44
STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONS
there are two classes of flexible mooring Systems in which stationary guy Unes
hang as catenary curves from the structures to the sea floor. The first is the
single line that constrains a vessel or buoy. The second is the multi-line System
that constrains vessels, semisubmersible platforms, and compilant towers. The
lines in présent use are ropes of metallic wire or of synthetic fiber such as
nylon. Dacron, or Kevlar; and Steel chains with solid or hollow links. Practical
aspects of guy line design are discussed in the U.S. Navy publication NAVFAC
DM-26 (1968), in the four papers by Childers (1973-1975), and in the work of
Niedzwecki and Casarella (1975). These references include analyses of multiline Systems with inextensible or nonstretching lines. Refined analyses which
include clumped weights and additional anchors along the cables, as well as the
effects of elastic cable stretching, are presented by Adrezin et al. (1996), Ansari
(1980), and Wilson and Orgill (1984).
For a taut cable with negligible sag, the longitudinal extension <5 dépends
on both the applied longitudinal force F„ and the material properties of the
cable. For instance, Wilson (1959) used the following power law to correlate the
load-extension behavior of both Steel wire and synthetic fiber line employed in
mooring ships:
Fe = C06n
(2.48)
Here, Cq and n are constants depending on the material, its length, and its
cross-sectional area. If the deflections are sufficiently small, n = 1 and the
force-deflection relationship based on elementary theory is given by
Ft = ^5 = k,6
(2.49)
where Aq is the cross-sectional area, £ is the length, and Ee is the équivalent
Young’s modulus for longitudinal extension of the line. In this case, équation
(2.49) defines the longitudinal stiffness constant k\ = AqE,JI.
A more convenient form of équation (2.48), which also includes équation
2.49) and approximately represents the behavior of an assembly of taut cables
tied to a common point whose deflection is <5 under load Fe, is
Fe = kxè + k2\6\ô + k3^+ ■■■
(2.50)
where /q, k3,... define the stiffness. The absolute value sign on each even-order
term in 6 forces Fe to be antisymmetrical about 5 = 0, assuring that the restraint
stiffness is the same for loading and unloading. Example problems will show
that the approximation of équation (2.50) facilitâtes the dynamic analysis of
offshore structures with both extensible and inextensible supporting cables.
One should keep in mind that équations (2.48)-(2.50) apply only when cable
dynamics can be neglected; that is, in cases where the fundamental cable frequency in both longitudinal and transverse vibration is much higher than the
free vibration frequency of the structure that it restrains. In applications, this
frequency criterion should always be checked. Methods to calculate the structural frequency for single degree of freedom Systems are given in Chapter 5; and
methods to calculate cable frequencies are discussed in Chapter 10.
STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONS
there are two classes of flexible mooring Systems in which stationary guy Unes
hang as catenary curves from the structures to the sea floor. The first is the
single line that constrains a vessel or buoy. The second is the multi-line System
that constrains vessels, semisubmersible platforms, and compilant towers. The
lines in présent use are ropes of metallic wire or of synthetic fiber such as
nylon. Dacron, or Kevlar; and Steel chains with solid or hollow links. Practical
aspects of guy line design are discussed in the U.S. Navy publication NAVFAC
DM-26 (1968), in the four papers by Childers (1973-1975), and in the work of
Niedzwecki and Casarella (1975). These references include analyses of multiline Systems with inextensible or nonstretching lines. Refined analyses which
include clumped weights and additional anchors along the cables, as well as the
effects of elastic cable stretching, are presented by Adrezin et al. (1996), Ansari
(1980), and Wilson and Orgill (1984).
For a taut cable with negligible sag, the longitudinal extension <5 dépends
on both the applied longitudinal force F„ and the material properties of the
cable. For instance, Wilson (1959) used the following power law to correlate the
load-extension behavior of both Steel wire and synthetic fiber line employed in
mooring ships:
Fe = C06n
(2.48)
Here, Cq and n are constants depending on the material, its length, and its
cross-sectional area. If the deflections are sufficiently small, n = 1 and the
force-deflection relationship based on elementary theory is given by
Ft = ^5 = k,6
(2.49)
where Aq is the cross-sectional area, £ is the length, and Ee is the équivalent
Young’s modulus for longitudinal extension of the line. In this case, équation
(2.49) defines the longitudinal stiffness constant k\ = AqE,JI.
A more convenient form of équation (2.48), which also includes équation
2.49) and approximately represents the behavior of an assembly of taut cables
tied to a common point whose deflection is <5 under load Fe, is
Fe = kxè + k2\6\ô + k3^+ ■■■
(2.50)
where /q, k3,... define the stiffness. The absolute value sign on each even-order
term in 6 forces Fe to be antisymmetrical about 5 = 0, assuring that the restraint
stiffness is the same for loading and unloading. Example problems will show
that the approximation of équation (2.50) facilitâtes the dynamic analysis of
offshore structures with both extensible and inextensible supporting cables.
One should keep in mind that équations (2.48)-(2.50) apply only when cable
dynamics can be neglected; that is, in cases where the fundamental cable frequency in both longitudinal and transverse vibration is much higher than the
free vibration frequency of the structure that it restrains. In applications, this
frequency criterion should always be checked. Methods to calculate the structural frequency for single degree of freedom Systems are given in Chapter 5; and
methods to calculate cable frequencies are discussed in Chapter 10.