FREQUENCIES FOR NONLINEAR STRUCTURES
111
Several important features about the two System models can be deduced
from their respective frequencies as given by équations (5.36) and (5.40). Those
features are summarized as follows:
1. The Euler buckling load for this three-legged structure is PE - 3n2EI/t2.
The Rayleigh model frequency of équation (5.36) shows that increasing the deck
load reduces the effective bending stiffness of the legs. In fact, the structure
buckles if the deck load is sufficiently high, a critical condition for which luq — 0
and mdg — PE.
2. If PE :•> mdg, then the bending stiffnesses for the two models are nearly
the same, a resuit deduced by comparing the numerators of équations (5.36)
and (5.40). That is, the coefficients of El/C in these respective équations are
3tt4/8 = 36.5 and 36.
3. If PE
mdg and the deck mass is much greater than that of the legs,
then the System mass is simply md for both models. However, such a design is
probably unrealistic.
4. If the deck mass is of comparable magnitude to that of the legs, then the
Rayleigh model shows that the virtual mass of the legs has a significant influence
on wq. For instance, in the spécial case where I ~ d, then the parameters of
équations (5.37) and (5.38) become: f = 31/8, t!' — 0, and the virtual mass of
the legs becomes m — 3mfiE When the denominators of équations (5.36) and
(5.40) are compared, then 3m£' — 9mt/8 = 3mfi£, or /i = 0.375. This justifies
the assumption in Example Problem 2.8 in which 37.5 percent of the virtual
mass for the submerged legs, together with 37.5 percent of the actual leg mass
above water, was lumped with the total mass of the deck to formulate the single
degree of freedom model. The recommended frequency expression for this case
is thus
(3tt2EI/!2 — mdg) tt2/(8I)
W° =
3(0.375)m£ + md
in which the virtual mass for each of the three legs is given by équation (5.41).
Realistic numerical data for a jackup platform with three steel pipes for legs
are listed in Table 5.2. Listed also is the System frequency computed from équation (5.36): u>o = 1-36 rad/sec or /0 — 0.217 Hz. From the numerical values in
the numerator of u>q, it is observed that the deck weight has a srnall effect on
the stiffness bending stiffness. However, numerical values in the denominator
indicate that the leg mass is a significant portion of the System mass. In conclusion, it is noted that the fundamental period of oscillation of this platform is
To = 1//q = 4.61 sec, which is well below the 12 to 15 sec periods of the highest
energy offshore waves in a typical sea State.
5.2 FREQUENCIES FOR NONLINEAR STRUCTURES
In the last section it was demonstrated that for linear structural models with
small motion, the free vibration frequency was independent of the amplitude
111
Several important features about the two System models can be deduced
from their respective frequencies as given by équations (5.36) and (5.40). Those
features are summarized as follows:
1. The Euler buckling load for this three-legged structure is PE - 3n2EI/t2.
The Rayleigh model frequency of équation (5.36) shows that increasing the deck
load reduces the effective bending stiffness of the legs. In fact, the structure
buckles if the deck load is sufficiently high, a critical condition for which luq — 0
and mdg — PE.
2. If PE :•> mdg, then the bending stiffnesses for the two models are nearly
the same, a resuit deduced by comparing the numerators of équations (5.36)
and (5.40). That is, the coefficients of El/C in these respective équations are
3tt4/8 = 36.5 and 36.
3. If PE
mdg and the deck mass is much greater than that of the legs,
then the System mass is simply md for both models. However, such a design is
probably unrealistic.
4. If the deck mass is of comparable magnitude to that of the legs, then the
Rayleigh model shows that the virtual mass of the legs has a significant influence
on wq. For instance, in the spécial case where I ~ d, then the parameters of
équations (5.37) and (5.38) become: f = 31/8, t!' — 0, and the virtual mass of
the legs becomes m — 3mfiE When the denominators of équations (5.36) and
(5.40) are compared, then 3m£' — 9mt/8 = 3mfi£, or /i = 0.375. This justifies
the assumption in Example Problem 2.8 in which 37.5 percent of the virtual
mass for the submerged legs, together with 37.5 percent of the actual leg mass
above water, was lumped with the total mass of the deck to formulate the single
degree of freedom model. The recommended frequency expression for this case
is thus
(3tt2EI/!2 — mdg) tt2/(8I)
W° =
3(0.375)m£ + md
in which the virtual mass for each of the three legs is given by équation (5.41).
Realistic numerical data for a jackup platform with three steel pipes for legs
are listed in Table 5.2. Listed also is the System frequency computed from équation (5.36): u>o = 1-36 rad/sec or /0 — 0.217 Hz. From the numerical values in
the numerator of u>q, it is observed that the deck weight has a srnall effect on
the stiffness bending stiffness. However, numerical values in the denominator
indicate that the leg mass is a significant portion of the System mass. In conclusion, it is noted that the fundamental period of oscillation of this platform is
To = 1//q = 4.61 sec, which is well below the 12 to 15 sec periods of the highest
energy offshore waves in a typical sea State.
5.2 FREQUENCIES FOR NONLINEAR STRUCTURES
In the last section it was demonstrated that for linear structural models with
small motion, the free vibration frequency was independent of the amplitude