1.Introduction
Many countries have been faced with the land scarcity in the process of pier extension and Very Large Floating Structure (VLFS) has been utilized as one of the effective substitute instead of reclamation of sea spaces. As a pier of VLFS is exposed to external wave forces for long periods at sea, it is essential to analyze the response characteristics according to the wave loads. Dynamic response of VLFS due to the wave loads gives rise to the change of pressure of fluid and motion of structure. Such motion of structure comprising elastic deformation means hydroelasticity (Wang et al., 2008).
In order to solve the hydroelastic problem, many researchers use analytical/semianalytical approach and numerical approach. Voluminous papers related to the hydroelastic response of VLFS have been studied by authors at home and abroad. To analyze the problem of structure part and fluid part, modal method and direct method are commonly used in the frequency domain (Watanabe et al., 2004). We et al. (1995) treated linear twodimensional problem and extended the eigenfunction expansionmatching method using the modal expansions to analyze the waveinduced responses of VLFS. Takagi et al. (2000) proposed an antimotion device for VLFS and studied theoretically and experimentally using the eigenfunction expansion method. Kim and Ertkin (1998) also introduced this method for predicting linear hydroelastic behavior a shallowdraft VLFS. Hong et al. (2003) extended Kim and Ertkin (1998)’s eigenfunction expansion method to three dimensions considering the effect of nonzero draft. Because of simplicity and numerical efficiency, Hamamoto and Fujita (2002) used the wetmode approach to study hydroelastic response of VLFS with arbitrary shape.
In the direct mode, the response of structure is not represented through a superposition of the global modal responses and is determined by directly solving the equation of motion (Watanabe et al., 2004; Wang et al., 2008). This method uses full modes of structure and has a high accuracy for complicated distribution of stiffness (Kim et al., 2006). Yago et al. (1996) modified a direct method using pressure distribution method and compared with the experiment results of a zerodraft VLFS. Yasuzawa et al. (1997) developed a numerical code for dynamic response of mattype VLFS in regular waves using the direct method. Similar code was applied to container yard and marine pier of VLFS (Park et al., 2003; Lee, 2011).
In this paper, we suggest a pontoontype pier of VLFS with the length of 500m, breadth of 200m and height of 2m in Yeosu new port. In order to give foundational results for establishing the criteria of VLFS pier, several factors such as the wavelength, water depth, wave direction and flexural rigidity of structure are considered. For the purpose of high accuracy than efficiency, direct method is used to analyze the fluidstructure problem. Dynamic responses caused by the elastic deformation and rigid motion of the pier are analyzed by using numerical calculation. Fluid part is analyzed by using boundary element method (BEM) and structural part is calculated by using finite element method (FEM).
2.Background Theory
Analysis theory is based on the potential theory widely used in terms of the wave loads and estimation of responses. The fluid is assumed inviscid, incompressible and irrotational. Under this assumption, fluid velocity is obtained by derivative of spatial coordinates of velocity potential. Governing equation is the continuity equation and is written by Laplace equation of velocity potentials.
As shown in Fig. 1, we analyze the response characteristics of a floating pier in regular waves, which installed in infinite widespread sea with constant water depth. We formulate respectively the fluid part and structural part, and induce the coupled motion of equation. Analysis should be done under the following assumption (Park et al., 2003; Yasuzawa et al., 1997).

1) The fluid is inviscid, incompressible and the motion of fluid is irrotational and velocity potential is defined.

2) Draft is floating structure can be ignored.

3) Sea bottom is assumed to be flat and sea domain is infinitely extended.

4) Motion of fluid is governed by the linear vibration theory and with regard to deformation, only deflection is considered.

5) Damping effect of floating structure can be ignored.
2.1Analysis of fluid part
1)diffraction problem
Fluid part around VLFS is formulated using boundary element method (BEM) and fluid motion is represented using velocity potential. In the analysis model, components of fluid force are potentials by incident wave, diffraction wave and radiation wave. Total potential can be written :
where ω is the circular frequency of harmonic motion and φ_{i} , φ_{d}, φ_{r} are the incident, diffraction, radiation potentials, respectively. When the structure have no motion and deformation, we seek φ_{d} by diffraction wave. We assume the structure have no motion and deformation as a rigid body, pure boundary value problems of fluid field. As shown in Fig. 1, the boundary value problems can be represented :
where n is the outwarddirected normal to the fluid domain, g is the acceleration of gravity, k is the wave number and h is the water depth, Γ_{B} , Γ_{H} , Γ_{F} ,Γ_{∞} are boundaries of bottom, body surface, free surface and farfield, respectively.
Laplace equation (2) is governed equation in fluid domain Ω. Boundary condition (3) is represent in constant water depth h The equation (4) is the surface boundary condition in undersurface of structure, equation (5) is the linearized free surface condition, and equation (6) is the radiation condition indicating behavior in the farfield.
2)radiation problem
We regard structure as an elastic body and seek velocity potential φ_{r} generated by motion and deformation of structure. Boundary value problems can be made :
Above problems are same as the diffraction problems, excluding equation (9). Velocity potential φ_{r} can be obtained using boundary element method and integral equation.
3)boundary integral equation
Diffraction problem and radiation problem can be formulated by finite element method using Green function :
Applying the boundary conditions of (2)∼(11) in the equation (12), boundary integral equation can be derived :
4)Formulation by boundary element method
After dividing integral boundary aspect of (13) and (14) into boundary element, equation is formulated :
where M is the total number of boundary elements. Coordinates of elements and φ, $\frac{\partial \phi}{\partial n}$ are expressed using linear combination as follows :
where N_{k} (ξ_{1}, ξ_{2} ) is shape function. Substituting these into (15) gives
where N is the total nodal point (i = 1 ∼N) and φ_{jk} is the jth element and kth nodal velocity potential. Arranging these equation concerning corresponding relation of nodal point, matrix equation is derived :
Substituting each potentials into (20) gives
Equation (21) is the final equation of motion in the fluid part.
2.2Analysis of structural part
Structural part of VLFS is formulated using finite element method. The equation of motion is induced from the principle of virtual work. We regard finite elements as the rectangle plate elements generating bending deformation. Bending displacement of structure can be made :
where N_{w} is shape function and {ν}_{e} is the nodal point displacement vector. Because this finite element method is used bending vibration of general plate, we omit detailed theorem (Petyt, 2010).
When the finite element method is applied to the plate generating bending vibration and the equation of motion is induced by principle of virtual work, following equation can be obtained :
where [K] is the stiffness matrix of structure, [M] is the mass matrix of structure, ν is the displacement vector and {f} is the external force vector.
2.3Coupled motion equation of fluid and structural part
Coupled motion equation of vibration be induced by connecting the equation of fluid and structural part. Fluctuation pressure ΔP can be obtained by Bernoulli’s theorem :
where ρ_{f} is the density of fluid. Total matrix equation can be written :
Collecting equations (21) and (25), simultaneous equation is finally obtained :
Equation (26) is the final coupled equation of motion in the fluid part and structural part. By solving this equation, nodal displacement vector and nodal velocity potential can be obtained (Park et al., 2004; Petyt, 2010).
3.Numerical analysis of wave responses
Pontoontype VFLS typically has large horizontal dimensions whereas the height is several meters. Fig. 2 shows the concept of a pontoontype pier of VLFS with breakwater. In order to use as marine pier having a capacity for two cargo ships and small yachts, we suggest the VLFS with the length of 500m, breadth of 200m, and height of 2m in Yeosu new port.
We analyzed the wave responses of model A in Table 1. To investigate the response characteristics, this study examined the several factors such as the wavelength, water depth, wave direction and flexural rigidity of structure. We divided the model with 126 nodes and 100 elements. Wave response program is developed and improved by two researchers (Park and Park, 2000).
3.1Response characteristics according to the wavelength
We examined the effect of incident waves by changing the ratio of structure length and the wavelength. Fig. 3 shows the corelation between the response amplitude and 3 representative points on the centerline of structure (incident point, middle point, penetration point). The abscissa L/λ on Fig. 3 indicates the ratio of structure length and the wavelength, and the ordinate w/(H/2) represents the ratio of vertical displacement amplitude and wave amplitude. Bending amplitude w for L/λ have been divided by the half wavelength. Nondimensional distribution for the direction of structure length is represented on Fig. 3. The response changed at the point of L/λ=1.5. Due to the large incident wave force, the left side of L/λ=1.5, responses show large amplitude like sea wave. On the contrary, the right side of L/λ=1.5, the penetration side of structure indicates the decreasing curve and elastic responses.
3.2Response characteristics according to the water depth
In this part, we examined the responses according to the ratio of water depth and the wavelength. Shin et al. (2000) studied that the vertical displacement amplitude in the depth of 8m is smaller than the depth of 58.5m. Kyoung et al. (2005) investigated that the hydroelastic responses are influenced by the sea bottom topographies rather than the change of sea depths.
We calculated by changing h/λ and compared the results as shown in Fig. 4. Wave responses are not considerably influenced by the change of depths in case of L/λ=4.0 whereas the responses show increasing tendency as the depths increase at the incident point in case of L/λ=8.0. The results feature the elastic responses due to the influence of seabed and slight variation of vibration mode due to the wavelength.
3.3Response characteristics according to the direction of incident waves
This part deals with the response characteristics according to various incident wave direction. We analyzed the responses by changing the incident wave angles like 0 degree (a), 30 degree (b), 50 degree (c), 80 degree (d) in case of L/λ=4.0 waves.
Fig. 5 represents the torsional phenomenon according to the incident waves. The maximum point of displacement varies with the wave directions. It was found that the maximum point of response does not correspond with the incident wave angles.
3.4Response characteristics according to the Flexural rigidity of structure
This part analyzed the response characteristics by changing flexural rigidity of structure. Responses of model B are shown on Fig. 6 and D is the flexural rigidity of structure. The flexural rigidity (D) of (a) is 100 times larger than model A, (b) is 10 times smaller than model A, and (c) is 100 times smaller than model A. When the flexural rigidity of structure increases, peak point of L/λ moves left side and responses decrease at the large L/ λ ratio. On the contrary, peak point of L/λ moves right side in case the flexural rigidity of structure decreases. Peak point of vertical displacement amplitude moves from side to side according to flexural rigidity of structure. When the flexural rigidity of structure increases, elastic response is dominant. On the contrary, lower flexural rigidity of structure shows a form of riding waves despite decrease of the wavelength.
4.Conclusion
We analyzed the wave response characteristics of a suggested pontoontype pier of VLFS in Yesou new port using the direct method and obtained following results according to the several factors such as the wavelength, water depth, wave direction and flexural rigidity of structure.

(1) Wave response changed at the point of L/λ=1.5. Responses show large amplitude like sea wave on the left side of L/λ=1.5 and the penetration side of structure indicates the decreasing curve and elastic responses.

(2) The responses show increasing tendency as the depths increase at the incident point in case of L/λ=8.0. The results feature the elastic responses due to the influence of seabed and slight variation of vibration mode due to the wavelength.

(3) Torsional phenomenon are shown according to the various incident waves. The maximum point of displacement varies with the wave directions.

(4) Peak point of vertical displacement amplitude moves from side to side as the flexural rigidity of structure changes. Elastic response is dominant in case flexural rigidity increases whereas lower flexural rigidity of structure shows a form of riding waves despite decrease of the wavelength.
The effect of changing depths is debatable point because this study is investigated in the range of comparatively shallow depths considering local area of Yeosu new port. Also, future study is needed about overall safety evaluation of the pier of VLFS for actuality to the domestic port.