Page 484 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

454 SLENDER STRUCTURES AND AXIAL FLOW

to within 2.5%. Calculations were done with both in-plane and out-of-plane versions of
the theory, which for straight pipes should give identical results. The in-plane version was
nevertheless found to give superior agreement with analytical results for the same number
of elements, presumably because of the use of quintic as opposed to cubic interpolation
functions.
Sample Argand diagrams for in-plane and out-of-plane motions of a semi-circular pipe
obtained by the modified inextensible theory are shown in Figures 6.19 and 6.20, where
they are compared with those obtained by the conventional inextensible theory. Both
theories predict divergence followed by flutter at higher P for in-plane motions, and only
flutter for out-of-plane motions (although divergence in the first mode almost occurs).
The critical flow velocity for in-plane divergence is approximately the same (Figure 6.19)
according to the two theories, ii$ 2: 0.7, in contrast to the results for clamped-clamped
pipes. However, the critical flow velocities for flutter are much lower according to the
modified inextensible theory: 2:f cx 1.3 for in-plane motions and Z:f E 0.8 for out-
of-plane motions, versus 4.2 and 3.5, respectively. The differences are large but not
surprising, in view of the dramatic effect that accounting for the steady fluid forces has
been found to have on the dynamics of pipes with both ends supported (Section 6.4).

0 2 4 6 8 10
Re (a*)
Figure 6.19 Argand diagram for the lowest four eigenfrequencies for in-plane motion of a
cantilevered semi-circular pipe conveying fluid for /? = 0.75: -0-, conventional inextensible theory
(l7 = 0); - - , modified inextensible theory (I7 # 0). The two sets of results sensibly coincide
0
for the first mode, so only one is shown (Misra et al. 1988a).