Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 6482527, 14 pages

https://doi.org/10.1155/2017/6482527

## Analysis of the Hydroelastic Performance of Very Large Floating Structures Based on Multimodules Beam Theory

^{1}PLA University of Science and Technology, Nanjing, China^{2}Wuxi First Scientific Research Institute, Wuxi, China^{3}College of Shipbuilding Engineering, Harbin Engineering University, Heilongjiang, China^{4}Center for Offshore Foundation Systems, School of Civil, Environmental and Mining Engineering, University of Western Australia, Perth, WA, Australia^{5}State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China

Correspondence should be addressed to Da Lu; moc.qq@8813375401

Received 30 December 2016; Revised 30 March 2017; Accepted 2 May 2017; Published 24 May 2017

Academic Editor: Yuri Vladimirovich Mikhlin

Copyright © 2017 Jin Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 three-dimensional 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.

#### 1. Introduction

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, time-varying 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 [1, 2] or the displacement response of a VLFS in waves with a moving load point [3, 4].

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 [5, 6]. Wu [7] extended the work of Bishop and Price to three dimensions, and Chen et al. (2003a) developed three-dimensional nonlinear hydroelastic theory. Hydroelastic theory can be also extended to the time domain [8, 9].

To account for the effects of moving point masses, simulations must be carried out in the time domain. Chakrabarti [10] and Jacobsen and Clauss [11] analyzed the hydrodynamic interactions of a multibody system based on the impulse response function (IRF) method, but the bodies are not connected to each other. Shen et al. [12] presented procedures for numerical solutions of system motion responses of a multi-rigid-body in the time domain. A moving load on a rigid body has also been studied [13–15].

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. [16]. In this method, the VLFS is divided into submodules that are connected by a beam with specific stiffness through the gravity center of the submodules. 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. Then, the final hydroelastic response equations can be established based on multibody hydrodynamic theory and beam theory. The displacement along the VLFS can be obtained by solving these equations, and then the structural deformation as well as the section loads can be computed.

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. [16] and calculate the displacement response of the structure by the method of matrix transformation. To obtain the complete bending moment distribution, the high-order difference is obtained for a few accurate moment values. This method is more rigorous, and the displacement and bending moment along the whole VLFS can be obtained. This paper avoids the resonance problem caused by small gap reasonably when calculating the hydrodynamic coefficient [17].

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 [18, 19], and the analysis method of moving load on rigid floating body. MBT and IRF are used in the time domain analysis of a VLFS; the floating body in the moving load is treated as a rigid body, and the inertia force introduced by the moving load is added to the wave force of this floating body. Although the method is very approximate, the results can reflect the response characteristics of the structure.

#### 2. Mathematical Formulation

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.

##### 2.1. Hydrodynamic Model

Based on the assumptions of an ideal fluid and linearity, the velocity potential can be decomposed into three parts as follows:where , , and denote, respectively, the incident wave potential, diffraction wave potential, and radiation wave potential. The incident, diffraction potential, and radiation potential satisfy the following boundary conditions:

As shown in Figure 1, is the fluid domain, and , , , and are the free surface, bottom surface, wetted body surface of the* k*th module, and boundary surface at infinity of the fluid, respectively. represents the outward-directed unit vector normal to the wetted surface of the* k*th module. is the speed on the wetted surface of the* k*th module. is the velocity potential. After the velocity potential is obtained, the added mass and the radiation damping of the bodies, as well as the wave excitation force, can be calculated.