Page 120 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

102 SLENDER STRUCTURES AND AXIAL FLOW

& Issid (1974), and for the effect of externally applied tension to Bolotin (1956) and Plaut
& Huseyin (1975).

3.4.3 Pipes on an elastic foundation

An elastic foundation represents the distributed support provided to long pipes resting
on a generally elastic medium, e.g. in the case of pipelines laid on the ocean floor.
For pipes with supported ends the additional stiffness supplied by the elastic foundation
simply renders the system stiffer [see equation (3.70)], and hence the qualitative effect
on stability is predictable.
The critical flow velocity for divergence, ucd, or more generally Vcd as per the second of
equations (3.100), may be obtained by the method of Section 3.3.6(a) in a similar manner
as used to obtain equations (3.90a-c); indeed, as first obtained by Roth (1964),+

(3.10 1 a)

However, if k is sufficiently large, e.g. k = lo3, Vcd as given by (3.101a) is overesti-
mated, because divergence can be associated with a higher mode at a lower value of Vcd,
obtained from

(3.101b)

where the mode number n is identified with the beam eigenfunction z/z sin(nrrx/l).*
The mode to become unstable is that leading to the smallest Vcd, and is thus associated
with the smallest n satisfying
k
n2(n + 112 2 -; (3.102)
IT4
e.g. fork = 300 one obtains n = 1, whereas for k = 500, n = 2. What happens physically
is that the support provided by the elastic foundation can be thought of as providing added
supports along the length, making the first divergence with one or more nodes within the
span feasible.
For a clamped-clamped pipe, by Gderkin’s method (Roth 1964), one obtains

and (3.103)
+ ) 1/2
Vcd=n n2+ for k 2 (84/11)n4
n4(n2 + 1)

?Roth’s excellent work, written in German, is unfortunately hardly ever cited in the English-language
r4 6n2 +
+
literature. The interested reader is encouraged to refer to Roth (1965a,b, 1966) also.
$It is of interest that for all the solutions given by (3.101b), and also (3.103), the condition &/k z 4 is
satisfied, so that the discriminant of (3.82) is positive (or zero, when k = n4), and hence real values of the ai
are obtained.