STRUCTURAL MASS, DAMPING, AND RESTRAINT
43
small by comparison to v. Now lump a fraction fa of the virtual mass of ail three
flexible legs with the rigid deck mass, md. From the coefficient of v in équation
(2.23), the équivalent virtual mass for this structural System is deduced as
m — 3 ( rôo + Ca p D' ) fad + 3m0(^ - d)/i + md
(2.44)
in which mo is the actual mass per unit length of a single leg, d is the water
depth, and (£ — d) is the length of a leg between the still water line and the
bottom of the deck. (Example Problem 5-4 will show that fa = 0.375.) Assume
that structural damping is mainly produced by the submerged portions of the
legs. From the coefficient of v in équation (2.23), the damping coefficient is
deduced as
cj = 3cd + 3C'Dpd^
(2.45)
From classical beam theory, the restoring force constant is calculated as
ri
fci = 36-^(2.46)
which is the magnitude of the horizontal force at deck level that produces a unit
deflection (v = 1) at that point. (For a single leg, fci = Y2EI / f3 ). The value
of ki given by équation (2.46) is an upper bound value for two reasons. First,
this stiffness is decreased as the full fixity conditions on the legs are relaxed
at the mud line or at the deck. Second, this leg stiffness is also reduced as the
magnitude of the deck load approaches the Euler buckling load for this structure,
an effect that will be explored in a subséquent example problem.
In summary, the jackup drilling rig of Figure 2.17a and Figure 2.17b is
modeled as a single degree of freedom System whose free body sketch is shown
in Figure 2.17c. When Newton’s second law is applied to the équivalent virtual
mass in this latter sketch, the governing équation (2.43) is obtained. With the
respective values of m, ci, and fci given by équations (2.44), (2.45), and (2.46),
the explicit form of équation (2.43) becomes
jmd + 3 (m0 + Ca P^^2) M + 3bio(f - d)fa j v
3cd +
pdD^
Z
■
El
û + 36-^-v=pi(t)
(2-47)
The modeling of pi (t) its deferred to later chapters.
Cable Restraints
Cables or guy lines are employed to restrain the motion for several types of
offshore structures. The simplest cable configuration is the vertical one used
to restrain floating, tension leg platforms such as in Figure 1.1e. In addition,
43
small by comparison to v. Now lump a fraction fa of the virtual mass of ail three
flexible legs with the rigid deck mass, md. From the coefficient of v in équation
(2.23), the équivalent virtual mass for this structural System is deduced as
m — 3 ( rôo + Ca p D' ) fad + 3m0(^ - d)/i + md
(2.44)
in which mo is the actual mass per unit length of a single leg, d is the water
depth, and (£ — d) is the length of a leg between the still water line and the
bottom of the deck. (Example Problem 5-4 will show that fa = 0.375.) Assume
that structural damping is mainly produced by the submerged portions of the
legs. From the coefficient of v in équation (2.23), the damping coefficient is
deduced as
cj = 3cd + 3C'Dpd^
(2.45)
From classical beam theory, the restoring force constant is calculated as
ri
fci = 36-^(2.46)
which is the magnitude of the horizontal force at deck level that produces a unit
deflection (v = 1) at that point. (For a single leg, fci = Y2EI / f3 ). The value
of ki given by équation (2.46) is an upper bound value for two reasons. First,
this stiffness is decreased as the full fixity conditions on the legs are relaxed
at the mud line or at the deck. Second, this leg stiffness is also reduced as the
magnitude of the deck load approaches the Euler buckling load for this structure,
an effect that will be explored in a subséquent example problem.
In summary, the jackup drilling rig of Figure 2.17a and Figure 2.17b is
modeled as a single degree of freedom System whose free body sketch is shown
in Figure 2.17c. When Newton’s second law is applied to the équivalent virtual
mass in this latter sketch, the governing équation (2.43) is obtained. With the
respective values of m, ci, and fci given by équations (2.44), (2.45), and (2.46),
the explicit form of équation (2.43) becomes
jmd + 3 (m0 + Ca P^^2) M + 3bio(f - d)fa j v
3cd +
pdD^
Z
■
El
û + 36-^-v=pi(t)
(2-47)
The modeling of pi (t) its deferred to later chapters.
Cable Restraints
Cables or guy lines are employed to restrain the motion for several types of
offshore structures. The simplest cable configuration is the vertical one used
to restrain floating, tension leg platforms such as in Figure 1.1e. In addition,