wael 3 floating bridges

7/30/2019 Wael 3 Floating Bridges

1/14

- 1 -

Dynamic Analysis of a Pontoon-Separated Floating Bridge

Subjected to a Moving Load

WANG Cong ( )a, FU Shi-xiao ()

a,b,

LI Ning ( )c, CUI Wei-cheng ()dand LIN Zhu-ming ()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 floating bridge, under the effect of a moving load. In the paper, a brief summary of the dynamic analysis of the

floating bridge is first introduced. The motion equations for a pontoon-separated floating bridge, considering the nonlinear

properties of connectors and vehicles inertia effects, are proposed. The super-element method is applied to reduce the 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 floating bridge; moving load; 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, 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]


2/14

- 2 -

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 DOFs 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, 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:

[ ] [ ] extint RRDCDM =++ 1

where { }D

and { }D

are respectively the structural velocity and acceleration vectors; [ ]M and [ ]C arethe structural mass and damping matrices; { }intR and { }extR are the internal force and external load vectors.


3/14

- 3 -

Structural external load { }extR consists of the body force { }ext1R , the surface force { }ext2R and theconcentrated force { }ext3R , and is variable with time. As for a floating bridge with uniformly mass distribution,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.

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, which can be written as:

{ } [ ] { } weS S

SNRw we

dT

1

ext

2 = 2

where wS is the total area of the wetted surface, [ ]N is the shape function, and { } is the prescribedsurface traction force and has the form as:

{ } { }uS

M

we

ea= . 3

Here, eaM is the element added mass of the wetted surface obtained by using Todds method (Todd, 1961),

weS is the element area of the wetted surface, { }u is the acceleration vector for any point in the wettedelement, and can be known from the nodal acceleration vector { }d ,

{ } [ ]{ }dNu = . 4

Substituting Eqs. (3) and (4) into Eq. (2) and assuming wet to be the uniform thickness of the wetted

element will produce:

{ } [ ]{ }DMR a =1ext

2 5

with

[ ] [ ]=wS

eaa mM , 6

[ ] [ ] [ ]=weV

we

wewe

eaea VN

tS

MNm d

T, 7

where wewewe tSV is the volume of the wetted element, { }D is constructed by standard FEM 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 theconsistent added mass

matrix. The diagonal matrix form of the added mass matrix is computed by evenly assigning eaM to the


4/14

- 4 -

translational DOFs of the node in each wetted element; however, structural added mass matrix [ ]aM must becalculated by standard FEM procedures with non-zeros on the corresponding DOFs between the 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

{ } [ ] { } =w we

S S

we

we

b SuS

pNR d

T

2

ext

2 8

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

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

displacement vector { }d ,

{ } [ ]{ }dNu = . 9

Then similar to Eq. (5), we have

{ } [ ]{ }DKR b=2ext

2 10

with

[ ] [ ]=wS

bb kK , 11

[ ] [ ] [ ] eT

d wV wewe

bb VN

tS

pNk

we

= . 12

The expression of [ ]bk in Eq. (12) has the same form as Eq. (7) and is specified as the consistent form ofthe hydrostatic force. Similarly, the distributed element force is evenly assigned to the wetted surface nodes to

form the element diagonal hydrostatic force matrix, and [ ]bK is the hydrostatic force coefficients matrix withnon-zeros on the element nodal DOFs of the wetted 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:

{ } ( ) ( )( ) [ ] ( ){ } =vall ve

E E

veVvtve StpNEnumEnumR dT

3

ext

2 13

where, vallE and vtE respectively represent all the loaded elements in the procedure of the motion and theelements subjected to the moving load at time t; and veE is one of the vtE elements; ( )num denotes


5/14

- 5 -

the element number; is the Dirac function; veS indicates the area of veE ; ( ){ } ( )tAPtp VV = impliesthe gravity distribution density of all loaded elements at time t, with VP the weight of the vehicle and ( )tA the loaded area at time t.

The structural inertial force due to the moving vehicle can be given by

{ } ( ) ( )( ) [ ] ( )[ ]{ } =vall ve

E E

veVvtve SdNtNEnumEnumR dT

4

ext

2 14

where gtpt VV )()( = is the surface density of mass distribution on all vehicle-loaded elements. Provided

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

[ ]DtMR V )(4ext

2 = 15

with

[ ] ( ) ( )( )[ ] =vallE

VvtveV mEnumEnumtM )( , 16

[ ] [ ]( )

[ ]=veV

ve

veve

VV VN

tS

tNm d

T , 17

where veveve tSV is the volume of the vehicle-loaded elements; [ ])(tMV is the moving mass matrix dueto 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 vectorintR is related to the nodal displacement,

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

{ } [ ]{ }DKR =int 18

where [ ]K is the stiffness matrix for a linear system and assembled by the standard FEM procedures tooverlap the element stiffness matrix [ ]k . Substituting Eq. (18) into Eq. (1), the dynamic equilibrium equationof a linear system can be transformed into:

[ ]{ } [ ]{ } [ ]{ } { }extRDKDCDM =++ . 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:

{ } [ ]{ } { }intconint RDKR += 20


6/14

- 6 -

where [ ]{ }DK is the linear internal force, { }intconR is the part of the nonlinear connectors which can besummed by the force { }

irintcon of each nonlinear connector.

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

two nodes, 0L is the initial length in case of non-stress condition, and pG is the initial gap.

Fig.1 Tension-only truss element with initial gap ( 0pG ).

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

{ }{ }{ }

{ }{ }

=

=k

j

k

j

id

d

CC

CC

L

AE

F

Fr

000000

000000

0000

000000

000000

0000

11

11

int

con 21

where, { } { })()( kjkj

dzdydxd = is the nodal translation vector along x , y and z direction; A is

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

222 )()()( kkjjkkjjkkjj dzzdzzdyydyydxxdxxL +++++= . 22

For the tension-only element,


7/14

- 7 -

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:

[ ] [ ] [ ]MKC += 24

where and are called, respectively, the stiffness and mass proportional damping constants, which can

be associated with the fraction of critical damping by

( ) /5.0 += . 25

Therefore, and can be determined by choosing the fractions of critical damping ( 1 and 2 ) at two

different frequencies ( 1 and 2 ):

( ) ( )( ) ( )

==

2

1

2

2122121

2

1

2

21122

22

. 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.

[ ] [ ]( ) [ ] [ ] [ ]( ){ } { } [ ] intcon)()( RDtMtPDKKDCDMM VVba =++++ (27)

where

{ } 3ext

2)( RtPV= .

2. 4 Condensation of the Equation

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:

[ ]{ } [ ]{ } [ ]{ } { } [ ]{ } { }intcon000 )()( RDtMtPDKDCDM VV =++ (28)

where [ ]0M , [ ]0C and [ ]0K are, respectively, condensed mass, damping and stiffness matrices of thesystem, 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).


8/14

- 8 -

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 anchoringraft 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.

Fig.3 Schematic view of a floating bridge.

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 beamelements meshed on the corresponding lines of the shell elements. The nonlinear connecting components

Pontoon-Separated Floating Bridge

Anchoring Raft

Ribbon Bridge

Bridge Pontoon

Bridge Span


9/14

- 9 -

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

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 KN588

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).


10/14

- 10 -

0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .0-0 .8

-0 .7

-0 .6

-0 .5

-0 .4

-0 .3

-0 .2

-0 .1

0 .0

0 .1

x /L

Verticaldisplacemen

t(m)

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

moving load of KNPV 588= with different velocities. It is demonstrated that the maximum magnitudes of

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, theoscillatory 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 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .0

-0 .8

-0 .7

-0 .6

-0 .5

-0 .4

-0 .3

-0 .2

-0 .1

0 .0

0 .1

x /L

Verticaldisplacement(m)

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


11/14

- 11 -

0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .0

-0 .8

-0 .7

-0 .6

-0 .5

-0 .4

-0 .3

-0 .2

-0 .1

0 .0

0 .1

x /L

Verticaldisplacemen

t(m)

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

0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .0

-2 0 0

0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

1 4 0 0

x /L

Connectionforce(KN)

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

0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .0

- 2 0 0

0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

1 4 0 0

1 6 0 0

x /L

Connectionforce(KN)

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


12/14

- 12 -

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 action of KNPV 588= with 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 KN200 ; 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, 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.

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 Journal, 3 (2):380.

Seif M. S. and Inoue Y., 1998. Dynamic analysis of floating bridges,Marine Structures, 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,Progress in Structural Engineering and Materials,

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 Association for Bridge and Structural

Engineering (IABSE), 13 (2): 128-132.


13/14

- 13 -

Wu J. J., Whittaker A. R. and Cartmell M. P., 2001. Dynamic responses of structures to moving bodies using

combined finite element and analytical methods, International Journal of Mechanical Sciences, 43(11):

2555-2579.

Wu J. J., Whittaker A. R. and Cartmell M. P., 2000. The use of finite element techniques for calculating the

dynamic response of structures to moving loads, Computers and Structures, 78(6): 789-799.

Wu J. S. and Sheu J. J., 1996. An exact solution for a simplified model of the heave and pitch motions of a ship

hull due to a moving load and experimental results,Journal of Sound and Vibration, 192 (2): 495-520.

Wu J.S. and Shih P. Y., 1998. Moving-load-induced vibrations of a moored floating bridge, Computers and

Structures,66 (4): 435-461.

Appendix

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

[ ]{ } { } 0=+ xkxm A1

with [ ] [ ] [ ]ammm~~ += and [ ] bkkk

~~+= , where [ ]m~ , [ ]am

~ , k~

and bk~

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:

[ ]{ } [ ]{ } 0=+ xkxm . A2

Suppose the nodal coordinate { }x can be represented as the master coordinate { }1x and slave one { }2x ,or expressed as the vector form { } { }T21 xxx = , Eq. (A2) becomes:

=

+

0

0

2

1

2221

1211

2

1

2221

1211

x

x

kk

kk

x

x

mm

mm A3

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

( ) ( ) 01212122222 =+ xmkxmk . A4

Hence,

( ) ( ) 121211

22222 xmkmkx =

. A5

Neglecting the inertial forces, then


14/14

- 14 -

[ ]{ }12

1xT

x

x=

A6

where

[ ]

=

21

1

22 kk

IT A7

where, [ ]I is a unit matrix of the same order as the dimension of { }1x . Introducing Eq. (A6) into Eq. (A2)and multiplying [ ]TT to both sides of the equation, the reduced stiffness and mass matrices can be simplydescribed as:

[ ] [ ] [ ][ ] [ ] [ ][ ] [ ]211

221211

T

0 kkkkTkTk== A8

and

[ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ][ ] [ ]211

2222

1

221221

1

221221

1

221211

T

0 kkmkkkkmmkkmTmTm

+== . A9

Similarly, the damping matrix is

[ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ][ ] [ ]211

2222

1

221221

1

221221

1

221211

T

0 kkckkkkcckkcTcTc

+== . A10

wael 3 floating bridges

Documents