effects of elastic joints on 3-d nonlinear responses of a deep-ocean pipe_modeling and boundary...
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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
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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
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International Journal of Offshore and Polar Engineering
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).
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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
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International Journal of Offshore and Polar Engineering
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
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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
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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
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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
-
8/11/2019 Effects of Elastic Joints on 3-D Nonlinear Responses of a Deep-Ocean Pipe_Modeling and Boundary Conditions, Jin
9/9
International Journal of Offshore and Polar Engineering
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.
REFERENCES
Bathe, KJ (1982). Finite Element Procedure in Engineering Analysis
Prentice-Hall.
Caldwell, JB, and Gammage, WE (1976). A Method for Analysis of a
Prototype Articulated Multiline Marine Production Riser System, J
Pressure Vessel Tech ASME; also presented at Petroleum Mech
2/1
Eng and Pressure Vessels and Piping Conf
Mexico City, June 21,
1976.
Cheng, B-R, Chung, JS, and Huttelmaier, H-P (1994). Three
Dimensional Coupled Responses of a Deep-Ocean Pipe: Part II.
Excitation at Pipe Top and External Torsion,
/nt
J
Offshore and
Polar Eng
ISOPE, Vol 4, No 4, pp 331-339.
Cheng, B, Chung, JS, and Zheng, ZC (1995). Effects of Elastic Joints
on the 3-D Nonlinear Coupled Responses of a Long Vertical Pipe,
Proc /nt Offshore and Polar Eng Conf
The Hague, ISOPE, Vol 2,
pp 236-243.
Cheng, B, and Chung, JS (1996). Effects of Axial Dampers and
Elastic Joints on the 3-D Dynamic Responses of a Deep-Ocean Pipe
With Torsional Coupling, Proc 6th /nt Offshore and Polar Eng
Conf Los Angeles, ISOPE, Vol I, pp 37-45.
Chung, JS, and Whitney, AK (1981, 1983). Dynamic Vertical
Stretching Oscillation of a Deep-Ocean Mining Pipe, Proc
Offshore Tech Conf Houston, Paper No. 4092, and J Energy
Resources Tech ASME, Vol 105,pp 195-200.
Chung, JS, and Whitney, AK (1993). Flow-Induced Moment and Lift
for a Circular Cylinder with Cable Arrangement, /nt
J
Offshore and
Polar Eng ISOPE, Vol 3, No 4, in press, pp 280-287.
Chung, JS (1994). Deep-Ocean Cobalt-Rich Crust Mining System
Concepts, Proc MTS-94 Conf Washington, DC.
Chung, JS, Cheng, B-R, and Huttelmaier, H-P (1994). Three
Dimensional Coupled Responses of a Deep-Ocean Pipe: Part I.
Excitation at Pipe Ends and External Torsion, /nt J Offshore and
Polar Eng ISOPE, Vol 4, No 4, pp 320-330.
Chung, JS, and Cheng, B-R (1995). Eigenvalues for a Long Vertical
Deep-Ocean Pipe with Elastic Joints, Proc Flow-/nduced
Vibrations Symp
ASME, Honolulu, July 24-28, PVP-Vol 298, pp
153-160.
Ortloff, IE, Caldwell, JB, and Teers, ML (1976). An Articulated
Multiline Production Riser for Deepwater Application, J Pressure
Vessel Tech
ASME; also presented at
Petroleum Mech Eng and
Pressure Vessels and Piping Conf
Mexico City, June 21,1976.
Zheng, Z-C, and Cheng, B-R (1991). Introduction for a Computer
Code of Engineering Structure Analysis of Offshore Structure
(ENSA-OS88 Code), Proc Asia Pacific Conf Comput Mech Hong
Kong, pp 143-148.
Zheng, Z-C, and Xie, G (1992). A New Approach of Dynamic
Substructure Method,
Acta Solid Mechanica
Beijing, No 4, pp
407-417.