (FT-LB)

L

=

4,000 FT

ka = 106 LB/FT

kb = 107 FT-LB/RAD

112J12J12J

412J12J12J

112J12J12J12J

~

N 212J12J12J

J:

l-

..

0

312J12J12J

10 10

2

10 10 10 10 10

7

10 10 10 ,.

kb (FT-LB/RAD)

L = 4,000 FT

1 JOINT; Z

=

-1,750 FT

ka (LB/FT)

= 00.

kb

=

107 FT-LB/RAD

6000.0

1

S000.0

-7.5

2 5

5 0

Fig. 4 Effect of elastic joint on bending moment distribution

along pipe A free bottom end, a buffer and

=

4,000 ft): I joint

at z] = -1,750 ft;

ka

= lx1061b/ft and

kb

= lx107 ft-Ib/rad

Effect of bending stiffness.

With axial stiffness of ka

=

00, the

bending stiffness, kb, is varied from 1.0x I 010 ft-Ib/rad nearly

rigid connection) to 0 pinned connection). With a proper value of

kb, the bending moments may be reduced along the pipe, and the

maximum bending moment can be reduced lower than those with

the other kb values. For example, the kb =

I.OxlO7

ft-Ib/rad

reduces the maximum bending moment to 6,230 ft-Ib from 7,310

ft-lb with

kb

= 1.Ox10

10

ft-Ib/rad Fig. 2). The results also show

that the variation in the bending stiffness of the joint has little

effect on the static bending and axial displacements.

ka (LB/FTl

L = 4.000 FT

1

JOINT,

Z

=

-1.750

FT

ka

=

106 LB/FT, kb

=

107 FT-LB/RAD

0 0

~

o

1=

o

III

~

I

Z

w

: ;

w

u

-J

D..

en

o

-J

~ -10.0

10

Fig. 3 Effect of axial stiffness of joint on static axial z-) dis

placement of pipe A free bottom end a buffer and

= 4,000 ft) at

bottom end of pipe: I joint at z] = -1,750 ft; kb =

107

ft-Ib/rad

STATIC RESPONSES

The pipe system is assumed to encounter a steady current Vcx

# 0) profile in the x-direction. For = 4,000 ft, the current profile

consists of Vcxt = 3 ftls for the top 2,000 ft and Vcxb = lftls for the

bottom 2,000 ft. Corresponding steady drag induced by the current

is 14.64 Ib/ft and 1.624 Ib/ft, respectively. The subscripts,

c, x,

t

and b, represent current, x-direction, pipe top and pipe bottom,

respectively. Torsional moments due to the cable-pipe asymmetry

to the flow or nodal torsion,

MT)N

are 9.212 ft-Ib/ft and 1.048 ft

Ib/ft, respectively. The additional weight of WB

=

448,000 Ib

accounts for the buffer at the bottom end of the pipe. The corre

sponding current force on the buffer is 600 Ib, and the associated

torsional moment or buffer torsion, MT)B is 1800 ft-Ib.

4,OOO-ftPipe with Buffer and Free Bottom End: Pipe A

Computation of the nonlinear coupled static responses of a

4,000-ft pipe with no joint gives the maximum bending moment

of 7,310 ft-Ib at z

= -

1,750 ft from the pipe top. Placing a single

elastic joint at this position, effects of the axial and bending stiff

ness are investigated.

Effect of Axial Stiffness. With the bending stiffness,

kb

1.0x107 ft-Ib/rad, where the smallest value of the maximum bend

ing moment occurs, the axial stiffness, ka is varied. The results

show that the axial stiffness of the joint has a great effect only on

the static axial displacement Fig. 3). Although it has little influ

ence on the pipe deflection, internal axial force and bending

moment, an optimum axial stiffness of a joint can greatly change

the fundamental axial frequency of the pipe system, while keep

ing the axial displacement within a practical range Chung and

Cheng, 1995). In this example, ka ~1.OxI06Ib/ft is found to be the

optimum value.

The comparison of the static pipe responses between no joint

No joint

joint

Fundamental Axial Frequency Hz)

0.5430

.4894

Fundamental Bending Frequency Hz)

0.01478

.01478

Bending Deflection at Bottom ft)

59.78

9.78

Biaxial Bending Deflection at Bottom ft)

0.0001203.0001167

Axial Displacement at Bottom ft)

-2.133

2.768

Twis t Angle at Bottom r ad)

0.1309

.1309

Maximum Internal Axial Force lb)

771000

71000

Maximum Bending Moment ft-Ib)

7310

230

Table 2 Comparison of fundamental frequencies and static

responses of pipe A free bottom end and

=

4000 ft) between no

joint and 1 elastic joint at 1,750 ft from top end; a buffer of WB =

448,000 Ib; = 1.0xI06Ib/ft; =

I.OxI07

ft-Ib/rad


5/9


207

/3JOINTS

- 1,500 FT

-1.750FT

_ ..._ ~ - 2,000 FT

1 JOINT

-Y::f;-

NO JOINT

Z=-1.750FT

-3000

-1000

1=

:

N

:i

-2000~

a..

w

o

Effect of multijoints.

For the purpose of further reducing the

maximum bending moment on the pipe, as this is the primary pur

pose of the present study, effects of the multiple joints are investi

gated. For the present example, 3 joints with

ka

= 1.0x1061b/ft

and

kb

= 2.0x106 ft-1b/rad are placed along the pipe at 1,500-ft,

1,750-ft and 2,000-ft levels. This further reduces the maximum

bending moments. The comparison of the fundamental frequen

cies and static responses of the pipes with no joint, 1 j oint and 3

joints is made (Table 3). The effectiveness of reducing the corre

sponding bending moment distributions along the pipe is shown

in Fig. 7.

Effect of position of 1 joint.

If the elastic joint is moved from

1,750 ft to 1,500 or 2,000 ft, which are measured from the top

end, the bending stiffness of the joint does not reduce the maxi

mum bending moment; on the contrary, it increases the maximum

bending moment (Fig. 6). This indicates that the position of the

elastic joint has a greater influence on reducing the maximum

bending moment than its stiffness. The joint should generally be

placed at the position of the pipe where the maximum bending

moment occurs.

18,OOO-ftPipe with Buffer and Free Bottom End: Pipe C

The steady current velocity

Vcx oF

0) profile on the pipe as

used is

Vext =

3 ftls for the top 2,000 ft and

Vcxb =

I ftls for the

bottom 16,000 ft. In establishing a static equilibrium state, the

flow-induced torsional moment along the pipe induced by the

cable-pipe arrangement asymmetric to the current flow or nodal

torsional,

MT N,

and buffer torsional moment at the pipe s bot

tom end, MT B, are accounted for (Chung and Whitney, 1993).

For a pipe of L = 18,000 ft, the influence of the elastic joints on

the static responses is small, except for bending (x) moment (Fig .

8). However, their influence on the dynamic responses is substan

tial.

In order to compare the responses of the present free bottom

end to that of the pinned bottom end, the axial stiffness is kept

rigid

ka

= 00), while the rotational bending stiffness

kb

is non-

1.5E+4

1.5E+4

1 JOINT

Z = -1,750 FT

0.0E+0

0.0E+0

-1.5E+4

-1.5E+4

BENDING MOMENT. M (FT-LB)

BENDING MOMENT, M (FT-LB)

G::~,~

0-_

NO JOINT -./- - --

- 1,500 FT7

~:,t~

z = -1,750 FT

0


6/9

208 Effects of Elastic Joints on 3 D Nonlinear Responses of a Deep Ocean Pipe: Modeling and Boundary Conditions

18,000-ft Pipe with Pinned Bottom End and No Buffer: Pipe D

0.015

1.0E 3

NO JOINT

0.010

0.0E 0

1 JOINT: -1,000 FT

PIPE0 PINNED BOTTOM)

L=1B,000FT

ka

= OX

kb = 5x105 LB-FT/RAD

PRETENSION= 2.05X106 LB

0.005

;--------------------~

---.-----

\

.

.

\, 3 JOINTS

~,~ -1,000 FT

~ -9,000FT

\ -17,000 FT

'\

0.000

/ NOJOINT

0...::.::: - == - - ~ - - - ~ - --

PIPEC WITH BUFFER

L

=

18,000 FT

co

kb = 1x106 L8-FT/RAD

o

6000

3000

18000

0.005

15000

12000

18000

3.0E 3 2.0E 3 1.0E 3

0

3000 6000

i=

o

:i

9000

t-

o..

Q

12000

15000

:i 9000

Ii:

w

Q

BENDINGMOMENT, M FT-LB)

Fig. 8 Effect of I or 3 elastic joints of pipe C (free bottom end, a

buffer and L

=

18,000 ft) on bending moment distribution: I joint

at

zJ =

-1,000 ft, 3 joints at

zJ =

-1,000, -9,000 and -17,000 ft;

ka

=

00

and

kb

= lxlO6lb-ftlrad

I Joint 3 Joints

106 106

5xlO6 2xlO6

0.9385 0.7874

0.01612 0.01612

174.2 174.2

0.002345 0.001853

4.430 4.057

0.1012 0.1011

3.23xl


7/9


209

Fig. 9b Effect of 3 elastic joints of pipe D pinned bottom end, no

buffer and L

=

18,000 ft) on bending moment distribution: 3

joints at z} = -1,000, -9,000 and -17,000 ft, ka =

00

and kb =

5x1051b-ft/rad; pretension at top end = 2.05x1061b

300

5000

50

NOJOINT~:

TIME 5

1000

PIPE C WITH BUFFER

L = 18,000 FT

ka = 00

kb = lxl06 FT-LB/RAD

0.66128

0.66126

0.66124

o

~

0.05

)-

en

0.00 ~

~~A_

JOINTS

z

a

~

-0.05

II:

aI

j

\::;::::: ::\ 1 JOINT

:>

-0.10

)

Z

PIPE C WITH BUFFER

z

-0.15

L = 18,000 FT

w

al

ka = 00

....

kb

=

lxl06 FT-LB/RAD

I:

-0.20

:;;i::

i:

NO JOINT

X


8/9

210 Effects of Elastic Joints on 3 D Nonlinear Responses of a Deep Ocean Pipe: Modeling and Boundary Conditions

0.1

-0.1

PIPE 0 PINNED BOTTOM)

L

=

18,000 FT

ka = 00

kb =

5xl05

LB-FT/RAD

PRETENSION = 2.05X106 LB

AT

ZJ

=

-14,000

FT

140

3 JOINTS

NO JOINT

120

0

TIME 5)

60

20

-0.2

o

0 2

1=

: :.

>-

en

z

o

i=

a::

a:l .

:>

-0.0 c.

Cl

Z

C

z

w

a:l

..J

~

iii

Furthermore, 3 joints reduce mean values of torsional, axial and

bending vibrations. It is noted that the fundamental axial, bending

and torsional periods are identical with or without joints.

In the previous investigation (Cheng, Chung and Zheng, 1995),

the I-joint case moved

TA

from 5.07 s to 7.56 s. When the excita

tion at

T

7.4 s, which is close to T = 7.56 s of pipe with a joint,

was applied to the pipe with 1joint, the amplitudes of axial stress

es at

Z

= -1,000 ft are larger than the excitation with T = 5 s,

while the biaxial (y-) and torsional

8z-)

vibrations show little dif

ference between the excitation periods of

T

= 5 sand 7.4 s, and

both bending (x-) vibrations at the bottom end still reach a steady

state.

Thus proper selection of stiffness and the number and locations

of joints can significantly reduce the coupled dynamic responses

of long pipes, even in the case when the excitation period is close

to the fundamental axial period.

Number of Elastic Joints

0

1.833xlO

.833xlO.737xlO

8.lxlO

.3xIO

.7xlO

4.47IxlO

.929xlO

.742xlO

I.lxlO

.85xlO

.85xlO

Fig. 12 Comparison of biaxial bending (y-) vibrations of pipe D

(pinned bottom end, no buffer and L = 18,000 ft) at zJ = -14,000

ft: no joint and 3 j oints at zJ

=

-1,000, -9,000 and -17,000 ft;

ka

00 Ib/ft and

kb

= 5x105 ft-Ib/rad; pretension at top end = 2.05x106

Ib

125

00

L = 18,000 FT

_ WB = 448,000 LB

1 JOINT; Z = -1,000 FT

ka

=

105 LB/FT, kb

=

0

Xo

=

5 FT, Zo

=

3 FT

T = 7.4 5

6t

=

0.001 5

6t

=

0.00025 5

25

i= 172 8

: :.

'i

173 0

~

J

I-

173 3

0

=

::

5 173 5

CJ

z

jj 173 8

0

the elastic joints with proper values of their stiffness and position

along the pipe can change the static responses. The bending stiff

ness of the joints can change the distribution of the bending

moment along the pipe, while having little effect on the corre

sponding bending deflection. However, the axial stiffness can sig

nificantly affect the axial displacement of the pipe, while not

changing the corresponding internal axial forces.

If a joint material of optimum stiffness is used, the elastic joint

can substantially change the axial fundamental frequency and

effectively reduce the maximum bending moment, while keeping

the axial displacements within a certain practical range.

Similarly, the optimum positioning of the joints along the pipe is

also very effective in reducing maximum bending moments. If the

joint is placed at a wrong position on the pipe, it can increase the

maximum bending moment instead of reducing it. In general, the

joint should be placed at the position on the pipe where the bending

moment is maximum. Multiple joints can be designed to be more

effective in achieving these purposes than the single joint.

Moreover, the elastic joint can effectively improve the dynamic

50 75

TIME 5)

Fig. 13 Effect of time-step size on bending (x-) vibrations of pipe

C (free bottom end, a buffer and L = 18,000 ft) at bottom end: LIt

= 0.001 sand 0.00025 s at

T

= 7.4 s; 1 joint at zJ = -1,000 ft, and

ka = 105 Ib/ft and = 0

Static analysis of both pipes A and B of L = 4,000-ft shows that

Time-Step Size

The results presented above are obtained with a time-step size

of LIt = 0.001 s. In order to test an optimum LIt size, computations

for pipe C are also made with LIt = 0.00025 s for the above numer

ical examples of both the excitation periods of

T

= 5 sand 7.4 s.

There is little difference in the x-, y-, z- and 8z- responses

between the two time-step sizes. An example is shown in Fig. 13

for the bending (x-) vibrations.

18,000-ft Pipe with Pinned Bottom End and No Buffer: Pipe D

Because the fundamental axial period is T = 4.27 s, the top

end of the pipe is excited at T = 4.2 s, close to TA with

Zo

= 3.0 ft

and

Xo

= 5.0. The period of both axial and bending vibrations is

the same as the excitation period.

For a pipe with no joints, the amplitudes of the axial stresses

are about 8 times larger than the static stress. The maximum

bending (x-) deflection occurs at Z = -17,000 ft, and there the

amplitude of the bending vibration is

Xo

12 ft, and the corre

sponding mean bending deflection increases. The torsional vibra

tion period is the same as its fundamental period

T

T = 3.37 s, and

the amplitude decays with time.

However, the biaxial bending (y-) vibration occurs at different

pipe position and period. Its amplitude of

Yo

= 0.1 ft, which is

much larger than the maximum static deflection of y = 0.0103 ft

at Z = -17,000 ft, occurs at z = -14,000 ft, and it occurs at a posi

tion different from the maximum static deflection. Its period is 8.4

s, about twice of

TA

(Fig. 12). Its amplitude can be reduced about

40 times when the 3 elastic joints are installed as in Table 5.

However, these 3 joints have few effects on the dynamic axial or

bending stresses.

CONCLUSIONS

Table 6 Effects of elastic joints on dynamic axial and bending

stresses of pipe C (free bottom end and

L

= 18,000 ft:

ka

= 00 and

kb

= lx106 ft-lb/rad


9/9


responses of the pipe. For an example of pipe C (an 18,000-ft pipe

with free bottom end with no joint), amplitudes of the torsional

Bz- ,

as well as the biaxial (y-) bending vibrations to the excita

tion near the fundamental axial period grow with time, the vibra

tions do not reach a steady state and show beatings. and the

amplitudes of the axial stresses were large. When 3 elastic joints

are placed at proper positions on the pipe, the axial stress and

bending stress amplitudes decrease 10 and 20 , respectively,

and the torsional as well as biaxial bending vibrations change

their characteristics, reducing the corresponding stresses, causing

their beatings to disappear and the corresponding vibrations to

reach steady state. Even in the case where the excitation period is

close to the fundamental axial period of the pipe with a joint, all

dynamic responses reach a steady state. The 3-joint case improves

the pipe responses more effectively than the I-joint case Zj =

-1,000 ft).

For pipe D (an 18,000-ft pipe with pinned bottom end) with no

buffer, the elastic joints improve the response characteristics of

the pipe. The biaxial bending (y-) vibration amplitude can be

reduced about 40 times when the 3 elastic joints are installed.

However, these 3 joints have few effects on the dynamic axial or

bending stresses. These 3 joints work better for pipe C than for

pipeD.

Also, the results show that multiple elastic joints are very effec

tive in improving and controlling the coupled responses of a deep

ocean pipe. Further design application study of multiple elastic

joints and axial dampers is provided in the recent work by Cheng

and Chung (1996).

ACKNOWLEDGEMENT

The authors gratefully acknowledge the support of The

National Science Foundation, Arlington, Virginia, under Research

Grant BCS 9207967 and the partial support of the first author

while he was in China from The National Natural Science

Foundation of China.

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