PROBLEMS
97
4.3
Shown in Figure 4.6 is a simplified model of a cantilevered monopod
structure in water of depth d. The deck mass is M, modeled as a point mass
at height £q above the sea floor. The leg has a diameter D, a uniform mass
per unit length of m0, and a bending stiffness El. The leg is subjected to
a simple plane wave whose respective horizontal water particle velocity and
accélération are much larger than those of the leg. Neglect drag forces and also
neglect any réduction in the leg bending stiffness due to the deck mass. Dérivé
for this structure a single degree of freedom model for the motion of the deck,
an équation similar to équation (4.5). Define explicitly the System mass and
stiffness. Carry out the intégrations needed to describe the wave load, pi(t).
4.4
Solve Problem 4.3, but now include drag forces on the leg. Where
possible, carry out the intégrations for wave loading.
4.5
Suppose that the structure of Figure 4.6 is flexible enough so that
its horizontal water particle velocity and accélération are of the same order as
those of a simple, incident wave. Deduce the corresponding équation of motion that includes drag forces. Identify the mass, damping, stiffness, and fluid
loading terms for the linearized form similar to équation (4.8). Carry out the
intégrations where possible. Then write down a brief outline of a numerical,
computer-aided procedure to calculate the time history of the horizontal displacement for the platform.
4.6
Deduce the form of the wave loading term of Example Problem 4-1 for
which there are N simple waves. The z-th values of frequency, wave height, and
wave phase are uji, Hi and Ei respectively, where i = 1,2,... , N. The phase
is added to the argument of the harmonie terms in u and û of Table 3.1.
Figure 4.7 Platform structure for Problem 4.7.
4.7
The unbraced platform shown in Figure 4.7 has four legs, each of diameter D and length
separated by distance £. The legs are subjected to a
single, simple plane wave for which A 3> D. Neglect the fluid drag forces and
also neglect the motion (û,û) of the structure compared to the respective horizontal motion parameters (u,û) of the wave particles. Dérivé the expression for
wrave loading in each leg, accounting for the relative magnitude of the séparation
length £ compared to the wavelength A. Then discuss the net wave-induced force
97
4.3
Shown in Figure 4.6 is a simplified model of a cantilevered monopod
structure in water of depth d. The deck mass is M, modeled as a point mass
at height £q above the sea floor. The leg has a diameter D, a uniform mass
per unit length of m0, and a bending stiffness El. The leg is subjected to
a simple plane wave whose respective horizontal water particle velocity and
accélération are much larger than those of the leg. Neglect drag forces and also
neglect any réduction in the leg bending stiffness due to the deck mass. Dérivé
for this structure a single degree of freedom model for the motion of the deck,
an équation similar to équation (4.5). Define explicitly the System mass and
stiffness. Carry out the intégrations needed to describe the wave load, pi(t).
4.4
Solve Problem 4.3, but now include drag forces on the leg. Where
possible, carry out the intégrations for wave loading.
4.5
Suppose that the structure of Figure 4.6 is flexible enough so that
its horizontal water particle velocity and accélération are of the same order as
those of a simple, incident wave. Deduce the corresponding équation of motion that includes drag forces. Identify the mass, damping, stiffness, and fluid
loading terms for the linearized form similar to équation (4.8). Carry out the
intégrations where possible. Then write down a brief outline of a numerical,
computer-aided procedure to calculate the time history of the horizontal displacement for the platform.
4.6
Deduce the form of the wave loading term of Example Problem 4-1 for
which there are N simple waves. The z-th values of frequency, wave height, and
wave phase are uji, Hi and Ei respectively, where i = 1,2,... , N. The phase
is added to the argument of the harmonie terms in u and û of Table 3.1.
Figure 4.7 Platform structure for Problem 4.7.
4.7
The unbraced platform shown in Figure 4.7 has four legs, each of diameter D and length
separated by distance £. The legs are subjected to a
single, simple plane wave for which A 3> D. Neglect the fluid drag forces and
also neglect the motion (û,û) of the structure compared to the respective horizontal motion parameters (u,û) of the wave particles. Dérivé the expression for
wrave loading in each leg, accounting for the relative magnitude of the séparation
length £ compared to the wavelength A. Then discuss the net wave-induced force