Page 51 - Marine Structural Design

28 Part IStructural Design Principles

speed corrections and the lack of viscous effects. All these methods assume the ship to be rigid
beam. Bishop and Price (1979) developed a flexible beam strip theory that accounts for
bending and shear stiffness of the hull when solving for compatibility between strips. This
kind of theory can estimate the distortional higher frequency responses of a hull to slamming
and whipping excitation. However, it is still linear analysis and extreme response is not well
modeled.
2.3.2 Wave-Induced Forces
Jensen and Pedersen (1979) proposed a second order strip theory for hydro-elastic analysis in
fiequency-domain. Their theory is based on a perturbational expression of the hydrodynamic
and the hydrostatic coefficients around the still water line and includes the incident pressure
field from second order Stokes’ waves. The equation used to evaluate the forces acting on a
ship in such an analysis is similar to:
F(X,t)= FH(X>t)+FB(X,t) (2.23)
The procedure for actually working out the above equation is rather complicated due to the
non-linear nature of some of the parameters. The following explanation is only to give a basic
understanding of the parameters present in Eq. (2.23).
The right hand side of Eq. (2.23) consists of two parts. The second part is the buoyancy force
known as the Froude-Krylov buoyancy force:

(2.24)

where,
B = Breadth of the ship
Y = Distance along an axis starting from the bottom of the hull and moving
vertically upwards
v = Instantaneous vertical displacement of the hull

rl = Distance from the calm water surface to the local elevation of the ocean wave
X = Distance along an axis starting fiom the aft of the ship and travelling forward
along a horizontal axis
t = Time
T = Still-water draught
P = Pressure given by Bernoulli’s equation:

(2.25)

where,
p = Fluid density
4 = Velocity potential which is made up of first- and second- order terms. The
derivation of 4 is well described by Jensen and Pedersen (1979)
g = Acceleration due to gravity