Hydroelastic analysis of a nonlinearly connected floating bridge subjected to moving loads

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Marine Structures 18 (2005) 85–107 www.elsevier.com/locate/marstruc

Hydroelastic analysis of a nonlinearly connected floating bridge subjected to moving loads$ Fu Shixiaoa, Cui Weichenga,b,, Chen Xujunb,c, Wang Conga a

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, China b China Ship Scientific Research Center, Wuxi 214082, China c Engineering Institute of Engineering Corps, PLA University of Science and Technology, Nanjing 210007, China

Received 17 December 2004; received in revised form 26 May 2005; accepted 31 May 2005

Abstract In this paper, the motion equations for the nonlinearly connected floating bridge, considering the nonlinear properties of connectors and vehicles’ inertia effects, are proposed. The super-element method is used to condense the whole calculation scale, and the direct integration and Newton–Raphson iteration method are applied to solve the reduced equations. Based on the modal and static analyses, the dynamic displacement and connection forces characteristics of a floating bridge with nonlinear connectors subjected to moving loads are investigated. It is found that nonlinearity and initial gap of the connectors are important for the hydroelastic response of a nonlinearly connected floating bridge. r 2005 Elsevier Ltd. All rights reserved. Keywords: Floating bridges; Moving loads; Dynamic responses; Initial gap; Nonlinear; FEM

$

This project was supported by the National Natural Science Foundation of China (Grant no. 50309018). Corresponding author. Tel.: +86 21 62932081; fax: +86 21 62933160. E-mail address: [email protected] (Cui Weicheng). 0951-8339/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.marstruc.2005.05.001

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1. Introduction Since 1912 the first modern Galata steel floating bridge was completed in Istanbul, many civil floating bridges have been constructed in countries such as USA, Norway, UK, Japan and Canada [1–3]. Until now, there are basically two different structural forms for floating bridges [4]: continuous concrete pontoon-type floating bridges have been utilized in the USA for many years, and a steel truss deck one which is supported by discrete pontoons has been constructed in Norway. Dynamic response analysis of flexible structures induced by moving loads is an important aspect in structural dynamic problems. Fryba [5] has summarized various analytical models used in the vehicle–bridge coupled system. However, with the development of the computer and finite element method (FEM), the numerical method, in which the structures and vehicles can be described more accurately [6–10], has gradually substituted the traditional analytical method. In a sequence, the methods used for the flexible structures are applied to the analyses of the dynamic responses of floating bridges subjected to moving loads. Regarding the structural dynamic responses of a floating bridge subjected to moving loads, Virchis [11] studied the dynamic response problems of a military floating bridge by Runge–Kutta method, where the initial conditions of the wheels, the variety of the speed and the separation between vehicles and bridge were taken into account. Wu and Sheu [12] investigated the coupled heave and pitch motions of a simplified non-uniform ship hull floating on a still water surface and subjected to a moving load, considering the ship hull as a rigid body supported by an elastic foundation with distributed springs and dampers. Wu and Shih [13] 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. To simulate the characteristics of the rigid- or hingeconnected floating bridge, the stiffness and mass matrices of two-node beam element with different nodal DOFs are derived. Hydroelastic theories [14,15] have been applied to the design and research works related to marine structures for several decades, e.g. Chen et al. [16,17], and they are also used to analyze the hydrodynamics of a floating bridge. As for the hydroelastic response analysis of floating bridges in waves in the frequency domain, Ueda et al. [18–20], Oka et al. [21] and Ikegami et al. [22] have reported and obtained the numerical results verified experimentally by a large-scale detailed elastic model test in the wave tank. In these references the structure has been modeled by 3D linear finite elements and the fluid effect has been determined by the solution of 3D water wave problem on the basis of boundary element method taking the free surface, the water depth, the hull vibrations (the motions of the floating units) and the interaction among the floating units within the framework of the linear theory. While for that in the time domain, Watanabe et al. [23] have analyzed its hydroelastic behavior and done the calculation taking the memory effect to the wave damping term into consideration. However, Refs. [18–23] have adopted the linear theory to predict the hydroelastic responses of the floating bridge, which cannot deal with the nonlinear properties of the structures. This is the

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main emphasis of the present investigation which results in the difficulty of the nonlinearly connected floating bridge subjected to moving loads. Ertekin et al. [24] analyzed the hydroelastic responses of the mechanically interconnected military floating bridge under the combined effects of the stationary moving and/or static loads acted on the bridge deck, the current loads and other external loads, and found that the drag forces were comparatively smaller than the model tests due to the neglection of the drag components other than the skin friction and the form drag. Fleischer and Park [25] calculated the hydroelastic vibrations of a beam with rectangular cross-section under the effect of a uniformly moving single axle vehicle by using the modal analysis and two-dimensional potential flow theory of the fluid and neglecting the effect of surface waves aside the beam. For most cases, the floating bridge was modeled as a simple beam without taking account of the nonlinearity of connectors and inertia effects of vehicles by assuming the loads as moving concentrated forces. However, for the nonlinearly connected floating bridges, each module cannot be easily simplified to be a linear hinge- or rigid-connected [26] system. In this study, considering the nonlinear properties of the connectors and inertia effects of loads, the three-dimensional dynamic response governing equations are proposed based on the nonlinear FEM. The equations are solved jointly by the direct integration method and Newton–Raphson iteration method by reducing the scale of equations with substructure method. And the characteristics of three-dimensional dynamic responses of nonlinearly connected floating bridges subjected to a moving load are investigated.

2. Nonlinear motion equations The nonlinear motion equations governing the dynamic response of a structure will 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 element can be written by € þ ½cfdg _ þ frint g ¼ frext g, ½mfdg (1) _ € where fdg and fdg are, respectively, the nodal velocity and acceleration vectors, ½m and ½c are the element mass and damping matrices and can be defined by Z ½m ¼ r½NT ½N dV e , (2) Ve

Z

cd ½NT ½N dV e .

½c ¼

(3)

Ve

where ½N, r and V e are shape functions, mass density of the material, and volume of the elements respectively, cd is a material-damping parameter analogous to viscosity.

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The vectors frint g and frext g in Eq. (1) are the element internal forces and external loads vectors, and can be defined by Z int fr g ¼ ½BT fsg dV e , (4) Ve

frext g ¼

Z

½NT fF g dV þ

Ve

Z

½NT fFg dS e þ Se

n X

fpgi ,

(5)

i¼1

where Se are the surface of the elements, ½B is the strain matrix, fsg is the stress matrix, fF g are the body forces, fFg are prescribed surface traction forces which are typically non-zero over only a portion of surface Se , and fpgi are concentrated loads that acting at total of n points on the element. The global structural nonlinear equilibrium equations can be obtained by assembling Eq. (1): € þ ½CfDg _ þ fRint g ¼ fRext g, ½MfDg (6) _ and fDg, € structural matrices ½M and ½C, and loads where structural vectors fDg vectors fRint g and fRext g are constructed by standard FEM procedures, i.e. conceptual _ fdg, € ½m, ½c, frint g and frext g to ‘‘structure size’’ expansion of element matrices fdg, followed by addition of overlapping coefficients. Structural external loads fRext g ext ext including body forces fRext 1 g, surface forces fR2 g and concentrated forces fR3 g are functions of time. Regarding the floating body with uniform mass distribution, the body forces, namely the weight of the floating body, and the static buoyancy, i.e. the integration of static pressure over undisturbed wetted surface, are equilibrated with each other. Thus, only the moving distribution forces and the unbalanced forces originated from the dynamic buoyancy will be considered in Eq. (6). 2.1. Fluid forces The oscillating body in the fluid will rise the movement of surrounding water, inversely in the inertial forces of water will induce the reaction forces to the wetted surface of the body can be expressed by XZ fRext g ¼ ½NT fFg dSwe , (7) 2 1 Sw

S we

where S w is the entire area of the wetted surface, and fFg ¼ 

M ea € fug. Swe

(8)

Here, M ea is the element added mass of the wetted surface estimated according to € is the method given by Todd [27], S we is the element area of the wetted surface, fug the acceleration vector for any point in the wetted element, and can be evaluated by the nodal acceleration, that is € € ¼ ½Nfdg. fug

(9)

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Substituting Eqs. (8) and (9) into Eq. (7) and assuming twe is the uniform thickness of the wetted element will produce € fRext 2 g1 ¼ ½M a fDg

(10)

with ½M a  ¼

X

(11)

½mea ;

Sw

Z

½NT

½mea  ¼ V we

M ea ½N dV we , S we twe

(12)

where V we ¼ Swe  twe is the volume of each wetted element. The mass matrix can roughly be divided into two categories: consistent mass matrix and lumped mass matrix. Eq. (12) can be termed as the consistent added mass matrix and is obviously analogous to the former one. The diagonal matrix form of the added mass matrix is computed by evenly assigning M ea to each element nodal translational DOFs of the wetted surface. However, structural added mass matrix ½M a  can be obtained by standard FEM procedures with non-zeros only on the corresponding DOFs between the interfaces and the fluid while 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 acted upon the structural wetted surface, one can obtain XZ p ext fR2 g4 ¼ ½NT b fug dS we , (13) S we S we S w

where pb is the hydrostatic force caused by unit displacement and can be obtained by the draft–displacement curve, fug are the displacements of any point on the wetted element and can be expressed as a function of the element interpolation functions and nodal displacements (14)

fug ¼ ½Nfdg.

Substituting Eq. (14) into Eq. (13) and assuming twe is the uniform thickness of the element will lead to fRext 2 g4 ¼ ½K b fDg

(15)

with ½K b  ¼

X

(16)

½kb ;

Sw

Z

½NT

½kb  ¼ V we

pb ½N dV we . Swe twe

(17)

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Eq. (17) is the hydrostatic force matrix of a single wetted element and has the same form with Eq. (12). The expression of ½kb  is specified as the consistent form of the hydrostatic force. It is similar to the lumped mass matrix that the distributed element force is evenly assigned to the wetted surface nodes to form the element diagonal hydrostatic force matrix. However, ½K b  is the hydrostatic force coefficients matrix with non-zeros only on the element nodal DOFs of the wetted surface while zeros on the remaining. 2.2. Distribution forces of moving vehicles With the assumption that the vehicles are always in contact with the surface of the floating bridge, surface distributed loads will be produced due to the gravitation and inertial forces of the moving vehicles, where the elastic and damping characteristics are neglected. In addition, the gravitation loads of the vehicles can be written by Z X fRext g ¼ dðnumðE Þ  numðE ÞÞ ½NT fpðtÞg dS ve , (18) ve vt 2 2 E ve

E move

where Emove are the elements that will be subjected to the moving loads during the whole procedures, num( ) denotes the element number of the element in the parenthesis, Evt are the elements that subjected to loads at time t, Eve is one of the Evt elements and Sve is the area of Eve; d is the Kroneker delta function; fpðtÞg ¼ PV =AvðtÞ is the gravity distribution density of all loaded elements at time t, PV is the weight of the vehicle, and Av(t) is the loaded area at time t. The structural inertial forces due to the moving vehicle can be given by Z X ext € dS ve , fR2 g3 ¼ dðnumðE ve Þ  numðE vt ÞÞ ½NT rV ðtÞ½Nfdg (19) E ve

E move

where rV ðtÞ ¼ pðtÞ=g is the surface density of mass distribution on all vehicle-loaded elements. By assuming all vehicle-loaded elements are of the uniform thickness twe, Eq. (19) becomes € fRext 2 g3 ¼ ½M v ðtÞfDg

(20)

with ½M v ðtÞ ¼

X

dðnumðE ve Þ  numðE vt ÞÞ½mv ;

(21)

E move

Z

½NT

½mv  ¼ V ve

rV ðtÞ ½N dV ve , Sve tve

(22)

where V ve ¼ Sve tve is the volume of the vehicle-loaded elements, ½M v ðtÞ is the moving mass matrix due to the inertial forces of the vehicle and only the DOFs of the vehicle-loaded elements at time t are non-zeros, the remaining are zeros.

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2.3. Nonlinear connection forces For the nonlinear system, the structural internal force vector fRint g is related to the nodal displacement, velocity and acceleration, whereas the system is of linear stress–strain relation, the element internal force can be expressed by Z frint g ¼ ½kfdg ¼ ½BT ½Ds ½Bfdg dV , (23) Ve

where ½Ds  is the stress–strain relation matrix, ½B is the strain matrix, fdg is the element nodal displacement vector. Therefore, the internal force for the linear system can be written as the linear expression of nodal displacement fRint g ¼ ½KfDg,

(24)

where ½K is the stiffness matrix for linear system and assembled by overlapping the element stiffness matrix ½k by following the standard FEM procedures. Replacing Eq. (24) into Eq. (6), the dynamic equilibrium equation of the linear system is obtained by € þ ½CfDg _ þ ½KfDg ¼ fRext g. ½MfDg

(25)

For the nonlinear connected floating bridge, the nonlinear internal force is featured by the tension-only or compression-only connectors between the modules, while the linear one represented by other linear structures of the modules. Hence, the internal force of the floating body can be divided into fRint g ¼ ½KfDg þ fRint con g,

(26)

where ½KfDg is the internal force of the linear structures, fRint con g is the force of the nonlinear connectors which can be derived by addition of the force frint con gi for each nonlinear connectors. 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 connectors. Conversely, the element illustrated in Fig. 2 can provide the internal force only when the extension between the two nodes is less than initial gap, which can simulate the mechanical characteristics of the compression-only connectors. The notation L is the present length between two nodes, L0 is the preliminary length under the non-stress condition, and Gp is the initial gap. j

k

z y x

Gp

L L0

Fig. 1. Tension-only truss element with initial gap (Gp o0).

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j

z y x

Gp

k

Gp

L0 L

Fig. 2. Compression-only truss element with initial gap (Gp 40).

Thus, the internal force of the ith nonlinear connector can be given by 3 2 C 1 0 0 C 1 0 0 6 0 0 0 0 0 07 7( 6 ( ) ) 7 6 fF g 6 0 0 0 0 0 7 fd j g j AE 6 0 int 7 frcon gi ¼ ¼ , 7 fF k g L 6 6 C 1 0 0 C 1 0 0 7 fd k g 7 6 4 0 0 0 0 0 05 0 0 0 0 0 0

(27)

where fdgjðkÞ ¼ f dx dy dz gjðkÞ is the translational displacement vector of the two nodes along x, y, z direction, A is the section area, E is Young’s modulus, L is the present length of element and can be defined by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L ¼ ðxj þ dxj  xk  dxk Þ2 þ ðyj þ dyj  yk  dyk Þ2 þ ðzj þ dzj  zk  dzk Þ2 . (28) For the tension-only element, ( 1:0; DlX0; C1 ¼ 0:0; Dlo0 and for the compression-only element, ( 0:0; DlX0; C1 ¼ 1:0; Dlo0

(29)

(30)

with Dl ¼ L  L0 . 2.4. Damping Considering the nonlinear connectors of the floating bridge, the damping features are actually nonlinear and pretty difficult to evaluate. In the practical structural dynamic problems, various linear and nonlinear damping are usually simplified as the viscous damping, which is comparatively easy to deal with mathematically. A popular spectral damping scheme, called Rayleigh or proportional damping [12,13] is often used to form the damping matrix as a linear combination of the stiffness and

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mass matrices of the structure, that is ½C ¼ a½K þ b½M,

(31)

where a and b are called, respectively, the stiffness and mass proportional damping constants, which can be associated with the fraction of critical damping x by x ¼ 0:5ðao þ b=oÞ.

(32)

Therefore, a and b can be determined by choosing the fractions of critical damping (x1 and x2) at two different frequencies (o1 and o2), and can be solved by the following equations: a ¼ 2ðx2 o2  x1 o1 Þ=ðo22  o21 Þ, b ¼ 2o1 o2 ðx1 o2  x2 o1 Þ=ðo22  o21 Þ.

ð33Þ

The damping factor a applied to the stiffness matrix ½K increases with increasing frequency, whereas the damping factor b applied to the mass matrix ½M increases with decreasing frequency. In this paper, the Rayleigh damping factors a and b are computed on the basis of the first two frequencies corresponding to the first- and second-order bending modes of the floating bridge, and the critical damping x is chosen as 5%. Substituting Eqs. (10), (15), (20) and (26) into Eq. (6), one can obtain the governing equation of hydroelastic responses of the nonlinear connected floating bridge subjected to moving loads € þ ½CfDg _ þ ð½K b  þ ½KÞfDg ð½M þ ½M a ÞfDg €  fRint g ¼ fPV ðtÞg  ½M V ðtÞfDg con

with fPV ðtÞg ¼

ð34Þ

fRext 2 g2 .

3. Condensation of the motion equations Due to the nonlinearity of the internal force, iteration method must be applied to solve Eq. (34). However, the solution time may sharply increase with the increase of DOF numbers, which even makes the solution impossible to obtain. Therefore, it is of significance to condense the equation scale before the solution starts. In the super element method [28], the structure will be divided into several substructures. Some DOFs of a certain substructure are chosen as master DOFs and the remaining slaves, and the properties of the substructure are condensed to the master DOFs. The substructures without slave DOFs are termed as super element. By combining the super and non-super elements via standard FEM procedures, we can obtain the mass, damping, stiffness and external loads matrices of the system. The equation of the free vibration of one substructure for a single module can be written by ¯ € þ ½kfxg ½mf ¼ 0, ¯ xg

(35)

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~ þ ½k~b , where ½m, ~ and ½k~b  are the mass ¯ ¼ ½k ~ þ ½m ~ a  and ½k ~ ½m ~ a , ½k with ½m ¯ ¼ ½m matrix, the added mass matrix, the stiffness matrix and the hydrostatic force matrix for a single module, respectively. Thus, the eigenvalue equation of the substructure is derived by ¯ l½mfxg þ ½kfxg ¼ 0, ¯

(36)

Divide the nodal coordinate fxg into the master one fx1 g and the slave fx2 g, or fxg ¼ f x1 x2 gT , Eq. (36) has the form " #( ) " #( )   m x1 x1 ¯ 12 ¯ 11 m k¯ 11 k¯ 12 0 l þ ¯ ¼ . (37) m x2 x2 ¯ 21 m ¯ 22 k21 k¯ 22 0 From the second row of Eq. (37), one can obtain ðk¯ 22  lm ¯ 22 Þx2 þ ðk¯ 21  lm ¯ 21 Þx1 ¼ 0.

(38)

Thus, x2 ¼ ðk¯ 22  lm ¯ 22 Þ1 ðk¯ 21  lm ¯ 21 Þx1 . Neglecting the inertial forces, one obtains ( ) x1 ¼ ½Tfx1 g, x2

(39)

(40)

where " ½T ¼

I

#

1 . k¯ 22 k¯ 21

(41)

Here, ½I is a unit matrix of the same order as the dimension of fx1 g. Introducing Eq. (40) into Eq. (36) and multiplying with ½TT on both sides of the equation, the reduced stiffness and mass matrices can be simply expressed by ¯ ¼ ½k¯ 11   ½k¯ 12 ½k¯ 22 1 ½k¯ 21 , ½k0  ¼ ½TT ½k½T

(42)

¼ ½m ½m0  ¼ ½TT ½m½T ¯ ¯ 11   ½k¯ 12 ½k¯ 22 1 ½m ¯ 21   ½m ¯ 12 ½k¯ 22 1 ½k¯ 21  þ ½k¯ 12 ½k¯ 22 1 ½m ¯ 22 ½k¯ 22 1 ½k¯ 21 .

ð43Þ

Similarly, the damping matrix will be expressed by ½c0  ¼ ½TT ½¯c½T ¼ ½c11   ½k12 ½k22 1 ½c21   ½c12 ½k22 1 ½k21  þ ½k12 ½k22 1 ½c22 ½k22 1 ½k21 .

ð44Þ

½k0 , ½m0 , ½c0  in Eqs. (42)–(44) are, respectively, called as condensed stiffness, mass, damping matrix, which can be used as ordinary elements in the proceeding analyses, and finally the governing equation of hydroelastic responses of the nonlinear connected floating bridge subjected to moving loads, consisting of super and

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non-super elements, can be derived by € þ ½C 0 fDg _ þ ½K 0 fDg ¼ fPV ðtÞg  ½M V ðtÞfDg €  fRint g, ½M 0 fDg con

(45)

where ½M 0 , ½C 0  and ½K 0  are, respectively, condensed mass, damping and stiffness matrix of the system, which can be obtained by addition of ½m0 , ½c0  and ½k0  via standard FEM procedures. Then direct integration and Newton–Raphson iteration method can be applied to solve the nonlinear Eq. (45).

4. Numerical example and discussion 4.1. Floating bridge model As shown in Fig. 3(a), the floating bridge is composed of 15 modules, with all the left translational displacements and those at right along y and z constrained. Nos. 1–16 are the position numbers referred to the two ends of each module, and Ps is the static loads. The connection forms are illustrated in Fig. 3(b), with the bottom linked by the rotational hinge, whereas the top jointed by tension-only and compressiononly connectors. The tension-only connector has the initial gap G p ¼ Gap T, and the compression-only one has the initial gap Gp ¼ Gap P. The particular dimensions of each module are: A ¼ 0:001 m2 , E ¼ 2:0  105 MPa, Lunit ¼ 6:7 m, B ¼ 8:082 m, Ddep ¼ 1:07 m, Ps ¼ 230 300 N, Gap T ¼ 0:028 m, and Gap P ¼ 0:0055 m. 4.2. Finite element model The global bridge is discreted by the combination of the shell and beam elements, with the beam elements meshed on the corresponding lines of the shell elements. The nonlinear connectors between the modules are modeled by the tension-only and compression-only truss element with initial gap, and the hydrostatic forces and the y B 1 2

3

4

5

6

7

Ps

8

9

10

11

12

13

14

15

16

x

Lunit 15 × L unit

(a)

z

5

4

z

Dep x (b)

− Hinge

−−− Compression - only Connector

y −− Tension - only Connector

Fig. 3. General view of global floating bridge: (a) general arrangement (b) illustration of connection.

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Fig. 4. Finite element model of global floating bridge.

Fig. 5. Finite element model of bridge module.

Fig. 6. Super-element model of bridge module.

inertial effect matrices are simulated by the linear spring elements and point mass elements respectively. Figs. 4 and 5 illustrate the finite element models of the floating bridge. In the finite element model for a certain module, the nodes located on the interface between the Emove, the elements to be subjected to moving loads, and other elements, as well as the nodes between the different modules and those the inertial and hydrostatic forces are applied to, are selected to be master nodes; whereas the remaining slave ones. Condensation can be carried out to the whole module except for the Emove, and the super elements of the module are obtained, as shown in Figs. 6 and 7.

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Fig. 7. Super-element model of the global floating bridge.

Table 1 Modes of a single module and the entire floating bridge Description of mode shape

Single module Heave Roll Pitch The global floating bridge First-order vertical bending Second-order vertical bending Third-order vertical bending First-order torsion Fourth-order vertical bending First-order horizontal bending + torsion Fifth-order vertical bending Second-order torsion Sixth-order vertical bending Seventh-order vertical bending Third-order torsion Eighth-order vertical bending Second-order horizontal bending + torsion Ninth-order vertical bending

Full finite element model (Hz)

Super-element model (Hz)

Errors (%)

0.355 0.353 0.354

0.355 0.353 0.354

0.0 0.0 0.0

0.362 0.410 0.535 0.726 0.749 0.882 1.040 1.354 1.395 1.800 1.968 2.241 2.359

0.357 0.390 0.498 0.609 0.701 0.767 0.988 1.343 1.346 1.759 1.932 2.216 2.019

1.4 4.9 6.9 16.1 6.4 13.0 5.0 0.8 3.5 2.3 1.8 1.1 14.4

2.702

2.700

0.1

4.3. Modal analyses The modes of a single module and the whole floating bridge, calculated with full finite element and super-element models, are listed in Table 1, and the first six order flexible mode shapes of the whole bridge are shown in Fig. 8, where all the nonlinearities are neglected. As seen in Table 1, for the vertical bending modes, based on the precise simulation of the inertia effects among the master DOFs along the vertical direction the natural frequencies, as well corresponding vibration shapes predicted by full finite element models and super-element models have a good agreement, which implies that the super-element models can describe the vertical direction stiffness and mass properties of the floating bridge well. However, for the horizontal bending and torsion modes, there exist some errors between the two

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Fig. 8. Mode shapes of the entire floating bridge: (a) first-order vertical bending; (b) second-order vertical bending; (c) third-order vertical bending; (d) first-order torsion; (e) fourth-order vertical bending; (f) firstorder horizontal bending + torsion; (g) fifth-order vertical bending; (h) second-order torsion; (i) secondorder horizontal bending + torsion.

models, which may come from the neglect of inertia effects between master and master DOFs/ master and slave DOFs in horizontal and all rotational directions. the vertical bending modes will be the principal modes excited by the moving loads. By balancing the cost of calculation and simulation errors, the super element method is regarded to be able to simulate the stiffness and mass properties of the floating bridge.

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0.15

Vertical Displacement (m)

0.10 0.05 0.00 -0.05 -0.10 -0.15 -0.20 -0.25

Linearly Connected GapP=0.0055m GapP=0.0m Experiment

-0.30 -0.35 -0.40 0

1

2

3

4

5

6

(a)

7

8

9

10

11

12

13

14

15

16

17

Position Number 0.15

Vertical Displacement (m)

0.10 0.05 0.00 -0.05 -0.10 -0.15 -0.20

Linearly Connected GapT=0.028m GapT=0.005m GapT=0.0m Experiment

-0.25 -0.30 -0.35 -0.40 0

1

2

(b)

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

Position Number

Fig. 9. Vertical displacement of the floating bridge subjected to static loads with: (a) different Gap P, (b) different Gap T.

4.4. Static analyses Fig. 9 shows the vertical displacement of the floating bridge with static loads at position number 7. From the figure, we can find that, neglecting all the nonlinearities of the connectors results in a smooth deflection curve with maximum displacement 14.1 cm, and the influential length is 74 m. For this case, the full-scale test has been carried out and the experimental results are 21.0 cm and 47 m, respectively [29]. It shows unacceptable errors existing in the simplifications. When the initial gap and nonlinearities of the tension-only and compression-only connectors are taken into account, the deflection shape of the floating bridge differs from the linearly connected one. The upward displacement occurs at two ends of influential lengths, and the maximum calculated and experimental displacements are in a good agreement. While only the nonlinearities are considered, the static deflection shape is different from the one with the initial gap effect: although the load influential length is the same, the maximum vertical displacement exists larger errors and the upward

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displacement changed apparently. Comparing (a) and (b) of Fig. 9, one can find that the maximum static deflection of the floating bridge is more sensitive to the initial gap of the compression-only connectors than that of the tension-only ones. Based on the above comparison, one can draw the conclusion that both the initial gap and nonlinearities of the connectors have a considerable effect on the nonlinearly connected floating bridge subjected to static loads. 4.5. Dynamic response analyses From experiments [29], the time histories for the vertical displacements of the position 7 of the full-scale floating bridge subjected to moving loads PV ¼ 132 300 N with speed V ¼ 6:09 m=s were obtained and are shown in Fig. 10. With the same parameters, a numerical analysis was carried out and the results are shown in Fig. 11. From Figs. 10 and 11 one can find that when moving loads passing the dynamic response of position 7 has a time delay occurred in the abscissa due to the different damping properties between the physical and mathematical models. Similarly, the difference between gap and Gap T may lead to the upward displacement of the experiment being greater than that of the simulation. However, both the time histories characters and the maximum values of the vertical

Vertical Displacement (m)

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6.10s

0.05 0.00 -0.05 -0.10 -0.15 -0.20

Fig. 10. Time histories of vertical displacement at position 7 obtained from experiments.

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Fig. 11. Time histories of vertical displacement at position 7 obtained from numerical simulation.

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Fu Shixiao et al. / Marine Structures 18 (2005) 85–107 0.20 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 -0.20 -0.25 -0.30 0

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Fig. 12. Vertical dynamic responses of the floating bridge with constant initial gap of tension-only connectors subjected to uniform moving loads with velocity: (a) 3 m/s, (b) 7 m/s, (c) 10 m/s, (d) 15 m/s.

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Fig. 13. Vertical dynamic responses of the floating bridge with constant initial gap of compression-only connectors subjected to uniform moving loads with velocity: (a) 3 m/s, (b) 7 m/s, (c) 10 m/s, (d) 15 m/s.

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displacements obtained experimentally and numerically have a good agreement with each other. Figs. 12 and 13 show the vertical displacements responses history at position 7 with constant initial gap of tension-only and compression-only connectors, respectively, subjected to moving load PV ¼ 230 300 N with different velocity. As we can see from Figs. 12 and 13, the magnitude of the hydroelastic response hump for the floating bridge subjected to moving loads is almost the same with that in the former static analysis; when the loads arrive or depart the scope of a certain point the corresponding response is of symmetry about the arrival time, which implies the dynamic effect is not apparent and the resulting phenomenon has a good agreement with the experiment [29]. However, with the increase of the moving speed, the response is no longer symmetric and the dynamic response hump rises, which indicates the dynamic effect of the floating bridge matters and is not negligible, for instance, when the speed reaches 15 m/s, the hump becomes 1.5 times of that with the speed of 3 m/s. Furthermore, as the load arrives or departs within the scope of the certain point the upward dynamic displacements may be generated, while beyond the scope the oscillation with declined amplitude at the equilibrium position may occur, and both of which go up with the increase of the moving speed. The dynamic characteristics of the linearly connected floating bridge are greatly different from the nonlinearly connected one: the linearly one has much smaller dynamic displacement response, the longer period and smaller amplitude of the free declined vibration than the nonlinearly one. The magnitude of the gap also plays an important role: the increase of the gap in the compression-only connectors will lead to the increase of the dynamic displacement amplitude, the upward displacements and amplitude of the declined vibration; while for the tension-only ones, the increase only results in the increase of the upward displacements. Figs. 14 and 15 show the dynamic connection forces response history in the connector of position number 7 with constant initial gap of tension-only and compression-only connectors respectively, subjected to moving loads PV ¼ 230 300 N with different velocities. The negative value in the figures means the connectors is under the state of compression, while the positive is for the tension state. As depicted in Figs. 14 and 15, the amplitude of the connection forces for the nonlinearly connected floating bridge increases with the moving vehicles velocity increasing. Furthermore, the compression state of the connectors retains constant within the scope of the influential length, while the magnitude changes with the load position. When the moving loads have passed the scope of the influential length around the position, the connectors will endure the impact forces produced by the declined vibration of the modules of the floating bridge, which is much smaller than the maximum connection forces that moving loads ever caused. It is of great significance to take the initial gaps into account by considering the fact that the amplitude of the connection forces decreases with the initial gap of compression-only connectors increase under a low vehicle speed and increases under a high speed, as always increases with the initial gap of tension-only connectors increasing, which shows that the dynamic effect is remarkable and agrees with the aforementioned

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Fig. 14. Dynamic response history of the connection forces in the connector of position number 7 with constant initial gap of tension-only connectors subjected to uniform moving loads with velocity: (a) 3 m/s, (b) 7 m/s, (c) 10 m/s, (d) 15 m/s.

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Fig. 15. Dynamic response history of the connection forces in the connector of position number 7 with constant initial gap of compression-only connectors subjected to uniform moving loads with velocity: (a) 3 m/s, (b) 7 m/s, (c) 10 m/s, (d) 15 m/s.

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displacement response analysis. Comparing with the linear and nonlinear connectors, we can find that the existence of the initial gaps in the connectors can influence the state of the connectors. From Fig. 14, we can find that, for the floating bridge with constant larger initial gap of tension-only connectors, the tops are mainly subjected to compression, and tension only when the loads move fast; while the linearly connected one, the connectors are always under the conditions of tension or compression (compression only within the load scope and tension only beyond that scope). As shown in Fig. 15, the decrease of initial gap of tension-only connectors changed the compression state of the connectors to tension states after the loads moving out the influential scope of the connector.

5. Summary and conclusions A governing equation of three-dimensional dynamic response analyses for the nonlinear connected floating bridge subjected to moving loads has been presented. The nonlinear motion equations have been condensed by the super-element method, and solved with direct integration and iteration method. The static and dynamic characters of the floating bridge with nonlinear connectors have been studied, and the following conclusions can be drawn: (1) Super-element method can precisely describe the features of the stiffness and mass distribution for the floating bridge. (2) The properties of the nonlinear connectors can be well and truly simulated by the nonlinear truss element with initial gap. (3) Nonlinearities and initial gap of the connectors must be taken into account for the nonlinear connected floating bridge since they have an obvious impact on the static deflection, dynamic displacement responses and dynamic connection forces of the bridge.

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