Page 463 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CURVED PIPES CONVEYING FLUID 435

and [11]-[1111 are given by

while al, bl, etc. are defined in equations (6.87).
(cl Calculation of no

In the discretization procedure described earlier, the pipe is divided into a series of
constant-curvature elements, each of which may be treated as an incomplete circular pipe.
Here, the steady combined force no acting on an incomplete circular pipe subtending an
angle 0 and conveying fluid at a constant nondimensional velocity U is determined.
If the gravity effect is neglected, the solution to equation (6.52) may be written as

=
-
no(<) C, sin(<@ + ~2) ii2, (6.94)
where CI and C2 are two constants of integration which can be determined from the
boundary conditions. The boundary conditions, in turn, are determined from equilibrium
considerations and the application of Castigliano’s theorem. They can be shown to be

(6.95)

for a clamped-free incomplete circular pipe, and
n”(1) -2, n”’(1) 0 (6.96)
=
=
for a clamped-clamped, clamped-pinned or pinned-pinned incomplete circular pipe. In
equations (6.95), nP = (Aipi - A,p,)L2/EI and represents the steady-state nondimen-
sional force due to the pressures of the internal and external fluids.
Using the boundary values (6.95) and (6.96) in equation (6.94) one obtains

(6.97)

where
(6.98)

for a clamped-free pipe and
no = -$ (6.99)

if both ends of the pipe are supported (clamped or pinned).