Page 259 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

240 SLENDER STRUCTURES AND AXIAL FLOW

theory. It is seen that both ucf and mcf obtained by the Timoshenko theories agree better
with the experimental data for A < 75 than the results obtained by the Euler-Bernoulli
plug-flow theory; but surprisingly not for ucf when A > 100. This last paradox may be
explained in terms of nonlinear theory (Chapter 5).+ Comparing the results obtained by
the TPF and TRF theories, it is seen that they are very close. Nevertheless, for very short
pipes, TRF theory displays superior agreement with the experimental data.$

4.4.9 Concluding remarks on short pipes and refined-flow models
In general, for short pipes clamped at both ends the use of Timoshenko rather than
Euler-Bernoulli beam theory results in lower critical flow velocities for divergence,
u,d - substantially lower for A < 1000 (Figure 4.19) - as a consequence of the pipe
being effectively less stiff since it deforms not only by bending but also by transverse
shear. The use of refined versus plug-flow fluid-dynamic modelling, on the other hand,
has a less pronounced effect on the dynamics of the system: the refined model gives
slightly higher values of the eigenfrequencies, as well as for the critical flow velocities
for divergence. This is consistent with the concept of smaller-than-ideal virtual mass of the
enclosed fluid, according to the refined three-dimensional fluid-mechanics model devel-
oped in this theory, as discussed in the foregoing. However, the differences in dynamical
behaviour, both qualitative and quantitative, in terms of the refined and simple (plug-flow)
Timoshenko theories are small; hence, from the practical point of view, down to A = IO2,
the simple (plug-flow) Timoshenko theory is good enough for predicting the dynamical
behaviour of short clamped-clamped pipes conveying fluid.
In the case of short cantilevered pipes conveying fluid, the Euler-Bernoulli plug-flow
model is adequate provided A > 1000 approximately. Once again, differences between
refined and plug-flow Timoshenko theory are small, unless A < 25 approximately - an
even lower A than for clamped-clamped pipes.
Finally, by comparison with experiments with cantilevered elastomer pipes, it was
shown that the refined (TRF) theory is necessary for describing adequately the dynamical
behaviour of short pipes (LID < 5 approximately), although Timoshenko beam theory
together with a plug-flow model (TPF theory) is quite satisfactory for relatively longer
pipes; for ‘long’ pipes (LID > 15), Euler-Bernoulli beam theory and the plug-flow model
are perfectly adequate.
There is no question, however, that if one is interested in the dynamics of the system
in its higher modes, e.g. for forced vibration analysis rather than stability (usually lost in
one of the lower modes), then the differences between the three theories become larger,
as may be appreciated from Figures 4.18, 4.20 and 4.21. Thus, although the first-mode
behaviour is adequately predicted by EBPF theory down to A = 1000, third- and fourth-
mode behaviour, and more so for higher modes, requires the use of Timoshenko theory
and refined fluid mechanics (TRF theory) even at much larger values of A.

+The Hopf bifurcation for low A (hence low L/a) may be subcritical, while for higher A it is supercritical.
Hence, for low A, the measured thresholds tend to be lower than would otherwise be the case. In this light,
both the degree of excellence of the agreement with TRF theory for A < 75 and the better agreement with
EBPF theory for A > 100 may be wholly fortuitous.
*With the Timoshenko plug-flow theory, the shortest cantilevered pipe for which calculations have been
conducted corresponds to A = 13.07. In the case of A 5 lO(p = 0.155, p = 0.02, y 5 0.01), TPF theory, or
at least the computer program utilized, fails to give a convergent solution.