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The prediction of the dynamic and structural motions of a floating-pier system in waves
ARTICLE IN PRESS
Ocean Engineering 34 (2007) 1044–1059 www.elsevier.com/locate/oceaneng
The prediction of the dynamic and structural motions of a floating-pier system in waves Hsien Hua Leea,, Liang-Yin Chena, Wen-Kai Wengb, S.-W. Shyuea a
Department of Marine Environment and Engineering, National Sun Yat-sen University (NSYSU), Kaohsiung, Taiwan b Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan Received 14 August 2005; accepted 30 January 2006 Available online 27 September 2006
Abstract In this study, a two-dimensional floating pier consists of single rectangular impermeable pontoon with side supporting pile-columns is studied. The purpose of this study is to present a theoretical solution for the linearized problem of incident waves exerting on a floating pier with pile-restrained. All boundary conditions are linearized in the problem, which is incorporated into a scattering problem and radiation problem with unit displacement. The method of separation of variables is used to solve for velocity potentials. For the radiation problem with unit heave and pitch amplitude, the boundary value problem with non-homogeneous boundary condition beneath the structure is solved by using a solution scheme. By calculating the wave force from velocity potential and solving the equation of motion of the floating structure simultaneously a close form theoretical solution for the problem is developed. The finite element method was also applied to calculate the dynamic responses on the supporting piles subjected to the pontoon motions and incident waves. r 2006 Elsevier Ltd. All rights reserved. Keywords: Floating-pier; Wave structure interaction; Wave force; Pier with guide-pile; Dynamic response
1. Introduction This paper provides an analytical study for the responses of a floating pier structural system confined horizontally with pile-columns. Floating pier structural system by using semi-submerged buoyant pontoons providing vertical supports for the deck is a widely used structural system because it is easy to construct and convenient to use for offshore deep water application. In the floating pier structural system the pontoon-pier is either anchored by the tension legs to the seabed or supported by pile-columns on the sides for the so-called bridge type floating pier system. The pile-columns not only work as an anchoring system but also provide a confinement to the pontoon-pier from moving around. A complete study for the dynamic behavior of a floating pier structural system is rather complicated, particularly in analytical methods because the analysis must account for not only the interactions between the structure (pontoon-pier) and the waves, but also the Corresponding author.
E-mail address: [email protected] (H.H. Lee). 0029-8018/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2006.01.015
interactions between waves, pontoon-pier and the anchoring systems. The interactions between the floating structure and the incident waves are the well-known scattering problem incorporated with radiation problem. In 1969, Black and Mei first studied the reflection and transmission of waves encountering the floating structure and identified it as a scattering problem. Later on Black et al. (1971) then studied a moving floating structure making waves in the flow field that was identified as a radiation problem. However, to solve these two problems simultaneously the nonlinearity of structural motion and the non-homogeneous boundary conditions on the free surface and on the structural bottom (Garrison, 1974) must be overcome first. This problem has been solved in numerical methods since the 1970s (Mei, 1978; Yamamoto et al., 1982), and also studied through experimental testing (Yamamoto et al., 1982) until Sarpakaya and Isaacson (1981) linearized the boundary conditions. By using the scheme to linearize the boundary conditions, the close form theoretical solution could be obtained by incorporating the scattering problem with the radiation problem.
ARTICLE IN PRESS H.H. Lee et al. / Ocean Engineering 34 (2007) 1044–1059
To examine the dynamic motion of floating pier structural system, besides the interactions between the structure (pontoon-pier) and waves the behavior of anchoring system, either a tension-leg or supporting pile-column for the pier is also essential. The other problem is that most analytical studies on the interactions between floating structure and waves incorporating with both the scattering problem and radiation problem are focused either on the surge radiation (Lee and Lee, 1993) or heave radiation (Lee, 1995), where the anchoring systems are either ignored or simplified. In the later studies for an offshore floating structure (Lee et al., 1999; Lee and Wang, 2000a, b) the properties of the anchoring tension-legs were taken into accounts along with the wave-structure interactions, where the large body effects from the pontoon-pier was also under the consideration. The results showed that for the same wave properties, the dynamic behavior of both the tether of the tension leg and the platform itself are closely related to the material property and the tether dimension. Although these studies accounted for the interaction between the wave and the pontoon-pier structure, the drag effect of tethers on the platform structures was ignored. By ignoring the effect of drag motion of the tethers on the platform, the response of the pontoon-pier could not be estimated accurately. However, after overcoming the complexities of multi-interactions in the previous analysis, the interactions between the pontoon-pier (large body), waves, and tethers (small body) have been taken into accounts in the analytical solutions (Lee and Wang, 2001). It was shown that the inclusion of the tether drag effect drastically reduced the platform amplitude under the same wave conditions. In this study the interactions between waves and structures are also taken into accounts and moreover, besides the radiation problem induced from surge, the radiation induced from heave and pitch motions of the pontoon-pier is also under consideration. For the sidesupporting floating pier system, the motion of the pontoonpier, basically the surge motion is confined by the pilecolumns on two sides of the floating structure. Therefore, the dynamic behavior of the pile-column must be taken into calculation simultaneously along with the response of the pontoon-pier, which in the same time interacts with waves and influence the wave properties that provide exerting pressure on the pile-columns. Based on previous studies (Lee and Wang, 2001, 2003), the equations for the scattering problem and the radiation problem were established at the related boundaries with variables of velocity potential obtained from the Laplace equation. The forcing function related to the velocity potentials was used in the equation of motion for the pontoon-pier and the pile-columns. The analytical solution for the unknown coefficients and the responses of both the pontoon-pier and the pile-columns were found by solving all of the equations derived on the boundaries simultaneously. The purpose of this study is to investigate the response of a floating pier structural system subjected to incident waves
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and flow drags on the pile-columns, including the multiinteraction among waves, pontoon-pier structure, and pilecolumns. Numerical examples are carried out, and the results are discussed focusing on both the reflection and transmission coefficients and on the motion of the pontoon-pier subjected to waves of various periods. The influence of parameters such as the material property of the pile-columns, dimension of the pontoon and the submerged depth on the pontoon was also studied. It is found that the response of the floating pier structure is significantly dependant on the properties of the pilecolumns, the wave conditions and the interactions between the pontoons and the waves. 2. General theory of wave and flow field 2.1. Basic provisions on the problems Illustrated in Fig. 1 is a two-dimensional schematic view for a typical floating pier structural system, where a set of floatable pontoon are mounted under the deck of the pier while two ends of the deck are connected to a pair of pilecolumns fixed to the seabed. The pier-decks are allowed to vertical and rotational motions because the connection joints between the pile-column and the deck are vertically movable hinges. However, even though due to the connection joints the relative horizontal motions between the deck and pile-column are not allowable, the surge motions for the whole floating pier structural system, which includes the pile-columns as well, are allowable and studied. It is assumed that the water depth, where the floating pier structure is located is h, draught of the pontoon is d and the width of the pontoon is 2b. In the study, both the large body and small body theorems for the applications of wave force are utilized. For the pile-columns the small body theorem is assumed because the dimension of the pile ds is usually small compared to the wave length L (d s =Lo0:2) while the large body theorem must be applied for the pontoons since the dimension for the pontoon of the floating pier could be larger than 1/5 of the wavelength. Therefore, the interactions between waves and
Fig. 1. Sketch of a typical floating pier structural system subjected to waves.
ARTICLE IN PRESS H.H. Lee et al. / Ocean Engineering 34 (2007) 1044–1059
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pontoons of the floating pier are taken into accounts while the back-influence to the response of pile-columns is also considered as well. In the analysis the inviscid and incompressible fluid and irrotational flow field are assumed and a single-valued velocity potential f can be defined as ~ u ¼ rf, where ~ u is the velocity and r is the gradient operator. The velocity potential satisfies the Laplace equation: r2 f ¼ 0 and the Bernoulli equation in the flow field as qf 1 p þ rfrf þ þ gz ¼ 0, qt 2 rw
(1)
where p is the pressure, rw the water density, g the gravitational constant, and z the variable of depth. The nonlinear term in Eq. (1) is ignored when the linear small amplitude wave is assumed. A two-dimensional floating pier structural system constrained on a pair of pile-columns interacting with a monochromatic small amplitude wave propagating in the +x-direction is considered here as shown in Fig. 1. The waveform and the associated velocity potential are given accordingly as ZI ðx; tÞ ¼ RefAeðK 0 xþstÞ g
(2)
and fI ðx; z; tÞ ¼ iA
g cos K 0 ðz þ hÞ iðK 0 xþstÞ e , s cos K 0 h
(3)
where A is the wave amplitude, g is the gravitational constant and h is the water depth. s ¼ 2p=T is the angular frequency with period T, K 0 ¼ ik and k ¼ 2p=L is the wave number with wavelength L. Coefficient K0 satisfies the dispersion relation s2 ¼ gK 0 tanðK 0 hÞ. For the floating pontoon, three degrees of freedom including surge, heave and pitch motions are assumed to be harmonically corresponding to the motion of waves when the exerting waves are periodic with small amplitude. Therefore, the displacement functions corresponding to surge x, heave Z and pitch y are given by x ¼ S 1 eist ,
(4)
2.2. Boundary value problems As shown in Fig. 1, the floating pier structural system is divided into three regions with two artificial boundaries at x ¼ b and x ¼ þb. Since the whole flow field is composed of incident waves, scattered waves and radiated waves, the total velocity potential for each region of the flow field may be represented as follows: f ¼ fI þ fs þ S 1 fw þ S 2 fh þ S3 fp
where the superscripts I, s, w, h and p indicate velocity potential corresponding to incident waves, scattered waves and radiated waves from the surge, heave and pitch motion of the pontoon structures respectively. In region I: the total velocity potential consists of velocity potentials of incident waves, scattered waves and radiated waves of surge, heave and pitch motions as f1 ¼ fI þ fs1 þ S1 fw1 þ S2 fh1 þ S 3 fp1 . In regions II and III: only scattered waves and radiated waves are considered as f2 ¼ fs2 þ S 1 fw2 þ S 2 fh2 þ S3 fp2 and f3 ¼ fs3 þ S 1 fw3 þ S2 fh3 þ S 3 fp3 . All of the velocity potentials satisfy the Laplace equation. Furthermore, on the infinite boundary 1 in region I and þ1 in region III, the Sommerfeld’s radiation condition is satisfied for unique solutions. Now the solutions for the interaction between the pontoon and waves include the scattering problem and radiation problem. For the radiation problem, all motions from the pontoon-pier structure must be taken into account thus including surge, heave and pitch. 2.2.1. Scattering problems In the scattering problem the incident wave is diffracted by the pontoon-pier structure, which is presumably fixed. The corresponding boundary conditions for each region of the flow field in the scattering problem are shown in Fig. 2. By applying the method of separation of variables, matching the horizontal boundary conditions for the free water surface and sea bed in each regions, and applying the Sommerfeld’s condition to regions I and III, the velocity potentials in each region are found as fs1 ¼
1 X
As1n
g cos½K n ðz þ hÞ K n ðxbÞ ist e e , s cosðK n hÞ
(8)
As3n
g cos½K n ðz þ hÞ K j ðxþbÞ ist e e , s cosðK n hÞ
(9)
n¼0
Z ¼ S 2 eist ,
(5) fs3 ¼
y ¼ S3 e
ist
,
(6)
where S i represents the amplitude corresponding to surge, heave and pitch motions, respectively. For the interaction problems between the pontoons and incident waves, from the linearity of the problem, the problem can be incorporated into a scattering and a radiation problem (Black et al., 1971). The wave force calculated from the scattering problem provides the force function in the radiation problem, and the forced oscillation of the structure then generates outgoing waves.
(7)
1 X n¼0
where the eigenvalue K n is solved from the dispersion equation s2 ¼ gK n tan K n h. g s A20 x þ Bs20 eist fs2 ¼ s 1 X g s K 2n ðxbÞ A2n e þ þ Bs2n eK 2n ðxþbÞ s n¼1 cos K 2n ðz þ hÞeist , where K 2n ¼ np=h d.
ð10Þ
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Fig. 2. Illustration of the scattering boundary-value problem for the floating pier structural system.
The series of 4N unknowns As1n , As2n , As3n and Bs2n in the velocity potential of each region can be further solved through the boundaries between two regions as indicated as follow: Kinematic boundary condition (KBC) on x ¼ b between regions I and II: 8 0 dpzp0; s qðf1 þ fI Þ < s ¼ qf2 (11) ; hpzp d: : qx qx Dynamic boundary condition (DBC) on x ¼ b between regions I and II: qðfs1 þ fI Þ qfs2 ¼ ; qt qt
hpzp d .
(12)
(13)
DBC on x ¼ b between regions II and III: hpzp d .
(14)
Now by substituting velocity potentials in each region as shown in Eqs. (8)–(10) into kinematic and dynamic conditions as shown in Eqs. (11)–(14), and employing the orthogonality of the velocity potential functions, the 4N series of unknowns are ready to be solved in the following equations. E m As1m þ C m As20 ¼ id0m E m Aexp
1 X
n¼1 ikb
,
K 2n Omn As2n Bs2n e2K 2n
G mn As1n þ ½bðh dÞAs20 þ ðh dÞBs20
n¼0
þ
ðh dÞ s A2m þ Bs2m e2K 2m b ¼ iAG m0 eikb , 2
C m As20
1 X
ð16Þ
K 2n Omn As2n e2K 2n b Bs2n þ E m As3m ¼ 0,
n¼1
(17) ðh dÞ s 2K b A2m e 2m þ Bs2m ðh dÞbAs20 ðh dÞBs20 2 1 X þ G mn As3n ¼ 0; m ¼ 021, ð18Þ n¼0
where the parametersE m ; C m ; Omn ; G mn are defined in Appendix A.
KBC on x ¼ b between regions II and III: 8 0; dpzp0; qfs3 < s ¼ qf2 ; hpzp d: : qx qx qfs3 qfs2 ¼ ; qt qt
1 X
2.2.2. Radiation problems due to unit surge motion For the radiation problems due to unit surge motion of the pontoon structure, the corresponding boundary conditions in each region are shown in Fig. 3 such as the free water surface condition, impermeable conditions on seabed and the bottom surface of the pontoon structures. Again by applying the method of separation of variables, matching the horizontal boundary conditions in each regions, and applying the Sommerfeld’s condition to regions I and III, the velocity potentials in each region are found as fw1 ¼
1 X
Aw1n
g cos½K n ðz þ hÞ K n ðxbÞ ist e e , s cosðK n hÞ
(19)
Aw3n
g cos½K n ðz þ hÞ K j ðxþbÞ ist e e , s cosðK n hÞ
(20)
n¼0
b ð15Þ
fw3 ¼
1 X n¼0
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Fig. 3. Illustration of the radiation boundary-value problem for the floating pier structural system with unit surge motion.
fw2 ¼
1 X g w g w K 2n ðxbÞ ðA20 x þ Bw20 Þeist þ A2n e s s n¼1 þBw2n eK 2n ðxþbÞ cos K 2n ðz þ hÞ ist.
Aw1n ,
Aw2n ,
Aw3n
E m Aw1m þ C m Aw20 þ ð21Þ
qt
¼
qt
;
hpzp d,
¼ i 1 X
s S1 F m , g
hd ð2b 1ÞAw2m þ Bw2m 2 n¼0 ðh dÞ w A2m þ Bw2m e2K 2m b ¼ 0, 2 Gmn Aw1n d0m
C m Aw20
1 X
ð27Þ
K 2n Omn Aw2n e2K 2n b Bw2n þ E m Aw3m
n¼1
s2 ¼ i S1 F m , g (23)
ð26Þ
hd ð2b þ 1ÞAw2m Bw2m 2 ðh dÞ w 2K 2m b A2m e þ Bw2m 2 1 X þ G mn Aw3n ¼ 0; m ¼ 021,
ð28Þ
d0m
KBC on x ¼ b between regions II and III: 8 ist < isS1 e ; dpzp0; w qf3 w ¼ qf2 : ; hpzp d; qx qx
(24)
hpzp d.
ð29Þ
n¼0
where parameters E m ; C m ; Omn ; G mn ; F m are also presented in Appendix A.
DBC on x ¼ b between regions II and III: qfw3 qfw ¼ 2; qt qt
2
Bw2n
DBC on x ¼ b between regions I and II: qfw2
K 2n Omn Aw2n Bw2n e2K 2n b
n¼1
and in Again the series of 4N unknowns the velocity potential of each region can be further solved through the boundary conditions between two regions as indicated as follows: KBC on x ¼ b between regions I and II: 8 < isS1 ; dpzp0; qfw1 w (22) ¼ @f2 ; hpzp d; : qx @x
qfw1
1 X
(25)
Now after the substitution of velocity potentials in each region into kinematic and dynamic conditions and after the rearrangement, Eqs. (22)–(25) become the form of Eqs. (26)–(31). By employing the orthogonality of the velocity potential functions, the 4N series of unknowns are ready to be solved from these equations.
2.2.3. Radiation problems due to unit heave motion For the radiation problems due to unit heave motion of the pontoon structure, the corresponding boundary conditions in each region are shown in Fig. 4. Again by applying the method of separation of variables, matching the horizontal boundary conditions in each region, and applying the Sommerfeld’s condition to region I and III,
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Fig. 4. Illustration of the radiation boundary-value problem for the floating pier structural system with unit heave.
the velocity potentials in each region are found as fh1 ¼
1 X
Ah1n
g cos½K n ðz þ hÞ K n ðxbÞ ist e e , s cosðK n hÞ
Ah3n
g cos½K n ðz þ hÞ K j ðxþbÞ ist e e , s cosðK n hÞ
n¼0
fh3
¼
1 X
where (30)
(34)
and
n¼0
(31)
where the eigenvalue Kn is solved from the dispersion equation s2 ¼ gK n tan K n h. It is noticed that since the boundary is vertically movable for the bottom of the pontoon structure during the heave motion the nonhomogeneous boundary conditions are obtained as qfh2 ¼ isS 2 eist ; qZ
z ¼ d.
(32)
The method of separation of variables cannot be directly applied for the solution in this case and therefore, the velocity potential in region II due to unit heave motion of the pontoon is obtained through a scheme (Lee, 1995) that incorporates the non-homogeneous boundary conditions into two homogeneous boundary conditions, namely the vertical and horizontal one. Then the boundary value problem becomes two homogeneous boundary value problems. The velocity potential is subsequently obtained as the sum of the solution from the vertical boundary value problem and horizontal boundary problem in a form as g fh2 ¼ ðAh2n x þ Bh2n Þeist s 1 X g h K 2n ðxbÞ þ þ Bh2n eK 2n ðxþbÞ Þ cos K 2n ðz þ hÞeist ðA2n e s n¼1 þ
2 ¯ h ¼ is S2 ½cosð2ln bÞ 1 A 2n gbl2n sinh½ln ðh dÞ
1 X
h g A2n sin ln ðx bÞ cosh ln ðz þ hÞeist ; s n¼1
ð33Þ
K 2n ¼
np ¼ ln . 2b
(35)
The boundary conditions between two regions including both the kinematic and dynamic type are presented as follows. KBC on x ¼ b between regions I and II: 8 < 0; ¼ qfh2 : qx ; qx
qfh1
dpzp0; hpzp d;
(36)
DBC on x ¼ b between regions I and II: qfh1 qfh2 ¼ ; qt qt
hpzp d,
(37)
KBC on x ¼ b between regions II and III: 8 0; qfh3 < h ¼ qf2 : qx ; qx
dpzp0; hpzp d;
(38)
DBC on x ¼ b between regions II and III: qfh3 qfh2 ¼ ; qt qt
hpzp d.
(39)
Now after the substitution of velocity potentials in each region back into kinematic and dynamic conditions and after the rearrangement, Eqs. (36)–(39) become the form of Eqs. (40)–(43).
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E m Ah1m þ C m Ah20 þ
1 X
K 2n Omn ðAh2n Bh2n e2K 2n b Þ
n¼1
¼
1 X
h A2n ln Pmn ,
ð40Þ
n¼1 1 X
hd ð2b 1ÞAh2m þ Bh2m d0m 2 n¼0 ðh dÞ h A2m þ Bh2m e2K 2m b ¼ 0, 2
motion from the pontoon structure. Similarly, by applying the method of separation of variables, matching the horizontal boundary conditions in each regions, and applying the Sommerfeld’s condition to regions I and III, the velocity potentials in each region are found as fp1 ¼
G mn Ah1n
1 X
Ap1n
g cos½K n ðz þ hÞ K n ðxbÞ ist e e , s cosðK n hÞ
(44)
Ah3n
g cos½K n ðz þ hÞ K j ðxþbÞ ist e e . s cosðK n hÞ
(45)
n¼0
ð41Þ
fp3 ¼
1 X n¼0
C m Ah20
1 X
K 2n Omn ðAh2n e2K 2n b Bh2n Þ þ E m Ah3n
n¼1
¼
1 X
h
A2n ln Pmn cos 2ln b,
ð42Þ
n¼1
hd ðh dÞ h 2K 2m b ½ð2b þ 1ÞAh2m Bh2m ðA2m e þ Bh2m Þ 2 2 1 X G mn Aw3n ¼ 0; m ¼ 021, ð43Þ þ
d0m
n¼0
where parameters E m ; C m ; Omn ; G mn ; Pmn are also presented in Appendix A. By solving theses equations simultaneously and employing the orthogonality of the velocity potential functions, the 4N series of unknowns are ready to be solved. 2.2.4. Radiation problems due to unit pitch motion When the motion of the pontoon structure is small the boundaries are assumed on the position of equilibrium and then the pitching kinematics of the pontoon structure will be similar to a positively moving gradient boundary (Lee, 1995). Illustrated in Fig. 5 is the boundary condition for the radiation problems subjecting to an unit pitching
Similar to the radiation problem of the heave motion, where the boundary is vertically movable for the bottom of the pontoon structure during the pitch motion, the nonhomogeneous boundary conditions are obtained as qfp2 ¼ isS 3 eist x; qZ
z ¼ d.
(46)
The method of separation of variables is not applicable for the solution directly and the velocity potential in region II due to unit pitch motion of the pontoon is obtained through a method as presented in heave-radiation problem, in a form as g fp2 ¼ ðAp2n x þ Bp2n Þeist s þ
þ
1 X g
s n¼1
Ap2n eK 2n ðxbÞ þ Bp2n eK 2n ðxþbÞ cos K 2n ðz þ hÞeist
1 X
p g A2n sin ln ðx bÞ cosh ln ðz þ hÞeist ; s n¼1
ð47Þ
where 2
¯ p2n ¼ is S3 ½cosð2ln bÞ þ 1 A gl2n sinh½ln ðh dÞ
Fig. 5. Illustration of the radiation boundary-value problem for the floating pier structural system with unit pitch.
(48)
ARTICLE IN PRESS H.H. Lee et al. / Ocean Engineering 34 (2007) 1044–1059
and K 2n ¼
3. Equations of motion of the floating-pier structure with side supports
np ¼ ln . hd
(49) 3.1. Equations of motion of the pontoon structure
The boundary conditions of both kinematic and dynamic type are presented as follows. KBC on x ¼ b between regions I and II: 8 isS 3 ðz z0 Þ; dpzp0; qfp1 < p (50) ¼ qf2 : ; hpzp d; qx qx DBC on x ¼ b between regions I and II: qfp1 qfp2 ¼ ; qt qt
hpzp d,
(51)
KBC on x ¼ b between regions II and III: 8 isS 3 ðz z0 Þ; dpzp0; qfp3 < p ¼ qf2 : ; hpzp d; qx qx
hpzp d.
(52)
(53)
Now after the substitution of velocity potentials in each region back into kinematic and dynamic conditions and after the rearrangement, Eqs. (50)–(53) become the form of Eqs. (54)–(57). E m Ap1m C m Ap20 ¼
1 X
K 2n Omn Ap2n Bp2n e2K 2n b
n¼1 2
p
A2n ln Pmn þ i
n¼1 1 X
1 X
s S 3 Qm , g
ð54Þ
where the mass matrix 2 m 6 0 ½M ¼ 4
0 m
3 mðzc z0 Þ mðxc x0 Þ 7 5,
(59)
I yy
and the stiffness matrix 2 3 0 0 0 6 7 rgI A ½K ¼ 4 0 rgð2bÞ 5. x A V 0 rgI A rgðI þ I Þ mgðz z Þ c 0 x xx z
(60)
It is noticed in Eq. (60) that for the surge motion of the pontoon structure there is no stiffness or constraints from the system in this formulation. In the mass matrix ½M, m is the mass of the pontoon structure that can be obtained from the displaced water as rð2bÞd; I yy is the rotational inertia in the y-direction with respect to the mass center; I A xx and I V z are increments of the moment of the buoyancy due to the rotational angle S 3 , which may be presented as Z b A ðx x0 Þ2 dx, (61) I xx ¼ b
hd ð2b 1ÞAp2m þ Bp2m d0m 2 n¼0 ðh dÞ p A2m þ Bp2m e2K 2m b ¼ 0, 2 G mn Ap1n
C m Ap20
When a floating structure, which is free of any constraint subjected to incident waves, the equations of motion for the surge, heave and pitch may be presented as (Mei, 1983) 8 9 8 9 8 9 x> x> > > Z = < = = < < n1 > 2 > d (58) ½M 2 Z þ ½K Z ¼ r f;t n2 dS > > dt > S ; : > ; ; : > :n > y y 3
mðzc z0 Þ mðxc x0 Þ
DBC on x ¼ b between regions II and III: qfp3 qfp2 ¼ ; qt qt
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1 X
IV z ð55Þ
K 2n Omn Ap2n e2K 2n b Bp2n
1 X n¼1
p
A2n ln Pmn cos 2ln b þ i
s2 S3 Qm , g
0
ð56Þ
ðh dÞ p 2K b hd ð2b þ 1ÞAp2m Bp2m A2m e 2m þ Bp2m 2 2 1 X p þ G mn A3n ¼ 0; m ¼ 021, ð57Þ
d0m
n¼0
where parameters E m ; C m ; Omn ; G mn ; Pmn ; Qm are presented in Appendix A. By solving theses equations simultaneously and employing the orthogonality of the velocity potential functions, the 4N series of unknowns are to be solved from these equations.
Z
b
ðz z0 Þ dxdz.
¼ d
n¼1
þ E m Ap3n ¼
Z
(62)
b
R The wave forces r S f;t dS induced from incident waves, scattering waves and radiated waves can be obtained after the total velocity potential f is found from the scattering and radiation problems as indicated in previous sections. The wave forces on the right-hand side include the horizontal, vertical forces and rotational moment. The horizontal forces are obtained by carrying out the time differentiation of velocity potentials in all three regions and further written as Z 0 Z r f;t n1 dS ¼ isr ½fs3 jx¼b ðfI þ fs1 Þjx¼b dz S
Z
d 0
þ S1 d Z 0
þ S3 d
½fw3 jx¼b fw1 jx¼b dz ½fp3 jx¼b fp1 jx¼b dz .
ð63Þ
It is noticed that all of the waves including the incident wave, scattering waves and radiated waves from surge and
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pitch motions are taken into accounts in the calculation. For the vertical forces both the scattering waves and radiated waves due to heave motions are taken into accounts while the resulting vertical force from the radiated waves due to surge and pitch motion is zero. The vertical force is presented as Z b Z Z b s h r f;t n2 dS ¼ isr f2 jz¼d dx þ S 2 f2 jz¼d dx . S
b
b
(64) The rotational moments from the incident waves, scattering waves and radiated waves of surge and pitch motion are calculated Z 0 Z r f;t n3 dS ¼ isr ðz z0 Þ½fs3 jx¼b S
d
Z b ðfI þ fs1 Þjx¼b dz x½fs2 jz¼d dx b Z 0 w þ S1 ðz z0 Þðf3 jx¼b fw1 jx¼b Þ dz Z
d b
xðfw2 jz¼d Þ dx
b
Z
0
þ S3 Z
d
ðz z0 Þðfp3 jx¼b fp1 jx¼b Þ dz
b
xðfw2 jz¼d Þ dx
.
ð65Þ
presented. Eq. (66) may be derived by combining the following equation of motions in each degree of freedom. Z d2 (67) m 2 ½x þ ðzc z0 Þy þ kx þ k dy ¼ r f;t n1 dS, dt S Z d2 A m 2 ½Z ðxc x0 Þy þ rgð2bÞZ rgI x y ¼ r f;t n2 dS, dt S (68) d2 x d 2Z d2y A V I z 2 I x 2 þ I yy 2 þ k dx þ ðrgðI xx þ I z Þ dt dt dt Z 2 rgI A x Z þ kd Þ mgðzc z0 Þy ¼ r
3.2. Equations of motion of the pontoon structure with side supports As observed in Eq. (60), the surge motion of a free constrained pontoon will be only governed by the mass of itself and if the surge from the wave force is large the surge motion of the pontoon will be large too. When the side supports of pile-columns are applied to the floating pier structural system, the behavior of pontoon subjected to waves will be constrained, particularly in the horizontal direction and therefore, the equations of motion for the pontoon are rewritten into a form as 8 9 8 9 8 9 8 9 Tx > x> x> > > > Z = < = = = < < < n1 > 2 > d ½M 2 Z þ ½K Z þ T Z ¼ r f;t n2 dS. > > dt > S ; > : > ; ; ; : > :T > :n > y y y 3 (66) Compared to Eq. (58), it is noticed that an additional vector of third term in the left-hand side representing the constraints from the side supports of the pile-columns is
ð69Þ
By combining Eqs. (67), (68) and (69) and employing the displacement functions, the additional vector of constraints is combined into the stiffness matrix and Eq. (66) may be rewritten into a form in terms of the amplitudes of surge, heave and pitch as 8 9 8 9 8 9 f > > > = = = < S1 > < S1 > < 1> 2 S S þ ½K ¼ isr f 2 . (70) s ½M 2 2 > > > ; ; ; :S > :S > :f > 3 3 3 Assuming that the mass center and rotational center are collocated and then ðzc z0 Þ ¼ 0 and ðxc x0 Þ ¼ 0, the mass matrix [M] and stiffness matrix [K] can be reformulated as 2 3 m 0 0 6 7 ½M ¼ 4 0 m 0 5 (71) 0
b
To solve for the unknowns of the amplitudes of the surge, heave and pitch motions, which are implicitly contained in the velocity potential, Eq. (58) must be solved simultaneously with the velocity potentials in the scattering and radiation problems.
f;t n3 dS. S
I yy
0
and 2 6 ½K ¼ 6 4
4EI ðhþd=4Þ3
0
0
rgð2bÞ
4EI ðhþd=4Þ3
0
4EI ðhþd=4Þ3
3
d
0 V rgðI A xx þ I z Þ þ
4EI ðhþd=4Þ3
d2
7 7. 5 (72)
The wave forces corresponding to surge, heave and pitch as shown in the force vector of Eq. (70) are further written as: Z 0 ½fs3 jx¼b ðfI þ fs1 Þjx¼b dz f1 ¼ d
Z
0
½fw3 jx¼b fw1x¼b dz
þ S1 d Z 0
þ S3 d
Z
½fp3 jx¼b fp1 jx¼b dz,
b
fs2 jz¼d dx þ S 2
f2 ¼
Z
b
Z
0
f3 ¼
b
fh2 jz¼d dx,
b
ðz z0 Þ fs3 jx¼b ðfI þ fs1 Þjx¼b dz
d
Z
b
x fs2 jz¼d dx
b
Z
0
þ S1 d
ð73Þ
ðz z0 Þðfw3 jx¼b fw1 jx¼b Þ dz
(74)
ARTICLE IN PRESS H.H. Lee et al. / Ocean Engineering 34 (2007) 1044–1059
Z
b
Q33 representing the resultant couple for the unit pitch motion is given as Z 0 Z b p p Q33 ¼ ðz z0 Þ f3 jx¼b f1 jx¼b dz x fp2 jz¼d dx
xðfw2 jz¼d Þ dx
b
Z
o
þ S3 Z
d b
b
ðz z0 Þðfp3 jx¼b fp1 jx¼b Þ dz
xðfp2 jz¼d Þ dx
.
d
After regrouping the coefficients for the vector of amplitude, Eq. (70) then can be rewritten as 8 9 8 s9 f > S1 > > = < 1> 2 < = s S 2 ¼ isr f 2 , (76) s ½M þ ½K þ isr½Q > > ; ; :S > :fs > 3
where 2
Q11
0
6 ½Q ¼ 4 0 Q31
Q22 0
Q13
3
0 7 5. Q33
(77)
Q13 representing the horizontal resultant force term for the unit pitch motion is given as Z 0 p Q13 ¼ f3 jx¼b fp1 jx¼b dz d 1 X ðAp3n Ap1n Þ g sin K n h sin K n ðh dÞ ¼ , ð79Þ Kn cos K n h s n¼0 Q22 representing the vertical resultant force term for the unit heave motion is given as Z b ½fh2 jz¼d dx Q22 ¼
¼
cos K n ðh dÞ , Kn
n¼1
ð80Þ
b
s
ð82Þ
where parameters Ln , M n , N n are defined in Appendix A. The force vector on the right-hand side now is solely related to the scattering waves and presented as Z 0 s f s1 ¼ f3 jx¼b ðfI þ fs1 Þjx¼b dz 1 X g ðAs3n As1n Þ sin K n sin K n ðh dÞ ¼ s cos K n h Kn n¼0
iAeiK 0 b g sinh kh sinh kðh dÞ , k cos K o h s
f s2
Z
ðAw2n Bw2n Þ cos K n ðh dÞM n ,
ð83Þ
b
½fs2 jz¼d dx
¼ b
1 X g g s ðA2n þ Bs2n Þ ¼ ð2bÞBs20 þ s s n¼1
ð1 e2K 2n b Þ
f s3 ¼
Z
cos K 2n ðh dÞ , K 2n
ð84Þ
0
ðz z0 Þ½fs3 jx¼b ðfI þ fs1 Þjx¼b dz
d
Z
b
x½fs2 jz¼d dx
b
1 X g iAeikb g ðAs1n As3n Þ g 2b3 L0 Ln ¼ As20 s cosh kh s cos K n h s 3 n¼0 1 X g n¼1
s
ðAs2n Bs2n Þ cos K 2n ðh dÞM n .
ð85Þ
To solve for the amplitude of the pontoon structure S1, S2 and S3 in Eq. (76), equations in the scattering and radiation problems are also solved simultaneously. The velocity potentials in each region of the flow field will be obvious with known amplitude of the wave motions.
4. Analytical examples of the floating pier structural system subjected to waves
1 X g ðAw3n Aw1n Þ g 2b3 Ln Aw20 ¼ s cos K n h s 3 n¼0 1 X g
ðAp2n Bp2n Þ cos K n ðh dÞM n
p g A2n cosh ln ðh dÞN n , s n¼1
Q31 representing the resultant couple for the unit surge motion is given as Z 0 Z b Q31 ¼ ðz z0 Þ½fw3 jx¼b fw1 jx¼b dz x½fw2 jz¼d dx
s n¼1
1 X
b 1 X g g h ðA2n þ Bh2n Þð1 eK 2n b Þ ¼ ð2bÞBh20 þ s s n¼1
d
1 X g
d
In [Q] matrix, Q11 representing the horizontal resultant force term for the unit surge motion is defined as Z 0 w Q11 ¼ f3 jx¼b fw1 jx¼b dz d 1 X ðAw3n Aw1n Þ g sin K n h sin K n ðh dÞ ¼ , ð78Þ cos K n h s Kn n¼0
b
p p 1 X g ðA3n A1n Þ g 2b3 Ln Ap20 ¼ s cos K n h s 3 n¼0
ð75Þ
3
1053
ð81Þ
Analytical examples for the floating pier structure with vertical pile system are carried out in this section. The investigation was focused on the displacement responses of
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the pile-column system, the influence on the flow field and the response of pontoon-pier structure subjected to waves with various periods. The influence of the flexural rigidity of the piles, dimension of the pontoon structure and the draught of the pontoons on the pontoon-pier motion and the pile-column responses was also studied. Before carrying out the calculation for the analytical examples the convergence of the analytical solution was examined and verified first. 4.1. The verification and convergence of the analytical solution Since the solution of the response in this study is in an analytical form, the convergence test was performed for the reflection coefficient in terms of the mode number needed for a stable solution of series form. Presented in Fig. 6(a) and (b) is the reflection coefficient corresponding to the mode numbers when the wave period is ranged from 2 to 10 s and 12 to 20 s, respectively. It is observed that for most cases when the mode number is more than 10 the reflection coefficient will converge to a constant value. It is also
Fig. 6. The convergence of the reflection coefficient corresponding to the number of mode for the series solution: (a) wave period from 2 to 10 s, and (b) Wave period from 12 to 20 s.
known that the mode number needed for the convergence is related to the water depth and the draught and dimension of the structure. To further verify that this study may accurately model the motion of the pontoon-pier structure subjected to waves, a case study with only accounting for the heave radiation was also performed to compare to Lee’s study (1995). By adopting the same conditions for the calculation, the amplitude of the incident waves A ¼ 0.1 m; draught of the pontoon structure is 7.5 m; the dimension of the pontoon structure 2b ¼ 10 m while the diameter of the pile-column is 0.05 m and the elastic modulus of the material is E ¼ 196 Gpa. Compared to Lee’s solution, the dimensionless amplitude of the waves generated by the pontoon structure under the same conditions as mentioned above is illustrated in Fig. 7, where good agreement to each other is observed in the solution. 4.2. The influence of the pontoon draught on the response of structure and flow field Presented in Fig. 8(a) and (b) are the reflection and transmission coefficients for the floating pier structural system with various pontoon draught, respectively. The variation of the pontoon-draught was taken into accounts as 5.5, 6.5 and 7.5 m. It is observed that when the water depth is constant, corresponding to the increase of the pontoon-draught the reflection coefficient is larger associated with a lower resonant frequency. For the dimensionless amplitude response of the pontoon structure, Fig. 9(a–c) illustrate the dimensionless amplitude of surge, heave and pitch of the pontoon-pier with respect to dimensionless wave frequency. It is found that corresponding to the increase of the pontoon-draught, the response of the pontoon structure becomes more drastic. The motion of the pitch is basically in accordance with the motion of surge but with smaller amplitudes. It is more obvious when observing from Fig. 10, where the peak amplitudes from
Fig. 7. Comparison of the dimensionless amplitude of the waves generated by the pontoon structure to Lee’s solution.
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Fig. 8. The reflection and transmission coefficients for the floating pier structural system with various pontoon draughts: (a) the refection coefficient corresponding to the wave frequency, and (b) the transmission coefficient corresponding to the wave frequency.
the frequency response were presented against dimensionless draught of the pontoon structure. It shows that the increasing trend of the response of all motions including surge, heave and pitch is with respect to the increase of the draught. 4.3. The influence of the pontoon dimension on the response of structure and flow field Fig. 11(a) and (b) present the reflection and transmission coefficient for the floating pier structural system corresponding to dimensionless frequency while the dimension of the pontoon structure is varied as 8, 12 and 16 m. It is noticed from these figures that the variation of the dimension of the pontoon structure influences generally the responding frequency. When the dimension of the pontoon structure is larger the reflection coefficient fluctuates in a smaller range of frequency. For the dimensionless amplitude response of the pontoon structure as illustrated in Fig. 12(a–c) corresponding to surge, heave and pitch of the pontoon structure. It is found that
Fig. 9. The dimensionless response of the pontoon-pier with various draughts in the pontoon-pier: (a) the dimensionless amplitude of the surge motion corresponding to the wave frequency. (b) The dimensionless amplitude of the heave motion corresponding to the wave frequency. (c) The dimensionless amplitude of the pitch motion corresponding to the wave frequency.
corresponding to the increase of the dimension, the response of the pontoon structure in the surge and pitch motion becomes more significant. However, for the heave
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H.H. Lee et al. / Ocean Engineering 34 (2007) 1044–1059
Fig. 10. Response peaks of the pontoon-pier corresponding to the variation of the draught.
Fig. 11. The reflection and transmission coefficients for the floating pier structural system with various pontoon dimensions: (a) the refection coefficient corresponding to the wave frequency, and (b) the transmission coefficient corresponding to the wave frequency.
motion as shown in Fig. 12(b), the peak-value of the amplitude becomes smaller with respect to the increase of the dimension of structure. The peak amplitudes from the
Fig. 12. The dimensionless response of the pontoon-pier with various dimensions of pontoon: (a) the dimensionless amplitude of the surge motion corresponding to the wave frequency, (b) the dimensionless amplitude of the heave motion corresponding to the wave frequency, and (c) the dimensionless amplitude of the pitch motion corresponding to the wave frequency.
frequency response were drawn against the dimensionless width of the pontoon structure as shown in Fig. 13. It shows that the most significant response is the heave
ARTICLE IN PRESS H.H. Lee et al. / Ocean Engineering 34 (2007) 1044–1059
1057
Fig. 13. Response peaks of the pontoon-pier corresponding to the variation of the pontoon dimension.
motion, to which the increase of the pontoon dimension has inverse influence than the surge and pitch motions. 4.4. The influence of the flexural rigidity of the pile-column on the structural response The flexural rigidity of the pile-column is taken into consideration here as a various parameter. The dimensionless flexural rigidity of the column-pile is defined as EI=rgh5 and varied as 0.026, 0.046 and 0.070. The responses of the pontoon structure as presented in Fig. 14(a) and (b) are the dimensionless amplitude of surge and heave motion of the pontoon structure corresponding to the dimensionless frequency. In the surge motion the response amplitude seems to be suppressed by the increasing flexural rigidity of the pile-column. However, as expected, the heave motion as was shown in Fig. 14(b) is not influenced by the variation of flexural rigidity. All three curves of different flexural rigidity are matched into one. The behavior of the pile-column in the time domain is shown in Fig. 15 as the time-history of transverse response for the pile-column corresponding to various flexural rigidities of the pile and shown in Fig. 16 as the response comparison between the pile-column and the pontoon structure. Observed from these time-history responses, it is found that two frequencies are dominant for the dynamic behavior of pile-column and obviously, one dominant frequency is from the wave as shown in Fig. 16, where during the lower frequency motion the pile-column is in accordance with the surge motion of the pontoon. The other higher frequency motion is due to the property of the pile-column itself, which can be seen from Fig. 15, where as the flexural rigidity of the pile increases the amplitudes of the response becomes smaller. The variation of two dominant frequencies for the pile-column responses can be observed clearly from Fig. 17 the response spectrum of the pile-column, where as the flexural rigidity increases the
Fig. 14. The dimensionless response of the pontoon-pier due to the variation of the pile-column property: (a) the dimensionless amplitude of the surge motion corresponding to the wave frequency, and (b) the dimensionless amplitude of the heave motion corresponding to the wave frequency.
Fig. 15. The time-history of transverse response for the pile-column corresponding to various flexural rigidity of the pile.
higher portion of frequency also increases associated with the reduction of responses while the lower portion of frequency remains invariant.
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H.H. Lee et al. / Ocean Engineering 34 (2007) 1044–1059
Fig. 16. The response comparison between the pile-column and the pontoon structure.
frequencies but for the variation of the pontoon dimension, the influence on the wave motion is mainly on the frequencies of the wave. The influence on the wave field due to variation of the flexural rigidity of the pile-column is mainly the amplitudes of the waves, which is actually directly affected from the motion of the pontoon-piers while the surge motion of pier is confined by the pilecolumns. It is observed from the response of the floating pier system that the most significant response is the heave motion since the confinement from the pile-columns is only for the surge motion. Also because of the pile-column confinement even though the surge and pitch motion of the pontoon-pier is increased when the dimension and draught of the pontoon-pier increases, the increments are still confined into a small range. For the increase of the dimension of the pontoon-pier, the amplitudes of all motions for the pontoon-pier are increased correspondingly. However, when parameter of the draught of the pontoon-pier is under consideration, the increase of the pontoon dimension has inverse influence on the heave motions than the surge and pitch motions. For the time-history responses of the pile-columns, two frequencies are dominant for the dynamic behavior. One dominant frequency, the lower one is from the wave that forces the motion of the pontoon and then exerts on the pile-column. The other higher frequency motion is due to the property of the pile-column. As the flexural rigidity increases the higher portion of frequency also increases associated with the reduction of responses while the lower portion of frequency remains invariant.
Acknowledgments Fig. 17. The response spectrum of the pile-column with various flexural rigidity of the pile.
This study has been partially supported by the National Science Council, R.O.C. under grant No. NSC 93-2611-E110-003. This support is gratefully acknowledged.
5. Conclusions In this study, a two-dimensional floating pier consists of single rectangular impermeable pontoon with side supporting pile-columns was studied and a set of equations for the response of the pontoon and the pile-column were derived and solved analytically and numerically respectively while interactions among the wave, pontoon structure and pilecolumn were taken into accounts. Based on the numerical examples for the floating pier structural systems subjected to waves, it was found that with side supporting pilecolumns, the motion of both the pontoon-piers and the wave field is related to the dimension and draught of the pontoon-pier and the mechanic properties of the pilecolumn as well. The wave motions due to the interactions between the wave and pontoon-pier are also obtained. For the variation of the draught of the pontoon, the influence on the wave motion as observed from the coefficient of reflection and transmission is on both the amplitudes and
Appendix A Z 0 Km Em ¼ cos K n ðz þ hÞ cos K m ðz þ hÞdZ cos K m h h Km sin 2K m h h ¼ þ , 4K m 2 cos K m h Z
d
cos K m ðz þ hÞdZ ¼
Cm ¼ h
Z
sin K m ðh dÞ Km
d
Omn ¼
cos K 2n cos K m ðz þ hÞdZ h
1 sinðK 2n þ K m Þðh dÞ sinðK 2n K m Þðh dÞ ¼ þ , 2 K 2n þ K m K 2n K m
ARTICLE IN PRESS H.H. Lee et al. / Ocean Engineering 34 (2007) 1044–1059
Z d 1 cos K n cos K 2m dZ cos K n h h 1 sinðK n þ K 2m Þðh dÞ ¼ cos K n h K n þ K 2m sinðK n K 2m Þðh dÞ þ , K n K 2m
G mn ¼
m ¼ 0,
d0m ¼ 1; Z
d
Pmn ¼
cosh ln ðz þ hÞ cos K m ðz þ hÞ dZ h
¼
l2n
1 ½K m cosh ln ðh dÞ sin K m ðh dÞ þ K 2m
þ ln sinh ln ðh dÞ cos K m ðh dÞ, Z
0
ðz z0 Þ cos K n ðz þ hÞ dz
Ln ¼ d
cos K n h cos K n ðh dÞ K 2n d þ ðsin K n ðh dÞ þ 3 sin K n hÞ for z0 ¼ 43d 4K n Z b Mn ¼ x Aw2n eK 2n ðxbÞ þ Bw2n eK 2n ðxþbÞ dx " b # b 1 2K 2n b 2K 2n b ¼ ð1 þ e Þ þ 2 ðe 1Þ ðAw2n Bw2n Þ, K 2n K 2n ¼
Z
b
x sin ln ðx bÞ dx ¼
Nn ¼ b
Z
b ð1 þ cos 2ln bÞ, ln
0
ðz z0 Þ cos K m ðz þ hÞ dx
Qm ¼ d
¼
d ½sin K m ðh dÞ þ 3 sin K m h 4K m 1 þ 2 ½cos K m h cos K m ðh dÞ. Km
1059
References Black, J.L., Mei, C.C., 1969. Scattering of surface waves by rectangular obstacles in water of finite depth. Journal of Fluid Mechanics 38 (Part. 3), 499–511. Black, J.L., Mei, C.C., Bray, M.C.G., 1971. Radiation and scattering of water waves by rigid bodies. Journal of Fluid Mechanics 46, 151–164. Garrison, C.J., 1974. Dynamic response of floating bodies. OTC 2067, 365–378. Lee, C.P., Lee, J.F., 1993. Wave-induced surge motion of a tension leg structure. Ocean Engineering 20 (2), 171–186. Lee, H.H., Wang, P.-W., 2000a. Analytical solution on the surge motion of tension leg twin-platform structural system. Ocean Engineering 27, 393–415. Lee, H.H., Wang, P.-W., 2000b. Dynamic behavior of tension-leg platform with net-cage system subjected to wave forces. Ocean Engineering 28, 179–200. Lee, H.H., Wang, P.-W., Lee, C.P., 1999. Dragged surge motion of tension leg platforms and strained elastic tethers. Ocean Engineering 26, 575–594. Lee, H.H., Wang, W.-S., 2001. Analytical solution on the dragged surge vibration of TLPs with wave large body and small body multiinteractions. Journal of Sound and Vibration 248 (3), 533–556. Lee, H.H., Wang, W.-S., 2003. On the dragged surge vibration of twin TLP systems with multi-interactions of wave and structures. Journal of Sound and Vibration 263, 743–774. Lee, J.F., 1995. On the heave radiation of rectangular structure. Ocean Engineering 22 (1), 19–34. Mei, C.C., 1978. Numerical methods in water wave diffraction and radiation. Annual Reviews in Fluid Mechanics 10, 393. Mei, C.C., 1983. The Applied Dynamics of Ocean Surface Waves. Wiley, New York. Sarpakaya, T., Isaacson, M., 1981. Mechanics of Wave Forces on Offshore Structures. Van Nostrand Reihold, New York. Yamamoto, T., Yoshida, A., Ijima, T., 1982. Dynamics of elastically moored floating objects. In: Kirk, C.L. (Ed.), Dynamics Analysis of Offshore Structures. CML, Southampton.
Ocean Engineering 34 (2007) 1044–1059 www.elsevier.com/locate/oceaneng
The prediction of the dynamic and structural motions of a floating-pier system in waves Hsien Hua Leea,, Liang-Yin Chena, Wen-Kai Wengb, S.-W. Shyuea a
Department of Marine Environment and Engineering, National Sun Yat-sen University (NSYSU), Kaohsiung, Taiwan b Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan Received 14 August 2005; accepted 30 January 2006 Available online 27 September 2006
Abstract In this study, a two-dimensional floating pier consists of single rectangular impermeable pontoon with side supporting pile-columns is studied. The purpose of this study is to present a theoretical solution for the linearized problem of incident waves exerting on a floating pier with pile-restrained. All boundary conditions are linearized in the problem, which is incorporated into a scattering problem and radiation problem with unit displacement. The method of separation of variables is used to solve for velocity potentials. For the radiation problem with unit heave and pitch amplitude, the boundary value problem with non-homogeneous boundary condition beneath the structure is solved by using a solution scheme. By calculating the wave force from velocity potential and solving the equation of motion of the floating structure simultaneously a close form theoretical solution for the problem is developed. The finite element method was also applied to calculate the dynamic responses on the supporting piles subjected to the pontoon motions and incident waves. r 2006 Elsevier Ltd. All rights reserved. Keywords: Floating-pier; Wave structure interaction; Wave force; Pier with guide-pile; Dynamic response
1. Introduction This paper provides an analytical study for the responses of a floating pier structural system confined horizontally with pile-columns. Floating pier structural system by using semi-submerged buoyant pontoons providing vertical supports for the deck is a widely used structural system because it is easy to construct and convenient to use for offshore deep water application. In the floating pier structural system the pontoon-pier is either anchored by the tension legs to the seabed or supported by pile-columns on the sides for the so-called bridge type floating pier system. The pile-columns not only work as an anchoring system but also provide a confinement to the pontoon-pier from moving around. A complete study for the dynamic behavior of a floating pier structural system is rather complicated, particularly in analytical methods because the analysis must account for not only the interactions between the structure (pontoon-pier) and the waves, but also the Corresponding author.
E-mail address: [email protected] (H.H. Lee). 0029-8018/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2006.01.015
interactions between waves, pontoon-pier and the anchoring systems. The interactions between the floating structure and the incident waves are the well-known scattering problem incorporated with radiation problem. In 1969, Black and Mei first studied the reflection and transmission of waves encountering the floating structure and identified it as a scattering problem. Later on Black et al. (1971) then studied a moving floating structure making waves in the flow field that was identified as a radiation problem. However, to solve these two problems simultaneously the nonlinearity of structural motion and the non-homogeneous boundary conditions on the free surface and on the structural bottom (Garrison, 1974) must be overcome first. This problem has been solved in numerical methods since the 1970s (Mei, 1978; Yamamoto et al., 1982), and also studied through experimental testing (Yamamoto et al., 1982) until Sarpakaya and Isaacson (1981) linearized the boundary conditions. By using the scheme to linearize the boundary conditions, the close form theoretical solution could be obtained by incorporating the scattering problem with the radiation problem.
ARTICLE IN PRESS H.H. Lee et al. / Ocean Engineering 34 (2007) 1044–1059
To examine the dynamic motion of floating pier structural system, besides the interactions between the structure (pontoon-pier) and waves the behavior of anchoring system, either a tension-leg or supporting pile-column for the pier is also essential. The other problem is that most analytical studies on the interactions between floating structure and waves incorporating with both the scattering problem and radiation problem are focused either on the surge radiation (Lee and Lee, 1993) or heave radiation (Lee, 1995), where the anchoring systems are either ignored or simplified. In the later studies for an offshore floating structure (Lee et al., 1999; Lee and Wang, 2000a, b) the properties of the anchoring tension-legs were taken into accounts along with the wave-structure interactions, where the large body effects from the pontoon-pier was also under the consideration. The results showed that for the same wave properties, the dynamic behavior of both the tether of the tension leg and the platform itself are closely related to the material property and the tether dimension. Although these studies accounted for the interaction between the wave and the pontoon-pier structure, the drag effect of tethers on the platform structures was ignored. By ignoring the effect of drag motion of the tethers on the platform, the response of the pontoon-pier could not be estimated accurately. However, after overcoming the complexities of multi-interactions in the previous analysis, the interactions between the pontoon-pier (large body), waves, and tethers (small body) have been taken into accounts in the analytical solutions (Lee and Wang, 2001). It was shown that the inclusion of the tether drag effect drastically reduced the platform amplitude under the same wave conditions. In this study the interactions between waves and structures are also taken into accounts and moreover, besides the radiation problem induced from surge, the radiation induced from heave and pitch motions of the pontoon-pier is also under consideration. For the sidesupporting floating pier system, the motion of the pontoonpier, basically the surge motion is confined by the pilecolumns on two sides of the floating structure. Therefore, the dynamic behavior of the pile-column must be taken into calculation simultaneously along with the response of the pontoon-pier, which in the same time interacts with waves and influence the wave properties that provide exerting pressure on the pile-columns. Based on previous studies (Lee and Wang, 2001, 2003), the equations for the scattering problem and the radiation problem were established at the related boundaries with variables of velocity potential obtained from the Laplace equation. The forcing function related to the velocity potentials was used in the equation of motion for the pontoon-pier and the pile-columns. The analytical solution for the unknown coefficients and the responses of both the pontoon-pier and the pile-columns were found by solving all of the equations derived on the boundaries simultaneously. The purpose of this study is to investigate the response of a floating pier structural system subjected to incident waves
1045
and flow drags on the pile-columns, including the multiinteraction among waves, pontoon-pier structure, and pilecolumns. Numerical examples are carried out, and the results are discussed focusing on both the reflection and transmission coefficients and on the motion of the pontoon-pier subjected to waves of various periods. The influence of parameters such as the material property of the pile-columns, dimension of the pontoon and the submerged depth on the pontoon was also studied. It is found that the response of the floating pier structure is significantly dependant on the properties of the pilecolumns, the wave conditions and the interactions between the pontoons and the waves. 2. General theory of wave and flow field 2.1. Basic provisions on the problems Illustrated in Fig. 1 is a two-dimensional schematic view for a typical floating pier structural system, where a set of floatable pontoon are mounted under the deck of the pier while two ends of the deck are connected to a pair of pilecolumns fixed to the seabed. The pier-decks are allowed to vertical and rotational motions because the connection joints between the pile-column and the deck are vertically movable hinges. However, even though due to the connection joints the relative horizontal motions between the deck and pile-column are not allowable, the surge motions for the whole floating pier structural system, which includes the pile-columns as well, are allowable and studied. It is assumed that the water depth, where the floating pier structure is located is h, draught of the pontoon is d and the width of the pontoon is 2b. In the study, both the large body and small body theorems for the applications of wave force are utilized. For the pile-columns the small body theorem is assumed because the dimension of the pile ds is usually small compared to the wave length L (d s =Lo0:2) while the large body theorem must be applied for the pontoons since the dimension for the pontoon of the floating pier could be larger than 1/5 of the wavelength. Therefore, the interactions between waves and
Fig. 1. Sketch of a typical floating pier structural system subjected to waves.
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pontoons of the floating pier are taken into accounts while the back-influence to the response of pile-columns is also considered as well. In the analysis the inviscid and incompressible fluid and irrotational flow field are assumed and a single-valued velocity potential f can be defined as ~ u ¼ rf, where ~ u is the velocity and r is the gradient operator. The velocity potential satisfies the Laplace equation: r2 f ¼ 0 and the Bernoulli equation in the flow field as qf 1 p þ rfrf þ þ gz ¼ 0, qt 2 rw
(1)
where p is the pressure, rw the water density, g the gravitational constant, and z the variable of depth. The nonlinear term in Eq. (1) is ignored when the linear small amplitude wave is assumed. A two-dimensional floating pier structural system constrained on a pair of pile-columns interacting with a monochromatic small amplitude wave propagating in the +x-direction is considered here as shown in Fig. 1. The waveform and the associated velocity potential are given accordingly as ZI ðx; tÞ ¼ RefAeðK 0 xþstÞ g
(2)
and fI ðx; z; tÞ ¼ iA
g cos K 0 ðz þ hÞ iðK 0 xþstÞ e , s cos K 0 h
(3)
where A is the wave amplitude, g is the gravitational constant and h is the water depth. s ¼ 2p=T is the angular frequency with period T, K 0 ¼ ik and k ¼ 2p=L is the wave number with wavelength L. Coefficient K0 satisfies the dispersion relation s2 ¼ gK 0 tanðK 0 hÞ. For the floating pontoon, three degrees of freedom including surge, heave and pitch motions are assumed to be harmonically corresponding to the motion of waves when the exerting waves are periodic with small amplitude. Therefore, the displacement functions corresponding to surge x, heave Z and pitch y are given by x ¼ S 1 eist ,
(4)
2.2. Boundary value problems As shown in Fig. 1, the floating pier structural system is divided into three regions with two artificial boundaries at x ¼ b and x ¼ þb. Since the whole flow field is composed of incident waves, scattered waves and radiated waves, the total velocity potential for each region of the flow field may be represented as follows: f ¼ fI þ fs þ S 1 fw þ S 2 fh þ S3 fp
where the superscripts I, s, w, h and p indicate velocity potential corresponding to incident waves, scattered waves and radiated waves from the surge, heave and pitch motion of the pontoon structures respectively. In region I: the total velocity potential consists of velocity potentials of incident waves, scattered waves and radiated waves of surge, heave and pitch motions as f1 ¼ fI þ fs1 þ S1 fw1 þ S2 fh1 þ S 3 fp1 . In regions II and III: only scattered waves and radiated waves are considered as f2 ¼ fs2 þ S 1 fw2 þ S 2 fh2 þ S3 fp2 and f3 ¼ fs3 þ S 1 fw3 þ S2 fh3 þ S 3 fp3 . All of the velocity potentials satisfy the Laplace equation. Furthermore, on the infinite boundary 1 in region I and þ1 in region III, the Sommerfeld’s radiation condition is satisfied for unique solutions. Now the solutions for the interaction between the pontoon and waves include the scattering problem and radiation problem. For the radiation problem, all motions from the pontoon-pier structure must be taken into account thus including surge, heave and pitch. 2.2.1. Scattering problems In the scattering problem the incident wave is diffracted by the pontoon-pier structure, which is presumably fixed. The corresponding boundary conditions for each region of the flow field in the scattering problem are shown in Fig. 2. By applying the method of separation of variables, matching the horizontal boundary conditions for the free water surface and sea bed in each regions, and applying the Sommerfeld’s condition to regions I and III, the velocity potentials in each region are found as fs1 ¼
1 X
As1n
g cos½K n ðz þ hÞ K n ðxbÞ ist e e , s cosðK n hÞ
(8)
As3n
g cos½K n ðz þ hÞ K j ðxþbÞ ist e e , s cosðK n hÞ
(9)
n¼0
Z ¼ S 2 eist ,
(5) fs3 ¼
y ¼ S3 e
ist
,
(6)
where S i represents the amplitude corresponding to surge, heave and pitch motions, respectively. For the interaction problems between the pontoons and incident waves, from the linearity of the problem, the problem can be incorporated into a scattering and a radiation problem (Black et al., 1971). The wave force calculated from the scattering problem provides the force function in the radiation problem, and the forced oscillation of the structure then generates outgoing waves.
(7)
1 X n¼0
where the eigenvalue K n is solved from the dispersion equation s2 ¼ gK n tan K n h. g s A20 x þ Bs20 eist fs2 ¼ s 1 X g s K 2n ðxbÞ A2n e þ þ Bs2n eK 2n ðxþbÞ s n¼1 cos K 2n ðz þ hÞeist , where K 2n ¼ np=h d.
ð10Þ
ARTICLE IN PRESS H.H. Lee et al. / Ocean Engineering 34 (2007) 1044–1059
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Fig. 2. Illustration of the scattering boundary-value problem for the floating pier structural system.
The series of 4N unknowns As1n , As2n , As3n and Bs2n in the velocity potential of each region can be further solved through the boundaries between two regions as indicated as follow: Kinematic boundary condition (KBC) on x ¼ b between regions I and II: 8 0 dpzp0; s qðf1 þ fI Þ < s ¼ qf2 (11) ; hpzp d: : qx qx Dynamic boundary condition (DBC) on x ¼ b between regions I and II: qðfs1 þ fI Þ qfs2 ¼ ; qt qt
hpzp d .
(12)
(13)
DBC on x ¼ b between regions II and III: hpzp d .
(14)
Now by substituting velocity potentials in each region as shown in Eqs. (8)–(10) into kinematic and dynamic conditions as shown in Eqs. (11)–(14), and employing the orthogonality of the velocity potential functions, the 4N series of unknowns are ready to be solved in the following equations. E m As1m þ C m As20 ¼ id0m E m Aexp
1 X
n¼1 ikb
,
K 2n Omn As2n Bs2n e2K 2n
G mn As1n þ ½bðh dÞAs20 þ ðh dÞBs20
n¼0
þ
ðh dÞ s A2m þ Bs2m e2K 2m b ¼ iAG m0 eikb , 2
C m As20
1 X
ð16Þ
K 2n Omn As2n e2K 2n b Bs2n þ E m As3m ¼ 0,
n¼1
(17) ðh dÞ s 2K b A2m e 2m þ Bs2m ðh dÞbAs20 ðh dÞBs20 2 1 X þ G mn As3n ¼ 0; m ¼ 021, ð18Þ n¼0
where the parametersE m ; C m ; Omn ; G mn are defined in Appendix A.
KBC on x ¼ b between regions II and III: 8 0; dpzp0; qfs3 < s ¼ qf2 ; hpzp d: : qx qx qfs3 qfs2 ¼ ; qt qt
1 X
2.2.2. Radiation problems due to unit surge motion For the radiation problems due to unit surge motion of the pontoon structure, the corresponding boundary conditions in each region are shown in Fig. 3 such as the free water surface condition, impermeable conditions on seabed and the bottom surface of the pontoon structures. Again by applying the method of separation of variables, matching the horizontal boundary conditions in each regions, and applying the Sommerfeld’s condition to regions I and III, the velocity potentials in each region are found as fw1 ¼
1 X
Aw1n
g cos½K n ðz þ hÞ K n ðxbÞ ist e e , s cosðK n hÞ
(19)
Aw3n
g cos½K n ðz þ hÞ K j ðxþbÞ ist e e , s cosðK n hÞ
(20)
n¼0
b ð15Þ
fw3 ¼
1 X n¼0
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1048
Fig. 3. Illustration of the radiation boundary-value problem for the floating pier structural system with unit surge motion.
fw2 ¼
1 X g w g w K 2n ðxbÞ ðA20 x þ Bw20 Þeist þ A2n e s s n¼1 þBw2n eK 2n ðxþbÞ cos K 2n ðz þ hÞ ist.
Aw1n ,
Aw2n ,
Aw3n
E m Aw1m þ C m Aw20 þ ð21Þ
qt
¼
qt
;
hpzp d,
¼ i 1 X
s S1 F m , g
hd ð2b 1ÞAw2m þ Bw2m 2 n¼0 ðh dÞ w A2m þ Bw2m e2K 2m b ¼ 0, 2 Gmn Aw1n d0m
C m Aw20
1 X
ð27Þ
K 2n Omn Aw2n e2K 2n b Bw2n þ E m Aw3m
n¼1
s2 ¼ i S1 F m , g (23)
ð26Þ
hd ð2b þ 1ÞAw2m Bw2m 2 ðh dÞ w 2K 2m b A2m e þ Bw2m 2 1 X þ G mn Aw3n ¼ 0; m ¼ 021,
ð28Þ
d0m
KBC on x ¼ b between regions II and III: 8 ist < isS1 e ; dpzp0; w qf3 w ¼ qf2 : ; hpzp d; qx qx
(24)
hpzp d.
ð29Þ
n¼0
where parameters E m ; C m ; Omn ; G mn ; F m are also presented in Appendix A.
DBC on x ¼ b between regions II and III: qfw3 qfw ¼ 2; qt qt
2
Bw2n
DBC on x ¼ b between regions I and II: qfw2
K 2n Omn Aw2n Bw2n e2K 2n b
n¼1
and in Again the series of 4N unknowns the velocity potential of each region can be further solved through the boundary conditions between two regions as indicated as follows: KBC on x ¼ b between regions I and II: 8 < isS1 ; dpzp0; qfw1 w (22) ¼ @f2 ; hpzp d; : qx @x
qfw1
1 X
(25)
Now after the substitution of velocity potentials in each region into kinematic and dynamic conditions and after the rearrangement, Eqs. (22)–(25) become the form of Eqs. (26)–(31). By employing the orthogonality of the velocity potential functions, the 4N series of unknowns are ready to be solved from these equations.
2.2.3. Radiation problems due to unit heave motion For the radiation problems due to unit heave motion of the pontoon structure, the corresponding boundary conditions in each region are shown in Fig. 4. Again by applying the method of separation of variables, matching the horizontal boundary conditions in each region, and applying the Sommerfeld’s condition to region I and III,
ARTICLE IN PRESS H.H. Lee et al. / Ocean Engineering 34 (2007) 1044–1059
1049
Fig. 4. Illustration of the radiation boundary-value problem for the floating pier structural system with unit heave.
the velocity potentials in each region are found as fh1 ¼
1 X
Ah1n
g cos½K n ðz þ hÞ K n ðxbÞ ist e e , s cosðK n hÞ
Ah3n
g cos½K n ðz þ hÞ K j ðxþbÞ ist e e , s cosðK n hÞ
n¼0
fh3
¼
1 X
where (30)
(34)
and
n¼0
(31)
where the eigenvalue Kn is solved from the dispersion equation s2 ¼ gK n tan K n h. It is noticed that since the boundary is vertically movable for the bottom of the pontoon structure during the heave motion the nonhomogeneous boundary conditions are obtained as qfh2 ¼ isS 2 eist ; qZ
z ¼ d.
(32)
The method of separation of variables cannot be directly applied for the solution in this case and therefore, the velocity potential in region II due to unit heave motion of the pontoon is obtained through a scheme (Lee, 1995) that incorporates the non-homogeneous boundary conditions into two homogeneous boundary conditions, namely the vertical and horizontal one. Then the boundary value problem becomes two homogeneous boundary value problems. The velocity potential is subsequently obtained as the sum of the solution from the vertical boundary value problem and horizontal boundary problem in a form as g fh2 ¼ ðAh2n x þ Bh2n Þeist s 1 X g h K 2n ðxbÞ þ þ Bh2n eK 2n ðxþbÞ Þ cos K 2n ðz þ hÞeist ðA2n e s n¼1 þ
2 ¯ h ¼ is S2 ½cosð2ln bÞ 1 A 2n gbl2n sinh½ln ðh dÞ
1 X
h g A2n sin ln ðx bÞ cosh ln ðz þ hÞeist ; s n¼1
ð33Þ
K 2n ¼
np ¼ ln . 2b
(35)
The boundary conditions between two regions including both the kinematic and dynamic type are presented as follows. KBC on x ¼ b between regions I and II: 8 < 0; ¼ qfh2 : qx ; qx
qfh1
dpzp0; hpzp d;
(36)
DBC on x ¼ b between regions I and II: qfh1 qfh2 ¼ ; qt qt
hpzp d,
(37)
KBC on x ¼ b between regions II and III: 8 0; qfh3 < h ¼ qf2 : qx ; qx
dpzp0; hpzp d;
(38)
DBC on x ¼ b between regions II and III: qfh3 qfh2 ¼ ; qt qt
hpzp d.
(39)
Now after the substitution of velocity potentials in each region back into kinematic and dynamic conditions and after the rearrangement, Eqs. (36)–(39) become the form of Eqs. (40)–(43).
ARTICLE IN PRESS H.H. Lee et al. / Ocean Engineering 34 (2007) 1044–1059
1050
E m Ah1m þ C m Ah20 þ
1 X
K 2n Omn ðAh2n Bh2n e2K 2n b Þ
n¼1
¼
1 X
h A2n ln Pmn ,
ð40Þ
n¼1 1 X
hd ð2b 1ÞAh2m þ Bh2m d0m 2 n¼0 ðh dÞ h A2m þ Bh2m e2K 2m b ¼ 0, 2
motion from the pontoon structure. Similarly, by applying the method of separation of variables, matching the horizontal boundary conditions in each regions, and applying the Sommerfeld’s condition to regions I and III, the velocity potentials in each region are found as fp1 ¼
G mn Ah1n
1 X
Ap1n
g cos½K n ðz þ hÞ K n ðxbÞ ist e e , s cosðK n hÞ
(44)
Ah3n
g cos½K n ðz þ hÞ K j ðxþbÞ ist e e . s cosðK n hÞ
(45)
n¼0
ð41Þ
fp3 ¼
1 X n¼0
C m Ah20
1 X
K 2n Omn ðAh2n e2K 2n b Bh2n Þ þ E m Ah3n
n¼1
¼
1 X
h
A2n ln Pmn cos 2ln b,
ð42Þ
n¼1
hd ðh dÞ h 2K 2m b ½ð2b þ 1ÞAh2m Bh2m ðA2m e þ Bh2m Þ 2 2 1 X G mn Aw3n ¼ 0; m ¼ 021, ð43Þ þ
d0m
n¼0
where parameters E m ; C m ; Omn ; G mn ; Pmn are also presented in Appendix A. By solving theses equations simultaneously and employing the orthogonality of the velocity potential functions, the 4N series of unknowns are ready to be solved. 2.2.4. Radiation problems due to unit pitch motion When the motion of the pontoon structure is small the boundaries are assumed on the position of equilibrium and then the pitching kinematics of the pontoon structure will be similar to a positively moving gradient boundary (Lee, 1995). Illustrated in Fig. 5 is the boundary condition for the radiation problems subjecting to an unit pitching
Similar to the radiation problem of the heave motion, where the boundary is vertically movable for the bottom of the pontoon structure during the pitch motion, the nonhomogeneous boundary conditions are obtained as qfp2 ¼ isS 3 eist x; qZ
z ¼ d.
(46)
The method of separation of variables is not applicable for the solution directly and the velocity potential in region II due to unit pitch motion of the pontoon is obtained through a method as presented in heave-radiation problem, in a form as g fp2 ¼ ðAp2n x þ Bp2n Þeist s þ
þ
1 X g
s n¼1
Ap2n eK 2n ðxbÞ þ Bp2n eK 2n ðxþbÞ cos K 2n ðz þ hÞeist
1 X
p g A2n sin ln ðx bÞ cosh ln ðz þ hÞeist ; s n¼1
ð47Þ
where 2
¯ p2n ¼ is S3 ½cosð2ln bÞ þ 1 A gl2n sinh½ln ðh dÞ
Fig. 5. Illustration of the radiation boundary-value problem for the floating pier structural system with unit pitch.
(48)
ARTICLE IN PRESS H.H. Lee et al. / Ocean Engineering 34 (2007) 1044–1059
and K 2n ¼
3. Equations of motion of the floating-pier structure with side supports
np ¼ ln . hd
(49) 3.1. Equations of motion of the pontoon structure
The boundary conditions of both kinematic and dynamic type are presented as follows. KBC on x ¼ b between regions I and II: 8 isS 3 ðz z0 Þ; dpzp0; qfp1 < p (50) ¼ qf2 : ; hpzp d; qx qx DBC on x ¼ b between regions I and II: qfp1 qfp2 ¼ ; qt qt
hpzp d,
(51)
KBC on x ¼ b between regions II and III: 8 isS 3 ðz z0 Þ; dpzp0; qfp3 < p ¼ qf2 : ; hpzp d; qx qx
hpzp d.
(52)
(53)
Now after the substitution of velocity potentials in each region back into kinematic and dynamic conditions and after the rearrangement, Eqs. (50)–(53) become the form of Eqs. (54)–(57). E m Ap1m C m Ap20 ¼
1 X
K 2n Omn Ap2n Bp2n e2K 2n b
n¼1 2
p
A2n ln Pmn þ i
n¼1 1 X
1 X
s S 3 Qm , g
ð54Þ
where the mass matrix 2 m 6 0 ½M ¼ 4
0 m
3 mðzc z0 Þ mðxc x0 Þ 7 5,
(59)
I yy
and the stiffness matrix 2 3 0 0 0 6 7 rgI A ½K ¼ 4 0 rgð2bÞ 5. x A V 0 rgI A rgðI þ I Þ mgðz z Þ c 0 x xx z
(60)
It is noticed in Eq. (60) that for the surge motion of the pontoon structure there is no stiffness or constraints from the system in this formulation. In the mass matrix ½M, m is the mass of the pontoon structure that can be obtained from the displaced water as rð2bÞd; I yy is the rotational inertia in the y-direction with respect to the mass center; I A xx and I V z are increments of the moment of the buoyancy due to the rotational angle S 3 , which may be presented as Z b A ðx x0 Þ2 dx, (61) I xx ¼ b
hd ð2b 1ÞAp2m þ Bp2m d0m 2 n¼0 ðh dÞ p A2m þ Bp2m e2K 2m b ¼ 0, 2 G mn Ap1n
C m Ap20
When a floating structure, which is free of any constraint subjected to incident waves, the equations of motion for the surge, heave and pitch may be presented as (Mei, 1983) 8 9 8 9 8 9 x> x> > > Z = < = = < < n1 > 2 > d (58) ½M 2 Z þ ½K Z ¼ r f;t n2 dS > > dt > S ; : > ; ; : > :n > y y 3
mðzc z0 Þ mðxc x0 Þ
DBC on x ¼ b between regions II and III: qfp3 qfp2 ¼ ; qt qt
1051
1 X
IV z ð55Þ
K 2n Omn Ap2n e2K 2n b Bp2n
1 X n¼1
p
A2n ln Pmn cos 2ln b þ i
s2 S3 Qm , g
0
ð56Þ
ðh dÞ p 2K b hd ð2b þ 1ÞAp2m Bp2m A2m e 2m þ Bp2m 2 2 1 X p þ G mn A3n ¼ 0; m ¼ 021, ð57Þ
d0m
n¼0
where parameters E m ; C m ; Omn ; G mn ; Pmn ; Qm are presented in Appendix A. By solving theses equations simultaneously and employing the orthogonality of the velocity potential functions, the 4N series of unknowns are to be solved from these equations.
Z
b
ðz z0 Þ dxdz.
¼ d
n¼1
þ E m Ap3n ¼
Z
(62)
b
R The wave forces r S f;t dS induced from incident waves, scattering waves and radiated waves can be obtained after the total velocity potential f is found from the scattering and radiation problems as indicated in previous sections. The wave forces on the right-hand side include the horizontal, vertical forces and rotational moment. The horizontal forces are obtained by carrying out the time differentiation of velocity potentials in all three regions and further written as Z 0 Z r f;t n1 dS ¼ isr ½fs3 jx¼b ðfI þ fs1 Þjx¼b dz S
Z
d 0
þ S1 d Z 0
þ S3 d
½fw3 jx¼b fw1 jx¼b dz ½fp3 jx¼b fp1 jx¼b dz .
ð63Þ
It is noticed that all of the waves including the incident wave, scattering waves and radiated waves from surge and
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1052
pitch motions are taken into accounts in the calculation. For the vertical forces both the scattering waves and radiated waves due to heave motions are taken into accounts while the resulting vertical force from the radiated waves due to surge and pitch motion is zero. The vertical force is presented as Z b Z Z b s h r f;t n2 dS ¼ isr f2 jz¼d dx þ S 2 f2 jz¼d dx . S
b
b
(64) The rotational moments from the incident waves, scattering waves and radiated waves of surge and pitch motion are calculated Z 0 Z r f;t n3 dS ¼ isr ðz z0 Þ½fs3 jx¼b S
d
Z b ðfI þ fs1 Þjx¼b dz x½fs2 jz¼d dx b Z 0 w þ S1 ðz z0 Þðf3 jx¼b fw1 jx¼b Þ dz Z
d b
xðfw2 jz¼d Þ dx
b
Z
0
þ S3 Z
d
ðz z0 Þðfp3 jx¼b fp1 jx¼b Þ dz
b
xðfw2 jz¼d Þ dx
.
ð65Þ
presented. Eq. (66) may be derived by combining the following equation of motions in each degree of freedom. Z d2 (67) m 2 ½x þ ðzc z0 Þy þ kx þ k dy ¼ r f;t n1 dS, dt S Z d2 A m 2 ½Z ðxc x0 Þy þ rgð2bÞZ rgI x y ¼ r f;t n2 dS, dt S (68) d2 x d 2Z d2y A V I z 2 I x 2 þ I yy 2 þ k dx þ ðrgðI xx þ I z Þ dt dt dt Z 2 rgI A x Z þ kd Þ mgðzc z0 Þy ¼ r
3.2. Equations of motion of the pontoon structure with side supports As observed in Eq. (60), the surge motion of a free constrained pontoon will be only governed by the mass of itself and if the surge from the wave force is large the surge motion of the pontoon will be large too. When the side supports of pile-columns are applied to the floating pier structural system, the behavior of pontoon subjected to waves will be constrained, particularly in the horizontal direction and therefore, the equations of motion for the pontoon are rewritten into a form as 8 9 8 9 8 9 8 9 Tx > x> x> > > > Z = < = = = < < < n1 > 2 > d ½M 2 Z þ ½K Z þ T Z ¼ r f;t n2 dS. > > dt > S ; > : > ; ; ; : > :T > :n > y y y 3 (66) Compared to Eq. (58), it is noticed that an additional vector of third term in the left-hand side representing the constraints from the side supports of the pile-columns is
ð69Þ
By combining Eqs. (67), (68) and (69) and employing the displacement functions, the additional vector of constraints is combined into the stiffness matrix and Eq. (66) may be rewritten into a form in terms of the amplitudes of surge, heave and pitch as 8 9 8 9 8 9 f > > > = = = < S1 > < S1 > < 1> 2 S S þ ½K ¼ isr f 2 . (70) s ½M 2 2 > > > ; ; ; :S > :S > :f > 3 3 3 Assuming that the mass center and rotational center are collocated and then ðzc z0 Þ ¼ 0 and ðxc x0 Þ ¼ 0, the mass matrix [M] and stiffness matrix [K] can be reformulated as 2 3 m 0 0 6 7 ½M ¼ 4 0 m 0 5 (71) 0
b
To solve for the unknowns of the amplitudes of the surge, heave and pitch motions, which are implicitly contained in the velocity potential, Eq. (58) must be solved simultaneously with the velocity potentials in the scattering and radiation problems.
f;t n3 dS. S
I yy
0
and 2 6 ½K ¼ 6 4
4EI ðhþd=4Þ3
0
0
rgð2bÞ
4EI ðhþd=4Þ3
0
4EI ðhþd=4Þ3
3
d
0 V rgðI A xx þ I z Þ þ
4EI ðhþd=4Þ3
d2
7 7. 5 (72)
The wave forces corresponding to surge, heave and pitch as shown in the force vector of Eq. (70) are further written as: Z 0 ½fs3 jx¼b ðfI þ fs1 Þjx¼b dz f1 ¼ d
Z
0
½fw3 jx¼b fw1x¼b dz
þ S1 d Z 0
þ S3 d
Z
½fp3 jx¼b fp1 jx¼b dz,
b
fs2 jz¼d dx þ S 2
f2 ¼
Z
b
Z
0
f3 ¼
b
fh2 jz¼d dx,
b
ðz z0 Þ fs3 jx¼b ðfI þ fs1 Þjx¼b dz
d
Z
b
x fs2 jz¼d dx
b
Z
0
þ S1 d
ð73Þ
ðz z0 Þðfw3 jx¼b fw1 jx¼b Þ dz
(74)
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Z
b
Q33 representing the resultant couple for the unit pitch motion is given as Z 0 Z b p p Q33 ¼ ðz z0 Þ f3 jx¼b f1 jx¼b dz x fp2 jz¼d dx
xðfw2 jz¼d Þ dx
b
Z
o
þ S3 Z
d b
b
ðz z0 Þðfp3 jx¼b fp1 jx¼b Þ dz
xðfp2 jz¼d Þ dx
.
d
After regrouping the coefficients for the vector of amplitude, Eq. (70) then can be rewritten as 8 9 8 s9 f > S1 > > = < 1> 2 < = s S 2 ¼ isr f 2 , (76) s ½M þ ½K þ isr½Q > > ; ; :S > :fs > 3
where 2
Q11
0
6 ½Q ¼ 4 0 Q31
Q22 0
Q13
3
0 7 5. Q33
(77)
Q13 representing the horizontal resultant force term for the unit pitch motion is given as Z 0 p Q13 ¼ f3 jx¼b fp1 jx¼b dz d 1 X ðAp3n Ap1n Þ g sin K n h sin K n ðh dÞ ¼ , ð79Þ Kn cos K n h s n¼0 Q22 representing the vertical resultant force term for the unit heave motion is given as Z b ½fh2 jz¼d dx Q22 ¼
¼
cos K n ðh dÞ , Kn
n¼1
ð80Þ
b
s
ð82Þ
where parameters Ln , M n , N n are defined in Appendix A. The force vector on the right-hand side now is solely related to the scattering waves and presented as Z 0 s f s1 ¼ f3 jx¼b ðfI þ fs1 Þjx¼b dz 1 X g ðAs3n As1n Þ sin K n sin K n ðh dÞ ¼ s cos K n h Kn n¼0
iAeiK 0 b g sinh kh sinh kðh dÞ , k cos K o h s
f s2
Z
ðAw2n Bw2n Þ cos K n ðh dÞM n ,
ð83Þ
b
½fs2 jz¼d dx
¼ b
1 X g g s ðA2n þ Bs2n Þ ¼ ð2bÞBs20 þ s s n¼1
ð1 e2K 2n b Þ
f s3 ¼
Z
cos K 2n ðh dÞ , K 2n
ð84Þ
0
ðz z0 Þ½fs3 jx¼b ðfI þ fs1 Þjx¼b dz
d
Z
b
x½fs2 jz¼d dx
b
1 X g iAeikb g ðAs1n As3n Þ g 2b3 L0 Ln ¼ As20 s cosh kh s cos K n h s 3 n¼0 1 X g n¼1
s
ðAs2n Bs2n Þ cos K 2n ðh dÞM n .
ð85Þ
To solve for the amplitude of the pontoon structure S1, S2 and S3 in Eq. (76), equations in the scattering and radiation problems are also solved simultaneously. The velocity potentials in each region of the flow field will be obvious with known amplitude of the wave motions.
4. Analytical examples of the floating pier structural system subjected to waves
1 X g ðAw3n Aw1n Þ g 2b3 Ln Aw20 ¼ s cos K n h s 3 n¼0 1 X g
ðAp2n Bp2n Þ cos K n ðh dÞM n
p g A2n cosh ln ðh dÞN n , s n¼1
Q31 representing the resultant couple for the unit surge motion is given as Z 0 Z b Q31 ¼ ðz z0 Þ½fw3 jx¼b fw1 jx¼b dz x½fw2 jz¼d dx
s n¼1
1 X
b 1 X g g h ðA2n þ Bh2n Þð1 eK 2n b Þ ¼ ð2bÞBh20 þ s s n¼1
d
1 X g
d
In [Q] matrix, Q11 representing the horizontal resultant force term for the unit surge motion is defined as Z 0 w Q11 ¼ f3 jx¼b fw1 jx¼b dz d 1 X ðAw3n Aw1n Þ g sin K n h sin K n ðh dÞ ¼ , ð78Þ cos K n h s Kn n¼0
b
p p 1 X g ðA3n A1n Þ g 2b3 Ln Ap20 ¼ s cos K n h s 3 n¼0
ð75Þ
3
1053
ð81Þ
Analytical examples for the floating pier structure with vertical pile system are carried out in this section. The investigation was focused on the displacement responses of
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the pile-column system, the influence on the flow field and the response of pontoon-pier structure subjected to waves with various periods. The influence of the flexural rigidity of the piles, dimension of the pontoon structure and the draught of the pontoons on the pontoon-pier motion and the pile-column responses was also studied. Before carrying out the calculation for the analytical examples the convergence of the analytical solution was examined and verified first. 4.1. The verification and convergence of the analytical solution Since the solution of the response in this study is in an analytical form, the convergence test was performed for the reflection coefficient in terms of the mode number needed for a stable solution of series form. Presented in Fig. 6(a) and (b) is the reflection coefficient corresponding to the mode numbers when the wave period is ranged from 2 to 10 s and 12 to 20 s, respectively. It is observed that for most cases when the mode number is more than 10 the reflection coefficient will converge to a constant value. It is also
Fig. 6. The convergence of the reflection coefficient corresponding to the number of mode for the series solution: (a) wave period from 2 to 10 s, and (b) Wave period from 12 to 20 s.
known that the mode number needed for the convergence is related to the water depth and the draught and dimension of the structure. To further verify that this study may accurately model the motion of the pontoon-pier structure subjected to waves, a case study with only accounting for the heave radiation was also performed to compare to Lee’s study (1995). By adopting the same conditions for the calculation, the amplitude of the incident waves A ¼ 0.1 m; draught of the pontoon structure is 7.5 m; the dimension of the pontoon structure 2b ¼ 10 m while the diameter of the pile-column is 0.05 m and the elastic modulus of the material is E ¼ 196 Gpa. Compared to Lee’s solution, the dimensionless amplitude of the waves generated by the pontoon structure under the same conditions as mentioned above is illustrated in Fig. 7, where good agreement to each other is observed in the solution. 4.2. The influence of the pontoon draught on the response of structure and flow field Presented in Fig. 8(a) and (b) are the reflection and transmission coefficients for the floating pier structural system with various pontoon draught, respectively. The variation of the pontoon-draught was taken into accounts as 5.5, 6.5 and 7.5 m. It is observed that when the water depth is constant, corresponding to the increase of the pontoon-draught the reflection coefficient is larger associated with a lower resonant frequency. For the dimensionless amplitude response of the pontoon structure, Fig. 9(a–c) illustrate the dimensionless amplitude of surge, heave and pitch of the pontoon-pier with respect to dimensionless wave frequency. It is found that corresponding to the increase of the pontoon-draught, the response of the pontoon structure becomes more drastic. The motion of the pitch is basically in accordance with the motion of surge but with smaller amplitudes. It is more obvious when observing from Fig. 10, where the peak amplitudes from
Fig. 7. Comparison of the dimensionless amplitude of the waves generated by the pontoon structure to Lee’s solution.
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Fig. 8. The reflection and transmission coefficients for the floating pier structural system with various pontoon draughts: (a) the refection coefficient corresponding to the wave frequency, and (b) the transmission coefficient corresponding to the wave frequency.
the frequency response were presented against dimensionless draught of the pontoon structure. It shows that the increasing trend of the response of all motions including surge, heave and pitch is with respect to the increase of the draught. 4.3. The influence of the pontoon dimension on the response of structure and flow field Fig. 11(a) and (b) present the reflection and transmission coefficient for the floating pier structural system corresponding to dimensionless frequency while the dimension of the pontoon structure is varied as 8, 12 and 16 m. It is noticed from these figures that the variation of the dimension of the pontoon structure influences generally the responding frequency. When the dimension of the pontoon structure is larger the reflection coefficient fluctuates in a smaller range of frequency. For the dimensionless amplitude response of the pontoon structure as illustrated in Fig. 12(a–c) corresponding to surge, heave and pitch of the pontoon structure. It is found that
Fig. 9. The dimensionless response of the pontoon-pier with various draughts in the pontoon-pier: (a) the dimensionless amplitude of the surge motion corresponding to the wave frequency. (b) The dimensionless amplitude of the heave motion corresponding to the wave frequency. (c) The dimensionless amplitude of the pitch motion corresponding to the wave frequency.
corresponding to the increase of the dimension, the response of the pontoon structure in the surge and pitch motion becomes more significant. However, for the heave
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Fig. 10. Response peaks of the pontoon-pier corresponding to the variation of the draught.
Fig. 11. The reflection and transmission coefficients for the floating pier structural system with various pontoon dimensions: (a) the refection coefficient corresponding to the wave frequency, and (b) the transmission coefficient corresponding to the wave frequency.
motion as shown in Fig. 12(b), the peak-value of the amplitude becomes smaller with respect to the increase of the dimension of structure. The peak amplitudes from the
Fig. 12. The dimensionless response of the pontoon-pier with various dimensions of pontoon: (a) the dimensionless amplitude of the surge motion corresponding to the wave frequency, (b) the dimensionless amplitude of the heave motion corresponding to the wave frequency, and (c) the dimensionless amplitude of the pitch motion corresponding to the wave frequency.
frequency response were drawn against the dimensionless width of the pontoon structure as shown in Fig. 13. It shows that the most significant response is the heave
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Fig. 13. Response peaks of the pontoon-pier corresponding to the variation of the pontoon dimension.
motion, to which the increase of the pontoon dimension has inverse influence than the surge and pitch motions. 4.4. The influence of the flexural rigidity of the pile-column on the structural response The flexural rigidity of the pile-column is taken into consideration here as a various parameter. The dimensionless flexural rigidity of the column-pile is defined as EI=rgh5 and varied as 0.026, 0.046 and 0.070. The responses of the pontoon structure as presented in Fig. 14(a) and (b) are the dimensionless amplitude of surge and heave motion of the pontoon structure corresponding to the dimensionless frequency. In the surge motion the response amplitude seems to be suppressed by the increasing flexural rigidity of the pile-column. However, as expected, the heave motion as was shown in Fig. 14(b) is not influenced by the variation of flexural rigidity. All three curves of different flexural rigidity are matched into one. The behavior of the pile-column in the time domain is shown in Fig. 15 as the time-history of transverse response for the pile-column corresponding to various flexural rigidities of the pile and shown in Fig. 16 as the response comparison between the pile-column and the pontoon structure. Observed from these time-history responses, it is found that two frequencies are dominant for the dynamic behavior of pile-column and obviously, one dominant frequency is from the wave as shown in Fig. 16, where during the lower frequency motion the pile-column is in accordance with the surge motion of the pontoon. The other higher frequency motion is due to the property of the pile-column itself, which can be seen from Fig. 15, where as the flexural rigidity of the pile increases the amplitudes of the response becomes smaller. The variation of two dominant frequencies for the pile-column responses can be observed clearly from Fig. 17 the response spectrum of the pile-column, where as the flexural rigidity increases the
Fig. 14. The dimensionless response of the pontoon-pier due to the variation of the pile-column property: (a) the dimensionless amplitude of the surge motion corresponding to the wave frequency, and (b) the dimensionless amplitude of the heave motion corresponding to the wave frequency.
Fig. 15. The time-history of transverse response for the pile-column corresponding to various flexural rigidity of the pile.
higher portion of frequency also increases associated with the reduction of responses while the lower portion of frequency remains invariant.
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Fig. 16. The response comparison between the pile-column and the pontoon structure.
frequencies but for the variation of the pontoon dimension, the influence on the wave motion is mainly on the frequencies of the wave. The influence on the wave field due to variation of the flexural rigidity of the pile-column is mainly the amplitudes of the waves, which is actually directly affected from the motion of the pontoon-piers while the surge motion of pier is confined by the pilecolumns. It is observed from the response of the floating pier system that the most significant response is the heave motion since the confinement from the pile-columns is only for the surge motion. Also because of the pile-column confinement even though the surge and pitch motion of the pontoon-pier is increased when the dimension and draught of the pontoon-pier increases, the increments are still confined into a small range. For the increase of the dimension of the pontoon-pier, the amplitudes of all motions for the pontoon-pier are increased correspondingly. However, when parameter of the draught of the pontoon-pier is under consideration, the increase of the pontoon dimension has inverse influence on the heave motions than the surge and pitch motions. For the time-history responses of the pile-columns, two frequencies are dominant for the dynamic behavior. One dominant frequency, the lower one is from the wave that forces the motion of the pontoon and then exerts on the pile-column. The other higher frequency motion is due to the property of the pile-column. As the flexural rigidity increases the higher portion of frequency also increases associated with the reduction of responses while the lower portion of frequency remains invariant.
Acknowledgments Fig. 17. The response spectrum of the pile-column with various flexural rigidity of the pile.
This study has been partially supported by the National Science Council, R.O.C. under grant No. NSC 93-2611-E110-003. This support is gratefully acknowledged.
5. Conclusions In this study, a two-dimensional floating pier consists of single rectangular impermeable pontoon with side supporting pile-columns was studied and a set of equations for the response of the pontoon and the pile-column were derived and solved analytically and numerically respectively while interactions among the wave, pontoon structure and pilecolumn were taken into accounts. Based on the numerical examples for the floating pier structural systems subjected to waves, it was found that with side supporting pilecolumns, the motion of both the pontoon-piers and the wave field is related to the dimension and draught of the pontoon-pier and the mechanic properties of the pilecolumn as well. The wave motions due to the interactions between the wave and pontoon-pier are also obtained. For the variation of the draught of the pontoon, the influence on the wave motion as observed from the coefficient of reflection and transmission is on both the amplitudes and
Appendix A Z 0 Km Em ¼ cos K n ðz þ hÞ cos K m ðz þ hÞdZ cos K m h h Km sin 2K m h h ¼ þ , 4K m 2 cos K m h Z
d
cos K m ðz þ hÞdZ ¼
Cm ¼ h
Z
sin K m ðh dÞ Km
d
Omn ¼
cos K 2n cos K m ðz þ hÞdZ h
1 sinðK 2n þ K m Þðh dÞ sinðK 2n K m Þðh dÞ ¼ þ , 2 K 2n þ K m K 2n K m
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Z d 1 cos K n cos K 2m dZ cos K n h h 1 sinðK n þ K 2m Þðh dÞ ¼ cos K n h K n þ K 2m sinðK n K 2m Þðh dÞ þ , K n K 2m
G mn ¼
m ¼ 0,
d0m ¼ 1; Z
d
Pmn ¼
cosh ln ðz þ hÞ cos K m ðz þ hÞ dZ h
¼
l2n
1 ½K m cosh ln ðh dÞ sin K m ðh dÞ þ K 2m
þ ln sinh ln ðh dÞ cos K m ðh dÞ, Z
0
ðz z0 Þ cos K n ðz þ hÞ dz
Ln ¼ d
cos K n h cos K n ðh dÞ K 2n d þ ðsin K n ðh dÞ þ 3 sin K n hÞ for z0 ¼ 43d 4K n Z b Mn ¼ x Aw2n eK 2n ðxbÞ þ Bw2n eK 2n ðxþbÞ dx " b # b 1 2K 2n b 2K 2n b ¼ ð1 þ e Þ þ 2 ðe 1Þ ðAw2n Bw2n Þ, K 2n K 2n ¼
Z
b
x sin ln ðx bÞ dx ¼
Nn ¼ b
Z
b ð1 þ cos 2ln bÞ, ln
0
ðz z0 Þ cos K m ðz þ hÞ dx
Qm ¼ d
¼
d ½sin K m ðh dÞ þ 3 sin K m h 4K m 1 þ 2 ½cos K m h cos K m ðh dÞ. Km
1059
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