The mooring force of a floating structure subjected to waves or seismic motion is usually determined by means of a numerical simulation. The equation of motions of a moored floating structure takes on a second-order differential equation with six degrees of freedom. The time-domain numer- ical simulation is done in consideration of forces from irregular waves, the second order wave force or so-called steady or fluctuating drift force, current force, fluctuating wind force, seismic force and the nonlinear load- deformation characteristics including hysteresis of mooring facilities (see Figure 5.3).
After the statistical analysis of the time-domain numerical simulation results, quantities such as the maximum value, the maximum amplitude, the significant value and the mean of motions, mooring forces, the response acceleration and velocity are calculated.
5.3.2 Numerical simulation method
The numerical simulation of a floating structure is a time-domain analysis of second-order differential equation described by the following equation
involving six degrees of freedom, namely surge, sway, heave, roll, pitch and yaw: 6 j=1 (Mij+ mij(∞)) ¨Xj(t) + 6 j=1 ! t −∞Lij(t − τ) ˙Xj(τ)dτ + Di(t) ' + 6 j=1 (Cij+ Gij)Xj(t) = Fi(t) (5.2) where Mij is the mass of floating structure; mij(∞) the added mass; Xj(t) the motion of floating structure at time t; Lij(t) the retardation function at time t; Di(t) the damping force due to mooring lines and viscosity at time t; Cij the restoring force coefficient; Gij the mooring force coefficient; Fi(t) the external forces at time t; i, j the mode of motions of floating structure (i, j= 1 to 6).
The retardation function and the added mass are calculated by using the following relations: Lij(t) =σ2 ∞ 0 Bij(σ ) cos σt dσ (5.3) mij(∞) = Aij(σ ) + 1 σ ∞ 0 Lij(t) sin σ t dt (5.4)
where σ is the angular frequency; Aij(σ ) the added mass at angular frequencyσ ; and Bij(σ ) the damping coefficient at angular frequency σ .
The motions of the floating structure are determined by the predom- inant angular frequency of external forces. At the natural-angular fre- quency, the frequency characteristics are related to the natural period of the moored floating structure. Therefore, the radiation forces are usually represented by the value at a significant wave period or at a natural period of motion.
When the spectrumof external forces is not wide band, motions of floating structure are calculated by the following equation:
6 j=1 (Mij+ Aij(σ0)) ¨Xj(t) + 6 j=1 ! t −∞Bij(σ0) ˙Xj(τ)dτ + Di(t) ' + 6 j=1 (Cij+ Gij)Xj(t) = Fi(t) (5.5) where the added mass and the damping coefficient for radiation forces are calculated at the angular frequencyσ0such as the predominant frequency of external forces or the natural frequency.
Design of station-keeping systems 103 5.3.3 Loads and forces act on a floating structure
Loads and external forces acting on a floating structure are the selfweight, buoyancy and external forces such as wave force, wind force, current force, seismic force, and so on. With the action of those loads and forces, motions of a floating structure are developed, the mooring system deformed and reaction forces generated. The load action is given on the right-hand side of Eq. (5.5). The motions of a floating structure and mooring forces of the station-keeping system are calculated by numerical simulations.
As loads such as wind force, wave force and seismic force are irregular and periodic, the frequency characteristics must be considered in the numerical simulation. Below, is an outline of the treatment of wind force, wave force and seismic force.
Wind force
Generally wind speed is given as the average wind speed. A wind speed varies with respect to time and space and the maximum instantaneous wind speed is usually larger than the average wind speed. The ratio of the maximum instantaneous wind speed and the average wind speed of a certain point is called the gust ratio.
Though it is appropriate that the frequency spectrumof wind shall be determined according to the observed data in a construction site, how- ever, if there is no observed data, the frequency spectrumof wind speed by Davenport (1967), Hino (1976) or any other type can be used in the numeri- cal simulations as an irregular and periodic wind speed. Equations (5.6) and (5.7) are the frequency spectrumof wind speed by Davenport and Hino, respectively, fSu(f ) = 4KrU102 X2 (1 + X2)4/3, X= 1200f U10 (5.6) Su(f ) = 2.856KrβU10 1+ f β 2−5/6 β = 1.169 × 10−3 U 10α √ Kr z 10 2mα−1 (5.7) Wind forces that are acting on a floating structure can be calculated by using the following equations:
RX= 12ρaU2ATCX (5.8)
RY= 12ρaU2ALCY (5.9)
where CX is the drag coefficient in the X direction (i.e. fromthe front of a floating structure); CY the drag coefficient in the Y direction (i.e. from the side of a floating structure); CM the pressure-moment coefficient about the center of gravity of a floating structure; RX the component of resul- tant wind force (kN) in the X direction; RY the component of resultant wind force (kN) in the Y direction RM the moment (kJ) of resultant wind force about the center of gravity of a floating structure; ρa the density of air= 1.23 × 10−3(t/m3); U the wind speed (m/s); A
T the front-projected area above the water surface (m2); A
L the side-projected area above the water surface (m2); and L the length of a floating structure (m). Note that the coefficients CX, CY, and CMare determined by a wind tunnel test. Wave force
The wave force is a force exerted by the incident waves on a floating struc- ture when the floating structure is considered to be fixed in the water. It is composed of a linear force that is proportional to the amplitude of the inci- dent waves and a nonlinear force that is proportional to the square of the amplitude of the incident waves. The linear force is the force that the floating structure receives fromthe incident waves as the reaction when it deforms the incident waves. It is expressed as the sumof the Froude–Krylov force and the diffracted wave force. As sea waves are irregular and periodic, the wave force is given as a time series of irregular and periodic value. For the numerical simulations, one may use the frequency spectrum by Bretshneider (1968) and Mituyasu (1970) given by:
S(f ) = 0.257H1/32T1/3(T1/3f)−5exp
−1.03(T1/3f)−4
(5.11) Where S(f ) is the frequency spectrum; f the frequency; H1/3 the signifi- cant wave height and T1/3is the significant wave period. The equation was originally proposed by Bretshneider and was modified by Mituyasu on the condition that there was the relation of Tp ∼= 1.0T1/3 between the peak period and significant wave period. Goda (1987) proposed Eq. (5.12) in correcting the relation between the peak period and significant wave period as Tp∼= 1.1T1/3according to observed data.
S(f ) = 0.205H1/32T1−4/3f−5exp
−0.75(T1/3f)−4
(5.12) Those wave spectrums of Eqs (5.11) and (5.12) may be applied when the wind field is stationary. However, JONSWAP spectrumis applied to wind waves which grow in rather short fetch-length under strong winds. Other types of frequency spectrummay also be used.
Radiation forces are induced accordingly to develop motions of a float- ing structure. The radiation force is divided into two components that are
Design of station-keeping systems 105
proportional to the acceleration and velocity. They are, respectively, treated as added mass and damping coefficient in the equation of motions.
The wave-drift force which is proportional to the square of the wave height must be considered when the length of a floating structure becomes equal or larger than the wavelength. By assuming that the floating structure is two- dimensional and the wave energy is not dissipated, the wave-drift force is given by: Fd =18ρwgHi2R R= KR2 1+ 4πh/L sinh(4πh/L) (5.13)
where Fd is the wave drift force per unit length (kN);ρw is the density of the seawater (kg/m3); H
i is the wave height of incident wave (m); KR is the ratio of reflection and R is the coefficient of wave drift force.
Seismic load
When a floating structure is located in a seismic region, the station keeping systemhas to be designed with consideration of seismic forces. In such places, the floating structure is subjected to seismic forces which are proportional to its acceleration and mass as well as the interaction force between itself and the station-keeping system.
The relative response acceleration of the floating structure is small because the natural period of a moored floating structure may be larger than the predominant period of the seismic motion. However, since the relative dis- placement with the mooring system is large, it is recommended that mooring systems with appropriate stiffness be selected with the condition of use in mind.
5.3.4 Numerical simulation and statistical analysis of results
Motions and mooring forces calculated by means of numerical simulations and results of the hydraulic experiment areofirregular and periodic value. Yet, the expected value of the variables is to be estimated in the duration of rough weather condition because generally the elapsed time of the numerical simulation or the hydraulic experiment is limited. Results from the computa- tional analysis and the hydraulic experiment shall be analyzed and presented in the form of time histories with the maximum values, the significant values, the average values, the frequency spectrum, the expected values, and so on. In consideration of both frequency characteristics of external forces and the response characteristics of the moored floating structure, it is
recommended that the time step of the numerical simulation be set at about 1/8 of the minimum period of the external forces or less. It is also recom- mended that the numerical simulation be performed as long as possible when more than one hundred effective amplitudes of motions and mooring forces are to be obtained in order to calculate the expected values accurately.