86
WAVE FORCES ON STRUCTURES
The total horizontal force pi(t) is given by integrating équation (4.1) over
the length of the cylinder. For Cd = 0, this is
r(f-d) _
Pi(0 =
(ldz
J-d
_
r^-d}
= CMi-pD2
ûdz + CM2-pD2 I
(ù — v)dz
(4.3)
4
J-d
4
J-d
The stiffness of the cylinder, equal in magnitude to the horizontal static force
applied to the top which will produce a unit horizontal deflection, is easily
shown to be fci = SEI/P. The équivalent lumped mass, computed in chapter
5, is rriQ = O.227moF For a conservative (or high) estimate of v, lump pi(t)
at the top of the cylinder, or the coordinate point, z — l - d. For a material
damping constant of Cj, the équation of motion becomes
mov + Ciû + kpv — pi(t)
(4-4)
It is further assumed that ù is much larger than v along the cylinder and that ü
is independent of the coordinate z, consistent with this single degree of freedom
model. Using équation (4.3) with the mass and stiffness results just obtained,
équation (4.4) becomes
f0.22“vioï + (7W2-pZ)2ÉQ u + C]û H—~v — Ckf^-pD2
ûdz
(4.5)
v
4
/
4
J_d
Using û from Table 3.1 (with x — 0), the intégral in équation (4.5) is easily
evaluated. It is seen from this last resuit that C,m2 is actually the added mass
coefficient C&. The virtual mass, or the coefficient of ü in équation (4.5), thus
arises in a natural way in this formulation.
Example Problem 4-2. Solve Example Problem 4-1 shown in Figure 4.1, but
this time include the fluid drag term. Then linearize the resulting équation of
motion following the method discussed by Berge and Penzien (1974).
The steps in the solution are summarized as follows. The équation of motion
is
Af-d}
”><)!' + ci v + kpv = /
qdz —pi(t)
J-d
The loading term is
9 — Cm —pD2ù — Cm2~pD2 v + Cd y/^/npD (it — û) cr
in which a, the standard déviation of the relative velocity (u — û), is
(4.6)
(4.7a)
(4-7b)