The natural frequency of a cantilever with effective mass meper unit
length is
fn0.56(EI/mel^4 )1/2 (14.76)
whereEis the modulus of elasticity, and Iis the second moment of area: l
is the ‘effective length’ of the pile, equal to the water depth plus the appar-
ent fixity depth (Fig. 14.21) to allow for the stiffness of the pile–soil inter-
action. For stiff clay, the apparent fixity depth is 3.5Dto 4.5D, where Dis
the diameter of the pile. For very soft silts it is 7Dto 8.5D.
The resonance condition in the cross-flow direction occurs when the
forcing frequency fvfn. For the cylinder, with the vortex shedding fre-
quency given by S0.2, resonance takes place when the flow speed is as
follows:
for cross-flow motion, V 5 fnD; (14.77)
for in-line motion, V2.5fnD. (14.78)
The amplitude, , of oscillations of the vertical pile in water may be
expressed functionally as
~me,fn, ,V, ,μ,D.
Dimensional considerations show that
~, , ,.
The effect of the Reynolds number may be considered insignificant.
Therefore,
~, ,.
The first term on the right-hand side of the above equation is usually
called the reduced velocity. The last two terms are combined to form a
stability parameter Ksdefined as
Ks 2 me / D^2.
Thus
~Ks,. (14.79)
V
fnD
D
me
D^2
V
fnD
D
VD
μ
me
D^2
V
fnD
D
614 WAVES AND OFFSHORE ENGINEERING
