40
STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONS
calculated through a dynamic response analysis. Magnification factors will be
discussed in chapter 5. The alternative is to use Cs — 5.5 if no dynamic response
analysis is made for the flexible cylinder.
Wave slamming on an offshore structure in which the waves are underneath
the deck and in the vertical direction, can be a further design considération. One
way to calculate the resulting sudden uplift force on the deck is to use équation
(2.38) in which u is the vertical wave particle velocity and the product DI is
replaced by the deck’s area of impact. However, further research is still needed
to détermine the range of the fluid coefficient Cs for this particular type of wave
impact. For a detailed analysis of wave forces on decks of offshore platforms,
see Bea, et al. (1999).
2.4 STRUCTURAL MASS, DAMPING, AND RESTRAINT
Structural Mass and Stiffness
The following two example problems illustrate the modeling of single and
multiple beams as a point mass located by the single coordinate v = v(t).
Classical beam theory gives the bending stiffness as k\ = CEI/£3. For a single
beam, El is the flexural stiffness, £ is the length, and C is a constant that
dépends on the end fixity of the beam (C — 3 for a cantilevered beam with
full fixity at the base and no moment at its tip). For a tubular beam with an
outside diameter D and an inside diameter Dt, then I = tt(D4 — Z?4)/64.
Example Problem 2.7. Consider the horizontal motion of a tubular cross
brace welded to the relatively rigid and stationary legs of a jacket platform.
This structural element, defined in Figure 2.16a, has full fixity at its ends and
is subjected to the uniform horizontal load per unit length q, given by the right
side of équation (2.23). The total horizontal load, modeled as a single point
load at midspan, is qt — pi(t), or
D
D2
Pi(f) = Cq
u + Cai plîr-^-ü
(2.39)
The dominant mode of motion ^(x) is shown by the broken lines of Figure
2.16b. The midspan coordinate is v = vit), which locates the lumped, virtual
mass m of the massless cross brace of bending stiffness El, as shown in Figure
2.16c. The virtual mass is deduced from équation (2.23), or
(
D2\
m = l^mo + CA pir— j
£
(2.401
in which rho is the actual mass per unit length. Note that
= 1 gives an upper
bound for m, but this is a bad choice since it is obvious that ail of the mass
along ihe length does not hâve the same displacement as that of the lumped
mass. In Chapter 5, Example Problem 5.3, it is shown that A = 0.370 for this
STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONS
calculated through a dynamic response analysis. Magnification factors will be
discussed in chapter 5. The alternative is to use Cs — 5.5 if no dynamic response
analysis is made for the flexible cylinder.
Wave slamming on an offshore structure in which the waves are underneath
the deck and in the vertical direction, can be a further design considération. One
way to calculate the resulting sudden uplift force on the deck is to use équation
(2.38) in which u is the vertical wave particle velocity and the product DI is
replaced by the deck’s area of impact. However, further research is still needed
to détermine the range of the fluid coefficient Cs for this particular type of wave
impact. For a detailed analysis of wave forces on decks of offshore platforms,
see Bea, et al. (1999).
2.4 STRUCTURAL MASS, DAMPING, AND RESTRAINT
Structural Mass and Stiffness
The following two example problems illustrate the modeling of single and
multiple beams as a point mass located by the single coordinate v = v(t).
Classical beam theory gives the bending stiffness as k\ = CEI/£3. For a single
beam, El is the flexural stiffness, £ is the length, and C is a constant that
dépends on the end fixity of the beam (C — 3 for a cantilevered beam with
full fixity at the base and no moment at its tip). For a tubular beam with an
outside diameter D and an inside diameter Dt, then I = tt(D4 — Z?4)/64.
Example Problem 2.7. Consider the horizontal motion of a tubular cross
brace welded to the relatively rigid and stationary legs of a jacket platform.
This structural element, defined in Figure 2.16a, has full fixity at its ends and
is subjected to the uniform horizontal load per unit length q, given by the right
side of équation (2.23). The total horizontal load, modeled as a single point
load at midspan, is qt — pi(t), or
D
D2
Pi(f) = Cq
u + Cai plîr-^-ü
(2.39)
The dominant mode of motion ^(x) is shown by the broken lines of Figure
2.16b. The midspan coordinate is v = vit), which locates the lumped, virtual
mass m of the massless cross brace of bending stiffness El, as shown in Figure
2.16c. The virtual mass is deduced from équation (2.23), or
(
D2\
m = l^mo + CA pir— j
£
(2.401
in which rho is the actual mass per unit length. Note that
= 1 gives an upper
bound for m, but this is a bad choice since it is obvious that ail of the mass
along ihe length does not hâve the same displacement as that of the lumped
mass. In Chapter 5, Example Problem 5.3, it is shown that A = 0.370 for this