Page 92 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 75

discharge to atmosphere, in which case there may be a mean pressure F at x = L, over
and above that expended to overcome friction (see also Section 3.4.2). Thus, T and p
would act uniformly over the total length of the pipe. Now, if the downstream end is
completely fixed, i.e. the system of Figure 3.l(a) rather than (b,c), internal pressurization
induces an additional tensile force, which for a thin pipe is equal to 2u7A, where u
is the Poisson ratio, as first introduced by Naguleswaran & Williams (1968); i.e. the
tendency of the pipe to expand radially and hence to become shorter, induces this tensile
force. One may derive this in terms of (i) an axial stress distribution a, = T/A, and
(ii) the stress distribution due to 7, a,.,. + am = 27A/A,,, where A,, is the cross-sectional
area of the pipe material (Sechler 1952); these two are then superposed to give the
axial strain = [a, - ~(a,.,. + a@)] /E. Now, since no axial movement is allowed at the
ends, s,” dx = 0, which yields T = 2 UTA. Hence, in general, equation (3.36) may be
written as

(3.37)

where S = 0 signifies that there is no constraint to axial motion at x = L, and 6 = 1 if
there is. Of course, it could be argued that, in practice, 7 and p can only be imposed
if S = 1, so that one should really write A[?= - pA(1 - 2u)l; still, one can conceive of
ingenious theoretical ways in which T and p may be applied, even for the system of
Figure 3.l(b) - e.g. by strings and pulleys and bellows - and hence the form of equa-
tion (3.37) will be retained. Now, substitution of (3.37) into (3.34) gives the equation of
small lateral motions:

MU2-T+FA(l -24- (M+m)g-M%] (L-x)}$
dt

If gravity, dissipation, tensioning and pressurization effects are either absent or
neglected and U is constant, this simplifies to equation (3.1). The derivation given
here follows Paldoussis & Issid’s (1974). Earlier derivations of the simpler form,
equation (3.1), for pipes with supported ends, were made by Feodos’ev (195 1), Housner
(1952) and Niordson (1953), and for cantilevered pipes by Benjamin (1961a) and Gregory
& Pa’idoussis (1966a). The equation derived by Ashley & Haviland (1950) is wrong,
missing the all-important MU2(a2w/ax2) term. Similarly, an equation derived by Chen
(1971 b) for the case of harmonically perturbed flow is partly wrong, in that the first term
of equation (3.28) or (3.321, i.e. the axial acceleration effect, is missing, although the last
term in (3.29) is present; as a result, instead of the M(dU/dt)(L - x)(a2w/ax2), a term
M(dU/dr)(aw/ax) is found in Chen’s equation of motion.
There are some subtleties in this derivation that are not quite obvious. This is partly
the reason for the derivation of Appendix A.
In several calculations in the following, dissipation in the material of the pipe will be
modelled not by the Kelvin-Voigt viscoelastic model as in equation (3.38), but by the so-
called hysteretic or striictiiruf damping model. As shown by Bishop & Johnson (1960) for