The hydroelastic behavior of very large floating structures (VLFSs) is investigated based on the proposed multimodules beam theory (MBT). To carry out the analysis, the VLFS is first divided into multiple submodules that are connected through their gravity center by a spatial beam with specific stiffness. The external force exerted on the submodules includes the wave hydrodynamic force as well as the beam bending force due to the relative displacements of different submodules. The wave hydrodynamic force is computed based on threedimensional potential theory. The beam bending force is expressed in the form of a stiffness matrix. The motion response defined at the gravity center of the submodules is solved by the multibody hydrodynamic control equations; then both the displacement and the structure bending moment of the VLFS are determined from the stiffness matrix equations. To account for the moving point mass effects, the proposed method is extended to the time domain based on impulse response function (IRF) theory. The method is verified by comparison with existing results. Detailed results through the displacement and bending moment of the VLFS are provided to show the influence of the number of the submodules and the influence of the moving point mass.
A very large floating structure (VFLS) is a unique type of oceanic structure that embraces a range of unprecedented parameters. VLFSs are designed primarily for floating airports and as a dock for calm waters on open seas.
Because it is larger than existing floating structures, a VLFS’s flexibility must be taken into account. Also, for a VLFS to be used as an airport or bridge, timevarying moving loads must be considered, for example, the analysis of the displacement and bending moment of a very large floating bridge in calm water with some moving point masses [
Simulations of the displacement and bending moment of a VLFS are mostly based on hydroelastic theory to date. In the frequency domain, hydroelastic theory was initially developed in two dimensions [
To account for the effects of moving point masses, simulations must be carried out in the time domain. Chakrabarti [
An approximation theory for the analysis of the displacement and bending moment of a VLFS, based on multimodules beam theory (MBT), was recently developed by Lu et al. [
In this paper, Lu’s methodology for the computation of the displacement along the VLFS has been improved, and some new methods have been developed for the computation of the section loads. We discard the beam bending method of Lu et al. [
The methodology has been extended to time domain analysis based on IRF theory, and this paper attempts to simulate the hydroelastic response of the VLFS in waves with some moving point mass. Study on the moving load of an elastic floating body under wave action is the foundation of research on landing pontoons, but there is little literature. This paper combines MBT (Lu, 2015), IRF theory [
Lu’s method is based on potential theory and multibody theory. For the hydrodynamic aspect, the ideal fluid assumption is adopted; that is, the fluid is inviscid, irrotational, and incompressible. The incident wave amplitude is assumed to be small relative to a characteristic wavelength and body dimension. For the structural aspect, a stiffness matrix is introduced between two adjacent modules to consider the flexible structure.
Based on the assumptions of an ideal fluid and linearity, the velocity potential can be decomposed into three parts as follows:
As shown in Figure
Definition of the fluid and structure boundaries.
In the frequency domain, the excitation forces are related to the incident and diffracted potential as follows:
The added mass and damping coefficient is given by
According to Lu’s method, a VLFS is divided into many modules, and the equation of sixdegreeoffreedom (DOF) motion in the frequency domain for a given wave amplitude and wave frequency
Further, considering all modules, (
As a linear system, the coefficient matrix of unknowns
The sixDOF motion of a module’s center is restricted by the deformation condition of the equivalent beam between two modules.
As Lu [
The relationship between
To illustrate this methodology, we use the model mentioned by Yokosuka [
Details of the VLFS model.
Parameters  Units  Value 

Length  L/m  300 
Width  B/m  60 
Depth  D/m  2 
Draft  d/m  0.5 
Young’s modulus  E/N/m^{2}  1.19 
Poisson ratio 

0.13 
Density 

256.25 
Water depth  H/m  58.5 
Lu (2015) [
It can be seen from (
Equation (
The end interface loads
To get the displacement
Lu’s method (Lu, 2015) can only calculate the vertical displacement response. Equation (
The present results are compared with experimental results [
Vertical response contrast of VLFS.
To obtain the moment of the VLFS, Lu [
Schematic diagram of the equilibrium forces exerted on a free beam.
Lu adopted this method and computed the bending moment along the VLFS, where eight submodules were used, and the numerical results were compared to the values calculated by the hydroelastic theory [
Moment distribution of the model.
To demonstrate the influence of the number of submodules, Figure
Moment distributions with different numbers of submodules.
It can be seen that there is a big difference between the bending moment distributions for five submodules and eight submodules. To overcome this problem, some improvements have been proposed in the following section, specifically to use fewer submodules to obtain high accuracy of the bending moment distribution.
In the multisubmodule beam theory of Lu (2015) [
However, the continuous distribution pressure has to be taken into account when solving for the displacement and the bending moment, as seen in the following equations:
In Lu’s original work, only the exact bending moment is used to perform the spline interpolation. However, it can be seen from (
Since the current method takes into account both the bending moment and the shear force when solving the bending moment distribution along the VLFS, the interpolation accuracy must be greater than that of the original method of Lu. Figure
Moment distributions computed by the highorder interpolation method.
Though the trend of the values computed by the HOI method is the same as for the spline interpolation scheme, there is still some difference in the absolute values of the bending moment distribution. Therefore, the displacementtodisplacement to moment (DDM) method is developed to improve the accuracy of the absolute values. Figure
Displacement distributions with different submodules.
Figure
Bending moment distribution with different interpolation schemes.
In the actual situation, considering the computational efficiency, fewer modules are used. The DDM method is useful to obtain highprecision calculation results.
To take into account the effects of moving mass points, the simulation must be carried out in the time domain. In terms of the convolution of body motions according to the Cummins equations [
The delay function and the added mass with infinite coefficient can be computed via the following equations:
Based on Lu’s theory, for the time domain hydroelastic simulations, (
If there exists some other force, for example, moving point mass, then the above equations should be modified as follows:
Section
Comparisons of the time domain values with frequency domain results.
To describe the position of the moving point mass, two coordinate systems are introduced in Figure
Demonstration of the absolute acceleration of the moving point mass.
If the relative velocity of the moving point mass with respect to the VLFS is taken as
For the linear case, (
Therefore, the external force exerted on the VLFS coming from the moving point mass can be computed by
Here, we use a nonrigorous method to study the hydroelastic response of the VLFS with a moving mass point.
Through the combination of time domain hydroelastic theory (
The main purpose of this paper is to discuss the applications of the proposed method for the computation of the hydroelastic response of the VLFS, with or without the moving point mass. Therefore, only one wave condition (wave length
Division of the total VLFS.
Figure
Bending moment and vertical response amplitude in midpoint.
Contour of the displacement of the VLFS.
Contour of the bending moment of the VLFS.
The hydrodynamic response of VLFS in different wave direction.
This paper first describes a new method, based on both multibody hydrodynamics and the EulerBernoulli assumption, to calculate the response of the VLFS. Then two methods, the HOI method and DDM method, are proposed to improve the accuracy of the computation of the bending moment distributions along the VLFS. Via Fourier transformations, the method has been extended to time domain analysis and then used to investigate the hydroelastic response of the VLFS with a moving point mass on it.
Some test cases have been carried out. All the results are compared with the values obtained by threedimensional hydroelastic theory and the model experiment [
The authors declare that they have no conflicts of interest.