Page 316 - Marine Structural Design

292 Pari II Ultimate Strength

k=- kSkp = the equivalent spring stifmess for the system
ks +k,
The motion of the mass-spring system may then be expressed as
d2x
T?Z-i+kX=O ( 14.1 3)
dt
Considering the initial condition (mass move at velocity v, x=O when t=O, the solution to the
above differential equation is,
t
x=v -sinwt (14.14)

where the natural frequency is

(1 4.15)

From Eq.(14.14), the maximum impact force is obtained as,

F,, =kx,, =v& (14.
and impact duration (the time from the initiation of the impact to the peak impact force) is

(14. 7)

The impact durationTo is typically 1-2 seconds, (see also Figures 14.16 and 14.17), and it is
much longer than the natural frequency of the main hit member and structure system. Hence,
ship impact is usually handled in a quasi-static way. The time history of the impact force is
further illustrated in Figure 14.16 (e) and Figure 14.17 (b). When the impact force is the
maximum, the velocity of the motions for the ship and platform is zero and the deformation
energies in the ship and platform are as follows:

(14.19)

The maximum impact force expressed in Eq.(14.16) may also be obtained from the following,.

1
E, -k E, =-mv2 (14.20)
2
14.4.2 Basic Numerical Examples

In the following section, a number of simple numerical examples, which serve to demonstrate
the accuracy and the efficiency of the developed three-dimensional beam-column elements,
will be presented (Bai and Pedersen, 1993).