Page 240 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 22 1

for the fluid flow, following closely the work by Pa’idoussis et al. (1986). This will be
referred to as the Timoshenko refined-flow theory, or TRF for short. The pipes are either
clamped at both ends or cantilevered; in the latter case, special ‘outflow models’ are
introduced to describe the boundary conditions on the fluid exiting from the free end.

4.4.1 Equations of motion

The system under consideration consists of a tubular beam of length L, flexural rigidity
EI,, and shear rigidity GAP, conveying fluid with an axial velocity which in the unde-
formed, straight pipe is equal to U. Here, with no loss of generality, the pipe is supposed
to hang vertically, with the fluid flowing down, so that the x-axis is in the direction of
gravity.
In contrast to the Euler-Bernoulli beam theory, the Timoshenko beam theory takes into
account the deformation due to transverse shear. If + denotes the slope of the deflection
curve by bending and x the angle of shear at the neutral axis in the same cross-section
(Figure 4.151, then the total slope (dw/dx) is given by

dw
-=++x, (4.30)
dx
with

and
(4.3 1 b)

where .ht is the bending moment, Q the transverse shearing force, E Young’s modulus and
G the shear modulus; Ap is the cross-sectional area of the pipe (i.e. of the pipe material;
as distinct from Af, the flow area), and Zp the area-moment of inertia of the empty pipe
cross-section; k’ is the shear coefficient, which depends on the cross-sectional shape of
the beam; for the circular cross-section of the tubular beam here under consideration, it
is approximately given (Cowper 1966) by

6(1 + u)(l + cx2)2
k‘ = (4.32)
(7 + 6v)( 1 + cx2)2 + (20 + 12u)a2 ’
in which u is Poisson’s ratio and cx is the ratio of internal to external radius of the
pipe.
In general, an element 6x of the pipe is subjected to a fluid-dynamic force, the compo-
nents of which, for steady flow and to first-order magnitude, are respectively zero and
FA 6x in the x and z directions (cf. Section 3.3.2). FA, the lateral inviscid fluid-dynamic
force (per unit length), the main concern of this work, is discussed in Sections 4.4.3 and
4.4.4; the subscript A denotes that it is related to the total acceleration of the fluid.
An element of the pipe and the forces and moments acting on it are considered next
(Figure 4.15). By writing down the equations of dynamic equilibrium and neglecting
terms of second-order magnitude, one can obtain the equations of motion of the system.