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the case of VLFS, the elastic modes can be considered as the manner as in Ma & Hirayama, 1997. In
addition, hydrodynamic force and wave exciting force are evaluated directly; the motions and
deflections can be obtained without many difficulties. The detail formulations are given in our
previous paper (Hirayama and Ma (1995a), (1 995b))
In head wave condition, deflection of the structure is mainly caused by vertical bending deformation,
thus, the structure can be treated as an elastic uniform beam with two free ends. Then, vertical
displacement is expressed as following equation.
1;1~1d~Z 8'2
+
+
kz
El
pAZ + - - - = q(x,t), (0 < x < L) (3)
w ax4 ax4
where p is the mass density, A is the sectional area, 11 is the structural damping coefficient, E1 is the
bending rigidity, k is spring coefficient of foundation. L is beam length. q(x, t) represents external load
acting at x coordinate. Over-dots denote differentiation with respect to time (t).
According to the principle of mode superposition, the deflection z can be represented as an aggregation
of the product of mode function Z, and principal coordinate p, as shown in Eqn. 2.
4x7 t) = 2 2, (4 Pr (t) (4)
*
r=l
In this study, we applied the analytic mode functions of beam for Z,. The motion equation of principal
coordinate of mode pr is un-coupled and could be expressed as following.
Where Q, brk, % are the generalized mass, structural damping and stiffness of mode, A&, Bt are the
generalized hydrodynamic added mass and damping coefficient respectively. C& is the generalized
hydrostatic restoring force coefficient. Finally, the displacement of beam can be obtained fiom Eqn. 4.
2.2 Steady Wave Drwt Forces

The steady drift forces in regular waves are the time averaged mean force due to second order pressure,
and the forces can be obtained fiom integration the pressure on wetted surface (near field method). On
the other hand, far field method is also available which is based on momentum theory (Maruo (1960)).
In that, the momentums of fluid motions in far field are considered in terms of Kochin functions.
However, the near field method provides straightforward way in solving the drift forces and the force
are resolved into each component. Thus, the near field method is applied in this study. By using the
perturbation approach, the final expression of horizontal drift force is given in Eqn.6 (Pinkster (1980)).
Here, the contribution of second order potentials and hydrostatic force due to second order
displacement are ignored.

In which, ii is normal vector of the point on wetted surface SO, 6, is relative wave elevation, x,
is the motion of CG in space-fixed coordinate system, +t is time differential of velocity potential.
T('),E(')are first order linear and rotational motion vectors respectively, which satisfied the following
relation.
p = jp +&I) )( X
g (7)
Where, X is position vector of the point on wetted surface.
Once the first order potentials and motions are determined, the drift force will be able to obtained
according to Eqn.6. For very large floating structure, the deformation due to elastic mode should be
included and the results of hydro-elasticity analysis fore-mentioned are utilized.