Page 220 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

202 SLENDER STRUCTURES AND AXIAL FLOW

(a) Free or free-to-slide-axially downstream end. In this case it is presumed that no
externally imposed tensioning is possible; it is also assumed that the internal fluid
discharges into the external fluid at x = L and that p;(L) 2: p,(L), equal to the hydrostatic
pressure at that point. Thus, T(L) = p(L)[A,(L) -A;@)], which may be rewritten in terms
of a drag coefficient
T(L) = ipiAiU:Cj-;; (4.17)
it is recognized that, since (A, - A~)L is small, T(L) will be small and may alternatively
be neglected.
(b) Supported end with no axial sliding. In this case,

T(L) = T + [T + peAe - p;Ail~, (4.18)

where T represents a possible externally applied tension. The second term is evaluated
by considering the flow-related terms by themselves and imposing the condition that the
axial strain E, satisfy s," E, dx = 0, as in the derivation of equation (3.37). It is noted that
E, = [a, - u(arr + am)]/E, in which a, = T(x)/A(x), where A(x) = A,(x) - A;(x)
(A, - A;)x, and u is the Poisson ratio; furthermore, a,, + am E 2(p;Ai - p,A,)/(A, -
Ai), by assuming that the tubular beam area variations are sufficiently gradual for the
stress distribution applicable to a uniform tubular beam subjected to uniform internal and
external pressure to hold true for each cross-section. Hence, one finds

from which (T + p,A, - piA;)~ may be obtained if the form of A,(x), A,(x) and the
pressure distributions are known. In general, one may write
+
[T + PeAe - PAIL = (1 - 2v)[PeAe - ~iAil~f1 ~i(Aiui>ui(L)f (4.20)
2,
in which fl and f 2 must be obtained via (4.19). It is of interest to note that for a uniform
tubular beam internally pressurized by pi and immersed in a uniform ambient pressure,
the second term in (4.20) vanishes while the first gives -(1 - 2u)piA;, thus retrieving the
results of Section 3.3.2. It should also be noted that, unless pressurization effects exist,
both fl and f2 are very small terms which may be neglected for slightly tapered tubular
beams.
Hence, the equation of small motions of the system, subject to all the assumptions and
approximations made, is

- { T(L) + p;AiUi[U; -