Ribbon bridge in waves based on hydroelasticity theory

Abstract

For the design and operation of a floating bridge, the understanding of its hydroelastic behavior in waves is of great importance. This paper investigated the hydroelastic performances of a ribbon bridge under wave action. A brief introduction on the estimation of dynamic responses of the floating bridge and the comparisons between the experiments and estimation were presented. Based on the 3D hydroelasticity theory, the hydroelastic behavior of the ribbon bridge modeled by finite element method (FEM) was analyzed by employing the mode superposition method. And the relevant comparisons between the numerical results and experimental data obtained from one tenth scale elastic model test in the ocean basin were made. It is found that the present method is applicable and adaptable for predicting the hydroelastic response of the floating bridge in waves.

Keywords

hydroelasticity / ribbon bridge / wave / finite element method (FEM)

Introduction

The floating bridge, as one of the important large floating structures, plays a significant role in national defense and civil engineering. However, with regards to the mobility and complexity of wind, wave and current that the floating bridge may encounter, the investigation into the dynamic response of floating bridges subjected to those loads is full of challenges; especially, the flexible deformation of a floating bridge due to its slender configuration can remarkably affect the dynamic response. Accordingly, the hydroelasticity theory, which links the interaction force and the solution to the fluid-structure coupled equations, must be employed.

The USA is the first country to conduct analyses and experiments on the dynamic response of floating bridges. Mukherji [1] studied the dynamic behavior of a continuous floating bridge under wave action in frequency and time domain by employing the beam theory, and compared it with the experimental results. On the basis of a 3D source distribution method, Seif et al. [2] simulated the hydrodynamic loads and derived the equations of motions taking into account all of the static and dynamic couplings from different parts of the bridge. Meanwhile, they investigated the effects of wave parameters such as height, period and direction on the behavior of the bridge. Ueda et al. [3-5] reported the hydroelastic response of floating bridge in frequency domain, in which all of the structures have been discretized by 3D finite element method (FEM), and the fluid effect was determined by the solution to the 3D water wave problem using boundary element method (BEM), and combined the free surface, water depth, hull vibrations (the motions of the floating units) and the interaction among the floating units within the framework of linear theory.

Based on the results above [6-8], the paper mainly studied the hydroelastic response of ribbon bridges in waves. Considering the feasibility and accuracy of the analysis, every bridge raft was modeled by 3D FEM and each was connected using linear element. Consequently, the 3D hydroelasticity theory is applied to the whole 15 bridge rafts to solve the dynamic response of ribbon bridges under wave action. Finally, the comparison of the results with the experimental data [9] is demonstrated.

Basic theory

Equations of flexible structure containing rigid motions

The equation of motion for a multiple degree-of-freedom structural system can be represented as

(1)\mathit{M}\ddot{\mathit{U}}+\mathit{C}\dot{\mathit{U}}+\mathit{K}\mathit{U}=\mathit{P}+\mathit{F}+\mathit{G},

where \mathit{M}, \mathit{C} and \mathit{K} are the overall mass, damping and stiffness matrices of the structure, respectively; \mathit{U} is the nodal displacement vector for the global structure; \mathit{P}, \mathit{F} and \mathit{G} are the equivalent vectors of the structural distributed, concentrated and body forces, respectively.

Neglecting all the external forces and damping, Eq. (1) is simplified as the free vibration form and can be written as

(2)\mathit{M}\ddot{\mathit{U}}+\mathit{K}\mathit{U}=\mathbf{0}.

Suppose Eq. (2) has a harmonic solution as

(3)\mathit{U}=\mathit{D}{\mathrm{e}}^{\mathrm{i}\omega t}.

Hence,

(4)(-{\omega }^{\mathrm{2}}\mathit{M}+\mathit{K})\mathit{D}=\mathbf{0}.

Obviously, it is a generalized eigenvalue problem. Regarding the constrained structure, \mathit{M} and \mathit{K} are the positive definite matrices, and Eq. (2) has a series of positive real roots {\omega }_r(r=\mathrm{1},\mathrm{2},\mathrm{...},n) defined as the r-th natural frequency and the corresponding eigenvector {\mathit{D}}_r indicated as the r-th natural mode:

(5){\mathit{D}}_r={[{\mathit{D}}_{r\mathrm{1}},{\mathit{D}}_{r\mathrm{2}},\mathrm{...},{\mathit{D}}_{rj},\mathrm{...},{\mathit{D}}_{rn}]}^{\mathrm{T}},

where n denotes the total n nodes of the structure. However, each node has six degrees of freedom: translations in the nodal x, y and z directions and rotations about x, y and z axes, then

(6){\mathit{D}}_{rj}=[u_r,v_r,w_r,{\theta }_{xr},{\theta }_{yr},{\theta }_{zr}],

where j represents the j-th node of the structure.

As for the partly constrained structure, i.e. the body can have rigid motions in certain directions, the stiffness matrix \mathit{K} is positive semidefinite, Eq. (2) has zero roots corresponding to the mode of rigid motion; while for the unconstrained free body, it has six zero roots corresponding to the six-DOF rigid motion.

In order to accord with the seakeeping theory, the rigid mode is described by the three translational components u_G, v_G, w_G and three rotational ones {\theta }_{xG}, {\theta }_{yG}, {\theta }_{zG} about the mass center in the global coordinate system coincident with the equilibrium one. Thus, the first six rigid modes of the j-th node of the free floating body can be expressed by

(7)\{\begin{array}{l}{\mathit{D}}_{\mathrm{1}j}={\left[\begin{array}{llllll}\mathrm{1},& \mathrm{0},& \mathrm{0},& \mathrm{0},& \mathrm{0},& \mathrm{0}\end{array}\right]}^{\mathrm{T}},\hfill \\ {\mathit{D}}_{\mathrm{2}j}={\left[\begin{array}{llllll}\mathrm{0},& \mathrm{1},& \mathrm{0},& \mathrm{0},& \mathrm{0},& \mathrm{0}\end{array}\right]}^{\mathrm{T}},\hfill \\ {\mathit{D}}_{\mathrm{3}j}={\left[\begin{array}{llllll}\mathrm{0},& \mathrm{0},& \mathrm{1},& \mathrm{0},& \mathrm{0},& \mathrm{0}\end{array}\right]}^{\mathrm{T}},\hfill \\ {\mathit{D}}_{\mathrm{4}j}={\left[\begin{array}{llllll}\mathrm{0},& -(z_j-z_G),& y_j-y_G,& \mathrm{1},& \mathrm{0},& \mathrm{0}\end{array}\right]}^{\mathrm{T}},\hfill \\ {\mathit{D}}_{\mathrm{5}j}={\left[\begin{array}{llllll}{z}_j-z_G,& \mathrm{0},& -(x_j-x_G),& \mathrm{0},& \mathrm{1},& \mathrm{0}\end{array}\right]}^{\mathrm{T}},\hfill \\ {\mathit{D}}_{\mathrm{6}j}={\left[-\begin{array}{llllll}(y_j-y_G),& x_j-x_G,& \mathrm{0},& \mathrm{0},& \mathrm{0},& \mathrm{1}\end{array}\right]}^{\mathrm{T}}.\hfill \end{array}

It corresponds to the six rigid motions: surge, sway, heave, roll, pitch and roll, in which (x_j,y_j,z_j) and (x_G,y_G,z_G) are the coordinates of the j-th node and those of mass center, respectively. Accordingly, the first six rigid displacement modes of any point (x,y,z) can be obtained

(8)\{\begin{array}{l}{\mathit{u}}_{\mathrm{1}}={\left[\begin{array}{lll}\mathrm{1},& \mathrm{0},& \mathrm{0}\end{array}\right]}^{\mathrm{T}},\hfill \\ {\mathit{u}}_{\mathrm{2}}={\left[\begin{array}{lll}\mathrm{0},& \mathrm{1},& \mathrm{0}\end{array}\right]}^{\mathrm{T}},\hfill \\ {\mathit{u}}_{\mathrm{3}}={\left[\begin{array}{lll}\mathrm{0},& \mathrm{0},& \mathrm{1}\end{array}\right]}^{\mathrm{T}},\hfill \\ {\mathit{u}}_{\mathrm{4}}={\left[\begin{array}{lll}\mathrm{0},& -(z-z_G),& y-y_G\end{array}\right]}^{\mathrm{T}},\hfill \\ {\mathit{u}}_{\mathrm{5}}={\left[\begin{array}{lll}z-z_G,& \mathrm{0},& -(x-x_G)\end{array}\right]}^{\mathrm{T}},\hfill \\ {\mathit{u}}_{\mathrm{6}}={\left[\begin{array}{lll}-(y-y_G),& x-x_G,& \mathrm{0}\end{array}\right]}^{\mathrm{T}}.\hfill \end{array}

Based on the linear mode superposition method, the real structural displacement \mathit{U} of each node can be precisely calculated by the accumulation of all modes. However, from the engineering point of view, in case of being subjected to external excitation, the first several oscillating modes usually dominate the structural dynamic response. In this way, it is assumed that the nodal displacement be superposed by the first m modes, then

(9)\mathit{U}={\displaystyle \sum _{r=\mathrm{1}}^m{\mathit{D}}_r{p}_r(t)},

where p_r(t) implies the r-th principal coordinate. When r=1-6, {\mathit{D}}_r represents the vector of the first six rigid modes, and p_r(t) the value of rigid displacement about mass center (x_G,y_G,z_G). Equation (9) can be rewritten as

(10)\mathit{U}=\mathit{D}\mathit{p}.

Substituting Eq. (10) into Eq. (1) and multiplying by {\mathit{D}}^{\mathrm{T}}, the generalized equation of motion can be given as follows:

(11)\mathit{a}\ddot{\mathit{p}}+\mathit{b}\dot{\mathit{p}}+\mathit{c}\mathit{p}=\mathit{Z}+\mathit{R}+\mathit{Q}

with

(12)\{\begin{array}{l}\mathit{a}={\mathrm{D}}^{\mathrm{T}}\mathit{M}\mathit{D},\hfill \\ \mathit{b}={\mathrm{D}}^{\mathrm{T}}\mathit{C}\mathit{D},\hfill \\ \mathit{c}={\mathrm{D}}^{\mathrm{T}}\mathit{K}\mathit{D},\hfill \\ \mathit{Z}={\mathrm{D}}^{\mathrm{T}}\mathit{P}=[Z_{\mathrm{1}},Z_{\mathrm{2}},\cdots Z_m],\hfill \\ \mathit{R}={\mathrm{D}}^{\mathrm{T}}\mathit{F}=[R_{\mathrm{1}},R_{\mathrm{2}},\cdots R_m],\hfill \\ \mathit{Q}={\mathrm{D}}^{\mathrm{T}}\mathit{G}=[Q_{\mathrm{1}},Q_{\mathrm{2}},\cdots Q_m],\hfill \end{array}

where \mathit{a}, \mathit{b} and \mathit{c} are the generalized mass, damping and stiffness matrices respectively; \mathit{Z}, \mathit{R} and \mathit{Q} are the distributed, concentrated and body forces respectively and can be further expressed as

(13)\{\begin{array}{l}{Z}_r=-{\displaystyle \underset{S}{\iint }{\mathit{n}}^{\mathrm{T}}{\mathit{u}}_{r}P\mathrm{d}S,}\hfill \\ R_r={\displaystyle \sum _i{\mathit{f}}_i^{\mathrm{T}}{\mathit{u}}_r},\hfill \\ Q_r=-{\displaystyle \underset{\mathrm{\Omega }}{\iiint }{\rho }_{\mathrm{b}}g{w}_r\mathrm{d}\mathrm{\Omega }},\hfill \end{array}

in which P is the pressure on the wetted surface S, {\rho }_{\mathrm{b}} the density of the structure, and w_r is the structural vertical displacement of the r-th mode.

For the practical structural dynamic problems, various linear and nonlinear damping are usually simplified as viscous damping, which is comparatively easy to deal with mathematically. A popular spectral damping scheme, called Rayleigh or proportional damping, is often used to form the damping matrix as a linear combination of the stiffness and mass matrices of the structure, that is

(14)\mathit{C}=\alpha \mathit{K}+\beta \mathit{M},

where \alpha and \beta are called, respectively, the stiffness and mass proportional damping constants, which can be associated with the fraction of critical damping \xi as

(15)\xi =\mathrm{0.5}(\alpha \omega +\beta /\omega ).

Therefore, \alpha and \beta can be determined by choosing the fractions of critical damping ({\xi }_{\mathrm{1}} and {\xi }_{\mathrm{2}}) at two different frequencies ({\omega }_{\mathrm{1}} and {\omega }_{\mathrm{2}}), and can be solved by the following equations:

(16)\{\begin{array}{l}\alpha =\mathrm{2}({\xi }_{\mathrm{2}}{\omega }_{\mathrm{2}}-{\xi }_{\mathrm{1}}{\omega }_{\mathrm{1}})/({\omega }_{\mathrm{2}}^{\mathrm{2}}-{\omega }_{\mathrm{1}}^{\mathrm{2}}),\hfill \\ \beta =\mathrm{2}{\omega }_{\mathrm{1}}{\omega }_{\mathrm{2}}({\xi }_{\mathrm{1}}{\omega }_{\mathrm{2}}-{\xi }_{\mathrm{2}}{\omega }_{\mathrm{1}})/({\omega }_{\mathrm{2}}^{\mathrm{2}}-{\omega }_{\mathrm{1}}^{\mathrm{2}}).\hfill \end{array}

The damping factor \alpha applied to the stiffness matrix \mathit{K} increases with increasing frequency, whereas \beta applied to the mass matrix \mathit{M} increases with decreasing frequency. In this paper, the Rayleigh damping factors \alpha and \beta were 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 \xi is chosen as 5%.

So Eq. (1), which is dependent on the nodal displacement vector \mathit{U}, is replaced by Eq. (11) which is dependent on the principal coordinate \mathit{P}, which decreases the scale of the motion equation and facilitates the numerical analysis. Finally, the dynamic response of the discretized structure can be calculated by solving Eq. (11) to obtain the principal coordinate value of each mode.

Equations of 3D linear hydroelasticity

When in the process of the motion and deformation of floating body, the fluid pressure can be evaluated by the Lagrangian integration on the mean structural wetted surface \overline{S} in the equilibrium coordinate system:

(17)P{(x,y,z,t)|}_{\overline{S}}=-\rho {\left[{\displaystyle \frac{\partial \varphi }{\partial t}}+{\displaystyle \frac{{(\nabla \varphi )}^{\mathrm{2}}}{\mathrm{2}}}+gz\right]}_{\overline{S}}.

However, the motion of the structure can affect the transformation from the mean wetted surface \overline{S} to the transient position S(t), which cause the difference of the pressure. Based on the theory of perturbation, the pressure can be expanded to the vicinity of \overline{S}:

(18)P{(x,y,z,t)|}_{S(t)}=\left[\mathrm{1}+\stackrel{\rightharpoonup }{u}\cdot \nabla +{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left(\stackrel{\rightharpoonup }{u}\cdot \nabla \right)\left(\stackrel{\rightharpoonup }{u}\cdot \nabla \right)+\cdots \right]P{(x,y,z,t)|}_{\overline{S}}.

Assume the rotation and unsteady velocity potential are comparatively small and neglect the second- and high-order part, then

(19)P{\left|{}_{S(t)}=-\rho \left\{{\displaystyle \frac{\partial \varphi }{\partial t}}+g{z}^{\prime }+gw\right\}\right|}_{\overline{S}},

where \rho, z^{\prime } and w are the density of fluid, the coordinate values along z^{\prime } axis in the body fixed reference frame, and structural vertical displacement, respectively.

Adopt the linear superposition approach [8], then

(20)\begin{array}{ll}{\displaystyle \sum _{k=\mathrm{1}}^m{a}_{rk}}{\ddot{p}}_k+{\displaystyle \sum _{k=\mathrm{1}}^m{b}_{rk}}{\dot{p}}_k+{\omega }_r^{\mathrm{2}}{a}_{rr}{p}_r=F_r+E_r+R_r+{\overline{R}}_r+{\mathit{\Delta }}_r,& r=\mathrm{1},\mathrm{2},\mathrm{...},m,\end{array}

where F_r, E_r, R_r, {\overline{R}}_r and {\mathit{\Delta }}_r are indicated as the r-th generalized excitation forces, generalized radiation forces, generalized restoring forces, static forces and generalized concentrated forces respectively.

Referring to the generalized restoring force, it takes the form of

(21)R_r=-{\displaystyle \sum _{k=\mathrm{1}}^m{p}_k{C}_{rk}}

with

(22)C_{rk}=-\rho {\displaystyle \underset{\overline{S}}{\iint }\mathit{n}{\mathit{n}}_r}(g{w}_k)\mathrm{d}S,

where C_{rk} denotes the generalized restoring coefficients. When r\le \mathrm{6} and k\le \mathrm{6}, it is reduced to the restoring coefficient matrix of rigid motion.

Similarly, the other generalized coefficients can also be obtained [10]. Taking all the above generalized coefficients into Eq. (11) yields the expression of the linear generalized hydroelastic motion equation:

(23)(\mathit{a}+\mathit{A})\ddot{\mathit{p}}+(\mathit{b}+\mathit{B})\dot{\mathit{p}}+(\mathit{c}+\mathit{C})\mathit{p}=\mathit{F}+\overline{\mathit{R}}+\mathit{\Delta },

where \mathit{a}, \mathit{b} and \mathit{c} are the generalized mass, damping and stiffness matrices of the floating structure respectively; \mathit{A}, \mathit{B} and \mathit{C} are the generalized added mass, added damping and restoring coefficient matrices of the fluid respectively; \mathit{F}, \overline{\mathit{R}} and \mathit{\Delta } are the generalized wave exciting force vectors, the generalized static force vectors, and the generalized concentrated force vectors, respectively.

Numerical model and discussion

Finite element model (FEM)

As shown in Fig.1, the ribbon bridge is composed of 15 rafts, with the translations of the two ends in the y and z directions of each pair of adjacent rafts being coupled together and Nos. 1-16 are the position numbers referred to the two ends of each module. The global bridge is discretized by the combination of shell and beam elements, with the beam elements meshed on the corresponding lines of the shell elements. The connectors between the rafts are modeled by the 3D spar element which is a uniaxial tension-compression element with three translations in the nodal x, y and z directions. The main dimensions of each raft are: A=0.001 m², E=2.0×10⁵ MPa, L_uint=6.7 m, B=8.082 m, D=0.74 m. Figure 2 illustrates the FEM of the ribbon bridge.

Since the vertical grids of wetted elements have little influence on the dynamic response of the floating structure [7], the vertical wetted elements are not taken into consideration, which can reduce the scale of the numerical calculation. Furthermore, to satisfy the requirements for the hydroelastic analysis, the numbering of the nodes of the wetted elements needs to be còntinuous [8].

Hydroelastic analysis

Figures 3-5 show the distribution features of the vertical displacement response along the centerline of the ribbon bridge and the corresponding experimental data in different waves. As for the different waves, the longitudinal, transversal and oblique waves are defined when the incident wave angle is equal to 0, 90 degrees, and 30 degrees, respectively.

From Figs. 3-5, it can be seen that the numerical results employing the 3D hydroelasticity theory have a comparatively good agreement with the experimental data both in the magnitude and distribution trend, which indicates that the theory is adaptable and accurate enough to predict the hydroelastic response of ribbon bridges.

In addition, for the same wavelength, the vertical displacement response along the centerline of ribbon bridges increases with the increase of the incident wave angle, i.e. from the longitudinal wave loads to the oblique wave loads till the transversal wave loads. Also regarding the same wave load condition, namely the same incident wave angle, the response amplitude goes up with the increasing wavelength. Especially, when in longitudinal wave load condition, the amplitude of the ribbon bridge apparently has the oscillatory performance and definitely reaches the same maximum value in the fore and aft of the ribbon bridge while the ratio of wavelength to the length of the floating bridge comes to a certain critical value, approximately 0.3.

Furthermore, the wide difference between the calculated results and experimental ones is made with the increase of the wavelength, which indicates that the accuracy of the prediction method decreases and the numerical error increases when the above-mentioned ratio rises. From the numerical analysis, this phenomenon can be explained from the following aspects: first, in the finite element analysis procedure, all the connectors of the ribbon bridge have been simplified as the 3D linear spar element without taking the nonlinear feature into consideration, which may bring the inaccurate information to the hydroelastic analysis and result in the final error. Second, when simulating the ribbon bridge, the gaps between each bay are ignored for the convenience of finite element modeling, which may lead to the increase of overall stiffness and the decrease of global flexibility. Third, during the wave experiment conducted in the lab, the whole ribbon bridge on water is connected with the ramp bay fixed on the bank of the tank, whereas it is neglected in the finite element analysis and the free boundary condition replaces the real boundary.

Concluding remarks

Based on the 3D hydroelasticity theory, an analysis of ribbon bridges under wave actions was presented in this paper. The proposed 3D hydroelastic method has a certain precision to predict the hydroelastic response; however, the nonlinear connectors, the gaps between each bay and the real boundary conditions are of significance in the analysis and much attention should be paid in future studies. Consequently, more satisfactory and precise results for the prediction of hydroelastic response amplitude of floating bridge can be obtained.

References

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