Page 468 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

440 SLENDER STRUCTURES AND AXIAL FLOW

that the critical flow velocity is also lower, reflecting the relative stiffness in the two
directions. These results are also very close to Chen’s (1973), except for the second mode
in one caset.
Before closing this discussion, it is remarked that, throughout this chapter, the modes are
numbered sequentially, strictly in ascending order of frequency, irrespective of whether
they are asymmetric or symmetric. For in-plane motions of a semi-circular pipe, modes
1-4 in Figure 6.5 correspond respectively to the modes in Figure 6.7(a-d), i.e. the
modes are numbered in ascending order of the number of nodes. Similarly, for out-of-
plane motions: the first mode would have no nodes, the second mode a node at mid-point,
and so on.

Figure 6.7 Schematics of (a,c) the asymmetric and (b,d) symmetric modes for in-plane motions
of an inextensible semi-circular pipe at ii = 0, and approximately for an extensible one.

6.4.2 Extensible theory

As in the previous case, a study of convergence was conducted, to determine what a
reasonable number of finite elements would be for accurate computation of the eigenfre-
quencies. Some results are presented in Figure 6.8 for U = 0 and various values of d.
It may be seen that convergence is very slow, and that it is affected by the slenderness
parameter d (i.e. ApL2/Z); convergence for the third mode is even slower (Misra et aE.
1988b). For a small number of elements (10 or so), the results for different values of
d are very different. For a larger number of elements (40 or so), the results are compa-
rable. In the curved beam theory used in this work, it has been assumed that the length
of the pipe is large in comparison with its radius. This implies that d must be large;
however, calculations with large d result in high computational cost. Therefore, a value
of d that provides a reasonable trade-off between cost and accuracy has been used in the
calculations to be presented, namely d = lo4.

+In this regard, it is noted that the value of ii* at which w* = 0 should be independent of ,9, as is the case
in Chen’s results but not in those of Misra et al. - either due to a plotting error in the latter or, more likely,
because of the use of an insufficient number of finite elements to ensure adequate accuracy.