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Dynamic Analysis Analysis of a Pontoon-Sepa Pontoon-Separated rated Floating Bridge Subjected to a Moving Load

 

a

a,b

WANG Cong (王  琮) , FU Shi-xiao (付世晓) , c



LI Ning (李  宁) , CUI Wei-cheng (崔维成)  and LIN Zhu-ming (林铸明)

e

a. School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, China; b. Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, Trondheim N-7491, Norway; c. Engineering Institute of Engineering Crops, PLA University of Science and Technology, Nanjing 210007, China; d. China Ship Scientific Research Center, Wuxi 214082, China; e. The First Engineers Scientific Research Institute of the General Armament Department, Wuxi 214035, China.

ABSTRACT

For the design and operation of a floating bridge, the understanding of its dynamic behavior under a moving load is of great importance. The purpose of this paper is to investigate the dynamic performances of a new type floating bridge, the  pontoon-separated  pontoonseparated floa floating ting bridge bridge,, under the eeffect ffect of a m moving oving load load.. In the pape paper, r, a brief ssummary ummary of th thee dynamic aanalysis nalysis of the floating bridge is first introduced. The motion equations for a pontoon-separated floating bridge, considering the nonlinear  properties of connectors connectors and vehicl vehicles’ es’ iner inertia tia effects, are proposed proposed.. The super-element super-element method is applied to reduce the numerical numerical analysis scale to solve the reduced equations. Based on the static analysis, the dynamic features of the new type floating bridge subjected to a moving load are investigated. It is found that the dynamic behavior of the pontoon-separated floating bridge is superior to that of the ribbon bridge by taking the nonlinearity of connectors into account. Keywords: pontoon-separated  pontoon-separated flo floating ating bridg bridgee; moving load ; dynamic  dynamic;; FEM ; nonlinear  

1. Introduction

Since the most significant feature of the moving loads is its mobility, the interaction between the vehicle and  bridge is very complicated, which can be classified as a coupled vibration problem. Therefore, much attention has been paid to the dynamic behavior of flexible structures under the effect of the moving loads. Fryba (1999) has widely discussed and analyzed the effects of moving loads on various structures, from 1D structural members to 3D structures, as well as the effect of variable speed of the load. In addition, theoretical formulations and mathematical solutions for all cases and their application to civil, mechanical and naval structures are summarized. Many large fixed bridges have been constructed across rivers as well as seas, however, floating bridges take advantage of the natural law that the buoyancy of water can support the dead and live loads on the bridge. Therefore, floating bridges have been constructed in many countries such as the USA, Norway, Norway, UK, Japan and Canada (Watanabe, 2003; Watanabe and Utsunomiya, 2003; Watanabe et al ., ., 2004). Until now, two different structural forms for floating bridges have been used (Seif and Inoue, 1998): Continuous concrete pontoon type



 This project was supported by the Commission of Science Technology and Industry for National Defense.

1 Corresponding author. E-mail: [email protected] -1-

 

floating bridges and the steel truss deck one which is supported by discrete pontoons. Regarding the structural dynamic responses of a floating bridge subjected to moving loads, Virchis (1979) has once performed the numerical calculation to obtain the dynamic response of a military floating bridge subjected to tracked or wheeled loads by Runge-Kutta method, taking into account the initial condition of wheels, the variable speed and the separation status in vehicle-bridge system. However, with the popularity of the computer and the development of the finite element method (FEM), it is possible for the researchers to simulate the main features of the vehicle and bridge models more clearly and accurately. Wu et al . (2000, 2001) presented a technique employing combined finite element and analytical methods to predict the dynamic response of an experimental mobile gantry crane structure due to the two-dimensional motion of the trolley. Wu and Sheu (1996) investigated the dynamic performances of a rigid ship hull subjected to a moving load by simplifying the hydrostatic forces as the action of linear springs and dampers, obtained the motion behavior based on the solution of the heave-pitch coupled equation, and compared it with the corresponding experiment. Wu and Shih (1998) studied the elastic vibration of a partial-catenary-moored floating bridge (in still water) subjected to a moving load by taking the entire pontoon as a slender beam resting on an elastic foundation, and the influence of hydrodynamic forces as constant added mass, respectively. Furthermore, the stiffness and mass matrices of two-node beam element with different nodal DOF’s have been derived to simulate the features of the rigid- or hinged-connection by using the FEM and the conservation of energy. Considering the nonlinear  properties of the connection, Fu et al . (2005) estimated the dynamic response of a military ribbon bridge subjected to a moving load by means of the super element method. The present investigation is a part of the project “Hydroelastic Response of the Floating Bridge Subjected to Moving Loads in High Speed Current”. Based on the previous researches (Fu et al ., ., 2005; Fu, 2005), the paper mainly focuses on the dynamic behavior of a new type floating bridge, the pontoon-separated one, which conspicuously differs from the aforementioned ribbon bridge in structural format. Consequently C onsequently,, it is necessary to investigate the features of dynamic responses of the pontoon-separated floating bridge subjected to a moving load. For a more precise dynamic analysis, the full floating bridge is modeled by 3D FEM and each ferry raft is connected by nonlinear elements, and to diminish the scale of calculation, the substructure method is applied to condense the DOFs of the system. 2. Description of the Equation of Motion

The nonlinear motion equations governing the dynamic response of a structure can be derived by assuming the work of external forces to be absorbed by the work of internal, inertial and viscous forces, for any small kinetically admissible motion. On the basis of the FEM and local separation of variables, the nonlinear equilibrium equations of the structure can be expressed as:

   +  R int =  R ext [ M ] D + [C   ] D where

   and

{ D}

(1)

 

   are respectively the structural velocity and acceleration vectors;

{ D}

  and

  are

C ] [ M rnal ] load[vectors. { }  and { R }  are the internal force and external exte

the structural mass and damping matrices;  R

int

ext

-2-

 

{ }  consists

Structural external load  R

ext

{ }, ext

of the body force  R1

{ }  and ext

the surface force  R2

the

{ }, and is variable with time. As for a floating bridge with uniformly mass distribution, ext

concentrated force  R3

the body forces (the weight of the floating bridge) can be balanced by the static buoyancy (the integration of static pressure over undisturbed wetted surface); therefore, only the unbalanced force originated from the dynamic buoyancy and the moving distribution force count.  count. 

2. 1 External Forces

The oscillating body in the fluid will cause the movement of surrounding water, inversely the inertial forces of water will induce the reaction forces to the wetted surface of the body body,, which can be written as:

{ R }  = ∑    ∫ [ N ]   {Φ}d S  T

ext 2 1

we

(2)

 

S w S  we

where S w   is the total area of the wetted surface, [ N ]   is the shape function, and

{Φ}   is

the prescribed

surface traction force and has the form as:

 M ea

{Φ} = −

S we

{u}.

(3)

Todd’s method (Todd, 1961), Here,  M ea   is the element added mass of the wetted surface obtained by using Todd’s

S we   is the element area of the wetted surface,

{u}  is the acceleration vector for any point in the wetted

element, and can be known from the nodal acceleration vector

{d },

  ]{d  }. {u}  =  [ N 

(4)

Substituting Eqs. (3) and (4) into Eq. (2) and assuming t we   to be the uniform thickness of the wetted element will produce:

{ R } =  −[ M  ]{ D} 

(5)

[ M a ] = ∑ [mea ] ,

(6)

ext 2 1

a

with

S w

[mea ] = ∫ [ N ]T V we

 M ea S we t we

[ N ]d V we ,

(7)

{ }

   is constructed by standard FEM procedures, where V we   =S wet we   is the volume of the wetted element,  D procedures, i.e. conceptual expansion of element matrices to “structure size”. On the basis of the classification of the mass matrix, Eq. (7) can be termed as the the   consistent added mass matrix. The diagonal matrix form of the added mass matrix is computed by evenly assigning  M ea   to the -3-

 

translational DOFs of the node in each wetted element; however, structural added mass matrix [ M a ]   must be calculated by standard FEM procedures with non-zeros on the corresponding DOFs between tthe he interfaces and the fluid and zeros on the remaining. The so-called hydrostatic force is defined as wetted surface distribution force produced by the buoyancy when the floating bridge is away from the equilibrium position. Assuming the force is linear to the vertical displacement of the floating bridge and uniformly acts upon the structural wetted surface, then

{ R } = ∑   ∫   − [ N ] ext

2

T

2

S w S  we

 pb

  {u}d S we  

(8)

S we

determined in the draft-displacement  draft-displacement  where  p b   is the hydrostatic pressure of unit displacement and can be determined

{u}  is the displacement of any point in the wetted element and can be obtained from the nodal displacement vector {d }, curve; and

{u}  =  [ N ]{d }.

(9)

Then similar to Eq. (5), we have

{ R } =  −[ K  ]{ D}  b

(10)

[ K b ] = ∑ [k b ] ,

(11)

ext 2 2

with

S w

[k b ] = ∫ [ N ]T V we

 pb S we t we

[ N ]d V we .

(12)

The expression of [k b ]   in Eq. (12) has the same form as Eq. (7) and is spec specified ified as the consistent form of the hydrostatic force. Similarly, the distributed element force is evenly assigned to the wetted surface nodes to coefficients matrix with with form the element diagonal hydrostatic force matrix, and [ K b ]   is the hydrostatic force coefficients non-zeros on the element element nodal DOFs DOFs of the wetted surface surface and

zeros on the remaining.

On the assumption that the vehicles are always in contact with the surface of the floating bridge, leaving out of consideration of elastic and damping features, we will have the surface distributed load due to the gravitation and the inertial force of the motion. Here, the gravitation load of the vehicles is:

{ R } =  ∑ δ (num( E  ) − num( E  )) ∫  [ N ] { p  (t )}d S  T

ext 2 3

ve

 E vall 

vt 



ve

 

(13)

 E ve

where,  E vall   and  E vt    respectively represent represent all the loaded elements in the procedure of the motion motion and the elements subjected to the moving load at time t ; and  E ve   is one of the  E vt    elements; num( )   denotes -4-

 

the element number; δ    is the Dirac Dirac function; S ve   indicates the area of  E ve ; { pV  (t  )} =  P  V   A(t )   implies vehicle and  A(t )   the gravity distribution density of all loaded elements at time t , with  P  V    the weight of the vehicle the loaded area at time t . The structural inertial force due to the moving vehicle can be given by

{ R } = ∑ −  δ (num( E   ) − num( E  )) ∫  [ N ]  ρ   (t )[ N ]{d }d S  T

ext 2 4

ve

vt 

 E vall 



ve

 

(14)

 E  ve

where  ρ V  (t ) 

=  pV  (t )  g   is the surface density of mass distribution on all vehicle vehicle-loaded -loaded elements. Provided Provided

that all the above elements have the uniform thickness t ve , Eq. (14) becomes:

    R2ext 4  = −[ M V  (t )] D

(15)

[ M V  (t )] = ∑ δ   (num  ( E ve ) − num( E vt  ))[mV  ] ,

(16)

with

 E vall 

  V  (t ) [ N ]d V ve , [mV  ] = ∫ [ N ]T   ρ  V ve

(17)

S ve t ve

where V ve=S vet ve   is the volume of the vehicle-loaded elements; [ M V  (t )]   is the “moving mass mass matrix” due to the inertial forces of the vehicle with non-zeros on the DOFs of the vehicle-loaded elements at time t   and zeros on the remaining. 2. 2 Internal Forces

For the nonlinear system, the structural internal force vector  R

int

  is related to the nodal displacement, displacement,

velocity and acceleration, whereas the linear stress-strain system has the following relation:

{ R   } =  [ K ]{ D}  int

(18)

where [ K ]   is the stiffness matrix for a linear system and assembled by the standard FEM procedures to overlap the element stiffness matrix

[k ] . Substituting Eq. (18) into Eq. (1), the dynamic equilibrium equation

of a linear system can be transformed into:

[ M ]{ D}+  [C   ]{ D  }+  [ K ]{ D} =  { R ext }.

(19)

Since the nonlinear properties of the connection of the floating bridge can be primarily featured by the tension-only or compression-only connectors, the internal force can be further decomposed into:

{ R } =  [ K    ]{ D} +  { R }  int

int con

(20) -5-

 

{ }

where [ K ]{ D}   is the linear internal force,  Rcon   is the part of the nonlinear connectors which can be int

summed by the force

{r  }   of each nonlinear connector. int con i

The element shown in Fig.1 can provide the internal force only when the extension between the two nodes is larger than initial slack gap, which can simulate the mechanical characteristics of the tension-only connector. Conversely, when the extension is less than the initial gap, the element illustrated in Fig.2 can generate the internal force that can describe the behavior of the compression-only one.  L   is the present length between gap.  two nodes,  L0   is the initial length in case of non-stress condition, and G p   is the initial gap.

Fig.1 Tension-only truss element with initial gap ( G p <0).

Fig.2 Compression-only truss element with initial gap ( G p >0).

Consequently,, the internal force of the i th nonlinear connector can be written as: Consequently

{r  }

int con i

where,

⎡ C  1 ⎢ 0 ⎢   ⎧{ F  j }⎫  AE ⎢ 0 =⎨ ⎢ ⎬=  L { }  F  ⎩ k  ⎭ ⎢− C 1 ⎢ 0 ⎢ ⎢⎣ 0

{d } j ( k ) = {dx

dy

0

0

− C  1

0

0⎤

0

0⎥

0

0

0

0

0

0

0

0

0

C 1

0

0

0

0

0

0

0

0

0

⎥ 0⎥ ⎧{d  j }⎫ ⎬  ⎥⎨ 0⎥ ⎩{d k }⎭ 0⎥ ⎥ 0⎥⎦

(21)

dz } j ( k )   is the nodal translation vector along  x ,  y   and  z   direction;  A   is

the sectional area;  E   is the Young’s modulus;  L   is the present length of element and can be defined as:

 L = ( x j + dx j − x k  − dx k  ) 2 + ( y j +  dy j −  y k  − dy k  ) 2 + ( z  j + dz  j − z k  − dz k  ) 2 .

(22)

For the tension-only element,

⎧1.0 C 1 = ⎨ ⎩0.0 where

Δl  ≥ 0   Δl  < 0

(23)

Δl  =  L − L0 ; while for the compression-only element, the value of C 1   is just reversed to that of the

tension-only one. 2. 3 Damping

Considering the nonlinear connectors of the floating bridge, it is difficult to estimate the real value of damping; however, various linear and nonlinear damping are usually simplified as the viscous damping in the -6-

 

structural dynamic problems, which is convenient to be dealt with numerically. A popular spectral damping scheme, called Rayleigh or proportional damping (Wu and Sheu, 1996; Wu and Shih, 1998) is often used to form the damping matrix as a linear combination of the stiffness and mass matrices of the structure, that is:

[C ] = α [  K    ] + β [ M ]  

(24)

where α    and  β    are called, respectively, respectively, the stiffness and mass proportional damping consta constants, nts, which can  be associated with the fraction fraction of critical damping damping ξ    by

ξ  = 0.5(α    +  β  /

).

(25)

Therefore, α    and  β    can be determined by choosing the the fractions of critical damping (ξ 1   and ξ 2 ) at two different frequencies ( ω 1   and ω 2 ):

⎧⎪α  = 2(ξ 2ω 2 − ξ 1ω 1 )   (ω 22 − ω 12 ) . ⎨ ⎪⎩ β  = 2ω 1ω 2 (ξ 1ω 2 − ξ 2ω 1 )   (ω 22 − ω 12 )

(26)

Substituting the Eqs. (5), (10), (15) and (20) into Eq. (1), one can derive the governing equation of the floating bridge subjected to a moving load by taking the nonlinear properties into account.

  ] ){ D} = { P  (t )} −  [ M  (t )] D  −  R int ([ M ] + [ M a ]) D  + [C ] D + ([ K b ]+ [ K  con V  V  where { P   )} =  R2 V  (t 

ext 3

 

(27)

.

2. 4 Condensation of the Equation E quation

Considering the nonlinearity of the internal force, iteration method must be applied to solve Eq. (27). However, large numbers of DOFs usually make the solution very time consuming and even computation unpractical, which inevitably results in the application of the super element method (Guyan, 1965). In this approach, the structural system is divided into different substructures, and some DOFs of the particular substructures are specified as master DOFs while the remaining as slave ones, whose properties are also related to the master ones. The substructure without slave DOFs are termed as the super element. By the combination of the super and non-super elements, the governing equation of the structure can be derived as: int [ M 0 ]{ D}+  [C 0 ]{ D }+  [ K 0 ]{ D} =  { P V  (t )} −  [ M   V  (t )]{ D}− { Rcon } 

(28)

where [ M 0 ] , [C 0 ]   and [ K 0 ]   are, respectively, condensed mass, damping and stiffness matrices of the system, which can be given by the composition of the corresponding element matrices (detailed please see Appendix). Then direct integration and Newton-Raphson iteration method can be applied to solve the nonlinear Eq. (28).

-7-

 

3. Analysis and Discussion 3. 1 Physical Model

As shown in Fig.3, the floating bridge is composed of the pontoon-separated floating bridge, the anchoring raft and the ribbon bridge. Regarding the research objective of the investigation, only the pontoon-separated  part is taken into consideration, which is rigidly connected by nine bridge rafts. Fig.4 shows the schematic view of a bridge raft with the bridge span being supported and rigidly connected with the board of the two  bridge pontoons.

Pontoon-Separated Floating Bridge 

Anchoring Raft  Ribbon Bridge

Fig.3 Schematic view of a floating bridge.

Bridge Span

Bridge Pontoon

Fig.4 Schematic view of a bridge raft of the pontoon-separated floating bridge.

3. 2 Finite Element Model

The floating bridge is discretized by the combination of the shell and beam elements, with the beam elements meshed on the corresponding lines of the shell elements. The nonlinear connecting components -8-

 

 between the ferries are modeled by the tension-only and/or compression-only truss element with initial gap, and the hydrostatic forces are simulated by the linear spring elements. Fig.5 is the finite element model of the  bridge raft.

Fig.5 Finite element model of the bridge raft.

In the finite element model for a bridge raft, the nodes of the vehicle-loaded elements, and those located at the connection of different bridge rafts are defined to be the master nodes; whereas the remaining to be the slave ones. Based on the condensation performed to the bridge raft exclusive of the vehicle-loaded elements, the super element model is shown in Fig.6.

Fig.6 Super element model of the bridge raft.

3. 3 Static Analysis Analysis

Since the floating bridge, over 300 meters, is connected by nine rafts, for the convenience and simplicity of computation, we just take the five-raft connected part into consideration. Fig.7 illustrates the vertical displacement of the vehicle-loaded substructures of the floating bridge with a constant static load of 588 KN   at the position amidst the longitudinal length of the five-connected bridge. It is found that the maximum displacement response is 65.5 cm and the influential length is nearly 51m. Furthermore, the deflection curve is oscillatory at the two ends of the influential length by taking the initial gap and the nonlinearities of the connectors into consideration, and this is not quite analogous to that of the nonlinearly connected ribbon bridge (Fu et al ., ., 2005).

-9-

 

0.1

0.0

- 0 .1

   )   m - 0 .2    (    t   n   e   m - 0 .3   e   c   a    l   p - 0 .4   s    i    d

 

   l   a   c - 0 .5    i    t   r   e    V - 0 .6 - 0 .7

- 0 .8 0 .0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1 .0

x/L

 

Fig.7 Static response of the floating bridge.

3. 4 Dynamic Response Analysis

Figs. 8 and 9 show the vertical displacement response history of the vehicle-loaded substructures of the floating bridge at the position amidst the longitudinal length of the five-connected bridge subjected to a V  moving load of  P  demonstrated that the maximum magnitudes of = 588 KN    with different velocities. It is demonstrated the dynamic response of the floating bridge for different velocities approximate 65cm, which is almost the

same as that in the abovementioned static analysis, and happens after the middle point of the overall length. However, as for the oscillation feature, the vertical displacement response of the floating bridge subjected to a moving load is much more remarkable than that to a static load; while in the state of dynamic loading, the oscillatory traits weaken in the fore part of the floating bridge and strengthen in the aft part with the increase of the speed of the moving load. Furthermore, there only exists the upward dynamic displacement in the vicinity of the end of the pontoon-separated floating bridge, which is totally different from that of the ribbon bridge (Fu et al ., ., 2005). The difference can be accounted by the fact that the pontoon-separated bridge has more self-weight and can endure larger moving load than the ribbon bridge. 0.1

0.0

-0 .1

   )   m    (    t   n   e   m   e   c   a    l   p   s    i    d    l   a   c    i    t   r   e    V

-0 .2

-0 .3

 

-0 .4

-0 .5

-0 .6

-0 .7

-0 .8 0 .0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1 .0

x /L

Fig.8 Dynamic response of the floating bridge subjected to a moving load with v=8.33m/s.

- 10 -

 

 

0.1

0.0

-0 .1

   )   m    (    t   n   e   m   e   c   a    l   p   s    i    d    l   a   c    i    t   r   e    V

-0 .2

-0 .3

 

-0 .4

-0 .5

-0 .6

-0 .7

-0 .8 0 .0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1 .0

x/L

 

Fig.9 Dynamic response of the floating bridge subjected to a moving load with v=13.89m/s.

1400

1200

1000

   )    N    K    (   e   c   r   o    f   n   o    i    t   c   e   n   n   o    C

800

600

 

400

200

0

-2 0 0 0 .0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1 .0

x /L

 

Fig.10 Dynamic response history of the connection force (v=8.33m/s).

1600 1400

1200

   )    N 1 0 0 0    K    (   e 800   c   r   o    f   n 600   o    i    t   c   e   n 400   n   o    C

 

200

0

-2 0 0 0 .0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1 .0

x /L

  Fig.11 Dynamic response history of the connection force (v=13.89m/s).

- 11 -

 

Figs. 10 and 11 show the dynamic connection force response histories in the lower connector of the middle  bridge span of the floating bridge under a moving load load action of  P  V 

= 588 KN    with different different velocities. The

 positive values in the figures stand for the state of tension, and the negative ones for the state of compression. As illustrated, the amplitude of the connection force goes up conspicuously with the increase of the speed of the moving load; for instance, the maximum difference of the hump value is over 200 KN  ; meanwhile, the hump value appears at the position of aft part for the low speed and fore part for high speed. In addition, the connector has been, to some extent, always in the tension state with the value varying along the longitudinal direction. When the moving load is not within the scope of the influential length, the connector will tolerate the impact produced by the declined vibration of the floating bridge, which is much smaller than the hump value that the moving load has ever produced, and the trend of decline will speed up with the rise of the moving velocity. 4. Summary

A governing equation of three-dimensional dynamic response analysis for a pontoon-separated floating  bridge subjected to a moving load, considering the nonlinear properties of connectors and vehicles’ inertia effects, has been introduced, in which the nonlinear motion equation has been condensed by the super element method and solved with the direct integration and iteration method. Moreover, the static and dynamic  particulars of the pontoon-separated floating bridge have been studied and compared with those of the ribbon  bridge. It is testified that the pontoon-separated type has more advantages than the ribbon one from the dynamic point of view by taking the nonlinearities into consideration. References

Fryba L., 1999. Vibration of Solids and Structures under Moving Loads, Loads, Telford, London. Fu S. X. et al., 2005. Hydroelastic analysis of a nonlinearly connected floating bridge subjected to moving loads,  Marine Structures,, 18 (1): 85-107. Structures Fu S. X., 2005. Nonlinear hydroelastic analyses of flexible moored structures and floating bridges, Ph. D. thesis, Shanghai Jiao Tong University (In Chinese). Guyan R. J., 1965. Reduction of stiffness and mass matrices, AIAA matrices,  AIAA Journal  Journal , 3 (2):380. Seif M. S. and Inoue Y., 1998. Dynamic analysis of floating bridges, Marine bridges,  Marine Structur Structures es,, 11(1): 29-46. Todd F. H., 1961. Ship Hull Vibration, Edward Arnold, London. Virchis V. J., 1979. Prediction of Impact Factor for Military Bridges, ISVR Technical Report No.107. Watanabe E. and Utsunomiya T., 2003. Analysis and design of floating bridges, Progr bridges,  Progress ess in Structural Structural Engineering Engineering a and nd Materia Materials ls,, 5: 127-144.

Watanabe E. et al., 2004. Very large floating structures: applications, analysis and design. Center for Offshore Research and Engineering National University of Singapore, Core Report No. 2004-02. Watanabe E., 2003. Floating bridges: past and present,  Journal of the International International Association for Bridge and Structu Structural ral  Engineering  Engineerin g (IABSE) (IABSE),, 13 (2): 128-132. - 12 -

 

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Appendix The equation of the free vibration of a single substructure can be written as:

[m ]{  x} + k  { x} = 0  

(A1)

~ ~ ~ ~ ~ ~ ~ ~ with [m ] = [m] + [ma ]   and k  = k  + k b , where [m] , [ma ] ,  k    and k b   are the mass matrix, the

[]

added mass matrix, the stiffness matrix and the hydrostatic force matrix respectively. Thus, Eq. (A1) has been changed into an eigen-value equation:  equation: 

− λ [m ]{ x} +  [k ]{ x} = 0 .

(A2)

Suppose the nodal coordinate { x}   can be represented represented as the ma master ster coordinate { x1 }  and slave slave one { x 2 } , or expressed as the vector form { x}  = { x1  x2 } , Eq. (A2) becomes: T

m12 ⎤ ⎧ x1 ⎫ ⎡ k 11 ⎡m − λ ⎢ 11 ⎥⎨ ⎬ + ⎢ ⎣m 21 m22 ⎦ ⎩ x 2 ⎭ ⎣k 21

k 12 ⎤ ⎧ x1 ⎫

⎧0⎫ ⎥⎨ ⎬ = ⎨ ⎬   k 22 ⎦ ⎩ x 2 ⎭ ⎩0⎭

(A3)

From the second row of Eq. (A3), one can obtain:

(k 

22

− λ m22 ) x  2 + ( k 21 − λ m21 ) x1 = 0 .

(A4)

Hence,

(

 

)− ( 1

)

 x 2 = − k 22 − λ  m22   k 21 − λ m 21  x1 .

(A5)

 Neglecting the inertial forces, forces, then

- 13 -

 

⎧ x1 ⎫ ⎨ ⎬ = [T ]{ x1 }   ⎩ x2 ⎭

(A6)

where

⎡  I  ⎤ [T ] = ⎢⎣−  k   22−1k 21 ⎥⎦  

(A7)

where, [ I ]   is a unit matrix of the same order order as the dimension of { x1 } . Introducing Eq. (A6) into Eq. (A2) and multiplying [T ]   to both sides of the equation, the reduced stiffness and ma mass ss matrices can be simply T

described as:   −1 [k 0 ] = [T ]T [k ][T ] =  [k 11 ]− [k 12 ][k 22 ] [k 21 ] 

(A8)

[m0 ] = [T ]T [m ]][[T ] = [m11 ] −  [k 12 ][k 22 ]− 1[m21 ]  − [m12 ][k 22 ]−1 [k 21 ]+ [k 12 ][k 22 ]− 1[m22 ][k 22 ]−1 [k 21 ].

(A9)

and

Similarly,, the damping matrix is Similarly

[c0 ] = [T ]T [c ][T ] = [c11 ] − [k 12 ][k 22 ]−1[c21 ] − [c12 ][k 22 ]−1[k 21 ] + [k 12 ][k 22 ]−1[c22 ][k 22 ]−1[k 21 ] .

- 14 -

(A10)

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