Page 168 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

150 SLENDER STRUCTURES AND AXIAL FLOW

along the length of the pipe; a corresponding term appears in the shear boundary condition
at the clamped end. The new term in the equation of motion opposes the centrifugal term,
U2(a2w/ax2), and this generates a strong stabilizing effect. Adding foundation damping
in this case is destabilizing, although these results were confined to B = only, and
hence should not be considered as general.
All the foregoing apply to uniform foundations. Unusual behaviour may be expected
for nonuniform ones, however, in view of Hauger & Vetter’s (1976) results for the system
with the follower load (pipe with j? = 0) and k = k(t). Thus, if k(t) is zero at = 0 and
1 and triangularly distributed in-between, with a maximum at 4 = ern of k,,, then the
effect is destabilizing for all k,,,. The opposite is true if k = 0 at 6 = tm, and kmaX at
[=Oand 1.

3.5.8 Effects of tension and refined fluid mechanics modelling

The system shown in Figure 3.57(a) was studied by Guran & Plaut (1994), in which
the compression P is conservative, i.e. it is constant and remains along the undeformed
axis of the pipe. The equation of motion is equation (3.98) with Tj = 0 and T = -P,
or r = -9 in dimensionless form, where 9 = PL2/EI. The boundary conditions, in
addition to ~(0, t) = 0 and (a2q/at2)1+, = 0 are

where K* = CL/EI. Clearly, in the limit of K* = 00 the pipe becomes cantilevered.
Typical results for K* = co are shown in Figure 3.57(b), and it is clear that they are
similar to those of Figure 3.33. Indeed, the physical systems are similar: in the case of
Figure 3.33 the pipe is subjected to conservative gravity-induced distributed compression,
but in this case to conservative uniform compression along the length.
Results for K* # 00 are similar. The influence of K* on stability is given in graph-
ical form, for both divergence and flutter, in Guran & Plaut (1994). The condition for
divergence is
u2 + 9 cos v - (Vp)/K*) sin u = 0, v = v’2TiF. (3.111)

As suggested by Figure 3.57(b) and as may be verified numerically with
equation (3.1 I l), if the system is subjected to conservative fension it cannot lose stability
by divergence, but only by flutter.
The effect of the small tension induced by the presence of a terminal nozzle on a
cantilevered pipe - refer to the discussion associated with equations (3.40)-(3.42) - was
taken into account by Bishop & Fawzy (1976), who found
TL = iMU2(~j - 1) 2 (3.1 12)

by taking a force balance across the nozzle, where a, = A/Aj is the ratio of pipe flow area
to nozzle terminal flow area. This tension was neglected in the theoretical calculations
to which the experiments of Gregory & Paldoussis (1966b) were compared. As seen
in (3.112) this is not necessarily negligible for crj substantially different from 1; it is
properly taken into account in the comparison in Figure 3.53. A more refined treatment,